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1

In principle, it is very simple and straightforward. The problem is to map out the region where the integer filling state is the ground state. Suppose you have $L$ sites. Take $N=L$ particles, find its ground state energy, which is denoted as $E_g(L)$. Note that here the Hamiltonian does not contain the $\mu $ term. Do it again for $N=L+1$, the ground ...


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You simply take $|\psi|^2=1/g[\mu-V(x)]$. This because now your time independent GPE is $\mu\psi=(V+g|\psi|^2)\psi$


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The answer to the second question is actually quite straightforward: by computing $\partial_t E$ and using what given in the Gross-Pitaevskii for $\dot{\psi}$ and $\dot{\psi^*}$ one can check that all the terms cancel out so that $\partial_tE=0$. To derive the expression for the energy one could also start considering a lagrangian giving the ...


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Hints to the sought-for formula (16) for $\hat{H}$: Use integration by parts in ${\bf r}$-space to remove derivatives from the Dirac delta distributions, cf. comment by user ACuriousMind. Work on the problem from both ends (15) and (16). Use Leibniz rule $$\tag{*}\nabla^2 (fg)~=~ g\nabla^2 f + f \nabla^2 g+ 2 \nabla f\cdot\nabla g,$$ so that $\nabla$ only ...


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As long as you consider the BEC without inter-particle interactions (because they are negligible for instance) you can simply use the Schrodinger equation. However, if you want to take interactions into account you may want to consider to take the Thomas-Fermi approximation. When interactions are dominant in the dynamics of the system and the kinetic energy ...


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Yes, Bose-Einstein condensates are affected by gravity. Most condensates are formed in laser traps and often (especially in the early experiments) the lasers must be turned off to get a good image of the condensate, with the consequence that many images of condensates (again, especially from the early experiments) show them falling. An example (source):


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Atomic condensates are subjected to gravity as every massive particle is. What do you mean by 'in sufficient quantity'? In any theoretical treatment of a BEC I know of gravity is not taken into considerations as obviously the effects on the dynamics to the atomic level are negligible. However, yes, there is always a gravitational attraction between the ...



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