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American Journal of Physics

A general expression for the multipole moments of the quantum harmonic oscillator

Victor Namias

Published Nov 1, 1987• 5 citations
Publisher

Abstract

Multipole moments for the harmonic oscillator as well as for other familiar quantum systems can be obtained without knowledge of the wavefunction by means of simple recursion relations. The third-order differential equation satisfied by the probability density is first obtained and from it, the recursion relation for the moments is directly deduced. A general and presumably novel formula giving all the multipole moments is obtained by solving the finite difference equation generated by the recursion relation.

DOI
10.1119/1.14924

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Multipole moments for the harmonic oscillator as well as for other familiar quantum systems can be obtained without knowledge of the wavefunction by means of simple recursion relations. The third-order differential equation satisfied by the probability density is first obtained and from it, the recursion relation for the moments is directly deduced. A general and presumably novel formula giving all the multipole moments is obtained by solving the finite difference equation generated by the recursion relation.
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The Schrödinger equation is a significant achievement on the development of quantum mechanics. By solving the Schrödinger equation, the fundamental behaviors and properties of a microscopic particle can be found in one- to three-dimensions. The article focuses on the derivations of quantum harmonic oscillators in one dimension by solving second order-differential equation. The wave functions, probability densities under different energy levels are presented. The results can be used to estimate different forms of continuous potential experienced by an oscillator. By calculating the uncertainty relation of the oscillator under one specific excited state, the general relation can be confirmed. Based on the output, by splitting the variables, the two-dimensional harmonic oscillator can also be derived. The degenerate energy levels are presented as . Numerical simulations are made to visualize those results and suggests the relation of energy levels and number of maxima of the probability densities.
Abstract This paper advocates writing the eigenfunctions of the harmonic oscillator without Hermite polynomials, by means of the following recurrence relation: Ψ n = 2 α / n x Ψ n − 1 − ( n − 1 ) / n Ψ n − 2 ( n > 0 ) , with Ψ 0 = ( α / π ) 1 / 4 e − α x 2 / 2 . It is easy to prove that these functions are eigenfunctions of the harmonic oscillator. It is shown that many of the exercises related to the harmonic oscillator found in textbooks can be solved in a simple and elegant way by using this recurrence relation.
We consider properties of the ground state density for the [Formula: see text]-dimensional Fermi gas in an harmonic trap. Previous work has shown that the [Formula: see text]-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third-order linear differential equation, previously known in the literature but from other considerations. It is shown too how this implies a closed form expression for the [Formula: see text]th non-negative integer moments of the density, and a second-order recurrence. Both can be extended to general Re[Formula: see text]. The moments, and the smoothed density, permit expansions in [Formula: see text], where [Formula: see text], with [Formula: see text] denoting the shell label. The moment expansion substituted in the second-order recurrence gives a generalization of the Harer–Zagier recurrence, satisfied by the coefficients of the [Formula: see text] expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.
Abstract Moments are expectation values over wave functions (or averages over a set of classical particles) of products of powers of position and momentum. For the harmonic oscillator, the evolution in the quantum case is very closely related to that of the classical case. Here we consider the non-relativistic evolution of moments of all orders for the oscillator in one dimension and investigate invariant combinations of the moments. In particular, we find an infinite set of invariants that enable us to express the evolution of any moment in terms of sinusoids. We also find explicit expressions for the inverse of these relations, thus enabling the expression of the evolution of any moment in terms of the initial set of moments. More detailed attention is given to moments of the third and fourth order in terms of the invariant combinations.