It is shown how to extract the odds for getting different values of momentum from a generic wave function by writing it as a sum over functions of definite momentum. A recipe is given for finding states of definite energy, which requires solving a differential equation that depends on what potential the particle is experiencing. The particle in a box is considered and the allowed energies derived.
The fact that the wave function provides the complete description of a particle’s location and momentum is emphasized. Measurement collapses the wave function into a spike located at the measured value. The quantization of momentum for a particle on a ring is deduced.
Lecture begins with a detailed review of the double slit experiment with electrons. The fate of an electron traversing the double slit is determined by a wave putting an end to Newtonian mechanics. The momentum and position of an electron cannot both be totally known simultaneously. The wave function is used to describe a probability density function for an electron. Heuristic arguments are given for the wave function describing a particle of definite momentum.
The double slit experiment, which implies the end of Newtonian Mechanics, is described. The de Broglie relation between wavelength and momentum is deduced from experiment for photons and electrons. The photoelectric effect and Compton scattering, which provided experimental support for Einstein’s photon theory of light, are reviewed. The wave function is introduced along with the probability interpretation. The uncertainty principle is shown to arise from the fact that the particle’s location is determined by a wave and that waves diffract when passing a narrow opening.
Young’s double slit experiment shows clearly that light is a wave. (In order to observe the wave behavior of light, the slit size and separation should be comparable or smaller than the wavelength of light.) Interference is described using real and complex numbers (in anticipation of quantum mechanics). Grating and crystal diffraction are analyzed.
Ray diagrams are used to investigate the behavior of light incident on mirrors and lenses. The principle of least time is used to show that all rays from an object in front of a concave mirror focus on the image point if they are not too far from the axis. The experiments describing the breakdown of geometric optics are discussed.
Geometric optics is discussed as an approximation to wave theory when the wavelength is very small compared to other lengths in the problem (such as the size of openings). Many results of geometric optics involving reflection, refraction (mirrors and lenses) are derived in a unified way using Fermat’s Principle of Least Time.
The physical meaning of the components of the wave equation and their applications are discussed. The power carried by the wave is derived. The fact that, unlike Newton’s laws, Maxwell’s equations are already consistent with relativity is discussed. The existence of magnetism is deduced from a thought experiment using relativity.
Midterm Exam covers the first half of the course.
Waves on a string are reviewed and the general solution to the wave equation is described. Maxwell’s equations in their final form are written down and then considered in free space, away from charges and currents. It is shown how to verify that a given set of fields obeys Maxwell’s equations by considering them on infinitesimal cubes and loops. A simple form of the solutions is assumed and the parameters therein fitted using Maxwell’s equations.
