Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

The one-dimensional harmonic oscillator in the presence of a dipole-like interaction

Guillermo Palma, Ulrich Raff

Published Feb 13, 2003• 28 citations
Publisher

Abstract

The Schrödinger equation for a one-dimensional harmonic oscillator is solved in the presence of an external dipole field. The solution is obtained in closed form and resembles the solution of the three-dimensional harmonic oscillator in spherical coordinates. Due to the existence of a singularity at the origin and the absence of tunneling, this one-dimensional system has a twofold degeneracy. Therefore, the nondegeneracy of the eigenvalues breaks down as has been observed in other systems.

DOI
10.1119/1.1526131

Related papers

The singular harmonic oscillator revisited

Douglas R.M. Pimentel, Antonio S. de Castro

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.

arXiv:1304.0492•Apr 1, 2013•2 citations

Solutions of Schrödinger’s equation in one dimension

A.P. French, Edwin F. Taylor

This chapter examines some explicitly one-dimensional systems, including the finite square well, the harmonic oscillator, and the application of harmonic oscillator theory to molecular vibrations. It analyzes the computer solutions of the Erwin Schrodinger equation. Although there are no real one-dimensional square wells in nature, many real systems have potentials that resemble the square well. Near the center of force the different wave functions must differ from one another or else they would not satisfy the Schrodinger equation for different values of energy. In contemporary research almost all the manipulation of the Schrodinger equation is done not analytically but rather by computer using numerical methods. A major advantage of the dimensionless form of the Schrodinger equation is that a single solution for one kind of system is trivially easy to extend to all similar systems. The dimensionless form of the Schrödinger equation can be solved numerically if it is approximated by a so-called difference equation.

Routledge eBooks•May 11, 2018•0 citations

Wave Mechanics in Quantum Phase Space: Harmonic Oscillator

Jun Lu

In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.

Physica Scripta•Jan 1, 2004•4 citations

Harmonic oscillator Green functions

D. B. Khrebtukov, J. H. Macek

The Green function for the Schrödinger equation with an isotropic, three-dimensional harmonic-oscillator potential is given in closed form. A similar closed form is obtained when the Schrödinger equation also contains a magnetic interaction and the magnetic field is such that the precession and oscillation frequencies are equal. The latter Green function is used to obtain energy and Sturmian eigenvalues that occur in the theory of atom-atom collisions.

Journal of Physics A Mathematical and General•Mar 27, 1998•10 citations

Classical and quantum description of the two-dimensional simple harmonic oscillator in elliptic coordinates

A. J. Fendrik, M. Bernath

The Schr\"odinger equation for the two-dimensional simple harmonic oscillator is solved using elliptic coordinates where it is separable. The separability of the problem in such coordinates is independent of the selection of the focal distance. The solutions are labeled by the total number of quanta N and by a set of characteristic values b corresponding to the eigenvalues of an observable B^, which does not commute with L^, the total angular momentum or H${^}_{x}$, the energy associated with the x degree of freedom. The well-known quantum energies as well as the characteristic values are obtained by imposing physical polynomial solutions.

Physical Review A•Oct 1, 1989•5 citations

Exact solutions of the spherically symmetric multidimensional isotropic harmonic oscillator

K. J. Oyewumi, Amos Wale Ogunsula

The complete orthonormalised energy eigenfunctions and the energy eigenvalues of the spherically symmetric isotropic harmonic oscillator in N dimensions, are obtained through the methods of separation of variables. Also, the degeneracy of the energy levels are examined. KEY WORDS: - Schrödinger Equation, Isotropic harmonic oscillator, Hyperspherical harmonics, Multidimensional space. Global Jnl Pure & Applied Sciences Vol.10(2) 2004: 343-349

Global Journal of Pure and Applied Sciences•Jun 2, 2004•2 citations
The Schrödinger equation for a one-dimensional harmonic oscillator is solved in the presence of an external dipole field. The solution is obtained in closed form and resembles the solution of the three-dimensional harmonic oscillator in spherical coordinates. Due to the existence of a singularity at the origin and the absence of tunneling, this one-dimensional system has a twofold degeneracy. Therefore, the nondegeneracy of the eigenvalues breaks down as has been observed in other systems.
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.
This chapter examines some explicitly one-dimensional systems, including the finite square well, the harmonic oscillator, and the application of harmonic oscillator theory to molecular vibrations. It analyzes the computer solutions of the Erwin Schrodinger equation. Although there are no real one-dimensional square wells in nature, many real systems have potentials that resemble the square well. Near the center of force the different wave functions must differ from one another or else they would not satisfy the Schrodinger equation for different values of energy. In contemporary research almost all the manipulation of the Schrodinger equation is done not analytically but rather by computer using numerical methods. A major advantage of the dimensionless form of the Schrodinger equation is that a single solution for one kind of system is trivially easy to extend to all similar systems. The dimensionless form of the Schrödinger equation can be solved numerically if it is approximated by a so-called difference equation.
In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.
The Schr\"odinger equation for the two-dimensional simple harmonic oscillator is solved using elliptic coordinates where it is separable. The separability of the problem in such coordinates is independent of the selection of the focal distance. The solutions are labeled by the total number of quanta N and by a set of characteristic values b corresponding to the eigenvalues of an observable B^, which does not commute with L^, the total angular momentum or H${^}_{x}$, the energy associated with the x degree of freedom. The well-known quantum energies as well as the characteristic values are obtained by imposing physical polynomial solutions.