Use of the WKB Method for Obtaining Energy Eigenvalues

Abstract

A simple, general method is derived for evaluating the second- and third-order WKB energy integrals by rewriting the integrals having nonintegrable singularities in terms of derivatives, with respect to the energy, of integrals having integrable singularities. As an example, it is shown that the higher-order WKB integrals vanish for the one-dimensional linear harmonic oscillator. A calculation of some eigenvalues using this method is made for potentials of the form V(x)=λx2ν and the results are compared to the ``exact'' results obtained from a numerical integration of the Schrödinger equation. It is observed that inclusion of the third-order integral improves the accuracy of WKB eigenvalues.