Nonlinear Schrodinger equation containing the time derivative of the probability density: A numerical study

TL;DR

This study investigates a nonlinear Schrödinger equation with a time derivative of probability density, demonstrating that it preserves stationary solutions, unitarity, and exhibits state evolution akin to wave-function reduction through numerical simulations.

Contribution

It introduces and numerically analyzes a nonlinear Schrödinger equation with a unique term, showing its solutions mimic quantum state transitions and maintain key physical properties.

Findings

Wave functions evolve into eigenstates of the linear Hamiltonian.

Solutions show spontaneous and stimulated transition behaviors.

The equation preserves unitarity and stationary solutions.

Abstract

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.