Healing length (${\ell }_h=\hslash /\sqrt{2mg{n}_0}$)

The healing length of a repulsive homogeneous condensate is the length at which density-gradient kinetic energy and interaction energy balance. The displayed convention sets the dimensionless Laplacian coefficient to one in the stationary Gross–Pitaevskii equation. A hard-wall number density profile has width $\sqrt{2}{\ell }_h$; definitions without the factor $\sqrt{2}$ occur, so the equation and convention must be specified.

A repulsive condensate next to an impenetrable planar wall has Dirichlet boundary condition $\psi (0)=0$. Writing $\psi =\sqrt{n_0}f$ reduces its stationary Gross–Pitaevskii equation to ${\ell }_h^2{f}^{\prime \prime }+(1-f^2)f=0$. The bulk conditions $f\to 1,f^{\prime }\to 0$ give ${\ell }_h^2(f^{\prime }{)}^2=(1-f^2{)}^2/2$ and hence the displayed number density. Omitting $n_0$ describes normalized number density, not dimensional number density.