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

The Journal of Chemical Physics

Matrix Elements of General Potentials in the Harmonic-Oscillator Representation

Ralph M. Wilcox

Published Nov 1, 1966• 23 citations
Publisher

Abstract

The m, n matrix element of an arbitrary potential function V(q) in the one-dimensional harmonic-oscillator representation is shown to be given by 〈m | V(q) | n〉= ∑ r=0mα(m!n!)12r!(m−r)!(n−r)! ∫ −∞∞dyg(αy)e−12y2(iy)m+n−2r,where α≡(2ω/ℏ)½, and g(αy) is the Fourier transform of V(q). This formula is specialized to the cases where V(q) is given by qj, qje−½γq2, ejαzq, and q−1 sin(αλq), where j is a nonnegative integer and γ, z, and λ are real parameters. Results are compared, where possible, with previous work.

DOI
10.1063/1.1728107

Related papers

Matrix elements in harmonic oscillator representations I. General solution and application to one dimension

Stephen G. Bell

General expressions have been obtained for the matrix elements of linear or radial perturbations in the representations of the one-, two-, and three-dimensional harmonic oscillators. The solutions are also general for the two types of matrix element where the two wave functions belong to the same basis set or to two different basis sets. In this paper these expressions are applied to one- dimensional problems.

Journal of Physics B Atomic and Molecular Physics•Jun 1, 1970•13 citations

Matrix Elements for the one-dimensional Harmonic Oscillator

José Luis López-Bonilla, Gerardo Ovando

The properties of the Hermite polynomials are used to obtain the matrix elements m|e -γx |n for the harmonic oscillator in one dimension in an easy way.

Irish Mathematical Society Bulletin•Jan 1, 2000•6 citations

General formula and recurrence formula for radial matrix elements ofN-dimensional isotropic harmonic oscillator

Chen Chang-Yuan

In this paper, the general formulas and the recurrence formulas for radial matrix elements of N -dimensional isotropic harmonic oscillator are obtained. The relevant results of 2- dimensional and 3- dimensional isotropic harmonic oscillators reported in the reference papers are contained in a more general equations derived in this paper as special cases.

Chinese Physics•Oct 1, 2000•10 citations

Transformation properties of wave functions of a two-dimensional harmonic oscillator

I. E. Mazets

A transformation formula for wave functions of type $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}$$ , where ψn is the eigenfunction of a harmonic oscillator for the nth energy level, was found for an arbitrary rotation of the (x 1, x 2) Cartesian system. By way of example, matrix elements for contact interaction of two particles in an external harmonic potential field were calculated.

Technical Physics•Feb 1, 2001•0 citations

Matrix elements of X xk and e αX x k in the harmonic oscillator basis

Włodzisław Duch

General and compact formulae for matrix elements of x k and X k X k e αX are derived using the second quantisation technique.

Journal of Physics A Mathematical and General•Dec 21, 1983•15 citations

Multidimensional harmonic oscillator matrix elements

Francisco Marcelo Fernández, Richard H. Tipping

Multidimensional harmonic oscillator matrix elements of arbitrary products of exponentials for operators containing only linear and quadratic terms in the creation and annihilation operators are obtained. A simple and straightforward method is presented that leads to recurrence relations and generating functions for such matrix elements. Results are shown to be general enough to cover the cases of Franck–Condon overlaps, transition probabilities for time-dependent harmonic oscillators, and correlation functions for electronic spectra.

The Journal of Chemical Physics•Nov 1, 1989•6 citations
The m, n matrix element of an arbitrary potential function V(q) in the one-dimensional harmonic-oscillator representation is shown to be given by 〈m | V(q) | n〉= ∑ r=0mα(m!n!)12r!(m−r)!(n−r)! ∫ −∞∞dyg(αy)e−12y2(iy)m+n−2r,where α≡(2ω/ℏ)½, and g(αy) is the Fourier transform of V(q). This formula is specialized to the cases where V(q) is given by qj, qje−½γq2, ejαzq, and q−1 sin(αλq), where j is a nonnegative integer and γ, z, and λ are real parameters. Results are compared, where possible, with previous work.
Multidimensional harmonic oscillator matrix elements of arbitrary products of exponentials for operators containing only linear and quadratic terms in the creation and annihilation operators are obtained. A simple and straightforward method is presented that leads to recurrence relations and generating functions for such matrix elements. Results are shown to be general enough to cover the cases of Franck–Condon overlaps, transition probabilities for time-dependent harmonic oscillators, and correlation functions for electronic spectra.