Quantum systems in contact with a thermal environment: Rigorous treatment of a simple model
type: Item
instance of: scholarly article
title: Quantum systems in contact with a thermal environment: Rigorous treatment of a simple model
zbMATH Open document ID: 0657.60085
DOI: 10.1007/BF01217962
author: S. H. Smith
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: We study the quantum dynamics of a particle of mass M in an external potential V(Q), interacting with a simple model environment - a harmonic chain of 2N particles with mass m and spring constant k. The classical version of this model was studied by \textit{R. J. Rubin} [see J. Math. Phys. 1, 309 ff (1960) and ibid. 2, 373 ff (1961)] and is equivalent to standard models of a particle interacting with a phonon bath. Setting \(m=m^*/L\) and \(k=k^*L\), we prove that for a suitable class of potentials V and initial states \(\omega_ 0\), the time evolution of the mass M particle converges, when \(N\to \infty\) and \(L\to \infty\), to the time evolution governed by the quantum Langevin equation (QLE) which has been found by \textit{G. W. Ford}, \textit{M. Kac} and \textit{P. Mazur} [ibid. 6, 504 ff (1965)]. Furthermore we show that, for this class of potentials, the QLE has a unique solution for all positive times, such a solution can be expressed as a convergent expansion in the deviation of V(Q) from a harmonic potential. The equilibrium properties of the particle with mass M can be expressed in terms of an integral, over path space, with a Gaussian measure which has mean zero and covariance proportional to \([-\Delta +\eta h/M\sqrt{-\Delta}]^{-1}\); where \(\eta =2\sqrt{km}\) is the friction constant, and h is the Plancks' constant (divided by \(2\pi)\).
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 81P20
zbMATH DE Number: 4074123
zbMATH Keywords: quantum systems in a dissipative environment
zbMATH Keywords: particle interacting with a phonon bath
zbMATH Keywords: quantum Langevin equation
zbMATH Keywords: harmonic potential
MaRDI profile type: Publication
cites work: Tunnelling in the equilibrium state of a spin-boson model
cites work: A mechanical model of Brownian motion
cites work: Stochastic Problems in Physics and Astronomy
cites work: Statistical Mechanics of Assemblies of Coupled Oscillators
cites work: Return to thermal equilibrium by the solution of a quantum Langevin equation
cites work: On the quantum Langevin equation
cites work: Q3854185
cites work: Q3872705
cites work: Quantum stochastic processes
cites work: Q5683807
cites work: Q3909512
cites work: Q4106816
full work available at URL: https://doi.org/10.1007/bf01217962
OpenAlex ID: W2024128136
description: scientific article; zbMATH DE number 4074123