Four-term WKBJ approximation and its application to Varshni V and Lennard-Jones potentials

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Abstract
An expression is derived for the fourth non-zero term in the WKBJ quantization condition. The four-term WKBJ approximation is applied to the Varshni V potential for obtaining the eigenvalues, and the relevant integrals are analytically evaluated. Numerical results are obtained for the Lennard-Jones potential, which is a special case of the Varshni V potential, and are compared to the quantum mechanical eigenvalues obtained by Kirschner and Le Roy for η = 50. It is shown that the inclusion of the fourth term leads to an improvement in the eigenvalues over the three-term WKBJ eigenvalues. The results also indicate that for calculating extremely precise eigenvalues for any given potential, a high-order WKBJ quantization condition provides a good alternative to direct numerical quantum mechanical calculations.

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