arXiv:quant-ph/9506033

Nonlinear Gauge Transformations and Exact Solutions of the Doebner-Goldin Equation

P. Nattermann, Wolfgang Scherer

Published Jun 21, 1995• 8 citations

Abstract

Invariants of nonlinear gauge transformations of a family of nonlinear Schrödinger equations proposed by Doebner and Goldin are used to characterize the behaviour of exact solutions of these equations.

DOI: 10.48550/arxiv.quant-ph/9506033

Related papers

Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations

H. D. Doebner, Gerald A. Goldin

In earlier work we proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary group representations [Doebner and Goldin, Phys. Lett. A 162, 397 (1992); J. Phys. A 27, 1771 (1994)]. Such nonlinear Schr\"odinger equations are expected to describe irreversible and dissipative quantum systems. Here we introduce and justify physically the group of nonlinear gauge transformations necessary to interpret our equations. We determine the parameters that are actually gauge invariant and describe some of their properties. Our conclusions contradict, at least in part, the view that any nonlinearity in quantum mechanics leads to unphysical predictions. We also show how time-dependent nonlinear gauge transformations connect our equations to those proposed by Kostin [J. Chem. Phys. 57, 3589 (1972)] and by Bialynicki-Birula and Mycielski [Ann. Phys. 100, 62 (1976)]. We believe our approach to be a fundamental generalization of the usual notions about gauge transformations in quantum mechanics. \textcopyright{} 1996 The American Physical Society.

Physical Review ANov 1, 1996134 citations

Gauge transformations in quantum mechanics and the unification of nonlinear Schrödinger equations

H. D. Doebner, Gerald A. Goldin, P. Nattermann

Beginning with ordinary quantum mechanics for spinless particles, together with the hypothesis that all experimental measurements consist of positional measurements at different times, we characterize directly a class of nonlinear quantum theories physically equivalent to linear quantum mechanics through nonlinear gauge transformations. We show that under two physically motivated assumptions, these transformations are uniquely determined: they are exactly the group of time-dependent, nonlinear gauge transformations introduced previously for a family of nonlinear Schrödinger equations. The general equation in this family, including terms considered by Kostin, by Bialynicki-Birula and Mycielski, and by Doebner and Goldin, with time-dependent coefficients, can be obtained from the linear Schrödinger equation through gauge transformation and a subsequent process we call gauge generalization. We thus unify, on fundamental grounds, a rather diverse set of nonlinear time evolutions in quantum mechanics.

arXiv:quant-ph/9709036Journal of Mathematical PhysicsJan 1, 199961 citations

Generalized nonlinear Doebner-Goldin Schrödinger equation and the relaxation of quantum systems

V. V. Dodonov, Salomon Sylvain Mizrahi

Physica A Statistical Mechanics and its ApplicationsApr 1, 19958 citations

Dissipative quantum mechanics. A special Doebner-Goldin equation, its properties and exact solutions

A.G. Ushveridze

Physics Letters AFeb 1, 199414 citations

Nonlinear gauge transformation for a class of Schrödinger equations containing complex nonlinearities

G. Kaniadakis, A.M. Scarfone

arXiv:quant-ph/0209075Reports on Mathematical PhysicsAug 1, 20008 citations

Generalized Gauge Invariants for Certain Nonlinear Schrodinger Equations

Gerald A. Goldin

tions #N (x 1,. xN ,t), defined on the (pos=6Tfiflx# configurationsonfig in a way that is s trictly local ands atis#+ as eparation condition for products tates [6]. The jusT#6flx=Fj forcons6Fflx=F them to be gaugetrans##FWflx=Fj is as follows For all of the nonlinear quantumtheories under dis#+jflx=j we interpret # = = R 2as the probability densfi y in configurationsnfigu We adopt as a working hypothes6 the view (taken by many theorisx= that all meas++flx= ts in ordinary quantum mechanics can be regarded as asTFW6=fl of pos#666flx measx=++T ts made at di#erent times where externalfields exerting forces may be impos= on thesefl=fi between measjfi6flx ts [7, 8]. Then for any wave function # obeying aSchrodinger equation (linear or nonlinear) in our family, the wave function # #transflx==W by (1.1) obeys a trans=flx## Schrodinger equation,suat in the family, withgauge-transFTfls externalfields while theoutcomes of all phys#Ffi measF#Wflx ts remain invariant. To be explicit, with #

Jan 1, 20001 citation