  In the mean-field approximation, the number density \rho(r) for the ground
state of a Bose-Einstein condensate trapped by an external potential V(r)
satisfies a classical field equation called the Gross-Pitaevskii equation. We
show that quantum corrections to \rho are dominated by quantum fluctuations
with wavelengths of order 1/\sqrt{\rho a}, where a is the S-wave scattering
length. By expanding the equations for the Hartree-Fock approximation to second
order in the gradient expansion, we derive local correction terms to the
Gross-Pitaevskii equation that take into account the dominant effects of
quantum fluctuations. We also show that the gradient expansion for the density
breaks down at fourth order.
