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Al-Fiziya Journal of Materials Science Geophysics Instrumentation and Theoretical Physics

Radial wave function of 2D and 3D quantum harmonic oscillator

Adam B. Cahaya

Published Jun 4, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.15408/fiziya.v5i2.26172

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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 purpose of this paper is two-fold. First, we would like to write down algebraic expression for the wave function of general excited state of harmonic oscillator which doesn't include derivative signs (this is to be contrasted with typical physics textbook which only gets rid of derivative signs for first few excited states, while leaving derivatives in when it comes to Hermite polynomial for general n). Secondly, we would like to write similar expression for two dimensional case as well. In the process of tackling two dimensions, we will highlight the interplay between Cartesian and polar coordinates in 2D in the context of an oscillator. All of the above mentioned results have probably been derived by others but unfortunately they are not easily available. The purpose of this paper is to make it easier for both students and general public to look up said results and their derivations, should the need arise. We also attempt to illustrate different angles from which one could look at the problem and this way encourage students to think more deeply about the material.
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.
The one-dimensional harmonic oscillator is one of the few models in quantum mechanics that can be solved analytically, and it is used to exam the theories and methods of quantum mechanics, possessing rich connotations. Using the Fourier transform and the generating function of Hermite polynomials, the transformation from the eigenfunction in the coordinate representation to that in the momentum representation for a one-dimensional harmonic oscillator has been achieved. The equivalence and connection between the two representations for describing a one-dimensional harmonic oscillator are verified. The eigenenergies and eigenfunctions of the one-dimensional harmonic oscillator in the coordinate representation and momentum representation are also obtained by using the algebraic solution method, respectively. The equivalence of the two representations is further discussed. The idea of psolving physical problems in higher dimensional spaceq is emphasized, which deepens students' understanding of quantum mechanics and stimulates their interest in learning.
The cylindrically and the spherically symmetric harmonic oscillators are considered by means of matrix techniques in a basis that is appropriate for each symmetry. For example, the spherically symmetric oscillator is described by the radial and angular momentum quantum numbers. The energy eigenvalues and the matrix elements of the promotion-demotion operators are found in each case. By expressing the operators in curvilinear coordinates, the promotion-demotion operators are used to deduce the form of the corresponding wave functions.
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.
The angular momentum states may be realized by the wave functions of two dimensional harmonic oscillators in polar coordinates. Using the generating function of these wave functions, Schwinger's generating function of Clebsh-Gordan in Fock space and Gauss integral, we determine the generating function of the couplings function of two angular momentums in coordinates representations. We deduce from the last function the connection of the wave function of the hydrogen atom and the four dimensional harmonic oscillator. We also find the expression of the relation between the spherical and parabolic wave functions of hydrogen atom. In addition, we find a new bilinear generating function involving the products of Laguerre polynomials.