Judah L. Schwartz

Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena

We describe the details involved in presenting the time development of one-dimensional quantum-mechanical systems in the form of computer-generated motion pictures intended for pedagogic purposes. Concentrating on reflection-transmission phenomena, we formulate the problem in terms of a Gaussian wave packet impinging on a square well or barrier and being reflected and transmitted. The wave equation is solved numerically by methods discussed in detail and photographs of the wave packet vs position at a variety of times and for a range of projectile energies are given.

Integration of the Schrödinger equation in imaginary time

Abstract Numerical techniques for the integration of the Schrodinger equation in imaginary time are investigated. Because the spatial dependence of the solution in the limit of large imaginary time is that of the ground state of the Hamiltonian, the method can be applied to bound states of quantum-mechanical three-body systems. Knowledge of the analytic dependence of the asymptotic form of the wavefunction on the trial eigenvalue is not required. In this paper, the first of a series, we have confined our attention to the problems arising from the imaginary time integration and reserve for a later date a discussion of the problems that are due to a multidimensional spatial grid.

Computer Generated Films for Solid-State Physics.

A series of computer-generated films is described, intended to serve as pedagogic aids in presenting ideas concerning the motion of electrons in periodic potentials. Three films have been produced which illustrate the following phenomena: (A) propagation of wave packets of several different mean energies through the ideal crystal lattice; (B) reflection and transmission of a wave packet incident on a crystal from free space, for mean energy values lying (1) in an allowed energy band of the crystal and (2) in a forbidden gap of the crystal; and (C) scattering of a wave packet by an impurity atom in the lattice.

A Quantum-Mechanical Ripple Tank

The partial wave expansion for a quantum-mechanical plane-wave scattering from a hard sphere is evaluated numerically, and various graphical presentations of the probability density are made.