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Journal of Wenzhou University

Eigenvalues and Eigenfunction of N Modes Coupled Harmonic Oscillators

Long Zheng-Wen

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Through coordinate transformation,Hamiltonian of N modes coupled harmonic oscillators could be expressed as the standard quadric form in new bases to eliminate the coupled items.As a result of calculation,a fact could be found that the new bases satisfy quasi-canonical commutation relation.By introducing quasi-particle’s creation and annihilation operators,the Hamiltonian could be simplified as N modes independent harmonic oscillators.Furthermore,the corresponding eigenvalues and eigenfunction could be obtained.

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Through coordinate transformation,Hamiltonian of N modes coupled harmonic oscillators could be expressed as the standard quadric form in new bases to eliminate the coupled items.As a result of calculation,a fact could be found that the new bases satisfy quasi-canonical commutation relation.By introducing quasi-particle’s creation and annihilation operators,the Hamiltonian could be simplified as N modes independent harmonic oscillators.Furthermore,the corresponding eigenvalues and eigenfunction could be obtained.
Hamiltonian of two modes coupled harmonic oscillators is general quadric form of generalized coordinates and momenta.After coordinate transformation,it can be expressed as standard quadric form under new bases.Because canonical commutation relation is satisfied among new coordinates,creation operator and annihilation operator are introduced.Thus,coupling between modes is eliminated and Hamiltonian can be simplified as two mode independent harmonic oscillators,problem is solved.This method can be generalized in solving non-identical n modes coupled harmonic oscillators.
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.
Commuting translations in the phase plane are shown to be observables that are canonically conjugate to the annihilation operator of the harmonic oscillator states. The eigenfunctions and the eigenvalues of these commuting operators form the kq-representation. The eigenfunctions of the annihilation operator are the coherent states. The kq-eigenfunctions and the coherent states are the limiting cases of quantum-mechanical and classical wavefunctions, respectively, and it is satisfying to discover that they relate to canonically conjugate operators.
By means of quadratic theory,we present a general method to solve the energy eigenvalue of coordinate and momentum coupling of n-dimensional harmonic oscillator.To apply this method to solve the energy eigenvalue of two-dimensional and three-dimensional coupling harmonic oscillator,our conclusion is the same with the others.Since the present method does not require a specific form of transformation matrix,this makes it be simple to solve the eigenvalue of a symmetrical form Hamilton and easy to calculate the results.This method is a universal and very effective algebraic method.