Quantum theory of a dissipative oscillator

I. R. Senitzky

The development of a quantum-mechanical formalism for systems with dissipation that was presented in an earlier article, and intended mainly for application to the electromagnetic field in a cavity, is extended. The problem of the harmonic oscillator with dissipation is shown to be the same as that of a harmonic oscillator coupled to a thermal reservoir, and the need of the formalism to contain the appropriate statistical mechanics is discussed. The derivation of relationships which permit the calculation of all moments of the oscillator coordinate and momentum provides the necessary extension of the theory. The formal resemblance of the completed theory to that of classical Brownian motion, some differences due to quantum mechanics, and the fact that certain fundamental relationships which are assumed in the latter are derived in the present analysis, are pointed out. The application of the theory is illustrated by the consideration of three problems: the proof of Ott's formula, and the derivation of both the probability density and energy distribution of the oscillator in equilibrium with a thermal reservoir.

Ilya Prigogine

Vinay Ambegaokar

Emmanouil George Thrapsaniotis

We study the dynamics of a quantum dissipative system. Besides its linear coupling with a harmonic bath modelling the dissipation, we suppose that it is coupled with an oscillator with an interaction of the form s 2 x 2 . In our study, we integrate over the bath and the oscillator, extract the corresponding influence functionals and then solve the system’s sign problem. We apply the theory to the case of a double well and study the time evolution of the expectation value of the position.

Aritra Ghosh, Jasleen Kaur, Malay Bandyopadhyay

José M. Cerveró, Javier Villarroel

The longstanding problem of the physical interpretation of the one-dimensional quantum damped oscillator is analysed here from the viewpoint of group theory. It is shown how to transform the system into a quantum oscillator with variable frequency. However, the main features of the final system are rather those of a simple stationary oscillator with 'renormalised' constant frequency. The quantum mechanical system cannot be at all interpreted as a 'dissipative' quantum system as has been claimed by some authors.