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Chinese Physics

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

Chen Chang-Yuan

Published Oct 1, 2000• 10 citations
Publisher

Abstract

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.

DOI
10.1088/1009-1963/9/10/003

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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.
The isotropic harmonic oscillator is one of the potential fields with strict solution.There is important significance to discuss and study it in theory and application.In this paper the recurrence formula for radial matrix elements 〈n_rl｜r~k｜n′_rl′〉 of N-dimensional isotropic harmonic oscillator is derived and on the basis of it we obtain the recurrence formula for the expectation values.We discuss the common property of the recurrence formula for radial matrix elements,however,in Refs[3] and [4] the recurrence formula for radial matrix elements are only the examples of the result in this paper.
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.