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# 10 31 bogoliubov theory 5 and relative uctuations in

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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 p=0 in the ground state. The importance of diﬀerent terms is determined by the 1 /2 number of b± factors, since each of them carries a large factor N0 . The p=0 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 N0 U0 + 2V ( p (b† b† p + bp b−p ) p− + 2n0 U0 − µ)(b† bp + b† p b−p ) + n0 U0 p − p=0 p=0 (3.12) In summations p=0 momentum pairs p, -p should be...
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