School of Mathematical and Physical Sciences PHYS1220 21 August, 20021 PHYS1220 – Quantum...

21 August, 2002 1

School of Mathematical and Physical Sciences PHYS1220School of Mathematical and Physical Sciences PHYS1220

PHYS1220 – Quantum Mechanics

Lecture 2August 21, 2002

Dr J. QuintonOffice: PG 9 ph [email protected]

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Einstein (1906)Light is truly relativistic – it travels with speed c

Relativistic momentumSince v = c, The rest mass of light must be zero (otherwise the momentum would be infinite!)

Relativistic energy is given by

But m0=0 and therefore momentum is given by

Therefore, to summarise, the kinetic energy and momentum of light ‘quanta’ are given by

Light Momentum

2 2

1;

1 /p mv

v c

2 2 2 2 40E p c m c

E hf hp

c c

;hc h

E hf p

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Compton EffectCompton (1923) performed scattering experiments with X-rays and a carbon block.

X-rays scatter from electrons and have a longer wavelength than beforehand, therefore an energy loss must occurThe greater the angle through which the X-ray is scattered, the greater the wavelength shift (and hence the greater the energy loss).

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Compton Effect IICompton showed that the effect could only be explained by an elastic collision between light (acting as a particle) and electrons

Compton coined the name ‘photon’ to represent the light ‘particle’

Analysis of the Compton EffectBefore collision, the photon energy and momentum are given by

After the collision, the photon energy and momentum (Nb v is still equal to c) are

The electron is assumed to be initially at rest but free to move when struck and recoils at an angle

;hc h

E hf p

' '' ';

hc hE hf p

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Compton Effect IIIElectron

Conservation of Energy

Conservation of momentum

x-component

y-component

Eliminating v and (Tutorial Exercise: Giancoli Chapter 38, P25) leads to an expression for the wavelength shift

21e eKE m c e ep m v

2'

( 1) e

hc hcm c

'cos cose

h hm v

'0 sin sine

hm v

1

2

3

' (1 cos )e

h

m c Compton Shift

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Compton Effect IV

Compton shift

The characteristic quantity of this equation (in this case for a free electron) is defined as

For the record, classical wave theory predicts that an incoming EM wave with frequency f should set electrons into oscillation with frequency f. The electron should then re-emit light with the same frequency. Therefore, no wavelength shift should happen

Thus the Compton effect further supports the particle theory of light.

Compton won the 1927 Nobel Prize in Physics for this work

0

C

2C

' (1 cos )

e

h

m c

Ce

h

m c Compton wavelength

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ExampleAfter a 0.8nm x-ray photon scatters from a free electron, the electron recoils at 1.4x106 m.s-1. What was the Compton shift in the photon’s wavelength?

Through what angle was the photon scattered?

34 8 1

-19 -1 10

6.626x10 . 3x10 .1.55

1.602x10 . 8x10

hc J s m sE h f keV

J eV m

2 2 19

2 2

11 1 8.92x10 5.57

1 /e e eKE m c m c J eV

v c

' 2'

34 8' '

' 19

1 1.550 5.57 1544.43

6.626x10 3x100.8034 0.0034

1544.43 1.602x10

e

hc hcE m c keV eV eV

hcnm nm

E

31 8

1 1 9 034

9.1x10 3x10cos 1 cos 1 0.0034x10 113.63

6.626x10em c

h

2 1918.918x10 5.57

2e eKE m v J eV

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Pair ProductionThe Photoelectric effect dominates at low photon energies (IR-UV) and Compton effect at intermediate energies (X-rays), but at high energies (-rays) an entirely different mechanism can occurIf a photon has sufficiently high energy, it can create a matter-antimatter pair such as an electron and an anti-electron (called a positron, which has the same mass but a charge of +e)

This is an example of pure energy-mass conversionA photon cannot create a lone electron, otherwise charge would not be conserved

20E m c

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Pair Production IICloud chamber - Wilson (1895)A bath of superheated liquid hydrogen, in a magnetic field

Dirac first predicted the existence of the positron in 1931Anderson (1932) discovered the positron in cosmic rays experiments, won 1936 Nobel Prize (for first antimatter discovery)

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20

22 31 8 140

34 813

6 19

2 2

9.109x10 3x10 8.1981x10 511.7

2 511.7 2(500 ) 2.023

6.626x10 3x106.13x10

2.023x10 1.602x10

ph

ph

ph

hcE m c KE

m c kg J keV

E keV keV MeV

hcm

E

Pair Production IIIIf the electron and positron meet, they will annihilate one another to produce energy (ie a photon or photons)

Positrons do not normally last very long in nature!

Note that photon induced pair production cannot occur in empty space because momentum and energy cannot be simultaneously conserved. A heavy nucleus is needed to carry away some momentum.Example: Calculate the wavelength of a photon that is needed to create an electron-positron pair, each with a KE of 500 keV.Answer:

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So is Light a Wave or a Particle?The topics discussed so far illustrate a particle nature of light, but don’t forget that light has been shown to illustrate wave behaviour as well (diffraction, interference).

Aren’t these two descriptions incompatible? so which is correct?

The answer is that both are correct. Light has a dual nature, it can behave as a wave, or as a particle. This phenomenon is called wave-particle duality

When measurements involving light are made, one type of behaviour will dominate, but it depends upon both the interaction involved and the method used to observe it!

The Principle of Complimentarity – Bohr In order to understand any given experiment, we must use

either the wave or the photon theory, but not both A full understanding of light, however, requires awareness of

both aspects, but is impossible to visualise

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de Broglie’s HypothesisLouis de Broglie (1923), doctoral thesisIf photons have wave and particle characteristics, then perhaps all forms of matter have wave as well as particle properties!

Every particle has a characteristic wavelength that is dependent upon its momentum. This wavelength is called its de Broglie wavelength, and is given by

Furthermore, they obey the Planck relationship, so the frequency of these matter waves is

At the time, no experimental evidence supported this

h

p

Ef

h

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ExampleQuestion: If everyday objects possess particle-like properties, then why don’t people experience diffraction or interference?

Calculate the de Broglie wavelength of a 75kg person who is walking with a speed of 1m/s

and so the wavelength of ordinary objects are much too small to be detected (and even if the speed were 20 orders smaller)

What about a 100eV electron? (non-relativistic)

3436

1

6.626x10 .8.8x10

75 1 .

h h J sm

p mv kg m s

6 1 1025.9x10 . 1.2x10

e e

E hv m s m

m m v

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Davisson-Germer ExperimentThe wavelength of electrons is small, but large enough to detect (typical interatomic distances in crystalline solids is ~ 0.3 nm = 3 Å = 3x10-10 m)In 1927, Davisson and Germer scattered electrons from aluminium foil and observed diffractionThe measured wavelength was precisely that predicted by de Broglie,

Who was then awarded the Physics Nobel Prize (1929), Davisson and G.P Thomsen in (1937)

Electron Diffraction: Example 38-11 in GiancoliBeam is incident at 90 degrees to surfaceSmallest diffraction angle (m=1) at 240

2 2

20.123

2 2 2

0.123sin 0.30

sin sin 24

e e e

p h hKE nm

m m m KE

nmd d nm

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Young’s Experiment RevisitedOf course, an undisputable ‘test’ of electron wave-like behaviour is by performing Young’s double slit interference experiment.

The interference pattern will not appear unless the electrons truly exhibit wave-like behaviour

Many discussions and thought experiments were made Richard Feynman – if a machine gun was shot at an iron plate

with two slits in it and a concrete wall behind it, what kind of pattern would the bullets make?

www.colorado.edu/physics/2000/schroedinger/two-slit3.html

In Japan, 1989 the experiment was done for the first time with controlled electron flux

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Electron MicroscopesElectron microscopes are based on the wave nature of electrons. Resolution depends on wavelength of radiationElectrons accelerated with ~105 V give a wavelength ~ 0.004 nm. The practical resolution limited to ~0.1-0.5 nm.103 times better than an optical microscopeMax magnification about 106 times

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Putting Perspective on ‘Duality’We have seen that both waves and particles are really ‘wavicles’

they exhibit both wave-like and particle-like behaviour

This is not consistent with our everyday experience, why not? We see a wave or a particle, but never both together

But think for a moment about the mechanism of sight We can only ‘see’ light by absorbing it And we only ‘see’ particles by absorbing light from them Light interacts with matter (especially electrons) on microscopic

scales

So the behaviour that we see macroscopically depends very much upon how we detect it.For particles to exhibit wave-like behaviour they must have very small momenta (because h is so small)

Question: What would the universe be like if Planck’s constant, h was equal to 1 J.s (ie 34 orders of magnitude larger)?

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