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

Thermal Schrödinger equation

Morton D. Kostin

Published Jul 1, 1995• 3 citations
Publisher

Abstract

A form of the Schr\"odinger equation is given in which the time variable is replaced by a complex time-temperature variable and the interaction with the heat bath is represented by a stochastic interaction operator with Gaussian white noise. It is shown that at large times the ensemble averaged solution yields the equilibrium spatial probability density of quantum statistical mechanics. As equilibrium is approached, we also show how the master equation for a canonical ensemble can be derived from this form of the Schr\"odinger equation.

DOI
10.1103/physreve.52.229

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A form of the Schr"odinger equation is given in which the time variable is replaced by a complex time-temperature variable and the interaction with the heat bath is represented by a stochastic interaction operator with Gaussian white noise. It is shown that at large times the ensemble averaged solution yields the equilibrium spatial probability density of quantum statistical mechanics. As equilibrium is approached, we also show how the master equation for a canonical ensemble can be derived from this form of the Schr"odinger equation.
We present a microscopic derivation of a stochastic Schrödinger equation for an oscillator valid at finite temperatures. Damping and noise arise from interaction with a heat bath. Vacuum and thermal noise drive the wavevector in rather different ways which we illuminate by treating the fate of an initial Schrödinger cat state. While the decoherence of the two macroscopically distinct components of such a state is thermally enhanced, the rise of a classically interpretable signal indicating `life' or `death' is not. Suitable averages over the noises could be obtained from a well known master equation and demand interpretation in terms of ensembles, but results contingent on single realizations of the noises may be related to single runs of experiments.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.
We solve the time-dependent Schrodinger equation for the combination of a spin system interacting with a spin bath environment. In particular, we focus on the time development of the reduced density matrix of the spin system. Under normal circumstances we show that the environment drives the reduced density matrix to a fully decoherent state, and furthermore the diagonal elements of the reduced density matrix approach those expected for the system in the canonical ensemble. We show one exception to the normal case is if the spin system cannot exchange energy with the spin bath. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the canonical ensemble is a state that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics.
A master equation for a quantum system coupled to a heat bath with stochastically fluctuating energy levels is derived by making use of the ensemble averaging and the averaging with respect to a stochastic process in the bath. Relaxation terms are determined in the Born approximation with respect to the system-bath interaction and the damping parameters related to a relaxation kernel are specified. In parallel with the spectral strength of the bath, the damping parameters determine the transient times for the Markovian description creating the physical origin of the slippage [A. Suarez, R. Sibey, and I. Oppenheim, J. Chem. Phys. 97, 5101 (1992)]. The influence of energy fluctuations of the bath is analyzed for a two-level system including the solution of the corresponding non-Markovian equation for the level population difference. The conditions for the formation of Boltzmann's thermal ratio between steady-state populations are evaluated as well.