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Ehrenfest theorem, Galilean invariance and nonlinear Schr"odinger equations

German Kälbermann

Published Jul 3, 2003• 9 citations
OpenAlex

Abstract

Galilean invariant Schr"odinger equations possessing nonlinear terms coupling the amplitude and the phase of the wave function can violate the Ehrenfest theorem. An example of this kind is provided. The example leads to the proof of the theorem: A Galilean invariant Schr"odinger equation derived from a lagrangian density obeys the Ehrenfest theorem. The theorem holds for any linear or nonlinear lagrangian.

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Galilean invariant Schr"odinger equations possessing nonlinear terms coupling the amplitude and the phase of the wave function can violate the Ehrenfest theorem. An example of this kind is provided. The example leads to the proof of the theorem: A Galilean invariant Schr"odinger equation derived from a lagrangian density obeys the Ehrenfest theorem. The theorem holds for any linear or nonlinear lagrangian.
The authors recently derived a family of nonlinear Schrodinger equations on R 3 from fundamental considerations of generalized symmetry: ih(cross) delta t psi =-(h(cross) 2 /2m) Del 2 psi +F( psi , psi ) psi +ih(cross)D( Del 2 psi +( mod Del psi mod 2 / mod psi mod 2 ) psi ), where F is an arbitrary real functional and D a real, continuous quantum number. These equations, descriptive of a quantum mechanical current that includes a diffusive term, correspond to unitary representations of the group Diff(M) parametrized by D, where M=R 3 is the physical space. In the present paper we explore the most natural ansatz for F, which is labelled by five real coefficients. We discuss the invariance properties, describe the stationary states and some non-stationary solutions, and determine the extra, dissipative terms that occur in the Ehrenfest theorem. We identify an interesting, Galilean-invariant subfamily whose properties we investigate, including the case where the dissipative terms vanish.
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.
Hierarchies of nonlinear Schrödinger equations were investigated for multiparticle systems, satisfying the separation property, i.e., where product wave functions evolve by the separate evolution of each factor. Such a hierarchy defines a nonlinear derivation on tensor products of the single-particle wave-function space, and satisfies a certain homogeneity property characterized by two new universal physical constants. A canonical construction of hierarchies is derived that allows the introduction, at any particular ‘‘threshold’’ number of particles, of truly new physical effects absent in systems having fewer particles. In particular, if single quantum particles satisfy the usual (linear) Schrödinger equation, a system of two particles can evolve by means of a fairly simple nonlinear Schrödinger equation without violating the separation property. Examples of Galilean-invariant hierarchies are given.
Phase and amplitude of the complex wave function are not independent of each other, but coupled, which becomes obvious looking at Madelung's hydrodynamic formulation of quantum mechanics. In the time-independent case, this leads to a kind of conservation law that allows for the reformulation of the linear Schrodinger equation in terms of a nonlinear Ermakov equation which is equivalent to a complex Riccati equation where the quadratic term in this equation explains the origin of the phase-amplitude coupling. A similar conservation law and corresponding nonlinear equations can also be found in the time-dependent case. The gain from the nonlinear formulations emerges when open systems with dissipation and irreversibility are considered. Describing this kind of systems by an effective nonlinear Schrodinger equation leads to a modification of the above-mentioned equations with new qualitative effects like Hopf bifurcations.
Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.