Solutions for a q-generalized Schr"odinger equation of entangled interacting particles

We report on the time dependent solutions of the $q-$generalized Schr"odinger equation proposed by Nobre et al. [Phys. Rev. Lett. 106, 140601 (2011)]. Here we investigate the case of two free particles and also the case where two particles were subjected to a Moshinsky-like potential with time dependent coefficients. We work out analytical and numerical solutions for different values of the parameter $q$ and also show that the usual Schr"odinger equation is recovered in the limit of $q\rightarrow 1$. An intriguing behavior was observed for $q=2$, where the wave function displays a ring-like shape, indicating a bind behavior of the particles. Differently from the results previously reported for the case of one particle, frozen states appear only for special combinations of the wave function parameters in case of $q=3$.

💡 Research Summary

The paper investigates time‑dependent solutions of the q‑generalized nonlinear Schrödinger equation introduced by Nobre, Rego‑Monteiro and Tsallis (often called the NRT equation) for a system of two interacting particles. Starting from the Tsallis entropy S_q, the authors write the NRT equation (Eq. 2) which reduces to the ordinary linear Schrödinger equation when the deformation parameter q approaches 1. The study focuses on two scenarios: (i) two free particles and (ii) two particles subjected to a time‑dependent Moshinsky‑like potential V(x₁,x₂,t)=α(t)x₁²+β(t)x₂²+γ(t)x₁x₂+η(t).

For both cases the authors adopt a q‑Gaussian ansatz (Eq. 5) for the wavefunction ψ(x₁,x₂,t), introducing four time‑dependent coefficients a(t), b(t), c(t) and d(t). Substituting the ansatz into the NRT equation yields a coupled set of ordinary differential equations (Eqs. 8‑11 for the free case, and Eqs. 45‑48 when the external potential is present). The equations are highly nonlinear, and analytical solutions are only obtainable under restrictive initial conditions; otherwise the authors rely on numerical integration.

In the linear limit q→1 the coupled equations simplify to a set of complex Riccati‑type equations whose solutions decay as 1/t (a, b, d) and 1/t² (c) for large times. Numerical simulations (Fig. 2) show that |ψ|² evolves into a product of two Gaussian profiles, indicating that the non‑linear coupling disappears asymptotically and the particles behave as independent free particles.

For the strongly nonlinear case q=2 the authors impose a(t)=b(t) to make the problem tractable. Analytic expressions involving tangent and secant functions (Eqs. 34‑36) are derived. The resulting dynamics display an initial splitting into two peaks which later merge into a ring‑shaped probability density (Fig. 3). This ring indicates a self‑trapping or “binding” effect generated solely by the nonlinearity, a phenomenon previously reported for a single particle but now observed for an entangled pair.

When q=3 the equations further simplify: c(t) becomes constant and a(t)=b(t). The solution (Eqs. 41‑43) contains oscillatory exponential factors. The authors find that “frozen” states—time‑independent probability densities—appear only for very special parameter choices satisfying 4a(t)b(t)=c(t)² and d(t)=0. In the generic two‑particle situation the non‑linear coupling prevents true frozen states, although quasi‑stationary pulsations with period 2π/(αc₀) can occur for particular initial conditions.

The paper then extends the analysis to a Moshinsky‑like potential with time‑dependent coefficients. By setting α(t)=β(t)=η(t)=1 and γ(t)=e⁻ᵗ (strong initial coupling that decays), the authors solve the modified coefficient equations numerically. For q=1 the dynamics reduce to the linear case and the wavefunction retains the free‑particle shape (Fig. 6). For q=2 the same ring‑shaped binding emerges despite the presence of the external potential (Fig. 7). For q=3 the same restrictions on frozen states apply, and no permanent localization is observed.

Overall, the study demonstrates that the deformation parameter q controls qualitatively different behaviors: q=1 recovers standard quantum mechanics; q=2 induces self‑binding ring structures; q=3 allows only transient or highly constrained frozen configurations. The non‑linear term generates effective entanglement between the particles even in the absence of explicit interaction potentials. The work highlights both the analytical challenges and the utility of numerical methods for exploring non‑linear quantum dynamics, and suggests future extensions to many‑body systems, external fields, and possible experimental realizations.