Quantum dynamics of vibrational relaxation in condensed media

Abstract

The Zwanzig−Mori projection−operator formalism is employed to describe the dynamics of relaxation in a prototypic model well known for its application to a variety of physical problems: a harmonic oscillator coupled to a heat bath. First an exact equation of motion in generalized Langevin form is obtained for G (t), the expectation value of the occupation number of the oscillator, known to be in a definite initial nonequilibrium state. This equation is then solved in the appropriate van Hove weak−coupling, long−time limit for the usual case of physical interest in which the oscillator−bath interaction is linear in the oscillator coordinate. In this case G (t) is found to decay exponentially with a time constant τ_l = τ_d/2, where τ_d is the so called ’’dephasing’’ time, i.e., the time constant associated with the exponentially decaying time correlation function of the oscillator coordinate, 〈Q (0) Q (t) 〉. The approach taken here, which may be easily generalized, leads in a rather natural and straightforward way not only to relations between different relaxation times but also to useful explicit expressions for these times.