Higher-order JWKB approximations for radial problems. I. Modification of the effective potential

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Abstract
In the context of higher-order JWKB approximations for radial problems, the need for modifying the strength of the centrifugal barrier is considered. For spherically symmetric potentials V(r) satisfying the condition r2V(r) to 0 as r to 0, it is shown how to determine the modification required in an arbitrary order n that will ensure that the nth-order JWKB wavefunction has the correct behaviour ( approximately rl+1) near the origin. The second-order modification of Beckel and Nakhleh (1963) is a special case of the proposed nth-order modification, as are those of Froman and Froman (1974). It is demonstrated that, with the correct modification, the JWKB series truncated at any order n leads to the exact energy spectrum for both the harmonic oscillator and the Coloumb potentials.