SWKB Approach to Confined Isospectral Potentials

Abstract

Recently we had formulated the supersymmetric Wentzel–Kramers–Brillouin (SWKB) quantization rule for one-dimensional confined quantum systems and applied the same to two trigonometric potentials, tangentially limited by infinite walls at x=0 and x=L, viz., V(x)=V ₀ cot²(πx/L) and the Pöschl–Teller potential, V(x)=V ₀₁ cosec²({πx/(2L))}+V ₀₂sec ²(πx/(2L)). Both the potentials have received quite a lot of attention by various authors because of their importance in molecular physics. Though these potentials have been studied in the framework of WKB, BS (Bohr–Sommerfeld), mBS (matrix formulation of BS) formalisms, it was observed that the supersymmetric approach not only rendered the calculations simpler and more transparent, it also reproduced the exact analytical energies in both the cases.

In this study, we shall generate isospectral Hamiltonians of the above potentials with the help of a modified form of Darboux's theorem. We shall show that though the new potentials look different from the original ones, and have different eigenfunctions, they too, are confined in the same region of space, and share the same energy spectrum as their original counterparts. This may be of substantial importance in determining the energy spectrum of highly non-trivial systems.