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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Alternative model of dissipation in quantum mechanics

David R. Reichman, Minhaeng Cho

Published Apr 1, 1996• 2 citations
Publisher

Abstract

The model of dissipation studied by Castro Neto and Caldeira [Phys. Rev. Lett. 67, 1960 (1991)] is reexamined. The main results for transport are derived in a simple manner without the use of Feynman path integrals. By means of a unitary transformation, a connection is made between the model and a model previously studied in the field of quantum dissipation. Lastly, extensions of the model are discussed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.53.4184

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The model of dissipation studied by Castro Neto and Caldeira [Phys. Rev. Lett. 67, 1960 (1991)] is reexamined. The main results for transport are derived in a simple manner without the use of Feynman path integrals. By means of a unitary transformation, a connection is made between the model and a model previously studied in the field of quantum dissipation. Lastly, extensions of the model are discussed. \textcopyright{} 1996 The American Physical Society.
We suggest taking the fluctuation-dissipation theorem of Callen and Welton [Phys. Rev. 83, 34 (1951)] as a basis to study quantum dissipative phenomena (such as macroscopic quantum tunneling) in a manner analogous to the Nambu-Goldstone theorem for spontaneous symmetry breakdown [Phys. Rev. 122, 345 (1961); Nuovo Cimento 19, 154 (1961)]. It is shown that the essential physical contents of the Caldeira-Leggett model [Phys. Rev. Lett. 46, 211 (1981); Ann. Phys. (N.Y.) 149, 374 (1983)] such as the suppression of quantum coherence by Ohmic dissipation are derived from general principles only, namely, the fluctuation-dissipation theorem and unitarity and causality (i.e., dispersion relations), without referring to an explicit form of the Lagrangian. An interesting connection between quantum tunneling with Ohmic dissipation and Anderson's orthogonality theorem is also noted.
This chapter demonstrates the equivalence between Feynman–Vernon influence functional approach and Keldysh diagram technique developed to describe non-equilibrium phenomena in quantum many-body systems with interactions. Path integrals appearing in both quantum mechanics and quantum field theory may be viewed as a formal representation of the exact infinite series of Feynman diagrams. In other words, path integral and diagram techniques are formally equivalent to each other. The chapter discusses a single degree of freedom, introducing interactions with a bath of harmonic oscillators. In a way, to employ the Caldeira–Leggett model of a bath of harmonic oscillators can be considered just as a physical trick helping to derive the influence functional in the case of linear Ohmic damping. Gaussian influence functionals are exact provided the environment consists of an arbitrary number of harmonic oscillators coupled linearly to the system of interest q .
A systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last 15 years, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given which will serve as a quick reference for solving path integrals. Explicit formulas for the so-called basic path integrals are presented on which the general scheme to classify and calculate path integrals in quantum mechanics is based.
In earlier work we proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary group representations [Doebner and Goldin, Phys. Lett. A 162, 397 (1992); J. Phys. A 27, 1771 (1994)]. Such nonlinear Schr\"odinger equations are expected to describe irreversible and dissipative quantum systems. Here we introduce and justify physically the group of nonlinear gauge transformations necessary to interpret our equations. We determine the parameters that are actually gauge invariant and describe some of their properties. Our conclusions contradict, at least in part, the view that any nonlinearity in quantum mechanics leads to unphysical predictions. We also show how time-dependent nonlinear gauge transformations connect our equations to those proposed by Kostin [J. Chem. Phys. 57, 3589 (1972)] and by Bialynicki-Birula and Mycielski [Ann. Phys. 100, 62 (1976)]. We believe our approach to be a fundamental generalization of the usual notions about gauge transformations in quantum mechanics. \textcopyright{} 1996 The American Physical Society.
The translation from classical to quantum mechanics using the hamiltonian formalism for a non-dissipative systems is very well known. If in the classical picture there is dissipation, the equation of motion is given in terms of newtonian mechanics whose quantum equivalent is not obvious. Using results of previous works we show a way of obtaining a corresponding hamiltonian and discuss the wave function in physical space. We then proceed to analyze the quantum mechanical motion of gaussian wave packets with dissipation.