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I did some calculations and I came up with <E> = (2/3)*Fermi-energy.

Is this correct? I could post my calculations but my Latech-skills are very poor and the calculation involves some long integrals.

I used <E> = Etotal / N and calculated N and E by using the Fermi-Dirac distribution.

The other question is the same but then for bosons. In that case the average energie would be 0 right?