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Electronic journal of theoretical physics

Rayleigh process and matrix elements for the one-dimensional harmonic oscillator

J. H. Caltenco, José Luis López-Bonilla, J. Morales, Escuela Superior de Ingenier

Published Jun 1, 2006• 3 citations
OpenAlex

Abstract

Abstract: We show that, the matrix elements 〈m |e−γ x |n 〉 for the one-dimensional harmonic oscillator have application in Markov process theory, permitting thus to resolve the Fokker-Planck equation for the two-dimensional probability density corresponding to Rayleigh case. c © Electronic Journal of Theoretical Physics. All rights reserved.

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Abstract: We show that, the matrix elements 〈m |e−γ x |n 〉 for the one-dimensional harmonic oscillator have application in Markov process theory, permitting thus to resolve the Fokker-Planck equation for the two-dimensional probability density corresponding to Rayleigh case. c © Electronic Journal of Theoretical Physics. All rights reserved.
Abstract In both classical and quantum physics the harmonic oscillator is one of the most important physical systems [1–4]. Furthermore, because of the simplicity and symmetry of the Hamiltonian operator, the quantum mechanical oscillator is particularly suited to analysis by abstract operator methods. For this reason the study of the one dimensional oscillator is a useful introduction to the application of operator methods in quantum mechanics. In Sections 2.1–2.4 we use these methods to determine energy eigenvalues and eigenkets, and matrix elements; the analysis is extended to isotropic multidimensional oscillators in Section 2.10. Quasiclassical states of the oscillator are discussed in Section 2.5. In Section 2.6 shift operators are used to calculate wave functions. In Sections 2.7–2.11 we discuss some applications of the theory of the oscillator.
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.
In 1959, March and Young ( Nucl. Phys. 12 237) rewrote the equation of motion for the Dirac density matrix γ( x , x 0 ) in terms of sum and difference variables. Here, γ( , 0 ) for the d -dimensional isotropic harmonic oscillator for an arbitrary number of closed shells is shown to satisfy, using the variables ∣ + 0 ∣/2 and ∣ − 0 ∣/2, a generalized partial differential equation embracing the March–Young equation for d = 1. As applications, we take in turn the cases d = 1, 2, 3 and 4, and obtain both the density matrix γ( , 0 ) and the diagonal density ϱ( r ) = γ( , 0 )∣ 0 = , this diagonal element already being known to satisfy a third-order linear homogeneous differential equation for d = 1 through 3. Some comments are finally made on the d -dimensional kinetic energy density, which is important for first-principles density functional theory in allowing one to bypass one-particle Schrödinger equations (the so-called Slater–Kohn–Sham equations).
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.