On the consistent description of quantum relaxation processes: The harmonic oscillator

Abstract

The quantum-mechanical description of a relaxation process is critically reviewed. The initially reversible “universe” consists of a harmonic oscillator (system) interacting with N further harmonic oscillators (heat bath). The coupling is linear in both, system- and bath-coordinates. The underlying dynamical equations are exactly solvable. A reasonable quantum-mechanical treatment of relaxation processes should guarantee that for all times the canonical commutation relations deviate only in a higher order of the coupling constant from their reversible values. A hierarchy of descriptions with decreasing quantum-mechanical consistency is formulated and discussed: 1. (i) the exact irreversible solution in connection with a decoupling of system and bath at t = 0; 2. (ii) the exact irreversible solution in connection with a conditional thermal average at t = 0; 3. (iii) a semi-Markovian stochastic differential equation (SDE), where the dissipation is described by a single parameter while the fluctuation (i.e. the correlation function of the stochastic force includes the information on the continuous bath-spectrum; 4. (iv) a complete Markovian SDE; dissipation and fluctuation are characterized by one single parameter. For the Drude model the descriptions (i)–(iii) are equivalent in relation to consistent results of first order in the dissipative parameters. On the contrary, description (iv) is unsuitable in the quantum-mechanical regime.