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Unformatted text preview: nstead of (3.3) is that it has the correct number of particles on the average
Ψ0 |N |Ψ0 = N0 (3.10) 3.1. BOGOLIUBOV THEORY 5 and relative ﬂuctuations in the number of particles are negligible in the thermodynamic limit.
ΨN |( N − N0 )2 |ΨN ∆N = 1/2 1/2 = N0 (3.11) The beneﬁt of using the state (3.8) is that it dramatically simpliﬁes calculations.
To continue perturbation theory in U we apply the traditional methodology
of mean-ﬁeld approaches. We replace b± operators by their expectation values
in the ground state. The importance of diﬀerent terms is determined by the
number of b± factors, since each of them carries a large factor N0 . The
most important terms, where all operators are at p = 0, are given by equation
(3.4). The next contribution comes from terms that have two operators at
non-zero momentum, which gives us the mean ﬁeld Hamiltonian
HMF = − 2
2V ( p (b† b† p + bp b−p )
p− + 2n0 U0 − µ)(b† bp + b† p b−p ) + n0 U0
p=0 p=0 (3.12)
In summations p=0 momentum pairs p, -p should be...
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- Fall '10