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Perturbed solution
We start from the three-dimensional energy functional (1.110) of a homogeneous
weakly-interacting Bose gas in the presence of a -function perturbation (an
impurity) moving with a constant velocity
|
(7.1) |
where is the condensate wave function, is the chemical potential,
mass of a particle in the condensate,
and
are particle-particle and particle-impurity coupling constants, with and
being the respective scattering lengths7.1. We will assume that the
interaction with impurity is small and we will use perturbation theory. By splitting
the wave function into a sum of the unperturbed solution and a small correction
and linearizing the time-dependent GP equation
with respect to , we obtain an equation describing the time
evolution of
|
|
|
(7.2) |
In a homogeneous system is a constant fixed by the particle density
and .
The perturbation follows the moving impurity, i.e. is a function of
, so the coordinate derivative is related to
the time derivative
|
|
|
(7.3) |
We shall work in the frame moving with the impurity
and the subscript over will be dropped.
Eq. (7.2) for a perturbation in a homogeneous system can be
conveniently solved in momentum space. In order to do this we introduce the
Fourier transform of the wave function
. Eq. (7.2
) becomes
|
(7.4) |
Here the second equation is obtained by doing the substitution
and and complex conjugation. We also use property of the Fourier
transformation
. The system
of linear equations (7.4) can be easily solved
|
(7.5) |
Next: Total energy
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G.E. Astrakharchik
15-th of December 2004