Linear Invariants and the Solution of a Time-Dependent Coupled Quantal Oscillator.
S. Bös, U. B. Dub
Summary. - We obtain here the solution of a system of two coupled oscillators under the influence of damping force and time-dependent external force. The solution is obtained by first constructing linear invariants for this system. The exact dynamics of time-dependent quantal harmonic oscillator with timedependent frequency has been extensively studied in recent years exploiting the explicitly time-dependent invariants (1-5). The fact that there exists a simple relationship between the eigenstates of the invariant and the solution of the corresponding SehrSclinger equation has prompted people to find general form of invariants for time-dependent oscillator-type Hamiltonian. In these approaches quadratic type of invariants have mostly been considered. Further, these invariants involve a simple function ~(t), related to the amplitude of the classical free oscillator, and satisfies a nonlinear equation which should be solved for practical application. However, there has been attempt so far to study the dynamics of a time-dependent coupled-oscillator system. We, therefore, consider in the present work a system of two coupled oscillators, both damped and under time-dependent external force. To get the solution of the SchrSdinger equation for this system, we first obtain explicitly timedependent invariants which are linear in position and momentum variables. The two advantages with the linear invariants is that the eigenvalues of the invariant can be made continuous with suitable initial condition on the motion of oscillators and, further, that it involves only the amplitude of the classical oscillator and thus can be easily diagonalized.
•Jan 1, 1985•0 citations