Gross–Pitaevskii equation

The Gross- Pitaevskii equation ( to Eugene P. Gross and Lev Petrovich Pitajewski ) is in quantum mechanics a nonlinear generalization of the Schrödinger equation; they describe the time evolution of a macroscopic wave function:

The wave function is of the order parameter of the phase transition. The parameter describes whether the interaction is attractive ( ) or repulsive (). For eliminates the nonlinear interaction term and one obtains the Schrödinger equation.

The Gross- Pitaevskii equation plays an important role in the theoretical treatment of bosonic quantum liquids such as Bose - Einstein condensates (BEC ), superconductors and superfluids. It contains among other things, solitary solutions (nonlinear waves) and vortices ( quantized vortex ). It corresponds to a molecular field, with the interaction with the mean field of the other bosons in the nonlinear Term

Taking into account also electrically charged particles (charge, vector potential ), so you have to replace the momentum operator. In this case, from the Gross- Pitaevskii equation, the Ginzburg- Landau equation, which is the phenomenological description of superconductors.

Energy and dispersion

The dispersion relation is: