A Simple Hartree SCF Calculation on a One-Dimensional Model of the He Atom

Abstract

It has been noted that it is difficult for undergraduates to come to grips with the fundamentals of quantum theory. Undoubtedly, this difficulty is related to the fact that there are not very many examples of meaningful quantum calculations that can be performed by undergraduates. The use of a computer in undergraduate physics courses opens the door to a large number of instructive problems which were once precluded due to the necessity of numerical analysis. Once such example for elementary quantum mechanics is the problem of a bound system of more than two particles which may be solved by the method of the self-consistent field(SCF). In this case, am important technique for teaching introductory wave mechanics is provided by the numerical calculation of stationary state solution of the Schrodinger equation. Realistic problems in wave mechanics that do not require sophisticated computer techniques are very valuable for the instruction of undergraduates. In the central-field model for atoms, the calculation of electron states reduces to the solution of a one-dimensional differential equation whose numerical solutions are no more difficult to obtain than those for a finite square well. A method used by the student is discussed in this work for the calculation of the electron wave functions in a central-field potential. By working this problem, it is obtained meaningful results that give them insight into the theory of atomic structure. Results of this experience are included, and possibilities of extending this example are discussed.