11

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

22

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

33

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

44

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

55

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

66

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 to such that the electrons are tuned to such that = = / / dd = ( = (hh / /pdpd)) is experimentally is experimentally discernable. Here electrons behave discernable. Here electrons behave like wave. In this case, the effect of like wave. In this case, the effect of hh is not negligible, hence quantum effect is not negligible, hence quantum effect sets insets in

77

EssentaillyEssentailly

hh characterised the scale at which characterised the scale at which quantum nature of particles starts to take quantum nature of particles starts to take over from macroscopic physicsover from macroscopic 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 (pp x x dd in the previous in the previous example), particle behaves like wave; example), particle behaves like wave; whenever whenever hh is negligible, particle behave is negligible, particle behave like particlelike particle

88

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

99

Principle of ComplementarityPrinciple of Complementarity

The complete description of a physical entity The complete description of a physical entity such as proton or electron cannot be done in such as proton or electron cannot be done in terms of particles or wave exclusively, but that terms of particles or wave exclusively, but that both aspect must be consideredboth aspect must be considered

The aspect of the behaviour of the system that The aspect of the behaviour of the system that we observe depends on the kind of experiment we observe depends on the kind of experiment we are performingwe are performing

e.g. in Double slit experiment we see the wave e.g. in Double slit experiment we see the wave nature of electron, but in Milikan’s oil drop nature of electron, but in Milikan’s oil drop experiment we observe electron as a particleexperiment we observe electron as a particle

1010

Davisson and Gremer Davisson and Gremer experimentexperiment

DG confirms the wave nature of DG confirms the wave nature of electron in which it undergoes electron in 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 onto the accelerated and focused onto the target (all apparatus is in vacuum target (all apparatus is in vacuum condition)condition)

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

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

Strong scattered e- beam is Strong scattered e- beam is detected at detected at = 50 degree for V = = 50 degree for V = 54 V54 V

1111

Electron’s de Broglie wave Electron’s de Broglie wave undergoes Bragg’s diffractionundergoes Bragg’s diffraction

Explained as (first order, n= 1) Explained as (first order, n= 1) constructive interference of wave constructive interference of wave scattered by the atoms in the crystalline scattered by the atoms in the crystalline lattice: 2dsinlattice: 2dsin = =

Geometry: Geometry: = 90 = 90oo ––/2 /2 Feed in experimental data that Feed in experimental data that d d = 0.91 = 0.91

Amstrong (obtained from a x-ray Amstrong (obtained from a x-ray Bragg’s diffraction experiment done Bragg’s diffraction experiment done independently)independently)

and and = 65 degree, the wavelength of = 65 degree, the wavelength of the electron wave is the electron wave is = = 2dsin2dsin = = 1.65 1.65 Angstrom Angstrom

Here, 1.65 Here, 1.65 Angstrom is the Angstrom is the experimentally inferred value, which experimentally inferred value, which could be checked against the value could be checked against the value theoretically predicted by de Broglietheoretically predicted by de Broglie

1212

Theoretical value of Theoretical value of of the of the electronelectron

The kinetic energy of the electron is The kinetic energy of the electron is KK = = 54 eV (non-relativistic treatment is suffice 54 eV (non-relativistic treatment is suffice because K << mbecause K << meecc22 = 0.51 MeV) = 0.51 MeV)

According to de Broglie, the wavelength of According to de Broglie, the wavelength of an electron accelerated to kinetic energy an electron accelerated to kinetic energy of of KK = = pp22/2/2mmee = 54 eV has a equivalent = 54 eV has a equivalent

matter wave wavelength matter wave wavelength = = hh//pp = = hh/(2/(2KmKmee))-1/2 -1/2 = 1.67 Amstrong= 1.67 Amstrong

1313

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))

1414

Application of electrons as wave: Application of electrons as wave: scanning electron microscopescanning electron microscope

1515

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

1616

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 wavesparticles in term of waves

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

1717

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//

1818

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.

1919

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

2020

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

2121

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

2222

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

2323

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

{{ppxx,,xx}, {}, {EE,,tt} are called } are called conjugate variablesconjugate variables

The conjugate variables cannot in principle The conjugate variables cannot in principle be measured (or known) to infinite be measured (or known) to infinite precision simultaneously precision simultaneously

2424

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 value of 5.00 10 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 = 4.561010-27-27 Ns; Ns; pp = 0.003% = 0.003% pp = 1.37 = 1.371010-27-27 Ns Ns Hence, Hence, x x ≥ ≥ hh/4/4pp = 0.38 mm = 0.38 mm

x

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

0

x

2525

ExampleExample

A charged A charged meson has rest energy of 140 MeV meson has rest energy of 140 MeV and a lifetime of 26 ns. Find the energy and a lifetime of 26 ns. Find the energy uncertainty of the uncertainty of the meson, expressed in MeV meson, expressed in MeV and also as a function of its rest energyand also as a function of its rest energy

SolutionSolution Given Given EEmmcc

2 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

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ExampleExample 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

SolutionSolutionFor For xx ~ 1 m, we have ~ 1 m, we have

pp ≥ ≥hh/4/4xx = 5.3 = 5.31010-35-35 Ns, Ns, So So vv = ( = (pp)/)/mm ≥ 5.3 ≥ 5.31010-34-34 m/s m/s One can consider One can consider vv 5.3 5.31010-34-34 m/s (extremely m/s (extremely tiny) is the speed of the billard ball at tiny) is the speed of the billard ball at anytime caused by quantum effectsanytime caused by quantum effects

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

1 m long billard table

A billard ball of 100 g, size ~ 2 cm

vv 5.3 5.31010-34-34 m/s m/s

2727

A particle contained in a finite A particle contained in a finite box must has some minimal KEbox must has some minimal KE

One of the most dramatic consequence of One of the most dramatic consequence of the uncertainty principle is that a particle the uncertainty principle is that a particle confined in a small region of finite width confined in a small region of finite width a a (=(=x) x) cannot be exactly at restcannot be exactly at rest

Why? Because…Why? Because… ...if it were, its momentum would be ...if it were, its momentum would be

precisely zero, (meaning precisely zero, (meaning pp = 0) which = 0) which would in turn violate the uncertainty would in turn violate the uncertainty principleprinciple

2828

Estimation of Estimation of KKave ave of a particle in a of a particle in a

box due to Uncertainty Principlebox 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 p p 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

~

2

ave 222 mam

p

m

pK

av

2929

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 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).

3030

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