Quantum Mechanics Using Maple

Marko Horbatsch

Marko Horbatsch

Peter Reinecker, Michael Schulz, Beatrix M. Schulz

Yutaka Abe

This article discusses a set of Maple programs developed to reinforce understanding of basic concepts presented in an introductory quantum mechanics course. It was designed so that the solutions of the problems were analytically tractable as well as numerically obtainable. We note that there exist various approaches to a simple quantum mechanical problem. For example, for the problem of bound-states with a one-dimensional potential, the transfer matrix method, the Laplace transform method, and the Feynman path integral method can equally be applied. We believe that this type of program is extremely useful for igniting the student’s interest and widening his or her viewpoint. Several examples, such as the solution of various one-dimensional potential problems and the solution of the one-dimensional Schroedinger wave equation, are discussed in order to indicate how constructive reinforcement is established.

A. V. Zorin, N. P. Tretyakov

Abstract In the Maple computer algebra system, an algorithm is implemented for symbolic and numerical computations for finding the transition probabilities for hydrogen-like atoms in quantum mechanics with a nonnegative quantum distribution function (QDF). Quantum mechanics with a nonnegative QDF is equivalent to the standard theory of quantum measurements. However, the presence in it of a probabilistic quantum theory in the phase space gives additional possibilities for calculating and interpreting the results of quantum measurements. The methods of computer algebra seem to be necessary for the relevant calculations. The calculation of the matrix elements of operators is necessary for determining the energy levels, oscillator strengths, and radiation transition parameters for atoms and ions with an open shell. Transition probabilities are calculated and compared with experimental data. They are calculated using the Galerkin method with the Sturm functions of the hydrogen atom as coordinate functions. The verification of the model showed good agreement between the calculated and experimentally measured transition probabilities.