On the application of a gauss transformation in nonlinear quantum mechanics. II: Problems with spin

Abstract

The nonlinear and nonlocal Schrodinger equation with Gaussian kernel, generating in the local-limit case the cubic Schrodinger equation, can only be applied to spin-0 fields. There arises the question in which way the considerations can be extended to spin-1/2 fields. Taking account of the nonlocality and nonlinearity of such a field we have to be aware of an internal spin-orbit coupling in order to describe transitions between different multiplicity. The Pauli equation did not turn out to be a sufficient starting point, as the spin-orbit coupling is incorrectly obtained in a nonrelativistic approach. However, the assumption that the nonlinear term acts as internal potential is sufficient to derive the correct spin-orbit term in the nonlinear and nonlocal case. It is also possible to obtain an approximation method in the nonlocal domain, basing on a self-interacting 3-dimensional harmonic oscillator with spin. In the local limit, where the Gaussian distribution becomes a δ-distribution, the existence of solitonic solutions and the relationship to the superconductivity are discussed.