In the research of one-dimensional harmonic oscillaton as the object in Heisenberg’s picture. We have got two kinds of solutions of its energy level and wave fanction by algebraic approach of using matrix mechanics and factorization method.
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.
A quantum state corresponds to a specific wave function. We adopt a new mathematical method [1] to improve Dirac's ladder operator method. A set of orthonormal wave functions will be used to associate the operator with the square matrix corresponding to it. These allow us to determine the matrix elements by using the operator relations without having to know the specific wave functions. As a result, we can get the direct results of matrix mechanics and wave mechanics on one-dimensional Harmonic oscillator and their descriptions will be also unified.