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Chongqing Shifan Daxue xuebao. Ziran kexue ban

Algebraic Approach to Wave Function of Harmonic Oscillator and Recursion Relations of Hermite Polynomial

HU Xian-quan

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

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.

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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.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.
Quantum mechanics is one of the branches of physics. In this mechanics, physical problems are solved by algebraic and analytic methods. By applying a simple procedure we can find the general solutions of Schrodinger’s time - independent wave equation of one dimensional quantum mechanical harmonic oscillator without making any approximation. In this paper, we will discuss the Eigenenergy values and Eigenfunctions of one of the most important physical models of quantum mechanics, namely the one-dimensional Quantum mechanical Harmonic Oscillator by modifying the Hermite differential equation.
This study presents a complete algebraic formulation of the quantum harmonic oscillator, demonstrating that the Hermite polynomials, H n ( x ) , emerge as the fundamental solution when the power-series method is applied to the Schrödinger equation. The physical requirement of square integrability enforces the truncation of the series, leading precisely to these polynomials, which constitute the polynomial part of the energy eigenfunctions, ψ n ( x )= N n H n ( αx ) e − α 2 x 2 /2 , where n is the quantum number. This quantum number directly determines the quantized energy levels through E n =ℏω( n+ 1 2 ) . The analysis based on creation and annihilation operators further reveals that the action of the creation operator on the ground state generates the excited states along with their corresponding Hermite functions. The orthogonality of these polynomials ensures the orthonormality of the complete set of eigenfunctions. Additionally, the geometric Berry phase emerges as a natural extension when the system undergoes adiabatic cyclic changes, revealing a deeper topological structure that complements the conventional algebraic and analytic descriptions. Therefore, Hermite polynomials encapsulate both the mathematical structure and the physical essence of quantization in this fundamental system.
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 harmonic oscillator is one of the most studied systems in Physics with a myriad of applications. One of the first problems solved in a Quantum Mechanics course is calculating the energy spectrum of the simple harmonic oscillator with analytic and algebraic approaches. In the algebraic solution, creation and annihilation operators are introduced to factorize the Hamiltonian. This work presents an algebraic solution for the simple harmonic oscillator in the context of Classical Mechanics, exploring the Hamiltonian formalism. In this solution, similarities between the canonical coordinates in a convenient basis for the classical problem and the corresponding operators in Quantum Mechanics are highlighted. Moreover, the presented algebraic solution provides a straightforward procedure for the quantization of the classical harmonic oscillator, motivating and justifying some operator definitions commonly used to solve the correspondent problem in Quantum Mechanics.
This chapter begins discussing the classical harmonic oscillator and its solution as a second-order differential equation of the general type. Next, the one-particle quantum harmonic oscillator is presented, its corresponding Hamiltonian functionand wave equation followed by the undimensionalization of this wave equation yielding a second-order differential equation of the general type which leads to the general Hermite equation and Hermite polinomials as partial solution of the wave function. The energy for each wave function is obtained from the recursion relation of the Hermite equation. Afterwards, the numerical analysis of the one-particle harmonic oscillator is given by using the Numerov numerical method. Finally, a corresponding Fortran code for this problem is presented in order to plot the wave function according to a pre-determined quantum number. Plots of the wave function obtained from the Fortran code are also given. In the end, some exercises are given to include other information in the Fortran code.(153words).