1997/10/02 by Waldemar Puszkarz, Puszkarz, Waldemar
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Quantum Physics (quant-ph) #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/9710010
Latex, 21 pages, extended and slightly modified, new references added
openalex publication_date 1997/10/02 · arxiv created 1999/05/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in Δmln (Ψ*/Ψ) multiplied by Ψ*Ψ. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase S rather than the amplitude R of the wave function Ψ=Rexp (iS). From this general scheme we choose the simplest minimal model defined in some reasonable way. The model in question offers the simplest way to modify the Bohm formulation of quantum mechanics so as to allow a leading phase contribution to the quantum potential and a leading quantum contribution to the probability current removing asymmetries present in Bohm's original formulation. It preserves most of physically relevant properties of the Schrödinger equation including stationary states of quantum-mechanical systems. It can be thought of as the simplest model of nonlinear quantum mechanics of extended objects among other such models that also emerge within the general scheme proposed. The extensions of this model to n particles and the question of separability of compound systems are studied. It is noted that there exists a weakly separable extension in addition to a strongly separable one. The place of the general modification scheme in a broader spectrum of nonlinear modifications of the Schrödinger equation is discussed. It is pointed out that the models it gives rise to have a unique definition of energy in that the field-theoretical energy functional coincides with the quantum-mechanical one. It is found that the Lagrangian for its simplest variant represents the Lagrangian for a restricted version of the Doebner-Goldin modification of this equation.