The Oscillator in Quantum Mechanics

Abstract

One of the basic problems of nonrelativistic quantum mechanics is to find the energy spectrum and eigenfunctions of a microsystem described by the Schrödinger equation with an appropriate potential. Exact solutions of this equation are found [1]–[4] for a limited class of potentials such as the harmonic oscillator, the Coulomb potential, and some others. Most quantum systems are described by potentials for which the Schrödinger equation cannot be solved analytically. Thus, the solution of the Schrödinger equation with a sufficiently arbitrary potential represents the main mathematical task. With this aim many approximate analytical and numerical methods have been worked out. Great progress in the development of computer techniques and effective algorithms for the numerical solution of differential equations enables us to obtain numerical solutions for the energy spectrum and wave functions with quite a high accuracy, although practical calculations are usually very laborious and require powerful computers.