Part II deBroglie and his matter waves, and its consequences for physics and our concept of reality

an integer number of wavelengths fits into the circular orbit where l is the de Broglie wavelength

ParticleValue of l Electrons of kinetic energy Protons of kinetic energy Thermal neutrons (300K) Neutrons of kinetic energy He atoms at 300K 1 eV 100 eV eV 12.2 A 1.2 A 0.12A 1 keV 1 MeV 1 GeV A 28.6 F 0.73 F 1.5 A 9.0 F 0.75 A me, walking to the student union for lunch at 2 miles per hour

Electrons have a wavelength that is much shorter than visible light. The smallest detail that can be resolved is equal to one wavelength.

Human hair Red Blood Cells Table Salt

bulk (thick) foil (thin) Specimen interactions involved in forming an image Backscattered electrons can be used to identify elements in the material. Auger electrons also can give compositional information. 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. X-rays are produced by de-excited atoms. 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. Elastically scattered Bragg electrons give info about atomic spacing, crystal orientations, etc. Loss of energy by electrons is characteristic of bulk composition.

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.

Uncertainty on optics:

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. wavefunction of the electron amplitude of wave with wavenumber k=2 p / l adding varying amounts of an infinite number of waves sinusoidal expression for harmonics

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. dx probability for an electron to be found between x and x+dx Y( x,t ) 2 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. Assuming this particle exists, at any given time it must be somewhere. This requirement can be expressed mathematically as: If you search from hither to yon you will find your particle once, not twice (that would be two particles) but once.

Let’s try a typical classical wavefunction: 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). Similarly, for a particle propagating in the –x direction: For a particle propagating in the +x direction. Oops, the particle vanishes at integer multiples of p /2, 2 p /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 f -f x Clearly, we need to express this in a different way.