Higher-order WKB approximations in supersymmetric quantum mechanics

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.