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Journal of Physics A Mathematical and General

Quantum fluid dynamics description of a multiterm potential nonlinear Schrodinger-Langevin equation

Antonio Boulhosa Nassar

Published Jun 1, 1985• 5 citations
Publisher

Abstract

Via quantum fluid dynamics the author presents a general method to obtain a solution of the nonlinear Schrodinger-Langevin equation with a multiterm potential for the description of interactions in non-conservative systems. The nonlinear complete set of equations forming the basis of the model are derived. The corresponding solutions are exhibited in terms of auxiliary ordinary differential equations which provide relations between the potential coefficients and other coefficients prescribing the fluid-particle behaviour.

DOI
10.1088/0305-4470/18/8/004

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Via quantum fluid dynamics the author presents a general method to obtain a solution of the nonlinear Schrodinger-Langevin equation with a multiterm potential for the description of interactions in non-conservative systems. The nonlinear complete set of equations forming the basis of the model are derived. The corresponding solutions are exhibited in terms of auxiliary ordinary differential equations which provide relations between the potential coefficients and other coefficients prescribing the fluid-particle behaviour.
Quantum fluid dynamics description of a multiterm potential nonlinear Schroumloumloumldinger-Langevin equation
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.
We describe recent development of quantum hydrodynamics for ultracold Bose particle studying and consider different kinds of interactions. The method of derivation of equations describing the evolution of the neutral Bose particle system at low temperatures is described. Despite the fact that we consider the neutral particles we account the short-range interaction between particles. We consider the particles in the Bose-Einstein condensate (BEC) state. This method is called the method of quantum hydrodynamics, because natural for of the quantum mechanics rewritten in terms of material fields of observable quantities in three dimensional space is the set of equations, which look like the hydrodynamics equations. It can be shown that from the quantum hydrodynamics equations can be derived macroscopic non-linear Schrodinger equation. Most famous non-linear Schrodinger equation is the Gross-Pitaevskii (GP) equation, which contains nonlinearity of the third degree. There are generalizations of the GP equation. New term appears in the GP equation at account of the three-particle interaction. This term contains nonlinearity of the fifth degree. At more detailed account of the two particle interaction we come to the nonlocal non-linear Schrodinger equation. This equation contains spatial derivatives of the order parameter in the non-linear terms caused by the interaction. In this terminology the GP equation corresponds to the first order by the interaction radius. For the BEC of the neutral particles with anisotropic long-range dipole-dipole interaction the generalization of the GP equation was also suggested. Detailed analyses of the applicability conditions shows that this equation valid for the system of dipoles parallel to each other, which do not change their direction, and where the dipole-dipole interaction interferences translational motion of particles.
Quantum Fluids follow Quantum Dynamical Equation(s), which were not known till date. There exist a set of two equations that is semi classical approach to Quantum Fluids called Madelung’s Equations. But a new fully quantum variant of Madelung’s Equations when embedded in the Schrodinger Equation is gives full description of evolution of Quantum fluid with respect to time and position. The equation presented in this article has two unknown variables, one is density and other is velocity field as a function of spatial and time coordinates. The equation presented in this article, is derived from Schrodinger Equation, obeying Continuity equation, and Navier Strokes Equation. Bohm’s potential was externally added in Madeline’s equation. But the new equation which is fully quantum mechanical in nature; Bohm’s potential appears out of the equation, which is interesting to observe. Astrophysical cold stellar dynamics and condensed fluids have the main application of this equation. Quantum fluids show strange behaviour when compared to normal fluids. It is also shown that quantum fluid also has spins which has no classical analog.
A simple scheme of unified description of different physical phenomena by using the Langevin type equations is reviewed. Within this approach much attention is being paid to the study of Brownian and quantum motions. Stochastic equations with a white noise term give all characteristics of the Brownian motion. Some generalization of the Langevin type equations allows us to obtain nonlinear equations of particles' motion, which are formally equivalent to the Schrödinger equation. Thus, we establish Nelson's stochastic mechanics on the basis of the Langevin equation.