Quantum Statistical Mechanics: Equilibrium and non-equilibrium theory from first principles by Phil Attard

Author:Phil Attard
Language: eng
Format: epub, pdf
ISBN: 9780750311885
Publisher: IOP Publishing
Published: 2015-10-12T16:00:00+00:00

4.2 Stochastic time propagator

4.2.1 General features

As mentioned above, this chapter will finally focus upon the canonical equilibrium system. However, it is worthwhile initially giving a more general treatment for the case of a time-varying Hamiltonian operator, since these results are useful in later chapters as well. The present section is concerned with going beyond the adiabatic time propagator to take into account the interactions with the reservoir (environment, heat bath) and their effect on the evolution of the sub-system (open system).

The sub-system is a projection of the total system. In general mathematical terms, a projection operation gives rise to stochastic behavior. Even though the evolution of the total system is deterministic (the total evolution is adiabatic, the equation is Schrödingerʼs), the evolution of the sub-system is not determined by the sub-system alone. Since one has no knowledge of the specific quantum state of the reservoir at a given instant, one has to account for the influence of the environment probabilistically, based upon statistical thermodynamic considerations.

The stochastic dissipative equation of motion for the rate of change of the wave function of the sub-system may be written in a form analogous to the Schrödinger equation

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