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Physical Review A

Anisotropic time-dependent coupled oscillators

M. Sebawe Abdalla

Published Apr 1, 1990• 16 citations
Publisher

Abstract

The problem of a two-dimensional anisotropic oscillator coupled through a complex parameter that oscillates at the frequency \ensuremath{\omega} is considered. The treatment has been applied to the case when \ensuremath{\omega} is not equal to the frequency difference of the oscillators. An exact solution for the Schr\"odinger wave function has been obtained. The Green's function as well as the expectation value of the energy are calculated.

DOI
10.1103/physreva.41.3775

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The problem of a two-dimensional anisotropic oscillator coupled through a complex parameter that oscillates at the frequency \ensuremath{\omega} is considered. The treatment has been applied to the case when \ensuremath{\omega} is not equal to the frequency difference of the oscillators. An exact solution for the Schr"odinger wave function has been obtained. The Green's function as well as the expectation value of the energy are calculated.
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.
Summary. -- From the poir.t of view of quar.tum ~echanics, the problem of two coupled oscillators with two different coupling parameters is considered. By using the accurate definition of Dirac operators the wave function in both coherent state and number state (SchrSdinger picture) are obtained. The Green's function and the expectation value of the energy are calculated; the transition amplitude between the coherent states when the coupling parameters are different and equal, as well as the eigenstates, are given. The constants of the motion for such a system have been also considered.