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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 non-interacting 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 self-consistently according to the instantaneous value of the density. Equation describing such self-consistent dynamics is
called the Gross-Pitaevskii (GP) equation.
We rewrite Hamiltonian (3.1) in real space rather than momentum space 3.2. GROSS-PITAEVSKII EQUATION 9 representations. We also add external potential Vext (r, t) for generality
2m d3 r | Ψ|2 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