Application of the Second-Order WBK Approximation to Radial Problems

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Abstract
When the WBK approximation is applied to radial problems, difficulties are encountered at the origin. Previous work has led to the conclusion that the difficulties are removed by substitution of (J+½)2 for J(J+1) in the effective potential. The use of (J+½)2 is here shown to be justified for the first‐order WBK approximation only. The second‐order WBK wavefunction is found to approximate the exact one as r→0, if one substitutes K for J(J+1), where K obeys the equation K+1/(64K)=J(J+1). Moreover, the second‐order WBK energy levels can be made to coincide with the exact levels for the hydrogen atom and for the radial harmonic oscillator if the same substitution is made. Implications with regard to the rotating vibrator problem are discussed.

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