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IOP Conference Series Earth and Environmental Science

Study on two - dimensional linear harmonic oscillator characteristics based on MATLAB software

Xijin Fu

Published Jul 1, 2019• 1 citation
Publisher

Abstract

Abstract Based on the theory of quantum mechanics, this paper systematically analyzes the basic characteristics of n-dimensional linear harmonic oscillator in quantum mechanics, focuses on the eigenfunction and probability density of one-dimensional harmonic oscillator, and simulates the eigenfunction and probability density of some energy levels with MATLAB software. Finally, MATLAB software was used to compare the probability distribution of linear harmonic oscillator in classical mechanics and quantum mechanics. The results indicate that the number of points of intersection between a wave function and a φ=0 line is n; The probability distribution satisfies the normalization condition; Taking different values of n, the probability distribution function of harmonic oscillator in quantum mechanics has n different nodes, and the amplitude of harmonic oscillator in classical mechanics also changes accordingly. φ0 by the ground state probability distribution of quantum mechanics and classical mechanics distribution probability of the simulation image can be seen that the shape of the two distribution curve is the opposite, but when n is large, the probability density of quantum mechanics |ϕn (ξ)|2 local average and classical probability distribution P(ξ), that is, classical mechanics and quantum mechanics ZhongZhen gradually increase the probability distribution of similarity. These results are reflected in the image, and the characteristics shown in the image are consistent with the theoretical results.

DOI
10.1088/1755-1315/295/3/032042

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The manner in which probability amplitudes of paths sum up to form wave functions of a harmonic oscillator, as well as other, simple 1-dimensional problems, is described. Using known, closed-form, path-based propagators for each problem, an integral expression is written that describes the wave function. This expression conventionally takes the form of an integral over initial locations of a particle, but it is re-expressed here in terms of a characteristic momentum associated with motion between the endpoints of a path. In this manner, the resulting expression can be analyzed using a generalization of stationary-phase analysis, leading to distributions of paths that exactly describe each eigenstate. These distributions are valid for all travel times, but when evaluated for long times they turn out to be real, non-negative functions of the characteristic momentum. For the harmonic oscillator in particular, a somewhat broad distribution is found, peaked at value of momentum that corresponds to a classical energy which in turn equals the energy eigenvalue for the state being described.
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