Everything is a wave deBroglie and his matter waves, and its consequences for physics and our concept of reality De Broglie late in his aristrocratic career decided to study physics and he wrote a doctoral thesis trying to get a more reasonable picture for Bohr. De Broglie went back to an earlier notion that discreteness came from waves – like the modes of a string – so he wondered if electrons in atoms had anything to do with the waves of a string

Why quantization of angular momentum? an integer number of wavelengths fits into the circular orbit where And he realized something very simple. If the electron were a wave (or had wave-like character) and the allowed orbits were standing waves of the electrons in a circle. Then Bohrs condition of angular momentum quantization states that the wave-length of the electron must be related to its momentum. Incidentally it is the same relation for photons. l is the de Broglie wavelength

Magnitudes of deBroglie wavelengths Particle Value of l Electrons of kinetic energy 12.2 A 1.2 A 0.12A 1 eV 100 eV 10000 eV Protons of kinetic energy 1 keV 1 MeV 1 GeV 0.009 A 28.6 F 0.73 F Thermal neutrons (300K) 1.5 A Neutrons of kinetic energy 9.0 F So what is this length-scale that we are talking about – really small for even the smallest things……But we have measured this scale before – anyone remember when? Diffraction!! 0.75 A He atoms at 300K you, walking to the student union for lunch at 2 miles per hour

The Davisson Germer experiment Davisson and Germer at Bell labs (and son Thomson) here tried to do exactly that and they invented electron diffraction. But Davisson and Germer didn’t know/care about de Broglie (which was about the same time) – they were just shooting electrons into nickel to study the atomic structure of its surface. They expected electrons to undergo diffuse reflection as if from a rough surface and that is what they were seeing. But then they accidentally annealed the crystal and they saw this pattern and they thought were interesting. Max Born was who realized that this verified de Broglies theory.

Transmission Electron Microscope As a by-product we had a rather interesting application – electrons have much smaller wave-lengths then light. Electrons have a wavelength that is much shorter than visible light. The smallest detail that can be resolved is equal to one wavelength.

Scanning Electron Microscope Human hair Table Salt Red Blood Cells

Specimen interactions involved in forming an image bulk (thick) foil (thin) Thin Specimens Thick Specimens Unscattered electrons are those which are transmitted through the material. Since the probability of transmission is proportional to thickness, it can give a reading of thickness variations. Backscattered electrons can be used to identify elements in the material. Auger electrons [emission of e in level drop] also can give compositional information. Elastically scattered Bragg electrons give info about atomic spacing, crystal orientations, etc. Secondary electrons are low in energy and thus can’t escape from the interior of the material. They mostly give information about the surface topography. Loss of energy by electrons is characteristic of bulk composition. X-rays are produced by de-excited atoms.

Double Slit experiment for electrons

Double Slit experiment 2 Slits, 1 electron 2 Slits, arbitrary energy electrons 1 Right Slit, many electrons 2 Slits, tuned energy electrons 1 Left Slit, many electrons

What does it mean? A large number of electrons going through a double slit will produce an interference pattern, like a wave. However, each electron makes a single impact on a phosphorescent screen-like a particle. Electrons have indivisible (as far as we know) mass and electric charge, so if you suddenly closed one of the slits, you couldn’t chop the electron in half-because it clearly is a particle. A large number of electrons fired at two simultaneously open slits, however, will eventually, once you have enough statistics, form an intereference pattern. Their cumulative impact is wavelike. This leads us to believe that the behavior of electrons is governed by probabilistic laws. --The wavefunction describes the probability that an electron will be found in a particular location.

adding varying amounts of an infinite number of waves Given the Uncertainty Principle, how do you write an equation of motion for a particle? First, remember that a particle is only a particle sort of, and a wave sort of, and it’s not quite like anything you’ve encountered in classical physics. We need to use Fourier’s Theorem to represent the particle as the superposition of many waves. adding varying amounts of an infinite number of waves sinusoidal expression for harmonics wavefunction of the electron amplitude of wave with wavenumber k=2p/l

probability for an electron to be found between x and x+dx Y(x,t) We saw a hint of probabilistic behavior in the double slit experiment. Maybe that is a clue about how to describe the motion of a “particle” or “wavicle” or whatever. We can’t write a deterministic equation of motion as in Newtonian Mechanics, however, we know that a large number of events will behave in a statistically predictable way. dx probability for an electron to be found between x and x+dx Y(x,t) 2 Assuming this particle exists, at any given time it must be somewhere. This requirement can be expressed mathematically as: you will find your particle once, not twice (that would be two particles) but once. If you search from hither to yon

Wavefunction requirements

What is the form of this wavefunction? Let’s try a typical classical wavefunction: For a particle propagating in the +x direction. Similarly, for a particle propagating in the –x direction: We also know that if Y1 and Y2 are both allowed waves, then Y1+Y2 must also be allowed (this is called the superposition principle). Oops, the particle vanishes at integer multiples of p/2, 2p/3, etc. and we know our particle is somewhere.

Graphical representation of a complex number z as a point in the complex plane. The horizontal and vertical Cartesian components give the real and imaginary parts of z respectively.

Why do we express the wavefunction as a complex number instead of sines and cosines like classical waves?? Consider another typical classical wavefunction: Now imagine k -> -k and w -> -w Clearly, we need to express this in a different way. f x -f