The Hamiltonian of a system of spinless bosons, interacting through the pair
potential and immersed in the external field is given,
in second quantization, by
(1.1)
here is the mass of a particle,
and
are the boson field operators that
annihilate and create a particle at the position .
If the gas is dilute and cold, then the two-body potential can be
replaced by the pseudopotential
which is fixed by a single parameter, the -wave
scattering length , through the coupling constant
The field operator can decomposed as
, where
are
single-particle wavefunctions with quantum number . The
bosonic creation and annihilation operators and
are defined in Fock space through the relations
(1.4)
(1.5)
where are the eigenvalues of the operator
giving the number of atoms
in the single-particle state . The operators
and
obey the usual bosonic commutation
rules
(1.6)
Bose-Einstein condensation occurs when the number of particles in
one particular single-particle state becomes very large
. In this limit the states with and
correspond to the same physical
configuration and, consequently, the operators
and can be treated as complex numbers
(1.7)
For a uniform gas in a volume the good single-particle states
correspond to momentum states and BEC occurs in the single-particle
state
having zero momentum. Thus, the field
operator
can be decomposed in the form
.
The generalization for the case of nonuniform and time-dependent
configurations is given by
(1.8)
where the Heisenberg representation for the field operators is
used. Here
is a complex function defined as the
expectation value of the field operator
.
The function
is a classical field having the
meaning of an order parameter and is often called the
wave-function of the condensate. The mean-field theory, which
describes the behavior of the classical field
and
ignores the fluctuations
is contained in the
Gross-Pitaevskii theory. A more refined approach, which takes into
account the fluctuations of the field operator was proposed by
Bogoliubov.
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