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Physical Review A

Dissipation of energy in the damped harmonic oscillator

R. K. Colegrave, Effat Kheyrabady

Published Nov 1, 1986• 8 citations
Publisher

Abstract

A canonical description of a harmonic oscillator with energy dissipation is sought which combines the advantages of the Kanai-Caldirola Hamiltonian and a simple model of strangulation previously considered by the authors. The treatment is in the Heisenberg picture of quantum mechanics or alternatively in classical mechanics. A strangulation \ensuremath{\beta} is superimposed on the damping (or growth) \ensuremath{\gamma}, represented by an oscillator mass m(t)=${m}_{0}$exp(2\ensuremath{\gamma}t). In general, the expected dissipation of energy occurs only if \ensuremath{\beta}>\ensuremath{\Vert}\ensuremath{\gamma}\ensuremath{\Vert}. If \ensuremath{\beta}<\ensuremath{\Vert}\ensuremath{\gamma}\ensuremath{\Vert} the attempted strangulation induces a long-term growth in energy unless a special initial motion is chosen. A slight deviation from this initial motion results in a temporary decay in energy followed by growth. \ensuremath{\beta}<0 always induces a growth in energy.

DOI
10.1103/physreva.34.3634

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A canonical description of a harmonic oscillator with energy dissipation is sought which combines the advantages of the Kanai-Caldirola Hamiltonian and a simple model of strangulation previously considered by the authors. The treatment is in the Heisenberg picture of quantum mechanics or alternatively in classical mechanics. A strangulation \ensuremath{\beta} is superimposed on the damping (or growth) \ensuremath{\gamma}, represented by an oscillator mass m(t)=m0{m}_{0}m0​exp(2\ensuremath{\gamma}t). In general, the expected dissipation of energy occurs only if \ensuremath{\beta}>\ensuremath{\Vert}\ensuremath{\gamma}\ensuremath{\Vert}. If \ensuremath{\beta}<\ensuremath{\Vert}\ensuremath{\gamma}\ensuremath{\Vert} the attempted strangulation induces a long-term growth in energy unless a special initial motion is chosen. A slight deviation from this initial motion results in a temporary decay in energy followed by growth. \ensuremath{\beta}<0 always induces a growth in energy.
We begin by revisiting the so-called Caldirola - Kanai Hamiltonian and discuss its inherent ambiguity: does it represent a dissipative harmonic oscillator (HO) subject to a friction force, or does it describe an HO with a time-dependent mass (TDM)? Although classically both descriptions do coexist, in the quantum domain the solution of the Schrödinger equation (or Heisenberg equations of motion) with a TDM does not present inconsistencies, however, the dissipative Hamiltonian shows violation of the Heisenberg uncertainty principle. This violation is avoided by introducing a stochastic force in the equations of motion, which will take care of the fluctuations due to the environment. Once the distinction between the dissipative and amplifying Hamiltonian is made clear, we consider the problem of the quantum TDM HO subject to dissipation, showing that both phenomena may be merged and described by a single Hamiltonian, the amplifying - dissipative Hamiltonian . We obtain the solutions of the Heisenberg equations of motion for the canonical momentum and position; next, we specialize on the weak damping limit and analyse the effects of the amplifying - dissipative process on the mean values of the physical variables.
The usual treatment of damping forces in quantum mechanics starts from the introduction of the explicitly time dependent Kanai Hamiltonian, which actually represents a variable mass particle, and the misinterpretation of this Hamiltonian as representing a particle of constant mass leads to certain physical difficulties. However the Hamiltonian can be modified so that it can be reinterpreted as describing a constant mass particle. Here we explicitly introduce the mass as a new dynamical variable, which allows us to write a linear, time independent Hamiltonian for the system, which can be solved by conventional methods. The damped harmonic oscillator and damped free particle are treated in detail, both for the Kanai Hamiltonian and for our case, and the solutions are compared. Our solution can be reduced to the Kanai one in appropriate circumstances, but in general it has a much greater versatility, as a result of which it can be more easily reinterpreted as describing a constant mass particle subject to a damping force, which reinterpretation is of course necessary if the method is to have practical applicability. We also show how such a reinterpretation can be carried out in detail by introducing a ’’dissipation variable’’, in terms of which one may avoid the concept of a variable mass altogether.
A harmonic oscillator subject to the combined effects of damping and pulsating is represented by a Kanai–Caldirola Hamiltonian. The equations of motion are solved in the Heisenberg picture in the case of weak pulsation. The rotating-wave approximation (RWA) is used to obtain the motion in the neighborhood of the principal resonance. The RWA Schrödinger equation is solved exactly and pseudostationary and quasicoherent states are described. The transition probability between quasicoherent and coherent states is obtained and the gain in energy is discussed.
The need for a quantum-mechanical formalism for systems with dissipation which is applicable to the radiation field of a cavity is discussed. Two methods that have been used in this connection are described. The first, which starts with the classical Newtonian equation of motion for a damped oscillator and applies the conventional formal quantization techniques, leads to an exact solution; but subsequent discussion shows that this method is invalid, the results being unacceptable from a quantum-mechanical viewpoint. The second method, which considers the interaction of two systems, the lossless oscillator and the loss mechanism, is adopted in the present article. No special model is used for the loss mechanism, but this mechanism is assumed to have a large number of densely-spaced energy states.The approximations with respect to the loss mechanism that underlie the concept of dissipation are discussed. These approximations are then applied to the analysis, and a differential equation for a coordinate operator of the harmonic oscillator is obtained which has the formal appearance of the Newtonian equation of motion for a driven damped harmonic oscillator, the driving term being an operator referring to the loss mechanism. The presence of the driving term is responsible for the difference between the present theory and that of the first method mentioned above. A solution of the differential equation for the coordinate operator is given explicitly. An examination of the physical significance of the solution shows that the driving term is responsible not only for the thermal fluctuations which are due to the loss mechanism, but also for the proper commutation relationship of the conjugate coordinates of the oscillator and for its zero-point fluctuations.A generalization of the solution to provide for a classical driving force and coupled atomic systems is given. The results are then restated in a form that refers to the loss mechanism only through the two parameters by which it is usually described---the dissipation constant and the temperature.