Modified WKB approximation for phase shifts of an attractive singular potential

Abstract

The conventional WKB method for phase-shift calculations is known to fail for singular potentials at low incident energies. The modified WKB method can be applied successfully to singular potentials even at low energies since it is an extreme generalization of the conventional WKB method. As a special case, the modified WKB method can be reduced to the conventional WKB approximation and then agrees with the conventional WKB method in the high-energy region where the conventional WKB approximation is valid. Unlike the conventional WKB method, the modified WKB method can be applied to multi-turning-point problems without any difficulty. It is for these reasons that we present this method for determining the phase shifts of the attractive singular potential using a potential of the form $U(r)=\ensuremath{-}{g}^{2}{r}^{\ensuremath{-}4}$ as an example. We restrict ourselves to the low-energy, nonzero-orbital-angular-momentum case where there are three classical turning points in order to demonstrate the ability of this method to handle many turning points. The phase shifts obtained by this method agree with the numerical results.