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Quantum BayesianismBasic QED

ContentsArticles

Bell's theorem 1EPR paradox 15Eigenvalues and eigenvectors 25Quantum Bayesianism 44Wave function collapse 48Relational quantum mechanics 51Quantum tunnelling 61Planck constant 70Maxwell's equations 81

ReferencesArticle Sources and Contributors 95Image Sources, Licenses and Contributors 97

Article LicensesLicense 98

Bell's theorem 1

Bell's theoremBell's theorem is a no-go theorem famous for drawing an important line in the sand between quantum mechanics(QM) and the world as we know it classically. In its simplest form, Bell's theorem states:[]

No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.When introduced in 1927, the philosophical implications of the new quantum theory were troubling to manyprominent physicists of the day, including Albert Einstein. In a well known 1935 paper, Einstein and co-authorsBoris Podolsky and Nathan Rosen (collectively EPR) demonstrated by a paradox that QM was incomplete. Thisprovided hope that a more complete (and less troubling) theory might one day be discovered. But that conclusionrested on the seemingly reasonable assumptions of locality and realism (together called "local realism" or "localhidden variables", often interchangeably). In the vernacular of Einstein: locality meant no instantaneous ("spooky")action at a distance; realism meant the moon is there even when not being observed. These assumptions were hotlydebated within the physics community, notably between Nobel laureates Einstein and Niels Bohr.In his groundbreaking 1964 paper, "On the Einstein Podolsky Rosen paradox", physicist John Stewart Bell presentedan analogy (based on spin measurements on pairs of entangled electrons) to EPR's hypothetical paradox. Using theirreasoning, he said, a choice of measurement setting here should not affect the outcome of a measurement there (andvice versa). After providing a mathematical formulation of locality and realism based on this, he showed specificcases where this would be inconsistent with the predictions of QM.In experimental tests following Bell's example, now using quantum entanglement of photons instead of electrons,John Clauser and Stuart Freedman (1972) and Alain Aspect et al. (1981) convincingly demonstrated that thepredictions of QM are correct in this regard. While this does not demonstrate QM is complete, one is forced to rejecteither locality or realism (or both).Cornell solid-state physicist David Mermin has described the various appraisals of the importance of Bell's theoremwithin the physics community as ranging from "indifference" to "wild extravagance".[1] Lawrence Berkeley particlephysicist Henry Stapp declared: “Bell’s theorem is the most profound discovery of science.”.[2]

OverviewBell’s theorem states that the concept of local realism, favoured by Einstein,[3] yields predictions that disagree withthose of quantum mechanical theory. Because numerous experiments agree with the predictions of quantummechanical theory, and show correlations that are, according to Bell, greater than could be explained by local hiddenvariables, the experimental results have been taken by many as refuting the concept of local realism as anexplanation of the physical phenomena under test. For a hidden variable theory, if Bell's conditions are correct, thenthe results which are in agreement with quantum mechanical theory appear to evidence superluminal effects, incontradiction to the principle of locality.

Bell's theorem 2

Illustration of Bell test for particles such as photons. A source produces a singletpair, one particle is sent to one location, and the other is sent to another location. A

measurement of the entangled property is performed at various angles at eachlocation.

The theorem applies to any quantum systemof two entangled qubits. The most commonexamples concern systems of particles thatare entangled in spin or polarization.

Following the argument in theEinstein–Podolsky–Rosen (EPR) paradoxpaper (but using the example of spin, as inDavid Bohm's version of the EPRargument[4][5]), Bell considered anexperiment in which there are "a pair of spinone-half particles formed somehow in thesinglet spin state and moving freely inopposite directions."[4] The two particles travel away from each other to two distant locations, at whichmeasurements of spin are performed, along axes that are independently chosen. Each measurement yields a result ofeither spin-up (+) or spin-down (−); it means, spin in the positive or negative direction of the chosen axis.

The probability of the same result being obtained at the two locations varies, depending on the relative angles atwhich the two spin measurements are made, and is subject to some uncertainty for all relative angles other thanperfectly parallel alignments (0° or 180°). Bell's theorem thus applies only to the statistical results from many trialsof the experiment. For this reason, the terms "correlated", "anti-correlated", and "uncorrelated" apply only to sets ofseveral pairs of measurements. The correlation of two binary variables can be defined as the average of the productof the two outcomes of the pairs of measurements. This definition is in accordance with the definition of covariancebetween real-valued random variables. Using this definition, if the pairs of outcomes are always the same, thecorrelation will be +1, no matter which same value each pair of outcomes have. If the pairs of outcomes are alwaysopposite, the correlation will be -1. Finally, if the pairs of outcomes are perfectly balanced, being 50% of the times inaccordance, and 50% of the times opposite, the correlation, being an average, will be 0. Measuring the spin of theseentangled particles along anti-parallel directions, i.e. along the same axis but in opposite directions, the set of allresults will be correlated. On the other hand, if the measurements are performed along parallel directions will alwaysyield opposite results, and the set of measurements will show perfect anti-correlation. Finally, measurement atperpendicular directions will have a 50% chance of matching, and the total set of measurement will be uncorrelated.These basic cases are illustrated in the table below.

Anti-parallel Pair 1 Pair 2 Pair 3 Pair 4 … Pair n

Alice, 0° + − + + … −

Bob, 180° + − + + … −

Correlation = ( +1 +1 +1 +1 … +1 ) / n = +1

(100% identicall)

Parallel Pair 1 Pair 2 Pair 3 Pair 4 … Pair n

Alice, 0° + − − + … +

Bob, 0° or 360° − + + − … −

Correlation = ( -1 -1 -1 -1 … -1 ) / n = -1

(100% opposite)

Orthogonal Pair 1 Pair 2 Pair 3 Pair 4 … Pair n

Alice, 0° + − + − … −

Bob, 90° or 270° − − + + … −

Bell's theorem 3

Correlation = ( −1 +1 +1 −1 … +1 ) / n = 0

(50% identical, 50% opposite)

The local realist prediction (solid lines) for quantum correlation for spin (assuming100% detector efficiency). The quantum mechanical prediction is the dotted

(cosine) curve. In this plot the angle is taken between the positive direction of oneaxis and the negative direction of the other axis.

With the measurements oriented atintermediate angles between these basiccases, the existence of local hidden variablescould agree with a linear dependence of thecorrelation in the angle but, according toBell inequality, could not agree with thedependence predicted by quantummechanical theory, namely, that thecorrelation is the cosine of the angle.Experimental results match the curvepredicted by quantum mechanics.[]

Bell achieved his breakthrough by firstderiving the results that he posits localrealism would necessarily yield. Bellclaimed that, without making anyassumptions about the specific form of thetheory beyond requirements of basic consistency, the mathematical inequality he discovered was clearly at odds withthe results (described above) predicted by quantum mechanics and, later, observed experimentally. If correct, Bell'stheorem appears to rule out local hidden variables as a viable explanation of quantum mechanics (though it stillleaves the door open for non-local hidden variables). Bell concluded:

In a theory in which parameters are added to quantum mechanics to determine the results of individualmeasurements, without changing the statistical predictions, there must be a mechanism whereby the setting ofone measuring device can influence the reading of another instrument, however remote. Moreover, the signalinvolved must propagate instantaneously, so that a theory could not be Lorentz invariant.—[4]

Over the years, Bell's theorem has undergone a wide variety of experimental tests. However, various commondeficiencies in the testing of the theorem have been identified, including the detection loophole[6] and thecommunication loophole.[6] Over the years experiments have been gradually improved to better address theseloopholes, but no experiment to date has simultaneously fully addressed all of them.[6] However, it is generallyconsidered unreasonable that such an experiment, if conducted, would give results that are inconsistent with the priorexperiments. For example, Anthony Leggett has commented:

[While] no single existing experiment has simultaneously blocked all of the so-called ‘‘loopholes’’, eachone of those loopholes has been blocked in at least one experiment. Thus, to maintain a local hiddenvariable theory in the face of the existing experiments would appear to require belief in a very peculiarconspiracy of nature.[]

To date, Bell's theorem is generally regarded as supported by a substantial body of evidence and is treated as afundamental principle of physics in mainstream quantum mechanics textbooks.[7][8]

Bell's theorem 4

Importance of the theoremBell's theorem, derived in his seminal 1964 paper titled On the Einstein Podolsky Rosen paradox,[4] has been called,on the assumption that the theory is correct, "the most profound in science".[9] Perhaps of equal importance is Bell'sdeliberate effort to encourage and bring legitimacy to work on the completeness issues, which had fallen intodisrepute.[10] Later in his life, Bell expressed his hope that such work would "continue to inspire those who suspectthat what is proved by the impossibility proofs is lack of imagination."[11]

The title of Bell's seminal article refers to the famous paper by Einstein, Podolsky and Rosen[12] that challenged thecompleteness of quantum mechanics. In his paper, Bell started from the same two assumptions as did EPR, namely(i) reality (that microscopic objects have real properties determining the outcomes of quantum mechanicalmeasurements), and (ii) locality (that reality in one location is not influenced by measurements performedsimultaneously at a distant location). Bell was able to derive from those two assumptions an important result, namelyBell's inequality, implying that at least one of the assumptions must be false.In two respects Bell's 1964 paper was a step forward compared to the EPR paper: firstly, it considered more hiddenvariables than merely the element of physical reality in the EPR paper; and Bell's inequality was, in part, liable to beexperimentally tested, thus raising the possibility of testing the local realism hypothesis. Limitations on such tests todate are noted below. Whereas Bell's paper deals only with deterministic hidden variable theories, Bell's theoremwas later generalized to stochastic theories[] as well, and it was also realised[13] that the theorem is not so much abouthidden variables as about the outcomes of measurements which could have been done instead of the one actuallyperformed. Existence of these variables is called the assumption of realism, or the assumption of counterfactualdefiniteness.After the EPR paper, quantum mechanics was in an unsatisfactory position: either it was incomplete, in the sensethat it failed to account for some elements of physical reality, or it violated the principle of a finite propagation speedof physical effects. In a modified version of the EPR thought experiment, two hypothetical observers, nowcommonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons, preparedat a source in a special state called a spin singlet state. It is the conclusion of EPR that once Alice measures spin inone direction (e.g. on the x axis), Bob's measurement in that direction is determined with certainty, as being theopposite outcome to that of Alice, whereas immediately before Alice's measurement Bob's outcome was onlystatistically determined (i.e., was only a probability, not a certainty); thus, either the spin in each direction is anelement of physical reality, or the effects travel from Alice to Bob instantly.In QM, predictions are formulated in terms of probabilities — for example, the probability that an electron will bedetected in a particular place, or the probability that its spin is up or down. The idea persisted, however, that theelectron in fact has a definite position and spin, and that QM's weakness is its inability to predict those valuesprecisely. The possibility existed that some unknown theory, such as a hidden variables theory, might be able topredict those quantities exactly, while at the same time also being in complete agreement with the probabilitiespredicted by QM. If such a hidden variables theory exists, then because the hidden variables are not described byQM the latter would be an incomplete theory.Two assumptions drove the desire to find a local realist theory:1.1. Objects have a definite state that determines the values of all other measurable properties, such as position and

momentum.2. Effects of local actions, such as measurements, cannot travel faster than the speed of light (in consequence of

special relativity). Thus if observers are sufficiently far apart, a measurement made by one can have no effect on ameasurement made by the other.

In the form of local realism used by Bell, the predictions of the theory result from the application of classicalprobability theory to an underlying parameter space. By a simple argument based on classical probability, he showedthat correlations between measurements are bounded in a way that is violated by QM.

Bell's theorem 5

Bell's theorem seemed to put an end to local realism. This is because, if the theorem is correct, then either quantummechanics or local realism is wrong, as they are mutually exclusive. The paper noted that "it requires littleimagination to envisage the experiments involved actually being made",[4] to determine which of them is correct. Ittook many years and many improvements in technology to perform tests along the lines Bell envisaged. The testsare, in theory, capable of showing whether local hidden variable theories as envisaged by Bell accurately predictexperimental results. The tests are not capable of determining whether Bell has accurately described all local hiddenvariable theories.The Bell test experiments have been interpreted as showing that the Bell inequalities are violated in favour of QM.The no-communication theorem shows that the observers cannot use the effect to communicate (classical)information to each other faster than the speed of light, but the ‘fair sampling’ and ‘no enhancement’ assumptionsrequire more careful consideration (below). That interpretation follows not from any clear demonstration ofsuper-luminal communication in the tests themselves, but solely from Bell's theory that the correctness of thequantum predictions necessarily precludes any local hidden-variable theory. If that theoretical contention is notcorrect, then the "tests" of Bell's theory to date do not show anything either way about the local or non-local natureof the phenomena.

Bell inequalitiesBell inequalities concern measurements made by observers on pairs of particles that have interacted and thenseparated. According to quantum mechanics they are entangled, while local realism would limit the correlation ofsubsequent measurements of the particles.Different authors subsequently derived inequalities similar to Bell´s original inequality, and these are herecollectively termed Bell inequalities. All Bell inequalities describe experiments in which the predicted result fromquantum entanglement differs from that flowing from local realism. The inequalities assume that each quantum-levelobject has a well-defined state that accounts for all its measurable properties and that distant objects do not exchangeinformation faster than the speed of light. These well-defined states are typically called hidden variables, theproperties that Einstein posited when he stated his famous objection to quantum mechanics: "God does not playdice."Bell showed that under quantum mechanics, the mathematics of which contains no local hidden variables, the Bellinequalities can nevertheless be violated: the properties of a particle are not clear, but may be correlated with thoseof another particle due to quantum entanglement, allowing their state to be well defined only after a measurement ismade on either particle. That restriction agrees with the Heisenberg uncertainty principle, a fundamental concept inquantum mechanics.In Bell's words:

Theoretical physicists live in a classical world, looking out into a quantum-mechanical world. The latter wedescribe only subjectively, in terms of procedures and results in our classical domain. (…) Now nobody knowsjust where the boundary between the classical and the quantum domain is situated. (…) More plausible to meis that we will find that there is no boundary. The wave functions would prove to be a provisional orincomplete description of the quantum-mechanical part. It is this possibility, of a homogeneous account of theworld, which is for me the chief motivation of the study of the so-called "hidden variable" possibility.(…) A second motivation is connected with the statistical character of quantum-mechanical predictions. Oncethe incompleteness of the wave function description is suspected, it can be conjectured that random statisticalfluctuations are determined by the extra "hidden" variables — "hidden" because at this stage we can onlyconjecture their existence and certainly cannot control them.(…) A third motivation is in the peculiar character of some quantum-mechanical predictions, which seem almost to cry out for a hidden variable interpretation. This is the famous argument of Einstein, Podolsky and

Bell's theorem 6

Rosen. (…) We will find, in fact, that no local deterministic hidden-variable theory can reproduce all theexperimental predictions of quantum mechanics. This opens the possibility of bringing the question into theexperimental domain, by trying to approximate as well as possible the idealized situations in which localhidden variables and quantum mechanics cannot agree.[14]

In probability theory, repeated measurements of system properties can be regarded as repeated sampling of randomvariables. In Bell's experiment, Alice can choose a detector setting to measure either or and Bob canchoose a detector setting to measure either or . Measurements of Alice and Bob may be somehowcorrelated with each other, but the Bell inequalities say that if the correlation stems from local random variables,there is a limit to the amount of correlation one might expect to see.

Original Bell's inequalityThe original inequality that Bell derived was:[4]

where ρ is the "correlation" of the particle pairs and A, B and C settings of the apparatus. This inequality is not usedin practice. For one thing, it is true only for genuinely "two-outcome" systems, not for the "three-outcome" ones(with possible outcomes of zero as well as +1 and −1) encountered in real experiments. For another, it applies only toa very restricted set of hidden variable theories, namely those for which the outcomes on both sides of theexperiment are always exactly anticorrelated when the analysers are parallel, in agreement with the quantummechanical prediction.Nevertheless, a simple limit of Bell's inequality has the virtue of being quite intuitive. If the result of three differentstatistical coin-flips A, B, and C have the property that:1.1. A and B are the same (both heads or both tails) 99% of the time2.2. B and C are the same 99% of the time,then A and C are the same at least 98% of the time. The number of mismatches between A and B (1/100) plus thenumber of mismatches between B and C (1/100) are together the maximum possible number of mismatches betweenA and C (a simple Boole–Fréchet inequality).In quantum mechanics, however, by letting A, B, and C be the values of the spin of two entangled particles measuredrelative to some axis at 0 degrees, θ degrees, and 2θ degrees respectively, the overlap of the wavefunction betweenthe different angles is proportional to cos(Sθ) ≈ 1–S2θ2/2. The probability that A and B give the same answer is1–ε2, where ε is proportional to θ. This is also the probability that B and C give the same answer.But A and C are the same 1 – (2ε)2 of the time. Choosing the angle so that ε=0.1, A and B are 99% correlated, B andC are 99% correlated, but now A and C are only 96% correlated!Imagine that two entangled particles in a spin singlet are shot out to two distant locations, and the spins of both aremeasured in the direction A. The spins are 100% correlated (actually, anti-correlated, but for this argument that isequivalent). The same is true if both spins are measured in directions B or C. It is safe to conclude that any hiddenvariables that determine the A, B, and C measurements in the two particles are 100% correlated, and can be usedinterchangeably. If A is measured on one particle and B on the other, the correlation between them is 99%. If B ismeasured on one and C on the other, the correlation is 99%. This allows us to conclude that the hidden variablesdetermining A and B are 99% correlated, and B and C are 99% correlated.But if A is measured in one particle and C in the other, the quantum mechanical results are only 96% correlated,which is a contradiction. This intuitive formulation is due to David Mermin, while the small-angle limit isemphasized in Bell's original article.

Bell's theorem 7

CHSH inequalityIn addition to Bell's original inequality,[4] the form given by John Clauser, Michael Horne, Abner Shimony and R. A.Holt,[] (the CHSH form) is especially important,[] as it gives classical limits to the expected correlation for the aboveexperiment conducted by Alice and Bob:

where C denotes correlation.Correlation of observables X, Y is defined as

Where represents the expected or average value of This is a non-normalized form of the correlation coefficient considered in statistics (see Quantum correlation).To formulate Bell's theorem, we formalize local realism as follows:1. There is a probability space and the observed outcomes by both Alice and Bob result by random sampling of

the parameter .2.2. The values observed by Alice or Bob are functions of the local detector settings and the hidden parameter only.

Thus• Value observed by Alice with detector setting is • Value observed by Bob with detector setting is

Implicit in assumption 1) above, the hidden parameter space has a probability measure and the expectation of arandom variable X on with respect to is written

where for accessibility of notation we assume that the probability measure has a density.Bell's inequality. The CHSH inequality (1) holds under the hidden variables assumptions above.For simplicity, let us first assume the observed values are +1 or −1; we remove this assumption in Remark 1 below.Let . Then at least one of

is 0. Thus

and therefore

Remark 1

The correlation inequality (1) still holds if the variables , are allowed to take on any real valuesbetween −1 and +1. Indeed, the relevant idea is that each summand in the above average is bounded above by 2. Thisis easily seen as true in the more general case:

Bell's theorem 8

To justify the upper bound 2 asserted in the last inequality, without loss of generality, we can assume that

In that case

Remark 2Though the important component of the hidden parameter in Bell's original proof is associated with the source andis shared by Alice and Bob, there may be others that are associated with the separate detectors, these others beingconditionally independent given the first, and with conditional probability distributions only depending on thecorresponding local setting (if dependent on the settings at all). This argument was used by Bell in 1971, and againby Clauser and Horne in 1974,[] to justify a generalisation of the theorem forced on them by the real experiments, inwhich detectors were never 100% efficient. The derivations were given in terms of the averages of the outcomesover the local detector variables. The formalisation of local realism was thus effectively changed, replacing A and Bby averages and retaining the symbol but with a slightly different meaning. It was henceforth restricted (in mosttheoretical work) to mean only those components that were associated with the source.However, with the extension proved in Remark 1, CHSH inequality still holds even if the instruments themselvescontain hidden variables. In that case, averaging over the instrument hidden variables gives new variables:

on , which still have values in the range [−1, +1] to which we can apply the previous result.

Bell inequalities are violated by quantum mechanical predictionsIn the usual quantum mechanical formalism, the observables X and Y are represented as self-adjoint operators on aHilbert space. To compute the correlation, assume that X and Y are represented by matrices in a finite dimensionalspace and that X and Y commute; this special case suffices for our purposes below. The von Neumann measurementpostulate states: a series of measurements of an observable X on a series of identical systems in state produces adistribution of real values. By the assumption that observables are finite matrices, this distribution is discrete. Theprobability of observing λ is non-zero if and only if λ is an eigenvalue of the matrix X and moreover the probabilityis

where EX (λ) is the projector corresponding to the eigenvalue λ. The system state immediately after the measurementis

From this, we can show that the correlation of commuting observables X and Y in a pure state is

We apply this fact in the context of the EPR paradox. The measurements performed by Alice and Bob are spin measurements on electrons. Alice can choose between two detector settings labelled a and a′; these settings

Bell's theorem 9

correspond to measurement of spin along the z or the x axis. Bob can choose between two detector settings labelled band b′; these correspond to measurement of spin along the z′ or x′ axis, where the x′ – z′ coordinate system is rotated135° relative to the x – z coordinate system. The spin observables are represented by the 2 × 2 self-adjoint matrices:

These are the Pauli spin matrices normalized so that the corresponding eigenvalues are +1, −1. As is customary, wedenote the eigenvectors of Sx by

Let be the spin singlet state for a pair of electrons discussed in the EPR paradox. This is a specially constructedstate described by the following vector in the tensor product

Now let us apply the CHSH formalism to the measurements that can be performed by Alice and Bob.

Illustration of Bell test for spin 1/2 particles. Source produces spin singlet pairs, oneparticle of each pair is sent to Alice and the other to Bob. Each performs one of the two

spin measurements.

The operators , correspond to Bob's spin measurements along x′ and z′. Note that the A operatorscommute with the B operators, so we can apply our calculation for the correlation. In this case, we can show that theCHSH inequality fails. In fact, a straightforward calculation shows that

and

so that

Bell's Theorem: If the quantum mechanical formalism is correct, then the system consisting of a pair of entangled electrons cannot satisfy the principle of local realism. Note that is indeed the upper bound for quantum mechanics called Tsirelson's bound. The operators giving this maximal value are always isomorphic to the Pauli

Bell's theorem 10

matrices.

Practical experiments testing Bell's theorem

Scheme of a "two-channel" Bell testThe source S produces pairs of "photons", sent in opposite directions. Each photon

encounters a two-channel polariser whose orientation (a or b) can be set by theexperimenter. Emerging signals from each channel are detected and coincidences of four

types (++, −−, +− and −+) counted by the coincidence monitor.

Experimental tests can determinewhether the Bell inequalities requiredby local realism hold up to theempirical evidence.Bell's inequalities are tested by"coincidence counts" from a Bell testexperiment such as the optical oneshown in the diagram. Pairs ofparticles are emitted as a result of aquantum process, analysed withrespect to some key property such aspolarisation direction, then detected.The setting (orientations) of theanalysers are selected by the

experimenter.Bell test experiments to date overwhelmingly violate Bell's inequality. Indeed, a table of Bell test experimentsperformed prior to 1986 is given in 4.5 of Redhead, 1987.[15] Of the thirteen experiments listed, only two reachedresults contradictory to quantum mechanics; moreover, according to the same source, when the experiments wererepeated, "the discrepancies with QM could not be reproduced".

Nevertheless, the issue is not conclusively settled. According to Shimony's 2004 Stanford Encyclopedia overviewarticle:[6]

Most of the dozens of experiments performed so far have favored Quantum Mechanics, but not decisivelybecause of the 'detection loopholes' or the 'communication loophole.' The latter has been nearly decisivelyblocked by a recent experiment and there is a good prospect for blocking the former.

To explore the 'detection loophole', one must distinguish the classes of homogeneous and inhomogeneous Bellinequality.The standard assumption in Quantum Optics is that "all photons of given frequency, direction and polarization areidentical" so that photodetectors treat all incident photons on an equal basis. Such a fair sampling assumptiongenerally goes unacknowledged, yet it effectively limits the range of local theories to those that conceive of the lightfield as corpuscular. The assumption excludes a large family of local realist theories, in particular, Max Planck'sdescription. We must remember the cautionary words of Albert Einstein[16] shortly before he died: "Nowadays everyTom, Dick and Harry ('jeder Kerl' in German original) thinks he knows what a photon is, but he is mistaken".Those who maintain the concept of duality, or simply of light being a wave, recognize the possibility or actuality thatthe emitted atomic light signals have a range of amplitudes and, furthermore, that the amplitudes are modified whenthe signal passes through analyzing devices such as polarizers and beam splitters. It follows that not all signals havethe same detection probability.[17]

Bell's theorem 11

Two classes of Bell inequalitiesThe fair sampling problem was faced openly in the 1970s. In early designs of their 1973 experiment, Freedman andClauser[] used fair sampling in the form of the Clauser–Horne–Shimony–Holt (CHSH[]) hypothesis. However,shortly afterwards Clauser and Horne[] made the important distinction between inhomogeneous (IBI) andhomogeneous (HBI) Bell inequalities. Testing an IBI requires that we compare certain coincidence rates in twoseparated detectors with the singles rates of the two detectors. Nobody needed to perform the experiment, becausesingles rates with all detectors in the 1970s were at least ten times all the coincidence rates. So, taking into accountthis low detector efficiency, the QM prediction actually satisfied the IBI. To arrive at an experimental design inwhich the QM prediction violates IBI we require detectors whose efficiency exceeds 82% for singlet states, but havevery low dark rate and short dead and resolving times. This is well above the 30% achievable[18] so Shimony’soptimism in the Stanford Encyclopedia, quoted in the preceding section, appears over-stated.

Practical challengesBecause detectors don't detect a large fraction of all photons, Clauser and Horne[] recognized that testing Bell'sinequality requires some extra assumptions. They introduced the No Enhancement Hypothesis (NEH):

A light signal, originating in an atomic cascade for example, has a certain probability of activating a detector.Then, if a polarizer is interposed between the cascade and the detector, the detection probability cannotincrease.

Given this assumption, there is a Bell inequality between the coincidence rates with polarizers and coincidence rateswithout polarizers.The experiment was performed by Freedman and Clauser,[] who found that the Bell's inequality was violated. So theno-enhancement hypothesis cannot be true in a local hidden variables model. The Freedman–Clauser experimentreveals that local hidden variables imply the new phenomenon of signal enhancement:

In the total set of signals from an atomic cascade there is a subset whose detection probability increases as aresult of passing through a linear polarizer.

This is perhaps not surprising, as it is known that adding noise to data can, in the presence of a threshold, help revealhidden signals (this property is known[19] as stochastic resonance). One cannot conclude that this is the onlylocal-realist alternative to Quantum Optics, but it does show that the word loophole is biased. Moreover, the analysisleads us to recognize that the Bell-inequality experiments, rather than showing a breakdown of realism or locality,are capable of revealing important new phenomena.

Theoretical challengesMost advocates of the hidden variables idea believe that experiments have ruled out local hidden variables. They areready to give up locality, explaining the violation of Bell's inequality by means of a non-local hidden variable theory,in which the particles exchange information about their states. This is the basis of the Bohm interpretation ofquantum mechanics, which requires that all particles in the universe be able to instantaneously exchange informationwith all others. A 2007 experiment ruled out a large class of non-Bohmian non-local hidden variable theories.[20]

If the hidden variables can communicate with each other faster than light, Bell's inequality can easily be violated.Once one particle is measured, it can communicate the necessary correlations to the other particle. Since in relativitythe notion of simultaneity is not absolute, this is unattractive. One idea is to replace instantaneous communicationwith a process that travels backwards in time along the past Light cone. This is the idea behind a transactionalinterpretation of quantum mechanics, which interprets the statistical emergence of a quantum history as a gradualcoming to agreement between histories that go both forward and backward in time.[21]

A few advocates of deterministic models have not given up on local hidden variables. For example, Gerard 't Hoofthas argued that the superdeterminism loophole cannot be dismissed.[22][23]

Bell's theorem 12

The quantum mechanical wavefunction can also provide a local realistic description, if the wavefunction values areinterpreted as the fundamental quantities that describe reality. Such an approach is called a many-worldsinterpretation of quantum mechanics. In this view, two distant observers both split into superpositions whenmeasuring a spin. The Bell inequality violations are no longer counterintuitive, because it is not clear which copy ofthe observer B observer A will see when going to compare notes. If reality includes all the different outcomes,locality in physical space (not outcome space) places no restrictions on how the split observers can meet up.This implies that there is a subtle assumption in the argument that realism is incompatible with quantum mechanicsand locality. The assumption, in its weakest form, is called counterfactual definiteness. This states that if the resultsof an experiment are always observed to be definite, there is a quantity that determines what the outcome would havebeen even if you don't do the experiment.Many worlds interpretations are not only counterfactually indefinite, they are factually indefinite. The results of allexperiments, even ones that have been performed, are not uniquely determined.E. T. Jaynes[24] pointed out two hidden assumptions in Bell Inequality that could limit its generality. According tohim:1.1. Bell interpreted conditional probability P(X|Y) as a causal inference, i.e. Y exerted a causal inference on X in

reality. However, P(X|Y) actually only means logical inference (deduction). Causes cannot travel faster than lightor backward in time, but deduction can.

2.2. Bell's inequality does not apply to some possible hidden variable theories. It only applies to a certain class oflocal hidden variable theories. In fact, it might have just missed the kind of hidden variable theories that Einsteinis most interested in.

Final remarksThe violations of Bell's inequalities, due to quantum entanglement, just provide the definite demonstration ofsomething that was already strongly suspected, that quantum physics cannot be represented by any version of theclassical picture of physics.[25] Some earlier elements that had seemed incompatible with classical pictures includedapparent complementarity and (hypothesized) wavefunction collapse. Complementarity is now seen not as anindependent ingredient of the quantum picture but rather as a direct consequence of the Quantum decoherenceexpected from the quantum formalism itself. The possibility of wavefunction collapse is now seen as one possibleproblematic ingredient of some interpretations, rather than as an essential part of quantum mechanics. The Bellviolations show that no resolution of such issues can avoid the ultimate strangeness of quantum behavior.[26]

The EPR paper "pinpointed" the unusual properties of the entangled states, e.g. the above-mentioned singlet state,which is the foundation for present-day applications of quantum physics, such as quantum cryptography; oneapplication involves the measurement of quantum entanglement as a physical source of bits for Rabin's oblivioustransfer protocol. This strange non-locality was originally supposed to be a Reductio ad absurdum, because thestandard interpretation could easily do away with action-at-a-distance by simply assigning to each particle definitespin-states. Bell's theorem showed that the "entangledness" prediction of quantum mechanics has a degree ofnon-locality that cannot be explained away by any local theory.In well-defined Bell experiments (see the paragraph on "test experiments") one can now falsify either quantummechanics or Einstein's quasi-classical assumptions: currently many experiments of this kind have been performed,and the experimental results support quantum mechanics, though some believe that detectors give a biased sample ofphotons, so that until nearly every photon pair generated is observed there will be loopholes.What is powerful about Bell's theorem is that it doesn't refer to any particular physical theory. What makes Bell'stheorem unique and powerful is that it shows that nature violates the most general assumptions behind classicalpictures, not just details of some particular models. No combination of local deterministic and local random variablescan reproduce the phenomena predicted by quantum mechanics and repeatedly observed in experiments.[27]

Bell's theorem 13

Notes[1] N. David Mermin, "Is the moon there when nobody looks? Reality and the quantum theory" in Physics Today, April, 38-47 (1985). PDF

(http:/ / www. phy. duke. edu/ undergraduate/ physics-articles/ mermin-is-the-moon-there-when-nobody-looks-physics-today-1985. pdf).[2] Henry Stapp, Nuovo Cimento 40B, 191 (1977). PDF (http:/ / www-physics. lbl. gov/ ~stapp/ NCimento. pdf).[5] Bohm, David Quantum Theory. Prentice−Hall, 1951.[6] Article on Bell's Theorem (http:/ / plato. stanford. edu/ entries/ bell-theorem) by Abner Shimony in the Stanford Encyclopedia of Philosophy,

(2004).[9][9] Stapp, 1975[10] Bell, JS, "On the impossible pilot wave." Foundations of Physics (1982) 12:989–99. Reprinted in Speakable and unspeakable in quantum

mechanics: collected papers on quantum philosophy. CUP, 2004, p. 160.[11] Bell, JS, "On the impossible pilot wave." Foundations of Physics (1982) 12:989–99. Reprinted in Speakable and unspeakable in quantum

mechanics: collected papers on quantum philosophy. CUP, 2004, p. 161.[14] Bell, JS, Speakable and unspeakable in quantum mechanics: Introduction remarks at Naples–Amalfi meeting., 1984. Reprinted in Speakable

and unspeakable in quantum mechanics: collected papers on quantum philosophy. CUP, 2004, p. 29.[15] M. Redhead, Incompleteness, Nonlocality and Realism, Clarendon Press (1987)[16] A. Einstein in Correspondance Einstein–Besso, p.265 (Herman, Paris, 1979)[17] Marshall and Santos, Semiclassical optics as an alternative to nonlocality (http:/ / www. crisisinphysics. co. uk/ optrev. pdf) Recent Research

Developments in Optics 2:683–717 (2002) ISBN 81-7736-140-6

References• A. Aspect et al., Experimental Tests of Realistic Local Theories via Bell's Theorem, Phys. Rev. Lett. 47, 460

(1981)• A. Aspect et al., Experimental Realization of Einstein–Podolsky–Rosen–Bohm Gedankenexperiment: A New

Violation of Bell's Inequalities, Phys. Rev. Lett. 49, 91 (1982).• A. Aspect et al., Experimental Test of Bell's Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804

(1982).• A. Aspect and P. Grangier, About resonant scattering and other hypothetical effects in the Orsay atomic-cascade

experiment tests of Bell inequalities: a discussion and some new experimental data, Lettere al Nuovo Cimento 43,345 (1985)

• B. D'Espagnat, The Quantum Theory and Reality (http:/ / www. sciam. com/ media/ pdf/ 197911_0158. pdf),Scientific American, 241, 158 (1979)

• J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966)• J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 3, 195–200 (1964)• J. S. Bell, Introduction to the hidden variable question, Proceedings of the International School of Physics 'Enrico

Fermi', Course IL, Foundations of Quantum Mechanics (1971) 171–81• J. S. Bell, Bertlmann’s socks and the nature of reality, Journal de Physique, Colloque C2, suppl. au numero 3,

Tome 42 (1981) pp C2 41–61• J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press 1987) [A collection

of Bell's papers, including all of the above.]• J. F. Clauser and A. Shimony, Bell's theorem: experimental tests and implications, Reports on Progress in Physics

41, 1881 (1978)• J. F. Clauser and M. A. Horne, Phys. Rev D 10, 526–535 (1974)• E. S. Fry, T. Walther and S. Li, Proposal for a loophole-free test of the Bell inequalities, Phys. Rev. A 52, 4381

(1995)• E. S. Fry, and T. Walther, Atom based tests of the Bell Inequalities — the legacy of John Bell continues, pp

103–117 of Quantum [Un]speakables, R.A. Bertlmann and A. Zeilinger (eds.) (Springer, Berlin-Heidelberg-NewYork, 2002)

• R. B. Griffiths, Consistent Quantum Theory', Cambridge University Press (2002).• L. Hardy, Nonlocality for 2 particles without inequalities for almost all entangled states. Physical Review Letters

71 (11) 1665–1668 (1993)

Bell's theorem 14

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press(2000)

• P. Pearle, Hidden-Variable Example Based upon Data Rejection, Physical Review D 2, 1418–25 (1970)• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993.• P. Pluch, Theory of Quantum Probability, PhD Thesis, University of Klagenfurt, 2006.• B. C. van Frassen, Quantum Mechanics, Clarendon Press, 1991.• M.A. Rowe, D. Kielpinski, V. Meyer, C.A. Sackett, W.M. Itano, C. Monroe, and D.J. Wineland, Experimental

violation of Bell's inequalities with efficient detection,(Nature, 409, 791–794, 2001).• S. Sulcs, The Nature of Light and Twentieth Century Experimental Physics, Foundations of Science 8, 365–391

(2003)• S. Gröblacher et al., An experimental test of non-local realism,(Nature, 446, 871–875, 2007).• D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, and C. Monroe, Bell Inequality Violation with Two

Remote Atomic Qubits, Phys. Rev. Lett. 100, 150404 (2008).• The comic Dilbert, by Scott Adams, refers to Bell's Theorem in the 1992-09-21 (http:/ / www. dilbert. com/ strips/

comic/ 1992-09-21/ ) and 1992-09-22 (http:/ / www. dilbert. com/ strips/ comic/ 1992-09-22/ ) strips.

Further readingThe following are intended for general audiences.• Amir D. Aczel, Entanglement: The greatest mystery in physics (Four Walls Eight Windows, New York, 2001).• A. Afriat and F. Selleri, The Einstein, Podolsky and Rosen Paradox (Plenum Press, New York and London, 1999)• J. Baggott, The Meaning of Quantum Theory (Oxford University Press, 1992)• N. David Mermin, "Is the moon there when nobody looks? Reality and the quantum theory", in Physics Today,

April 1985, pp. 38–47.• Louisa Gilder, The Age of Entanglement: When Quantum Physics Was Reborn (New York: Alfred A. Knopf,

2008)• Brian Greene, The Fabric of the Cosmos (Vintage, 2004, ISBN 0-375-72720-5)• Nick Herbert, Quantum Reality: Beyond the New Physics (Anchor, 1987, ISBN 0-385-23569-0)• D. Wick, The infamous boundary: seven decades of controversy in quantum physics (Birkhauser, Boston 1995)• R. Anton Wilson, Prometheus Rising (New Falcon Publications, 1997, ISBN 1-56184-056-4)• Gary Zukav "The Dancing Wu Li Masters" (Perennial Classics, 2001, ISBN 0-06-095968-1)

External links• An explanation of Bell's Theorem (http:/ / www. ncsu. edu/ felder-public/ kenny/ papers/ bell. html), based on N.

D. Mermin's article, Mermin, N. D. (1981). "Bringing home the atomic world: Quantum mysteries for anybody".American Journal of Physics 49 (10): 940. Bibcode: 1981AmJPh..49..940M (http:/ / adsabs. harvard. edu/ abs/1981AmJPh. . 49. . 940M). doi: 10.1119/1.12594 (http:/ / dx. doi. org/ 10. 1119/ 1. 12594).

• Mermin: Spooky Actions At A Distance? Oppenheimer Lecture (http:/ / www. youtube. com/watch?v=ta09WXiUqcQ)

• Quantum Entanglement (http:/ / www. ipod. org. uk/ reality/ reality_entangled. asp) Includes a simple explanationof Bell's Inequality.

• Bell's theorem on arXiv.org (http:/ / xstructure. inr. ac. ru/ x-bin/ theme3. py?level=2& index1=369244)• Interactive experiments with single photons: entanglement and Bell´s theorem (http:/ / www. didaktik. physik.

uni-erlangen. de/ quantumlab/ english/ index. html)• Bell's Inequalities: Obscurantist Obfuscation or Condign Confabulation? (http:/ / groups. google. com/ groups/

profile?hl=en& show=more& enc_user=8YcXCQ4AAABUc-oUoA1Uy7yFEaUY6YXQ& group=sci. physics)

EPR paradox 15

EPR paradox

Albert Einstein

Quantummechanics

IntroductionGlossary · History

The EPR paradox is an early and influential critique leveled against quantum mechanics. Albert Einstein and hiscolleagues Boris Podolsky and Nathan Rosen (known collectively as EPR) designed a thought experiment intendedto reveal what they believed to be inadequacies of quantum mechanics. To that end they pointed to a consequence ofquantum mechanics that its supporters had not noticed.According to quantum mechanics, under some conditions, a pair of quantum systems may be described by a singlewave function, which encodes the probabilities of the outcomes of experiments that may be performed on the twosystems, whether jointly or individually.At the time the EPR article was written, it was known from experiments that the outcome of an experimentsometimes cannot be uniquely predicted. An example of such indeterminacy can be seen when a beam of light isincident on a half-silvered mirror. One half of the beam will reflect, the other will pass. But what happens when wekeep decreasing the intensity of the beam, so that only one photon is in transit at any time? Whether any one photonwill reflect or transmit cannot be predicted quantum mechanically.The routine explanation of this effect was, at that time, provided by Heisenberg's uncertainty principle.[citation needed]

Physical quantities come in pairs which are called conjugate quantities. Examples of such conjugate pairs areposition and momentum of a particle and components of spin measured around different axes. When one quantitywas measured, and became determined, the conjugated quantity became indeterminate. Heisenberg explained this asa disturbance caused by measurement.The EPR paper, written in 1935, was intended to illustrate that this explanation is inadequate. It considered twoentangled particles, referred to as A and B, and pointed out that measuring a quantity of a particle A will cause theconjugated quantity of particle B to become undetermined, even if there was no contact, no classical disturbance.

EPR paradox 16

Heisenberg's principle was an attempt to provide a classical explanation of a quantum effect sometimes callednon-locality. According to EPR there were two possible explanations. Either there was some interaction between theparticles, even though they were separated, or the information about the outcome of all possible measurements wasalready present in both particles.The EPR authors preferred the second explanation according to which that information was encoded in some 'hiddenparameters'. The first explanation, that an effect propagated instantly, across a distance, is in conflict with the theoryof relativity.They then concluded that quantum mechanics was incomplete since, in its formalism, there was no space for suchhidden parameters.Bell's theorem is generally understood to have demonstrated that the EPR authors' preferred explanation was notviable. Most physicists who have examined the matter concur that experiments, such as those of Alain Aspect andhis group, have confirmed that physical probabilities, as predicted by quantum theory, do show the phenomena ofBell-inequality violations that are considered to invalidate EPR's preferred "local hidden-variables" type ofexplanation for the correlations to which EPR first drew attention.[1][]

History of EPR developmentsThe article that first brought forth these matters, "Can Quantum-Mechanical Description of Physical Reality BeConsidered Complete?" was published in 1935.[] Einstein struggled to the end of his life for a theory that could bettercomply with his idea of causality, protesting against the view that there exists no objective physical reality other thanthat which is revealed through measurement interpreted in terms of quantum mechanical formalism. However, sinceEinstein's death, experiments analogous to the one described in the EPR paper have been carried out, starting in 1976by French scientists Lamehi-Rachti and Mittig[2] at the Saclay Nuclear Research Centre. These experiments appearto show that the local realism idea is false.[3]

Quantum mechanics and its interpretationSince the early twentieth century, quantum theory has proved to be successful in describing accurately the physicalreality of the mesoscopic and microscopic world, in multiple reproducible physics experiments.Quantum mechanics was developed with the aim of describing atoms and explaining the observed spectral lines in ameasurement apparatus. Although disputed, it has yet to be seriously challenged. Philosophical interpretations ofquantum phenomena, however, are another matter: the question of how to interpret the mathematical formulation ofquantum mechanics has given rise to a variety of different answers from people of different philosophicalpersuasions (see Interpretations of quantum mechanics).Quantum theory and quantum mechanics do not provide single measurement outcomes in a deterministic way. According to the understanding of quantum mechanics known as the Copenhagen interpretation, measurement causes an instantaneous collapse of the wave function describing the quantum system into an eigenstate of the observable that was measured. Einstein characterized this imagined collapse in the 1927 Solvay Conference. He presented a thought experiment in which electrons are introduced through a small hole in a sphere whose inner surface serves as a detection screen. The electrons will contact the spherical detection screen in a widely dispersed manner. Those electrons, however, are all individually described by wave fronts that expand in all directions from the point of entry. A wave as it is understood in everyday life would paint a large area of the detection screen, but the electrons would be found to impact the screen at single points and would eventually form a pattern in keeping with the probabilities described by their identical wave functions. Einstein asks what makes each electron's wave front "collapse" at its respective location. Why do the electrons appear as single bright scintillations rather than as dim washes of energy across the surface? Why does any single electron appear at one point rather than some alternative point? The behavior of the electrons gives the impression of some signal having been sent to all possible points of

EPR paradox 17

contact that would have nullified all but one of them, or, in other words, would have preferentially selected a singlepoint to the exclusion of all others.[4]

Einstein's oppositionEinstein was the most prominent opponent of the Copenhagen interpretation. In his view, quantum mechanics isincomplete. Commenting on this, other writers (such as John von Neumann[5] and David Bohm[6]) hypothesized thatconsequently there would have to be 'hidden' variables responsible for random measurement results, somethingwhich was not expressly claimed in the original paper.The 1935 EPR paper [7] condensed the philosophical discussion into a physical argument. The authors claim thatgiven a specific experiment, in which the outcome of a measurement is known before the measurement takes place,there must exist something in the real world, an "element of reality", that determines the measurement outcome.They postulate that these elements of reality are local, in the sense that each belongs to a certain point in spacetime.Each element may only be influenced by events which are located in the backward light cone of its point inspacetime (i.e. the past). These claims are founded on assumptions about nature that constitute what is now known aslocal realism.Though the EPR paper has often been taken as an exact expression of Einstein's views, it was primarily authored byPodolsky, based on discussions at the Institute for Advanced Study with Einstein and Rosen. Einstein later expressedto Erwin Schrödinger that, "it did not come out as well as I had originally wanted; rather, the essential thing was, soto speak, smothered by the formalism."[8] In 1936 Einstein presented an individual account of his local realistideas.[9]

Description of the paradoxThe original EPR paradox challenges the prediction of quantum mechanics that it is impossible to know both theposition and the momentum of a quantum particle. This challenge can be extended to other pairs of physicalproperties.

EPR paperThe original paper purports to describe what must happen to "two systems I and II, which we permit to interact ...",and, after some time, "we suppose that there is no longer any interaction between the two parts." In the words ofKumar (2009), the EPR description involves "two particles, A and B, [which] interact briefly and then move off inopposite directions."[] According to Heisenberg's uncertainty principle, it is impossible to measure both themomentum and the position of particle B exactly. However, according to Kumar, it is possible to measure the exactposition of particle A. By calculation, therefore, with the exact position of particle A known, the exact position ofparticle B can be known. Also, the exact momentum of particle B can be measured, so the exact momentum ofparticle A can be worked out. Kumar writes: "EPR argued that they had proved that ... [particle] B can havesimultaneously exact values of position and momentum. ... Particle B has a position that is real and a momentum thatis real."

EPR appeared to have contrived a means to establish the exact values of either the momentum or theposition of B due to measurements made on particle A, without the slightest possibility of particle Bbeing physically disturbed.[]

EPR tried to set up a paradox to question the range of true application of Quantum Mechanics: Quantum theorypredicts that both values cannot be known for a particle, and yet the EPR thought experiment purports to show thatthey must all have determinate values. The EPR paper says: "We are thus forced to conclude that thequantum-mechanical description of physical reality given by wave functions is not complete."[]

The EPR paper ends by saying:

EPR paradox 18

While we have thus shown that the wave function does not provide a complete description of thephysical reality, we left open the question of whether or not such a description exists. We believe,however, that such a theory is possible.

Measurements on an entangled stateWe have a source that emits electron–positron pairs, with the electron sent to destination A, where there is anobserver named Alice, and the positron sent to destination B, where there is an observer named Bob. According toquantum mechanics, we can arrange our source so that each emitted pair occupies a quantum state called a spinsinglet. The particles are thus said to be entangled. This can be viewed as a quantum superposition of two states,which we call state I and state II. In state I, the electron has spin pointing upward along the z-axis (+z) and thepositron has spin pointing downward along the z-axis (−z). In state II, the electron has spin −z and the positron hasspin +z. Therefore, it is impossible (without measuring) to know the definite state of spin of either particle in the spinsinglet.[]:421-422

The EPR thought experiment, performed with electron–positron pairs. A source (center) sends particles toward two observers, electrons toAlice (left) and positrons to Bob (right), who can perform spin measurements.

Alice now measures the spin along the z-axis. She can obtain one of two possible outcomes: +z or −z. Suppose shegets +z. According to the Copenhagen interpretation of quantum mechanics, the quantum state of the systemcollapses into state I. The quantum state determines the probable outcomes of any measurement performed on thesystem. In this case, if Bob subsequently measures spin along the z-axis, there is 100% probability that he will obtain−z. Similarly, if Alice gets −z, Bob will get +z.There is, of course, nothing special about choosing the z-axis: according to quantum mechanics the spin singlet statemay equally well be expressed as a superposition of spin states pointing in the x direction.[]:318 Suppose that Aliceand Bob had decided to measure spin along the x-axis. We'll call these states Ia and IIa. In state Ia, Alice's electronhas spin +x and Bob's positron has spin −x. In state IIa, Alice's electron has spin −x and Bob's positron has spin +x.Therefore, if Alice measures +x, the system 'collapses' into state Ia, and Bob will get −x. If Alice measures −x, thesystem collapses into state IIa, and Bob will get +x.Whatever axis their spins are measured along, they are always found to be opposite. This can only be explained if the particles are linked in some way. Either they were created with a definite (opposite) spin about every axis—a "hidden variable" argument—or they are linked so that one electron "feels" which axis the other is having its spin measured along, and becomes its opposite about that one axis—an "entanglement" argument. Moreover, if the two particles have their spins measured about different axes, once the electron's spin has been measured about the x-axis (and the positron's spin about the x-axis deduced), the positron's spin about the z-axis will no longer be certain, as if (a) it knows that the measurement has taken place, or (b) it has a definite spin already, about a second axis—a hidden variable. However, it turns out that the predictions of Quantum Mechanics, which have been confirmed by

EPR paradox 19

experiment, cannot be explained by any hidden variable theory. This is demonstrated in Bell's theorem.[10]

In quantum mechanics, the x-spin and z-spin are "incompatible observables", meaning the Heisenberg uncertaintyprinciple applies to alternating measurements of them: a quantum state cannot possess a definite value for both ofthese variables. Suppose Alice measures the z-spin and obtains +z, so that the quantum state collapses into state I.Now, instead of measuring the z-spin as well, Bob measures the x-spin. According to quantum mechanics, when thesystem is in state I, Bob's x-spin measurement will have a 50% probability of producing +x and a 50% probability of-x. It is impossible to predict which outcome will appear until Bob actually performs the measurement.Here is the crux of the matter. You might imagine that, when Bob measures the x-spin of his positron, he wouldget an answer with absolute certainty, since prior to this he hasn't disturbed his particle at all. But Bob's positron hasa 50% probability of producing +x and a 50% probability of −x—so the outcome is not certain. Bob's positron"knows" that Alice's electron has been measured, and its z-spin detected, and hence B's z-spin calculated, so itsx-spin is uncertain.Put another way, how does Bob's positron know which way to point if Alice decides (based on informationunavailable to Bob) to measure x (i.e. to be the opposite of Alice's electron's spin about the x-axis) and also how topoint if Alice measures z, since it is only supposed to know one thing at a time? The Copenhagen interpretation rulesthat say the wave function "collapses" at the time of measurement, so there must be action at a distance(entanglement) or the positron must know more than it's supposed to (hidden variables).Here is the paradox summed up:

It is one thing to say that physical measurement of the first particle's momentum affects uncertainty in its ownposition, but to say that measuring the first particle's momentum affects the uncertainty in the position of the other isanother thing altogether. Einstein, Podolsky and Rosen asked how can the second particle "know" to have preciselydefined momentum but uncertain position? Since this implies that one particle is communicating with the otherinstantaneously across space, i.e. faster than light, this is the "paradox".Incidentally, Bell used spin as his example, but many types of physical quantities—referred to as "observables" inquantum mechanics—can be used. The EPR paper used momentum for the observable. Experimental realisations ofthe EPR scenario often use photon polarization, because polarized photons are easy to prepare and measure.

Locality in the EPR experimentThe principle of locality states that physical processes occurring at one place should have no immediate effect on theelements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seemsto be a consequence of special relativity, which states that information can never be transmitted faster than the speedof light without violating causality. It is generally believed that any theory which violates causality would also beinternally inconsistent, and thus useless.[]:427-428[]

It turns out that the usual rules for combining quantum mechanical and classical descriptions violate the principle oflocality without violating causality.[]:427-428[] Causality is preserved because there is no way for Alice to transmitmessages (i.e. information) to Bob by manipulating her measurement axis. Whichever axis she uses, she has a 50%probability of obtaining "+" and 50% probability of obtaining "−", completely at random; according to quantummechanics, it is fundamentally impossible for her to influence what result she gets. Furthermore, Bob is only able toperform his measurement once: there is a fundamental property of quantum mechanics, known as the "no cloningtheorem", which makes it impossible for him to make a million copies of the electron he receives, perform a spinmeasurement on each, and look at the statistical distribution of the results. Therefore, in the one measurement he isallowed to make, there is a 50% probability of getting "+" and 50% of getting "−", regardless of whether or not hisaxis is aligned with Alice's.However, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen were unwilling to abandon it. Einstein derided the quantum mechanical predictions as "spooky action at a distance". The

EPR paradox 20

conclusion they drew was that quantum mechanics is not a complete theory.[]

In recent years, however, doubt has been cast on EPR's conclusion due to developments in understanding localityand especially quantum decoherence. The word locality has several different meanings in physics. For example, inquantum field theory "locality" means that quantum fields at different points of space do not interact with oneanother. However, quantum field theories that are "local" in this sense appear to violate the principle of locality asdefined by EPR, but they nevertheless do not violate locality in a more general sense. Wavefunction collapse can beviewed as an epiphenomenon of quantum decoherence, which in turn is nothing more than an effect of theunderlying local time evolution of the wavefunction of a system and all of its environment. Since the underlyingbehaviour doesn't violate local causality, it follows that neither does the additional effect of wavefunction collapse,whether real or apparent. Therefore, as outlined in the example above, neither the EPR experiment nor any quantumexperiment demonstrates that faster-than-light signaling is possible.

Resolving the paradox

Hidden variablesThere are several ways to resolve the EPR paradox. The one suggested by EPR is that quantum mechanics, despiteits success in a wide variety of experimental scenarios, is actually an incomplete theory. In other words, there issome yet undiscovered theory of nature to which quantum mechanics acts as a kind of statistical approximation(albeit an exceedingly successful one). Unlike quantum mechanics, the more complete theory contains variablescorresponding to all the "elements of reality". There must be some unknown mechanism acting on these variables togive rise to the observed effects of "non-commuting quantum observables", i.e. the Heisenberg uncertainty principle.Such a theory is called a hidden variable theory.To illustrate this idea, we can formulate a very simple hidden variable theory for the above thought experiment. Onesupposes that the quantum spin-singlet states emitted by the source are actually approximate descriptions for "true"physical states possessing definite values for the z-spin and x-spin. In these "true" states, the positron going to Bobalways has spin values opposite to the electron going to Alice, but the values are otherwise completely random. Forexample, the first pair emitted by the source might be "(+z, −x) to Alice and (−z, +x) to Bob", the next pair "(−z, −x)to Alice and (+z, +x) to Bob", and so forth. Therefore, if Bob's measurement axis is aligned with Alice's, he willnecessarily get the opposite of whatever Alice gets; otherwise, he will get "+" and "−" with equal probability.Assuming we restrict our measurements to the z- and x-axes, such a hidden variable theory is experimentallyindistinguishable from quantum mechanics. In reality, there may be an infinite number of axes along which Aliceand Bob can perform their measurements, so there would have to be an infinite number of independent hiddenvariables. However, this is not a serious problem; we have formulated a very simplistic hidden variable theory, and amore sophisticated theory might be able to patch it up. It turns out that there is a much more serious challenge to theidea of hidden variables.

Bell's inequality

In 1964, John Bell showed that the predictions of quantum mechanics in the EPR thought experiment aresignificantly different from the predictions of a particular class of hidden variable theories (the local hidden variabletheories). Roughly speaking, quantum mechanics has a much stronger statistical correlation with measurementresults performed on different axes than do these hidden variable theories. These differences, expressed usinginequality relations known as "Bell's inequalities", are in principle experimentally detectable. Later work byEberhard showed that the key properties of local hidden variable theories which lead to Bell's inequalities are localityand counter-factual definiteness. Any theory in which these principles apply produces the inequalities. Arthur Finesubsequently showed that any theory satisfying the inequalities can be modeled by a local hidden variable theory.

EPR paradox 21

After the publication of Bell's paper, a variety of experiments were devised to test Bell's inequalities (experimentswhich generally rely on photon polarization measurement). All the experiments conducted to date have foundbehavior in line with the predictions of standard quantum mechanics theory.However, Bell's theorem does not apply to all possible philosophically realist theories. It is a common misconceptionthat quantum mechanics is inconsistent with all notions of philosophical realism, but realist interpretations ofquantum mechanics are possible, although, as discussed above, such interpretations must reject either locality orcounter-factual definiteness. Mainstream physics prefers to keep locality, while striving also to maintain a notion ofrealism that nevertheless rejects counter-factual definiteness. Examples of such mainstream realist interpretations arethe consistent histories interpretation and the transactional interpretation. Fine's work showed that, taking locality asa given, there exist scenarios in which two statistical variables are correlated in a manner inconsistent withcounter-factual definiteness, and that such scenarios are no more mysterious than any other, despite the inconsistencywith counter-factual definiteness seeming 'counter-intuitive'.Violation of locality is difficult to reconcile with special relativity, and is thought to be incompatible with theprinciple of causality. On the other hand the Bohm interpretation of quantum mechanics keeps counter-factualdefiniteness while introducing a conjectured non-local mechanism in form of the 'quantum potential', defined as oneof the terms of the Schrödinger equation. Some workers in the field have also attempted to formulate hidden variabletheories that exploit loopholes in actual experiments, such as the assumptions made in interpreting experimental data,although no theory has been proposed that can reproduce all the results of quantum mechanics.There are also individual EPR-like experiments that have no local hidden variables explanation. Examples have beensuggested by David Bohm and by Lucien Hardy.

Einstein's hope for a purely algebraic theoryThe Bohm interpretation of quantum mechanics hypothesizes that the state of the universe evolves smoothly throughtime with no collapsing of quantum wavefunctions. One problem for the Copenhagen interpretation is to preciselydefine wavefunction collapse. Einstein maintained that quantum mechanics is physically incomplete and logicallyunsatisfactory. In "The Meaning of Relativity," Einstein wrote, "One can give good reasons why reality cannot at allbe represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finitesystem of finite energy can be completely described by a finite set of numbers (quantum numbers). This does notseem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory forthe representation of reality. But nobody knows how to find the basis for such a theory." If time, space, and energyare secondary features derived from a substrate below the Planck scale, then Einstein's hypothetical algebraic systemmight resolve the EPR paradox (although Bell's theorem would still be valid). Edward Fredkin in the Fredkin FiniteNature Hypothesis has suggested an informational basis for Einstein's hypothetical algebraic system. If physicalreality is totally finite, then the Copenhagen interpretation might be an approximation to an information processingsystem below the Planck scale.

"Acceptable theories" and the experimentAccording to the present view of the situation, quantum mechanics flatly contradicts Einstein's philosophicalpostulate that any acceptable physical theory must fulfill "local realism".In the EPR paper (1935) the authors realised that quantum mechanics was inconsistent with their assumptions, butEinstein nevertheless thought that quantum mechanics might simply be augmented by hidden variables (i.e. variableswhich were, at that point, still obscure to him), without any other change, to achieve an acceptable theory. Hepursued these ideas for over twenty years until the end of his life, in 1955.In contrast, John Bell, in his 1964 paper, showed that quantum mechanics and the class of hidden variable theoriesEinstein favored[11] would lead to different experimental results: different by a factor of 3⁄2 for certain correlations.So the issue of "acceptability", up to that time mainly concerning theory, finally became experimentally decidable.

EPR paradox 22

There are many Bell test experiments, e.g. those of Alain Aspect and others. They support the predictions ofquantum mechanics rather than the class of hidden variable theories supported by Einstein.[] According to KarlPopper these experiments showed that the class of "hidden variables" Einstein believed in is erroneous. [citation needed]

Implications for quantum mechanicsMost physicists today believe that quantum mechanics is correct, and that the EPR paradox is a "paradox" onlybecause classical intuitions do not correspond to physical reality. How EPR is interpreted regarding locality dependson the interpretation of quantum mechanics one uses. In the Copenhagen interpretation, it is usually understood thatinstantaneous wave function collapse does occur. However, the view that there is no causal instantaneous effect hasalso been proposed within the Copenhagen interpretation: in this alternate view, measurement affects our ability todefine (and measure) quantities in the physical system, not the system itself. In the many-worlds interpretationlocality is strictly preserved, since the effects of operations such as measurement affect only the state of the particlethat is measured.[] However, the results of the measurement are not unique—every possible result is obtained.The EPR paradox has deepened our understanding of quantum mechanics by exposing the fundamentallynon-classical characteristics of the measurement process. Prior to the publication of the EPR paper, a measurementwas often visualized as a physical disturbance inflicted directly upon the measured system. For instance, whenmeasuring the position of an electron, one imagines shining a light on it, thus disturbing the electron and producingthe quantum mechanical uncertainties in its position. Such explanations, which are still encountered in popularexpositions of quantum mechanics, are debunked by the EPR paradox, which shows that a "measurement" can beperformed on a particle without disturbing it directly, by performing a measurement on a distant entangled particle.In fact, Yakir Aharonov and his collaborators have developed a whole theory of so-called Weak measurement.[citation

needed]

Technologies relying on quantum entanglement are now being developed. In quantum cryptography, entangledparticles are used to transmit signals that cannot be eavesdropped upon without leaving a trace. In quantumcomputation, entangled quantum states are used to perform computations in parallel, which may allow certaincalculations to be performed much more quickly than they ever could be with classical computers.

Mathematical formulationThe above discussion can be expressed mathematically using the quantum mechanical formulation of spin. The spindegree of freedom for an electron is associated with a two-dimensional complex vector space V, with each quantumstate corresponding to a vector in that space. The operators corresponding to the spin along the x, y, and z direction,denoted Sx, Sy, and Sz respectively, can be represented using the Pauli matrices:[]:9

where stands for Planck's constant divided by 2π.The eigenstates of Sz are represented as

and the eigenstates of Sx are represented as

The vector space of the electron-positron pair is , the tensor product of the electron's and positron's vectorspaces. The spin singlet state is

EPR paradox 23

where the two terms on the right hand side are what we have referred to as state I and state II above.From the above equations, it can be shown that the spin singlet can also be written as

where the terms on the right hand side are what we have referred to as state Ia and state IIa.To illustrate how this leads to the violation of local realism, we need to show that after Alice's measurement of Sz (orSx), Bob's value of Sz (or Sx) is uniquely determined, and therefore corresponds to an "element of physical reality".This follows from the principles of measurement in quantum mechanics. When Sz is measured, the system state ψcollapses into an eigenvector of Sz. If the measurement result is +z, this means that immediately after measurementthe system state undergoes an orthogonal projection of ψ onto the space of states of the form

For the spin singlet, the new state is

Similarly, if Alice's measurement result is −z, the system undergoes an orthogonal projection onto

which means that the new state is

This implies that the measurement for Sz for Bob's positron is now determined. It will be −z in the first case or +z inthe second case.It remains only to show that Sx and Sz cannot simultaneously possess definite values in quantum mechanics. One mayshow in a straightforward manner that no possible vector can be an eigenvector of both matrices. More generally,one may use the fact that the operators do not commute,

along with the Heisenberg uncertainty relation

References

Selected papers• P.H. Eberhard, Bell's theorem without hidden variables. Nuovo Cimento 38B1 75 (1977).• P.H. Eberhard, Bell's theorem and the different concepts of locality. Nuovo Cimento 46B 392 (1978).• A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered

complete? [12] Phys. Rev. 47 777 (1935). [7]• A. Fine, Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett. 48, 291 (1982).[13]• A. Fine, Do Correlations need to be explained?, in Philosophical Consequences of Quantum Theory: Reflections

on Bell's Theorem, edited by Cushing & McMullin (University of Notre Dame Press, 1986).• L. Hardy, Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett. 71

1665 (1993).[14]• M. Mizuki, A classical interpretation of Bell's inequality. Annales de la Fondation Louis de Broglie 26 683

(2001).•• P. Pluch, "Theory for Quantum Probability", PhD Thesis University of Klagenfurt (2006)

EPR paradox 24

• M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe and D. J. Wineland, Experimentalviolation of a Bell's inequality with efficient detection, Nature 409, 791–794 (15 February 2001). [15]

• M. Smerlak, C. Rovelli, Relational EPR [16]

Notes

[1] Bell, John. On the Einstein–Poldolsky–Rosen paradox (http:/ / www. drchinese. com/ David/ Bell_Compact. pdf), Physics 1 3, 195-200, Nov.1964

[2] Advances in atomic and molecular physics, Volume 14 By David Robert Bates (http:/ / books. google. com. au/ books?id=dkaCKHKLo3gC&pg=PA330& lpg=PA330& dq="Saclay"+ "Bell's+ inequality"& source=bl& ots=u-b4s3klA0& sig=1P7sX78b-I9TKtT15KvRSADgLlo&hl=en& ei=VJ7aTpn-FMW8iAeJs-jsDQ& sa=X& oi=book_result& ct=result& resnum=2& ved=0CCEQ6AEwAQ#v=onepage& q="Saclay""Bell's inequality"& f=false)

[3][3] .[4] http:/ / plato. stanford. edu/ entries/ qt-epr/[5] von Neumann, J. (1932/1955). In Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, translated into English by Beyer, R.T.,

Princeton University Press, Princeton, cited by Baggott, J. (2004) Beyond Measure: Modern physics, philosophy, and the meaning of quantumtheory, Oxford University Press, Oxford, ISBN 0-19-852927-9, pages 144–145.

[6] Bohm, D. (1951). Quantum Theory (http:/ / books. google. com. au/ books?id=9DWim3RhymsC& printsec=frontcover& dq=david+ bohm+quantum+ theory& source=bl& ots=6G-2u1wtav& sig=Q1GcoVDLFRmKOmDYFAJte6LzrZU& hl=en& ei=Pv45TNSnLYffcfnS6foO&sa=X& oi=book_result& ct=result& resnum=7& ved=0CEEQ6AEwBg#v=onepage& q& f=false), Prentice-Hall, Englewood Cliffs, page 29,and Chapter 5 section 3, and Chapter 22 Section 19.

[7] http:/ / prola. aps. org/ abstract/ PR/ v47/ i10/ p777_1[8] Quoted in Kaiser, David. "Bringing the human actors back on stage: the personal context of the Einstein–Bohr debate," British Journal for

the History of Science 27 (1994): 129–152, on page 147.[9] English translation by Jean Piccard, pp 349–382 in the same issue, doi: 10.1016/S0016-0032(36)91047-5 (http:/ / dx. doi. org/ 10. 1016/

S0016-0032(36)91047-5)).[10] George Greenstein and Arthur G. Zajonc, The Quantum Challenge, p. "[Experiments in the early 1980s] have conclusively shown that

quantum mechanics is indeed orrect, and that the EPR argument had relied upon incorrect assumptions."[12] http:/ / www. drchinese. com/ David/ EPR. pdf[13] http:/ / prola. aps. org/ abstract/ PRL/ v48/ i5/ p291_1[14] http:/ / prola. aps. org/ abstract/ PRL/ v71/ i11/ p1665_1[15] http:/ / www. nature. com/ nature/ journal/ v409/ n6822/ full/ 409791a0. html[16] http:/ / arxiv. org/ abs/ quant-ph/ 0604064

Books• John S. Bell (1987) Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press. ISBN

0-521-36869-3.• Arthur Fine (1996) The Shaky Game: Einstein, Realism and the Quantum Theory, 2nd ed. Univ. of Chicago Press.• J.J. Sakurai, J. J. (1994) Modern Quantum Mechanics. Addison-Wesley: 174–187, 223–232. ISBN

0-201-53929-2.• Selleri, F. (1988) Quantum Mechanics Versus Local Realism: The Einstein–Podolsky–Rosen Paradox. New

York: Plenum Press. ISBN 0-306-42739-7• Leon Lederman, L., Teresi, D. (1993). The God Particle: If the Universe is the Answer, What is the Question?

Houghton Mifflin Company, pages 21, 187 to 189.• John Gribbin (1984) In Search of Schrödinger's Cat. Black Swan. ISBN 978-0-552-12555-0

EPR paradox 25

External links• The Einstein–Podolsky–Rosen Argument in Quantum Theory; 1.2 The argument in the text;

http:/ / plato. stanford. edu/ entries/ qt-epr/ #1. 2• The original EPR paper. (http:/ / prola. aps. org/ abstract/ PR/ v47/ i10/ p777_1)• Stanford Encyclopedia of Philosophy: " The Einstein–Podolsky–Rosen Argument in Quantum Theory (http:/ /

plato. stanford. edu/ )" by Arthur Fine.• Internet Encyclopedia of Philosophy: " The Einstein-Podolsky-Rosen Argument and the Bell Inequalities (http:/ /

www. iep. utm. edu/ epr/ )".• Abner Shimony (2004) " Bell’s Theorem. (http:/ / plato. stanford. edu/ entries/ bell-theorem/ )"• EPR, Bell & Aspect: The Original References. (http:/ / www. drchinese. com/ David/ EPR_Bell_Aspect. htm)• Does Bell's Inequality Principle rule out local theories of quantum mechanics? (http:/ / math. ucr. edu/ home/

baez/ physics/ Quantum/ bells_inequality. html) From the Usenet Physics FAQ.• Theoretical use of EPR in teleportation. (http:/ / www. research. ibm. com/ journal/ rd/ 481/ brassard. html)• Effective use of EPR in cryptography. (http:/ / www. dhushara. com/ book/ quantcos/ aq/ qcrypt. htm)• EPR experiment with single photons interactive. (http:/ / www. QuantumLab. de)• Spooky Actions At A Distance?: Oppenheimer Lecture by Prof. Mermin. (http:/ / www. youtube. com/

watch?v=ta09WXiUqcQ)

Eigenvalues and eigenvectors

In this shear mapping the red arrow changes direction but the blue arrowdoes not. The blue arrow is an eigenvector of this shear mapping, and

since its length is unchanged its eigenvalue is 1.

An eigenvector of a square matrix is anon-zero vector that, when multiplied by ,yields the original vector multiplied by a singlenumber ; that is:

The number is called the eigenvalue of corresponding to .[1]

In analytic geometry, for example, a three-elementvector may be seen as an arrow inthree-dimensional space starting at the origin. Inthat case, an eigenvector of a 3×3 matrix is anarrow whose direction is either preserved orexactly reversed after multiplication by . Thecorresponding eigenvalue determines how thelength of the arrow is changed by the operation,and whether its direction is reversed or not.

In abstract linear algebra, these concepts are naturally extended to more general situations, where the set of real scalefactors is replaced by any field of scalars (such as algebraic or complex numbers); the set of Cartesian vectors isreplaced by any vector space (such as the continuous functions, the polynomials or the trigonometric series), andmatrix multiplication is replaced by any linear operator that maps vectors to vectors (such as the derivative fromcalculus). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as"eigenfunction", "eigenmode", "eigenface", or "eigenstate". Thus, for example, the exponential function

is an eigenfunction of the derivative operator " ", with eigenvalue , since its derivative is.

Eigenvalues and eigenvectors 26

The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is calledthe eigensystem of that matrix.[2] An eigenspace of a matrix is the set of all eigenvectors with the sameeigenvalue, together with the zero vector.[1] An eigenbasis for is any basis for the set of all vectors that consistsof linearly independent eigenvectors of . Not every real matrix has real eigenvalues, but every complex matrixhas at least one complex eigenvalue.The terms characteristic vector, characteristic value, and characteristic space are also used for these concepts.The prefix eigen- is adopted from the German word eigen for "self" or "proper".Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. They are used in matrixfactorization, in quantum mechanics, and in many other areas.

Definition

Eigenvectors and eigenvalues of a real matrix

Matrix acts by stretching the vector , not changing itsdirection, so is an eigenvector of .

In many contexts, a vector can be assumed to be a listof real numbers (called elements), written verticallywith brackets around the entire list, such as the vectorsu and v below. Two vectors are said to be scalarmultiples of each other (also called parallel orcollinear) if they have the same number of elements,and if every element of one vector is obtained bymultiplying each corresponding element in the othervector by the same number (known as a scaling factor,or a scalar). For example, the vectors

and

are scalar multiples of each other, because each elementof is −20 times the corresponding element of .

A vector with three elements, like or above, may represent a point in three-dimensional space, relative to someCartesian coordinate system. It helps to think of such a vector as the tip of an arrow whose tail is at the origin of thecoordinate system. In this case, the condition " is parallel to " means that the two arrows lie on the samestraight line, and may differ only in length and direction along that line.

If we multiply any square matrix with rows and columns by such a vector , the result will be anothervector , also with rows and one column. That is,

is mapped to

where, for each index ,

In general, if is not all zeros, the vectors and will not be parallel. When they are parallel (that is, whenthere is some real number such that ) we say that is an eigenvector of . In that case, the scalefactor is said to be the eigenvalue corresponding to that eigenvector.

Eigenvalues and eigenvectors 27

In particular, multiplication by a 3×3 matrix may change both the direction and the magnitude of an arrow inthree-dimensional space. However, if is an eigenvector of with eigenvalue , the operation may only changeits length, and either keep its direction or flip it (make the arrow point in the exact opposite direction). Specifically,the length of the arrow will increase if , remain the same if , and decrease it if .Moreover, the direction will be precisely the same if , and flipped if . If , then the length ofthe arrow becomes zero.

An example

The transformation matrix preserves the

direction of vectors parallel to (in blue)

and (in violet). The points that lie on the

line through the origin, parallel to an eigenvector,remain on the line after the transformation. The

vectors in red are not eigenvectors, therefore theirdirection is altered by the transformation. Seealso: An extended version, showing all four

quadrants.

For the transformation matrix

the vector

is an eigenvector with eigenvalue 2. Indeed,

On the other hand the vector

is not an eigenvector, since

and this vector is not a multiple of the original vector .

Eigenvalues and eigenvectors 28

Another example

For the matrix

we have

and

Therefore, the vectors , and are eigenvectors of corresponding to theeigenvalues 0, 3, and 2, respectively. (Here the symbol indicates matrix transposition, in this case turning the rowvectors into column vectors.)

Trivial cases

The identity matrix (whose general element is 1 if , and 0 otherwise) maps every vector to itself.Therefore, every vector is an eigenvector of , with eigenvalue 1.More generally, if is a diagonal matrix (with whenever ), and is a vector parallel to axis (that is, , and if ), then where . That is, the eigenvalues of a diagonalmatrix are the elements of its main diagonal. This is trivially the case of any 1 ×1 matrix.

General definitionThe concept of eigenvectors and eigenvalues extends naturally to abstract linear transformations on abstract vectorspaces. Namely, let be any vector space over some field of scalars, and let be a linear transformationmapping into . We say that a non-zero vector of is an eigenvector of if (and only if) there is ascalar in such that

.This equation is called the eigenvalue equation for , and the scalar is the eigenvalue of corresponding tothe eigenvector . Note that means the result of applying the operator to the vector , while meansthe product of the scalar by .[3]

The matrix-specific definition is a special case of this abstract definition. Namely, the vector space is the set ofall column vectors of a certain size ×1, and is the linear transformation that consists in multiplying a vector bythe given matrix .Some authors allow to be the zero vector in the definition of eigenvector.[4] This is reasonable as long as wedefine eigenvalues and eigenvectors carefully: If we would like the zero vector to be an eigenvector, then we mustfirst define an eigenvalue of as a scalar in such that there is a nonzero vector in with .We then define an eigenvector to be a vector in such that there is an eigenvalue in with .This way, we ensure that it is not the case that every scalar is an eigenvalue corresponding to the zero vector.

Eigenvalues and eigenvectors 29

Eigenspace and spectrumIf is an eigenvector of , with eigenvalue , then any scalar multiple of with nonzero is also aneigenvector with eigenvalue , since . Moreover, if and areeigenvectors with the same eigenvalue , then is also an eigenvector with the same eigenvalue .Therefore, the set of all eigenvectors with the same eigenvalue , together with the zero vector, is a linear subspaceof , called the eigenspace of associated to .[5][6] If that subspace has dimension 1, it is sometimes called aneigenline.[7]

The geometric multiplicity of an eigenvalue is the dimension of the eigenspace associated to , i.e.number of linearly independent eigenvectors with that eigenvalue. These eigenvectors can be chosen so that they arepairwise orthogonal and have unit length under some arbitrary inner product defined on . In other words, everyeigenspace has an orthonormal basis of eigenvectors.Conversely, any eigenvector with eigenvalue must be linearly independent from all eigenvectors that areassociated to a different eigenvalue . Therefore a linear transformation that operates on an -dimensionalspace cannot have more than distinct eigenvalues (or eigenspaces).[8]

Any subspace spanned by eigenvectors of is an invariant subspace of .The list of eigenvalues of is sometimes called the spectrum of . The order of this list is arbitrary, but thenumber of times that an eigenvalue appears is important.There is no unique way to choose a basis for an eigenspace of an abstract linear operator based only on itself,without some additional data such as a choice of coordinate basis for . Even for an eigenline, the basis vector isindeterminate in both magnitude and orientation. If the scalar field is the real numbers , one can order theeigenspaces by their eigenvalues. Since the modulus of an eigenvalue is important in many applications, theeigenspaces are often ordered by that criterion.

EigenbasisAn eigenbasis for a linear operator that operates on a vector space is a basis for that consists entirely ofeigenvectors of (possibly with different eigenvalues). Such a basis may not exist.

Suppose has finite dimension , and let be the sum of the geometric multiplicities over all distincteigenvalues of . This integer is the maximum number of linearly independent eigenvectors of , andtherefore cannot exceed . If is exactly , then admits an eigenbasis; that is, there exists a basis for that consists of eigenvectors. The matrix that represents relative to this basis is a diagonal matrix, whosediagonal elements are the eigenvalues associated to each basis vector.Conversely, if the sum is less than , then admits no eigenbasis, and there is no choice of coordinates thatwill allow to be represented by a diagonal matrix.Note that is at least equal to the number of distinct eigenvalues of , but may be larger than that.[9] Forexample, the identity operator on has , and any basis of is an eigenbasis of ; but its onlyeigenvalue is 1, with .

Eigenvalues and eigenvectors 30

Generalizations to infinite-dimensional spacesThe definition of eigenvalue of a linear transformation remains valid even if the underlying space is aninfinite dimensional Hilbert or Banach space. Namely, a scalar is an eigenvalue if and only if there is somenonzero vector such that .

EigenfunctionsA widely used class of linear operators acting on infinite dimensional spaces are the differential operators onfunction spaces. Let be a linear differential operator in on the space of infinitely differentiable realfunctions of a real argument . The eigenvalue equation for is the differential equation

The functions that satisfy this equation are commonly called eigenfunctions. For the derivative operator , aneigenfunction is a function that, when differentiated, yields a constant times the original function. If is zero, thegeneric solution is a constant function. If is non-zero, the solution is an exponential function

Eigenfunctions are an essential tool in the solution of differential equations and many other applied and theoreticalfields. For instance, the exponential functions are eigenfunctions of any shift invariant linear operator. This fact isthe basis of powerful Fourier transform methods for solving all sorts of problems.

Spectral theory

If is an eigenvalue of , then the operator is not one-to-one, and therefore its inverse isnot defined. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional ones. Ingeneral, the operator may not have an inverse, even if is not an eigenvalue.For this reason, in functional analysis one defines the spectrum of a linear operator as the set of all scalars forwhich the operator has no bounded inverse. Thus the spectrum of an operator always contains all itseigenvalues, but is not limited to them.

Associative algebras and representation theoryMore algebraically, rather than generalizing the vector space to an infinite dimensional space, one can generalize thealgebraic object that is acting on the space, replacing a single operator acting on a vector space with an algebrarepresentation – an associative algebra acting on a module. The study of such actions is the field of representationtheory.A closer analog of eigenvalues is given by the representation-theoretical concept of weight, with the analogs ofeigenvectors and eigenspaces being weight vectors and weight spaces.

Eigenvalues and eigenvectors 31

Eigenvalues and eigenvectors of matrices

Characteristic polynomialThe eigenvalue equation for a matrix is

which is equivalent to

where is the identity matrix. It is a fundamental result of linear algebra that an equation has anon-zero solution if and only if the determinant of the matrix is zero. It follows that the eigenvaluesof are precisely the real numbers that satisfy the equation

The left-hand side of this equation can be seen (using Leibniz' rule for the determinant) to be a polynomial functionof the variable . The degree of this polynomial is , the order of the matrix. Its coefficients depend on theentries of , except that its term of degree is always . This polynomial is called the characteristic

polynomial of ; and the above equation is called the characteristic equation (or, less often, the secular equation)of .For example, let be the matrix

The characteristic polynomial of is

which is

The roots of this polynomial are 2, 1, and 11. Indeed these are the only three eigenvalues of , corresponding tothe eigenvectors and (or any non-zero multiples thereof).

In the real domain

Since the eigenvalues are roots of the characteristic polynomial, an matrix has at most eigenvalues. If thematrix has real entries, the coefficients of the characteristic polynomial are all real; but it may have fewer than real roots, or no real roots at all.For example, consider the cyclic permutation matrix

This matrix shifts the coordinates of the vector up by one position, and moves the first coordinate to the bottom. Itscharacteristic polynomial is which has one real root . Any vector with three equal non-zeroelements is an eigenvector for this eigenvalue. For example,

Eigenvalues and eigenvectors 32

In the complex domain

The fundamental theorem of algebra implies that the characteristic polynomial of an matrix , being apolynomial of degree , has exactly complex roots. More precisely, it can be factored into the product of linear terms,

where each is a complex number. The numbers , , ... , (which may not be all distinct) are roots of thepolynomial, and are precisely the eigenvalues of .Even if the entries of are all real numbers, the eigenvalues may still have non-zero imaginary parts (and theelements of the corresponding eigenvectors will therefore also have non-zero imaginary parts). Also, the eigenvaluesmay be irrational numbers even if all the entries of are rational numbers, or all are integers. However, if theentries of are algebraic numbers (which include the rationals), the eigenvalues will be (complex) algebraicnumbers too.The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugate values,namely with the two members of each pair having the same real part and imaginary parts that differ only in sign. Ifthe degree is odd, then by the intermediate value theorem at least one of the roots will be real. Therefore, any realmatrix with odd order will have at least one real eigenvalue; whereas a real matrix with even order may have no realeigenvalues.

In the example of the 3×3 cyclic permutation matrix , above, the characteristic polynomial has twoadditional non-real roots, namely

and ,where is the imaginary unit. Note that , , and . Then

and

Therefore, the vectors and are eigenvectors of , with eigenvalues 1, , and ,respectively.

Algebraic multiplicities

Let be an eigenvalue of an matrix . The algebraic multiplicity of is its multiplicity as aroot of the characteristic polynomial, that is, the largest integer such that divides evenly thatpolynomial.Like the geometric multiplicity , the algebraic multiplicity is an integer between 1 and ; and the sum

of over all distinct eigenvalues also cannot exceed . If complex eigenvalues are considered, isexactly .It can be proved that the geometric multiplicity of an eigenvalue never exceeds its algebraic multiplicity

. Therefore, is at most .

Eigenvalues and eigenvectors 33

Example

For the matrix:

the characteristic polynomial of is

,

being the product of the diagonal with a lower triangular matrix.The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic multiplicity of each eigenvalue is2; in other words they are both double roots. On the other hand, the geometric multiplicity of the eigenvalue 2 is only1, because its eigenspace is spanned by the vector , and is therefore 1 dimensional. Similarly, thegeometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by . Hence, the totalalgebraic multiplicity of A, denoted , is 4, which is the most it could be for a 4 by 4 matrix. The geometricmultiplicity is 2, which is the smallest it could be for a matrix which has two distinct eigenvalues.

Diagonalization and eigendecompositionIf the sum of the geometric multiplicities of all eigenvalues is exactly , then has a set of linearlyindependent eigenvectors. Let be a square matrix whose columns are those eigenvectors, in any order. Then wewill have , where is the diagonal matrix such that is the eigenvalue associated to column of

. Since the columns of are linearly independent, the matrix is invertible. Premultiplying both sides bywe get . By definition, therefore, the matrix is diagonalizable.

Conversely, if is diagonalizable, let be a non-singular square matrix such that is some diagonalmatrix . Multiplying both sides on the left by we get . Therefore each column of must bean eigenvector of , whose eigenvalue is the corresponding element on the diagonal of . Since the columns of

must be linearly independent, it follows that . Thus is equal to if and only if isdiagonalizable.If is diagonalizable, the space of all -element vectors can be decomposed into the direct sum of theeigenspaces of . This decomposition is called the eigendecomposition of , and it is the preserved underchange of coordinates.A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvector can begeneralized to generalized eigenvectors, and that of diagonal matrix to a Jordan form matrix. Over an algebraicallyclosed field, any matrix has a Jordan form and therefore admits a basis of generalized eigenvectors, and adecomposition into generalized eigenspaces

Eigenvalues and eigenvectors 34

Further properties

Let be an arbitrary matrix of complex numbers with eigenvalues , , ... . (Here it isunderstood that an eigenvalue with algebraic multiplicity occurs times in this list.) Then• The trace of , defined as the sum of its diagonal elements, is also the sum of all eigenvalues:

.

• The determinant of is the product of all eigenvalues:

.

• The eigenvalues of the th power of , i.e. the eigenvalues of , for any positive integer , are

• The matrix is invertible if and only if all the eigenvalues are nonzero.• If is invertible, then the eigenvalues of are • If is equal to its conjugate transpose (in other words, if is Hermitian), then every eigenvalue is real.

The same is true of any a symmetric real matrix. If is also positive-definite, positive-semidefinite,negative-definite, or negative-semidefinite every eigenvalue is positive, non-negative, negative, or non-positiverespectively.

• Every eigenvalue of a unitary matrix has absolute value .

Left and right eigenvectorsThe use of matrices with a single column (rather than a single row) to represent vectors is traditional in manydisciplines. For that reason, the word "eigenvector" almost always means a right eigenvector, namely a columnvector that must placed to the right of the matrix in the defining equation

.There may be also single-row vectors that are unchanged when they occur on the left side of a product with a squarematrix ; that is, which satisfy the equation

Any such row vector is called a left eigenvector of .The left eigenvectors of are transposes of the right eigenvectors of the transposed matrix , since theirdefining equation is equivalent to

It follows that, if is Hermitian, its left and right eigenvectors are complex conjugates. In particular if is a realsymmetric matrix, they are the same except for transposition.

Eigenvalues and eigenvectors 35

Calculation

Computing the eigenvaluesThe eigenvalues of a matrix can be determined by finding the roots of the characteristic polynomial. Explicitalgebraic formulas for the roots of a polynomial exist only if the degree is 4 or less. According to theAbel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial withdegree 5 or more.It turns out that any polynomial with degree is the characteristic polynomial of some companion matrix of order

. Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicitalgebraic formula, and must therefore be computed by approximate numerical methods.In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of productsof matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to anyrequired accuracy.[] However, this approach is not viable in practice because the coefficients would be contaminatedby unavoidable round-off errors, and the roots of a polynomial can be an extremely sensitive function of thecoefficients (as exemplified by Wilkinson's polynomial).[]

Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until theadvent of the QR algorithm in 1961. [] Combining the Householder transformation with the LU decompositionresults in an algorithm with better convergence than the QR algorithm.[citation needed] For large Hermitian sparsematrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues andeigenvectors, among several other possibilities.[]

Computing the eigenvectorsOnce the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding non-zerosolutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients. Forexample, once it is known that 6 is an eigenvalue of the matrix

we can find its eigenvectors by solving the equation , that is

This matrix equation is equivalent to two linear equations

that is

Both equations reduce to the single linear equation . Therefore, any vector of the form , for anynon-zero real number , is an eigenvector of with eigenvalue .The matrix above has another eigenvalue . A similar calculation shows that the correspondingeigenvectors are the non-zero solutions of , that is, any vector of the form , for anynon-zero real number .Some numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectorsas a by-product of the computation.

Eigenvalues and eigenvectors 36

HistoryEigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arosein the study of quadratic forms and differential equations.Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrangerealized that the principal axes are the eigenvectors of the inertia matrix.[10] In the early 19th century, Cauchy sawhow their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.[11] Cauchyalso coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survivesin characteristic equation.[12]

Fourier used the work of Laplace and Lagrange to solve the heat equation by separation of variables in his famous1822 book Théorie analytique de la chaleur.[13] Sturm developed Fourier's ideas further and brought them to theattention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matriceshave real eigenvalues.[11] This was extended by Hermite in 1855 to what are now called Hermitian matrices.[12]

Around the same time, Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,[11] andClebsch found the corresponding result for skew-symmetric matrices.[12] Finally, Weierstrass clarified an importantaspect in the stability theory started by Laplace by realizing that defective matrices can cause instability.[11]

In the meantime, Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out oftheir work is now called Sturm–Liouville theory.[14] Schwarz studied the first eigenvalue of Laplace's equation ongeneral domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.[15]

At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators asinfinite matrices.[16] He was the first to use the German word eigen to denote eigenvalues and eigenvectors in 1904,though he may have been following a related usage by Helmholtz. For some time, the standard term in English was"proper value", but the more distinctive term "eigenvalue" is standard today.[17]

The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Von Misespublished the power method. One of the most popular methods today, the QR algorithm, was proposedindependently by John G.F. Francis[18] and Vera Kublanovskaya[19] in 1961.[20]

Applications

Eigenvalues of geometric transformationsThe following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues,and eigenvectors.

scaling unequal scaling rotation horizontal shear hyperbolic rotation

illustration

matrix

characteristicpolynomial

Eigenvalues and eigenvectors 37

eigenvalues ,

algebraic multipl.

geometricmultipl.

eigenvectors All non-zero vectors

Note that the characteristic equation for a rotation is a quadratic equation with discriminant ,which is a negative number whenever is not an integer multiple of 180°. Therefore, except for these special cases,the two eigenvalues are complex numbers, ; and all eigenvectors have non-real entries. Indeed,except for those special cases, a rotation changes the direction of every nonzero vector in the plane.

Schrödinger equation

The wavefunctions associated with the bound states of an electron in ahydrogen atom can be seen as the eigenvectors of the hydrogen atomHamiltonian as well as of the angular momentum operator. They areassociated with eigenvalues interpreted as their energies (increasing

downward: ) and angular momentum (increasingacross: s, p, d, ...). The illustration shows the square of the absolute value

of the wavefunctions. Brighter areas correspond to higher probabilitydensity for a position measurement. The center of each figure is the atomic

nucleus, a proton.

An example of an eigenvalue equation where thetransformation is represented in terms of adifferential operator is the time-independentSchrödinger equation in quantum mechanics:

where , the Hamiltonian, is a second-orderdifferential operator and , the wavefunction,is one of its eigenfunctions corresponding to theeigenvalue , interpreted as its energy.However, in the case where one is interested onlyin the bound state solutions of the Schrödingerequation, one looks for within the space ofsquare integrable functions. Since this space is aHilbert space with a well-defined scalar product,one can introduce a basis set in which and

can be represented as a one-dimensional arrayand a matrix respectively. This allows one torepresent the Schrödinger equation in a matrixform.Bra-ket notation is often used in this context. Avector, which represents a state of the system, inthe Hilbert space of square integrable functions isrepresented by . In this notation, theSchrödinger equation is:

where is an eigenstate of . It is a self adjoint operator, the infinite dimensional analog of Hermitianmatrices (see Observable). As in the matrix case, in the equation above is understood to be the vectorobtained by application of the transformation to .

Eigenvalues and eigenvectors 38

Molecular orbitalsIn quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, theatomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The correspondingeigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenvector is usedin a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and theireigenvalues. If one wants to underline this aspect one speaks of nonlinear eigenvalue problem. Such equations areusually solved by an iteration procedure, called in this case self-consistent field method. In quantum chemistry, oneoften represents the Hartree–Fock equation in a non-orthogonal basis set. This particular representation is ageneralized eigenvalue problem called Roothaan equations.

Geology and glaciologyIn geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a massof information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers.In the field, a geologist may collect such data for hundreds or thousands of clasts in a soil sample, which can only becompared graphically such as in a Tri-Plot (Sneed and Folk) diagram,[21][22] or as a Stereonet on a Wulff Net.[23]

The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectorsare ordered by their eigenvalues ;[24] then is the primary orientation/dip of clast,

is the secondary and is the tertiary, in terms of strength. The clast orientation is defined as the direction of theeigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valuedfrom 0° (no dip) to 90° (vertical). The relative values of , , and are dictated by the nature of thesediment's fabric. If , the fabric is said to be isotropic. If , the fabric is said tobe planar. If , the fabric is said to be linear.[25]

Principal components analysis

PCA of the multivariate Gaussian distributioncentered at with a standard deviation of

3 in roughly the direction

and of 1 in the orthogonal direction. The vectorsshown are unit eigenvectors of the (symmetric,positive-semidefinite) covariance matrix scaled

by the square root of the correspondingeigenvalue. (Just as in the one-dimensional case,

the square root is taken because the standarddeviation is more readily visualized than the

variance.

The eigendecomposition of a symmetric positive semidefinite (PSD)matrix yields an orthogonal basis of eigenvectors, each of which has anonnegative eigenvalue. The orthogonal decomposition of a PSDmatrix is used in multivariate analysis, where the sample covariancematrices are PSD. This orthogonal decomposition is called principalcomponents analysis (PCA) in statistics. PCA studies linear relationsamong variables. PCA is performed on the covariance matrix or thecorrelation matrix (in which each variable is scaled to have its samplevariance equal to one). For the covariance or correlation matrix, theeigenvectors correspond to principal components and the eigenvaluesto the variance explained by the principal components. Principalcomponent analysis of the correlation matrix provides an orthonormaleigen-basis for the space of the observed data: In this basis, the largesteigenvalues correspond to the principal-components that are associatedwith most of the covariability among a number of observed data.

Principal component analysis is used to study large data sets, such asthose encountered in data mining, chemical research, psychology, andin marketing. PCA is popular especially in psychology, in the field ofpsychometrics. In Q methodology, the eigenvalues of the correlation

Eigenvalues and eigenvectors 39

matrix determine the Q-methodologist's judgment of practical significance (which differs from the statisticalsignificance of hypothesis testing): The factors with eigenvalues greater than 1.00 are considered practicallysignificant, that is, as explaining an important amount of the variability in the data, while eigenvalues less than 1.00are considered practically insignificant, as explaining only a negligible portion of the data variability. Moregenerally, principal component analysis can be used as a method of factor analysis in structural equation modeling.

Vibration analysis

1st lateral bending (See vibration for more types of vibration)

Eigenvalue problems occur naturally in the vibration analysisof mechanical structures with many degrees of freedom. Theeigenvalues are used to determine the natural frequencies (oreigenfrequencies) of vibration, and the eigenvectorsdetermine the shapes of these vibrational modes. Inparticular, undamped vibration is governed by

or

that is, acceleration is proportional to position (i.e., we expect to be sinusoidal in time). In dimensions, becomes a mass matrix and a stiffness matrix. Admissible solutions are then a linear combination of solutions tothe generalized eigenvalue problem

where is the eigenvalue and is the angular frequency. Note that the principal vibration modes are differentfrom the principal compliance modes, which are the eigenvectors of alone. Furthermore, damped vibration,governed by

leads to what is called a so-called quadratic eigenvalue problem,

This can be reduced to a generalized eigenvalue problem by clever use of algebra at the cost of solving a largersystem.The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the systemcan be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is oftensolved using finite element analysis, but neatly generalize the solution to scalar-valued vibration problems.

Eigenvalues and eigenvectors 40

Eigenfaces

Eigenfaces as examples of eigenvectors

In image processing, processed images of faces can be seen asvectors whose components are the brightnesses of each pixel.[26]

The dimension of this vector space is the number of pixels. Theeigenvectors of the covariance matrix associated with a large set ofnormalized pictures of faces are called eigenfaces; this is anexample of principal components analysis. They are very usefulfor expressing any face image as a linear combination of some ofthem. In the facial recognition branch of biometrics, eigenfacesprovide a means of applying data compression to faces foridentification purposes. Research related to eigen vision systemsdetermining hand gestures has also been made.

Similar to this concept, eigenvoices represent the general directionof variability in human pronunciations of a particular utterance,such as a word in a language. Based on a linear combination ofsuch eigenvoices, a new voice pronunciation of the word can beconstructed. These concepts have been found useful in automaticspeech recognition systems, for speaker adaptation.

Tensor of moment of inertiaIn mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. The tensorof moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.

Stress tensorIn solid mechanics, the stress tensor is symmetric and so can be decomposed into a diagonal tensor with theeigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensorhas no shear components; the components it does have are the principal components.

Eigenvalues of a graphIn spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix , or(increasingly) of the graph's Laplacian matrix (see also Discrete Laplace operator), which is either (sometimes called the combinatorial Laplacian) or (sometimes called the normalized

Laplacian), where is a diagonal matrix with equal to the degree of vertex , and in , the thdiagonal entry is . The th principal eigenvector of a graph is defined as either the eigenvectorcorresponding to the th largest or th smallest eigenvalue of the Laplacian. The first principal eigenvector of thegraph is also referred to merely as the principal eigenvector.The principal eigenvector is used to measure the centrality of its vertices. An example is Google's PageRankalgorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the pageranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented bythe row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationarydistribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via spectralclustering. Other methods are also available for clustering.

Eigenvalues and eigenvectors 41

Basic reproduction numberSee Basic reproduction number

The basic reproduction number ( ) is a fundamental number in the study of how infectious diseases spread. Ifone infectious person is put into a population of completely susceptible people, then is the average number ofpeople that one infectious person will infect. The generation time of an infection is the time, , from one personbecoming infected to the next person becoming infected. In a heterogenous population, the next generation matrixdefines how many people in the population will become infected after time has passed. is then the largesteigenvalue of the next generation matrix.[27][28]

Notes[1] Wolfram Research, Inc. (2010) Eigenvector (http:/ / mathworld. wolfram. com/ Eigenvector. html). Accessed on 2010-01-29.[2] William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery (2007), [http://www.nr.com/ Numerical Recipes: The Art of

Scientific Computing, Chapter 11: Eigensystems., pages=563–597. Third edition, Cambridge University Press. ISBN 9780521880688[3][3] See ;[6][6] Lemma for the eigenspace[7] Schaum's Easy Outline of Linear Algebra (http:/ / books. google. com/ books?id=pkESXAcIiCQC& pg=PA111), p. 111[8] For a proof of this lemma, see ; ; ; ; and Lemma for linear independence of eigenvectors[10][10] See[11][11] See[12][12] See[13][13] See[14][14] See[15][15] See[16][16] See[17][17] See[18][18] and[19][19] . Also published in:[20][20] See ;[24] Stereo32 software (http:/ / www. ruhr-uni-bochum. de/ hardrock/ downloads. htm)

References• Korn, Granino A.; Korn, Theresa M. (2000), "Mathematical Handbook for Scientists and Engineers: Definitions,

Theorems, and Formulas for Reference and Review", New York: McGraw-Hill (1152 p., Dover Publications, 2Revised edition), Bibcode: 1968mhse.book.....K (http:/ / adsabs. harvard. edu/ abs/ 1968mhse. book. . . . . K),ISBN 0-486-41147-8.

• Lipschutz, Seymour (1991), Schaum's outline of theory and problems of linear algebra, Schaum's outline series(2nd ed.), New York, NY: McGraw-Hill Companies, ISBN 0-07-038007-4.

• Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (1989), Linear algebra (2nd ed.), Englewood Cliffs,NJ 07632: Prentice Hall, ISBN 0-13-537102-3.

• Aldrich, John (2006), "Eigenvalue, eigenfunction, eigenvector, and related terms" (http:/ / jeff560. tripod. com/ e.html), in Jeff Miller (Editor), Earliest Known Uses of Some of the Words of Mathematics (http:/ / jeff560. tripod.com/ e. html), retrieved 2006-08-22

• Strang, Gilbert (1993), Introduction to linear algebra, Wellesley-Cambridge Press, Wellesley, MA,ISBN 0-9614088-5-5.

• Strang, Gilbert (2006), Linear algebra and its applications, Thomson, Brooks/Cole, Belmont, CA,ISBN 0-03-010567-6.

• Bowen, Ray M.; Wang, Chao-Cheng (1980), Linear and multilinear algebra, Plenum Press, New York, NY,ISBN 0-306-37508-7.

Eigenvalues and eigenvectors 42

• Cohen-Tannoudji, Claude (1977), "Chapter II. The mathematical tools of quantum mechanics", Quantummechanics, John Wiley & Sons, ISBN 0-471-16432-1.

• Fraleigh, John B.; Beauregard, Raymond A. (1995), Linear algebra (3rd ed.), Addison-Wesley PublishingCompany, ISBN 0-201-83999-7 (international edition).

• Golub, Gene H.; Van Loan, Charles F. (1996), Matrix computations (3rd Edition), Johns Hopkins UniversityPress, Baltimore, MD, ISBN 978-0-8018-5414-9.

• Hawkins, T. (1975), "Cauchy and the spectral theory of matrices", Historia Mathematica 2: 1–29, doi:10.1016/0315-0860(75)90032-4 (http:/ / dx. doi. org/ 10. 1016/ 0315-0860(75)90032-4).

• Horn, Roger A.; Johnson, Charles F. (1985), Matrix analysis, Cambridge University Press, ISBN 0-521-30586-1(hardback), ISBN 0-521-38632-2 (paperback) Check |isbn= value (help).

• Kline, Morris (1972), Mathematical thought from ancient to modern times, Oxford University Press,ISBN 0-19-501496-0.

• Meyer, Carl D. (2000), Matrix analysis and applied linear algebra, Society for Industrial and AppliedMathematics (SIAM), Philadelphia, ISBN 978-0-89871-454-8.

• Brown, Maureen (October 2004), Illuminating Patterns of Perception: An Overview of Q Methodology.• Golub, Gene F.; van der Vorst, Henk A. (2000), "Eigenvalue computation in the 20th century", Journal of

Computational and Applied Mathematics 123: 35–65, doi: 10.1016/S0377-0427(00)00413-1 (http:/ / dx. doi. org/10. 1016/ S0377-0427(00)00413-1).

• Akivis, Max A.; Vladislav V. Goldberg (1969), Tensor calculus, Russian, Science Publishers, Moscow.• Gelfand, I. M. (1971), Lecture notes in linear algebra, Russian, Science Publishers, Moscow.• Alexandrov, Pavel S. (1968), Lecture notes in analytical geometry, Russian, Science Publishers, Moscow.• Carter, Tamara A.; Tapia, Richard A.; Papaconstantinou, Anne, Linear Algebra: An Introduction to Linear

Algebra for Pre-Calculus Students (http:/ / ceee. rice. edu/ Books/ LA/ index. html), Rice University, OnlineEdition, retrieved 2008-02-19.

• Roman, Steven (2008), Advanced linear algebra (3rd ed.), New York, NY: Springer Science + Business Media,LLC, ISBN 978-0-387-72828-5.

• Shilov, Georgi E. (1977), Linear algebra (translated and edited by Richard A. Silverman ed.), New York: DoverPublications, ISBN 0-486-63518-X.

• Hefferon, Jim (2001), Linear Algebra (http:/ / joshua. smcvt. edu/ linearalgebra/ ), Online book, St Michael'sCollege, Colchester, Vermont, USA.

• Kuttler, Kenneth (2007), An introduction to linear algebra (http:/ / www. math. byu. edu/ ~klkuttle/Linearalgebra. pdf) (PDF), Online e-book in PDF format, Brigham Young University.

• Demmel, James W. (1997), Applied numerical linear algebra, SIAM, ISBN 0-89871-389-7.• Beezer, Robert A. (2006), A first course in linear algebra (http:/ / linear. ups. edu/ ), Free online book under GNU

licence, University of Puget Sound.• Lancaster, P. (1973), Matrix theory, Russian, Moscow, Russia: Science Publishers.• Halmos, Paul R. (1987), Finite-dimensional vector spaces (8th ed.), New York, NY: Springer-Verlag,

ISBN 0-387-90093-4.• Pigolkina, T. S. and Shulman, V. S., Eigenvalue (in Russian), In:Vinogradov, I. M. (Ed.), Mathematical

Encyclopedia, Vol. 5, Soviet Encyclopedia, Moscow, 1977.• Greub, Werner H. (1975), Linear Algebra (4th Edition), Springer-Verlag, New York, NY, ISBN 0-387-90110-8.• Larson, Ron; Edwards, Bruce H. (2003), Elementary linear algebra (5th ed.), Houghton Mifflin Company,

ISBN 0-618-33567-6.• Curtis, Charles W., Linear Algebra: An Introductory Approach, 347 p., Springer; 4th ed. 1984. Corr. 7th printing

edition (August 19, 1999), ISBN 0-387-90992-3.• Shores, Thomas S. (2007), Applied linear algebra and matrix analysis, Springer Science+Business Media, LLC,

ISBN 0-387-33194-8.

Eigenvalues and eigenvectors 43

• Sharipov, Ruslan A. (1996), Course of Linear Algebra and Multidimensional Geometry: the textbook, arXiv:math/0405323 (http:/ / arxiv. org/ abs/ math/ 0405323), ISBN 5-7477-0099-5.

• Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2005), Indefinite linear algebra and applications,Basel-Boston-Berlin: Birkhäuser Verlag, ISBN 3-7643-7349-0.

External links• What are Eigen Values? (http:/ / www. physlink. com/ education/ AskExperts/ ae520. cfm) – non-technical

introduction from PhysLink.com's "Ask the Experts"• Eigen Values and Eigen Vectors Numerical Examples (http:/ / people. revoledu. com/ kardi/ tutorial/

LinearAlgebra/ EigenValueEigenVector. html) – Tutorial and Interactive Program from Revoledu.• Introduction to Eigen Vectors and Eigen Values (http:/ / khanexercises. appspot. com/ video?v=PhfbEr2btGQ) –

lecture from Khan Academy• Hill, Roger (2009). "λ – Eigenvalues" (http:/ / www. sixtysymbols. com/ videos/ eigenvalues. htm). Sixty

Symbols. Brady Haran for the University of Nottingham.Theory

• Hazewinkel, Michiel, ed. (2001), "Eigen value" (http:/ / www. encyclopediaofmath. org/ index. php?title=p/e035150), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Hazewinkel, Michiel, ed. (2001), "Eigen vector" (http:/ / www. encyclopediaofmath. org/ index. php?title=p/e035180), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Eigenvalue (of a matrix) (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=4397),PlanetMath.org.

• Eigenvector (http:/ / mathworld. wolfram. com/ Eigenvector. html) – Wolfram MathWorld• Eigen Vector Examination working applet (http:/ / ocw. mit. edu/ ans7870/ 18/ 18. 06/ javademo/ Eigen/ )• Same Eigen Vector Examination as above in a Flash demo with sound (http:/ / web. mit. edu/ 18. 06/ www/

Demos/ eigen-applet-all/ eigen_sound_all. html)• Computation of Eigenvalues (http:/ / www. sosmath. com/ matrix/ eigen1/ eigen1. html)• Numerical solution of eigenvalue problems (http:/ / www. cs. utk. edu/ ~dongarra/ etemplates/ index. html) Edited

by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst• Eigenvalues and Eigenvectors on the Ask Dr. Math forums: (http:/ / mathforum. org/ library/ drmath/ view/

55483. html), (http:/ / mathforum. org/ library/ drmath/ view/ 51989. html)Online calculators

• arndt-bruenner.de (http:/ / www. arndt-bruenner. de/ mathe/ scripts/ engl_eigenwert. htm)• bluebit.gr (http:/ / www. bluebit. gr/ matrix-calculator/ )• wims.unice.fr (http:/ / wims. unice. fr/ wims/ wims. cgi?session=6S051ABAFA. 2& + lang=en& + module=tool/

linear/ matrix. en)Demonstration applets

• Java applet about eigenvectors in the real plane (http:/ / scienceapplets. blogspot. com/ 2012/ 03/eigenvalues-and-eigenvectors. html)

Quantum Bayesianism 44

Quantum BayesianismQuantum Bayesianism (sometimes abbreviated "QBism", and referred to as the Radical Bayesianinterpretation[1]) is a "subjective Bayesian account of quantum probability"[] which attempts to provide anunderstanding of quantum mechanics and to derive modern quantum mechanics from informational considerations.Quantum Bayesianism has evolved primarily from the work of Carlton M. Caves, Christopher Fuchs and RüdigerSchack, and draws from the fields of quantum information and Bayesian probability. It deals with common questionsin the interpretation of quantum mechanics about the nature of wavefunction superposition, non-locality, andentanglement.[2][3] As the interpretation of quantum mechanics is important to philosophers of science, somecompare the idea of degree of belief and its application in Quantum Bayesianism with the idea of anti-realism[] fromphilosophy of science.On a technical level, QBism uses symmetric, informationally-complete, positive operator-valued measures(SIC-POVMs) to rewrite quantum states (either pure or mixed) as a set of probabilities defined over the outcomes ofa "Bureau of Standards" measurement.[4] That is, if one translates a density matrix into a probability distribution overthe outcomes of a SIC-POVM experiment, one can reproduce all the statistical predictions (normally computed byusing the Born rule) on the density matrix from the SIC-POVM probabilities instead. The Born rule then takes on thefunction of relating one valid probability distribution to another, rather than of deriving probabilities from somethingapparently more fundamental.[5] QBist foundational research stimulated interest in SIC-POVMs, which now haveapplications in quantum theory outside of foundational studies.[6] Likewise, a quantum version of the de Finettitheorem, introduced by Caves, Fuchs and Schack to provide a QBist understanding of the idea of an "unknownquantum state",[7] has found application elsewhere, in topics like quantum key distribution[8] and entanglementdetection.[9]

OriginIn the field of probability theory, there are different interpretations of probability and different forms of statisticalinference which influence the conclusions that can be made from analysis of uncertain phenomena. The twodominant approaches to statistical inference include the frequentist approach (called frequentist inference) and theBayesian approach (called Bayesian inference). The Bayesian approach upon which Quantum Bayesianism reliesgenerally refers to a mode of statistical inference originating in, and greatly extending, the work of Thomas Bayes instatistics and probability.Quantum Bayesianism tries to find a new understanding of quantum mechanics by applying Bayesian inference. Anynew insights into quantum mechanics are beneficial, especially in light of the recent attempts to combine quantummechanics and general relativity into a theory of quantum gravity and the interest in quantum computation. Quantummechanics is thought to be derivable from the principles of quantum information.In the book Lost Causes in and beyond Physics, Streater writes "[t]he first quantum Bayesian was von Neumann. InDie mathematischen Grundlagen der Quantenmechanik, he describes the measurement process of say the spinpolarization of an electron source ...".[10]

BackgroundQuantum Bayesianism applies the Bayesian approach to the fundamentals of quantum mechanics. The Bayesianapproach is a mode of statistical inference which is itself derived from information theory. It introduces the conceptof "degree of belief".[] Quantum bayesianism is used in quantum computer science for Quantum Bayesian networks,which find applications in "medical diagnosis, monitoring of processes, and genetics".[] A Bayesian framework isalso used for neural networks.[]

Quantum Bayesianism 45

An interpretation of quantum mechanics, Quantum Bayesianism attempts to find compatibility between differentunderstandings of quantum mechanics and their respective implications on metaphysics and philosophy. Mainly, itattempts to answer questions about the nature of the universe and the observer effect. [citation needed]

When the wavefunction of a system is written as a linear combination of the eigenstates of an observable such asposition, the square of the coefficient corresponding to the eigenstate also corresponds to the probability of thesystem being in that eigenstate with the particular observable value. Since this is probabilistic, this leads to thequestion of whether the universe is deterministic and how this is consistent with events being describedprobabilistically. Another idea which Quantum Bayesianism tries to address is whether quantum mechanicalprobabilities are objective or subjective, and the implications of the Born rule on either. [citation needed]

Other variationsQuantum Bayesianism is an alternative to the (more) popular Copenhagen interpretation of quantum mechanics,which is built upon the idea of wavefunction collapse. The Copenhagen interpretation of quantum mechanics doesnot assume a specific interpretation of probability.A related attempt is to derive physics from Fisher information, described in Roy Frieden's book titled Physics fromFisher Information. However, his claims do not stand up to a close examination.[11]

Other approaches to quantum mechanics are broadly related in that they also treat quantum states as expressions ofinformation, knowledge, belief, or expectation. All these approaches - including QBism - can be termed"psi-epistemic",[12] but they differ in what they consider quantum states to be information or expectations 'about', aswell as in the technical features of the mathematics they employ.[13]

Comparisons have also been made between QBism and the relational quantum mechanics espoused by Carlo Rovelliand others.[14]

References[3][3] Mermin (2012a), Mermin (2012b)[4][4] Fuchs and Schack (2011); Appleby, Ericsson and Fuchs (2011); Rosado (2011); Fuchs (2012)[5][5] Fuchs and Schack (2011); Appleby, Ericsson and Fuchs (2011); Fuchs (2012)[6] Scott (2006); Wootters and Sussman (2007); Fuchs (2012); Appleby et al. (2012)[7] Caves, Fuchs and Schack (2002); Baez (2007)[8][8] Renner (2005)[9] Doherty et al. (2005)[12][12] Harrigan and Spekkens (2010)[13][13] Bub and Pitowsky (2009)[14][14] Marlow (2006); Smerlak and Rovelli (2007); Fuchs (2012)

• Palge, Veiko; Konrad, Thomas (2008). "A remark on Fuchs’ Bayesian interpretation of quantum mechanics".Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 39(2): 273–287. doi: 10.1016/j.shpsb.2007.10.002 (http:/ / dx. doi. org/ 10. 1016/ j. shpsb. 2007. 10. 002).

• Plotnitsky, Arkady (2010). Epistemology and probability : Bohr, Heisenberg, Schrödinger and the nature ofquantum-theoretical thinking. New York: Springer. p. 12. ISBN 978-0-387-85333-8.

• C. M. Caves; C. A. Fuchs; R. Schack (2002). "Unknown quantum states: the quantum de Finetti representation".Journal of Mathematical Physics 43: 4537. arXiv: quant-ph/0104088 (http:/ / arxiv. org/ abs/ quant-ph/ 0104088).Bibcode: 2002JMP....43.4537C (http:/ / adsabs. harvard. edu/ abs/ 2002JMP. . . . 43. 4537C). doi:10.1063/1.1494475 (http:/ / dx. doi. org/ 10. 1063/ 1. 1494475).

• Renner, Renato (2005). "Security of Quantum Key Distribution (PhD thesis, ETH Zurich)". arXiv:quant-ph/0512258 (http:/ / arxiv. org/ abs/ quant-ph/ 0512258).

• A. C. Doherty; P. A. Parillo; F. M. Spedalieri (2005). "Detecting multipartite entanglement". Physical Review A 71 (3): 032333. arXiv: quant-ph/0407143 (http:/ / arxiv. org/ abs/ quant-ph/ 0407143). Bibcode:

Quantum Bayesianism 46

2005PhRvA..71c2333D (http:/ / adsabs. harvard. edu/ abs/ 2005PhRvA. . 71c2333D). doi:10.1103/PhysRevA.71.032333 (http:/ / dx. doi. org/ 10. 1103/ PhysRevA. 71. 032333).

• Marlow, Thomas (2006). "Relationalism vs. Bayesianism". arXiv: gr-qc/0603015 (http:/ / arxiv. org/ abs/ gr-qc/0603015).

• M. Smerlak; C. Rovelli (2007). "Relational EPR". Foundations of Physics 37: 427–445. arXiv: quant-ph/0604064(http:/ / arxiv. org/ abs/ quant-ph/ 0604064). Bibcode: 2007FoPh...37..427S (http:/ / adsabs. harvard. edu/ abs/2007FoPh. . . 37. . 427S). doi: 10.1007/s10701-007-9105-0 (http:/ / dx. doi. org/ 10. 1007/ s10701-007-9105-0).

• A. J. Scott (2006). "Tight Informationally Complete Quantum Measurements". Journal of Physics A 39: 13507.arXiv: quant-ph/0604049 (http:/ / arxiv. org/ abs/ quant-ph/ 0604049). Bibcode: 2006JPhA...3913507S (http:/ /adsabs. harvard. edu/ abs/ 2006JPhA. . . 3913507S). doi: 10.1088/0305-4470/39/43/009 (http:/ / dx. doi. org/ 10.1088/ 0305-4470/ 39/ 43/ 009).

• J. Baez (2007). "This Week's Finds in Mathematical Physics (Week 251)" (http:/ / math. ucr. edu/ home/ baez/week251. html). Retrieved 29 April 2012.

• W. K. Wootters; D. M. Sussman (2007). Discrete phase space and minimum-uncertainty states. arXiv: 0704.1277(http:/ / arxiv. org/ abs/ 0704. 1277). Bibcode: 2007arXiv0704.1277W (http:/ / adsabs. harvard. edu/ abs/2007arXiv0704. 1277W).

• J. Bub; I. Pitowsky (2009). "Two dogmas about quantum mechanics". Everett @ 50. arXiv: 0712.4258 (http:/ /arxiv. org/ abs/ 0712. 4258). Bibcode: 2007arXiv0712.4258B (http:/ / adsabs. harvard. edu/ abs/ 2007arXiv0712.4258B).

• C. Brukner; A. Zeilinger (2009). "Information Invariance and Quantum Probabilities". Foundations of Physics 39:677–689. arXiv: 0905.0653 (http:/ / arxiv. org/ abs/ 0905. 0653). Bibcode: 2009FoPh...39..677B (http:/ / adsabs.harvard. edu/ abs/ 2009FoPh. . . 39. . 677B). doi: 10.1007/s10701-009-9316-7 (http:/ / dx. doi. org/ 10. 1007/s10701-009-9316-7).

• Barnum, Howard (2010). "Quantum Knowledge, Quantum Belief, Quantum Reality: Notes of a QBist FellowTraveler". arXiv: 1003.4555 (http:/ / arxiv. org/ abs/ 1003. 4555).

• J. Bub (2010). "Quantum probabilities: an information-theoretic interpretation". Probabilities in Physics. arXiv:1005.2448 (http:/ / arxiv. org/ abs/ 1005. 2448). Bibcode: 2010arXiv1005.2448B (http:/ / adsabs. harvard. edu/abs/ 2010arXiv1005. 2448B).

• N. Harrigan; R. Spekkens (2010). "Einstein, incompleteness, and the epistemic view of quantum states".Foundations of Physics 40: 125. arXiv: 0706.2661 (http:/ / arxiv. org/ abs/ 0706. 2661). Bibcode:2010FoPh...40..125H (http:/ / adsabs. harvard. edu/ abs/ 2010FoPh. . . 40. . 125H). doi:10.1007/s10701-009-9347-0 (http:/ / dx. doi. org/ 10. 1007/ s10701-009-9347-0).

• Fuchs, C. A.; Schack, R. (2011). "A Quantum-Bayesian route to quantum-state space". Foundations of Physics 41(3): 345–56. arXiv: 0912.4252 (http:/ / arxiv. org/ abs/ 0912. 4252). Bibcode: 2011FoPh...41..345F (http:/ /adsabs. harvard. edu/ abs/ 2011FoPh. . . 41. . 345F). doi: 10.1007/s10701-009-9404-8 (http:/ / dx. doi. org/ 10.1007/ s10701-009-9404-8).

• Appleby, D.M.; A. Ericsson; and C. A. Fuchs (2011). "Properties of QBist state spaces". Foundations of Physics41 (3): 564–79. arXiv: 0910.2750 (http:/ / arxiv. org/ abs/ 0910. 2750). Bibcode: 2009arXiv0910.2750A (http:/ /adsabs. harvard. edu/ abs/ 2009arXiv0910. 2750A).

• Appleby, D.M.; S. T. Flammia; and C. A. Fuchs (2011). "The Lie algebraic significance of symmetricinformationally complete measurements". Journal of Mathematical Physics 52: 022202. arXiv: 1001.0004 (http:// arxiv. org/ abs/ 1001. 0004). Bibcode: 2010arXiv1001.0004A (http:/ / adsabs. harvard. edu/ abs/2010arXiv1001. 0004A).

• J. I. Rosado (2011). "Representation of quantum states as points in a probability simplex associated to aSIC-POVM". Foundations of Physics 41 (3): 1200–13. arXiv: 1007.0715 (http:/ / arxiv. org/ abs/ 1007. 0715).Bibcode: 2011FoPh...41.1200R (http:/ / adsabs. harvard. edu/ abs/ 2011FoPh. . . 41. 1200R). doi:10.1007/s10701-011-9540-9 (http:/ / dx. doi. org/ 10. 1007/ s10701-011-9540-9).

Quantum Bayesianism 47

• Leifer, M. S.; Spekkens, R. (2011). "Formulating Quantum Theory as a Causally Neutral Theory of BayesianInference". arXiv: 1107.5849 (http:/ / arxiv. org/ abs/ 1107. 5849).

• Leifer, M. S.; Spekkens, R. (2011). "A Bayesian approach to compatibility, improvement, and pooling ofquantum states". arXiv: 1110.1085 (http:/ / arxiv. org/ abs/ 1110. 1085).

• Bartlett, S. D.; T. Rudolph; R. Spekkens (2011). "Reconstruction of Gaussian quantum mechanics from Liouvillemechanics with an epistemic restriction". arXiv: 1111.5057 (http:/ / arxiv. org/ abs/ 1111. 5057).

• Fuchs, Christopher (2011). Coming of Age with Quantum Information: Notes on a Paulian Idea. CambridgeUniversity Press. ISBN 978-0-521-19926-1.

• X.-S. Ma et al. (2012). "Experimental delayed-choice entanglement swapping". Nature Physics. in press. arXiv:1203.4834 (http:/ / arxiv. org/ abs/ 1203. 4834). Bibcode: 2012NatPh...8..480M (http:/ / adsabs. harvard. edu/ abs/2012NatPh. . . 8. . 480M). doi: 10.1038/nphys2294 (http:/ / dx. doi. org/ 10. 1038/ nphys2294).

• D.M. Appleby; I. Bengtsson; S. Brierley; M. Grassl; D. Gross; J.-A. Larsson (2012). "The monomialrepresentations of the Clifford group". Quantum Information and Computation 12 (5&6): 0404–0431. arXiv:1102.1268 (http:/ / arxiv. org/ abs/ 1102. 1268). Bibcode: 2011arXiv1102.1268A (http:/ / adsabs. harvard. edu/abs/ 2011arXiv1102. 1268A).

• C. A. Fuchs (27 April 2012). "My Struggles with the Block Universe" (http:/ / www. perimeterinstitute. ca/personal/ cfuchs/ nSamizdat-2. pdf). Retrieved 29 April 2012.

• Mermin, N. David. "Quantum mechanics: Fixing the shifty split". Physics Today 65 (7): 8. doi:10.1063/PT.3.1618 (http:/ / dx. doi. org/ 10. 1063/ PT. 3. 1618).

• Mermin, N. David. "Measured responses to Quantum Bayesianism". Physics Today 65 (12): 12. doi:10.1063/PT.3.1803 (http:/ / dx. doi. org/ 10. 1063/ PT. 3. 1803).

• von Baeyer, Hans Christian (2013). "Quantum Weirdness? It's all in your mind". Scientific American 308 (6): 6.

Further readingvon Baeyer, Hans Christian (2013). "TCan Quantum Bayesianism Fix the Paradoxes of Quantum Mechanics?"(http:/ / www. scientificamerican. com/ article. cfm?id=can-quantum-beyesnism-fix-paradoxes-quantum-mechanics).Scientific American 2013 (June). Retrieved 6 June 2013.(subscription required)

External links• QBism, the Perimeter of Quantum Bayesianism (http:/ / arxiv. org/ abs/ 1003. 5209)• That the World Can Be Shaped: Quantum Bayesianism, Counterfactuals, Free Will (http:/ / www. templeton. org/

what-we-fund/ grants/ that-the-world-can-be-shaped-quantum-bayesianism-counterfactuals-free-will)• The Elegance of Enigma: Quantum Darwinism, Quantum Bayesianism (QBism) & Quantum Buddhism (http:/ /

www. jcer. com/ index. php/ jcj/ article/ view/ 189)• Quantum Mechanics as Quantum Information (and only a little more) (http:/ / perimeterinstitute. ca/ personal/

cfuchs/ VaccineQPH. pdf)• Being Bayesian in a Quantum World (http:/ / web. archive. org/ web/ 20090511060206/ http:/ / www.

uni-konstanz. de/ ppm/ events/ bbqw2005/ ) - 2005 conference at the University of Konstanz• Seeking SICs: A Workshop on Quantum Frames and Designs (http:/ / pirsa. org/ C08025) - 2008 conference at

the Perimeter Institute

Wave function collapse 48

Wave function collapse

Quantummechanics

IntroductionGlossary · History

In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in asuperposition of several eigenstates—appears to reduce to a single eigenstate after interaction with an observer.[] It isthe essence of measurement in quantum mechanics, and connects the wave function with classical observables likeposition and momentum. In classical terms, it is the reduction of all possible physical states to a single possibilitywhich is measured by the observer. Collapse is one of two processes by which quantum systems evolve in time; theother is continuous evolution via the Schrödinger equation.[1] However in this role, collapse is merely a black box forthermodynamically irreversible interaction with a classical environment.[] Calculations of quantum decoherencepredict apparent wave function collapse when a superposition forms between the quantum system's states and theenvironment's states. Significantly, the combined wave function of the system and environment continue to obey theSchrödinger equation.[]

When the Copenhagen interpretation was first expressed, Bohr postulated wave function collapse to cut the quantumworld from the classical.[2] This tactical move allowed quantum theory to develop without distractions frominterpretational worries. Nevertheless it was debated, for if collapse were a fundamental physical phenomenon,rather than just the epiphenomenon of some other process, it would mean nature were fundamentally stochastic, i.e.nondeterministic, an undesirable property for a theory.[][3] This issue remained until quantum decoherence enteredmainstream opinion after its reformulation in the 1980s.[][][4] Decoherence explains the perception of wave functioncollapse in terms of interacting large- and small-scale quantum systems, and is commonly taught at the graduatelevel (e.g. the Cohen-Tannoudji textbook).[5] The quantum filtering approach[][][] and the introduction of quantumcausality non-demolition principle[] allows for a classical-environment derivation of wave function collapse from thestochastic Schrödinger equation.

Mathematical descriptionBefore collapse, the wave function may be any square-integrable function. This function is expressible as a linearcombination of the eigenstates of any observable. Observables represent classical dynamical variables, and when oneis measured by a classical observer, the wave function is projected onto a random eigenstate of that observable. Theobserver simultaneously measures the classical value of that observable to be the eigenvalue of the final state.[]

Mathematical backgroundThe quantum state of a physical system is described by a wave function (in turn – an element of a projective Hilbertspace). This can be expressed in Dirac or bra-ket notation as a vector:

The kets , specify the different quantum "alternatives" available - a particular quantum state. Theyform an orthonormal eigenvector basis, formally

Where represents the Kronecker delta.An observable (i.e. measurable parameter of the system) is associated with each eigenbasis, with each quantum alternative having a specific value or eigenvalue, ei, of the observable. A "measurable parameter of the system"

Wave function collapse 49

could be the usual position r and the momentum p of (say) a particle, but also its energy E, z-components of spin(sz), orbital (Lz) and total angular (Jz) momenta etc. In the basis representation these are respectively .The coefficients c1, c2, c3... are the probability amplitudes corresponding to each basis . These arecomplex numbers. The moduli square of ci, that is |ci|

2 = ci*ci (* denotes complex conjugate), is the probability ofmeasuring the system to be in the state .For simplicity in the following, all wave functions are assumed to be normalized; the total probability of measuringall possible states is unity:

The process of collapseWith these definitions it is easy to describe the process of collapse. For any observable, the wave function is initiallysome linear combination of the eigenbasis of that observable. When an external agency (an observer,experimenter) measures the observable associated with the eigenbasis , the wave function collapses fromthe full to just one of the basis eigenstates, , that is:

The probability of collapsing to a given eigenstate is the Born probability, . Post-measurement,other elements of the wave function vector, , have "collapsed" to zero, and .More generally, collapse is defined for an operator with eigenbasis . If the system is in state , and

is measured, the probability of collapsing the system to state (and measuring ) would be . Note that this is not the probability that the particle is in state : that's complete nonsense. It is in state until cast to an eigenstate of .However, we never observe collapse to a single eigenstate of a continuous-spectrum operator (e.g. position,momentum, or a scattering Hamiltonian), because such eigenfunctions are non-normalizable. In these cases, thewave function will partially collapse to a linear combination of "close" eigenstates (necessarily involving a spread ineigenvalues) that embodies the imprecision of the measurement apparatus. The more precise the measurement, thetighter the range. Calculation of probability proceeds identically, except with an integral over the expansioncoefficient .[] This phenomenon is unrelated to the uncertainty principle, although increasingly precisemeasurements of one operator (e.g. position) will naturally homogenize the expansion coefficient of wave functionwith respect to another, incompatible operator (e.g. momentum), lowering the probability of measuring anyparticular value of the latter.

Quantum decoherenceWave function collapse is not fundamental from the perspective of quantum decoherence.[6] There are severalequivalent approaches to deriving collapse, like the density matrix approach, but each has the same effect:decoherence irreversibly converts the "averaged" or "environmentally traced over" density matrix from a pure stateto a reduced mixture, giving the appearance of wave function collapse.

History and contextThe concept of wavefunction collapse was introduced by Werner Heisenberg in his 1927 paper on the uncertaintyprinciple, "Über den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik", and incorporated intothe mathematical formulation of quantum mechanics by John von Neumann, in his 1932 treatise MathematischeGrundlagen der Quantenmechanik.[7] Consistent with Heisenberg, von Neumann postulated that there were twoprocesses of wave function change:

Wave function collapse 50

1. The probabilistic, non-unitary, non-local, discontinuous change brought about by observation and measurement,as outlined above.

2. The deterministic, unitary, continuous time evolution of an isolated system that obeys the Schrödinger equation(or a relativistic equivalent, i.e. the Dirac equation).

In general, quantum systems exist in superpositions of those basis states that most closely correspond to classicaldescriptions, and, in the absence of measurement, evolve according to the Schrödinger equation. However, when ameasurement is made, the wave function collapses—from an observer's perspective—to just one of the basis states,and the property being measured uniquely acquires the eigenvalue of that particular state, . After the collapse, thesystem again evolves according to the Schrödinger equation.By explicitly dealing with the interaction of object and measuring instrument, von Neumann[1] has attempted tocreate consistency of the two processes of wave function change.He was able to prove the possibility of a quantum mechanical measurement scheme consistent with wave functioncollapse. However, he did not prove the necessity of such a collapse. Although von Neumann's projection postulateis often presented as a normative description of quantum measurement, it was conceived by taking into accountexperimental evidence available during the 1930s (in particular the Compton-Simon experiment has beenparadigmatic), and many important present-day measurement procedures do not satisfy it (so-called measurements ofthe second kind).[8][9][10]

The existence of the wave function collapse is required in• the Copenhagen interpretation• the objective collapse interpretations• the transactional interpretation• the von Neumann interpretation in which consciousness causes collapse.On the other hand, the collapse is considered a redundant or optional approximation in• the Consistent histories approach, self-dubbed "Copenhagen done right"• the Bohm interpretation• the Many-worlds interpretation• the Ensemble InterpretationThe cluster of phenomena described by the expression wave function collapse is a fundamental problem in theinterpretation of quantum mechanics, and is known as the measurement problem. The problem is deflected by theCopenhagen Interpretation, which postulates that this is a special characteristic of the "measurement" process.Everett's many-worlds interpretation deals with it by discarding the collapse-process, thus reformulating the relationbetween measurement apparatus and system in such a way that the linear laws of quantum mechanics are universallyvalid; that is, the only process according to which a quantum system evolves is governed by the Schrödingerequation or some relativistic equivalent.Originating from Everett's theory, but no longer tied to it, is the physical process of decoherence, which causes anapparent collapse. Decoherence is also important for the consistent histories interpretation. A general description ofthe evolution of quantum mechanical systems is possible by using density operators and quantum operations. In thisformalism (which is closely related to the C*-algebraic formalism) the collapse of the wave function corresponds toa non-unitary quantum operation.The significance ascribed to the wave function varies from interpretation to interpretation, and varies even within aninterpretation (such as the Copenhagen Interpretation). If the wave function merely encodes an observer's knowledgeof the universe then the wave function collapse corresponds to the receipt of new information. This is somewhatanalogous to the situation in classical physics, except that the classical "wave function" does not necessarily obey awave equation. If the wave function is physically real, in some sense and to some extent, then the collapse of thewave function is also seen as a real process, to the same extent.

Wave function collapse 51

References[6] Wojciech H. Zurek, Decoherence, einselection, and the quantum origins of the classical,Reviews of Modern Physics 2003, 75, 715 or http:/ /

arxiv. org/ abs/ quant-ph/ 0105127[9][9] )[10][10] Discussions of measurements of the second kind can be found in most treatments on the foundations of quantum mechanics, for instance, ; ;

and .

Relational quantum mechanics

Quantummechanics

IntroductionGlossary · History

This article is intended for those already familiar with quantum mechanics and its attendant interpretationaldifficulties. Readers who are new to the subject may first want to read the introduction to quantummechanics.

Relational quantum mechanics (RQM) is an interpretation of quantum mechanics which treats the state of aquantum system as being observer-dependent, that is, the state is the relation between the observer and the system.This interpretation was first delineated by Carlo Rovelli in a 1994 preprint, and has since been expanded upon by anumber of theorists. It is inspired by the key idea behind Special Relativity, that the details of an observation dependon the reference frame of the observer, and uses some ideas from Wheeler on quantum information.[1]

The physical content of the theory is thus not to do with objects themselves, but the relations between them. AsRovelli puts it: "Quantum mechanics is a theory about the physical description of physical systems relative to othersystems, and this is a complete description of the world".[2]

The essential idea behind RQM is that different observers may give different accounts of the same series of events:for example, to one observer at a given point in time, a system may be in a single, "collapsed" eigenstate, while toanother observer at the same time, it may appear to be in a superposition of two or more states. Consequently, ifquantum mechanics is to be a complete theory, RQM argues that the notion of "state" describes not the observedsystem itself, but the relationship, or correlation, between the system and its observer(s). The state vector ofconventional quantum mechanics becomes a description of the correlation of some degrees of freedom in theobserver, with respect to the observed system. However, it is held by RQM that this applies to all physical objects,whether or not they are conscious or macroscopic (all systems are quantum systems). Any "measurement event" isseen simply as an ordinary physical interaction, an establishment of the sort of correlation discussed above. Theproponents of the relational interpretation argue that the approach clears up a number of traditional interpretationaldifficulties with quantum mechanics, while being simultaneously conceptually elegant and ontologicallyparsimonious.

History and developmentRelational Quantum Mechanics arose from a historical comparison of the quandaries posed by the interpretation of quantum mechanics with the situation after the Lorentz transformations were formulated but before Special Relativity. Rovelli felt that just as there was an "incorrect assumption" underlying the pre-relativistic interpretation of Lorentz's equations, which was corrected by Einstein's derivation of them from Lorentz covariance and the constancy of the speed of light in all reference frames, so a similarly incorrect assumption underlies many attempts to make sense of the quantum formalism, which was responsible for many of the interpretational difficulties posed

Relational quantum mechanics 52

by the theory. This incorrect assumption, he said, was that of an observer-independent state of a system, and he laidout the foundations of this interpretation to try to overcome the difficulty.[3] Since then, the idea has been expandedupon by Lee Smolin[4] and Louis Crane,[5] who have both applied the concept to quantum cosmology, and theinterpretation has been applied to the EPR paradox, revealing not only a peaceful co-existence between quantummechanics and Special Relativity, but a formal indication of a completely local character to reality.[6][7]

The problem of the observer observedThis problem was initially discussed in detail in Everett's thesis, The Theory of the Universal Wavefunction.Consider observer , measuring the state of the quantum system . We assume that has completeinformation on the system, and that can write down the wavefunction describing it. At the same time, thereis another observer , who is interested in the state of the entire - system, and likewise has completeinformation.To analyse this system formally, we consider a system which may take one of two states, which we shalldesignate and , ket vectors in the Hilbert space . Now, the observer wishes to make ameasurement on the system. At time , this observer may characterize the system as follows:

where and are probabilities of finding the system in the respective states, and obviously add up to 1. Forour purposes here, we can assume that in a single experiment, the outcome is the eigenstate (but this can besubstituted throughout, mutatis mutandis, by ). So, we may represent the sequence of events in this experiment,with observer doing the observing, as follows:

This is observer 's description of the measurement event. Now, any measurement is also a physical interactionbetween two or more systems. Accordingly, we can consider the tensor product Hilbert space , where

is the Hilbert space inhabited by state vectors describing . If the initial state of is , some degreesof freedom in become correlated with the state of after the measurement, and this correlation can take one oftwo values: or where the direction of the arrows in the subscripts corresponds to the outcome of themeasurement that has made on . If we now consider the description of the measurement event by the otherobserver, , who describes the combined system, but does not interact with it, the following gives thedescription of the measurement event according to , from the linearity inherent in the quantum formalism:

Thus, on the assumption (see hypothesis 2 below) that quantum mechanics is complete, the two observers andgive different but equally correct accounts of the events .

Relational quantum mechanics 53

Central principles

Observer-dependence of stateAccording to , at , the system is in a determinate state, namely spin up. And, if quantum mechanics iscomplete, then so is his description. But, for , is not uniquely determinate, but is rather entangled with thestate of — note that his description of the situation at is not factorisable no matter what basis chosen. But, ifquantum mechanics is complete, then the description that gives is also complete.Thus the standard mathematical formulation of quantum mechanics allows different observers to give differentaccounts of the same sequence of events. There are many ways to overcome this perceived difficulty. It could bedescribed as an epistemic limitation — observers with a full knowledge of the system, we might say, could give acomplete and equivalent description of the state of affairs, but that obtaining this knowledge is impossible inpractice. But whom? What makes 's description better than that of , or vice versa? Alternatively, we couldclaim that quantum mechanics is not a complete theory, and that by adding more structure we could arrive at auniversal description — the much vilified, and some would even say discredited, hidden variables approach. Yetanother option is to give a preferred status to a particular observer or type of observer, and assign the epithet ofcorrectness to their description alone. This has the disadvantage of being ad hoc, since there are no clearly defined orphysically intuitive criteria by which this super-observer ("who can observe all possible sets of observations by allobservers over the entire universe"[8]) ought to be chosen.RQM, however, takes the point illustrated by this problem at face value. Instead of trying to modify quantummechanics to make it fit with prior assumptions that we might have about the world, Rovelli says that we shouldmodify our view of the world to conform to what amounts to our best physical theory of motion.[9] Just as forsakingthe notion of absolute simultaneity helped clear up the problems associated with the interpretation of the Lorentztransformations, so many of the conundra associated with quantum mechanics dissolve, provided that the state of asystem is assumed to be observer-dependent — like simultaneity in Special Relativity. This insight follows logicallyfrom the two main hypotheses which inform this interpretation:• Hypothesis 1: the equivalence of systems. There is no a priori distinction that should be drawn between quantum

and macroscopic systems. All systems are, fundamentally, quantum systems.• Hypothesis 2: the completeness of quantum mechanics. There are no hidden variables or other factors which may

be appropriately added to quantum mechanics, in light of current experimental evidence.Thus, if a state is to be observer-dependent, then a description of a system would follow the form "system S is instate x with reference to observer O" or similar constructions, much like in relativity theory. In RQM it ismeaningless to refer to the absolute, observer-independent state of any system.

Information and correlationIt is generally well established that any quantum mechanical measurement can be reduced to a set of yes/noquestions or bits that are either 1 or 0. [citation needed] RQM makes use of this fact to formulate the state of a quantumsystem (relative to a given observer!) in terms of the physical notion of information developed by Claude Shannon.Any yes/no question can be described as a single bit of information. This should not be confused with the idea of aqubit from quantum information theory, because a qubit can be in a superposition of values, whilst the "questions" ofRQM are ordinary binary variables.Any quantum measurement is fundamentally a physical interaction between the system being measured and someform of measuring apparatus. By extension, any physical interaction may be seen to be a form of quantummeasurement, as all systems are seen as quantum systems in RQM. A physical interaction is seen as establishing acorrelation between the system and the observer, and this correlation is what is described and predicted by thequantum formalism.

Relational quantum mechanics 54

But, Rovelli points out, this form of correlation is precisely the same as the definition of information in Shannon'stheory. Specifically, an observer O observing a system S will, after measurement, have some degrees of freedomcorrelated with those of S. The amount of this correlation is given by log2k bits, where k is the number of possiblevalues which this correlation may take — the number of "options" there are.

All systems are quantum systemsAll physical interactions are, at bottom, quantum interactions, and must ultimately be governed by the same rules.Thus, an interaction between two particles does not, in RQM, differ fundamentally from an interaction between aparticle and some "apparatus". There is no true wave collapse, in the sense in which it occurs in the Copenhageninterpretation.Because "state" is expressed in RQM as the correlation between two systems, there can be no meaning to"self-measurement". If observer measures system , 's "state" is represented as a correlation between and . itself cannot say anything with respect to its own "state", because its own "state" is defined only relativeto another observer, . If the compound system does not interact with any other systems, then it willpossess a clearly defined state relative to . However, because 's measurement of breaks its unitaryevolution with respect to , will not be able to give a full description of the system (since it can onlyspeak of the correlation between and itself, not its own behaviour). A complete description of the

system can only be given by a further, external observer, and so forth.Taking the model system discussed above, if has full information on the system, it will know theHamiltonians of both and , including the interaction Hamiltonian. Thus, the system will evolve entirelyunitarily (without any form of collapse) relative to , if measures . The only reason that will perceive a"collapse" is because has incomplete information on the system (specifically, does not know its ownHamiltonian, and the interaction Hamiltonian for the measurement).

Consequences and implications

CoherenceIn our system above, may be interested in ascertaining whether or not the state of accurately reflects the stateof . We can draw up for an operator, , which is specified as:

with an eigenvalue of 1 meaning that indeed accurately reflects the state of . So there is a 0 probability of reflecting the state of as being if it is in fact , and so forth. The implication of this is that at time ,

can predict with certainty that the system is in some eigenstate of , but cannot say which eigenstateit is in, unless itself interacts with the system.An apparent paradox arises when one considers the comparison, between two observers, of the specific outcome of ameasurement. In the problem of the observer observed section above, let us imagine that the two experiments wantto compare results. It is obvious that if the observer has the full Hamiltonians of both and , he will be ableto say with certainty that at time , has a determinate result for 's spin, but he will not be able to say what

's result is without interaction, and hence breaking the unitary evolution of the compound system (because hedoesn't know his own Hamiltonian). The distinction between knowing "that" and knowing "what" is a common onein everyday life: everyone knows that the weather will be like something tomorrow, but no-one knows exactly whatthe weather will be like.

Relational quantum mechanics 55

But, let us imagine that measures the spin of , and finds it to have spin down (and note that nothing in theanalysis above precludes this from happening). What happens if he talks to , and they compare the results of theirexperiments? , it will be remembered, measured a spin up on the particle. This would appear to be paradoxical:the two observers, surely, will realise that they have disparate results.However, this apparent paradox only arises as a result of the question being framed incorrectly: as long as wepresuppose an "absolute" or "true" state of the world, this would, indeed, present an insurmountable obstacle for therelational interpretation. However, in a fully relational context, there is no way in which the problem can even becoherently expressed. The consistency inherent in the quantum formalism, exemplified by the "M-operator" definedabove, guarantees that there will be no contradictions between records. The interaction between and whatever hechooses to measure, be it the compound system or and individually, will be a physical interaction, aquantum interaction, and so a complete description of it can only be given by a further observer , who will havea similar "M-operator" guaranteeing coherency, and so on out. In other words, a situation such as that describedabove cannot violate any physical observation, as long as the physical content of quantum mechanics is taken torefer only to relations.

Relational networksAn interesting implication of RQM arises when we consider that interactions between material systems can onlyoccur within the constraints prescribed by Special Relativity, namely within the intersections of the light cones of thesystems: when they are spatiotemporally contiguous, in other words. Relativity tells us that objects have locationonly relative to other objects. By extension, a network of relations could be built up based on the properties of a setof systems, which determines which systems have properties relative to which others, and when (since properties areno longer well defined relative to a specific observer after unitary evolution breaks down for that observer). On theassumption that all interactions are local (which is backed up by the analysis of the EPR paradox presented below),one could say that the ideas of "state" and spatiotemporal contiguity are two sides of the same coin: spacetimelocation determines the possibility of interaction, but interactions determine spatiotemporal structure. The full extentof this relationship, however, has not yet fully been explored.

RQM and quantum cosmologyThe universe is the sum total of all that is in existence. Physically, a (physical) observer outside of the universewould require the breaking of gauge invariance,[10] and a concomitant alteration in the mathematical structure ofgauge-invariance theory. Similarly, RQM conceptually forbids the possibility of an external observer. Since theassignment of a quantum state requires at least two "objects" (system and observer), which must both be physicalsystems, there is no meaning in speaking of the "state" of the entire universe. This is because this state would have tobe ascribed to a correlation between the universe and some other physical observer, but this observer in turn wouldhave to form part of the universe, and as was discussed above, it is impossible for an object to give a completespecification of itself. Following the idea of relational networks above, an RQM-oriented cosmology would have toaccount for the universe as a set of partial systems providing descriptions of one another. The exact nature of such aconstruction remains an open question.

Relational quantum mechanics 56

Relationship with other interpretationsThe only group of interpretations of quantum mechanics with which RQM is almost completely incompatible is thatof hidden variables theories. RQM shares some deep similarities with other views, but differs from them all to theextent to which the other interpretations do not accord with the "relational world" put forward by RQM.

Copenhagen interpretationRQM is, in essence, quite similar to the Copenhagen interpretation, but with an important difference. In theCopenhagen interpretation, the macroscopic world is assumed to be intrinsically classical in nature, and wavefunction collapse occurs when a quantum system interacts with macroscopic apparatus. In RQM, any interaction, beit micro or macroscopic, causes the linearity of Schrödinger evolution to break down. RQM could recover aCopenhagen-like view of the world by assigning a privileged status (not dissimilar to a preferred frame in relativity)to the classical world. However, by doing this one would lose sight of the key features that RQM brings to our viewof the quantum world.

Hidden variables theoriesBohm's interpretation of QM does not sit well with RQM. One of the explicit hypotheses in the construction of RQMis that quantum mechanics is a complete theory, that is it provides a full account of the world. Moreover, theBohmian view seems to imply an underlying, "absolute" set of states of all systems, which is also ruled out as aconsequence of RQM.We find a similar incompatibility between RQM and suggestions such as that of Penrose, which postulate that someprocess (in Penrose's case, gravitational effects) violate the linear evolution of the Schrödinger equation for thesystem.

Relative-state formulationThe many-worlds family of interpretations (MWI) shares an important feature with RQM, that is, the relationalnature of all value assignments (that is, properties). Everett, however, maintains that the universal wavefunctiongives a complete description of the entire universe, while Rovelli argues that this is problematic, both because thisdescription is not tied to a specific observer (and hence is "meaningless" in RQM), and because RQM maintains thatthere is no single, absolute description of the universe as a whole, but rather a net of inter-related partial descriptions.

Consistent histories approachIn the consistent histories approach to QM, instead of assigning probabilities to single values for a given system, theemphasis is given to sequences of values, in such a way as to exclude (as physically impossible) all valueassignments which result in inconsistent probabilities being attributed to observed states of the system. This is doneby means of ascribing values to "frameworks", and all values are hence framework-dependent.RQM accords perfectly well with this view. However, where the consistent histories approach does not give a fulldescription of the physical meaning of framework-dependent value (that is it does not account for how there can be"facts" if the value of any property depends on the framework chosen). By incorporating the relational view into thisapproach, the problem is solved: RQM provides the means by which the observer-independent,framework-dependent probabilities of various histories are reconciled with observer-dependent descriptions of theworld.

Relational quantum mechanics 57

EPR and quantum non-locality

The EPR thought experiment, performed with electrons. A radioactive source (center)sends electrons in a singlet state toward two spacelike separated observers, Alice (left)and Bob (right), who can perform spin measurements. If Alice measures spin up on her

electron, Bob will measure spin down on his, and vice versa.

RQM provides an unusual solution tothe EPR paradox. Indeed, it managesto dissolve the problem altogether,inasmuch as there is no superluminaltransportation of information involvedin a Bell test experiment: the principleof locality is preserved inviolate for allobservers.

The problem

In the EPR thought experiment, aradioactive source produces twoelectrons in a singlet state, meaning that the sum of the spin on the two electrons is zero. These electrons are fired offat time towards two spacelike separated observers, Alice and Bob, who can perform spin measurements, whichthey do at time . The fact that the two electrons are a singlet means that if Alice measures z-spin up on herelectron, Bob will measure z-spin down on his, and vice versa: the correlation is perfect. If Alice measures z-axisspin, and Bob measures the orthogonal y-axis spin, however, the correlation will be zero. Intermediate angles giveintermediate correlations in a way that, on careful analysis, proves inconsistent with the idea that each particle has adefinite, independent probability of producing the observed measurements (the correlations violate Bell's inequality).

This subtle dependence of one measurement on the other holds even when measurements are made simultaneouslyand a great distance apart, which gives the appearance of a superluminal communication taking place between thetwo electrons. Put simply, how can Bob's electron "know" what Alice measured on hers, so that it can adjust its ownbehavior accordingly?

Relational solutionIn RQM, an interaction between a system and an observer is necessary for the system to have clearly definedproperties relative to that observer. Since the two measurement events take place at spacelike separation, they do notlie in the intersection of Alice' and Bob's light cones. Indeed, there is no observer who can instantaneously measureboth electrons' spin.The key to the RQM analysis is to remember that the results obtained on each "wing" of the experiment only becomedeterminate for a given observer once that observer has interacted with the other observer involved. As far as Aliceis concerned, the specific results obtained on Bob's wing of the experiment are indeterminate for her, although shewill know that Bob has a definite result. In order to find out what result Bob has, she has to interact with him at sometime in their future light cones, through ordinary classical information channels.[11]

The question then becomes one of whether the expected correlations in results will appear: will the two particlesbehave in accordance with the laws of quantum mechanics? Let us denote by the idea that the observer (Alice) measures the state of the system (Alice's particle).So, at time , Alice knows the value of : the spin of her particle, relative to herself. But, since theparticles are in a singlet state, she knows that

and so if she measures her particle's spin to be , she can predict that Bob's particle ( ) will have spin . All this follows from standard quantum mechanics, and there is no "spooky action at a distance" yet. From the "coherence-operator" discussed above, Alice also knows that if at she measures Bob's particle and then measures

Relational quantum mechanics 58

Bob (that is asks him what result he got) — or vice versa — the results will be consistent:

Finally, if a third observer (Charles, say) comes along and measures Alice, Bob, and their respective particles, hewill find that everyone still agrees, because his own "coherence-operator" demands that

and while knowledge that the particles were in a singlet state tells him that

Thus the relational interpretation, by shedding the notion of an "absolute state" of the system, allows for an analysisof the EPR paradox which neither violates traditional locality constraints, nor implies superluminal informationtransfer, since we can assume that all observers are moving at comfortable sub-light velocities. And, mostimportantly, the results of every observer are in full accordance with those expected by conventional quantummechanics.

DerivationA promising feature of this interpretation is that RQM offers the possibility of being derived from a small number ofaxioms, or postulates based on experimental observations. Rovelli's derivation of RQM uses three fundamentalpostulates. However, it has been suggested that it may be possible to reformulate the third postulate into a weakerstatement, or possibly even do away with it altogether.[12] The derivation of RQM parallels, to a large extent,quantum logic. The first two postulates are motivated entirely by experimental results, while the third postulate,although it accords perfectly with what we have discovered experimentally, is introduced as a means of recoveringthe full Hilbert space formalism of quantum mechanics from the other two postulates. The 2 empirical postulates are:• Postulate 1: there is a maximum amount of relevant information that may be obtained from a quantum system.• Postulate 2: it is always possible to obtain new information from a system.

We let denote the set of all possible questions that may be "asked" of a quantum system, which we shalldenote by , . We may experimentally find certain relations between these questions:

, corresponding to {intersection, orthogonal sum, orthogonal complement, inclusion, andorthogonality} respectively, where .

Structure

From the first postulate, it follows that we may choose a subset of mutually independent questions, whereis the number of bits contained in the maximum amount of information. We call such a question a complete

question. The value of can be expressed as an N-tuple sequence of binary valued numerals, which haspossible permutations of "0" and "1" values. There will also be more than one possible complete question.

If we further assume that the relations are defined for all , then is an orthomodular lattice,while all the possible unions of sets of complete questions form a Boolean algebra with the as atoms.[13]The second postulate governs the event of further questions being asked by an observer of a system , when

already has a full complement of information on the system (an answer to a complete question). We denote bythe probability that a "yes" answer to a question will follow the complete question . If is

independent of , then , or it might be fully determined by , in which case . There isalso a range of intermediate possibilities, and this case is examined below.If the question that wants to ask the system is another complete question, , the probability

of a "yes" answer has certain constraints upon it:

1.

Relational quantum mechanics 59

2.

3.

The three constraints above are inspired by the most basic of properties of probabilities, and are satisfied if

,

where is a unitary matrix.• Postulate 3 If and are two complete questions, then the unitary matrix associated with their probability

described above satisfies the equality , for all and .

This third postulate implies that if we set a complete question as a basis vector in a complex Hilbert space,we may then represent any other question as a linear combination:

And the conventional probability rule of quantum mechanics states that if two sets of basis vectors are in the relationabove, then the probability is

DynamicsThe Heisenberg picture of time evolution accords most easily with RQM. Questions may be labelled by a timeparameter , and are regarded as distinct if they are specified by the same operator but are performed atdifferent times. Because time evolution is a symmetry in the theory (it forms a necessary part of the full formalderivation of the theory from the postulates), the set of all possible questions at time is isomorphic to the set of allpossible questions at time . It follows, by standard arguments in quantum logic, from the derivation above that theorthomodular lattice has the structure of the set of linear subspaces of a Hilbert space, with the relationsbetween the questions corresponding to the relations between linear subspaces.It follows that there must be a unitary transformation that satisfies:

and

where is the Hamiltonian, a self-adjoint operator on the Hilbert space and the unitary matrices are an abeliangroup.

Notes[1][1] Wheeler (1990): pg. 3[2][2] Rovelli, C., 1996, "Relational quantum mechanics", International Journal of Theoretical Physics, 35: 1637-1678.[3][3] Rovelli (1996): pg. 2[4][4] Smolin (1995)[5][5] Crane (1993)[6][6] Laudisa (2001)[7] Rovelli & Smerlak (2006)[8][8] Page, Don N., "Insufficiency of the quantum state for deducing observational probabilities", Physics Letters B, Volume 678, Issue 1, 6 July

2009, 41-44.[9][9] Rovelli (1996): pg. 16[10][10] Smolin (1995), pg. 13[11][11] Bitbol (1983)[12][12] Rovelli (1996): pg. 14

Relational quantum mechanics 60

[13][13] Rovelli (1996): pg. 13

References• Bitbol, M.: "An analysis of the Einstein-Podolsky-Rosen correlations in terms of events"; Physics Letters 96A,

1983: 66-70• Crane, L.: "Clock and Category: Is Quantum Gravity Algebraic?"; Journal of Mathematical Physics 36; 1993:

6180-6193; arXiv:gr-qc/9504038 (http:/ / xxx. lanl. gov/ abs/ gr-qc/ 9504038).• Everett, H.: "The Theory of the Universal Wavefunction"; Princeton University Doctoral Dissertation; in DeWitt,

B.S. & Graham, R.N. (eds.): "The Many-Worlds Interpretation of Quantum Mechanics"; Princeton UniversityPress; 1973.

• Finkelstein, D.R. Quantum Relativity: A Synthesis of the Ideas of Einstein and Heisenberg; Springer-Verlag; 1996• Floridi, L.: "Informational Realism"; Computers and Philosophy 2003 - Selected Papers from the Computer and

Philosophy conference (CAP 2003), Conferences in Research and Practice in Information Technology, '37',2004, edited by J. Weckert. and Y. Al-Saggaf, ACS, pp. 7–12. (http:/ / crpit. com/ confpapers/ CRPITV37Floridi.pdf)

• Laudisa, F.: "The EPR Argument in a Relational Interpretation of Quantum Mechanics"; Foundations of PhysicsLetters, 14 (2); 2001: pp. 119–132; arXiv:quant-ph/0011016 (http:/ / lanl. arxiv. org/ abs/ quant-ph/ 0011016)

• Laudisa, F. & Rovelli, C.: "Relational Quantum Mechanics"; The Stanford Encyclopedia of Philosophy (Fall2005 Edition), Edward N. Zalta (ed.); online article (http:/ / plato. stanford. edu/ archives/ fall2005/ entries/qm-relational/ ).

• Mermin, N.D.: "What is Quantum Mechanics Trying to Tell us?"; American Journal of Physics, 66 (1998):753-767, arXiv:quant-ph/9801057 (http:/ / arxiv. org/ abs/ quant-ph/ 9801057).

• Rovelli, C. & Smerlak, M.: "Relational EPR"; Preprint: arXiv:quant-ph/0604064 (http:/ / xxx. lanl. gov/ abs/quant-ph/ 0604064).

• Rovelli, C.: "Relational Quantum Mechanics"; International Journal of Theoretical Physics 35; 1996: 1637-1678;arXiv:quant-ph/9609002 (http:/ / xxx. lanl. gov/ abs/ quant-ph/ 9609002).

• Smolin, L.: "The Bekenstein Bound, Topological Quantum Field Theory and Pluralistic Quantum Field Theory";Preprint: arXiv:gr-qc/9508064 (http:/ / xxx. lanl. gov/ abs/ gr-qc/ 9508064).

• Wheeler, J. A.: "Information, physics, quantum: The search for links"; in Zurek,W., ed.: "Complexity, Entropyand the Physics of Information"; pp 3–28; Addison-Wesley; 1990.

External links• Relational Quantum Mechanics (http:/ / plato. stanford. edu/ entries/ qm-relational/ ), The Stanford Encyclopedia

of Philosophy (Spring 2008 Edition)

Quantum tunnelling 61

Quantum tunnelling

Quantummechanics

IntroductionGlossary · History

Quantum tunneling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that itclassically could not surmount. This plays an essential role in several physical phenomena, such as the nuclear fusionthat occurs in main sequence stars like the sun.[1] It has important applications to modern devices such as the tunneldiode[2] and the scanning tunneling microscope. The effect was predicted in the early 20th century and itsacceptance, as a general physical phenomenon, came mid-century.[]

Tunneling is often explained using the Heisenberg uncertainty principle and the wave–particle duality of matter.Purely quantum mechanical concepts are central to the phenomenon, so quantum tunnelling is one of the novelimplications of quantum mechanics.

HistoryQuantum tunneling was developed from the study of radioactivity,[] which was discovered in 1896 by HenriBecquerel.[] Radioactivity was examined further by Marie and Pierre Curie, for which they earned the Nobel Prize inPhysics in 1903.[] Ernest Rutherford and Egon Schweidler studied its nature, which was later verified empirically byFriedrich Kohlrausch. The idea of the half-life and the impossibility of predicting decay was created from theirwork.[]

Friedrich Hund was the first to take notice of tunneling in 1927 when he was calculating the ground state of thedouble-well potential.[] Its first application was a mathematical explanation for alpha decay, which was done in 1928by George Gamow and independently by Ronald Gurney and Edward Condon.[3][4][5][] The two researcherssimultaneously solved the Schrödinger equation for a model nuclear potential and derived a relationship between thehalf-life of the particle and the energy of emission that depended directly on the mathematical probability oftunneling.After attending a seminar by Gamow, Max Born recognized the generality of tunnelling. He realized that it was notrestricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems.[]

Shortly thereafter, both groups considered the case of particles tunnelling into the nucleus. The study ofsemiconductors and the development of transistors and diodes led to the acceptance of electron tunnelling in solidsby 1957. The work of Leo Esaki, Ivar Giaever and Brian David Josephson predicted the tunnelling ofsuperconducting Cooper pairs, for which they received the Nobel Prize in Physics in 1973.[]

Quantum tunnelling 62

Introduction to the concept

Animation showing the tunnel effect and its application to STMmicroscope

Quantum tunnelling through a barrier. The energy of the tunnelled particle is thesame but the amplitude is decreased.

Quantum tunneling through a barrier. At the origin (x=0), there is a very high, butnarrow potential barrier. A significant Tunneling effect can be seen.

Quantum tunnelling falls under the domainof quantum mechanics: the study of whathappens at the quantum scale. This processcannot be directly perceived, but much of itsunderstanding is shaped by the macroscopicworld, which classical mechanics can notadequately explain. To understand thephenomenon, particles attempting to travelbetween potential barriers can be comparedto a ball trying to roll over a hill; quantummechanics and classical mechanics differ intheir treatment of this scenario. Classicalmechanics predicts that particles that do nothave enough energy to classically surmounta barrier will not be able to reach the otherside. Thus, a ball without sufficient energyto surmount the hill would roll back down.Or, lacking the energy to penetrate a wall, itwould bounce back (reflection) or in theextreme case, bury itself inside the wall(absorption). In quantum mechanics, theseparticles can, with a very small probability,tunnel to the other side, thus crossing thebarrier. Here, the ball could, in a sense,borrow energy from its surroundings totunnel through the wall or roll over the hill,paying it back by making the reflectedelectrons more energetic than they otherwisewould have been.[6]

The reason for this difference comes fromthe treatment of matter in quantummechanics as having properties of wavesand particles. One interpretation of thisduality involves the Heisenberg uncertaintyprinciple, which defines a limit on howprecisely the position and the momentum ofa particle can be known at the same time.[]

This implies that there are no solutions with a probability of exactly zero (or one), though a solution may approachinfinity if, for example, the calculation for its position was taken as a probability of 1, the other, i.e. its speed, wouldhave to be infinity. Hence, the probability of a given particle's existence on the opposite side of an intervening barrieris non-zero, and such particles will appear—with no indication of physically transiting the barrier—on the 'other' (asemantically difficult word in this instance) side with a frequency proportional to this probability.

Quantum tunnelling 63

An electron wavepacket directed at a potential barrier. Notethe dim spot on the right that represents tunnelling

electrons.

Quantum tunneling in the phase space formulation of quantummechanics. Wigner function for tunneling through the potential

barrier in atomic units (a.u.). The solid

lines represent the level set of the Hamiltonian.

The tunneling problem

The wave function of a particle summarizes everythingthat can be known about a physical system.[7]

Therefore, problems in quantum mechanics centeraround the analysis of the wave function for a system.Using mathematical formulations of quantummechanics, such as the Schrödinger equation, the wavefunction can be solved. This is directly related to theprobability density of the particle's position, whichdescribes the probability that the particle is at any givenplace. In the limit of large barriers, the probability oftunneling decreases for taller and wider barriers.

For simple tunneling-barrier models, such as therectangular barrier, an analytic solution exists.Problems in real life often do not have one, so"semiclassical" or "quasiclassical" methods have beendeveloped to give approximate solutions to theseproblems, like the WKB approximation. Probabilitiesmay be derived with arbitrary precision, constrained bycomputational resources, via Feynman's path integralmethod; such precision is seldom required inengineering practice.

Related phenomena

There are several phenomena that have the samebehavior as quantum tunneling, and thus can beaccurately described by tunneling. Examples includethe evanescent wave coupling (the application ofMaxwell's wave-equation to light) and the applicationof the non-dispersive wave-equation from acousticsapplied to "waves on strings". Evanescent wavecoupling, until recently, was only called "tunneling" inquantum mechanics; now it is used in other contexts.

These effects are modeled similarly to the rectangular potential barrier. In these cases, there is one transmissionmedium through which the wave propagates that is the same or nearly the same throughout, and a second mediumthrough which the wave travels differently. This can be described as a thin region of medium B between two regionsof medium A. The analysis of a rectangular barrier by means of the Schrödinger equation can be adapted to theseother effects provided that the wave equation has travelling wave solutions in medium A but real exponentialsolutions in medium B.

In optics, medium A is a vacuum while medium B is glass. In acoustics, medium A may be a liquid or gas andmedium B a solid. For both cases, medium A is a region of space where the particle's total energy is greater than itspotential energy and medium B is the potential barrier. These have an incoming wave and resultant waves in both

Quantum tunnelling 64

directions. There can be more mediums and barriers, and the barriers need not be discrete; approximations are usefulin this case.

ApplicationsTunnelling occurs with barriers of thickness around 1-3 nm and smaller,[8] but is the cause of some importantmacroscopic physical phenomena. For instance, tunnelling is a source of current leakage in very-large-scaleintegration (VLSI) electronics and results in the substantial power drain and heating effects that plague high-speedand mobile technology; it is considered the lower limit on how small computer chips can be made.[9]

Radioactive decayRadioactive decay is the process of emission of particles and energy from the unstable nucleus of an atom to form astable product. This is done via the tunnelling of a particle out of the nucleus (an electron tunnelling into the nucleusis electron capture). This was the first application of quantum tunnelling and led to the first approximations.

Spontaneous DNA MutationSpontaneous mutation of DNA occurs when normal DNA replication takes place after a particularly significantproton has defied the odds in quantum tunneling in what is called “proton tunneling.”[10] (quantum bio) A hydrogenbond joins normal base pairs of DNA. There exists a double well potential along a hydrogen bond separated by apotential energy barrier. It is believed that the double well potential is asymmetric with one well deeper than theother so the proton normally rests in the deeper well. For a mutation to occur, the proton must have tunneled into theshallower of the two potential wells. The movement of the proton from its regular position is called a tautomerictransition. If DNA replication takes place in this state, the base pairing rule for DNA may be jeopardized causing amutation.[11] Per-Olov Lowdin was the first to develop this theory of spontaneous mutation within the double helix(quantum bio). Other instances of quantum tunneling-induced mutations in biology are believed to be the cause ofaging and cancer.[citation needed]

Cold emissionCold emission of electrons is relevant to semiconductors and superconductor physics. It is similar to thermionicemission, where electrons randomly jump from the surface of a metal to follow a voltage bias because theystatistically end up with more energy than the barrier, through random collisions with other particles. When theelectric field is very large, the barrier becomes thin enough for electrons to tunnel out of the atomic state, leading to acurrent that varies approximately exponentially with the electric field.[] These materials are important for flashmemory and for some electron microscopes.

Quantum tunnelling 65

Tunnel junctionA simple barrier can be created by separating two conductors with a very thin insulator. These are tunnel junctions,the study of which requires quantum tunnelling.[12] Josephson junctions take advantage of quantum tunnelling andthe superconductivity of some semiconductors to create the Josephson effect. This has applications in precisionmeasurements of voltages and magnetic fields,[] as well as the multijunction solar cell.

A working mechanism of a resonant tunnelling diode device, based on the phenomenonof quantum tunneling through the potential barriers.

Tunnel diode

Diodes are electrical semiconductordevices that allow electric current flowin one direction more than the other.The device depends on a depletionlayer between N-type and P-typesemiconductors to serve its purpose;when these are very heavily doped thedepletion layer can be thin enough fortunnelling. Then, when a small forwardbias is applied the current due totunnelling is significant. This has amaximum at the point where thevoltage bias is such that the energylevel of the p and n conduction bandsare the same. As the voltage bias isincreased, the two conduction bands nolonger line up and the diode actstypically.[]

Because the tunnelling current drops off rapidly, tunnel diodes can be created that have a range of voltages for whichcurrent decreases as voltage is increased. This peculiar property is used in some applications, like high speed deviceswhere the characteristic tunnelling probability changes as rapidly as the bias voltage.[]

The resonant tunnelling diode makes use of quantum tunnelling in a very different manner to achieve a similar result.This diode has a resonant voltage for which there is a lot of current that favors a particular voltage, achieved byplacing two very thin layers with a high energy conductance band very near each other. This creates a quantumpotential well that have a discrete lowest energy level. When this energy level is higher than that of the electrons, notunnelling will occur, and the diode is in reverse bias. Once the two voltage energies align, the electrons flow like anopen wire. As the voltage is increased further tunnelling becomes improbable and the diode acts like a normal diodeagain before a second energy level becomes noticeable.[]

Quantum tunnelling 66

Tunnelling field effect transistorA European research project has demonstrated field effect transistors in which the gate (channel) is controlled viaquantum tunnelling rather than by thermal injection, reducing gate voltage from ~1 volt to 0.2 volts and reducingpower consumption by up to 100×. If these transistors can be scaled up into VLSI chips, they will significantlyimprove the performance per power of integrated circuits.[13]

Quantum conductivityWhile the Drude model of electrical conductivity makes excellent predictions about the nature of electronsconducting in metals, it can be furthered by using quantum tunnelling to explain the nature of the electron'scollisions.[] When a free electron wave packet encounters a long array of uniformly spaced barriers the reflected partof the wave packet interferes uniformly with the transmitted one between all barriers so that there are cases of 100%transmission. The theory predicts that if positively charged nuclei form a perfectly rectangular array, electrons willtunnel through the metal as free electrons, leading to an extremely high conductance, and that impurities in the metalwill disrupt it significantly.[]

Scanning tunnelling microscopeThe scanning tunnelling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer, allows imaging ofindividual atoms on the surface of a metal.[] It operates by taking advantage of the relationship between quantumtunnelling with distance. When the tip of the STM's needle is brought very close to a conduction surface that has avoltage bias, by measuring the current of electrons that are tunnelling between the needle and the surface, thedistance between the needle and the surface can be measured. By using piezoelectric rods that change in size whenvoltage is applied over them the height of the tip can be adjusted to keep the tunnelling current constant. Thetime-varying voltages that are applied to these rods can be recorded and used to image the surface of the conductor.[]

STMs are accurate to 0.001 nm, or about 1% of atomic diameter.[]

Faster than lightIt is possible for spin zero particles to travel faster than the speed of light when tunnelling.[] This apparently violatesthe principle of causality, since there will be a frame of reference in which it arrives before it has left. However,careful analysis of the transmission of the wave packet shows that there is actually no violation of relativity theory.In 1998, Francis E. Low reviewed briefly the phenomenon of zero time tunneling.[14] More recently experimentaltunneling time data of phonons, photons, and electrons are published by Günter Nimtz.[15]

Mathematical discussions of quantum tunnellingThe following subsections discuss the mathematical formulations of quantum tunnelling.

The Schrödinger equationThe time-independent Schrödinger equation for one particle in one dimension can be written as

or

where is the reduced Planck's constant, m is the particle mass, x represents distance measured in the direction ofmotion of the particle, Ψ is the Schrödinger wave function, V is the potential energy of the particle (measuredrelative to any convenient reference level), E is the energy of the particle that is associated with motion in the x-axis(measured relative to V), and M(x) is a quantity defined by V(x) - E which has no accepted name in physics.

Quantum tunnelling 67

The solutions of the Schrödinger equation take different forms for different values of x, depending on whether M(x)is positive or negative. When M(x) is constant and negative, then the Schrödinger equation can be written in the form

The solutions of this equation represent traveling waves, with phase-constant +k or -k. Alternatively, if M(x) isconstant and positive, then the Schrödinger equation can be written in the form

The solutions of this equation are rising and falling exponentials in the form of evanescent waves. When M(x) varieswith position, the same difference in behaviour occurs, depending on whether M(x) is negative or positive. It followsthat the sign of M(x) determines the nature of the medium, with positive M(x) corresponding to medium A asdescribed above and negative M(x) corresponding to medium B. It thus follows that evanescent wave coupling canoccur if a region of positive M(x) is sandwiched between two regions of negative M(x), hence creating a potentialbarrier.The mathematics of dealing with the situation where M(x) varies with x is difficult, except in special cases thatusually do not correspond to physical reality. A discussion of the semi-classical approximate method, as found inphysics textbooks, is given in the next section. A full and complicated mathematical treatment appears in the 1965monograph by Fröman and Fröman noted below. Their ideas have not been incorporated into physics textbooks, buttheir corrections have little quantitative effect.

The WKB approximationThe wave function is expressed as the exponential of a function:

, where

is then separated into real and imaginary parts:, where A(x) and B(x) are real-valued functions.

Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:

.

To solve this equation using the semiclassical approximation, each function must be expanded as a power series in. From the equations, the power series must start with at least an order of to satisfy the real part of the

equation; for a good classical limit starting with the highest power of Planck's constant possible is preferable, whichleads to

and

,

with the following constraints on the lowest order terms,

and

.At this point two extreme cases can be considered.

Case 1 If the amplitude varies slowly as compared to the phase and

Quantum tunnelling 68

which corresponds to classical motion. Resolving the next order of expansion yields

Case 2

If the phase varies slowly as compared to the amplitude, and

which corresponds to tunnelling. Resolving the next order of the expansion yields

In both cases it is apparent from the denominator that both these approximate solutions are bad near the classicalturning points . Away from the potential hill, the particle acts similar to a free and oscillating wave;beneath the potential hill, the particle undergoes exponential changes in amplitude. By considering the behaviour atthese limits and classical turning points a global solution can be made.

To start, choose a classical turning point, and expand in a power series about :

Keeping only the first order term ensures linearity:

.

Using this approximation, the equation near becomes a differential equation:

.

This can be solved using Airy functions as solutions.

Taking these solutions for all classical turning points, a global solution can be formed that links the limitingsolutions. Given the 2 coefficients on one side of a classical turning point, the 2 coefficients on the other side of aclassical turning point can be determined by using this local solution to connect them.Hence, the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits.The relationships between and are

and

With the coefficients found, the global solution can be found. Therefore, the transmission coefficient for a particletunnelling through a single potential barrier is

Quantum tunnelling 69

,

where are the 2 classical turning points for the potential barrier.

References[5] Interview with [[Hans Bethe (http:/ / www. aip. org/ history/ ohilist/ 4504_1. html)] by Charles Weiner and Jagdish Mehra at Cornell

University, 27 October 1966 accessed 5 April 2010][7][7] Bjorken and Drell, "Relativistic Quantum Mechanics", page 2. Mcgraw-Hill College, 1965.[9] "Applications of tunneling" (http:/ / psi. phys. wits. ac. za/ teaching/ Connell/ phys284/ 2005/ lecture-02/ lecture_02/ node13. html). Simon

Connell 2006.

Further reading• N. Fröman and P.-O. Fröman (1965). JWKB Approximation: Contributions to the Theory. Amsterdam:

North-Holland.• Razavy, Mohsen (2003). Quantum Theory of Tunneling. World Scientific. ISBN 981-238-019-1.• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.• James Binney and Skinner, D. (2010). The Physics of Quantum Mechanics: An Introduction (3rd ed.). Cappella

Archive. ISBN 1-902918-51-7.• Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.• Vilenkin, Alexander; Vilenkin, Alexander; Winitzki, Serge (2003). "Particle creation in a tunneling universe".

Physical Review D 68 (2): 023520. arXiv: gr-qc/0210034 (http:/ / arxiv. org/ abs/ gr-qc/ 0210034). Bibcode:2003PhRvD..68b3520H (http:/ / adsabs. harvard. edu/ abs/ 2003PhRvD. . 68b3520H). doi:10.1103/PhysRevD.68.023520 (http:/ / dx. doi. org/ 10. 1103/ PhysRevD. 68. 023520).

External links• Animation, applications and research linked to tunnel effect and other quantum phenomena (http:/ / www.

toutestquantique. fr/ #tunnel) (Université Paris Sud)• Animated illustration of quantum tunnelling (http:/ / molecularmodelingbasics. blogspot. com/ 2009/ 09/

tunneling-and-stm. html)• Animated illustration of quantum tunnelling in a RTD device (http:/ / nanohub. org/ resources/ 8799)

Planck constant 70

Planck constant

Values of h Units Ref.

6.62606957(29)×10−34 J·s [1]

4.135667516(91)×10−15 eV·s [1]

6.62606957(29)×10−27 erg·s [1]

2π EP·tPValues of ħ Units Ref.

1.054571726(47)×10−34 J·s [1]

6.58211928(15)×10−16 eV·s [1]

1.054571726(47)×10−27 erg·s [1]

1 EP·tP def

Values of hc Units Ref.

1.98644568×10−25 J·m

1.23984193 eV·μm

2π EP·ℓP

Plaque at the Humboldt University of Berlin:"Max Planck, discoverer of the elementary

quantum of action h, taught in this building from1889 to 1928."

The Planck constant (denoted h, also called Planck's constant) is aphysical constant that is the quantum of action in quantum mechanics.The Planck constant was first described as the proportionality constantbetween the energy (E) of a photon and the frequency (ν) of itsassociated electromagnetic wave. This relation between the energy andfrequency is called the Planck relation:

Since the frequency , wavelength λ, and speed of light c are relatedby λν = c, the Planck relation for a photon can also be expressed as

The above equation leads to another relationship involving the Planckconstant. Given p for the linear momentum of a particle, the de Brogliewavelength λ of the particle is given by

In applications where frequency is expressed in terms of radians persecond ("angular frequency") instead of cycles per second, it is oftenuseful to absorb a factor of 2π into the Planck constant. The resulting constant is called the reduced Planckconstant or Dirac constant. It is equal to the Planck constant divided by 2π, and is denoted ħ ("h-bar"):

The energy of a photon with angular frequency ω, where ω = 2πν, is given by

Planck constant 71

The reduced Planck constant is the quantum of angular momentum in quantum mechanics.The Planck constant is named after Max Planck, one of the founders of quantum theory, who discovered it in 1900.Classical statistical mechanics requires the existence of h (but does not define its value).[2] Planck discovered thatphysical action could not take on any indiscriminate value. Instead, the action must be some multiple of a very smallquantity (later to be named the "quantum of action" and now called Planck's constant). This inherent granularity iscounterintuitive in the everyday world, where it is possible to "make things a little bit hotter" or "move things a littlebit faster". This is because the quanta of action are very, very small in comparison to everyday macroscopic humanexperience. Hence, the granularity of nature appears smooth to us.Thus, on the macroscopic scale, quantum mechanics and classical physics converge at the classical limit.Nevertheless, it is impossible, as Planck discovered, to explain some phenomena without accepting the fact thataction is quantized. In many cases, such as for monochromatic light or for atoms, this quantum of action also impliesthat only certain energy levels are allowed, and values in-between are forbidden.[3] In 1923, Louis de Brogliegeneralized the Planck relation by postulating that the Planck constant represents the proportionality between themomentum and the quantum wavelength of not just the photon, but the quantum wavelength of any particle. Thiswas confirmed by experiments soon afterwards.

ValueThe Planck constant of action has the dimensionality of specific relative angular momentum (areal momentum) orangular momentum's intensity. In SI units, the Planck constant is expressed in joule seconds (J·s) or (N·m·s).The value of the Planck constant is:[1]

The value of the reduced Planck constant is:

The two digits inside the parentheses denote the standard uncertainty in the last two digits of the value. The figurescited here are the 2010 CODATA recommended values for the constants and their uncertainties. The 2010 CODATAresults were made available in June 2011[4] and represent the best-known, internationally-accepted values for theseconstants, based on all data available as of 2010. New CODATA figures are scheduled to be publishedapproximately every four years.

Significance of the valueThe Planck constant is related to the quantization of light and matter. Therefore, the Planck constant can be seen as asubatomic-scale constant. In a unit system adapted to subatomic scales, the electronvolt is the appropriate unit ofenergy and the Petahertz the appropriate unit of frequency. Atomic unit systems are based (in part) on the Planck'sconstant.The numerical value of the Planck constant depends entirely on the system of units used to measure it. When it isexpressed in SI units, it is one of the smallest constants used in physics. This reflects the fact that on a scale adaptedto humans, where energies are typically of the order of kilojoules and times are typically of the order of seconds orminutes, Planck's constant (the quantum of action) is very small.Equivalently, the smallness of Planck's constant reflects the fact that everyday objects and systems are made of a large number of particles. For example, green light with a wavelength of 555 nanometres (the approximate wavelength to which human eyes are most sensitive) has a frequency of 540 THz (540×1012 Hz). Each photon has an energy E of hν = 3.58×10−19 J. That is a very small amount of energy in terms of everyday experience, but everyday

Planck constant 72

experience is not concerned with individual photons any more than with individual atoms or molecules. An amountof light compatible with everyday experience is the energy of one mole of photons; its energy can be calculated bymultiplying the photon energy by the Avogadro constant, NA ≈ 6.022×1023 mol−1. The result is that green light ofwavelength 555 nm has an energy of 216 kJ/mol, a typical energy of everyday life.

Origins

Black-body radiation

Intensity of light emitted from a black body at any given frequency.Each color is a different temperature. Planck was the first to explain

the shape of these curves.

In the last years of the nineteenth century, Planck wasinvestigating the problem of black-body radiation firstposed by Kirchhoff some forty years earlier. It is wellknown that hot objects glow, and that hotter objectsglow brighter than cooler ones. The reason is that theelectromagnetic field obeys laws of motion just like amass on a spring, and can come to thermal equilibriumwith hot atoms. When a hot object is in equilibriumwith light, the amount of light it absorbs is equal to theamount of light it emits. If the object is black, meaningit absorbs all the light that hits it, then it emits themaximum amount of thermal light too.

The assumption that blackbody radiation is thermalleads to an accurate prediction: the total amount ofemitted energy goes up with the temperature accordingto a definite rule, the Stefan–Boltzmann law(1879–84). But it was also known that the colour of thelight given off by a hot object changes with thetemperature, so that "white hot" is hotter than "red hot". Nevertheless, Wilhelm Wien discovered the mathematicalrelationship between the peaks of the curves at different temperatures, by using the principle of adiabatic invariance.At each different temperature, the curve is moved over by Wien's displacement law (1893). Wien also proposed anapproximation for the spectrum of the object, which was correct at high frequencies (short wavelength) but not atlow frequencies (long wavelength).[] It still was not clear why the spectrum of a hot object had the form that it has(see diagram).

Planck hypothesized that the equations of motion for light are a set of harmonic oscillators, one for each possiblefrequency. He examined how the entropy of the oscillators varied with the temperature of the body, trying to matchWien's law, and was able to derive an approximate mathematical function for black-body spectrum.[5]

However, Planck soon realized that his solution was not unique. There were several different solutions, each ofwhich gave a different value for the entropy of the oscillators.[5] To save his theory, Planck had to resort to using thethen controversial theory of statistical mechanics,[5] which he described as "an act of despair … I was ready tosacrifice any of my previous convictions about physics."[] One of his new boundary conditions was

to interpret UN [the vibrational energy of N oscillators] not as a continuous, infinitely divisible quantity, but asa discrete quantity composed of an integral number of finite equal parts. Let us call each such part the energyelement ε;—Planck, On the Law of Distribution of Energy in the Normal Spectrum[5]

With this new condition, Planck had imposed the quantization of the energy of the oscillators, "a purely formal assumption … actually I did not think much about it…" in his own words,[6] but one which would revolutionize

Planck constant 73

physics. Applying this new approach to Wien's displacement law showed that the "energy element" must beproportional to the frequency of the oscillator, the first version of what is now termed "Planck's relation":

Planck was able to calculate the value of h from experimental data on black-body radiation: his result, 6.55 × 10−34

J·s, is within 1.2% of the currently accepted value.[5] He was also able to make the first determination of theBoltzmann constant kB from the same data and theory.[]

Note that the (black) Raleigh-Jeans curve never touches the Planck curve.

Prior to Planck's work, it had beenassumed that the energy of a bodycould take on any value whatsoever –that it was a continuous variable. TheRayleigh-Jeans law makes closepredictions for a narrow range ofvalues at one limit of temperatures, butthe results diverge more and morestrongly as temperatures increase. Tomake Planck's law, which correctlypredicts blackbody emissions, it wasnecessary to multiply the classicalexpression by a complex factor thatinvolves h in both the numerator andthe denominator. The influence of h inthis complex factor would notdisappear if it were set to zero or toany other value. Making an equationout of Planck's law that would reproduce the Rayleigh-Jeans law could not be done by changing the values of h, ofthe Boltzmann constant, or of any other constant or variable in the equation. In this case the picture given byclassical physics is not duplicated by a range of results in the quantum picture.

The black-body problem was revisited in 1905, when Rayleigh and Jeans (on the one hand) and Einstein (on theother hand) independently proved that classical electromagnetism could never account for the observed spectrum.These proofs are commonly known as the "ultraviolet catastrophe", a name coined by Paul Ehrenfest in 1911. Theycontributed greatly (along with Einstein's work on the photoelectric effect) to convincing physicists that Planck'spostulate of quantized energy levels was more than a mere mathematical formalism. The very first SolvayConference in 1911 was devoted to "the theory of radiation and quanta".[7] Max Planck received the 1918 NobelPrize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energyquanta".

Photoelectric effectThe photoelectric effect is the emission of electrons (called "photoelectrons") from a surface when light is shone onit. It was first observed by Alexandre Edmond Becquerel in 1839, although credit is usually reserved for HeinrichHertz,[8] who published the first thorough investigation in 1887. Another particularly thorough investigation waspublished by Philipp Lenard in 1902.[] Einstein's 1905 paper[9] discussing the effect in terms of light quanta wouldearn him the Nobel Prize in 1921,[8] when his predictions had been confirmed by the experimental work of RobertAndrews Millikan.[] The Nobel committee awarded the prize for his work on the photo-electric effect, rather thanrelativity, both because of a bias against purely theoretical physics not grounded in discovery or experiment, anddissent amongst its members as to the actual proof that relativity was real.[10]

Planck constant 74

Prior to Einstein's paper, electromagnetic radiation such as visible light was considered to behave as a wave: hencethe use of the terms "frequency" and "wavelength" to characterise different types of radiation. The energy transferredby a wave in a given time is called its intensity. The light from a theatre spotlight is more intense than the light froma domestic lightbulb; that is to say that the spotlight gives out more energy per unit time (and hence consumes moreelectricity) than the ordinary bulb, even though the colour of the light might be very similar. Other waves, such assound or the waves crashing against a seafront, also have their own intensity. However the energy account of thephotoelectric effect didn't seem to agree with the wave description of light.The "photoelectrons" emitted as a result of the photoelectric effect have a certain kinetic energy, which can bemeasured. This kinetic energy (for each photoelectron) is independent of the intensity of the light,[] but dependslinearly on the frequency;[] and if the frequency is too low (corresponding to a kinetic energy for the photoelectronsof zero or less), no photoelectrons are emitted at all, unless a plurality of photons, whose energetic sum is greaterthan the energy of the photoelectrons, acts virtually simultaneously (multiphoton effect) [11] Assuming the frequencyis high enough to cause the photoelectric effect, a rise in intensity of the light source causes more photoelectrons tobe emitted with the same kinetic energy, rather than the same number of photoelectrons to be emitted with higherkinetic energy.[]

Einstein's explanation for these observations was that light itself is quantized; that the energy of light is nottransferred continuously as in a classical wave, but only in small "packets" or quanta. The size of these "packets" ofenergy, which would later be named photons, was to be the same as Planck's "energy element", giving the modernversion of Planck's relation:

Einstein's postulate was later proven experimentally: the constant of proportionality between the frequency ofincident light (ν) and the kinetic energy of photoelectrons (E) was shown to be equal to the Planck constant (h).[]

Atomic structure

A schematization of the Bohr model of thehydrogen atom. The transition shown from then=3 level to the n=2 level gives rise to visible

light of wavelength 656 nm (red), as the modelpredicts.

Niels Bohr introduced the first quantized model of the atom in 1913, inan attempt to overcome a major shortcoming of Rutherford's classicalmodel.[] In classical electrodynamics, a charge moving in a circleshould radiate electromagnetic radiation. If that charge were to be anelectron orbiting a nucleus, the radiation would cause it to lose energyand spiral down into the nucleus. Bohr solved this paradox withexplicit reference to Planck's work: an electron in a Bohr atom couldonly have certain defined energies En

where R∞ is an experimentally-determined constant (the Rydbergconstant) and n is any integer (n = 1, 2, 3, …). Once the electronreached the lowest energy level (n = 1), it could not get any closer tothe nucleus (lower energy). This approach also allowed Bohr toaccount for the Rydberg formula, an empirical description of theatomic spectrum of hydrogen, and to account for the value of the Rydberg constant R∞ in terms of other fundamentalconstants.

Bohr also introduced the quantity h/2π, now known as the reduced Planck constant, as the quantum of angular momentum. At first, Bohr thought that this was the angular momentum of each electron in an atom: this proved incorrect and, despite developments by Sommerfeld and others, an accurate description of the electron angular momentum proved beyond the Bohr model. The correct quantization rules for electrons – in which the energy

Planck constant 75

reduces to the Bohr-model equation in the case of the hydrogen atom – were given by Heisenberg's matrixmechanics in 1925 and the Schrödinger wave equation in 1926: the reduced Planck constant remains thefundamental quantum of angular momentum. In modern terms, if J is the total angular momentum of a system withrotational invariance, and Jz the angular momentum measured along any given direction, these quantities can onlytake on the values

Uncertainty principleThe Planck constant also occurs in statements of Werner Heisenberg's uncertainty principle. Given a large number ofparticles prepared in the same state, the uncertainty in their position, Δx, and the uncertainty in their momentum (inthe same direction), Δp, obey

where the uncertainty is given as the standard deviation of the measured value from its expected value. There are anumber of other such pairs of physically measurable values which obey a similar rule. One example is time vs.energy. The either-or nature of uncertainty forces measurement attempts to choose between trade offs, and given thatthey are quanta, the trade offs often take the form of either-or (as in Fourier analysis), rather than the compromisesand gray areas of time series analysis.In addition to some assumptions underlying the interpretation of certain values in the quantum mechanicalformulation, one of the fundamental cornerstones to the entire theory lies in the commutator relationship between theposition operator and the momentum operator :

where δij is the Kronecker delta.

Dependent physical constantsThe following list is based on the 2006 CODATA evaluation;[] for the constants listed below, more than 90% of theuncertainty is due to the uncertainty in the value of the Planck constant, as indicated by the square of the correlationcoefficient (r2 > 0.9, r > 0.949). The Planck constant is (with one or two exceptions)[12] the fundamental physicalconstant which is known to the lowest level of precision, with a relative uncertainty ur of 5.0×10−8.

Rest mass of the electronThe normal textbook derivation of the Rydberg constant R∞ defines it in terms of the electron mass me and a varietyof other physical constants.

However, the Rydberg constant can be determined very accurately (ur = 6.6×10−12) from the atomic spectrum ofhydrogen, whereas there is no direct method to measure the mass of a stationary electron in SI units. Hence theequation for the calculation of me becomes

where c0 is the speed of light and α is the fine-structure constant. The speed of light has an exactly defined value in SI units, and the fine-structure constant can be determined more accurately (ur = 6.8×10−10) than the Planck constant: the uncertainty in the value of the electron rest mass is due entirely to the uncertainty in the value of the

Planck constant 76

Planck constant (r2 > 0.999).

Avogadro constantThe Avogadro constant NA is determined as the ratio of the mass of one mole of electrons to the mass of a singleelectron: The mass of one mole of electrons is the "relative atomic mass" of an electron Ar(e), which can bemeasured in a Penning trap (ur = 4.2×10−10), multiplied by the molar mass constant Mu, which is defined as0.001 kg/mol.

The dependence of the Avogadro constant on the Planck constant (r2 > 0.999) also holds for the physical constantswhich are related to amount of substance, such as the atomic mass constant. The uncertainty in the value of thePlanck constant limits the knowledge of the masses of atoms and subatomic particles when expressed in SI units. Itis possible to measure the masses more precisely in atomic mass units, but not to convert them more precisely intokilograms.

Elementary chargeSommerfeld originally defined the fine-structure constant α as:

where e is the elementary charge, ε0 is the electric constant (also called the permittivity of free space), and μ0 is themagnetic constant (also called the permeability of free space). The latter two constants have fixed values in theInternational System of Units. However, α can also be determined experimentally, notably by measuring the electronspin g-factor ge, then comparing the result with the value predicted by quantum electrodynamics.At present, the most precise value for the elementary charge is obtained by rearranging the definition of α to obtainthe following definition of e in terms of α and h:

Bohr magneton and nuclear magnetonThe Bohr magneton and the nuclear magneton are units which are used to describe the magnetic properties of theelectron and atomic nuclei respectively. The Bohr magneton is the magnetic moment which would be expected foran electron if it behaved as a spinning charge according to classical electrodynamics. It is defined in terms of thereduced Planck constant, the elementary charge and the electron mass, all of which depend on the Planck constant:the final dependence on h½ (r2 > 0.995) can be found by expanding the variables.

The nuclear magneton has a similar definition, but corrected for the fact that the proton is much more massive thanthe electron. The ratio of the electron relative atomic mass to the proton relative atomic mass can be determinedexperimentally to a high level of precision (ur = 4.3×10−10).

Planck constant 77

Determination

Method Value of h(10−34 J·s)

Relativeuncertainty

Ref.

Watt balance 6.62606889(23) 3.4×10−8 [13][14][]

X-ray crystal density 6.6260745(19) 2.9×10−7 [15]

Josephson constant 6.6260678(27) 4.1×10−7 [16][17]

Magnetic resonance 6.6260724(57) 8.6×10−7 [18][19]

Faraday constant 6.6260657(88) 1.3×10−6 [20]

CODATA 2010recommended value

6.62606957(29) 4.4×10−8 [1]

The nine recent determinations of the Planck constant cover five separate methods. Where there is more than one recent determination for a givenmethod, the value of h given here is a weighted mean of the results, as calculated by CODATA.

In principle, the Planck constant could be determined by examining the spectrum of a black-body radiator or thekinetic energy of photoelectrons, and this is how its value was first calculated in the early twentieth century. Inpractice, these are no longer the most accurate methods. The CODATA value quoted here is based on threewatt-balance measurements of KJ

2RK and one inter-laboratory determination of the molar volume of silicon,[] but ismostly determined by a 2007 watt-balance measurement made at the U.S. National Institute of Standards andTechnology (NIST).[] Five other measurements by three different methods were initially considered, but not includedin the final refinement as they were too imprecise to affect the result.There are both practical and theoretical difficulties in determining h. The practical difficulties can be illustrated bythe fact that the two most accurate methods, the watt balance and the X-ray crystal density method, do not appear toagree with one another. The most likely reason is that the measurement uncertainty for one (or both) of the methodshas been estimated too low – it is (or they are) not as precise as is currently believed – but for the time being there isno indication which method is at fault.The theoretical difficulties arise from the fact that all of the methods except the X-ray crystal density method rely onthe theoretical basis of the Josephson effect and the quantum Hall effect. If these theories are slightly inaccurate –though there is no evidence at present to suggest they are – the methods would not give accurate values for thePlanck constant. More importantly, the values of the Planck constant obtained in this way cannot be used as tests ofthe theories without falling into a circular argument. Fortunately, there are other statistical ways of testing thetheories, and the theories have yet to be refuted.[]

Josephson constantThe Josephson constant KJ relates the potential difference U generated by the Josephson effect at a "Josephsonjunction" with the frequency ν of the microwave radiation. The theoretical treatment of Josephson effect suggestsvery strongly that KJ = 2e/h.

The Josephson constant may be measured by comparing the potential difference generated by an array of Josephsonjunctions with a potential difference which is known in SI volts. The measurement of the potential difference in SIunits is done by allowing an electrostatic force to cancel out a measurable gravitational force. Assuming the validityof the theoretical treatment of the Josephson effect, KJ is related to the Planck constant by

Planck constant 78

Watt balanceA watt balance is an instrument for comparing two powers, one of which is measured in SI watts and the other ofwhich is measured in conventional electrical units. From the definition of the conventional watt W90, this gives ameasure of the product KJ

2RK in SI units, where RK is the von Klitzing constant which appears in the quantum Halleffect. If the theoretical treatments of the Josephson effect and the quantum Hall effect are valid, and in particularassuming that RK = h/e2, the measurement of KJ

2RK is a direct determination of the Planck constant.

Magnetic resonanceThe gyromagnetic ratio γ is the constant of proportionality between the frequency ν of nuclear magnetic resonance(or electron paramagnetic resonance for electrons) and the applied magnetic field B: ν = γB. It is difficult to measuregyromagnetic ratios precisely because of the difficulties in precisely measuring B, but the value for protons in waterat 25 °C is known to better than one part per million. The protons are said to be "shielded" from the applied magneticfield by the electrons in the water molecule, the same effect that gives rise to chemical shift in NMR spectroscopy,and this is indicated by a prime on the symbol for the gyromagnetic ratio, γ′p. The gyromagnetic ratio is related tothe shielded proton magnetic moment μ′p, the spin number I (I = 1⁄2 for protons) and the reduced Planck constant.

The ratio of the shielded proton magnetic moment μ′p to the electron magnetic moment μe can be measuredseparately and to high precision, as the imprecisely-known value of the applied magnetic field cancels itself out intaking the ratio. The value of μe in Bohr magnetons is also known: it is half the electron g-factor ge. Hence

A further complication is that the measurement of γ′p involves the measurement of an electric current: this isinvariably measured in conventional amperes rather than in SI amperes, so a conversion factor is required. Thesymbol Γ′p-90 is used for the measured gyromagnetic ratio using conventional electrical units. In addition, there aretwo methods of measuring the value, a "low-field" method and a "high-field" method, and the conversion factors aredifferent in the two cases. Only the high-field value Γ′p-90(hi) is of interest in determining the Planck constant.

Substitution gives the expression for the Planck constant in terms of Γ′p-90(hi):

Planck constant 79

Faraday constantThe Faraday constant F is the charge of one mole of electrons, equal to the Avogadro constant NA multiplied by theelementary charge e. It can be determined by careful electrolysis experiments, measuring the amount of silverdissolved from an electrode in a given time and for a given electric current. In practice, it is measured inconventional electrical units, and so given the symbol F90. Substituting the definitions of NA and e, and convertingfrom conventional electrical units to SI units, gives the relation to the Planck constant.

X-ray crystal densityThe X-ray crystal density method is primarily a method for determining the Avogadro constant NA but as theAvogadro constant is related to the Planck constant it also determines a value for h. The principle behind the methodis to determine NA as the ratio between the volume of the unit cell of a crystal, measured by X-ray crystallography,and the molar volume of the substance. Crystals of silicon are used, as they are available in high quality and purityby the technology developed for the semiconductor industry. The unit cell volume is calculated from the spacingbetween two crystal planes referred to as d220. The molar volume Vm(Si) requires a knowledge of the density of thecrystal and the atomic weight of the silicon used. The Planck constant is given by

Particle acceleratorThe experimental measurement of the Planck constant in the Large Hadron Collider laboratory was carried out in2011. The study called PCC using a giant particle accelerator helped to better understand the relationships betweenthe Planck constant and measuring distances in space. [needs citation]

FixationAs mentioned above, the numerical value of the Planck constant depends on the system of units used to describe it.Its value in SI units is known to 50 parts per billion but its value in atomic units is known exactly, because of the waythe scale of atomic units is defined. The same is true of conventional electrical units, where the Planck constant(denoted h90 to distinguish it from its value in SI units) is given by

with KJ–90 and RK–90 being exactly defined constants. Atomic units and conventional electrical units are very usefulin their respective fields, because the uncertainty in the final result does not depend on an uncertain conversionfactor, only on the uncertainty of the measurement itself.There are a number of proposals to redefine certain of the SI base units in terms of fundamental physicalconstants.[21] This has already been done for the metre, which is defined in terms of a fixed value of the speed oflight. The most urgent unit on the list for redefinition is the kilogram, whose value has been fixed for all science(since 1889) by the mass of a small cylinder of platinum–iridium alloy kept in a vault just outside Paris. Whilenobody knows if the mass of the International Prototype Kilogram has changed since 1889 – the value 1 kg of itsmass expressed in kilograms is by definition unchanged and therein lies one of the problems – it is known that oversuch a timescale the many similar Pt–Ir alloy cylinders kept in national laboratories around the world, have changedtheir relative mass by several tens of parts per million, however carefully they are stored, and the more so the morethey have been taken out and used as mass standards. A change of several tens of micrograms in one kilogram isequivalent to the current uncertainty in the value of the Planck constant in SI units.

Planck constant 80

The legal process to change the definition of the kilogram is already underway,[21] but it was decided that no finaldecision would be made before the next meeting of the General Conference on Weights and Measures in 2011.[22]

The Planck constant is a leading contender to form the basis of the new definition, although not the only one.[22]

Possible new definitions include "the mass of a body at rest whose equivalent energy equals the energy of photonswhose frequencies sum to 135,639,274×1042 Hz",[23] or simply "the kilogram is defined so that the Planck constantequals 6.62606896×10−34 J·s".The BIPM provided Draft Resolution A in anticipation of the 24th General Conference on Weights and Measuresmeeting (2011-10-17 through 2011-10-21), detailing the considerations "On the possible future revision of theInternational System of Units, the SI".[]

Watt balances already measure mass in terms of the Planck constant: at present, standard mass is taken as fixed andthe measurement is performed to determine the Planck constant but, were the Planck constant to be fixed in SI units,the same experiment would be a measurement of the mass. The relative uncertainty in the measurement wouldremain the same.Mass standards could also be constructed from silicon crystals or by other atom-counting methods. Such methodsrequire a knowledge of the Avogadro constant, which fixes the proportionality between atomic mass andmacroscopic mass but, with a defined value of the Planck constant, NA would be known to the same level ofuncertainty (if not better) than current methods of comparing macroscopic mass.

Notes[1] P.J. Mohr, B.N. Taylor, and D.B. Newell (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants" (Web

Version 6.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: http:/ / physics. nist. gov (http:/ / physics.nist. gov/ cgi-bin/ cuu/ Results?search_for=planck) [Thursday, 02-Jun-2011 21:00:12 EDT]. National Institute of Standards and Technology,Gaithersburg, MD 20899.

[5] . English translation: " On the Law of Distribution of Energy in the Normal Spectrum (http:/ / dbhs. wvusd. k12. ca. us/ webdocs/Chem-History/ Planck-1901/ Planck-1901. html)".

[8][8] See, e.g.,[10] Einstein, His Life and Universe, Walter Isaacson, pp. 309–314.[12] The main exceptions are the Newtonian constant of gravitation G and the gas constant R. The uncertainty in the value of the gas constant

also affects those physical constants which are related to it, such as the Boltzmann constant and the Loschmidt constant.[16][16] . .[21] 94th Meeting of the International Committee for Weights and Measures (2005). Recommendation 1: Preparative steps towards new

definitions of the kilogram, the ampere, the kelvin and the mole in terms of fundamental constants (http:/ / www. bipm. org/ utils/ en/ pdf/CIPM2005-EN. pdf)

[22] 23rd General Conference on Weights and Measures (2007). Resolution 12: On the possible redefinition of certain base units of theInternational System of Units (SI) (http:/ / www. bipm. org/ utils/ en/ pdf/ Resol23CGPM-EN. pdf).

References• Barrow, John D. (2002), The Constants of Nature; From Alpha to Omega – The Numbers that Encode the

Deepest Secrets of the Universe, Pantheon Books, ISBN 0-375-42221-8

External links• Quantum of Action and Quantum of Spin – Numericana (http:/ / www. numericana. com/ answer/ constants.

htm#h)• Moriarty, Philip; Eaves, Laurence; Merrifield, Michael (2009). "h Planck's Constant" (http:/ / www. sixtysymbols.

com/ videos/ planck. htm). Sixty Symbols. Brady Haran for the University of Nottingham.

Maxwell's equations 81

Maxwell's equations

Electromagnetism

•• Electricity•• Magnetism

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form thefoundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modernelectrical and communications technologies. Maxwell's equations describe how electric and magnetic fields aregenerated and altered by each other and by charges and currents. They are named after the Scottish physicist andmathematician James Clerk Maxwell who published an early form of those equations between 1861 and 1862.The equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and totalcurrent, including the complicated charges and currents in materials at the atomic scale; it has universal applicability,but may be unfeasible to calculate. The "macroscopic" set of Maxwell's equations defines two new auxiliary fieldsthat describe large-scale behavior without having to consider these atomic scale details, but it requires the use ofparameters characterizing the electromagnetic properties of the relevant materials.The term "Maxwell's equations" is often used for other forms of Maxwell's equations. For example, space-timeformulations are commonly used in high energy and gravitational physics. These formulations defined onspace-time, rather than space and time separately are manifestly[1] compatible with special and general relativity. Inquantum mechanics, versions of Maxwell's equations based on the electric and magnetic potentials are preferred.Since the mid-20th century, it has been understood that Maxwell's equations are not exact laws of the universe, butare a classical approximation to the more accurate and fundamental theory of quantum electrodynamics. In mostcases, though, quantum deviations from Maxwell's equations are immeasurably small. Exceptions occur when theparticle nature of light is important or for very strong electric fields.

Vector calculus formalismThroughout this article, symbols in bold represent vector quantities, and symbols in italics represent scalarquantities, unless otherwise indicated.To describe electromagnetism in the powerful language of vector calculus, the Lorentz force law defines the electricfield E, a vector field, and the magnetic field B, a pseudovector field, where each generally have time-dependence.The sources of these fields are electric charges and electric currents, which can be expressed as the total amounts ofelectric charge Q and current I within a region of space, or as local densities of these - namely charge density ρ andcurrent density J.In this language there are four equations. Two of them describe how the fields vary in space due to sources, if any; electric fields emanating from electric charges in Gauss's law, and magnetic fields as closed field lines not due to magnetic monopoles in Gauss's law for magnetism. The other two describe how the fields "circulate" around their respective sources; the magnetic field "circulates" around electric currents and time varying electric fields in Ampère's law with Maxwell's correction, while the electric field "circulates" around time varying magnetic fields in

Maxwell's equations 82

Faraday's law.The precise formulation of Maxwell's equations depends on the precise definition of the quantities involved.Conventions differ with the unit systems; because various definitions and dimensions are changed by absorbingdimensionfull factors like the speed of light c. This makes constants come out differently.

Conventional formulation in SI unitsThe equations in this section are given in the convention used with SI units. Other units commonly used areGaussian units based on the cgs system,[2] Lorentz–Heaviside units (used mainly in particle physics), and Planckunits (used in theoretical physics). See below for the formulation with Gaussian units.

Name Integral equations Differential equations

Gauss's law

Gauss's law for magnetism

Maxwell–Faraday equation (Faraday's law of induction)

Ampère's circuital law (with Maxwell's correction)

There are universal constants appearing in the equations; in this case the permittivity of free space ε0 and thepermeability of free space μ0, a general characteristic of fundamental field equations.In the differential equations, a local description of the fields, the nabla symbol ∇ denotes the three-dimensionalgradient operator, and from it ∇· is the divergence operator and ∇× the curl operator. The sources are appropriatelytaken to be as local densities of charge and current.In the integral equations; a description of the fields within a region of space, Ω is any fixed volume with boundarysurface ∂Ω, and Σ is any fixed open surface with boundary curve ∂Σ. Here "fixed" means the volume or surface donot change in time. Although it is possible to formulate Maxwell's equations with time-dependent surfaces andvolumes, this is not actually necessary: the equations are correct and complete with time-independent surfaces. Thesources are correspondingly the total amounts of charge and current within these volumes and surfaces, found byintegration. The volume integral of the total charge density ρ over any fixed volume Ω is the total electric chargecontained in Ω:

and the net electrical current is the surface integral of the electric current density J, passing through any open fixedsurface Σ:

where dS denotes the differential vector element of surface area S normal to surface Σ. (Vector area is also denotedby A rather than S, but this conflicts with the magnetic potential, a separate vector field).The "total charge or current" refers to including free and bound charges, or free and bound currents. These are usedin the macroscopic formulation below.

Maxwell's equations 83

Relationship between differential and integral formulationsThe differential and integral formulations of the equations are mathematically equivalent, by the divergence theoremin the case of Gauss's law and Gauss's law for magnetism, and by the Kelvin–Stokes theorem in the case of Faraday'slaw and Ampère's law. Both the differential and integral formulations are useful. The integral formulation can oftenbe used to simply and directly calculate fields from symmetric distributions of charges and currents. On the otherhand, the differential formulation is a more natural starting point for calculating the fields in more complicated (lesssymmetric) situations, for example using finite element analysis.[3]

Flux and divergence

Closed volume Ω and boundary ∂Ω, enclosing a source (+) andsink (−) of a vector field F. Here, F could be the E field with

source electric charges, but not the B field which has no magneticcharges as shown. The outward unit normal is n.

The "fields emanating from the sources" can be inferredfrom the surface integrals of the fields through the closedsurface ∂Ω, defined as the electric flux and magnetic fluxrespectively:

       as well as their divergences:

These surface integrals and divergences are connected bythe divergence theorem.

Circulation and curl

Maxwell's equations 84

Open surface Σ and boundary ∂Σ. F could be the E or B fields.Again, n is the unit normal. (The curl of a vector field doesn't

literally look like the "circulations", this is a heuristic depiction).

The "circulation of the fields" can be interpreted from theline integrals of the fields around the closed curve ∂Σ:

where dℓ is the differential vector element of path lengthtangential to the path/curve, as well as their curls:

These line integrals and curls are connected by Stokes'theorem, and are analogous to quantities in classical fluiddynamics: the circulation of a fluid is the line integral ofthe fluid's flow velocity field around a closed loop, andthe vorticity of the fluid is the curl of the velocity field.

Time evolution

The "dynamics" or "time evolution of the fields" is due tothe partial derivatives of the fields with respect to time:

These derivatives are crucial for the prediction of field propagation in the form of electromagnetic waves. Since thesurface is taken to be time-independent, we can make the following transition in Faraday's law:

see differentiation under the integral sign for more on this result.

Conceptual descriptions

Gauss's lawGauss's law describes the relationship between a static electric field and the electric charges that cause it: The staticelectric field points away from positive charges and towards negative charges. In the field line description, electricfield lines begin only at positive electric charges and end only at negative electric charges. 'Counting' the number offield lines passing though a closed surface, therefore, yields the total charge (including bound charge due topolarization of material) enclosed by that surface divided by dielectricity of free space (the vacuum permittivity).More technically, it relates the electric flux through any hypothetical closed "Gaussian surface" to the enclosedelectric charge.

Maxwell's equations 85

Gauss's law for magnetism: magnetic field linesnever begin nor end but form loops or extend to

infinity as shown here with the magnetic field dueto a ring of current.

Gauss's law for magnetism

Gauss's law for magnetism states that there are no "magneticcharges" (also called magnetic monopoles), analogous to electriccharges.[4] Instead, the magnetic field due to materials is generated bya configuration called a dipole. Magnetic dipoles are best representedas loops of current but resemble positive and negative 'magneticcharges', inseparably bound together, having no net 'magnetic charge'.In terms of field lines, this equation states that magnetic field linesneither begin nor end but make loops or extend to infinity and back. Inother words, any magnetic field line that enters a given volume mustsomewhere exit that volume. Equivalent technical statements are thatthe sum total magnetic flux through any Gaussian surface is zero, orthat the magnetic field is a solenoidal vector field.

Faraday's law

In a geomagnetic storm, a surge in the flux ofcharged particles temporarily alters Earth's

magnetic field, which induces electric fields inEarth's atmosphere, thus causing surges in

electrical power grids. Artist's rendition; sizes arenot to scale.

Faraday's law describes how a time varying magnetic field creates("induces") an electric field.[4] This dynamically induced electric fieldhas closed field lines just as the magnetic field, if not superposed by astatic (charge induced) electric field. This aspect of electromagneticinduction is the operating principle behind many electric generators:for example, a rotating bar magnet creates a changing magnetic field,which in turn generates an electric field in a nearby wire. (Note: thereare two closely related equations which are called Faraday's law. Theform used in Maxwell's equations is always valid but more restrictivethan that originally formulated by Michael Faraday.)

Ampère's law with Maxwell's correction

An Wang's magnetic core memory (1954) is anapplication of Ampère's law. Each core stores one

bit of data.

Ampère's law with Maxwell's correction states that magnetic fieldscan be generated in two ways: by electrical current (this was theoriginal "Ampère's law") and by changing electric fields (this was"Maxwell's correction").

Maxwell's correction to Ampère's law is particularly important: itshows that not only does a changing magnetic field induce an electricfield, but also a changing electric field induces a magnetic field.[4][5]

Therefore, these equations allow self-sustaining "electromagneticwaves" to travel through empty space (see electromagnetic waveequation).

Maxwell's equations 86

The speed calculated for electromagnetic waves, which could be predicted from experiments on charges andcurrents,[6] exactly matches the speed of light; indeed, light is one form of electromagnetic radiation (as are X-rays,radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861,thereby unifying the theories of electromagnetism and optics.

Vacuum equations, electromagnetic waves and speed of light

This 3D diagram shows a plane linearly polarizedwave propagating from left to right with the samewave equations where E = E0 sin(−ωt + k ⋅ r) and

B = B0 sin(−ωt + k ⋅ r)

In a region with no charges (ρ = 0) and no currents (J = 0), such as in avacuum, Maxwell's equations reduce to:

Taking the curl (∇×) of the curl equations, and using the curl of the curl identity ∇×(∇×F) = ∇(∇·F) − ∇2F weobtain the wave equations

which identify

with the speed of light in free space. In materials with relative permittivity εr and relative permeability μr,

can be slower. In addition, E and B are mutually perpendicular to each other and the direction of wave propagation,and are in phase with each other. A sinusoidal plane wave is one special solution of these equations. Maxwell'sequations explain how these waves can physically propagate through space. The changing magnetic field creates achanging electric field through Faraday's law. In turn, that electric field creates a changing magnetic field throughMaxwell's correction to Ampère's law. This perpetual cycle allows these waves, now known as electromagneticradiation, to move through space at velocity c.

Maxwell's equations 87

"Microscopic" versus "macroscopic"The microscopic variant of Maxwell's equation expresses the electric E field and the magnetic B field in terms of thetotal charge and total current present including the charges and currents at the atomic level. It is sometimes calledthe general form of Maxwell's equations or "Maxwell's equations in a vacuum". The macroscopic variant ofMaxwell's equation is equally general, however, with the difference being one of bookkeeping."Maxwell's macroscopic equations", also known as Maxwell's equations in matter, are more similar to those thatMaxwell introduced himself.

Name Integral equations Differential equations

Gauss's law

Gauss's law for magnetism

Maxwell–Faraday equation (Faraday's law of induction)

Ampère's circuital law (with Maxwell's correction)

Unlike the "microscopic" equations, the "macroscopic" equations factor out the bound charge Qb and current Ib toobtain equations that depend only on the free charges Qf and currents If. This factorization can be made by splittingthe total electric charge and current as follows:

The cost of this factorization is that additional fields, the displacement field D and the magnetizing field-H, aredefined that need to be determined. Phenomenological constituent equations relate the additional fields to the electricfield E and the magnetic B-field, often through a simple linear relation.For a detailed description of the differences between the microscopic (total charge and current including materialcontributes or in air/vacuum)[7] and macroscopic (free charge and current; practical to use on materials) variants ofMaxwell's equations, see below.

Maxwell's equations 88

Bound charge and current

Left: A schematic view of how an assembly of microscopic dipoles producesopposite surface charges as shown at top and bottom. Right: How an assembly ofmicroscopic current loops add together to produce a macroscopically circulating

current loop. Inside the boundaries, the individual contributions tend to cancel, butat the boundaries no cancelation occurs.

When an electric field is applied to adielectric material its molecules respond byforming microscopic electric dipoles – theiratomic nuclei move a tiny distance in thedirection of the field, while their electronsmove a tiny distance in the oppositedirection. This produces a macroscopicbound charge in the material even thoughall of the charges involved are bound toindividual molecules. For example, if everymolecule responds the same, similar to thatshown in the figure, these tiny movementsof charge combine to produce a layer ofpositive bound charge on one side of thematerial and a layer of negative charge onthe other side. The bound charge is mostconveniently described in terms of thepolarization P of the material, its dipole moment per unit volume. If P is uniform, a macroscopic separation ofcharge is produced only at the surfaces where P enter and leave the material. For non-uniform P, a charge is alsoproduced in the bulk.[8]

Somewhat similarly, in all materials the constituent atoms exhibit magnetic moments that are intrinsically linked tothe angular momentum of the components of the atoms, most notably their electrons. The connection to angularmomentum suggests the picture of an assembly of microscopic current loops. Outside the material, an assembly ofsuch microscopic current loops is not different from a macroscopic current circulating around the material's surface,despite the fact that no individual magnetic moment is traveling a large distance. These bound currents can bedescribed using the magnetization M.[9]

The very complicated and granular bound charges and bound currents, therefore can be represented on themacroscopic scale in terms of P and M which average these charges and currents on a sufficiently large scale so asnot to see the granularity of individual atoms, but also sufficiently small that they vary with location in the material.As such, the Maxwell's macroscopic equations ignores many details on a fine scale that can be unimportant tounderstanding matters on a gross scale by calculating fields that are averaged over some suitable volume.

Auxiliary fields, polarization and magnetizationThe definitions (not constitutive relations) of the auxiliary fields are:

where P is the polarization field and M is the magnetization field which are defined in terms of microscopic boundcharges and bound current respectively. The macroscopic bound charge density ρb and bound current density Jb interms of polarization P and magnetization M are then defined as

If we define the free, bound, and total charge and current density by

Maxwell's equations 89

and use the defining relations above to eliminate D, and H, the "macroscopic" Maxwell's equations reproduce the"microscopic" equations.

Constitutive relationsIn order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between displacementfield D and the electric field E, as well as the magnetizing field H and the magnetic field B. Equivalently, we have tospecify the dependence of the polarisation P (hence the bound charge) and the magnetisation M (hence the boundcurrent) on the applied electric and magnetic field. The equations specifying this response are called constitutiverelations. For real-world materials, the constitutive relations are rarely simple, except approximately, and usuallydetermined by experiment. See the main article for a fuller description.For materials without polarisation and magnetisation ("vacuum"), the constitutive relations are

for scalar constants and . Since there is no bound charge, the total and the free charge and current are equal.More generally, for linear materials the constitutive relations are

where ε is the permittivity and μ the permeability of the material. Even the linear case can have variouscomplications, however.• For homogeneous materials, ε and μ are constant throughout the material, while for inhomogeneous materials

they depend on location within the material (and perhaps time).• For isotropic materials, ε and μ are scalars, while for anisotropic materials (e.g. due to crystal structure) they are

tensors.• Materials are generally dispersive, so ε and μ depend on the frequency of any incident EM waves.Even more generally, in the case of non-linear materials (see for example nonlinear optics), D and P are notnecessarily proportional to E, similarly B is not necessarily proportional to H or M. In general D and H depend onboth E and B, on location and time, and possibly other physical quantities.In applications one also has to describe how the free currents and charge density behave in terms of E and B possiblycoupled to other physical quantities like pressure, and the mass, number density, and velocity of charge caryingparticles. E.g., the original equations given by Maxwell (see History of Maxwell's equations) included Ohms law inthe form .

Equations in Gaussian unitsGaussian units are a popular system of units, that is part of the centimetre–gram–second system of units (cgs). Whenusing cgs units it is conventional to use a slightly different definition of electric field Ecgs = c−1 ESI. This implies thatthe modified electric and magnetic field have the same units (in the SI convention this is not the case: e.g. for EMwaves in vacuum, |ESI| = c|B|, making dimensional analysis of the equations different). Then it uses a unit of chargedefined in such a way that the permittivity of the vacuum ε0 = 1/(4πc), hence μ0 = 4π/c. Using these differentconventions, the Maxwell equations become:[]

Maxwell's equations 90

Equations in Gaussian units

Name Microscopic equations Macroscopic equations

Gauss's law

Gauss's law for magnetism same as microscopic

Maxwell–Faraday equation (Faraday's law of induction) same as microscopic

Ampère's law (with Maxwell's extension)

Alternative formulationsFollowing is a summary of some of the numerous other ways to write Maxwell's equations in vacuum, showing theycan be collected together and formulated using different mathematical formalisms that describe the same physics.Often, they are also called the Maxwell equations. See the main articles for the details of each formulation. SI unitsare used throughout.

Formalism Formulation Homogeneous equations Non-homogeneous equations

Vectorcalculus

Fields 3D space+ time

Potentials (anygauge) 3D

space + time

Potentials(Lorenz gauge)

3D space +time

Tensorcalculus

Fields flatspace-time

Potentials (anygauge) flatspace-time

Potentials(Lorenz gauge)flat space-time

Differentialforms

Fields anyspace-time

Potentials (anygauge) anyspace-time

Potentials(Lorenz gauge)any space-time

Maxwell's equations 91

Geometriccalculus

Fields anyspace-time

Potentials (anygauge) anyspace-time

Potentials(Lorenz gauge)any space-time

where

• is the four-gradient with respect to coordinates in an inertial frame; ,

• is the D'Alembert operator,

• the square bracket [ ] denotes antisymmetrization of indices,• d is the exterior derivative, and is the Hodge star on forms defined by the Lorentzian metric of space-time

(in the case of defined on two forms depending only on the conformal class of the metric).• in geometric calculus, D is the covector derivative in any spacetime and reduces to ∇ in flat spacetime. Where ∇

in spacetime and is similar to ∇ in space and is related to the D'Alambertian by Other formulations include a matrix representation of Maxwell's equations. Historically, a quaternionicformulation[10][11] was used.

SolutionsMaxwell's equations are partial differential equations that relate the electric and magnetic fields to each other and tothe electric charges and currents. Often, the charges and currents are themselves dependent on the electric andmagnetic fields via the Lorentz force equation and the constitutive relations. These all form a set of coupled partialdifferential equations, which are often very difficult to solve. In fact, the solutions of these equations encompass allthe diverse phenomena in the entire field of classical electromagnetism. A thorough discussion is far beyond thescope of the article, but some general notes follow.Like any differential equation, boundary conditions[12][13][14] and initial conditions[15] are necessary for a uniquesolution. For example, even with no charges and no currents anywhere in spacetime, many solutions to Maxwell'sequations are possible, not just the obvious solution E = B = 0. Another solution is E = constant, B = constant, whileyet other solutions have electromagnetic waves filling spacetime. In some cases, Maxwell's equations are solvedthrough infinite space, and boundary conditions are given as asymptotic limits at infinity.[16] In other cases,Maxwell's equations are solved in just a finite region of space, with appropriate boundary conditions on that region:For example, the boundary could be an artificial absorbing boundary representing the rest of the universe,[17][18] orperiodic boundary conditions, or (as with a waveguide or cavity resonator) the boundary conditions may describe thewalls that isolate a small region from the outside world.[]

Jefimenko's equations (or the closely related Liénard–Wiechert potentials) are the explicit solution to Maxwell'sequations for the electric and magnetic fields created by any given distribution of charges and currents. It assumesspecific initial conditions to obtain the so-called "retarded solution", where the only fields present are the onescreated by the charges. Jefimenko's equations are not so helpful in situations when the charges and currents arethemselves affected by the fields they create.Numerical methods for differential equations can be used to approximately solve Maxwell's equations when an exactsolution is impossible. These methods usually require a computer, and include the finite element method andfinite-difference time-domain method.[12][14][19][20][21] For more details, see Computational electromagnetics.

Maxwell's equations 92

Maxwell's equations seem overdetermined, in that they involve six unknowns (the three components of E and B) buteight equations (one for each of the two Gauss's laws, three vector components each for Faraday's and Ampere'slaws). (The currents and charges are not unknowns, being freely specifiable subject to charge conservation.) This isrelated to a certain limited kind of redundancy in Maxwell's equations: It can be proven that any system satisfyingFaraday's law and Ampere's law automatically also satisfies the two Gauss's laws, as long as the system's initialcondition does.[22][23] Although it is possible to simply ignore the two Gauss's laws in a numerical algorithm (apartfrom the initial conditions), the imperfect precision of the calculations can lead to ever-increasing violations of thoselaws. By introducing dummy variables characterizing these violations, the four equations become notoverdetermined after all. The resulting formulation can lead to more accurate algorithms that take all four laws intoaccount.[24]

Limitations for a theory of electromagnetismWhile Maxwell's equations (along with the rest of classical electromagnetism) are extraordinarily successful atexplaining and predicting a variety of phenomena, they are not exact laws of the universe, but merelyapproximations. In some special situations, they can be noticeably inaccurate. Examples include extremely strongfields (see Euler–Heisenberg Lagrangian) and extremely short distances (see vacuum polarization). Moreover,various phenomena occur in the world even though Maxwell's equations predicts them to be impossible, such as"nonclassical light" and quantum entanglement of electromagnetic fields (see quantum optics). Finally, anyphenomenon involving individual photons, such as the photoelectric effect, Planck's law, the Duane–Hunt law,single-photon light detectors, etc., would be difficult or impossible to explain if Maxwell's equations were exactlytrue, as Maxwell's equations do not involve photons. For the most accurate predictions in all situations, Maxwell'sequations have been superseded by quantum electrodynamics.

VariationsPopular variations on the Maxwell equations as a classical theory of electromagnetic fields are relatively scarcebecause the standard equations have stood the test of time remarkably well.

Magnetic monopolesMaxwell's equations posit that there is electric charge, but no magnetic charge (also called magnetic monopoles), inthe universe. Indeed, magnetic charge has never been observed (despite extensive searches)[25] and may not exist. Ifthey did exist, both Gauss's law for magnetism and Faraday's law would need to be modified, and the resulting fourequations would be fully symmetric under the interchange of electric and magnetic fields.[26][27]

Notes[1][1] Maxwell's equations in any form are compatible with relativity. These space-time formulations, though, make that compatibility more readily

apparent.[4] J.D. Jackson, "Maxwell's Equations" video glossary entry (http:/ / videoglossary. lbl. gov/ 2009/ maxwells-equations/ )[5] Principles of physics: a calculus-based text (http:/ / books. google. com/ books?id=1DZz341Pp50C& pg=PA809), by R.A. Serway, J.W.

Jewett, page 809.[6] The quantity we would now call UNIQ-math-0-b827b7466afa665e-QINU , with units of velocity, was directly measured before Maxwell's

equations, in an 1855 experiment by Wilhelm Eduard Weber and Rudolf Kohlrausch. They charged a leyden jar (a kind of capacitor), andmeasured the electrostatic force associated with the potential; then, they discharged it while measuring the magnetic force from the current inthe discharge wire. Their result was , remarkably close to the speed of light. See The story of electrical and magnetic measurements: from 500B.C. to the 1940s, by Joseph F. Keithley, p115 (http:/ / books. google. com/ books?id=uwgNAtqSHuQC& pg=PA115)

[7] In some books—e.g., in U. Krey and A. Owen's Basic Theoretical Physics (Springer 2007)—the term effective charge is used instead of totalcharge, while free charge is simply called charge.

[8] See for a good description of how P relates to the bound charge.[9] See for a good description of how M relates to the bound current.

Maxwell's equations 93

[18] S. G. Johnson, Notes on Perfectly Matched Layers (http:/ / math. mit. edu/ ~stevenj/ 18. 369/ pml. pdf), online MIT course notes (Aug.2007).

[25] See magnetic monopole for a discussion of monopole searches. Recently, scientists have discovered that some types of condensed matter,including spin ice and topological insulators, which display emergent behavior resembling magnetic monopoles. (See (http:/ / www.sciencemag. org/ cgi/ content/ abstract/ 1178868) and (http:/ / www. nature. com/ nature/ journal/ v461/ n7266/ full/ nature08500. html).)Although these were described in the popular press as the long-awaited discovery of magnetic monopoles, they are only superficially related.A "true" magnetic monopole is something where ∇⋅B≠0, whereas in these condensed-matter systems, ∇⋅B=0 while only ∇⋅H≠0.

ReferencesFurther reading can be found in list of textbooks in electromagnetism

Historical publications• James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the

Royal Society of London 155, 459–512 (1865). (This article accompanied a December 8, 1864 presentation byMaxwell to the Royal Society.)

The developments before relativity• Joseph Larmor (1897) "On a dynamical theory of the electric and luminiferous medium", Phil. Trans. Roy. Soc.

190, 205–300 (third and last in a series of papers with the same name).• Hendrik Lorentz (1899) "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad.

Science Amsterdam, I, 427–43.• Hendrik Lorentz (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of

light", Proc. Acad. Science Amsterdam, IV, 669–78.• Henri Poincaré (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Néerlandaises, V, 253–78.• Henri Poincaré (1901) Science and Hypothesis• Henri Poincaré (1905) "Sur la dynamique de l'électron" (http:/ / www. soso. ch/ wissen/ hist/ SRT/ P-1905-1.

pdf), Comptes rendus de l'Académie des Sciences, 140, 1504–8.• James Clerk Maxwell, A Treatise on Electricity And Magnetism Vols 1 and 2 (http:/ / www. antiquebooks. net/

readpage. html#maxwell) 1904—most readable edition with all corrections—Antique Books Collection suitablefor free reading online.

• Maxwell, J.C., A Treatise on Electricity And Magnetism – Volume 1 – 1873 (http:/ / posner. library. cmu. edu/Posner/ books/ book. cgi?call=537_M46T_1873_VOL. _1) – Posner Memorial Collection – Carnegie MellonUniversity

• Maxwell, J.C., A Treatise on Electricity And Magnetism – Volume 2 – 1873 (http:/ / posner. library. cmu. edu/Posner/ books/ book. cgi?call=537_M46T_1873_VOL. _2) – Posner Memorial Collection – Carnegie MellonUniversity

• On Faraday's Lines of Force – 1855/56 (http:/ / blazelabs. com/ On Faraday's Lines of Force. pdf) Maxwell's firstpaper (Part 1 & 2) – Compiled by Blaze Labs Research (PDF)

• On Physical Lines of Force – 1861 Maxwell's 1861 paper describing magnetic lines of Force – Predecessor to1873 Treatise

• Maxwell, James Clerk, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of theRoyal Society of London 155, 459–512 (1865). (This article accompanied a December 8, 1864 presentation byMaxwell to the Royal Society.)

• Catt, Walton and Davidson. "The History of Displacement Current". Wireless World, March 1979. (http:/ / www.electromagnetism. demon. co. uk/ z014. htm)

•• Reprint from Dover Publications (ISBN 0-486-60636-8)• Full text of 1904 Edition including full text search. (http:/ / www. antiquebooks. net/ readpage. html#maxwell)

Maxwell's equations 94

• A Dynamical Theory Of The Electromagnetic Field – 1865 (http:/ / books. google. com/books?id=5HE_cmxXt2MC& vid=02IWHrbcLC9ECI_wQx& dq=Proceedings+ of+ the+ Royal+ Society+ Of+London+ Vol+ XIII& ie=UTF-8& jtp=531) Maxwell's 1865 paper describing his 20 Equations in 20 Unknowns –Predecessor to the 1873 Treatise

External links• Hazewinkel, Michiel, ed. (2001), "Maxwell equations" (http:/ / www. encyclopediaofmath. org/ index.

php?title=p/ m063140), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4• maxwells-equations.com (http:/ / www. maxwells-equations. com) — An intuitive tutorial of Maxwell's

equations.• Mathematical aspects of Maxwell's equation are discussed on the Dispersive PDE Wiki (http:/ / tosio. math.

toronto. edu/ wiki/ index. php/ Main_Page).

Modern treatments• Electromagnetism (http:/ / www. lightandmatter. com/ html_books/ 0sn/ ch11/ ch11. html), B. Crowell, Fullerton

College• Lecture series: Relativity and electromagnetism (http:/ / farside. ph. utexas. edu/ ~rfitzp/ teaching/ jk1/ lectures/

node6. html), R. Fitzpatrick, University of Texas at Austin• Electromagnetic waves from Maxwell's equations (http:/ / www. physnet. org/ modules/ pdf_modules/ m210. pdf)

on Project PHYSNET (http:/ / www. physnet. org).• MIT Video Lecture Series (36 x 50 minute lectures) (in .mp4 format) – Electricity and Magnetism (http:/ / ocw.

mit. edu/ OcwWeb/ Physics/ 8-02Electricity-and-MagnetismSpring2002/ VideoAndCaptions/ index. htm) Taughtby Professor Walter Lewin.

Other• Feynman's derivation of Maxwell equations and extra dimensions (http:/ / uk. arxiv. org/ abs/ hep-ph/ 0106235)• Nature Milestones: Photons – Milestone 2 (1861) Maxwell's equations (http:/ / www. nature. com/ milestones/

milephotons/ full/ milephotons02. html)

Article Sources and Contributors 95

Article Sources and ContributorsBell's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=557559371  Contributors: &Delta, A. di M., AC+79 3888, AdamSiska, AdamSolomon, Agge1000, Aldux, AmarChandra,Amareshdatta, Anakin101, Andrewthomas10, Android Mouse, Andyjsmith, Antonielly, Aranel, ArnoldReinhold, Arthur Rubin, Ashmoo, Avb, B9 hummingbird hovering, Ballhausflip,Barticus88, Baxxterr, BeatePaland, Bender235, BeteNoir, BiT, Bo Jacoby, Bob1960evens, BobKawanaka, Bobo192, Bongwarrior, Brad7777, Brazmyth, BrianWren, Bryan Derksen, Byrgenwulf,CRGreathouse, CSTAR, CYD, Cacycle, Calvero JP, Caroline Thompson, Catchanil, Cgingold, Chalst, Charles Matthews, Chkno, Chris Howard, Chris the speller, Christopher Cooper,Complexica, Count Iblis, Cremepuff222, Crocodealer, Curious1i, Cuzkatzimhut, DarwinPeacock, Dauto, Dave Runger, Deathphoenix, Deniz195, Dfrg.msc, Dirac66, Dmr2, DomenicDenicola,Don warner saklad, DrChinese, Drilnoth, Drmies, Dtgriscom, EcoMan, EdC, Eequor, Egg, Ekabhishek, Endlessmike 888, F=q(E+v^B), Fastily, FormerNukeSubmariner, Franl, Frish, Fulldecent,Fwappler, GangofOne, Gareth Griffith-Jones, Gene Nygaard, Geremia, Giftlite, Gill110951, Giraffedata, Gmusser, Gregbard, GregorB, Grok42, Groyolo, Gsjaeger, Guy Harris, Gzornenplatz,HaeB, Hairy Dude, Harald88, Headbomb, Henry David, Hmonroe, Hqb, IRWolfie-, IWillNeverLearn, Iconofiler, Incnis Mrsi, Informatorium, Interintel, Isaacdealey, Isocliff, J-Wiki, Jack whobuilt the house, Jcobb, Jess Riedel, Jim E. Black, Jim.belk, Jncraton, Jpittelo, Jules.LT, Jwpitts, KHamsun, Katherine Pendleton, Keenan Pepper, Keithbowden, Kingmundi, Kmmertes,KnowledgeOfSelf, Konstable, Ksolway, Kyrisch, LC, Laudaka, Leobold1, Lethe, Likebox, Linas, Loggerjack, Lumidek, Lycurgus, Malcohol, Mandolinface, Maniatis, Marek.zukowski,Mastertek, MathKnight, Maxw41, Mbweissman, Michael C Price, Michael Hardy, Mike mian, Mike2vil, Mister fu-ck you, MisterSheik, Mknjbhvg, Mmyotis, Mogism, Monslucis,MorphismOfDoom, Mujokan, Myrvin, NOrbeck, Nihil, Nilock, Nowakpl, NuclearWarfare, Oakwood, Oleg Alexandrov, Olek Bijok, Orangedolphin, Ott2, Paddu, Patrick0Moran, Patsobest, PaulAugust, Paulginz, Petri Krohn, Phoenix-antiscammer, Physicskev, Physis, Pjacobi, Plrk, Pokipsy76, Quibik, RJFJR, RK, Rbodgers, RekishiEJ, Rich Farmbrough, Richard75, Richwil, RickardVogelberg, Rl, Roadrunner, Robert1947, Rodasmith, Roke, Rror, Ryanmcdaniel, SJLPP, Sabri Al-Safi, Salsb, SchreiberBike, Scientific29, Seth Bresnett, Simetrical, Skeppy, Slawekb,SlimVirgin, Stdjmax, Steady unit, Stephen Poppitt, StevenJohnston, Stevertigo, Stirling Newberry, Suffusion of Yellow, Suslindisambiguator, Susvolans, Syncategoremata, Tethys sea, Thorseth,Tide rolls, Tothebarricades.tk, Tox, TriTertButoxy, Trusilver, Tsirel, Ueit, Ulrich Utiger, Uusijani, Vessels42, WMdeMuynck, Waleswatcher, Wfku, Wikikam, WookieInHeat, Wwoods, Ybband,Yecril, Ymblanter, Yuttadhammo, Zell2929, Zeycus, Zvis, 老 陳, 289 anonymous edits

EPR paradox  Source: http://en.wikipedia.org/w/index.php?oldid=557881880  Contributors: -- April, .:Ajvol:., 207.171.93.xxx, 2over0, ASCWiki, Agge1000, Agger, Alan McBeth, Alkivar,Amakuha, AmarChandra, Andejons, Apoc2400, Ark, Arpingstone, Astrofan7, AxelBoldt, B4hand, Ballhausflip, Bevo, Bigbluefish, Binksternet, Blauhart, Bob K31416, Brews ohare, Brianga,Brrk.3001, Bryan Derksen, CSTAR, CYD, Camrn86, Canadian-Bacon, Carbuncle, Cardmagic, Caroldermoid, Caroline Thompson, Chas zzz brown, Chjoaygame, Chris 73, Chris Howard,ChrisGualtieri, Clarityfiend, CobbSalad, Complexica, Cortonin, CosmiCarl, Cpiral, Craig Pemberton, Crowsnest, Cspan64, D0nj03, DAGwyn, DBooth, Dalit Llama, Darktaco, DaveBeal, DavidR. Ingham, David spector, Declare, Dogcow, Dpbsmith, Dr Smith, DrBob, DrChinese, DragonflySixtyseven, Dratman, Dryke, Dzhim, Długosz, EdH, Edward, Eequor, Egmontaz, Ehn, Eiffel,Ejrh, Elassint, Emvan, Esb, Etabackman, Ettrig, Euyyn, Falcorian, Finemann, Finlay McWalter, Fredkinfollower, Frish, Fulldecent, Fwappler, GeorgeMoney, Giftlite, Goldfritter, Graham87,Greedohun, GregorB, Gretyl, HCPotter, Hackwrench, Harald88, Headbomb, Hephaestos, Hhhippo, Hirak 99, Houftermann, Hugo Dufort, Hydnjo, IAdem, Iamthedeus, J-Star, JaGa,JamesMLane, Jan-Åke Larsson, Jcajacob, JerryFriedman, Jinxman1, Jnc, JocK, JohnBlackburne, Jperl66, Jpittelo, Jpowell, Jwrosenzweig, KHamsun, KLRajpal, Kapalama, Karada, KarolLangner, KasugaHuang, Keenan Pepper, Kuratowski's Ghost, Lapisphil, Larsobrien, Lethe, Lf89, LiDaobing, Linas, Linus M., Looxix, Lumidek, Maher27777, Marek.zukowski, Marie Poise,Mark Arsten, Mark J, Masudr, Maurice Carbonaro, Mav, Meiskam, Metron4, Michael C Price, Michael Hardy, Moink, MorphismOfDoom, MrJones, Msridhar, Myrvin, Naddy, Natevw,NathanHurst, Nellatnoj, Nickst, Nickyus, Noca2plus, ObsidianOrder, Orange Suede Sofa, Owen, Pace212, Paine Ellsworth, Paranoid, Pateblen, Patrick0Moran, Peashy, Pekka.virta, PenguiN42,Peter Erwin, PeterBFZ, Peterdjones, PetrGlad, Phancy Physicist, PierreAbbat, Previously ScienceApologist, Prezbo, Publicly Visible, Pwjb, Pérez, Quondum, RTreacy, Radiofriendlyunitshifta,Rama, Razimantv, Rentzepopoulos, Rich Farmbrough, Rjwilmsi, Roadrunner, Robert K S, Robertd, Robertefields, Ronjoseph, Rracecarr, Ryft, SGBailey, Sanders muc, Schneelocke, Seb,Shadanan, Shakyshake, Shalom Yechiel, Sidasta, Sigmundur, Skierpage, Snoyes, Sonicsuicide, Srleffler, Stain, Stephen Poppitt, Steve Quinn, Suisui, Sundar, Syko, Tarotcards, Tempshill, Tercer,Texture, ThomasK, Thrain2, Timwi, Tlabshier, Trilobitealive, Tsop, Ty8inf, TylerDurden8823, Uusijani, Vasiľ, Victor Gijsbers, Vodex, Voyajer, WMdeMuynck, Waleswatcher, Weekwhom,Widr, Wik, Wile E. Heresiarch, William M. Connolley, WillowW, Wing gundam, XJaM, Xgrrr, Xnquist, YUL89YYZ, Yill577, Zeycus, Zootm, Zymurgy, Александър, 老 陳, 338 anonymousedits

Eigenvalues and eigenvectors  Source: http://en.wikipedia.org/w/index.php?oldid=558703631  Contributors: 123Mike456Winston789, 2001:1680:10:502:3463:509A:DFEA:9314,2001:4898:E0:2061:E8B7:FFF9:1B16:7152, 2620:101:F000:9C00:29E4:AADD:4EFC:A974, 2A01:388:201:3340:F0FD:97CD:FE2A:A427, 336, 83d40m, A civilian, A930913,AManWithNoPlan, ANONYMOUS COWARD0xC0DE, Aacool, Aarongeller, Abstracte, Adam78, AdamSmithee, Adpadu, Adpete, Agthorr, Ahmad.tachyon, AhmedHan, Ajto8, Alainr345,AlanUS, Alansohn, Albmont, AlexCornejo, Aliekens, Almit39, Amberrock, Ancheta Wis, Anoko moonlight, Baccyak4H, Bdegfcunbbfv, BenFrantzDale, Bender2k14, Benzi455, Bevo, Bgwhite,BillC, Billymac00, Bissinger, Bjelleklang, Bknittel, Blanerhoads, Blotwell, Bobguy7, Bochev, Booyabazooka, Boris Alexeev, Bovlb, Boyzindahoos, Brad7777, Brian0918, Buster79,Butwhatdoiknow, CRGreathouse, Capagot, Chichui, Chire, ChrisGualtieri, Chrisbaird.ma, Christian75, CinchBug, Circeus, Cleared as filed, Colonel angel, Complexica, Connelly, Conscious,Cosmikz, Cotterr2, Crasshopper, Crowsnest, Crunchy Numbers, Crust, Curtdbz, Cwkmail, Cyde, D1ma5ad, DanBri, Danger, David Binner, David Eppstein, DavidFHoughton, Daytona2,Dcoetzee, Delirium, Denwid, Dependent Variable, Deryck Chan, Dfsisson, Dhollm, Dima373, Dirac66, Dmazin, Dmharvey, Dmn, Doleszki, Dotancohen, Dr. Nobody, DragonflySixtyseven,Dratman, Dwwaddell, Dysprosia, EconoPhysicist, Edinborgarstefan, Editor at Large, Edsanville, Ekwity, Ellliottt, Eric Forste, Etr52, Fgnievinski, Finell, Fintor, Foobarnix, Forbes72, Fortdj33,FrankFlanagan, FreplySpang, Fresheneesz, Frizzil, Gaius Cornelius, Gak, Gandalf61, Gareth Owen, Gbnogkfs, Gdorner, Giftlite, Gimbeline, Giuliopp, Grubber, Gtbanta, Guardian of Light,Gunderburg, GunnerJr, Gwideman, H1voltage, Haeleth, Hairy Dude, Hankel operator, HappyCamper, Haseldon, HcorEric X, Headbomb, Hede2000, Hetar, Hiiiiiiiiiiiiiiiiiiiii, Hitman012,Hongooi, Humanengr, Hunter.moseley, Hydrogravity, Iainscott, Ichakrab, Igny, Incnis Mrsi, InvictaHOG, Iridescent, Itsmine, J. Finkelstein, JMK, JPD, JYOuyang, JaGa, JabberWok, JahJah,Jakarr, Jakew, Jakob.scholbach, Jamesjhs, Javalenok, Jayden54, Jcarroll, Jeff560, JeffAEdmonds, JeffieAlex, Jefromi, Jheald, JinPan, Jitse Niesen, Joelr31, JohnBlackburne, Johnpacklambert,Jok2000, Jorge Stolfi, JosephCatrambone, Josh Cherry, Josp-mathilde, Jtwdog, JuPitEer, Justin W Smith, Jérôme, KHamsun, Kanie, Kanonkas, Kappa, Katefan0, Kausikghatak, Kedwar5,Keenan Pepper, Kevinj04, Kiefer.Wolfowitz, Kier07, Kimbly, Kjoonlee, Kmote, Kri, Krucraft, Kungfuadam, LBehounek, LOL, Lacatosias, Lalahuma, Landroni, Lantonov, Laplacian,Larryisgood, LkNsngth, LokiClock, Lone Isle, LordViD, Lowellian, LucasVB, Luk, Luna Santin, Luolimao, Lzyvzl, M4ry73, Madanor, Magister Mathematicae, Male1979, Mandolinface,MarSch, MarcelB612, Mark L MacDonald, Markus Schmaus, Martyulrich, Marudubshinki, Matthewmoncek, Mattrix, Maziar.irani, McKay, Mcstrother, Mct mht, Mebden, MedicineMan555,Metaeducation, Michael Hardy, Michael Slone, MichaelBillington, Mikhail Ryazanov, Moala, Moonraker12, Moriori, Ms2ger, Muhandes, Mushin, Mxipp, Myasuda, Napalm Llama, Natrij,NatusRoma, Nbarth, NewEconomist, Nichalp, Nick Number, Nickshanks, Nigellwh, Nihonjoe, NinjaDreams, NormDor, Not a dog, Ntjohn, ObsessiveMathsFreak, OlEnglish, Oleg Alexandrov,Oli Filth, Orderud, Orizon, Ouzel Ring, Oxygene123, PabloE, Paolo.dL, Patrick, Patrick0Moran, Pedant, Phyrexicaid, Piano non troppo, Plastikspork, Plrk, Pmanderson, Policron, Porejide,Protonk, Pt, Pushkar3, Qiangshiweiguan, R'n'B, RJFJR, RProgrammer, Raffamaiden, Rajah, Randomblue, Rbanzai, Rchandan, Reaverdrop, Red Act, Reddevyl, Repliedthemockturtle, Restu20,RexxS, Rgdboer, Rich257, Rick Norwood, Riteshsood, Rjwilmsi, Rlupsa, Rmbyoung, Rohitphy, Rspeer, Rubybrian, Rushiagr, Ruslan Sharipov, Ruud Koot, Rxnt, Saburr, Safalra, Saketh, Salixalba, Sameerkale, Sanchom, Schismata, Schutz, Scott Ritchie, Sebastian Klein, Severoon, Shai-kun, Sherif helmy, Shishir0610, Shizhao, Shreevatsa, Silly rabbit, Simon12, Skakkle, Skittleys,Slawekb, Smite-Meister, Somesh, Soultaco, Spikey, Srich32977, Sschongster, Ste4k, Stephen Poppitt, Stevelinton, StevenJohnston, Stevenj, Stevertigo, StradivariusTV, Sunray, Szabolcs Nagy,Sławomir Biały, TDogg310, TVilkesalo, Tabletop, Tac-Tics, Taco325i, Tarquin, Tator2, Tatpong, TeH nOmInAtOr, TeaDrinker, TerraNovawhatcanidotomakethisnottoosimilartosomeothername, Tesi1700, The Duke of Waltham, The suffocated, TheRealInsomnius, Thecheesykid, Thenub314, Thorfinn, Timeroot, Timhoooey,Timrollpickering, Tiny green, Titoxd, Tkuvho, Tobias Bergemann, Tomo, TomyDuby, TreyGreer62, Trifon Triantafillidis, Ttennebkram, TypoBoy, Tyraios, Urdutext, Urtis, User A1, Varuna,Vaughan Pratt, Vb, Veganaxos, Waldir, Wayp123, Wayward, WhiteHatLurker, Wikid77, Williampoetra, Winston Trechane, Woohookitty, Xantharius, Xnn, Yahya Abdal-Aziz, Yoshigev, Yurik,Zaslav, ZeroOne, Zinnmann, Zylinder, 790 ,سعی anonymous edits

Quantum Bayesianism  Source: http://en.wikipedia.org/w/index.php?oldid=558601887  Contributors: BD2412, Ceyockey, Dendron ch, Diberland, Editfromwithout, IRWolfie-, ItsZippy,JRSpriggs, Jakr, MorphismOfDoom, Northamerica1000, Remotelysensed, Rjwilmsi, RockMagnetist, Topbanana, Uploadvirus, Wctaiwan, Yasht101, 106 anonymous edits

Wave function collapse  Source: http://en.wikipedia.org/w/index.php?oldid=558360997  Contributors: Aarghdvaark, Afshar, Alessandro70, Antandrus, ArglebargleIV, Army1987, Arthur Rubin,AugPi, AxelBoldt, Belsazar, BenFrantzDale, Bryan Derksen, Bth, CSTAR, CecilWard, Charles Matthews, Chetvorno, Complexica, Dan Gluck, David R. Ingham, Dbfirs, Dratman, Eequor, Ettrig,F=q(E+v^B), Fredrik, GTBacchus, Giftlite, GreatWhiteNortherner, Headbomb, Henry Delforn (old), Hidaspal, J-Wiki, Joe Decker, John of Reading, JohnBlackburne, Juto20, Kevin aylward,Kinema, L33tminion, Lambiam, Lf89, Linas, Lordvolton, LucasVB, Lumidek, MagnaMopus, Marcus Brute, Maurice Carbonaro, Michael C Price, Mild Bill Hiccup, Miracle Pen, Neparis, Nixer,Nuadh, OlEnglish, Oleg Alexandrov, Overand, Pekka.virta, Peterdjones, Pfalstad, Physicsch, Pra1998, RG2, RayTomes, Robert K S, Runrider, Sabri Al-Safi, Samboy, Savidan, Sigmundur,Stanford96, Stephen Poppitt, Stevertigo, Tamorlan, The Wiki ghost, Thurth, Timwi, Toshikodo, Unara, WMdeMuynck, Wayiran, Wing gundam, Zowie, Zundark, Zylox, 66 anonymous edits

Relational quantum mechanics  Source: http://en.wikipedia.org/w/index.php?oldid=537931214  Contributors: Anville, Argumzio, Auspex1729, BD2412, Bender235, Black Falcon, Brewsohare, Byrgenwulf, Chuunen Baka, Davidkazuhiro, Duendeverde, Everyking, Fortdj33, GangofOne, Giftlite, Gregbard, Hairy Dude, Harizotoh9, Harold f, InXistant, Jonathan de Boyne Pollard,Jordgette, Linas, Maria Vargas, Maurice Carbonaro, Mbell, Michael C Price, Michael Hardy, MichalKotowski, NielsenGW, Peterdjones, Rich Farmbrough, SchreiberBike, Sina2, Situation, SkierDude, Tarotcards, Trovatore, V8rik, Woohookitty, 18 anonymous edits

Quantum tunnelling  Source: http://en.wikipedia.org/w/index.php?oldid=557876819  Contributors: .:Ajvol:., 5Q5, Aasi007onfire, Abaddon314159, Abieadja, Adrian-from-london, Adrignola, Aervanath, Albany NY, Alextangent, AndreasPraefcke, Anonymous Dissident, Arthena, Aspects, Asr1, Atkinson 291, Autoerrant, Axeman89, Bamse, Beatnik8983, Ben Webber, Bender235, Brad7777, Brent Perreault, Brothernight, Bryan Derksen, C h fleming, C.Fred, CBDroege, CapitalR, Careful With That Axe, Eugene, CecilWard, Cenarium, Charles Matthews, Chester Markel, Chris Howard, Christopherodonovan, Cj1340, CommonsDelinker, Complexica, Crossfire 7, Crowsnest, Cybercobra, Danko Georgiev, Dauto, Davidtheapple, Deamon138, DeepKling, Deltabeignet, Denysbondar, Destroyer 65, Digitalme, Dirac66, Dna-webmaster, Don Dueck, Download, Dr.K., DrTorstenHenning, Dtrention, ESkog, Eckler95, Elizgoiri, EoGuy, Euyyn,

Article Sources and Contributors 96

Evanovka, Evmore, Filou, Folajimi, Frau Holle, Freeboson, Fresheneesz, Frostedshreddedwheat, Fudoreaper, FurrySings, GNimtz, GaaraMsg, Gandalf61, GeeJo, Genya Avocado, GeorgeLouis,Georgelulu, Giftlite, Glenn, Gnuish, Golbez, GregRM, GregVolk, Gregmackin, Guoguo12, Gwideman, Götz, Headbomb, Hellisp, Heron, Hess88, Hmccasla, Igodard, Ivlatab, JIMCKEE, Ja 62,JabberWok, Jac16888, Jag123, Jakew, Jbergquist, Jef-Infojef, Jeremykemp, Jhjensen, Jimmilu, Jk408, John of Reading, Jrockley, Jubobroff, Karam.Anthony.K, KasugaHuang, Kim Dent-Brown,Klayborg, Korepin, Krator, LOL, LachlanA, Lady of the dead, Laurascudder, Lbs6380, Legare, Lethe, Light current, Littlealien182, LorenzoB, Lsustickgirl, M p w, Makelelecba, Mako098765,Manfroze, Martarius, Maurice Carbonaro, Mbell, Methecooldude, Mgiganteus1, Michael Hardy, Mister.wardrop, Mitsukai, Mnmngb, Momotaro, MotoTechUSA, Movado73, Mr0t1633, Muad,Napron, Natty1521, Nifky?, Nihiltres, Nikoladie, Nobbie, Oleg Alexandrov, Omegatron, Omnipaedista, Onco p53, Ontyx, Ortonmc, P0ckets0, P0quita, Paine Ellsworth, PaladinWhite, ParclyTaxel, Patrick0Moran, Peachypoh, PeterBFZ, Phancy Physicist, Phatboye, Phinnaeus, PhnomPencil, Phokat, Phuzion, PhySusie, Pmrobert49, Polyamorph, Poobarb, Previously ScienceApologist,Quantumor, Qubitting1993, R'n'B, RGForbes, RManka, RTG, Rdsmith4, Reaver789, Rfts, Rich Farmbrough, Rjwilmsi, Rob Hurt, RodC, Rorro, Rumiton, SAS22patrolmedic33, Salsb, Sargon3,Savidan, Sbroadwe, ScAvenger, Schzmo, Secret Squïrrel, Shaddack, Shashank rathore, Sheliak, Shoefly, Sholto Maud, Sintaku, Slightsmile, Snpoj, Stephan Leeds, Steve Wise, THEN WHOWAS PHONE?, Tarikur, Tarotcards, Technopilgrim, Tehskazsky, Tgeairn, That Guy, From That Show!, TheGerm, Threepounds, Tlabshier, Tls60, Toh, TonyMath, Tripodian, Trojanian, Twang,Tyco.skinner, Useight, Uvlr, Uxorion, V81, Vallor, Vnorris, Voorlandt, Wears Waldo, WereSpielChequers, Whitepaw, Widr, Wikiborg, Wild Wizard, WojciechSwiderski, Wydysh, Xanzzibar,Yguff88, Yoseph1998, Zeteg, Zundark, 345 ,دانشجوی گمنام anonymous edits

Planck constant  Source: http://en.wikipedia.org/w/index.php?oldid=558573220  Contributors: 129.128.164.xxx, 9p4gh9gkj, A little insignificant, A876, AManWithNoPlan, Abhinand.r,Adamrush, AdjustShift, Adoniscik, Ahoerstemeier, Alison, AlmostReadytoFly, Alvinwc, Alvktt, Ancheta Wis, Andres, Andris, AndyTheGrump, Animum, Apyule, Aqking, Army1987,Askewchan, AstroNox, Avb, Avia, AxelBoldt, Aymatth2, Bakkedal, Barbara Shack, BenRG, Bigly, Bkell, Blindman shady, Bo Jacoby, Boson, Brews ohare, Bryan Derksen, Bteunissen, CYD,Cablewoman, CapitalSasha, Cat2020, Catgut, Chali2, Chessofnerd, Chetvorno, Chink3tom, Chris the speller, Chuunen Baka, Ck lostsword, Cmichael, Colliohn, Colonies Chris, Conversionscript, Coroboy, Crazycomputers, Cuzkatzimhut, DAGwyn, Dadaist6174, Dah31, Dark Formal, David Gerard, Davidalbertosv, DefLog, Deklund, Der Golem, Devoutb3nji, Dicklyon, Dirac66,Djr32, Dmbeaster, Dmoulton, Dna-webmaster, Don Gosiewski, Drivenapart, Drmies, Duttakapil, ESkog, EddEdmondson, Edward, El, Eliasen, Emanspeaks, Eob, Epbr123, ErNa, Eric Kvaalen,Erkcan, Etoombs, Evercat, Evil saltine, Exert, Extransit, F-j123, Francine Rogers, Franco3450, Fratrep, FrenchIsAwesome, Fresheneesz, Gangstapope1, Gargantu, Gene Nygaard, GeorgeLouis,Gephart, GianniG46, Giftlite, Giuliof, Gnomepirate, GoShow, GoodBoy165, GraemeL, Graham87, Greg L, Guardian of Light, Gurchzilla, Götz, Hairy Dude, Headbomb, HelloAnnyong, Herbee,Hgrosser, Horiavulpe, Hroðulf, I do not exist, Io865we, Iridescent, Ironholds, ItsZippy, JBcallOnMe, JRSpriggs, JSquish, Jaan513, Jacksccsi, Jasonfward, Jdvelasc, Jimmy da tuna, Jlaire,Joedeshon, John, John Vandenberg, John of Reading, Johnny Assay, Jordgette, Jwoehr, Kae1is, Kaihsu, Kaikydelan, Kaldari, Katovatzschyn, Keraunos, Killer tadpole, Krash, L!-wgi, Leotohill,LikeLakers2, Likebox, Loadmaster, Loom91, Looxix, MK8, MPerel, Mac Davis, MagneticFlux, MaizeAndBlue86, MarkSweep, Materialscientist, MathKnight, Mathonius, Matteo, MauriceCarbonaro, Mav, Maxorz, McGeddon, Mcgibson, Mebden, Mfrosz, Michael Hardy, Michael.j.sykora, Michbich, Mike9110, Mindmatrix, Moe Epsilon, Momotaro, Mor, Moriel, Mpatel, Ms2ger,Mschlindwein, MuggsMcGinnis, Mxn, NBS525, NawlinWiki, Nbubis, Numbo3, Okiefromokla, Onlinetexts, Ospalh, Out of Phase User, P1415926535, PAR, PGWG, Palfrey, Palica,Patrick0Moran, Pawyilee, Pecos Joe, Peeter.joot, Pharaoh of the Wizards, Phoenixthebird, Phys, Physchim62, Pieter Kuiper, Pinethicket, Pit, Plasticup, Prhstark, Pt, Pvtyoyo, Q Science, Rajakhr,Randomblue, Raphaelmak, Rbj, RcktScientistX, Rdsmith4, Red Act, Renegade54, RetiredUser2, Rex711, Richard L. Peterson, Ringbang, Rjstott, Roadrunner, RockMagnetist, RogierBrussee,Ronewolf, Ronhjones, Ruste TTT, RyanEberhart, SJK, SarekOfVulcan, Scott Martin, SebastianHelm, Shiraun, Shoragan, Sikory, Skoch3, Slavik262, Smaines, Smalnercan, Snailwalker, Sodium,Sowlos, Spcebaby, Spinningspark, Spoon!, Spring243, Srich32977, Sriram sh, Srleffler, Stargazer84, Stevenj, Stizz, StradivariusTV, Sunev, Taargüs, Tbhotch, Thatscienceyguy, The Anome, TheFlying Spaghetti Monster, The Literate Engineer, The Original Wildbear, Thomasheedy, Tibbets74, TimothyRias, Timwi, Tls60, Tobias Bergemann, TootsieStanford, Tordail, Truthnlove,Tschwenn, UTn 4o2, Unquantum, Uriyan, Usien6, Vanished user 9i39j3, Vaughan Pratt, Voyajer, WadeSimMiser, Waffleguy4, Whirlwind a, Wholmestu, Winston Trechane, WojPob,Wtmitchell, Wwoods, Xaos, Xxxx00, Yelyos, Youandme, Zapvet, Zedshort, Zueignung, Zundark, Σ, सुभाष राऊत, 484 anonymous edits

Maxwell's equations  Source: http://en.wikipedia.org/w/index.php?oldid=558399267  Contributors: 130.225.29.xxx, 165.123.179.xxx, 16@r, 2002:8602:7D16:B:21F:5BFF:FEEB:2CC5,213.253.39.xxx, A.C. Norman, A930913, Acroterion, AgadaUrbanit, Ahoerstemeier, Alan Peakall, Alejo2083, Alexanderjt, Almeo, Ambuj.Saxena, Ancheta Wis, Andre Engels, Andrei Stroe,Andrew567, Andries, AndyBuckley, Anthony, Antixt, Anville, Anythingyouwant, Ap, Areldyb, Arestes, Arjayay, Army1987, Art Carlson, Asar, AugPi, Aulis Eskola, Avenged Eightfold,AxelBoldt, Axfangli, BAdhi, BD2412, Barak Sh, BehzadAhmadi, BenBaker, BenFrantzDale, Bender235, Berland, Bernardmarcheterre, Blaze Labs Research, Bob1817, Bora Eryilmaz,Brad7777, Brainiac2595, Brendan Moody, Brequinda, Brews ohare, Bryan Derksen, CYD, Calliopejen1, Can't sleep, clown will eat me, CanDo, Cardinality, Cassini83, Charles Matthews,Childzy, Chodorkovskiy, Chris Howard, Chris the speller, Cloudmichael, Coldwarrier, Colliand, Comech, Complexica, Conversion script, Cooltom95, Corkgkagj, Courcelles, Cpl.Luke, CraigPemberton, Cronholm144, Crowsnest, D6, DAGwyn, DGJM, DJIndica, Daniel.Cardenas, David spector, Davidiad, Delirium, DeltaIngegneria, Dgrant, Dicklyon, Dilwala314, Dmr2,Donarreiskoffer, Donreed, DrSank, Dratman, DreamsReign, Drkirkby, Drw25, Duckyphysics, Dxf04, Długosz, Ebehn, El C, Electrodynamicist, Eliyak, Enok.cc, Enormousdude, EoGuy,Eric.m.dzienkowski, F=q(E+v^B), FDT, Farhamalik, Fgnievinski, Fibonacci, Find the way, Finell, Fir-tree, Firefly322, First Harmonic, Fizicist, Fjomeli, Fledylids, Freepopcornonfridays,Fuhghettaboutit, Gaius Cornelius, Geek1337, Gene Nygaard, Geometry guy, George Smyth XI, GeorgeLouis, Ghaspias, Giftlite, Giorgiomugnaini, Giraffedata, Glicerico, Glosser.ca,GordonWatts, Graham87, Gremagor, Gseryakov, Gutworth, H.A.L., HaeB, Headbomb, Herbee, Heron, Hope I. Chen, Hpmv, Icairns, Ignorance is strength, Igor m, IronGargoyle, Ixfd64, Izno,JATerg, JNW, JRSpriggs, JTB01, JabberWok, Jaknelaps, Jakohn, Jan S., Janfri, Jao, Jasón, Jess Riedel, Jfrancis, Jj1236, Jmnbatista, Joconnor, Johann Wolfgang, JohnBlackburne, Johnlogic,Jordatech, Jordgette, Joseph Solis in Australia, Jpbowen, Jtir, JustinWick, Karada, Karl Dickman, Katterjohn, Keenan Pepper, Kelvie, Khazar2, KingofSentinels, Kissnmakeup, Kjak, Koavf,Kooo, Kragen, Kri, Kwamikagami, Kzollman, L-H, LAUBO, Lambiam, Larryisgood, Laurascudder, Lazcisco, Lcabanel, LedgendGamer, Lee J Haywood, Lethe, Light current, LilHelpa, Linas,Linuxlad, Lir, Lixo2, Lockeownzj00, Loom91, Looxix, Lseixas, Luk, LutzL, MFH, Mandarax, Manishearth, MarSch, Marek69, Mark viking, Marmelad, Marozols, Martin Hogbin, Maschen,Masudr, MathKnight, Maxellus, Maxim Razin, MaxwellSystem, Meier99, Melchoir, MeltBanana, Metacomet, Mets501, Mfrosz, Mgiganteus1, Michael Hardy, Michael Lenz, Michielsen, Micru,Miguel, Mild Bill Hiccup, Milikguay, Miserlou, Mjb, Mleaning, Modster, Mokakeiche, Monkeyjmp, Mpatel, Msablic, Msh210, Myasuda, NHRHS2010, Nabla, Nakon, Nebeleben, Neparis,NewEnglandYankee, Niteowlneils, Nmnogueira, Nousernamesleft, NuclearWarfare, Nudve, Oleg Alexandrov, Omegatron, One zero one, Opspin, Orbst, Out of Phase User, Paksam, Paolo.dL,Paquitotrek, Passw0rd, Patrick, Paul August, Paul D. Anderson, Peeter.joot, Pervect, Pete Rolph, Pete463251, Peterlin, Phlie, Phys, Physchim62, Pieter Kuiper, Pigsonthewing, Pinethicket,PiratePi, PlantTrees, Policron, Pouyan12, Pratyush Sarkar, Qrystal, Quibik, Quondum, RDBury, RG2, RK, RandomP, Ranveig, Rdrosson, Red King, Reddi, Reedy, Revilo314, Rgdboer, Rhtcmu,Rich Farmbrough, Rjwilmsi, Rklawton, Roadrunner, RogierBrussee, Rogper, Rossami, Rudchenko, Rudminjd, S7evyn, SJP, Sadi Carnot, SakseDalum, Salsb, Sam Derbyshire,Sameenahmedkhan, SamuelRiv, Sandb, Sanders muc, Sannse, Sbyrnes321, Scurvycajun, SebastianHelm, Selfstudier, Sgiani, Shamanchill, Shanes, Sheliak, Shlomke, Slakr, Sonygal, Sparkie82,Spartaz, Srleffler, Steve Quinn, Steve p, Steven Weston, Stevenj, StewartMH, Stikonas, StradivariusTV, TStein, Tahir mq, Tarquin, TeaDrinker, Template namespace initialisation script, Tercer,That Guy, From That Show!, The Anome, The Cunctator, The Original Wildbear, The Sanctuary Sparrow, The Wiki ghost, The undertow, TheObtuseAngleOfDoom, Thincat, Thingg, Tide rolls,Tim Shuba, Tim Starling, Tkirkman, Tlabshier, Tobias Bergemann, Tom.Reding, TonyMath, Toymontecarlo, Tpbradbury, Treisijs, Trelvis, Trusilver, Truthnlove, Tunheim, Urvabara, Warfvinge,Warlord88, Waveguy, Wavelength, Widr, Wik, Wikipelli, Wing gundam, Woodstone, Woohookitty, Wordsmith, WriterHound, Wtshymanski, Wurzel, Wwoods, XJaM, Xenonice, Xonqnopp,Yamaguchi先 生, Yevgeny Kats, Yinweichen, Youandme, Z = z² + c, Zhenyok 1, Zoicon5, ^musaz, 老 陳, 936 anonymous edits

Image Sources, Licenses and Contributors 97

Image Sources, Licenses and ContributorsImage:Bell's theorem.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Bell's_theorem.svg  License: Creative Commons Attribution 3.0  Contributors: Manwesulimo2004Image:StraightLines.svg  Source: http://en.wikipedia.org/w/index.php?title=File:StraightLines.svg  License: Public Domain  Contributors: Caroline ThompsonImage:Bells-thm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Bells-thm.png  License: GNU Free Documentation License  Contributors: Bdesham, Common Good, It Is Me Here,Joshbaumgartner, Karelj, Maksim, Mdd, Pieter Kuiper, Tano4595Image:Bell-test-photon-analyer.png  Source: http://en.wikipedia.org/w/index.php?title=File:Bell-test-photon-analyer.png  License: GNU Free Documentation License  Contributors: Chetvorno,Glenn, Joshbaumgartner, Karelj, Maksim, MddFile:Einstein.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Einstein.jpg  License: Public Domain  Contributors: Frank C. 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editsFile:GaussianScatterPCA.png  Source: http://en.wikipedia.org/w/index.php?title=File:GaussianScatterPCA.png  License: GNU Free Documentation License  Contributors: —Ben FrantzDale(talk) (Transferred by ILCyborg)File:beam mode 1.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Beam_mode_1.gif  License: GNU Free Documentation License  Contributors: Original uploader was Lzyvzl aten.wikipediaFile:Eigenfaces.png  Source: http://en.wikipedia.org/w/index.php?title=File:Eigenfaces.png  License: Attribution  Contributors: Laurascudder, Liftarn, Man vyi, YlebruFile:Quantum tunnel effect and its application to the scanning tunneling microscope.ogv  Source:http://en.wikipedia.org/w/index.php?title=File:Quantum_tunnel_effect_and_its_application_to_the_scanning_tunneling_microscope.ogv  License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:JubobroffImage:TunnelEffektKling1.png  Source: http://en.wikipedia.org/w/index.php?title=File:TunnelEffektKling1.png  License: Creative Commons Attribution-Sharealike 3.0  Contributors: FelixKlingFile:Quantum Tunnelling animation.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Quantum_Tunnelling_animation.gif  License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:YuvalrImage:EffetTunnel.gif  Source: http://en.wikipedia.org/w/index.php?title=File:EffetTunnel.gif  License: GNU Free Documentation License  Contributors: Original uploader was Jean-ChristopheBENOIST at fr.wikipediaFile:Wigner function for tunnelling.ogv  Source: http://en.wikipedia.org/w/index.php?title=File:Wigner_function_for_tunnelling.ogv  License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:DenysbondarFile:Rtd seq v3.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Rtd_seq_v3.gif  License: Creative Commons Attribution 3.0  Contributors: Saumitra R Mehrotra & Gerhard KlimeckFile:MaxPlanckWirkungsquantums20050815 CopyrightKaihsuTai.jpg  Source:http://en.wikipedia.org/w/index.php?title=File:MaxPlanckWirkungsquantums20050815_CopyrightKaihsuTai.jpg  License: GNU Free Documentation License  Contributors: Axel.Mauruszat,Captmondo, GeorgHH, JuTa, Kaihsu, Kilom691, Leit, Mdd, PAR, Steak, 2 anonymous editsFile:Wiens law.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wiens_law.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: 4CFile:Black body.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Black_body.svg  License: Public Domain  Contributors: Darth KuleFile:Bohr-atom-PAR.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Bohr-atom-PAR.svg  License: GNU Free Documentation License  Contributors: Original uplo:User:JabberWokFile:VFPt Solenoid correct2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:VFPt_Solenoid_correct2.svg  License: unknown  Contributors: Geek3File:OiintLaTeX.svg  Source: http://en.wikipedia.org/w/index.php?title=File:OiintLaTeX.svg  License: Creative Commons Zero  Contributors: User:MaschenFile:Divergence theorem in EM.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Divergence_theorem_in_EM.svg  License: Creative Commons Zero  Contributors: User:MaschenFile:Curl theorem in EM.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Curl_theorem_in_EM.svg  License: Creative Commons Zero  Contributors: User:MaschenImage:VFPt dipole magnetic1.svg  Source: http://en.wikipedia.org/w/index.php?title=File:VFPt_dipole_magnetic1.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:Geek3File:Magnetosphere rendition.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetosphere_rendition.jpg  License: Public Domain  Contributors: NASAImage:Magnetic core.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetic_core.jpg  License: Creative Commons Attribution 2.5  Contributors: Apalsola, Fayenatic london,Gribozavr, UberpenguinFile:Electromagneticwave3D.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Electromagneticwave3D.gif  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:LookangFile:Polarization and magnetization.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Polarization_and_magnetization.svg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: Marmelad

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