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College Physics

Exact wave function of the n-module coupled harmonic oscillators quantum system

Sun Li-hong

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

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.

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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.
The normal frequencies and its corresponding normal coordinates and conjugate momentums for a three-body coupled pendulum quantum system is derived in terms of the invariant eigen-operator method.With Hamiltonian of the system decoupled,the visible analytic solutions of normal frequencies in the system are obtained.The analytic solutions of exact wave function of the system in coordinate representation are deduced.Whereas,when calculating the Hamiltonian containing mechanical quantity with invariant eigen-operator method,the three power and more than three power terms of system Hamiltonian are very intricacy.
In this work we present the classical and quantum solutions of time-dependent coupled harmonic oscillators. In these systems the masses, frequencies and coupling parameter (k) are functions of time. Four systems are investigated. To obtain the classical solutions we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. We find the analytical expression for position and velocity of each oscillator of the systems. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld invariant method. The wave functions obtained are written in terms of a c-number quantity () which is solution of the Milne-Pinney equation. For each system we solve the respective Milne-Pinney equation and discuss how the quantum fluctuations and the uncertainty product evolve with time.
In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.