SWKB quantization rules for bound states in quantum wells

Abstract

In a recent paper by M A F Gomes and S K Adhikari(J.Phys.(B30) ,5987.(1997)), a matrix formulation of the Bohr-Sommerfeld (mBS) quantization rule has been applied to the study of bound states in one-dimensional quantum wells.They have observed that the usual Bohr-Sommerfeld (BS) and the WentzelKramers-Brillouin (WKB) quantization rules give poor estimates of the eigen energies of the two confined trigonometric potentials, viz.,, and the famous Pöschl-Teller potential,, the WKB approach being worse of the two, particularly for small values of n. They suggested a matrix formulation of the Bohr-Sommerfeld method (mBS). Though this technique improves the earlier results, it is not very accurate either. Here we study these potentials in the framework of supersymmetric Wentzel-Kramers-Brillouin (SWKB) approximation, and find that the SWKB quantization rule is superior to each one of the BS, mBS, and WKB approximations, as it reproduces the exact analytical results for the eigen energies. Its added advantage is that it gives the correct analytical ground state wave functions as well. *

Reply to comment on `Quantization rules for bound states in quantum wells'

“…In a comment to our work on the mBS quantization rule [5], Sinha and Roychoudhury [6] consider the supersymmetric version of the WKB (SWKB) quantization rule and apply it to the study of the Pöschl-Teller potential. For this potential, this approach yields exact analytic results for the wavefunction and eigenenergies.…”