Bound State Solutions of the Schrödinger Equation

Charles G. Wohl

H. T. Williams

I present a method in the spirit of Green’s functions for generating bound state solutions to the Schrödinger equation. The method is demonstrated by generating ground state and excited state solutions to the one-dimensional, time-independent Schrödinger equation. Its applicability to the radial equation for two and three dimensions is also discussed.

Roman Ya. Kezerashvili, Luo, Jianning, Claudio R. Malvino

Exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schrödinger equation is applied for a solution of a two--dimensional (2D) problem for two particles interacting via Kratzer, and modified Kratzer potentials. We found the exact bound state solutions of the two--dimensional Schrödinger equation with Kratzer--type potentials and present analytical expressions for the eigenvalues and eigenfunctions. The eigenfunctions are given in terms of the associated Laguerre polynomials.

M. A. F. Gomes, Sadhan K. Adhikari

An algebraic reformulation of the Bohr - Sommerfeld (BS) quantization rule is suggested and applied to the study of bound states in one-dimensional quantum wells. The energies obtained with the present quantization rule are compared to those obtained with the usual BS and WKB quantization rules and with the exact solution of the Schrödinger equation. We find that, in diverse cases of physical interest in molecular physics, the present quantization rule not only yields a good approximation to the exact solution of the Schrödinger equation, but yields more precise energies than those obtained with the usual BS and/or WKB quantization rules. Among the examples considered numerically are the Poeschl - Teller potential and several anharmonic oscillator potentials, which simulate molecular vibrational spectra and the problem of an isolated quantum well structure subject to an external electric field.