Rapidly rotating Bose-Einstein condensates∗ Alexander Fetter ...

• generalize to time-dependent GP equation
i ∂Ψ
∂t = (T + V tr + V H ) Ψ
• this result implies that stationary solutions have time
dependence exp(−iµt/)
Introduce hydrodynamic variables
• write Ψ(r,t) = |Ψ(r,t)| exp [iS(r,t)] with phase S
• condensate density is n(r,t) = |Ψ(r,t)| 2
• current is
j =
2Mi [Ψ∗ ∇Ψ − Ψ∇Ψ ∗ ] = |Ψ| 2 ∇S
M
• identify last factor as velocity v = ∇S/M
• note that v is irrotational so ∇ × v = 0
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