COMMENT: SWKB quantization rules for bound states in quantum wells

In a recent paper by Gomes and Adhikari (1997 J. Phys. B: At. Mol. Opt. Phys. 30 5987), a matrix formulation of the Bohr-Sommerfeld (mBS) quantization rule was applied to the study of bound states in one-dimensional quantum wells. They observed that the usual Bohr-Sommerfeld (BS) and the Wentzel-Kramers-Brillouin (WKB) quantization rules give poor estimates of the eigenenergies of the two confined trigonometric potentials, namely, V (x ) = V ₀ cot² (icons/Journals/Common/pi" ALT="pi" ALIGN="TOP"/> x /L ), and the famous Pöschl-Teller potential V (x ) = V ₀₁ cosec² (icons/Journals/Common/pi" ALT="pi" ALIGN="TOP"/> x /2L )+V ₀₂ sec² (icons/Journals/Common/pi" ALT="pi" ALIGN="TOP"/> x /2L ), the WKB approach being the worse of the two, particularly for small values of n . They suggested a matrix formulation of the Bohr-Sommerfeld method. Though this technique improves the earlier results, it is not very accurate either at least for the potentials discussed in their paper. Here we suggest that the supersymmetric Wentzel-Kramers-Brillouin (SWKB) quantization may be another formalism which can be applied for these and similar potentials. Supersymmetry makes the calculations simpler and more transparent. Its added advantage is that it also gives the correct analytical ground state wavefunctions.