Nonuniqueness of the Energy Correction in Application of the WKB Approximation to Radial Problems

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Abstract
An infinite set of transformations is given that convert a radial energy‐eigenvalue problem into a 1‐dimensional problem. Langer's transformations are included in this set. The transformations introduce no spurious, real‐axis singularities into the effective potential, and yield wavefunctions that meet 1‐dimensional boundary conditions and are asymptotically satisfactory for virtually all power law potentials. The first‐order radial WKB quantization condition is transformation dependent through two parameters. It may be distinct from the Langer‐Kemble condition obtained by substituting (l + ½)2 for l(l + 1) in the effective potential. Here, l is the rotational quantum number. For the vibrating rotator, a rough test of the usefulness of the (l + ½)2 substitution is described; flexibility advantages provided by the new transformations are pointed out.

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