4.1X-Ray Scattering 4.2De Broglie Waves 4.3Electron Scattering 4.4Wave Motion 4.6Uncertainty Principle 4.8Particle in a Box 4.7Probability, Wave Functions, and the Copenhagen Interpretation 4.5Waves or Particles? Wave Properties of Particles I thus arrived at the overall concept which guided my studies: for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. - Louis de Broglie, 1929 Louis de Broglie ( ) CHAPTER 4

Electron Scattering In 1925, Davisson and Germer observed electrons diffracting (much like x-rays) from nickel crystals. George P. Thomson (1892–1975), son of J. J. Thomson, saw electron diffraction from celluloid, gold, aluminum, and platinum. A randomly oriented polycrystalline sample of SnO 2 produces rings.

Beautiful Proof That Electrons are Waves: Imaging Using Them Imaging using light waves is well known. But optical microscopes’ resolution is only /2 ~ 200nm. Electrons have much smaller wavelengths, and electron microscopes can achieve resolutions of ~0.05nm. Electron micrograph of pollen grains with ~0.1nm resolution

Recall that waves diffract through slits. Fraunhofer diffraction patterns One slit Two slits In 1803, Thomas Young saw the two-slit pattern for light, confirming the wave nature of light. But particles are also waves. So they should exhibit similar patterns when passing through slits, especially pairs of slits.

Electron Double-Slit Experiment C. Jönsson of Tübingen, Germany, succeeded in 1961 in showing double-slit interference effects for electrons by constructing very narrow slits and using relatively large distances between the slits and the observation screen. This experiment demonstrated that precisely the same behavior occurs for both light (waves) and electrons (particles).

Shine light on the double slit and observe with a microscope. This should tell us which slit the electron went through. The photon momentum: The electron momentum: Which slit does an electron go through? Need ph < d (the slit separation) to distinguish the slits. Diffraction is significant only when the slit separation d ≤ el the wavelength of the e wave. So the photon momentum p ph > h/d ≥ p el. It’s enough to strongly modify the momentum of the electron, strongly deflecting it! The attempt to identify which slit the electron passes through changes the diffraction pattern, washing out the fringes! So we can’t tell which slit the electron went through.

Which slit does a photon go through? Dimming the light in Young’s two-slit experiment results in single photons at the screen. Since photons are particles, each can only go through one slit. So, at such low intensities, their distribution should become the single-slit pattern. Each photon actually goes through both slits! x

Two-Slit Experiment with Single Electrons The same is true for electrons!

Wave-particle-duality solution It’s very confusing that everything is both a particle and a wave. The wave-particle duality is a little less confusing if we think in terms of: Bohr’s Principle of Complementarity: It’s not possible to describe physical observables simultaneously in terms of both particles and waves. When we’re making a measurement, use the particle description, but when we’re not, use the wave description. When we’re looking, fundamental quantities are particles; when we’re not, they’re waves. In the two-slit problem, the electrons propagate as waves but are detected as particles.

The energy uncertainty of a wave packet is: Combined with the angular frequency relation we derived earlier: Uncertainty Principle: Energy Uncertainty Energy-Time Uncertainty Principle: Werner Heisenberg (1901–1976) In the Uncertainty Principle, we’ll henceforth use a width definition that yields an uncertainty product of ½.

The same mathematics relates x and k :  k  x ≥ ½ So it’s also impossible to measure simultaneously the precise values of k and x for a wave. Now the momentum can be written in terms of k : So the uncertainty in momentum is: But multiplying  k  x ≥ ½ by ħ : And we have Heisenberg’s Uncertainty Principle: Momentum Uncertainty Principle

How to think about Uncertainty The act of making one measurement perturbs the other. Precisely measuring the time disturbs the energy. Precisely measuring the position disturbs the momentum. The Heisenberg-mobile. The problem was that when you looked at the speedometer you got lost.

Since we’re always uncertain as to the exact position,, of a particle, for example, an electron somewhere inside an atom, the particle can’t have zero kinetic energy: Kinetic Energy Minimum so: The average of a positive quantity must always equal or exceed its uncertainty:

De Broglie matter waves should be described in a manner similar to light waves. The matter wave should also be a solution to a wave equation. Wave Motion and  (x,t) = A exp[i(kx –  t –  )] Define the wave number k and the angular frequency  as usual: x And it will often have a solution like:

Okay, if particles are also waves, what’s waving? Probability The wave function determines the likelihood (or probability) of finding a particle at a particular position in space at a given time: Probability, Wave Functions, and the Copenhagen Interpretation The total probability of finding the particle is 1. Forcing this condition on the wave function is called normalization. The probability of the particle being between x 1 and x 2 is given by:

The Copenhagen Interpretation 1. A system is completely described by a wave function , which represents an observer's knowledge of the system. (Heisenberg) 2. The description of nature is probabilistic. The probability of an event is the mag squared of the wave function related to it. (Max Born) 3. Heisenberg's Uncertainty Principle says it’s impossible to know the values of all of the properties of the system at the same time; properties not known with precision are described by probabilities. 4. Complementarity Principle: matter exhibits a wave-particle duality. An experiment can show the particle-like properties of matter, or wave-like properties, but not both at the same time. (Bohr) 5. Measuring devices are essentially classical devices, and they measure classical properties such as position and momentum. 6. The correspondence principle of Bohr and Heisenberg: the quantum mechanical description of large systems should closely approximate the classical description.

A particle (wave) of mass m is in a one-dimensional box of width ℓ. The box puts boundary conditions on the wave. The wave function must be zero at the walls of the box and on the outside. In order for the probability to vanish at the walls, we must have an integral number of half wavelengths in the box: The energy: The possible wavelengths are quantized and hence so are the energies: Particle in a Box

Probability of the particle vs. position Note that E 0 = 0 is not a possible energy level. The concept of energy levels, as first discussed in the Bohr model, has surfaced in a natural way by using waves. The probability of observing the particle between x and x + dx in each state is:

Bohr’s Quantization Condition revisited One of Bohr’s assumptions in his hydrogen atom model was that the angular momentum of the electron in a stationary state is nħ. This turns out to be equivalent to saying that the electron’s orbit consists of an integral number of electron de Broglie wavelengths: Multiplying by 2  /p, we find the circumference: Circumference electron de Broglie wavelengt h