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Technical Physics

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

I. E. Mazets

Published Feb 1, 2001• 0 citations
Publisher

Abstract

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.

DOI
10.1134/1.1349268

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A transformation formula for wave functions of type l\lower0.5em⌣\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}l\lower0.5em⌣ , 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.
In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.
One dimensional quantum harmonic oscillator is well studied in elementary textbooks of quantum mechanics. The wave function of one-dimensional oscillator harmonic can be written in term of Hermite polynomial. Due to the symmetry of the spring energy, the wave functions of two-dimensional and three-dimensional harmonic oscillators can be written as products of the one-dimensional case. Because of that, the wave functions of two- and three-dimensional cases are focused on cartesian coordinates. In this article, we utilize polar and spherical coordinates to describe the wave function of two- and three-dimensional harmonic oscillators, respectively. The radial part of the wave functions can be written in term of associated Laguerre polynomials.
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.
In variational calculations with harmonic oscillator wavefunctions as trial bases, the transformation coefficients that relate harmonic oscillator wavefunctions in two different sets of internal coordinates are convenient to the evaluation of some matrix elements. Here, we present the explicit expression of these transformation coefficients for five-body systems. These transformation coefficients can be collected in a matrix according to the quantum number N of harmonic oscillator shell and can be programmed for arbitrary N.