Unformatted text preview: is convenient to introduce
δρ =
δφ = √ n0 (δ Ψ + δ Ψ∗ )
1
√ (δ Ψ − δ Ψ∗ )
2i n0 (3.37) Then the last two equations can be written as
δn =
˙
˙
−δ φ = n0
m δφ ) U0 δn − 1
4mn0 −( (3.38)
2 δn (3.39) The ﬁrst equation can be understood as mass conservation. If we deﬁne the
superﬂuid current as js = (n0 /m) δ φ, we can rewrite equation (3.38) as δ n =
˙
˙
− js . Equation (3.39) is the socalled Josephson relation δ φ = δµ. Combining
the two equations we obtain
¨
δ φ = (U0 n0 − 2 2 ) 4m m δφ (3.40) Taking δφ ∼ φp eipx−iEp t we ﬁnd the collective mode dispersion given by equation (3.18). 3.3 Problems for Chapter 3 Problem 1
Let Ψ0 be the Bogoliubov ground state of a BEC. Consider a state obtained
from Ψ0 by creating l excitations with momentum +q
†l
α+q
√ Ψ0
l!
2
Verify by explicit calculation that this state contains lu2 + vq original (free)
q
2
particles with momentum +q and (l + 1)vq original (free) particles wi...
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 Fall '10
 EugeneDemler
 Physics, BoseEinstein condensation, Bogoliubov Theory, Bogoliubov quasiparticles

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