Quantum systems in contact with a thermal environment: Rigorous treatment of a simple model
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 \(2\pi)\).
quantum systems in a dissipative environment
scientific article; zbMATH DE number 4074123