Journal of Nonlinear Mathematical Physics

Nonlinear Schrödinger Equations and the Separation Property

Gerald A. Goldin, George Svetlichny

Published Jan 1, 1995• 0 citations

Abstract

We investigate hierarchies of nonlinear Schrödinger equations 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 Galileian-invariant hierarchies are given.

DOI: 10.2991/jnmp.1995.2.2.3

Related papers

Nonlinear Schrödinger equations and the separation property

Gerald A. Goldin, George Svetlichny

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.

Journal of Mathematical PhysicsJul 1, 199427 citations

Non-linear Schrödinger Equations, Separation and Symmetry

George Svetlichny

We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [2], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property).We show that there are obstructions to lifting symmetries existing at a certain number of particles to higher numbers.Such obstructions vanish for particles without internal degrees of freedom and the usual space-time symmetries.For particles with internal degrees of freedom, such as spin, these obstructions are present and their circumvention requires a choice of a new term in the equation for each particle number.A Lie-algebra approach for non-linear theories is developed.

Journal of Nonlinear Mathematical PhysicsJan 1, 19956 citations

Non-linear Schroedinger Equations, Separation and Symmetry

George Svetlichny

We investigate the symmetry properties of hierarchies of non-linear Schroedinger equations (introduced by Doebner and Goldin, and Goldin and Svetlichny), which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property). We show that there are obstructions to lifting symmetries existing at a certain number of particles to higher numbers. Such obstructions vanish for particles without internal degrees of freedom and the usual space-time symmetries. For particles with internal degrees of freedom, such as spin, these obstructions are present and their circumvention requires a choice of a new term in the equation for each particle number. A Lie-algebra approach for non-linear theories is developed.

arXiv:quant-ph/9906117Jun 29, 19990 citations

On linearity of separating multiparticle differential Schrödinger operators for identical particles

George Svetlichny

We show that hierarchies of differential Schrödinger operators for identical particles which are separating for the usual (anti-)symmetric tensor product, are necessarily linear, and offer some speculations on the source of quantum linearity.

arXiv:quant-ph/0308001Journal of Mathematical PhysicsFeb 12, 20045 citations

Derivation of the nonlinear Schrödinger equation from a many-body Coulomb system

László Erdős, Horng‐Tzer Yau

We consider the time evolution of iV bosonic particles interacting via a mean field Coulomb potential.Suppose the initial state is a product wavefunction.We show that at any finite time the correlation functions factorize in the limit N -> oo.Furthermore, the limiting one particle density matrix satisfies the nonlinear Hartree equation.The key ingredients are the uniqueness of the BBGKY hierarchy for the correlation functions and a new apriori estimate for the many-body Schrodinger equations.

arXiv:math-ph/0111042Advances in Theoretical and Mathematical PhysicsJan 1, 2001254 citations

Nonlinear Quantum Search / - eScholarship

Thomas G. Wong

Although quantum mechanics is linear, there are nevertheless quantum systems with multiple interacting particles in which the effective evolution of a single particle is governed by a nonlinear equation. This includes Bose-Einstein condensates, which are governed by the Gross-Pitaevskii equation, which is a cubic nonlinear Schrodinger equation with a term proportional to /[psi]/ ²[psi] scaling of Grover's algorithm. Since the Gross- Pitaevskii equation effectively approximates the multi- particle Schrodinger equation, for which Grover's algorithm is optimal, our result leads to a quantum information-theoretic bound on the number of particles needed for this approximation to hold, asymptotically. The Gross-Pitaevskii equation is not the only nonlinearity of the form f(/[psi]/²)[psi] that arises in effective equations for the evolution of real quantum physical systems, however : The cubic-quintic nonlinear Schrodinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schrodinger equation describes Bose liquids under certain conditions. Analysis of computation with such systems yields some surprising results; for example, when time-measurement precision is included in the resource accounting, searching a database'' when there is a single correct answer may be easier than searching when there are multiple correct answers. These results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically. Furthermore, strongly regular graphs, which have no global symmetry, are sufficiently complete for quantum search on them to asymptotically behave like unstructured search. Certain sufficiently complete graphs retain the improved runtime and resource scalings for some nonlinearities, so our scheme for nonlinear, analog quantum computation retains its benefits even when some structure is introduced

Jan 1, 20140 citations