Unformatted text preview: 0 )N vac
p
N! (3.3) where vac is the vacuum state. It is natural to take this state as zeroth
order approximation for ﬁnite but small interactions. Expectation value of the
3 4CHAPTER 3. BOSEEINSTEIN CONDENSATION OF WEAKLY INTERACTING ATOMIC GASES
Hamiltonian (3.1 in this state is
E = −µN0 + U0 2
N
2V 0 (3.4) Minimizing with respect to N0 we ﬁnd relation between the number of particles
and the chemical potential
N0
µ
=
V
U0 (3.5) To proceed to the next order in the interaction it is convenient to introduce the
idea of broken symmetry.
When we consider two point correlation functions for the state (3.3)
N0
V (3.6) bp eipr (3.7) ΨN Ψ† (r2 )Ψ(r1 )ΨN =
1
Ψ(r) = √
V p they do not depend on the relative distance between the two points. This
is the deﬁnition of the long range order. Naively one expects property (3.6)
to hold when individual expectation values of Ψ(r) and Ψ† (r) are equal to
(N0 /V )1/2 . However individual expectation values of these operators vanish
since they change the number of particles by one and state ...
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 Fall '10
 EugeneDemler
 Physics, BoseEinstein condensation, Bogoliubov Theory, Bogoliubov quasiparticles

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