Stationary and uniformly accelerated states in nonlinear quantum mechanics

Abstract

We consider two kinds of solutions of a recently proposed field theory leading to a nonlinear Schrödinger equation exhibiting solitonlike solutions of the power-law form eqi(kx-wt), involving the q exponential function naturally arising within nonextensive thermostatistics [eqz≡[1+(1-q)z]1/(1-q), with e1z=ez]. These fundamental solutions behave like free particles, satisfying p=k, E=ω, and E=p2/2m (1≤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 Schrödinger 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≤q<2, which should be appropriate for physical systems where one finds a low-energy particle localized inside a confining potential.