x y p z >0 p z

xx

yy

ppzz>0>0ppzz<0<0

ψψ

Quantum physics fromQuantum physics fromcoarse grained classical probabilitiescoarse grained classical probabilities

what is an atom ?what is an atom ?

quantum mechanics : isolated objectquantum mechanics : isolated object quantum field theory : excitation of quantum field theory : excitation of

complicated vacuumcomplicated vacuum classical statistics : sub-system of classical statistics : sub-system of

ensemble with infinitely many ensemble with infinitely many degrees of freedomdegrees of freedom

quantum mechanics quantum mechanics can be described by can be described by classical statistics !classical statistics !

quantum mechanics from quantum mechanics from classical statisticsclassical statistics

probability amplitudeprobability amplitude entanglemententanglement interferenceinterference superposition of statessuperposition of states fermions and bosonsfermions and bosons unitary time evolutionunitary time evolution transition amplitudetransition amplitude non-commuting operatorsnon-commuting operators violation of Bell’s inequalitiesviolation of Bell’s inequalities

essence of quantum essence of quantum mechanicsmechanics

description of appropriate description of appropriate subsystems of subsystems of

classical statistical ensemblesclassical statistical ensembles

1) equivalence classes of probabilistic 1) equivalence classes of probabilistic observablesobservables

2) incomplete statistics2) incomplete statistics 3) correlations between measurements based 3) correlations between measurements based

on conditional probabilitieson conditional probabilities 4) unitary time evolution for isolated 4) unitary time evolution for isolated

subsystemssubsystems

statistical picture of the statistical picture of the worldworld

basic theory is not deterministicbasic theory is not deterministic basic theory makes only statements basic theory makes only statements

about probabilities for sequences of about probabilities for sequences of events and establishes correlationsevents and establishes correlations

probabilism is fundamental , not probabilism is fundamental , not determinism !determinism !

quantum mechanics from classical statistics :quantum mechanics from classical statistics :not a deterministic hidden variable theorynot a deterministic hidden variable theory

Probabilistic realismProbabilistic realism

Physical theories and lawsPhysical theories and lawsonly describe probabilitiesonly describe probabilities

Physics only describes Physics only describes probabilitiesprobabilities

Gott würfeltGott würfelt

Physics only describes Physics only describes probabilitiesprobabilities

Gott würfeltGott würfelt Gott würfelt nichtGott würfelt nicht

Physics only describes Physics only describes probabilitiesprobabilities

Gott Gott würfelt würfelt

Gott würfelt nichtGott würfelt nicht

humans can only deal humans can only deal with probabilitieswith probabilities

probabilistic Physicsprobabilistic Physics

There is oThere is onene reality reality This can be described only by This can be described only by

probabilitiesprobabilities

one droplet of water …one droplet of water … 10102020 particles particles electromagnetic fieldelectromagnetic field exponential increase of distance exponential increase of distance

between two neighboring trajectoriesbetween two neighboring trajectories

probabilistic realismprobabilistic realism

The basis of Physics are The basis of Physics are probabilities probabilities

for predictions of real eventsfor predictions of real events

laws are based on laws are based on probabilitiesprobabilities

determinism as special case :determinism as special case :

probability for event = 1 or 0probability for event = 1 or 0

law of big numberslaw of big numbers unique ground state …unique ground state …

conditional probabilityconditional probability

sequences of sequences of events( measurements ) events( measurements )

are described by are described by

conditionalconditional probabilities probabilities

both in classical statisticsboth in classical statisticsand in quantum statisticsand in quantum statistics

w(tw(t11))

not very suitable not very suitable for statement , if here and nowfor statement , if here and nowa pointer falls downa pointer falls down

::

Schrödinger’s Schrödinger’s catcat

conditional probability :conditional probability :if nucleus decaysif nucleus decaysthen cat dead with wthen cat dead with wcc = 1 = 1 (reduction of wave function)(reduction of wave function)

one - particle wave one - particle wave functionfunction

from coarse graining from coarse graining of microphysical of microphysical

classical statistical classical statistical ensembleensemble

non – commutativity in classical statisticsnon – commutativity in classical statistics

microphysical ensemblemicrophysical ensemble

states states ττ labeled by sequences of labeled by sequences of

occupation numbers or bits noccupation numbers or bits ns s = 0 = 0 or 1or 1

ττ = [ n = [ nss ] = ] = [0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc.[0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc.

probabilities pprobabilities pττ > 0 > 0

function observablefunction observable

function observablefunction observable

ss

I(xI(x11)) I(xI(x44))I(xI(x22)) I(xI(x33))

normalized difference between occupied and empty normalized difference between occupied and empty bits in intervalbits in interval

generalized function generalized function observableobservable

normalizationnormalization

classicalclassicalexpectationexpectationvaluevalue

several species several species αα

positionposition

classical observable : classical observable : fixed value for every state fixed value for every state ττ

momentummomentum

derivative observablederivative observable

classical classical observable : observable : fixed value for every fixed value for every state state ττ

complex structurecomplex structure

classical product of classical product of position and momentum position and momentum

observablesobservables

commutes !commutes !

different products of different products of observablesobservables

differs from classical productdiffers from classical product

Which product describes Which product describes correlations of correlations of

measurements ?measurements ?

coarse graining of coarse graining of informationinformation

for subsystemsfor subsystems

density matrix from coarse density matrix from coarse graininggraining

• position and momentum observables position and momentum observables use only use only small part of the information contained small part of the information contained in in ppττ ,,

• relevant part can be described by relevant part can be described by density matrixdensity matrix

• subsystem described only by subsystem described only by information information which is contained in density matrix which is contained in density matrix • coarse graining of informationcoarse graining of information

quantum density matrixquantum density matrix

density matrix has the properties density matrix has the properties ofof

a quantum density matrixa quantum density matrix

quantum operatorsquantum operators

quantum product of quantum product of observablesobservables

the productthe product

is compatible with the coarse grainingis compatible with the coarse graining

and can be represented by operator productand can be represented by operator product

incomplete statisticsincomplete statistics

classical productclassical product

is not computable from information whichis not computable from information which

is available for subsystem !is available for subsystem ! cannot be used for measurements in the cannot be used for measurements in the

subsystem !subsystem !

classical and quantum classical and quantum dispersiondispersion

subsystem probabilitiessubsystem probabilities

in contrast :in contrast :

squared momentumsquared momentum

quantum product between classical observables :quantum product between classical observables :maps to product of quantum operatorsmaps to product of quantum operators

non – commutativity non – commutativity in classical statisticsin classical statistics

commutator depends on choice of product !commutator depends on choice of product !

measurement correlationmeasurement correlation

correlation between measurements correlation between measurements of positon and momentum is given of positon and momentum is given by quantum productby quantum product

this correlation is compatible with this correlation is compatible with information contained in subsysteminformation contained in subsystem

coarse coarse graininggraining

from fundamental from fundamental fermions fermions

at the Planck scaleat the Planck scaleto atoms at the Bohr to atoms at the Bohr

scalescale

p([np([nss])])

ρρ(x , x´)(x , x´)

quantumquantum particle from particle from classical probabilities in classical probabilities in

phase spacephase space

quantum particlequantum particleandand

classical particleclassical particle

quantum particle quantum particle classical particleclassical particle

particle-wave particle-wave dualityduality

uncertaintyuncertainty

no trajectoriesno trajectories

tunnelingtunneling

interference for interference for double slitdouble slit

particlesparticles sharp position and sharp position and

momentummomentum classical classical

trajectoriestrajectories

maximal energy maximal energy limits motionlimits motion

only through one only through one slitslit

double slit experimentdouble slit experiment

double slit experimentdouble slit experiment

one isolated particle ! one isolated particle ! no interaction between atoms passing through slitsno interaction between atoms passing through slits

probability –probability –distributiondistribution

double slit experimentdouble slit experiment

Is there a classical probability distribution in phase space ,Is there a classical probability distribution in phase space ,and a suitable time evolution , which can describe and a suitable time evolution , which can describe interference pattern ?interference pattern ?

quantum particle from quantum particle from classical probabilities in classical probabilities in

phase spacephase space probability distribution in phase space forprobability distribution in phase space for oonene particle particle

w(x,p)w(x,p) as for classical particle !as for classical particle ! observables different from classical observables different from classical

observablesobservables

time evolution of probability distribution time evolution of probability distribution different from the one for classical particledifferent from the one for classical particle

quantum mechanics quantum mechanics can be described by can be described by classical statistics !classical statistics !

quantum mechanics from quantum mechanics from classical statisticsclassical statistics

probability amplitudeprobability amplitude entanglemententanglement interferenceinterference superposition of statessuperposition of states fermions and bosonsfermions and bosons unitary time evolutionunitary time evolution transition amplitudetransition amplitude non-commuting operatorsnon-commuting operators violation of Bell’s inequalitiesviolation of Bell’s inequalities

quantum physicsquantum physics

wave functionwave function

probabilityprobability

phasephase

Can quantum physics be Can quantum physics be described by classical described by classical

probabilities ?probabilities ?“ “ No go “ theoremsNo go “ theorems

Bell , Clauser , Horne , Shimony , HoltBell , Clauser , Horne , Shimony , Holt

implicit assumption : use of classical correlation implicit assumption : use of classical correlation function for correlation between measurementsfunction for correlation between measurements

Kochen , SpeckerKochen , Specker

assumption : unique map from quantum assumption : unique map from quantum operators to classical observablesoperators to classical observables

zwitterszwitters

no different concepts for classical no different concepts for classical and quantum particles and quantum particles

continuous interpolation between continuous interpolation between quantum particles and classical quantum particles and classical particles is possible particles is possible

( not only in classical limit )( not only in classical limit )

classical particle classical particle without classical without classical

trajectorytrajectory

quantum particle quantum particle classical particleclassical particle

particle-wave particle-wave dualityduality

uncertaintyuncertainty

no trajectoriesno trajectories

tunnelingtunneling

interference for interference for double slitdouble slit

particle particle – wave – wave dualityduality

sharp position and sharp position and momentummomentum

classical trajectoriesclassical trajectories

maximal energy maximal energy limits motionlimits motion

only through one slitonly through one slit

no classical trajectoriesno classical trajectories

also for classical particles in microphysics :also for classical particles in microphysics :

trajectories with sharp position trajectories with sharp position

and momentum for each moment and momentum for each moment

in time are inadequate idealization !in time are inadequate idealization !

still possible formally as limiting casestill possible formally as limiting case

quantum particle classical quantum particle classical particleparticle

quantum - quantum - probability -probability -amplitude amplitude ψψ(x)(x)

Schrödinger - Schrödinger - equationequation

classical probability classical probability

in phase space w(x,p)in phase space w(x,p)

Liouville - equation for Liouville - equation for w(x,p)w(x,p)

( corresponds to Newton ( corresponds to Newton eq. eq.

for trajectories )for trajectories )

quantum formalism forquantum formalism forclassical particleclassical particle

probability distribution for probability distribution for oneone classical particle classical particle

classical probability distributionclassical probability distributionin phase spacein phase space

wave function for classical wave function for classical particleparticle

classical probability classical probability distribution in phase spacedistribution in phase space

wave function for wave function for classical classical particleparticle

depends on depends on positionposition and momentum ! and momentum !

CC

CC

wave function for wave function for oneoneclassicalclassical particle particle

realreal depends on position and momentumdepends on position and momentum square yields probabilitysquare yields probability

CC CC

similarity to Hilbert space for classical mechanicssimilarity to Hilbert space for classical mechanicsby Koopman and von Neumann by Koopman and von Neumann in our case : in our case : real real wave function permits computationwave function permits computationof wave function from probability distributionof wave function from probability distribution( up to some irrelevant signs )( up to some irrelevant signs )

quantum laws for quantum laws for observablesobservables

CC CC

xx

yy

ppzz>0>0ppzz<0<0

ψψ

time evolution of time evolution of classical wave functionclassical wave function

Liouville - equationLiouville - equation

describes describes classicalclassical time evolution of time evolution of classical probability distributionclassical probability distributionfor one particle in potential V(x)for one particle in potential V(x)

time evolution of time evolution of classical wave functionclassical wave function

CC

CC CC

wave equationwave equation

modified Schrödinger - equationmodified Schrödinger - equation

CC CC

wave equationwave equation

CC CC

fundamenal equation for classical fundamenal equation for classical particle in potential V(x)particle in potential V(x)replaces Newton’s equationsreplaces Newton’s equations

particle - wave dualityparticle - wave duality

wave properties of particles :wave properties of particles :

continuous probability distributioncontinuous probability distribution

particle – wave dualityparticle – wave duality

experiment if particle at position x – yes or no :experiment if particle at position x – yes or no :discrete discrete alternativealternative

probability distribution probability distribution for findingfor findingparticle at position x :particle at position x :continuouscontinuous

11

11

00

particle – wave dualityparticle – wave duality

All statistical properties of All statistical properties of classicalclassical particles particles

can be described in quantum formalism !can be described in quantum formalism !

no quantum particles yet !no quantum particles yet !

modification of Liouville modification of Liouville equationequation

evolution equationevolution equation

time evolution of probability has to time evolution of probability has to be specified as fundamental lawbe specified as fundamental law

not known a priori not known a priori Newton’s equations with trajectories Newton’s equations with trajectories

should follow only in should follow only in limiting caselimiting case

zwitterszwitters

same formalism for quantum and same formalism for quantum and classical particlesclassical particles

different time evolution of probability different time evolution of probability distributiondistribution

zwitters : zwitters :

between quantum and classical particle –between quantum and classical particle –

continuous interpolation of time continuous interpolation of time evolution equationevolution equation

quantum time evolutionquantum time evolution

modification of evolution modification of evolution forfor

classical probability classical probability distributiondistribution

HHWW

HHWW

CC CC

quantum particlequantum particle

evolution equationevolution equation

fundamental equation for fundamental equation for quantum particle in potential Vquantum particle in potential V

replaces Newton’s equationsreplaces Newton’s equations

CCCC CC

modified observablesmodified observables

Which observables should Which observables should be chosen ?be chosen ?

momentum: p or ?momentum: p or ?

position : x or ?position : x or ?

Different possibilities , should be Different possibilities , should be adapted to specific setting of adapted to specific setting of measurementsmeasurements

position - observableposition - observable

different observables according to different observables according to experimental situationexperimental situation

suitable observable for microphysis has suitable observable for microphysis has to be foundto be found

classical position observable : idealization classical position observable : idealization of infinitely precise resolutionof infinitely precise resolution

quantum – observable : remains quantum – observable : remains computable with coarse grained computable with coarse grained informationinformation

1515

quantum - observablesquantum - observables

observables for classical observables for classical position and momentumposition and momentum

observables for observables for quantum - quantum - position and momentumposition and momentum

… … do not commute ! do not commute !

uncertaintyuncertainty

quantum – observables containquantum – observables contain statistical partstatistical part ( similar to entropy , temperature )( similar to entropy , temperature )

Heisenberg’sHeisenberg’suncertainty relationuncertainty relation

Use quantum observables Use quantum observables for description of for description of measurements of measurements of

position and momentum of position and momentum of particles !particles !

quantum particlequantum particle

with evolution equationwith evolution equation

all expectation values and correlations forall expectation values and correlations forquantum – observables , as computed fromquantum – observables , as computed fromclassical probability distribution ,classical probability distribution ,coincide for all times precisely with coincide for all times precisely with

predictions predictions of quantum mechanics for particle in of quantum mechanics for particle in

potential Vpotential V

CCCC CC

classical probabilities – classical probabilities – not a deterministic classical not a deterministic classical

theorytheory

quantum paquantum particle from rticle from classical probabilities in phase space !classical probabilities in phase space !

quantum formalismquantum formalismfrom classical from classical probabilitiesprobabilities

pure statepure state

described by quantum wave functiondescribed by quantum wave function

realized by classicalrealized by classicalprobabilities in theprobabilities in theformform

time evolution described bytime evolution described bySchrödinger – equationSchrödinger – equation

density matrix anddensity matrix andWigner-transformWigner-transform

Wigner – transformed densityWigner – transformed densitymatrix in quantum mechanicsmatrix in quantum mechanics

permits simple computationpermits simple computationof expectation values ofof expectation values ofquantum mechanicalquantum mechanicalobservablesobservables

can be constructed from classical wave function !can be constructed from classical wave function !

CC CC

quantum observables quantum observables andand

classical observablesclassical observables

zwitterzwitter

difference between quantum and difference between quantum and classical particles only through classical particles only through different time evolutiondifferent time evolution

continuous continuous interpolatiointerpolationn

CLCL

QMQMHHWW

zwitter - Hamiltonianzwitter - Hamiltonian

γγ=0 : quantum – particle=0 : quantum – particle γγ==ππ/2 : classical particle/2 : classical particle

other interpolating Hamiltonians possible !other interpolating Hamiltonians possible !

HowHow good is quantum good is quantum mechanics ?mechanics ?

small parameter small parameter γγ can be tested can be tested experimentallyexperimentally

zwitter : no conserved microscopic zwitter : no conserved microscopic energyenergy

static state : orstatic state : or

CC

ground state for zwitterground state for zwitter

static state with loweststatic state with lowest

eigenstate for quantum energyeigenstate for quantum energy

zwitter –ground state has admixture of zwitter –ground state has admixture of excited excited

levels of quantum energylevels of quantum energy

quantum - energyquantum - energy

energy uncertainty ofenergy uncertainty ofzwitter ground statezwitter ground state

also small shift of energyalso small shift of energy

experiments for experiments for determination ordetermination orlimits on zwitter – limits on zwitter –

parameter parameter γγ ??

ΔΔE≠0E≠0

almost degenerate almost degenerate energy levels…?energy levels…?

lifetime of nuclear spin states > 60 hlifetime of nuclear spin states > 60 hγγ < 10 < 10-14 -14 ( Heil et al.) ( Heil et al.)

experiments for experiments for determination ordetermination orlimits on zwitter – limits on zwitter –

parameter parameter γγ ??

lifetime of nuclear spin states > 60 h ( Heil et al.) : lifetime of nuclear spin states > 60 h ( Heil et al.) : γγ < 10 < 10-14-14

sharpened observables –sharpened observables –between quantum and between quantum and

classicalclassical

ß=0 : quantum observables, ß=1 :classical ß=0 : quantum observables, ß=1 :classical observablesobservables

weakening of uncertainty weakening of uncertainty relationrelation

exp

eri

men

exp

eri

men

t ?

t ?

quantum particles and quantum particles and classical statisticsclassical statistics

common concepts and formalism for quantum common concepts and formalism for quantum and classical particles :and classical particles :

classical probability distribution and wave classical probability distribution and wave functionfunction

different time evolution , different Hamiltoniandifferent time evolution , different Hamiltonian quantum particle from “ coarse grained “ quantum particle from “ coarse grained “

classical probabilities ( only information classical probabilities ( only information contained in Wigner function is needed )contained in Wigner function is needed )

continuous interpolation between quantum continuous interpolation between quantum and classical particle : zwitterand classical particle : zwitter

conclusionconclusion

quantum statistics emerges from classical quantum statistics emerges from classical statisticsstatistics

quantum state, superposition, quantum state, superposition, interference, entanglement, probability interference, entanglement, probability amplitudeamplitude

unitary time evolution of quantum unitary time evolution of quantum mechanics can be described by suitable mechanics can be described by suitable time evolution of classical probabilitiestime evolution of classical probabilities

conditional correlations for measurements conditional correlations for measurements both in quantum and classical statisticsboth in quantum and classical statistics

experimental challengeexperimental challenge

quantitative tests , how accurate the quantitative tests , how accurate the predictions of quantum mechanics are predictions of quantum mechanics are obeyedobeyed

zwitterzwitter sharpened observables sharpened observables small parameter : small parameter : “ “ almost quantum mechanics “almost quantum mechanics “

endend

Quantenmechanik aus Quantenmechanik aus klassischen klassischen

WahrscheinlichkeitenWahrscheinlichkeitenklassische Wahrscheinlichkeitsverteilung kann klassische Wahrscheinlichkeitsverteilung kann

explizit angegeben werden für :explizit angegeben werden für :

quantenmechanisches Zwei-Zustands-quantenmechanisches Zwei-Zustands-System Quantencomputer : Hadamard System Quantencomputer : Hadamard gategate

Vier-Zustands-System ( CNOT gate )Vier-Zustands-System ( CNOT gate ) verschränkte Quantenzuständeverschränkte Quantenzustände InterferenzInterferenz

Bell’sche UngleichungenBell’sche Ungleichungen

werden verletzt durch werden verletzt durch bedingtebedingte Korrelationen Korrelationen

Bedingte Korrelationen für zwei EreignisseBedingte Korrelationen für zwei Ereignisse

oder Messungen reflektieren bedingte oder Messungen reflektieren bedingte WahrscheinlichkeitenWahrscheinlichkeiten

Unterschied zu klassischen Korrelationen Unterschied zu klassischen Korrelationen

( Klassische Korrelationen werden implizit zur ( Klassische Korrelationen werden implizit zur Herleitung der Bell’schen Ungleichungen Herleitung der Bell’schen Ungleichungen verwandt. )verwandt. )

Bedingte Dreipunkt- Korrelation nicht kommutativBedingte Dreipunkt- Korrelation nicht kommutativ

RealitätRealität

KorrelationenKorrelationen sind physikalische Realität , nicht sind physikalische Realität , nicht nur Erwartungswerte oder Messwerte einzelner nur Erwartungswerte oder Messwerte einzelner ObservablenObservablen

Korrelationen können Korrelationen können nicht-lokal nicht-lokal sein ( auch in klassischer Statistik ) ; sein ( auch in klassischer Statistik ) ; kausale Prozesse zur Herstellung kausale Prozesse zur Herstellung nicht-lokaler Korrelationen nicht-lokaler Korrelationen erforderlicherforderlich

Korrelierte Untersysteme sind nicht separabel in Korrelierte Untersysteme sind nicht separabel in unabhängige Teilsysteme – unabhängige Teilsysteme – Ganzes mehr als Ganzes mehr als Summe der TeileSumme der Teile

EPR - ParadoxonEPR - Paradoxon

Korrelation zwischen zwei Spins wird Korrelation zwischen zwei Spins wird bei Teilchenzerfall hergestelltbei Teilchenzerfall hergestellt

Kein Widerspruch zu Kausalität oder Kein Widerspruch zu Kausalität oder Realismus wenn Korrelationen als Teil der Realismus wenn Korrelationen als Teil der Realität verstanden werdenRealität verstanden werden

hat mal nicht Recht )hat mal nicht Recht )((

Untersystem und Untersystem und Umgebung:Umgebung:

unvollständige Statistikunvollständige Statistik

typische Quantensysteme sind typische Quantensysteme sind UntersystemeUntersysteme

von klassischen Ensembles mit unendlich vielenvon klassischen Ensembles mit unendlich vielen

Freiheitsgraden ( Umgebung )Freiheitsgraden ( Umgebung )

probabilistischeprobabilistische Observablen für Untersysteme : Observablen für Untersysteme :

Wahrscheinlichkeitsverteilung für Messwerte Wahrscheinlichkeitsverteilung für Messwerte

in Quantenzustandin Quantenzustand

conditional correlationsconditional correlations

example :example :two - level observables , two - level observables , A , B can take values ± 1A , B can take values ± 1

conditional probabilityconditional probability

probability to find value +1 for probability to find value +1 for product product

of measurements of A and Bof measurements of A and B

… … can be expressed in can be expressed in terms of expectation valueterms of expectation valueof A in eigenstate of Bof A in eigenstate of B

probability to find A=1 probability to find A=1 after measurement of B=1after measurement of B=1

measurement correlationmeasurement correlation

After measurement A=+1 the system After measurement A=+1 the system must be in eigenstate with this eigenvalue.must be in eigenstate with this eigenvalue.Otherwise repetition of measurement couldOtherwise repetition of measurement couldgive a different result !give a different result !

measurement changes statemeasurement changes statein all statistical systems !in all statistical systems !

quantum and classicalquantum and classical

eliminates possibilities that eliminates possibilities that are not realizedare not realized

physics makes statements physics makes statements about possibleabout possible

sequences of events sequences of events and their probabilitiesand their probabilities

M = 2 :M = 2 :

unique eigenstates for unique eigenstates for M=2M=2

eigenstates with A = 1eigenstates with A = 1

measurement preserves pure states if projectionmeasurement preserves pure states if projection

measurement correlation measurement correlation equals quantum correlationequals quantum correlation

probability to probability to measure measure A=1 and B=1 :A=1 and B=1 :

sequence of three sequence of three measurements measurements

and quantum commutatorand quantum commutator

two measurements commute , not threetwo measurements commute , not three