Quantum Mechanics of the Harmonic Oscillator

Thomas A. Garrity

Jacek Karwowski, Grzegorz Pestka

Carl W David

The classical harmonic oscillator and an elementary discussion of the quantum mechanical solutions for the harmonic oscillator are discussed.

<p>Sihui Yan¹, Jie Huang²</p>

Quantum mechanics holds a very important position as the theoretical foundation of the micro world. Harmonic oscillator system is a very classical model in quantum mechanics. In this paper, the wave function of a three-dimensional linear harmonic oscillator is derived based on the linear harmonic oscillator model in quantum mechanics. According to the probabilistic interpretation of quantum mechanics, we obtain the probability density function of the harmonic oscillator electron cloud. In this paper, Monte Carlo selection method is used for sampling, and Mathematica software is used to draw electron cloud images with different quantum number, so that people can intuitively see the distribution of electron cloud in three-dimensional space.

HU Xian-quan

Harmonic oscillator model,the physical method of which enjoys wide application in other related courses is the simplest but the most important model in quantum mechanics.On one hand,profound understanding of harmonic oscillator model through various ways is crucial to understand the essence of quantum mechanics and research into micro physical models by means of quantum mechanics;on the other hand,the application of algebraic approach to solve the eigenvalues and wave function of mechanics is the main means to solve the actual problems in quantum mechanics.This paper will take harmonic oscillator as an example,and apply algebraic approach to firstly present wave function of one-dimensional harmonic oscillator,and then multi-dimensional harmonic oscillator,and also combine corresponding commutator relations of operator to present Hermite polynomials and its recursion relations,which will avoid the process of using series to solve the equation of Heimite.Meanwhile,it will improve some shortcomings revealed in the same paper.