Numerical Methods for the Eigenvalue Determination of Central-Force-Field Problems in Quantum Mechanics

Abstract

An accurate method for the numerical solution of the eigenvalue problem of the second-order ordinary differential equation for the central-Force-Field problem in quantum mechanics is presented. Firstly, initial values for the eigenvalue and eigenfunction are obtained by using the discretized matrix eigenvalue method. Secondly, the eigenvalue and eigenfunction are solved by using the shooting method. Highly accurate solutions around zero are obtained by using the formal solution of power series expansion. Similarly, highly accurate solutions around infinity are obtained using asymptotic series expansion. These formal solutions are used for the initial value or guess for the shooting method. The initial value problem is solved highly accurately by using the higher-order linear multistep method based on the method of constructing the optimal operators. The eigenvalue is properly corrected by using Ridley’s formula and highly accurate numerical differentiation, integration, and a suitable choice of matching point. The efficiency of the present method is demonstrated by its application to bound states for the Coulomb potential, the Hulthén potential, the Yukawa potential and the Hellmann potential in the central-force-field problems.