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Physical Review A

Stationary and uniformly accelerated states in nonlinear quantum mechanics

Angel Ricardo Plastino, André M. C. Souza, Fernando Dantas Nobre, Constantino Tsallis

Published Dec 31, 2014• 23 citations
Publisher

Abstract

We consider two kinds of solutions of a recently proposed field theory leading to a nonlinear Schr\"odinger equation exhibiting solitonlike solutions of the power-law form ${e}_{q}^{i(kx\ensuremath{-}wt)}$, involving the $q$ exponential function naturally arising within nonextensive thermostatistics [${e}_{q}^{z}\ensuremath{\equiv}{[1+(1\ensuremath{-}q)z]}^{1/(1\ensuremath{-}q)}$, with ${e}_{1}^{z}={e}^{z}$]. These fundamental solutions behave like free particles, satisfying $p=\ensuremath{\hbar}k$, $E=\ensuremath{\hbar}\ensuremath{\omega}$, and $E={p}^{2}/2m$ $(1\ensuremath{\le}q<2)$. Here we introduce two additional types of exact, analytical solutions of the aforementioned field theory. As a first step we extend the theory to situations involving a potential energy term, thus going beyond the previous treatment concerning solely the free-particle dynamics. Then we consider both bound, stationary states associated with a confining potential and also time-evolving states corresponding to a linear potential function. These types of solutions might be relevant for physical applications of the present nonlinear generalized Schr\"odinger equation. In particular, the stationary solution obtained shows an increase in the probability for finding the particle localized around a certain position of the well as one increases $q$ in the interval $1\ensuremath{\le}q<2$, which should be appropriate for physical systems where one finds a low-energy particle localized inside a confining potential.

DOI
10.1103/physreva.90.062134

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We consider two kinds of solutions of a recently proposed field theory leading to a nonlinear Schr"odinger equation exhibiting solitonlike solutions of the power-law form eqi(kx\ensuremath−wt){e}_{q}^{i(kx\ensuremath{-}wt)}eqi(kx\ensuremath−wt)​, involving the qqq exponential function naturally arising within nonextensive thermostatistics [eqz\ensuremath≡[1+(1\ensuremath−q)z]1/(1\ensuremath−q){e}_{q}^{z}\ensuremath{\equiv}{[1+(1\ensuremath{-}q)z]}^{1/(1\ensuremath{-}q)}eqz​\ensuremath≡[1+(1\ensuremath−q)z]1/(1\ensuremath−q), with e1z=ez{e}_{1}^{z}={e}^{z}e1z​=ez]. These fundamental solutions behave like free particles, satisfying p=\ensuremathℏkp=\ensuremath{\hbar}kp=\ensuremathℏk, E=\ensuremathℏ\ensuremathωE=\ensuremath{\hbar}\ensuremath{\omega}E=\ensuremathℏ\ensuremathω, and E=p2/2mE={p}^{2}/2mE=p2/2m (1\ensuremath≤q<2)(1\ensuremath{\le}q<2)(1\ensuremath≤q<2). Here we introduce two additional types of exact, analytical solutions of the aforementioned field theory. As a first step we extend the theory to situations involving a potential energy term, thus going beyond the previous treatment concerning solely the free-particle dynamics. Then we consider both bound, stationary states associated with a confining potential and also time-evolving states corresponding to a linear potential function. These types of solutions might be relevant for physical applications of the present nonlinear generalized Schr"odinger equation. In particular, the stationary solution obtained shows an increase in the probability for finding the particle localized around a certain position of the well as one increases qqq in the interval 1\ensuremath≤q<21\ensuremath{\le}q<21\ensuremath≤q<2, which should be appropriate for physical systems where one finds a low-energy particle localized inside a confining potential.
We present some physically interesting, in general non-stationary, one-dimensional solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The solutions include a coherent state, a phase-modified Gaussian wave packet in the potential of harmonic oscillator whose strength varies in time, a free Gaussian soliton, and a similar soliton in the potential of harmonic oscillator comoving with the soliton. The last of these solutions implies that there exist an energy level in the spectrum of harmonic oscillator which is not predicted by the linear theory. The free solitonic solution can be considered a model for a particle aspect of the wave-particle duality embodied in the quantum theory. The physical size of this particle is naturally rendered equal to its Compton wavelength in the subrelativistic framework in which the self-energy of the soliton is assumed to be equal to its rest-mass energy. The solitonic solutions exist only for the negative coupling constant for which the Gaussian wave packets must be larger than some critical finite size if their energy is to be bounded, i.e., they cannot be point-like objects.
Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]
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.
We have previously formulated a simple criterion for deducing the intervals of oscillations in the solutions of second-order linear homogeneous differential equations. In this work, we extend analytically the same criterion to the cubic nonlinear Schrödinger equations that describe Bose-Einstein condensates. With this criterion guiding the search for solutions, we classify all types of solutions and we find new stationary solutions in the free-particle cases that were not noticed previously because of limited coverage in the adopted boundary conditions. The new solutions are produced by the nonlinear terms of the differential equations and they continue to exist when various external potentials are also incorporated. Surprisingly, these solutions appear when the nonlinearities are small.
The simplest nonlinear Schrodinger equation that contains the time derivative of the probability density is investigated. This equation has the same stationary solutions as its linear counterpart, and these solutions are the eigenstates of the corresponding linear Hamiltonian. The equation leads to the usual continuity equation and thus maintains the unitarity of the wave function. For the non-stationary solutions, numerical calculations are carried out for the one-dimensional infinite square-well potential and for several time-dependent potentials that tend to the former as time increases. Results show that for various initial states, the wave function always evolves into some eigenstate of the corresponding linear Hamiltonian of the one-dimensional infinite square-well potential. For a small time-dependent perturbation potential, solutions present the process similar to the spontaneous transition between stationary states. For a periodical potential with an appropriate frequency, solutions present the process similar to the stimulated transition. This nonlinear Schrodinger equation thus presents the state evolution similar to the wave-function reduction.
The dynamics of nonlinear solitary waves in studied by using the model of nonlinear Schrodinger equations(NSE) with an external harmonic potential. The model allows one to analyze on the general basis a variety of nonlinear phenomena appearing both in Bose-Einstein condensate in a magnetic trap, whose profile is describe by a quadratic function of coordinates, and an nonlinear optics, physics of lasers, and biophysics. It is shown that exact solutions for a quantum-mechanical particle in a harmonic potential and solutions obtained within the framework of the adiabatic perturbation theory of bright solitons in a parabolic trap are completely identical. This fact not only proves once more that solitons behave like particles but also that they can preserve such properties in different traps for the which the parabolic approximation is valid near potential energy minima. The conditions are found for formation of stable stationary states of antiphase soliton in a harmonic potential. The interaction dynamics of solitons in nonstationary potentials is studied and the possibility of the appearance of a soliton parametric resonance at with the amplitude of soliton oscillation in a trap exponentially increases with time is shown.