# Average energy of gas of fermions at T = 0

Consider a system of N (>>1) particles with mass m in a (big) volume V. What is the average energy per particle if the particles are fermions.

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?

## Answers and Replies

OlderDan
Science Advisor
Homework Helper
Henk said:
Consider a system of N (>>1) particles with mass m in a (big) volume V. What is the average energy per particle if the particles are fermions.

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?

Only you know exactly what you did, but I think your fermion result for <E> is OK. I got the same result using the Fermi-Dirac distribution with degeneracy proportional to the energy (spherically symmetric energy distribution in momentum space) without calculating a total energy or N. If you normalize the distrubtion for some assumed Fermi energy and do the expectation integral it comes out 2/3 of the assumed Fermi energy.

Why would the boson energy be zero. Isn't the minimum energy state for any confined particle greater than zero? The average energy for bosons should be the ground state energy, assuming no exitation.