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College Physics

Solving energy eigenvalue of n module coupling harmonic oscillators by virtue of quadratic form

Yunhai Zhang

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

A unitary matrix is constructed by virtue of quadratic form theory for coordinates and momentums transformation.The Hamiltonian of the n-D coupling harmonic oscillators is diagonalized by means of the transformation.The exact solution of energy spectrum of n-D coupling quantum harmonic oscillator is given.It also offers a general mathematic method solving problems of this kind.

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A unitary matrix is constructed by virtue of quadratic form theory for coordinates and momentums transformation.The Hamiltonian of the n-D coupling harmonic oscillators is diagonalized by means of the transformation.The exact solution of energy spectrum of n-D coupling quantum harmonic oscillator is given.It also offers a general mathematic method solving problems of this kind.
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.
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 normal frequencies and its corresponding normal coordinates and conjugate momentums of the n-module coupled harmonic oscillators quantum system are derived by using the invariant eigen-operator method.The Hamiltonian of the system is decoupled and then the visible analytic solution of normal frequencies in the system is obtained.The analytic solution of wave function of the system in the coordinate representation is deduced.Through the discussion of the coupling coefficients of different situations,we have concluded that the coupled harmonic oscillator systems with the n-module momentum or coordinates are just special cases of the system under study.