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Journal of Physics A Mathematical and General

A new class of nonlinear generalizations of the Schrodinger equation

V. V. Dodonov, Salomon Sylvain Mizrahi

Published Dec 7, 1993• 11 citations
Publisher

Abstract

We propose a new, nonlinear representation-independent generalized Schrodinger equation, satisfying the homogeneity condition, which is valid for arbitrary representations and arbitrary operators. This is a generalization of the equation recently proposed by Doebner and Goldin(1992) for the wavefunction written in Cartesian coordinates. The new model leads to simple exact solutions describing, for instance, the relaxation of the two-level system and the harmonic oscillator, both in Fock and in coherent states bases.

DOI
10.1088/0305-4470/26/23/050

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We propose a new, nonlinear representation-independent generalized Schrodinger equation, satisfying the homogeneity condition, which is valid for arbitrary representations and arbitrary operators. This is a generalization of the equation recently proposed by Doebner and Goldin(1992) for the wavefunction written in Cartesian coordinates. The new model leads to simple exact solutions describing, for instance, the relaxation of the two-level system and the harmonic oscillator, both in Fock and in coherent states bases.
Based on Gaussian wave packet solutions of the time-dependent Schrödinger equation, a generalization of the conventional creation and annihilation operators and the corresponding coherent states can be obtained. This generalization includes systems where also the width of the coherent states is time-dependent as they occur for harmonic oscillators with time-dependent frequency or systems in contact with a dissipative environment. The key point is the replacement of the frequency ω 0 that occurs in the usual definition of the creation/annihilation operator by a complex time-dependent function that fulfils a nonlinear Riccati equation. This equation and its solutions depend on the system under consideration and on the (complex) initial conditions. Formal similarities also exist with supersymmetric quantum mechanics. The generalized creation and annihilation operators also allow to construct exact analytic solutions of the free motion Schrödinger equation in terms of Hermite polynomials with time-dependent variable.
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)Δ^{m}\ln (Ψ^{*}/Ψ)Δmln(Ψ∗/Ψ) multiplied by Ψ∗ΨΨ^{*}ΨΨ∗Ψ. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase SSS rather than the amplitude RRR of the wave function Ψ=Rexp⁡(iS)Ψ=R\exp (iS)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 nnn 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.
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.