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Journal of Physics B Atomic and Molecular Physics

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

Stephen G. Bell

Published Jun 1, 1970• 13 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/3/6/001

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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.
A quantum state corresponds to a specific wave function. We adopt a new mathematical method [1] to improve Dirac's ladder operator method. A set of orthonormal wave functions will be used to associate the operator with the square matrix corresponding to it. These allow us to determine the matrix elements by using the operator relations without having to know the specific wave functions. As a result, we can get the direct results of matrix mechanics and wave mechanics on one-dimensional Harmonic oscillator and their descriptions will be also unified.
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.
For a class of problems it is shown that a solution to Newton's equation of motion may be interpreted as the generating function for the matrix elements of the coordinate operators in the Heisenberg representation. For three one-dimensional problems, the harmonic oscillator, the particle in a box, and the x4{x}^{4}x4 anharmonic oscillator, solutions to the quantum-mechanical problem are given in terms of the classical solution. A brief examination is made of a two-dimensional problem in which the angular momentum is conserved.