Approach to quantum Kramers’ equation and barrier crossing dynamics

Abstract

We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.

Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition

“…However, this QSE still assumes that the bath relaxes much faster than the system, and does not account for the non-Markovian effect that can have potentially important effects. Major applications of QFPE include quantum extension 17,27,28 of Kramers' barrier crossing problem and proton or electron transfer dynamics. 24,[29][30][31][32] In particular, for the latter case, there has been growing interest in the study of fast photo-induced reaction dynamics that can occur during time scales comparable to those of molecular relaxation and dephasing dynamics.…”

Quantum phase-space function formulation of reactive flux theory

“…However for a practical calculation we need a recipe for calculation of Q(t). This has been discussed earlier in several contexts 24,25,26,27,39,40 . For the present purpose we summarize it as follows:…”

“…(2) is an exact Langevin equation for which the noise properties ofF (t) can be defined using a suitable canonical initial distribution of bath co-ordinates and momentum. Our first task is to replace it by an equivalent generalized quantum Langevin equation (GQLE) in c-numbers 24,25,26,27 . To achieve this we proceed in two steps.…”

“…Over the decades the field has grown in various new directions, e.g. , extension of Kramers results to non-Markovian regime 4,5,6 , generalizations to higher dimensions 7,8 , inclusion of complex potentials 9,10 , generalization to open systems 11,12,13 , analysis of semiclassical and quantum effects 14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29 , thermal ratchet 30 and molecular motors 31 etc. These developments have been the subject of several reviews and monographs.We refer to 15,16,17,22 .…”

“…The essential requirement for such a theory is obviously the quantum phase space distribution function. Very recently, based on a coherent state representation of quantum noise operator and an ensemble averaging procedure, we have developed 24,25,26,27 a c-number quantum generalized Langevin equation. The theory is valid for arbitrary temperature and noise correlation.…”