Solution of Schrödinger Equation for a Particle Bound to More Than One Spherical Potential

Abstract

We consider the quantum-mechanical problem of a particle bound to a configuration of spherical potentials, each of finite range. If the Schrödinger equation can be solved for each potential by itself, then it is shown how to solve it for the configuration, provided the potentials do not overlap. The energy levels are the zeros of a determinant of formally infinite order, but in practice this is always well approximated by a finite determinant and often by one of small order. As illustrative examples we consider some states of a particle bound to three square wells with the configuration of an equilateral triangle, and to two truncated Coulomb potentials. The possibility of extending an approximate version of the method to overlapping potentials is pointed out.