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Journal of Bingtuan Education Institute

A Kind of Functional Relation of the Decline and Increased of Oscillator and Its Application

Ying Bao-xiang

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

When oscillator problems are solved in the quantum mechanics,the calculation of some special forms of the decline and rise of the oscillator operator plays a key role.However,the traditional methods of calculation is overelaborate.This article uses a formula derived from the quantum mechanics of a harmonic oscillator to reason a kind of functional relation related the decline and increased operator and applies it to concrete calculation and simplifies its calculating procedure.

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When oscillator problems are solved in the quantum mechanics,the calculation of some special forms of the decline and rise of the oscillator operator plays a key role.However,the traditional methods of calculation is overelaborate.This article uses a formula derived from the quantum mechanics of a harmonic oscillator to reason a kind of functional relation related the decline and increased operator and applies it to concrete calculation and simplifies its calculating procedure.
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.
Abstract In both classical and quantum physics the harmonic oscillator is one of the most important physical systems [1–4]. Furthermore, because of the simplicity and symmetry of the Hamiltonian operator, the quantum mechanical oscillator is particularly suited to analysis by abstract operator methods. For this reason the study of the one dimensional oscillator is a useful introduction to the application of operator methods in quantum mechanics. In Sections 2.1–2.4 we use these methods to determine energy eigenvalues and eigenkets, and matrix elements; the analysis is extended to isotropic multidimensional oscillators in Section 2.10. Quasiclassical states of the oscillator are discussed in Section 2.5. In Section 2.6 shift operators are used to calculate wave functions. In Sections 2.7–2.11 we discuss some applications of the theory of the oscillator.
The following text develops a new method of solving the eigenvalue problem of the quantum mechanical harmonic oscillator. Thereby are introduced new notions, like that of a reduced basis or that of a huge algebra. Furthermore are presented formulas for iterated commutation relations. Commutation Relation and Hamilton Function Oscillatory properties of a diatomic molecule, a carbon-monoxide molecule for example, can be modelled by a linear elastic oscillator. The observables for describing the oscillatory states of the oscillator are taken to be the dis- �� location from the equilibrium state and the relative momentum of the two masses.
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.
Now that we have learned some of the principles which apply in quantum mechanics, we move on to the next more difficult problem of a quantized harmonic oscillator. The goal here is to provide a rigorous application of the polynomial method of solving differential equations on a relatively simple case and to provide some insight into how the Schrödinger equation was first solved [1]. Then we proceed to application in the form of worked examples. We have tried to give sufficient details of this solution to allow a student to follow the derivation with pencil and paper but do not forget to ponder over the spectroscopic applications! This author would agree that it is more important to absorb the main conclusions of this material than to master the derivations. In fact, the highest recommendation of this author is to always ask ‘‘What does this mean?’’ and absorb the conclusions for a future activity called ‘‘thinking’’ rather than just memorizing facts.