Quantum Mechanical Liouville Equation for a System in Contact with a Reservoir

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Abstract
In this paper we derive the equations of motion satisfied by the density matrix of an arbitrary physical system in contact with an infinite thermal reservoir of quantum mechanical free particles. We show that for weak but sustained interactions between system and reservoir, the reservoir is represented by a linear time-independent kernel in the system density matrix space. We obtain the kernel for the three possible models of quantum mechanical free particles which are boson and fermion reservoirs in which the system-reservoir interaction conserves the number of particles of the reservoir and a boson reservoir in which reservoir "particles" may be created or destroyed. In order to derive the equations of motion without using the inconsistent "rerandomization of phases" after each interaction, we generalize the Wigner-Weisskopf theory of line broadening of spectral lines to the equations of motion for the density matrix. We next show that the kernel preserves the normalization, positive-definiteness, and Hermiticity of the system density matrix and causes the system to approach equilibrium monotonically. In the limit of a classical reservoir our results include a derivation, without the repeated use of random phasing, of Bloch's equations of motion for the density matrix of a spin system imbedded in a thermal reservoir.