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Journal of Physics Conference Series

Quantum properties of two-dimensional linear harmonic oscillator in polar coordinate system

Kaiqiang Xie, Xingrong Zheng, Jingtong Chen, Yujie Li

Published May 1, 2021• 1 citation
Publisher

Abstract

Abstract Using quantum theory and MATLAB software, the basic properties of two-dimensional linear harmonic oscillators in quantum mechanics are systematically studied in polar coordinate system, and obtain the visualized results. The results show that, in polar coordinate system, with the exception of special case nr =0, |m|=0, the degeneracy of two-dimensional linear harmonic oscillator is 2nr +|m|+1, and the corresponding energy eigenvalues is ħω (2nr +|m|+1). The number of intersection line between wave function and the plane with ψ=0 is 2nr +m. In the case of nr =0, the maximum number of probability density distributions is 2|m|. The results of this visualization are in complete agreement with the theoretical results. The visualization results in different coordinate systems can be verified with each other, which opens up a new research idea and also provides an idea for other quantum theoretical models to be studied.

DOI
10.1088/1742-6596/1906/1/012035

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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.