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College Physics

General solving method of two modes coupled harmonic oscillators

Chen Wen-xue

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

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.

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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.
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.
An algebraic (matrix) method is presented for canonically transforming the Hamiltonians quadratic in coordinate and momentum operators into the Hamiltonian of non-interacting harmonic oscillators. The method is illustrated by transforming the Hamiltonian of a harmonic oscillator in a constant magnetic field into the Hamiltonian of the two-dimensional, in general anisotropic, harmonic oscillator. The results are found to be in agreement with those obtained previously for the same Hamiltonian using a different technique.
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.
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.
In order to solve dynamic problems of double non-identical quantum harmonic oscillators with interaction potential and movement coupling, the new two mode composite coordinate representation |η1，η2〉 is proposed by the technique of integration within an ordered product of operators. The two mode squeezing operator U is constructed, and its characteristics are analyzed. Moreover, its application to double non-identical quantum harmonic oscillators with interaction potential and movement coupling is presented for solving some the dynamic problems. Approach of solving some the dynamic problems involving complicated quantum harmonic oscillators is proposed.