  In the mean-field approximation, a trapped Bose-Einstein condensate at zero
temperature is described by the Gross-Pitaevskii equation for the condensate,
or equivalently, by the hydrodynamic equations for the number density and the
current density. These equations receive corrections from quantum field
fluctuations around the mean field. We calculate the semiclassical corrections
to these equations for a general time-independent state of the condensate,
extending previous work to include vortex states as well as the ground state.
In the Thomas-Fermi limit, the semiclassical corrections can be taken into
account by adding a local correction term to the Gross-Pitaevskii equation. At
second order in the Thomas-Fermi expansion, the semiclassical corrections can
be taken into account by adding local correction terms to the hydrodynamic
equations.
