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Physica A Statistical Mechanics and its Applications

On the Schrödinger-Langevin picture and the master equation

Manuel O. Cáceres, Ana Karina Chattah

Published Dec 1, 1996• 9 citations
Publisher
DOI
10.1016/s0378-4371(96)00263-4

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We again note that Langevin equations written for observables (– operators in quantum mechanics) conform to the Heisenberg view. On the other hand, an entity that embodies the wave function of the Schrödinger picture is the von Neumann density operator, usually denoted by t( )ρ , whose classical counterpart is the probability distribution function (pdf) discussed earlier. It is helpful to introduce a formalism for the pdf completely at par with the Langevin description. The resultant Fokker-Planck (FP) equations serve as the subjects of this chapter. The simplest FP equation can be derived from the master equation (2.25) for a stationary Markov process, as shown below (Agarwal 1983, Risken 1996).
A consistent quantum formalism based on the localized basis of the Wannier functions in the Heisenberg and Schrödinger pictures to describe propagation of an electromagnetic field in a three-dimensional media including diffraction is presented. In the Schrödinger picture the Fokker-Planck equation for the Glauber-Sudarshan quasi-probability and the corresponding Langevin equations are given. As a result the spacetime description is obtained by a simple change of variables in the temporal master equation of the field. Using this formalism it is shown that the existence of integrals of motion in the propagation of light in a medium under the condition of non-degenerated parametric and two-photon interactions results in amplification of modes when non-classical properties of the light are conserved. Quantum propagation of light in a linear medium taking into account the diffraction is considered and its solution is found.