SWKB Quantization Rules for Bound States in Quantum Wells

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., V (x) = V0 cot 2 π x L , and the famous Pöschl-Teller potential, V (x) = V01 cosec 2 π x 2L + V02 sec 2 π x 2L , 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. ∗e-mail:[email protected] †e-mail:[email protected] Introduction : The study of confined quantum systems is of considerable importance in modern times as spatial confinement significantly alters the physical and chemical properties of the system [1-4]. It influences the bond formation and chemical reactivity inside the cavities to a great extent. Even the optical properties (absorption and emission of light in the visible or far infra-red range, Raman scattering) and electrical properties (capacitance and transport studies) change radically. Hence this branch of Science is extremely useful in the study of thermodynamic properties of non-ideal gases, investigation of atomic effects in solids, in atoms and molecules under high pressure, impurity binding energy in quantum wells, and even in the context of partially ionised plasmas. Various authors have empployed different techniques to study such systems. Fairly recently, M A F Gomes and S K Adhikari [5] have suggested a matrix formulation of the Bohr-Sommerfeld ( mBS) quantization rule to give an estimate of the eigen energies of the Schrödinger equation, for various one-dimensional quantum wells. They have compared the energies thus obtained with those by Wentzel-Kramers-Brillouin (WKB) and usual BohrSommerfeld ( BS) methods, as well as the exact analytical solution of the Schrödinger equation. They observed that in many cases the mBS quantization rule yields more precise energies than the WKB or BS quantization rules. For small n particularly, the WKB approximation gives the poorest estimates. In this short comment, we study spatial confinement in the framework of SWKB (supersymmetric version of WKB) approximation. The motivation for the SWKB approach arises from the fact that this gives exact results in case of shape-invariant potentials. In this work, we deal with two trigonometric potentials, discussed in ref. [5], viz., V (x) = V0 cot 2 π x L (1) and the famous Pöschl-Teller potential, V (x) = V01 cosec 2 π x 2L + V02 sec 2 π x 2L (2) Both the potentials are tangentially limited by infinite walls at x = 0 and x = L, ( L being the dimension of the confining box ) and are of tremendous