In the absence of the external potential , the
Hamiltonian (1.1) can be conveniently
expressed in momentum space
(1.19)
where the summation is carried out over all indices that appear
twice. By assuming that the relevant scattering processes involve
particles at low momenta, the matrix elements in the Hamiltonian
(1.19) can be replaced by their values at zero
momenta, then
(1.20)
In a dilute gas almost all particles are found in the condensed
state
,
then, as it was already discussed above (see eq.
(1.7)) the operators and
can be treated as ordinary numbers.
The application of perturbation theory means that the last term in
(1.20) should be decomposed in powers of the
small quantities
and
,
with . The zeroth term is
(1.21)
The first order terms are absent because they do not satisfy the
law of momentum conservation. The second order terms are
(1.22)
Here the
factor can be substituted with the total
number of particles , although in equation () it is necessary to use the more precise formula
(1.23)
As a result the sum of equations (1.21) and
(1.22) becomes equal to
The matrix element has to be expressed in terms of
the scattering length . In the second order terms this
can be done using the first Born approximation
, although in the zeroth order term
one should use the second Born approximation for collisions of
two particles from the condensate
(1.26)
or by introducing the speed of sound (see equation (1.18))
In order to calculate the energy levels of the system
one has to diagonalize the Hamiltonian (1.29).
This can be accomplished by using the Bogoliubov canonical transformation
of the field operators [15].
The operators
and
should be
expressed as a linear superposition of the qusiparticle operators
and
(1.30)
which have to satisfy the same commutation rules as the operators
(see eq.(1.6))
(1.31)
From the commutation rules (1.31) one can show that
the coefficients must satisfy the condition
.
The transformation (1.30) can be rewritten as
(1.32)
Let us substitute (1.32) into the Hamiltonian (1.25)
and set to zero the coefficient of the term proportional to
. This gives an equation for
(1.33)
which has two solutions
(1.34)
The solution with negative sign is unphysical, because the
term in the square root in (1.32) becomes
negative. Thus, the solution is
(1.35)
where stands for
(1.36)
The condition that the coefficient of the term proportional to
be zero gives the same
equation (1.33). Thus, if condition (1.35) is satisfied,
the Hamiltonian has been diagonalized and has the form
(1.37)
where
(1.38)
From the Hamiltonian (1.37) and the commutation rules
(1.31) one can identify
and
as the creation and annihilation operators of
quasiparticles with energy . The ground state energy is given
by which is the energy of the ``vacuum'' of quasiparticles
, where the ``vacuum'' state is
defined as
for any value of
The excited states are given by
and have energy and momentum .
It is interesting to note that the spectrum (1.36) of the
elementary excitations was already obtained in () from the Gross-Pitaevskii equations by considering small
oscillations of the order parameter around the stationary solution.
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