An Application of the Madelung Formalism for Dissipating and Decaying Systems

Maedeh Mollai,Seyed Majid Saberi Fathi

doi:10.3390/sym13050812

Abstract

This paper is concerned with the modeling and analysis of quantum dissipation and diffusion phenomena in the Schrödinger picture. We derive and investigate in detail the Schrödinger-type equations accounting for dissipation and diffusion effects. From a mathematical viewpoint, this equation allows one to achieve and analyze all aspects of the quantum dissipative systems, regarding the wave equation, Hamilton–Jacobi and continuity equations. This simplification requires the performance of “the Madelung decomposition” of “the wave function”, which is rigorously attained under the general Lagrangian justification for this modification of quantum mechanics. It is proved that most of the important equations of dissipative quantum physics, such as convection-diffusion, Fokker–Planck and quantum Boltzmann, have a common origin and can be unified in one equation.

Highlights

Since the early 20th century, the challenging problem of dissipation and diffusion modeling has been widely studied in quantum theory because a comprehensive understanding of dissipation in quantum mechanics is fundamental to the foundations of this theory [1,2]
Madelung showed that the two equations were mathematically equivalent [6,7], and if one writes the wave function in the form of eR+iS, the Schrödinger equation implies that, first, S is governed by a classical Hamilton–Jacobi-like equation, or alternatively that v = ∇S is formulated by a Newton-like equation; second, ρ (which is defined as ρ(x, t) = |ψ|2 = R(x, t)2) is governed by a classical continuity equation [8]
The only formal difference between these equations and their purely classical counterparts is the existence of an additional “quantum” potential. Since that time these equations have provided the basis for numerous classical interpretations of quantum mechanics, including the hydrodynamic interpretation first proposed by Madelung [6], the theory of stochastic mechanics due to Nelson and others [3,8,9,10,11,12], the hidden-variable and double-solution theories of Bohm and de Broglie [13,14,15] and quite possibly other interpretations as well [8,16,17]

Summary

Introduction

Since the early 20th century, the challenging problem of dissipation and diffusion modeling has been widely studied in quantum theory because a comprehensive understanding of dissipation in quantum mechanics is fundamental to the foundations of this theory [1,2]. The only formal difference between these equations and their purely classical counterparts is the existence of an additional “quantum” potential Since that time these equations have provided the basis for numerous classical interpretations of quantum mechanics, including the hydrodynamic interpretation first proposed by Madelung [6], the theory of stochastic mechanics due to Nelson and others [3,8,9,10,11,12], the hidden-variable and double-solution theories of Bohm and de Broglie [13,14,15] and quite possibly other interpretations as well [8,16,17]. Caldeira and Legget showed by using the influence-functional method [23] that dissipation tends to destroy quantum interference in a time scale shorter than the relaxation time of the system [22] This result has given justification for the use of logarithmic nonlinear wave equations [12,18,22,24] to describe quantum dissipation.

Madelung Decomposition of DSE

Applications

Convection–Diffusion Equation

Quantum Boltzmann Equation

Summery and Conclusions