An Expansion Method for Schrödinger Equation of Quantum Billiards with Arbitrary Shapes

Abstract

A numerical method for solving the time-independent Schrodinger equation of a particle moving freely in a three-dimen- sional axisymmetric region is developed. The boundary of the region is defined by an arbitrary analytic function. The method uses a coordinate transformation and an expansion in eigenfunctions. The effectiveness is checked and confirmed by applying the method to a particular example, which is a prolate spheroid. HE most frequently occurring equation in quantum me- Tchanics is the stationary Schrodinger equation, which is an ordinary differential equation(ODE) in the case of a one dimensional problem and a partial differential equation(PDE) if the corresponding physical system is higher dimensional. Unfortunately, the analytical solvability of this equation, even in one dimension is restricted to a few classes of potentials. Therefore, the use of numerical methods for the relevant problem gains a lot of significance. The stationary Schrodinger equation for a particle moving freely inside a closed region,