An amended Ehrenfest theorem for the Gross-Pitaevskii equation in one- and two-dimensional potential boxes

TL;DR

This paper derives and validates an amended Ehrenfest theorem for the Gross-Pitaevskii equation in 1D and 2D potential boxes, accounting for boundary effects and nonlinearity, with implications for Bose-Einstein condensate experiments.

Contribution

It introduces a modified Ehrenfest theorem for the nonlinear GPE in bounded potentials, including boundary-induced forces and confirming its validity through numerical simulations.

Findings

Amended ET includes boundary forces affecting COM motion.

Nonlinearity influences the irregularity of COM dynamics.

Validation through numerical comparison confirms the amended ET's accuracy.

Abstract

It is known that the usual form of the Ehrenfest theorem (ET), which couples the motion of the center of mass (COM) of the one-dimensional (1D) wave function to the respective classical equation of motion, is not valid in the case of the potential box, confined by the zero boundary conditions. A modified form of the ET was proposed for this case, which includes an effective force originating from the interaction of the 1D quantum particle with the box edges. In this work, we derive an amended ET for the Gross-Pitaevskii equation (GPE), which includes the cubic nonlinear term, as well as for the 2D square-shaped potential box. In the latter case, we derive an amended COM equation of motion with an effective force exerted by the edges of the rectangular box, while the nonlinear term makes no direct contribution to the 1D and 2D versions of the ET. Nonetheless, the nonlinearity affects the amended ET through the edge-generated force. As a result, the nonlinearity of the underlying GPE can make the COM motion in the potential box irregular. The validity of the amended ET for the 1D and 2D GPEs with the respective potential boxes is confirmed by the comparison of numerical simulations of the underlying GPE and the corresponding amended COM equation of motion. The reported findings are relevant to the ongoing experiments carried out for atomic Bose-Einstein condensates trapped in the box potentials.