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arXiv:0912.4933

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

Ji Luo

Published Dec 29, 2009• 0 citations

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.

DOI: 10.48550/arxiv.0912.4933

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The time-dependent Schrödinger equation with a time-dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wavefunction at the origin, one may derive the wavefunction everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the potential lead to the conservation of the normalization of the probability density.