Consider a state obtained from 0 by creating l

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online