A generalized langevin equation for wave functions. Intramolecular vibrational dephasing as a stochastic process

Abstract

A generalized Langevin equation satisfied by time-evolving wave functions is derived by adapting the Mori formalism to the Schrödinger time dependent equation. The equation obtained describes both deterministic and stochastic time evolution of wave functions. Memory kernel of the ”delayed friction” term of the equation obtained consists of the temporal correlation function (in the quantum mechanical sense) among non-stationary wave function, i.e., wave packets. The memory is likely to decline rapidly essentially due to the dispersion of the wave packets, so that a “Markov” limit is justified. An ensemble of probability amplitudes of different quantum states is considered, and the probability distribution of these probability amplitudes is discussed. This probability distribution is connected with spectroscopic observables: It can be derived from excitation profiles of Raman scattering amplitudes with many different final states. By assuming the “random force” term of the Langevin equation to be Gaussian white noise, a Fokker-Planck equation is derived. The solution of this Fokker-Planck equation describes how “randomization” proceeds in intramolecular vibrational dephasing. As time goes to infinity, all the accessible states are equally populated on the average, but fluctuation around the “microcanonical equilibrium” remains.