= Gaussian interaction energy of a trapped condensate
{c}
{title2=$E_I=N^2U_0/[2(2\pi)^{3/2}a_0^3]$}
For a normalized condensate orbital $\phi=\pi^{-3/4}a_0^{-3/2}e^{-r^2/(2a_0^2)}$ and $\psi=\sqrt N\phi$, the <contact interaction> functional is $E_I=(U_0/2)\int|\psi|^4d^3r$. The <Gaussian integral> gives $\int|\phi|^4d^3r=[(2\pi)^{3/2}a_0^3]^{-1}$. With $U_0=4\pi\hbar^2a/m$, this becomes $E_I/(N\hbar\omega)=Na/(\sqrt{2\pi}a_0)$. An exact number-state pair count replaces $N^2$ by $N(N-1)$.
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