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Physics Letters A

Exact and approximate solutions to the Schrödinger equation for the harmonic oscillator with a singular perturbation

Francisco Marcelo Fernández

Published Dec 1, 1991• 29 citations
Publisher
DOI
10.1016/0375-9601(91)91058-l

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In the previous chapters, we have shown how to solve the Schrodinger equation for several systems as a particle in an infinite and finite potential well, scattering, harmonic oscillator, and hydrogen atom. For all these systems, we were able to solve Schrodinger's equation exactly. However, there are complicated potentials for which Schrodinger's equation cannot be solved exactly. For such situations, we resort to approximation methods for finding approximate solutions to the Schrodinger equation. In this chapter, we discuss several approximate methods for solving Schrodinger's equation.
The one-dimensional Schrödinger equation with the singular harmonic oscillator is investigated. The Hermiticity of the operators related to observable physical quantities is used as a criterion to show that the attractive or repulsive singular oscillator exhibits an infinite number of acceptable solutions provided the parameter responsible for the singularity is greater than a certain critical value, in disagreement with the literature. The problem for the whole line exhibits a two-fold degeneracy in the case of the singular oscillator, and the intrusion of additional solutions in the case of a nonsingular oscillator. Additionally, it is shown that the solution of the singular oscillator can not be obtained from the nonsingular oscillator via perturbation theory.