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Canadian Journal of Physics

Statistical mechanics of quantum one-dimensional damped harmonic oscillators: II

O. N. Borges, L. A. Amarante Ribeiro

Published Jul 1, 1987• 1 citation
Publisher

Abstract

We show that the Caldirola–Kanai Hamiltonian of the one-dimensional damped oscillator describes in a specified basis a Gaussian, Markoffian process, in the weak coupling limit. We also discuss which assumption makes the process stationary and clarify the relationship with the master-equation approach.
DOI
10.1139/p87-104

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The quantum theory of a damped harmonic oscillator is developed by considering an exactly solvable model, previously discussed by Unruh and Zurek [Phys. Rev. D 40, 1071 (1989)]. A one-dimensional harmonic oscillator is coupled to a scalar field. The field is assumed to be initially in thermal equilibrium. The Heisenberg equations of motion are solved without approximation, and the first and second moments, 〈q(t)〉, 〈p(t)〉, 〈q2{\mathit{q}}^{2}q2(t)〉, 〈p2{\mathit{p}}^{2}p2(t)〉, and 〈q(t)p(t)+p(t)q(t)〉 of the coordinate and momentum of the oscillator are calculated. The high-temperature, zero-temperature, free-particle, and weak-damping limits are discussed. A Wigner function of Gaussian form is constructed. The density-matrix equation derived by Louisell $[---Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] is examined and shown to have a solution in agreement with our weak-damping limit.