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

Journal of Physics A Mathematical and General

On the radius of convergence of Rayleigh-Schrodinger perturbative solutions for quantum oscillators in circular and spherical boxes

Valdir C. Aguilera-Navarro, J. F. Gomes, Abraham Hirsz Zimerman, E. Ley‐Koo

Published Sep 11, 1983• 20 citations
Publisher

Abstract

The energy eigenvalues of harmonic oscillators in circular and spherical boxes are obtained through the Rayleigh-Schrodinger perturbative expansion, taking the free particle in a box as the non-perturbed system. The perturbative series is shown to be convergent for small boxes, and an upper bound for the radius of convergence is established. Pade-approximant solutions are also constructed for boxes of any size. Numerical comparison with the exact eigenvalues-which are obtained by constructing and diagonalising the Hamiltonian in the basis of the eigenfunctions of the free particle in a box-corroborates the accuracy and range of validity of the approximate solutions, particularly the convergence and the radius of convergence of the perturbative series.

DOI
10.1088/0305-4470/16/13/015

Related papers

Exact solutions for the D-dimensional spherical isotropic confined harmonic oscillator

Henry E Montgomery Jr, G. Campoy, N. Aquino

We study the size effect on the energy levels of the D-dimensional isotropic harmonic oscillator confined within a box of radius $r_c$ with impenetrable walls. Two different approaches are used to obtain the energy eigenvalues and eigenfunctions for D=1,2,...,5. In the first we solve the Schroedinger equation exactly. In the second we use a series expansion of the wave function. The numerical results obtained are extremely accurate; these values are reported with 50 decimal places.

arXiv:0803.4029•Mar 28, 2008•1 citation

The particle in a box with a perturbation

Francisco Marcelo Fernández

We apply perturbation theory to the particle in a box with a potential energy function and derive exact expressions for the perturbation corrections to the eigenfunctions and eigenvalues. We show closed-form results in terms of the quantum number for the linear potential and analyse the convergence properties of the perturbation series.

International Journal of Mathematical Education in Science and Technology•May 1, 1999•4 citations

Infinite-Order Perturbation Theory for Finite Systems

C. Alton Coulter

A perturbation theory is described which leads to two alternative methods for calculating the eigenvalues and normalized eigenstates of a perturbed Hamiltonian, the results being given to all orders in the perturbation. The expansions obtained are formulated in terms of two sets of partial sums which have simple, easily characterized forms, and they combine some of the advantages of both the Rayleigh-Schr\"odinger and Brillouin-Wigner perturbation methods while avoiding some of the difficulties of each. In particular, one form of the perturbation expansions given here will in most cases of interest have a larger radius of convergence than the Rayleigh-Schr\"odinger series. As is usual for infinite-order treatments, the explicit form of the final result applies only to nondegenerate levels. However, some of the intermediate relations obtained in the derivation are of general validity. Since certain classes of unlinked diagrams appear in the expansions, the method described here is not suitable for application to systems containing a large number of particles.

Physical Review•May 25, 1967•0 citations

New Developments in Time-Independent Quantum-Mechanical Perturbation Theory

Gerald I. Kerley

This report discusses two new ideas for using perturbation methods to solve the time-independent Schrödinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix diagonalization methods. That allows a simple, compact derivation of the standard (Rayleigh-Schrödinger) equations. But it also leads to a new iterative solution method that is not based on the usual power series expansion. The iterative method can also be used when the unperturbed system is two-fold degenerate. The second concept is quite different from the first but compatible with it. It is based on the fact that one can replace the true Hamiltonian with a synthetic Hamiltonian having the same eigenvalues. This approach allows one to cancel part of the perturbation and to reduce the size of the off-diagonal matrix elements, giving better convergence of the series. These methods are illustrated by application to three perturbed harmonic oscillator problems--a 1-D oscillator with a linear perturbation, a 1-D oscillator with a quartic perturbation, and a perturbed 2-D oscillator, which involves degenerate states. The iterative method and the synthetic Hamiltonian method are shown to give significantly better results than the standard method.

arXiv:1306.6912•Jun 24, 2013•0 citations

On approximating the free harmonic oscillator by a particle in a box

Kunle Adegoke, Adenike Olatinwo, Henry Otobrise +2 more

The main purpose of this paper is to demonstrate and illustrate, once again, the potency of the variational technique as an approximation procedure for the quantization of quantum mechanical systems. By choosing particle-in-a-box wavefunctions as trial wavefunctions, with the size of the box as the variation parameter, approximate eigenenergies and the corresponding eigenfunctions are obtained for the one dimensional free harmonic oscillator.

arXiv:1605.09374•May 29, 2016•0 citations

Energy eigenvalues of a bounded centrally located harmonic oscillator

Richard Vawter

In the study of the effects of finite boundaries on the magnetic properties of a solid, one encounters the problem of finding the energy eigenvalues of a one-dimensional harmonic oscillator located in a potential enclosure. Series expansion techniques are applied to solve this problem for a harmonic oscillator located at the center of an infinitely high potential well. An analytical expression for the energy eigenvalues is found as a function of the size potential enclosure L, the quantum state n, oscillator frequency ω, and the mass of the particle m. The first order approximation of this expression is given by E = Eosccoth(Eosc/Ebox) where Eosc = ℏ ω(n + 1/2) is the energy eigenvalues of an unbound harmonic oscillator and Ebox = (2m/ℏ2) (n + 1)2π2/L2 is the energy eigenvalues of a free particle in an infinitely high well. For the ground state this approximation is better than 1% for all values of L, ω, and m.

Journal of Mathematical Physics•Dec 1, 1973•25 citations
The energy eigenvalues of harmonic oscillators in circular and spherical boxes are obtained through the Rayleigh-Schrodinger perturbative expansion, taking the free particle in a box as the non-perturbed system. The perturbative series is shown to be convergent for small boxes, and an upper bound for the radius of convergence is established. Pade-approximant solutions are also constructed for boxes of any size. Numerical comparison with the exact eigenvalues-which are obtained by constructing and diagonalising the Hamiltonian in the basis of the eigenfunctions of the free particle in a box-corroborates the accuracy and range of validity of the approximate solutions, particularly the convergence and the radius of convergence of the perturbative series.
We study the size effect on the energy levels of the D-dimensional isotropic harmonic oscillator confined within a box of radius rcr_crc​ with impenetrable walls. Two different approaches are used to obtain the energy eigenvalues and eigenfunctions for D=1,2,...,5. In the first we solve the Schroedinger equation exactly. In the second we use a series expansion of the wave function. The numerical results obtained are extremely accurate; these values are reported with 50 decimal places.
A perturbation theory is described which leads to two alternative methods for calculating the eigenvalues and normalized eigenstates of a perturbed Hamiltonian, the results being given to all orders in the perturbation. The expansions obtained are formulated in terms of two sets of partial sums which have simple, easily characterized forms, and they combine some of the advantages of both the Rayleigh-Schr\"odinger and Brillouin-Wigner perturbation methods while avoiding some of the difficulties of each. In particular, one form of the perturbation expansions given here will in most cases of interest have a larger radius of convergence than the Rayleigh-Schr\"odinger series. As is usual for infinite-order treatments, the explicit form of the final result applies only to nondegenerate levels. However, some of the intermediate relations obtained in the derivation are of general validity. Since certain classes of unlinked diagrams appear in the expansions, the method described here is not suitable for application to systems containing a large number of particles.
This report discusses two new ideas for using perturbation methods to solve the time-independent Schrödinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix diagonalization methods. That allows a simple, compact derivation of the standard (Rayleigh-Schrödinger) equations. But it also leads to a new iterative solution method that is not based on the usual power series expansion. The iterative method can also be used when the unperturbed system is two-fold degenerate. The second concept is quite different from the first but compatible with it. It is based on the fact that one can replace the true Hamiltonian with a synthetic Hamiltonian having the same eigenvalues. This approach allows one to cancel part of the perturbation and to reduce the size of the off-diagonal matrix elements, giving better convergence of the series. These methods are illustrated by application to three perturbed harmonic oscillator problems--a 1-D oscillator with a linear perturbation, a 1-D oscillator with a quartic perturbation, and a perturbed 2-D oscillator, which involves degenerate states. The iterative method and the synthetic Hamiltonian method are shown to give significantly better results than the standard method.
In the study of the effects of finite boundaries on the magnetic properties of a solid, one encounters the problem of finding the energy eigenvalues of a one-dimensional harmonic oscillator located in a potential enclosure. Series expansion techniques are applied to solve this problem for a harmonic oscillator located at the center of an infinitely high potential well. An analytical expression for the energy eigenvalues is found as a function of the size potential enclosure L, the quantum state n, oscillator frequency ω, and the mass of the particle m. The first order approximation of this expression is given by E = Eosccoth(Eosc/Ebox) where Eosc = ℏ ω(n + 1/2) is the energy eigenvalues of an unbound harmonic oscillator and Ebox = (2m/ℏ2) (n + 1)2π2/L2 is the energy eigenvalues of a free particle in an infinitely high well. For the ground state this approximation is better than 1% for all values of L, ω, and m.