scholarly journals Solution of the Master Equation for Quantum Brownian Motion Given by the Schrödinger Equation

Mathematics ◽  
2016 ◽  
Vol 5 (1) ◽  
pp. 1
Author(s):  
R. Sinuvasan ◽  
Andronikos Paliathanasis ◽  
Richard Morris ◽  
Peter Leach
Keyword(s):  
Brownian Motion ◽  
Schrödinger Equation ◽  
Master Equation ◽  
Schrodinger Equation ◽  
Quantum Brownian Motion

scholarly journals SYMMETRY REDUCTION OF BROWNIAN MOTION AND QUANTUM CALOGERO–MOSER MODELS

2012 ◽  
Vol 13 (01) ◽  
pp. 1250007
Author(s):  
SIMON HOCHGERNER
Keyword(s):  
Brownian Motion ◽  
Schrödinger Equation ◽  
Jacobi Equation ◽  
Schrodinger Equation ◽  
Free Particle ◽  
Symmetry Reduction ◽  
Reduction Scheme ◽  
Polar Actions ◽  
Stochastic Setting ◽  
Hamilton Jacobi Equation

Let Q be a Riemannian G-manifold. This paper is concerned with the symmetry reduction of Brownian motion in Q and ramifications thereof in a Hamiltonian context. Specializing to the case of polar actions, we discuss various versions of the stochastic Hamilton–Jacobi equation associated to the symmetry reduction of Brownian motion and observe some similarities to the Schrödinger equation of the quantum–free particle reduction as described by Feher and Pusztai [10]. As an application we use this reduction scheme to derive examples of quantum Calogero–Moser systems from a stochastic setting.

SCHRÖDINGER EQUATION FOR QUANTUM FRACTAL SPACE–TIME OF ORDER n VIA THE COMPLEX-VALUED FRACTIONAL BROWNIAN MOTION

2001 ◽  
Vol 16 (31) ◽  
pp. 5061-5084 ◽  
Author(s):  
GUY JUMARIE
Keyword(s):  
Brownian Motion ◽  
Schrödinger Equation ◽  
Fractional Order ◽  
Schrodinger Equation ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Dimensional System ◽  
One Dimensional ◽  
Klein Gordon ◽  
Complex Valued

First remark: Feynman's discovery in accordance of which quantum trajectories are of fractal nature (continuous everywhere but nowhere differentiable) suggests describing the dynamics of such systems by explicitly introducing the Brownian motion of fractional order in their equations. The second remark is that, apparently, it is only in the complex plane that the Brownian motion of fractional order with independent increments can be generated, by using random walks defined with the complex roots of the unity; in such a manner that, as a result, the use of complex variables would be compulsory to describe quantum systems. Here one proposes a very simple set of axioms in order to expand the consequences of these remarks. Loosely speaking, a one-dimensional system with real-valued coordinate is in fact the average observation of a one-dimensional system with complex-valued coordinate: It is a strip modeling. Assuming that the system is governed by a stochastic differential equation driven by a complex valued fractional Brownian of order n, one can then obtain the explicit expression of the corresponding covariant stochastic derivative with respect to time, whereby we switch to the extension of Lagrangian mechanics. One can then derive a Schrödinger equation of order n in quite a direct way. The extension to relativistic quantum mechanics is outlined, and a generalized Klein–Gordon equation of order n is obtained. As a by-product, one so obtains a new proof of the Schrödinger equation.

Master equation in phase space applied to the quantum Brownian motion in a tilted periodic potential

2012 ◽  
Vol 45 (10) ◽  
pp. 105002 ◽  
Author(s):  
William T Coffey ◽  
Yuri P Kalmykov ◽  
Serguey V Titov ◽  
Liam Cleary ◽  
William J Dowling
Keyword(s):  
Brownian Motion ◽  
Phase Space ◽  
Master Equation ◽  
Periodic Potential ◽  
Quantum Brownian Motion
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