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Journal of Qinghai Normal University

Energy and wave function of the three-dimensional harmonic oscillator and hydrogen atom

Zhao Su-qin

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

Through the counterchanges of two methodologies of coordinates and special equations,the eigen equations of the three-dimensional harmonic oscillator and hydrogen atom are converted into the same equations in form.Then the relationship between the energy and wave function of the three-dimensional harmonic oscillator and hydrogen atom is given.

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Physica Scripta•Jan 1, 2004•4 citations
Through the counterchanges of two methodologies of coordinates and special equations,the eigen equations of the three-dimensional harmonic oscillator and hydrogen atom are converted into the same equations in form.Then the relationship between the energy and wave function of the three-dimensional harmonic oscillator and hydrogen atom is given.
In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.
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.