K. Someda et Rd. Levine, A GENERALIZED LANGEVIN EQUATION FOR WAVE-FUNCTIONS - INTRAMOLECULAR VIBRATIONAL DEPHASING AS A STOCHASTIC-PROCESS, Chemical physics, 184(1-3), 1994, pp. 187-212
A generalized Langevin equation satisfied by time-evolving wave functi
ons is derived by adapting the Mori formalism to the Schrodinger 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 te
mporal correlation function (in the quantum mechanical sense) among no
n-stationary wave functions, i.e., wave packets. The memory is likely
to decline rapidly essentially due to the dispersion of the wave packe
ts, so that a ''Markov'' limit is justified. An ensemble of probabilit
y amplitudes of different quantum states is considered, and the probab
ility 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 amplitu
des with many different final states. By assuming the ''random force''
term of the Langevin equation to be Gaussian white noise, a Fokker-Pl
anck 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 eq
ually populated on the average, but fluctuation around the ''microcano
nical equilibrium'' remains.