The present chapter is devoted to the theory of homogeneous
dilute Bose gases at zero temperature.
The mean-field theory for the dilute Bose gas is discussed in the
first part of the chapter. We derive the Gross-Pitaevskii equation
for the order parameter and we use it to calculate the ground-state
energy of the system. The small oscillations of the order parameter
around the equilibrium solution provide us with the elementary
excitation energies. The presentation of the material in this
section follows closely the review [6].
In the second part of the chapter we discuss the Bogoliubov model,
which is a theory beyond mean-field and takes into account the
fluctuations of the order parameter. We introduce Bogoliubov
effective Hamiltonian, discuss its diagonalization by means of the
Bogoliubov transformation and we calculate the corrections to the
of the ground state energy arising from the quantum fluctuations.
The excitation spectrum predicted by the Bogoliubov model agrees
with the one obtained from the time-dependent Gross-Pitaevskii
equation. Results for the number of noncondensed particles (quantum
depletion) and the one-body density matrix are also discussed. Much
of the treatment of this part of the chapter parallels closely the
one given in the book [11].
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