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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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This document was uploaded on 02/27/2014 for the course PHYS 284 at Harvard.
 Fall '10
 EugeneDemler
 Physics

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