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The simplest nonlinear Schrodinger equation that contains the time derivativeof the probability density is investigated. This equation has the samestationary solutions as its linear counterpart, and these solutions are theeigenstates of the corresponding linear Hamiltonian. The equation leads to theusual continuity equation and thus maintains the unitarity of the wavefunction. For the non-stationary solutions, numerical calculations are carriedout for the one-dimensional infinite square-well potential and for severaltime-dependent potentials that tend to the former as time increases. Resultsshow that for various initial states, the wave function always evolves intosome eigenstate of the corresponding linear Hamiltonian of the one-dimensionalinfinite square-well potential. For a small time-dependent perturbationpotential, solutions present the process similar to the spontaneous transitionbetween stationary states. For a periodical potential with an appropriatefrequency, solutions present the process similar to the stimulated transition.This nonlinear Schrodinger equation thus presents the state evolution similarto the wave-function reduction.

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