In the framework of the recently proposed supersymmetric WKB (SWKB) approximation scheme, we obtain an explicit expression for the quantization condition which contains all terms up to order ${ħ}^6$. For spherically symmetric potentials, we find that the SWKB approach automatically yields wave functions with the correct threshold behavior. This is in contrast to the usual WKB scheme, where proper r→0 behavior necessitates the use of cumbersome ‘‘Langer corrections.’’ Previous authors have shown that the leading-order (${ħ}^0$) SWKB quantization integral gives exact bound-state spectra for analytically solvable shape-invariant potentials. For these cases, we show that the higher-order correction terms vanish identically. Finally, for nonanalytically solvable potentials, a comparison of our results (comprising of higher-order corrections) with numerically determined eigenvalues reveals very good accuracy.

• Received 28 December 1987

DOI:https://doi.org/10.1103/PhysRevA.38.1679