In order to derive the equation for the wavefunction of the
condensate
one has to write the time evolution of
the field operator
using the Heisenberg
equation
(1.9)
Substitution of the Hamiltonian (1.3) into
(1.9) gives
(1.10)
We now replace the field operator
with the
classical field
. Then, the following equation for
the order parameter is obtained
(1.11)
This equation is called Gross-Pitaevskii equation [12,13,14] and describes the time evolution of the order
parameter.q