 Crossover from Gaussian to Thomas–Fermi condensate

Repulsive trapped condensates change smoothly from a nearly ideal quantum harmonic oscillator ground state for $\chi \ll 1$ to a broad Thomas–Fermi approximation for a condensate for $\chi \gg 1$. Comparing the Gaussian interaction energy of a trapped condensate with the total ideal energy gives $E_I/E_0=2\chi /(3\sqrt{2\pi })$. Thus $N\sim a_0/a$ is a crossover scale, not a sharply defined critical particle number. Total interaction energy must be compared with total ideal energy; comparing it with one oscillator quantum confuses total and per-particle scales.