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Journal of Mathematical Physics

Quasi-coherent states for damped and forced harmonic oscillator

Mustafa Dernek, Nuri Ünal

Published Sep 1, 2013• 3 citations
Publisher

Abstract

In this study, first we construct the quasi-coherent states for a damped and forced harmonic oscillator and discuss the transition of the system from the damped oscillations to forced steady state oscillations. Second, we generalize the Caldirola-Kanai Hamiltonian into the new systems such as the frequency and the mass parameters are time dependent and discuss three examples of them.

DOI
10.1063/1.4819261

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In this study, first we construct the quasi-coherent states for a damped and forced harmonic oscillator and discuss the transition of the system from the damped oscillations to forced steady state oscillations. Second, we generalize the Caldirola-Kanai Hamiltonian into the new systems such as the frequency and the mass parameters are time dependent and discuss three examples of them.
In this study, we discuss the harmonic oscillator with the time-dependent frequency, ω(t), and the mass, M(t), by generalizing the holomorphic coordinates for the harmonic oscillator. In general cases, we solve the Schrödinger equation by reducing it into the Riccati equation and discuss the uncertainties for the quasi-coherent states of the time-dependent harmonic oscillator. In special cases, we find the following results: First, for a time-dependent harmonic oscillator, if [ω(t)M(t)] is constant, then the coherent states will evolve as the coherent states. Second, for the driven harmonic oscillator, the coherent states will evolve as the coherent states with new eigenvalues. Third, we derive quasi-coherent states for the Caldirola–Kanai Hamiltonian and show that the product of uncertainties, ΔxΔp, is larger than minimum value; however, it is constant. We also discuss the classical equations of motion for the system.
We study the time evolution of SU(1,1) coherent states driven by a damped harmonic oscillator as described by the Kanai-Caldirola Hamiltonian. The coherence of these states is preserved and their time evolution is described by nonlinear ``classical'' equations. The trajectories of all SU(1,1) coherent states in phase space asymptotically approach a common point as t\ensuremath{\rightarrow}\ensuremath{\infty}. In all cases, this approach is accompanied by a damping of the associated energy expectation value to zero.