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Physica Scripta

The Adiabatic Approximation as Tool for Treating Dissipation in Quantum Mechanics

D. Sperber

Published Feb 1, 1984• 4 citations
Publisher

Abstract

It is suggested that frictional forces can be incorporated into quantum mechanics by using the adiabatic time dependent approximation. In particular it is shown that for a linear harmonoc oscillator the ratio between the square of the displacement of the damped and undamped harmonoc oscillator is the same as the ratio between the expectation values of the squares of the displacement of the damped and undamped harmonic oscillator.

DOI
10.1088/0031-8949/29/2/001

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It is suggested that frictional forces can be incorporated into quantum mechanics by using the adiabatic time dependent approximation. In particular it is shown that for a linear harmonoc oscillator the ratio between the square of the displacement of the damped and undamped harmonoc oscillator is the same as the ratio between the expectation values of the squares of the displacement of the damped and undamped harmonic oscillator.
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.
The damped harmonic oscillator is used to represent the damping processes in nature. Therefore it is necessary to find the Hamiltonian for the damped oscillator. The Hamiltonian is developed for the harmonic oscillator with frictional force proportional to velocity. The solution of the Schr\"odinger equation gives nonstationary states. The quantum friction coefficient is derived to be zero for massless particles. It is thought that velocity-dependent forces exist near the nucleus and therefore, the scattering amplitude for the deep inelastic process is derived.