School of Mathematical and Physical Sciences PHYS August, PHYS1220 – Quantum Mechanics Lecture 2 August 21, 2002 Dr J. Quinton Office: PG 9 ph

School of Mathematical and Physical Sciences PHYS August, 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 m 0 =0 and therefore momentum is given by Therefore, to summarise, the kinetic energy and momentum of light ‘quanta’ are given by Light Momentum

School of Mathematical and Physical Sciences PHYS August, 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).

School of Mathematical and Physical Sciences PHYS August, 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

School of Mathematical and Physical Sciences PHYS August, 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 Compton Shift

School of Mathematical and Physical Sciences PHYS August, 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 p 0 p/2 lClC 2lC2lC Dl Compton wavelength f 

School of Mathematical and Physical Sciences PHYS August, Example After a 0.8nm x-ray photon scatters from a free electron, the electron recoils at 1.4x10 6 m.s -1. What was the Compton shift in the photon’s wavelength? Through what angle was the photon scattered?

School of Mathematical and Physical Sciences PHYS August, 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

School of Mathematical and Physical Sciences PHYS August, 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)

School of Mathematical and Physical Sciences PHYS August, 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:

School of Mathematical and Physical Sciences PHYS August, 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

School of Mathematical and Physical Sciences PHYS August, 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

School of Mathematical and Physical Sciences PHYS August, 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)

School of Mathematical and Physical Sciences PHYS August, 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 = 3x 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 in Giancoli Beam is incident at 90 degrees to surface Smallest diffraction angle (m=1) at 24 0

School of Mathematical and Physical Sciences PHYS August, 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?  In Japan, 1989 the experiment was done for the first time with controlled electron flux

School of Mathematical and Physical Sciences PHYS August,

School of Mathematical and Physical Sciences PHYS August, Electron Microscopes Electron microscopes are based on the wave nature of electrons. Resolution depends on wavelength of radiation Electrons accelerated with ~10 5 V give a wavelength ~ nm. The practical resolution limited to ~ nm times better than an optical microscope Max magnification about 10 6 times

School of Mathematical and Physical Sciences PHYS August, 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)?