Quantum Langevin equation

Remark. Normally the name quantum Langevin equation is used for equations that are the direct analog of the classical Langevin equation e.g., in the case of a one-dimensional particle of unit mass in a potential V,... 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

The quantum Langevin equation formalism described here could be also employed for studying the conductance properties of disordered harmonic systems or chains combining impurities. In such systems, mode localization becomes a crucial factor [5,18]. 

Scheme for Numerical Simulation of Quantum Langevin Equation.193... 

The outlay of this chapter is as follows In Section 9.2, we introduce the system/ spin-bath model and derive the operator Langevin equation for the particle. This is followed by a discussion on stochastic dynamics in the presence of c-number noise, highlighting the role of the spectral density function in the high- and low-temperature regimes. A scheme for the generation of spin-bath noise as a superposition of several Ornstein-Uhlenbeck noise processes and its implementation in numerical simulation of the quantum Langevin equation are described in Section 9.3. Two examples have been worked out in Section 9.4 to illustrate the basic theoretical issues. This chapter is concluded in Section 9.5. 

The quantum Langevin Equation 9.26 may be expressed in a more convenient form for interpretation. We add the force term V q) on both sides. The resulting equation is given by... 

SCHEME FOR NUMERICAL SIMULATION OF QUANTUM LANGEVIN EQUATION... 

Baneijee, D., B. C. Bag, S. K. Banik, and D. S. Ray. 2004. Solution of quantum Langevin equation Approximations, theoretical and numerical aspects. Journal of Chemical Physics 120(19) 8960-8972. 

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

Cortes, E., West, B. J. and Lindenberg, K. On the generalized Langevin equation classical and quantum mechanical, J.Chem.Phys., 82 (1985), 2708-2717... 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

I. Generalized Langevin equation. Zwanzig s Hamiltonian, n. Evaluation of quantum rates for multi-dimensional systems, ni. Beyond the Langevin equation/quantum Kramers paradigm ... 

The Schrodinger-Langevin equation and the quantum master equation... 

Thus we have found the general form of the quantum master equation that corresponds to the Schrodinger-Langevin equation (5.3). 

We have arrived at the quantum master equation (5.6) heuristically via the Schrodinger-Langevin equation (5.3). It turns out, however, that (5.6) has a firmer foundation it can be proved mathematically on the basis of the following three general conditions concerning the evolution of p. 

First, the Langevin equation receives in a separate chapter the attention merited by its popularity. In this chapter also non-Gaussian and colored noise are studied. Secondly, a chapter has been added to provide a more complete treatment of first-passage times and related topics. Finally, a new chapter was written about stochasticity in quantum systems, in which the origin of damping and fluctuations in quantum mechanics is discussed. Inevitably all this led to an increase in the volume of the book, but I hope that this is justified by the contents. 

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