A professor told me that most physicists assume that all identical fermions are in completely orthogonal states. If that is true, then does that mean that that the total wave function is highly localized for a closed box of a huge number of electrons? I get confused between when your supposed multiply wavefunctions and when your supposed to add wavefunctions. I know that for one electron, adding all orthogonal states of that one electron in a closed box would yield a highly localized spatial wavefunction.
What you refer to is probably to ground state of a infinite potential wall. If this is the case, then there, in the ground state, the particles are not localized. You can find the solution of the orthogonal ground-states here: https://en.wikipedia.org/wiki/Infinite_potential_well
We can reduce the Problem to a one dimensional case, that doesn't change the Problem or solution to much.
With orthogonal states is meant that the integral over the wave function vanishes. So in the case of a one dimensional box this is fulfilled for the given sinus functions. That doesn't imply that the two particles may not be found at the some position in space. If that would be the case, the electrons would have to be localized. But this is not the requirement for orthogonal states.
I hope this helps to clarify this.