lecture3

# Consider a state obtained from 0 by creating l

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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 so-called 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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