Thus we can turn this operator equation into

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Unformatted text preview: r)Ψ(r)Ψ(r) d3 r Ψ† (r)Ψ(r) (3.31) We use canonical commutation relations of Ψ operators Ψ(r), Ψ† (r ) = δ (r − r ) [Ψ(r), Ψ(r )] = 0 (3.32) and write Heisenberg equations of motion i ˆ ∂ Ψ(r) ∂t ˆ = − H, Ψ(r) =− 1 2m 2ˆ ˆ ˆ ˆ ˆ ˆ Ψ(r) + Vext (r, t)Ψ(r) + U0 Ψ† (r)Ψ(r)Ψ(r) − µΨ(r) (3.33) ˆ We put Ψ to emphasize that at this point this is an exact operator equation of motion. However we want to describe states that have finite expectation values of Ψ(r, t) . Thus we can turn this operator equation into classical differential equations i 1 ∂ Ψcl (r) =− ∂t 2m 2 Ψcl (r) + Vext (r, t)Ψcl (r) + U0 Ψ† (r)Ψcl (r)Ψcl (r) − µΨcl (r) cl (3.34) Here Ψcl emphasizes that this is now differential equation on a classical field. Another way of thinking about the GP equation is to consider generalization of state (3.8) to time and space dependent wavefunction |Ψ(t) = C e R ˆ d3 r Ψcl (r,t)Ψ† (r ) |vac (3.35) Here C is normalization constant. We can think of state (3.35) as a time dependent variational...
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