# Correct way to do a Thomas-Fermi approximation for cold gases

I have calculated the total Gross-Pitaevskii energy for a 2D Bose-Einstein condensate in an harmonical trap, using a variational gaussian wave function with a variational parameter b. Now I want to compare the variational energy to the Thomas-Fermi result. I know that the Thomas-Fermi approximation means that you neglect the total kinetic energy in comparison to the interaction energy, but I was wondering how to do it specifically in this case. I namely have three different possibilities in mind:

1) Just remove the kinetic energy term from the energy expression I found with the variational wave function, and keep the value of the variational parameter b as it was before.

2) Remove the kinetic energy term from the energy expression I found with the variational wave function, and calculate a new value for the variational parameter b for this specific case.

3) Use the Thomas-Fermi approximation in the GP equation to find a new expression for the wave function (instead of the one I used before) and use this one to calculate the energy.

I can't seem to decide which of these three is the right one. Can anyone give a convincing argument as to which method I should use?

You simply take $|\psi|^2=1/g[\mu-V(x)]$.
$\mu\psi=(V+g|\psi|^2)\psi$