Department of Chemistry, Visva-Bharati, Shantiniketan-731 235, India indian Association for the Cultivation of Science, Jadavpur, Kolkata-700 032, India E-mail : pcdsr@mahendra.iacs.res.in Fax: 91-33-24722805 Max-Pianck-Institut fur Physik Komplexer Systeme, Nothnitzer Strabe 38, 01187 Dresden, Germany Manuscript received 12 November 2002 Arrhenius rate law is now more than hundred years old. The dynamical basis of this phenomenological law was provided by Kramers in early nineteenforties. Since late seventies, the extensive experimental investigation in chemical dynamics and allied areas centered around Kramers' dynamical rate theory. The present paper is an outline of further development of rate theory in quantum mechanical context which takes into account of thermal activation and tunneling within an unified description. The theory is based on a new scheme for quantum Brownian motion recently proposed by us to establish quantum analogues of Kramers', Smoluchowski and Fokker-Pianck equations with true probability distribution functions in c-number phase space.
Department of Chemistry, Visva-Bharati, Shantiniketan-731 235, India indian Association for the Cultivation of Science, Jadavpur, Kolkata-700 032, India E-mail : pcdsr@mahendra.iacs.res.in Fax: 91-33-24722805 Max-Pianck-Institut fur Physik Komplexer Systeme, Nothnitzer Strabe 38, 01187 Dresden, Germany Manuscript received 12 November 2002 Arrhenius rate law is now more than hundred years old. The dynamical basis of this phenomenological law was provided by Kramers in early nineteenforties. Since late seventies, the extensive experimental investigation in chemical dynamics and allied areas centered around Kramers' dynamical rate theory. The present paper is an outline of further development of rate theory in quantum mechanical context which takes into account of thermal activation and tunneling within an unified description. The theory is based on a new scheme for quantum Brownian motion recently proposed by us to establish quantum analogues of Kramers', Smoluchowski and Fokker-Pianck equations with true probability distribution functions in c-number phase space.
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.
Rapidly progressing laser technologies provide powerful tools to study potential barrier-passage dynamics in physical, chemical, and biological systems with unprecedented temporal and spatial resolution and a remarkable chemical and structural specificity. The available theories of barrier passage, however, operate with equations, potentials, and parameters that are best suited for a specific area of research and a specific class of systems and processes. Making connections among these theories is often anything but easy. Here, we address this problem by presenting a unified framework for the description of a vast variety of classical and quantum barrier-passage phenomena, revealing an innate connection between various types of barrier-passage dynamics and providing closed-form equations showing how the signature exponentials in classical and quantum barrier-passage rates relate to and translate into each other. In this framework, the Arrhenius-law kinetics, the emergence of the Gibbs distribution, Hund's molecular wave-packet well-to-well oscillatory dynamics, Keldysh photoionization, and Kramers' escape over a potential barrier are all understood as manifestations of a potential-driven Markovian dynamics whereby a system evolves from a state of local stability. Key to the irreducibility of quantum tunneling to thermally activated barrier passage is the difference in the ways the diffusion-driving potentials emerge in these two tunneling settings, giving rise to stationary states with a distinctly different structure.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
In only 150 pages, not counting appendices, references, or the index, this book is one author's perspective of the massive theoretical and philosophical hurdles in the no-man's-land separating the classical and quantum domains of physics. It ends with him emphasizing his own theoretical contribution to this area. In his own words, he has attempted to answer: 1. 'How can we obtain the quantum dynamics of open systems initially described by the equations of motion of classical physics (quantization process)? 2. 'How can we retrieve classical dynamics from the quantum mechanical equations of motion by means of a classical limiting process (dequantization process)?' However, this monograph seems overly ambitious. Although the publisher's description refers to this book as 'an accessible entrée', we find that this author scrambles too hastily over the peaks of information that are contained in his large collection of 272 references. Introductory motivating discussions are lacking. Profound ideas are glossed over superficially and shoddily. Equations morph. But no new convincing understanding of the physical world results. The author takes the viewpoint that physical systems are always in interaction with their environment and are thus not isolated and, therefore, not Hamiltonian. This impels him to produce a method of quantization of these stochastic systems without the need of a Hamiltonian. He also has interest in obtaining the classical limit of the quantized results. However, this reviewer does not understand why one needs to consider open systems to understand 'quantum-classical correspondence'. The author demonstrates his method using various examples of the Smoluchowski form of the Fokker--Planck equation. He then renders these equations in a Wigner representation, uses what he terms 'an infinitesimality condition', and associates with a constant having the dimensions of an action. He thereby claims to develop master equations, such as the Caldeira--Leggett equation, without use of a Hamiltonian. However, his procedure is seriously flawed. For example, he considers a case of anomalous Brownian motion using a Kolmogorov stochastic equation, and two cases of Brownian motion, the first with a constant force and the second with a linear force. In each of these three cases, his alleged solutions for a function F(x,p,t), which he wishes to interpret as a probability density, are not in fact solutions of their respective evolution equations. The author, in order to interpret these as probability densities, appears to have obtained the correct solutions of these equations, but then has normalized each of them with a time-dependent function, so that their integrals over the whole phase space remains unity for all time. However, an integration of the original equations over phase space demonstrates that an integral of their solutions over the whole phase space must be time-dependent. Despite the poor editing of this monograph, in which errors in equations abound, and in which the dimensionality of quantities is occasionally incorrect, this reviewer's interest in this subject was not extinguished. We are thus thankful for its extensive bibliography.
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Kolkata- 700 032, India E-Mail : pcdm@mahendra.iacs.res.in Present address: Department of Physical Science, R. K. M. Sikshanamandira, Belurmath, Howrah-711 202, India Present address : Department of Chemistry, R. K. M. Residential College, Narendrapur, Kolkata-700 103, India Also at : Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore-560 064, India Manuscript received 4 October 2002 We present here a non-perturbative cumulant expansion method of studying stochastically averaged dynamics of a TLS coupled to a gaussian colored noise bath. It uses a factorized cluster Ansatz for the evolution operator, U(t) suggested in an earlier formalism (Ref. 22). The factorization separates the operators involving only the system variables from those containing the intractions involving the bath variables. The stochastically averaged U(t) is obtained in a compact way by way of introducing bath-normal ordered products, which have zero bath averages for the bath-variables satisfying gaussian factorization property. We use a Wick-like operator reordering theorem (Ref. 25) to have a bath-normal ordered cluster expansion representation of U(t) and subsequently Novikov's theorem (Ref. 27) to get equations for the cluster amplitudes without free bath variables.The subdynamics of the system where the bath-variables are averaged out is governed in our method by a time-local effective hamiltonian involving system variables only. The method is efficacious when the bath is strongly colored and its coupling with the system is strong, e.g. when nonperturbative effects are prominent. The efficacy of the method is illustrated by computing power spectra from a model system which mimics nonadiabatic vibronic coupling simulated by colored noise.