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Science China Mathematics

ANALYTIC SOLUTION TO THE SCHRDINGER EQUATION WITH A HARMONIC OSCILLATOR POTENTIAL PLUS δ-POTENTIAL

Qinglin Peng

Published Jan 1, 1991• 0 citations
OpenAlex

Abstract

The analytic solution to the Schrodinger equation with a harmonic oscillator potentialplus δ-potential is presented in the coordinate representation. Its algebraic expression in theenergy representation is also given. The degeneracy of the energy levels is discussed and acomparison between the energy spectra of a harmonic oscillator and a particle in a δ-poten-tial is made.

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The analytic solution to the Schrodinger equation with a harmonic oscillator potentialplus δ-potential is presented in the coordinate representation. Its algebraic expression in theenergy representation is also given. The degeneracy of the energy levels is discussed and acomparison between the energy spectra of a harmonic oscillator and a particle in a δ-poten-tial is made.
A general solution of the Schrödinger equation in the potential representation has been obtained in the form of integral equations. In this representation, the wave function for positive and negative energies or bound states can be expressed as a product of the unperturbed solution for model potential and the function which depends on the additional potential or potential perturbation. Here we have proved that this method is equivalent to the method of variation of constants for negative energies. The linearly independent solutions of Schrödinger equation for harmonic oscillator potential have been obtained for derivation of integral equations, which are used for finding eigenfunctions and eigenvalues for Woods–Saxon potential. Eigenvalues obtained by numerical iterations of these integral equations are in good agreement with results obtained by the discretization method. The kernels of the obtained integral equations are proportional to the perturbation or difference of Woods–Saxon and harmonic oscillator potentials.
We derive the energy levels associated with the even-parity wavefunctions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the boundary conditions at the origin. This problem calls the attention of the students to an inaccurate statement in quantum mechanics textbooks often found in the context of the solution of the harmonic oscillator problem.
The exact solutions of the Schrödinger equation with a harmonic oscillator plus an inverse square potential are obtained in two dimensions. We construct the ladder operators directly from the radial wave functions and find that these operators satisfy the commutation relations of an SU (1, 1) group. We obtain the explicit expressions of the matrix elements for some related functions ρ and [Formula: see text] with ρ = r2. We also explore another symmetry between the eigenvalues E(r) and E(ir) by substituting r→ir.