11

Wave particle dualityWave particle duality

““Quantum nature of light” refers to the Quantum nature of light” refers to the particle attribute of lightparticle attribute of light

““Quantum nature of particle” refers to the Quantum nature of particle” refers to the wave attribute of a particlewave attribute of a particle

Light (classically EM waves) is said to Light (classically EM waves) is said to display “wave-particle duality” – it behave display “wave-particle duality” – it behave like wave in one experiment but as particle like wave in one experiment but as particle in others (c.f. a person with schizophrenia)in others (c.f. a person with schizophrenia)

22

33

Not only light does have “schizophrenia”, so are Not only light does have “schizophrenia”, so are other microscopic ``particle’’ such as electron, other microscopic ``particle’’ such as electron, (see later chapters), i.e. particle” also manifest (see later chapters), i.e. particle” also manifest wave characteristics in some experimentswave characteristics in some experiments

Wave-particle duality is essentially the Wave-particle duality is essentially the manifestation of the quantum nature of thingsmanifestation of the quantum nature of things

This is an very weird picture quite contradicts to This is an very weird picture quite contradicts to our conventional assumption with is deeply our conventional assumption with is deeply rooted on classical physics or intuitive notion on rooted on classical physics or intuitive notion on thingsthings

44

Whether light displays wave or particle Whether light displays wave or particle nature depends on the object it is interacting nature depends on the object it is interacting with, and also on the experimental set-up to with, and also on the experimental set-up to observe itobserve it

If an experiment is set-up to observe the If an experiment is set-up to observe the wave nature (such as in interference or wave nature (such as in interference or diffraction experiment), it displays wave diffraction experiment), it displays wave nature nature

If the experimental set-up has a scale that is If the experimental set-up has a scale that is corresponding to the quantum nature of corresponding to the quantum nature of radiation, then light will displays particle radiation, then light will displays particle behaviour, such as in Compton scatteringsbehaviour, such as in Compton scatterings

When is light wave and when is it When is light wave and when is it particle?particle?

55

Compton wavelength as a scale to Compton wavelength as a scale to set the quantum nature of light and set the quantum nature of light and

matter (electron)matter (electron) As an example of a ‘scale’ in a given As an example of a ‘scale’ in a given

experiment or a theory, let’s consider the experiment or a theory, let’s consider the Compton wavelength in Compton scatteringCompton wavelength in Compton scattering

Compton wavelength is the Compton wavelength is the lengthlength scalescale

which characterises the onset of quantum which characterises the onset of quantum nature of light (corpuscular nature) and nature of light (corpuscular nature) and electron (wave nature) in their interactionselectron (wave nature) in their interactions

66

Experimental scale vs Compton Experimental scale vs Compton wavelengthwavelength

If the wavelength of light is much larger than If the wavelength of light is much larger than the Compton wavelength of the electron it is the Compton wavelength of the electron it is interacting with, light behaves like wave (e.g. interacting with, light behaves like wave (e.g. in interference experiments with visible light). in interference experiments with visible light). Compton effect is negligible in this caseCompton effect is negligible in this case

On the other hand, if the wavelength of the On the other hand, if the wavelength of the radiation is comparable to the Compton radiation is comparable to the Compton wavelength of the interacting particle, light wavelength of the interacting particle, light starts to behave like particle and collides with starts to behave like particle and collides with the electron in an ‘particle-particle’ mannerthe electron in an ‘particle-particle’ manner

77

In short the identity manifested by In short the identity manifested by light depends on what it “sees” light depends on what it “sees”

(which in turns depend on its own (which in turns depend on its own wavelength) in a given experimental wavelength) in a given experimental

conditioncondition

Microscopic matter particle (such as Microscopic matter particle (such as electron and atoms) also manifest electron and atoms) also manifest

wave-particle dualitywave-particle duality

This will be the next agenda in our This will be the next agenda in our coursecourse

88

PYQ 1.16 Final Exam 2003/04PYQ 1.16 Final Exam 2003/04 Which of the following statements are true about Which of the following statements are true about

light?light? I.I. It propagates at the speed of It propagates at the speed of c c = 3 x 108 m/s in = 3 x 108 m/s in

all mediumall medium II.II. It’s an electromagnetic wave according to the It’s an electromagnetic wave according to the

Maxwell theoryMaxwell theory III.III. It’s a photon according to EinsteinIt’s a photon according to Einstein IV.IV. It always manifests both characteristics of wave It always manifests both characteristics of wave

and particle simultaneously in a given experimentand particle simultaneously in a given experiment A.A. I,IV I,IV B.B. II, III,IV II, III,IV C. C. I, II, III,IVI, II, III,IV D.D. I, II I, II E. E. II,IIIII,III ANS: E, my own questionANS: E, my own question

99

Wavelike properties of particleWavelike properties of particle In 1923, while still a graduate In 1923, while still a graduate

student at the University of student at the University of Paris, Louis de Broglie Paris, Louis de Broglie published a brief note in the published a brief note in the journal journal Comptes rendusComptes rendus containing an idea that was containing an idea that was to revolutionize our to revolutionize our understanding of the understanding of the physical world at the most physical world at the most fundamental level: fundamental level: That That particle has intrinsic wave particle has intrinsic wave propertiesproperties

For more interesting details:For more interesting details: http://www.davis-inc.com/http://www.davis-inc.com/

physics/index.shtmlphysics/index.shtml

Prince de Broglie, 1892-1987

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de Broglie’s postulate (1924)de Broglie’s postulate (1924)

The postulate: there should be a symmetry The postulate: there should be a symmetry between matter and wave. The wave aspect of between matter and wave. The wave aspect of matter is related to its particle aspect in exactly matter is related to its particle aspect in exactly the same quantitative manner that is in the case the same quantitative manner that is in the case for radiation. The total energy for radiation. The total energy EE and momentum and momentum p p of an entity, for both matter and wave alike, is of an entity, for both matter and wave alike, is related to the frequency related to the frequency of the wave of the wave associated with its motion via by Planck constantassociated with its motion via by Planck constant

E E = = hh; p ; p = = hh//

1111

A particle has wavelength!!!A particle has wavelength!!!

= = hh//pp is the de Broglie relation predicting the is the de Broglie relation predicting the

wave length of the matter wave wave length of the matter wave associated with the motion of a material associated with the motion of a material particle with momentum particle with momentum pp

Particle with linear momentum p

Matter wave with de Broglie wavelength

= p/h

A particle with momentum p is pictured as a wave

1212

A physical entity possess both A physical entity possess both aspects of particle and wave in a aspects of particle and wave in a

complimentary mannercomplimentary manner

BUT why is the wave nature of material particle

not observed?

Because …

1313

Because…we are too large and quantum effects are too Because…we are too large and quantum effects are too smallsmall

Consider two extreme cases: Consider two extreme cases: (i) an electron with kinetic energy K = 54 eV, de Broglie (i) an electron with kinetic energy K = 54 eV, de Broglie

wavelenght, wavelenght, = h/p = = h/p =

h / (2mh / (2meeK)K)1/21/2 = 1.65 Angstrom = 1.65 Angstrom

(ii) a billard (100 g) ball moving with momentum p = mv = (ii) a billard (100 g) ball moving with momentum p = mv = 0.1 kg x 10 m/s = 1 Ns, de Broglie wavelenght, 0.1 kg x 10 m/s = 1 Ns, de Broglie wavelenght, = h/p = = h/p = 1010-34-34 m, too small to be observed in any experiments m, too small to be observed in any experiments

1414

Matter wave is a quantum Matter wave is a quantum phenomenaphenomena

This also means that this effect is difficult to observe in This also means that this effect is difficult to observe in our macroscopic world (unless with the aid of some our macroscopic world (unless with the aid of some specially designed apparatus)specially designed apparatus)

The smallness of The smallness of hh in the relation in the relation = = hh//pp makes wave makes wave characteristic of particles hard to be observedcharacteristic of particles hard to be observed

The statement that when The statement that when h h 0, 0, becomes becomes vanishingly small means that vanishingly small means that

the wave nature will becomes effectively ``shut-off’’ the wave nature will becomes effectively ``shut-off’’ and there would appear to loss its wave nature and there would appear to loss its wave nature whenever the relevant scale (e.g. the whenever the relevant scale (e.g. the pp of the particle) of the particle) is too large in comparison with is too large in comparison with hh ~ 10-34 Js ~ 10-34 Js

In other words, the wave nature will of a particle will In other words, the wave nature will of a particle will only show up when the scale only show up when the scale pp is comparable (or is comparable (or smaller) to the size of smaller) to the size of hh

1515

Recap: de Broglie’s postulateRecap: de Broglie’s postulate Particles also have wave nature Particles also have wave nature The total energy The total energy EE and momentum and momentum p p of an entity, for of an entity, for

both matter and wave alike, is related to the frequency both matter and wave alike, is related to the frequency of the wave associated with its motion via by Planck of the wave associated with its motion via by Planck constantconstant

E E = = hh; ; = = hh//pp This is the de Broglie relation predicting the wave length This is the de Broglie relation predicting the wave length

of the matter wave of the matter wave associated with the motion of a associated with the motion of a material particle with momentum material particle with momentum pp

A free particle with linear momentum p

Matter wave with de Broglie wavelength

= p/h

A particle with momentum p is pictured as a wave

1616

What is the speed of the de Broglie What is the speed of the de Broglie wave?wave?

The momentum of a moving body at is The momentum of a moving body at is related to its measured speed via related to its measured speed via p = mvp = mv

On the other hand, de Broglie says a On the other hand, de Broglie says a moving body has momentum and moving body has momentum and wavelength related by wavelength related by pp = = hh//

Then logically the speed of the de Broglie Then logically the speed of the de Broglie wavelength (lets call it wavelength (lets call it vvpp) must be ) must be identified with identified with vv

Lets see if this is trueLets see if this is true

1717

The speed of de Broglie wave is related to the The speed of de Broglie wave is related to the wave’s frequency and de Broglie wavelength via wave’s frequency and de Broglie wavelength via vvpp==ff

where the de Broglie wavelength where the de Broglie wavelength is related to is related to the body’s measured speed via the body’s measured speed via = h/(= h/(mvmv))

The energy carried by a quantum of the de The energy carried by a quantum of the de Broglie wave is given by Broglie wave is given by EE==hfhf

The energy The energy EE must also be equal to the must also be equal to the relativistic energy of the moving body, relativistic energy of the moving body, EE = = mcmc22

1818

Equating both, Equating both, hf hf = = mcmc22

f f = = mcmc22//hh Substitute the de Broglie frequency into Substitute the de Broglie frequency into

vvpp==f f we obtain we obtain

vvpp=(=(hh//mvmv))mcmc22//hh ==cc22//v v > > c c !!!!!!!! We arrive at the unphysical picture that the We arrive at the unphysical picture that the

speed of the de Broglie wave speed of the de Broglie wave vvpp not only is not only is unequal to unequal to vv but also > but also > cc

So, something is going wrong here So, something is going wrong here

1919

Phase and group velocity of the de Phase and group velocity of the de Broglie waveBroglie wave

In the previous calculation we have failed to identify In the previous calculation we have failed to identify vvpp with with vv

The reason being that The reason being that vvpp is actually the PHASE is actually the PHASE velocity of the de Broglie wavevelocity of the de Broglie wave

By right we should have used the GROUP velocity By right we should have used the GROUP velocity We should picture the moving particle as a wave We should picture the moving particle as a wave

group instead of a pure wave with only single group instead of a pure wave with only single wavelengthwavelength

From the previous lecture, we have learned that the From the previous lecture, we have learned that the group velocity is given by group velocity is given by vvgg = d = d/d/dkk

We would like to see how We would like to see how vvgg is related to the moving is related to the moving object’s speedobject’s speed

2020

Indeed Indeed vvgg is identified with is identified with vv

22

0

20 0

3/ 22 2

03/ 22 2

22 2

2 2

1 /

22 2

1 /

/

g

g

dv

dk

mcf m c

h h

m c m vd d

dv h dv h v c

mmv dkk

h dv h v c

d d dkv v

dk dv dv

2121

The de Broglie’ group wave is The de Broglie’ group wave is identified with the moving body’s identified with the moving body’s vv

2222

ExampleExample

An electron has a de Broglie wavelength of 2.00 pm. Find its kinetic energy and the phase and the group velocity of its de Broglie waves.

You will do this example in your TuYou will do this example in your Tutorial 4 Please DIY!!!

2323

Matter wave (Matter wave ( = = hh//pp) is a quantum ) is a quantum phenomenaphenomena

The appearance of The appearance of hh is a theory generally means quantum is a theory generally means quantum effect is taking place (e.g. Compton effect, PE, pair-effect is taking place (e.g. Compton effect, PE, pair-production/annihilation)production/annihilation)

Quantum effects are generally difficult to observe due to the Quantum effects are generally difficult to observe due to the smallness of smallness of h h and is easiest to be observed in experiments and is easiest to be observed in experiments at the microscopic (e.g. atomic) scaleat the microscopic (e.g. atomic) scale

The wave nature of a particle (i.e. the quantum nature of The wave nature of a particle (i.e. the quantum nature of particle) will only show up when the linear momentum scale particle) will only show up when the linear momentum scale pp of the particle times the length dimension characterising of the particle times the length dimension characterising the experiment ( the experiment ( pp x x dd) is comparable (or smaller) to the ) is comparable (or smaller) to the quantum scale of quantum scale of hh

We will illustrate this concept with two examplesWe will illustrate this concept with two examples

2424

hh characterises the scale of characterises the scale of quantum physicsquantum physics

Example: shoot a beam Example: shoot a beam of electron to go though of electron to go though a double slit, in which the a double slit, in which the momentum of the beam, momentum of the beam, pp =(2 =(2mmeeKK))1/21/2, can be , can be controlled by tuning the controlled by tuning the external electric potential external electric potential that accelerates themthat accelerates them

In this way we can tune In this way we can tune the length the length [ = h [ = h /(2m/(2meeK)K)1/21/2 ]of the ]of the wavelength of the wavelength of the electronelectron

2525

Let Let dd = width between the double slits (= the = width between the double slits (= the length scale characterising the experiment)length scale characterising the experiment)

The parameter The parameter = = / / dd, (the ‘resolution , (the ‘resolution angle’ on the interference pattern) angle’ on the interference pattern) characterises the interference patterncharacterises the interference pattern

If we measure a non vanishing value of in an experiment, this means we have measures interference (wave)

d

2626

If there is no interference If there is no interference happening, the parameter happening, the parameter

= = / / d d becomes becomes 00

Wave properties of the incident beam is not revealed as no interference pattern is observed. We can picture the incident beam as though they all comprise of particles

2727

Electrons behave like particle when Electrons behave like particle when = = hh//pp << << d, d, like wave when like wave when = = hh//pp≈ ≈ dd If in an experiment the magnitude If in an experiment the magnitude

of of pdpd are such that are such that = = / d = ( / d = (hh / /pdpd) << 1 (too tiny to ) << 1 (too tiny to

be observed), electrons behave be observed), electrons behave like particles and no interference like particles and no interference is observed. In this scenario, the is observed. In this scenario, the effect of effect of h h is negligible is negligible

If If = = / /dd is not observationally is not observationally negligible, the wave nature is revealed negligible, the wave nature is revealed via the observed interference patternvia the observed interference pattern

This will happen if the momentum of This will happen if the momentum of the electrons are tuned in such a wat the electrons are tuned in such a wat that that = = / / dd = ( = (hh / /pdpd)) is is experimentally discernable. Here experimentally discernable. Here electrons behave like wave. In this electrons behave like wave. In this case, the effect of case, the effect of hh is not negligible, is not negligible, hence quantum effect sets inhence quantum effect sets in

Electron behave like particle

Electron behave like wave

2828

EssentiallyEssentially

hh characterised the scale at which quantum characterised the scale at which quantum nature of particles starts to take over from nature of particles starts to take over from macroscopic physicsmacroscopic physics

Whenever Whenever h h is not negligible compared to is not negligible compared to the characteristic scales of the the characteristic scales of the experimental setup (= experimental setup (= p dp d in the previous in the previous example), particle behaves like wave; example), particle behaves like wave; whenever whenever hh is negligible compared to is negligible compared to pdpd, , particle behave like just a conventional particle behave like just a conventional particleparticle

2929

Is electron wave or particleIs electron wave or particle?? They are both…but not They are both…but not

simultaneouslysimultaneously In any experiment (or In any experiment (or

empirical observation) only empirical observation) only one aspect of either wave or one aspect of either wave or particle, but not both can be particle, but not both can be observed simultaneously. observed simultaneously.

It’s like a coin with two faces. It’s like a coin with two faces. But one can only see one But one can only see one side of the coin but not the side of the coin but not the other at any instanceother at any instance

This is the so-called wave-This is the so-called wave-particle dualityparticle duality

Electron as particle

Electron as wave

3030

HomeworkHomework

Please read section 5.7 THE WAVE-Please read section 5.7 THE WAVE-PARTICLE DUALITY in page 179-185 to PARTICLE DUALITY in page 179-185 to get a more comprehensive answer to the get a more comprehensive answer to the question: is electron particle or wavequestion: is electron particle or wave

It’s a very interesting and highly It’s a very interesting and highly intellectual topic to investigateintellectual topic to investigate

3131

Davisson and Gremer Davisson and Gremer experimentexperiment

DG confirms the wave DG confirms the wave nature of electron in nature of electron in which it undergoes which it undergoes Bragg’s diffractionBragg’s diffraction

Thermionic electrons are Thermionic electrons are produced by hot filament, produced by hot filament, accelerated and focused accelerated and focused onto the target (all onto the target (all apparatus is in vacuum apparatus is in vacuum condition)condition)

Electrons are scattered at Electrons are scattered at an angle an angle into a movable into a movable detectordetector

3232

Pix of Davisson and Gremer

3333

Result of the DG experimentResult of the DG experiment

Distribution of electrons Distribution of electrons is measured as a is measured as a function of function of

Strong scattered e- Strong scattered e- beam is detected atbeam is detected at = = 50 degree for50 degree for V = V = 5454 V V

3434

How to interpret the result of How to interpret the result of DG?DG?

Electrons get diffracted by Electrons get diffracted by the atoms on the surface the atoms on the surface (which acted as diffraction (which acted as diffraction grating) of the metal as grating) of the metal as though the electron acting though the electron acting like they are WAVE like they are WAVE

Electron do behave like Electron do behave like wave as postulated by de wave as postulated by de BroglieBroglie

3535

Constructive Bragg’s diffractionConstructive Bragg’s diffraction

The peak of the diffraction pattern is The peak of the diffraction pattern is the m=1the m=1stst order constructive order constructive interference: interference: ddsin sin == 11

where where = = 50 degree for50 degree for V = V = 5454 V V From x-ray Bragg’s diffraction From x-ray Bragg’s diffraction

experiment done independentlyexperiment done independently we we know know d d = 2.15 Amstrong= 2.15 Amstrong

Hence the wavelength of the electron Hence the wavelength of the electron is is = = ddsinsin = = 1.65 Angstrom 1.65 Angstrom

Here, 1.65 Here, 1.65 Angstrom is the Angstrom is the experimentally inferred value, which experimentally inferred value, which is to be checked against the is to be checked against the theoretical value predicted by de theoretical value predicted by de BroglieBroglie

3636

Theoretical value of Theoretical value of of the of the electronelectron

An external potential An external potential V V accelerates the electron accelerates the electron via via eVeV==K K

In the DG experiment the kinetic energy of the In the DG experiment the kinetic energy of the electron is accelerated to electron is accelerated to KK = 54 eV (non- = 54 eV (non-relativistic treatment is suffice because K << mrelativistic treatment is suffice because K << meecc22 = 0.51 MeV) = 0.51 MeV)

According to de Broglie, the wavelength of an According to de Broglie, the wavelength of an electron accelerated to kinetic energy of electron accelerated to kinetic energy of KK = = pp22/2/2mmee = 54 eV has a equivalent matter wave = 54 eV has a equivalent matter wave wavelength wavelength = = hh//pp = = hh/(2/(2KmKmee))-1/2 -1/2 = 1.67 Amstrong= 1.67 Amstrong

In terms of the external potential, In terms of the external potential, = = hh/(2/(2eVmeVmee))-1/2-1/2

3737

Theory’s prediction matches Theory’s prediction matches measured valuemeasured value

The result of DG measurement agrees almost The result of DG measurement agrees almost perfectly with the de Broglie’s predictionperfectly with the de Broglie’s prediction: 1.65 : 1.65 Angstrom measured by DG experiment against Angstrom measured by DG experiment against 1.67 Angstrom according to theoretical 1.67 Angstrom according to theoretical predictionprediction

Wave nature of electron is hence experimentally Wave nature of electron is hence experimentally confirmedconfirmed

In fact, wave nature of microscopic particles are In fact, wave nature of microscopic particles are observed not only in e- but also in other particles observed not only in e- but also in other particles (e.g. neutron, proton, molecules etc. – most (e.g. neutron, proton, molecules etc. – most strikingly Bose-Einstein condensatestrikingly Bose-Einstein condensate))

3838

Application of electrons wave: Application of electrons wave: electron microscope, Nobel Prize electron microscope, Nobel Prize

1986 (Ernst Ruska)1986 (Ernst Ruska)

3939

Electron’s de Broglie Electron’s de Broglie wavelength can be wavelength can be tunned via tunned via

= = hh/(2/(2eVmeVmee))-1/2-1/2

Hence electron Hence electron microscope can magnify microscope can magnify specimen (x4000 times) specimen (x4000 times) for biological specimen or for biological specimen or 120,000 times of wire of 120,000 times of wire of about 10 atoms in widthabout 10 atoms in width

4040

Other manifestation of electron’s Other manifestation of electron’s wave naturewave nature

Experimentally it also seen to display diffraction Experimentally it also seen to display diffraction patternpattern

4141

Not only electron, other Not only electron, other microscopic particles also behave microscopic particles also behave

like wave at the quantum scalelike wave at the quantum scale The following atomic structural images provide insight into the The following atomic structural images provide insight into the

threshold between prime radiant flow and the interference threshold between prime radiant flow and the interference structures called matter.   structures called matter.   

In the right foci of the ellipse a real cobalt atom has been inserted. In the right foci of the ellipse a real cobalt atom has been inserted. In the left foci of the ellipse a phantom of the real atom has In the left foci of the ellipse a phantom of the real atom has appeared. The appearance of the phantom atom was not appeared. The appearance of the phantom atom was not expected.  expected. 

The ellipsoid coral was constructed by placing 36 cobalt atom on a The ellipsoid coral was constructed by placing 36 cobalt atom on a copper surface. This image is provided here to provide a visual copper surface. This image is provided here to provide a visual demonstration of the attributes of material matter arising from the demonstration of the attributes of material matter arising from the harmonious interference of background radiation. harmonious interference of background radiation. 

QUANTUM CORAL http://

home.netcom.com/~sbyers11/grav11E.htm

4242

Heisenberg’s uncertainty Heisenberg’s uncertainty principle (Nobel Prize,1932)principle (Nobel Prize,1932)

WERNER HEISENBERG (1901 - 1976) WERNER HEISENBERG (1901 - 1976) was one of the greatest physicists of was one of the greatest physicists of

the twentieth century. He is best known the twentieth century. He is best known as a founder of as a founder of quantum mechanicsquantum mechanics, , the new physics of the atomic world, the new physics of the atomic world, and especially for the and especially for the uncertainty principleuncertainty principle in quantum theory. in quantum theory. He is also known for his controversial He is also known for his controversial role as a leader of Germany's role as a leader of Germany's nuclear fissionnuclear fission research during World research during World War II. After the war he was active in War II. After the war he was active in elementary particle physics and West elementary particle physics and West German German science policyscience policy. .

http://www.aip.org/history/heisenberg/http://www.aip.org/history/heisenberg/p01.htmp01.htm

4343

A particle is represented by a wave A particle is represented by a wave packet/pulsepacket/pulse

Since we experimentally confirmed that Since we experimentally confirmed that particles are wave in nature at the quantum particles are wave in nature at the quantum scale scale hh (matter wave) we now have to describe (matter wave) we now have to describe particles in term of waves (relevant only at the particles in term of waves (relevant only at the quantum scale)quantum scale)

Since a real particle is localised in space (not Since a real particle is localised in space (not extending over an infinite extent in space), the extending over an infinite extent in space), the wave representation of a particle has to be in wave representation of a particle has to be in the form of wave packet/wave pulsethe form of wave packet/wave pulse

4444

As mentioned before, wavepulse/wave As mentioned before, wavepulse/wave packet is formed by adding many waves of packet is formed by adding many waves of different amplitudes and with the wave different amplitudes and with the wave numbers spanning a range of numbers spanning a range of k k (or (or equivalently, equivalently,

x

Recall that k = 2Recall that k = 2//, hence, hence

k/kk/k//

4545

Still remember the Still remember the uncertainty uncertainty relationships for classical waves?relationships for classical waves?

As discussed earlier, due to its nature, a wave packet must As discussed earlier, due to its nature, a wave packet must obey the uncertainty relationships for classical waves (which obey the uncertainty relationships for classical waves (which are derived mathematically with some approximations)are derived mathematically with some approximations)

2~

2

~ xkx 1 t

However a more rigorous mathematical treatment (without the However a more rigorous mathematical treatment (without the approximation) gives the exact relationsapproximation) gives the exact relations

2/14

2

xkx

4

1 t

To describe a particle with wave packet that is localised over a small region To describe a particle with wave packet that is localised over a small region xx requires a large range of wave number; that is, requires a large range of wave number; that is, kk is large. Conversely, a small is large. Conversely, a small range of wave number cannot produce a wave packet localised within a small range of wave number cannot produce a wave packet localised within a small distance.distance.

4646

Matter wave representing a particle Matter wave representing a particle must also obey similar wave must also obey similar wave

uncertainty relationuncertainty relation For matter waves, for which their For matter waves, for which their

momentum (energy) and wavelength momentum (energy) and wavelength (frequency) are related by (frequency) are related by pp = = hh// ( (EE = = hh), the uncertainty relationship of the ), the uncertainty relationship of the classical wave is translated into classical wave is translated into

2

xpx 2

tE

Where Where Prove these yourselves (hint: from Prove these yourselves (hint: from pp = = hh//, , pp//pp = = ))

2/h

4747

Heisenberg uncertainty relationsHeisenberg uncertainty relations

2

xpx

2

tE

The product of The product of the uncertainty the uncertainty in momentum in momentum (energy) and in (energy) and in position (time) is position (time) is at least as large at least as large as Planck’s as Planck’s constantconstant

4848

What meansWhat means

It sets the intrinsic lowest possible limits It sets the intrinsic lowest possible limits on the uncertainties in knowing the values on the uncertainties in knowing the values of of ppxx and and xx, no matter how good an , no matter how good an

experiments is madeexperiments is made It is impossible to specify simultaneously It is impossible to specify simultaneously

and with infinite precision the linear and with infinite precision the linear momentum and the corresponding position momentum and the corresponding position of a particleof a particle

2

xpx

4949

What meansWhat means

If a system is known to exist in a state of If a system is known to exist in a state of energy energy EE over a limited period over a limited period tt, then this , then this energy is uncertain by at least an amount energy is uncertain by at least an amount hh/(4/(4tt))

therefore, the energy of an object or therefore, the energy of an object or system can be measured with infinite system can be measured with infinite precision (precision (E=E=0) only if the object of 0) only if the object of system exists for an infinite time (system exists for an infinite time (tt→→∞∞))

2

tE

5050

Conjugate variables Conjugate variables (Conjugate observables)(Conjugate observables)

{{ppxx,,xx}, {}, {EE,,tt} are called } are called

conjugate variablesconjugate variables The conjugate The conjugate

variables cannot in variables cannot in principle be measured principle be measured (or known) to infinite (or known) to infinite precision precision simultaneouslysimultaneously

5151

ExampleExample The speed of an electron is measured to have a The speed of an electron is measured to have a

value of 5.00 x 10value of 5.00 x 1033 m/s to an accuracy of m/s to an accuracy of 0.003%. Find the uncertainty in determining the 0.003%. Find the uncertainty in determining the position of this electronposition of this electron

SOLUTIONSOLUTION Given Given vv = 5.00 = 5.00 10 1033 m/s; ( m/s; (vv)/)/v v = 0.003%= 0.003% By definition, By definition, pp = = mmeevv = 4.56 x 10 = 4.56 x 10-27-27 Ns; Ns; pp = 0.003% x = 0.003% x pp = 1.37x10 = 1.37x10-27-27 Ns Ns Hence, Hence, x x ≥ ≥ hh/4/4pp = 0.38 nm = 0.38 nm

x

p = (4.564.56±1.37)±1.37)1010-27-27 Ns Ns x = 0.38 nm

0

x

5252

ExampleExample A charged A charged meson has rest energy of 140 MeV and a meson has rest energy of 140 MeV and a

lifetime of 26 ns. Find the energy uncertainty of the lifetime of 26 ns. Find the energy uncertainty of the meson, expressed in MeV and also as a function of its meson, expressed in MeV and also as a function of its rest energyrest energy

SolutionSolution Given Given EEmmcc2 2 = 140 MeV, = 140 MeV, = 26 ns. = 26 ns. EE ≥ ≥hh/4/4.03.031010-27-27J J

= 1.27= 1.271010-14-14 MeV; MeV; EE//EE = 1.27 = 1.271010-14-14 MeV/140 MeV = 9 MeV/140 MeV = 91010-17-17

“Now you see it”

“Now you DONT”

Exist only for Exist only for = 26 ns = 26 nsE E ±±EE

5353

ExampleExampleestimating the quantum effect on a macroscopic particle

Estimate the minimum uncertainty velocity of a billard ball Estimate the minimum uncertainty velocity of a billard ball ((mm ~ 100 g) confined to a billard table of dimension 1 m ~ 100 g) confined to a billard table of dimension 1 m

SolutionSolutionForFor xx ~ 1 m, we have~ 1 m, we have

pp ≥ ≥hh/4/4xx = 5.3x10= 5.3x10-35-35 Ns, Ns, SoSo vv = ( = (pp)/)/mm ≥ 5.3x10≥ 5.3x10-34-34 m/s m/s One can considerOne can consider vv = 5.3x10= 5.3x10-34-34 m/s (extremely tiny) is m/s (extremely tiny) is

the speed of the billard ball at anytime caused by quantum the speed of the billard ball at anytime caused by quantum effectseffects

In quantum theory, no particle is absolutely at rest due to In quantum theory, no particle is absolutely at rest due to the Uncertainty Principlethe Uncertainty Principle

1 m long billard table

A billard ball of 100 g, size ~ 2 cm

v = 5.3 x 10v = 5.3 x 10-34-34 m/s m/s

5454

A particle contained within a A particle contained within a finite region must has some finite region must has some

minimal KEminimal KE

One of the most dramatic consequence of the One of the most dramatic consequence of the uncertainty principle is that a particle confined uncertainty principle is that a particle confined in a small region of finite width cannot be in a small region of finite width cannot be exactly at rest (as already seen in the exactly at rest (as already seen in the previous example)previous example)

Why? Because…Why? Because…

...if it were, its momentum would be precisely ...if it were, its momentum would be precisely zero, (meaning zero, (meaning pp = 0) which would in turn = 0) which would in turn violate the uncertainty principleviolate the uncertainty principle

5555

What is the What is the KKave ave of a particle in a of a particle in a

box due to Uncertainty Principle?box due to Uncertainty Principle? We can estimate the minimal KE of a particle confined in We can estimate the minimal KE of a particle confined in

a box of size a box of size aa by making use of the UP by making use of the UP Uncertainty principle requires that Uncertainty principle requires that pp ≥ ≥ ( (hh/2/2a a (we (we

have have ignored the factor 2 for some subtle statistical ignored the factor 2 for some subtle statistical reasons)reasons)

Hence, the magnitude of Hence, the magnitude of pp must be, on average, at least must be, on average, at least

of the same order as of the same order as pp: : Thus the kinetic energy, whether it has a definite value or Thus the kinetic energy, whether it has a definite value or

not, must on average have the magnitude not, must on average have the magnitude

2

2

~

2

~ave

2

ave 222 mam

p

m

pK

pp ||

5656

Zero-point energyZero-point energy

2

2

~

2

~

2

ave 222 mam

p

m

pK

av

This is the zero-point energy, the minimal possible kinetic energy for a quantum particle confined in a region of width a

Particle in a box of size a can never be at rest (e.g. has zero K.E) but has a minimal KE Kave (its zero-point energy)

We will formally re-derived this result again when solving for the Schrodinger equation of this system (see later).

a

5757

PYQ 3(d) KSCP 2003/04PYQ 3(d) KSCP 2003/04

Suppose that the Suppose that the xx-component of the -component of the velocity of a kg mass is measured to an velocity of a kg mass is measured to an accuracy of m/s. What is the limit of the accuracy of m/s. What is the limit of the accuracy with which we can locate the accuracy with which we can locate the particle along the particle along the xx-axis?-axis?

5858

PYQ 3(d) KSCP 2003/04PYQ 3(d) KSCP 2003/04

SolutionSolution

Gautreau and Savin, Schaum’s series Gautreau and Savin, Schaum’s series modern physics, pg.98, Q. 10.53modern physics, pg.98, Q. 10.53

m1063.242

2

;;2

25

vm

h

vmx

xvmxmv

mvpxp

5959

PYQ 2.11 Final Exam 2003/04PYQ 2.11 Final Exam 2003/04

Assume that the uncertainty in the position of a particle is equal to its de Broglie wavelength. What is the minimal uncertainty in its velocity, vx?

A. vx/4p B. vx/2p C. vx/8p

D. vx E. vx/p

ANS: A, Schaum’s 3000 solved problems, Q38.66, pg. 718

6060

RecapRecap

Measurement necessarily involves interactions between Measurement necessarily involves interactions between observer and the observed systemobserver and the observed system

Matter and radiation are the entities available to us for Matter and radiation are the entities available to us for such measurementssuch measurements

The relations The relations pp = = hh// and and EE = = hh are applicable to both are applicable to both matter and to radiation because of the intrinsic nature of matter and to radiation because of the intrinsic nature of wave-particle dualitywave-particle duality

When combining these relations with the universal When combining these relations with the universal waves properties, we obtain the Heisenberg uncertainty waves properties, we obtain the Heisenberg uncertainty relationsrelations

In other words, the uncertainty principle is a necessary In other words, the uncertainty principle is a necessary consequence of particle-wave dualityconsequence of particle-wave duality