= Condensate number current
{title2=$\mathbf j=2Dn\nabla S$}
For $i\psi_t=-D\nabla^2\psi+[V(n)+i\Gamma(n)]\psi$, with real $D,V,\Gamma$ and $\psi=\sqrt n e^{iS}$, the <continuity equation> is
$$
\partial_t n+\nabla\cdot\mathbf j=2\Gamma n,\qquad
\mathbf j=2D\operatorname{Im}(\psi^*\nabla\psi)=2Dn\nabla S.
$$
Multiply the equation by $\psi^*$ and subtract its <complex conjugate> multiplied by $\psi$ to obtain this identity; real interaction and potential terms cancel. Thus the current <velocity> is $\mathbf j/n=2D\nabla S$. With dimensional kinetic operator $-\hbar^2\nabla^2/(2m)$ and left side $i\hbar\psi_t$, this gives the physical <superfluid velocity> $(\hbar/m)\nabla S$. In normalized equations with $D=1/2$ or $D=1$, the current velocity is respectively $\nabla S$ or $2\nabla S$. The distinction fixes the radial-flow slope of a driven <quantum vortex>.
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