PHYS1220 – Quantum Mechanics

1. PHYS1220 – Quantum Mechanics Lecture 2 August 21, 2002 Dr J. Quinton Office: PG 9 ph 49-21-7025 phjsq@alinga.newcastle.edu.au

2. Light Momentum • Einstein (1906) • Light is truly relativistic – it travels with speed c • Relativistic momentum • Since v = c, g • 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

3. Compton Effect • Compton (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 occur • The greater the angle through which the X-ray is scattered, the greater the wavelength shift (and hence the greater the energy loss).

4. Compton Effect II • Compton 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 Effect Before 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 q

5. Compton Shift Compton Effect III • Electron • Conservation of Energy • Conservation of momentum • x-component • y-component • Eliminating v and q (Tutorial Exercise: Giancoli Chapter 38, P25) leads to an expression for the wavelength shift 1 2 3

6. Compton wavelength Compton Effect IV Dl • 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 2lC lC f 0 p/2 p 

7. Example • After 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?

8. Pair Production • The Photoelectric effect dominates at low photon energies (IR-UV) and Compton effect at intermediate energies (X-rays), but at high energies (g-rays) an entirely different mechanism can occur • If 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 conversion • A photon cannot create a lone electron, otherwise charge would not be conserved

9. Pair Production II • Cloud chamber - Wilson (1895) • A bath of superheated liquid hydrogen, in a magnetic field • Dirac first predicted the existence of the positron in 1931 • Anderson (1932) discovered the positron in cosmic rays experiments, won 1936 Nobel Prize (for first antimatter discovery)

10. Pair Production III • If 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:

11. 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

12. de Broglie’s Hypothesis • Louis de Broglie (1923), doctoral thesis • If 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

13. Example • Question: 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)

14. Davisson-Germer Experiment • The 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 diffraction • The 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 Giancoli • Beam is incident at 90 degrees to surface • Smallest diffraction angle (m=1) at 240

15. Young’s Experiment Revisited • Of 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

17. Electron Microscopes • Electron microscopes are based on the wave nature of electrons. • Resolution depends on wavelength of radiation • Electrons 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 microscope • Max magnification about 106 times

18. 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)?