A Nonlinear $q$-Deformed Schrödinger Equation

TL;DR

This work develops a q-deformed nonlinear Schrödinger framework by replacing the kinetic term with a nonlinear derivative operator $\mathbb{D}^{(q)}_{x}$, yielding a qNLSE that reduces to the standard Schrödinger equation as $q\to1$. A deformed Lagrangian and corresponding equation of motion are derived, establishing gauge invariance with covariant derivatives and a generalized continuity equation for the deformed density $\rho=(\Psi^{*}\Psi)^q$. The Hamiltonian formulation is obtained via discretization, confirming energy and momentum conservation with a $q$-dependent correction, and both perturbative (for $V=0$) and numerical analyses are performed. Numerical results in 1D reveal density patterns that transition from plane waves to oscillatory states and, for $q<0$, soliton-like profiles with finite norm, illustrating rich nonlinear quantum dynamics while recovering standard quantum mechanics at $q\to1$. The framework also accommodates electromagnetic interactions through an effective charge scaling and connects to nonextensive statistics through the $q$-derivative, offering a path to explore solitons and nonextensive effects in quantum systems.

Abstract

We construct a new nonlinear deformed Schrödinger structure using a nonlinear derivative operator which depends on a parameter $q$. This operator recovers Newton derivative when $q \rightarrow 1$. 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 $V(\vec{x})$. We analytically solve the nonlinear deformed Schrödinger equation for $V(\vec{x}) = 0$ and $q \simeq1$. 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 $q$.

Figures (8)

• Figure 1: $\rho$ versus $\xi = \vec{k}.\vec{x}-\omega t$ for $q = 1.4$.
• Figure 2: $\rho$ versus $\xi = \vec{k}.\vec{x}-\omega t$ for $q = 1.1$.
• Figure 3: $\rho$ versus $\xi = \vec{k}.\vec{x}-\omega t$ for $q = 1$.
• Figure 4: $\rho$ versus $\xi = \vec{k}.\vec{x}-\omega t$ for $q = 0.9$.
• Figure 5: $\rho$ versus $\xi = \vec{k}.\vec{x}-\omega t$ for $q = 0.6$.