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

A nonlinear Schrِdinger equation that contains the time-derivative of the probability density is investigated, which is motivated by the attempt to include the recoil effect of radiation. This equation has the same stationary solutions as its linear counterpart, and these solutions are the eigen-states of the corresponding linear Hamiltonian. The equation leads to the usual continuity equation and thus maintains the normalization of the wave function. For the non-stationary solutions, numerical calculations are carried out for the one-dimensional infinite square-well potential (1D ISWP) 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 eigen-state of the corresponding linear Hamiltonian of the 1D ISWP. 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 Schrِdinger equation thus presents the state evolution similar to the wave-function reduction.