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## Dilute Bose gas

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
 (1.2)

The Hamiltonian (1.1) takes the form
 (1.3)

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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