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Europhysics Letters (EPL)

A generalized nonlinear Schrödinger equation: Classical field-theoretic approach

Fernando Dantas Nobre, M. A. Rego-Monteiro, Constantino Tsallis

Published Feb 1, 2012• 57 citations
Publisher

Abstract

An exact classical field theory, for a recently proposed nonlinear generalization of the Schrödinger equation, is presented. In this generalization, a nonlinearity depending on an index q appears in the kinetic term, such that the free-particle linear Schrödinger equation is recovered in the limit q →1. It is shown that besides the usual , a new field must be introduced, which becomes only when q →1. In analogy to the linear case, is interpreted as the probability density for finding the particle at time t , in a given position inside an arbitrary finite volume V , for any q . The possible physical consequences are discussed, and, in particular, the important property that the fields and do not interact with light.

DOI
10.1209/0295-5075/97/41001

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An exact classical field theory, for a recently proposed nonlinear generalization of the Schrödinger equation, is presented. In this generalization, a nonlinearity depending on an index q appears in the kinetic term, such that the free-particle linear Schrödinger equation is recovered in the limit q →1. It is shown that besides the usual , a new field must be introduced, which becomes only when q →1. In analogy to the linear case, is interpreted as the probability density for finding the particle at time t , in a given position inside an arbitrary finite volume V , for any q . The possible physical consequences are discussed, and, in particular, the important property that the fields and do not interact with light.
A linear one-dimensional Schr\"odinger equation, defined by means of a non-Hermitian Hamiltonian characterized by position-dependent masses, was proposed lately. Herein we present an exact classical field theory for this equation, showing the need for an extra field $\ensuremath{\Phi}(x,t)$, in addition to the usual one, $\ensuremath{\Psi}(x,t)$, similar to what was done recently in the analysis of a class of nonlinear quantum equations. These generalizations of the Schr\"odinger equation depend on an index qqq, in such a way that the standard case is recovered in the limit $q\ensuremath{\rightarrow}1$. Particularly, the field $\ensuremath{\Phi}(x,t)$ becomes ${\ensuremath{\Psi}}^{*}(x,t)$ only when $q\ensuremath{\rightarrow}1$ and satisfies a similar Schr\"odinger equation for the Hermitian conjugate of the Hamiltonian operator. In terms of these two fields one may define a probability density following a standard continuity equation, leading to the preservation of probability in Cartesian space. Simple applications are performed by solving the equations for the two fields.
A Nonlinear qqq-Deformed Schrödinger Equation
We construct a new nonlinear deformed Schrödinger structure using a nonlinear derivative operator which depends on a parameter qqq. This operator recovers Newton derivative when q→1q \rightarrow 1q→1. Using this operator we propose a deformed Lagrangian which gives us a deformed nonlinear Schrödinger equation with a nonlinear kinetic energy term and a standard potential V(x⃗)V(\vec{x})V(x). We analytically solve the nonlinear deformed Schrödinger equation for V(x⃗)=0V(\vec{x}) = 0V(x)=0 and q≃1q \simeq1q≃1. This model has a continuity equation, the energy is conserved, as well as the momentum and also interacts with electromagnetic field. Planck relation remains valid and in all steps we easily recover the undeformed quantities when the deformation parameter goes to 1. Finally, we numerically solve the equation of motion for the free particle in any spatial dimension, which shows a solitonic pattern when the space is equal to one for particular values of qqq.
A possible generalization of quantum mechanics is examined by showing that the motion in phase space of a classical Brownian particle may be described by a complex probability amplitude depending on the phase-space coordinates and the time, and obeying a Schrödinger-like equation. However formal this result may seem, the usual dynamical operators may be defined whose physical meaning stems directly from the theory. An outstanding feature of the formalism is that ordinary quantum mechanics in configuration space may be recovered in a limiting process whereby the velocity variable, defined now through a statistical distribution, is eliminated. Therefore, it plays the role of a hidden variable. This result supports recent reinterpretations of von Neumann's theorem on the nonexistence of such variables in quantum mechanics and serves as a counterexample of the usual interpretation of his theorem.
Based on three principles taken from empiricism we establish a new nonlinear field theory which allows us to describe frictional effects in dissipative systems with the aid of a Schrödinger-type field equation with logarithmic nonlinearity. This nonlinear field equation corresponds to the classical Langevin equation and can be interpreted in different ways, taking the view of classical undulatory theory or probabilistic theories, respectively. Because of formal similarities, calculations can be performed independently of the accepted interpretation; only the occurring quantities have to be provided with the corresponding meaning. As an example, the nonlinear field equation for the damped harmonic oscillator is solved exactly. The solutions, a wavefunction and a wavepacketlike solution, a solution with Gaussian shape, exhibit reasonable properties, contain the correct reduced frequency Ω=(ω20−γ2/4)1/2 and are different from the solutions of the undamped problem. The properties of our nonlinear friction term are discussed and compared with those of similar nonlinear friction terms used by other authors. Finally, we derive a nonlinear friction term equivalent to our logarithmic nonlinearity but involving only combinations of suitably defined operators of position and linear momentum and mean values of these operators.
Quantization processes generally assume the Hamiltonian formulation of classical mechanics. The notion of dissipation does not fit easily into the Hamiltonian structure. There appeared in the literature a quantization process that circumvents the use of the Hamiltonian approach and derives the Schr\"odinger equation from first principles. Thus, the usual approach of assuming a dissipative force of the type fd=k(t)v{f}_{d}=k(t)vfd​=k(t)v, where k(t)k(t)k(t) is a scalar depending only on the time and vvv is the velocity, can be approached using this quantization process to mathematically derive a Schr\"odinger equation. The derivation for this simple case was already performed by the authors elsewhere. The full generalization of a dissipative force in the context of a linear response theory, as is usual in classical mechanics, would be to write this force as fi(x,t)=Kij(x,t)vj{f}_{i}(\mathbf{x},t)={\mathcal{K}}_{ij}(\mathbf{x},t){v}_{j}fi​(x,t)=Kij​(x,t)vj​, where K(x,t)\mathcal{K}(\mathbf{x},t)K(x,t) is a tensor giving the nonhomogeneity and nonisotropy of the process. In this paper, we present this generalization and connect our results with nonlinear Schr\"odinger-like equations already known from the literature.
Abstract A free particle may be characterized abstractly in terms of its invariance group. Both the Hamilton-Jacobi and Schrödinger equations for classical and quantum free particles are invariant under the same group, the Schrödinger group. The aim of this paper is to discuss possible restrictions on non-linear generalizations of quantum mechanics. The basic idea is to search for those non-linear generalization of quantum mechanics for which a description of the non-relativistic free particle includes a representation of the Schrödinger group, as in classical and quantum mechanics. This condition imposes strong constraints on possible models of free particles. To carry out such an investigation it is necessary to select an appropriate mathematical formalism, one which can describe physical systems of non-relativistic particles and allows for the incorporation of non-linearities in a natural way. Here the formalism of ensembles on configuration space is used, which is an approach that is capable of describing both classical and quantum systems and, in addition, allows for more general theories that describe mixed classical-quantum systems. Modifications of the equations involving derivatives of up to fourth order are considered