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Unformatted text preview: atoms into
a single state and then found the state by minimizing the energy. We can
also have macroscopic condensation of particles into a single state that is not
stationary but undergoes dynamic evolution. This form of dynamics is exact
for noninteracting particles, when all particles undergo identical evolution determined by the external ﬁelds (assuming that all atoms started in the same
state). It is also a good approximation for weakly interacting particles. The
role of interactions is to provide an eﬀective ﬁeld acting on the atoms. This
eﬀective ﬁeld needs to be computed selfconsistently according to the instantaneous value of the density. Equation describing such selfconsistent dynamics is
called the GrossPitaevskii (GP) equation.
We rewrite Hamiltonian (3.1) in real space rather than momentum space 3.2. GROSSPITAEVSKII EQUATION 9 representations. We also add external potential Vext (r, t) for generality
H= 1
2m d3 r  Ψ2 d3 r Vext (r, t)Ψ† (r)Ψ(r) + +
− µ U0
2 d3 r Ψ† (r)Ψ† (...
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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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