= Matched vortex energy in a parabolic condensate
{title2=$\Delta E_{\mathcal N}\simeq\pi n_c\hbar^2\mathcal N^2[\log(R/\xi)+L_{0|\mathcal N|}-1/2]/m$}
For a centered <quantum vortex> in a <two-dimensional Thomas–Fermi condensate>, use an intermediate radius $\xi_{\rm core}\ll L\ll R$. The approximately uniform core contributes $\pi n_c\hbar^2\mathcal N^2[\log(L/\xi)+L_{0|\mathcal N|}]/m$. The outer <superfluid velocity> is $\hbar\mathcal N/(mr)$, so its kinetic contribution is $(\pi\hbar^2\mathcal N^2/m)\int_L^R n_c(1-r^2/R^2)dr/r$. Integration cancels $L$ and produces the constant $-1/2$. The displayed result is the matched small-core approximation at fixed particle number, not a formula for arbitrarily large winding at fixed cloud size. The angular momentum is exactly $L_z=N\hbar\mathcal N$ for a normalized axisymmetric single-winding state.
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