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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Forced harmonic oscillator with damping and thermal effects

H. M. França, M. T. Thomaz

Published Mar 15, 1985• 9 citations
Publisher

Abstract

Nonperturbative quantum-mechanical solutions of the forced harmonic oscillator with radiation-reaction damping are obtained from a previous analysis based on stochastic electrodynamics. The transition to excited states is shown to be to coherent states which follows the classical trajectory. The quantum Wigner distribution in phase space is constructed. All the results are extended to fi- nite temperatures.

DOI
10.1103/physrevd.31.1337

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Nonperturbative quantum-mechanical solutions of the forced harmonic oscillator with radiation-reaction damping are obtained from a previous analysis based on stochastic electrodynamics. The transition to excited states is shown to be to coherent states which follows the classical trajectory. The quantum Wigner distribution in phase space is constructed. All the results are extended to fi- nite temperatures.
We apply the theory of stochastic electrodynamics to the study of the (nonrelativistic) harmonic oscillator by using the Fokker–Planck method. It is demonstrated that the equilibrium distribution in phase space is exactly equal to that given by quantum statistical mechanics, i.e., the corresponding Wigner distribution, and that analysis of this distribution by means of a decomposition in terms of canonical densities leads automatically to the usual description of quantum mechanics in terms of excited states. All fundamental equations of quantum mechanics are recovered as aprproximations to zero order in the radiation terms; the first-order terms lead to the radiative corrections predicted by quantum electrodynamics, namely, the decay of states and the Lamb shift of the energy levels. The necessary differences between both treatments of the oscillator and their implications are briefly discussed.
The physics of quantum electromagnetism in an absorbing medium is that of a field of damped harmonic oscillators. Yet until recently the damped harmonic oscillator was not treated with the same kind of formalism used to describe quantum electrodynamics in a arbitrary medium. Here we use the techniques of macroscopic QED, based on the Huttner--Barnett reservoir, to describe the quantum mechanics of a damped oscillator. We calculate the thermal and zero-point energy of the oscillator for a range of damping values from zero to infinity. While both the thermal and zero-point energies decrease with damping, the energy stored in the oscillator at fixed temperature increases with damping, an effect that may be experimentally observable. As the results follow from canonical quantization, the uncertainty principle is valid for all damping levels.
This paper is concerned with the dynamics of coupled quantum oscillators in the presence of a classical harmonic driving force. By means of a theorem which describes the evolution of Gaussian distributions of coherent states, the solutions of the equations of motion are presented in a very compact form. Specific applications deal with forced oscillation and parametric amplification. The theoretical models include systems of damping oscillators with a continuous frequency distribution. The approach to thermal equilibrium in the absence of a driving force is discussed as a special case.
We consider a forced harmonic oscillator in one-dimension. Using coherent states, we show that the treatment of the system is simplified, that the relationship between the classical and quantum solutions becomes transparent, and that the evolution operator of the system can be calculated easily as the free evolution operator of the harmonic oscillator followed by a displacement operator that depends on the classical solution. In addition, we consider the system in the rotating-wave-approximation (RWA), an application of the Averaging Theorem. We determine the relationship between the exact solution and the one in the RWA, test the accuracy of the RWA, and explain why the RWA gives accurate results in the realm of cavity quantum electrodynamics. Finally, we apply the results to a charged particle interacting with an electromagnetic field.
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.