Average energy of gas of fermions at T = 0

[Henk]

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?

 

[OlderDan]

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.

 

[thepatientmental]

Yes, your calculation for the average energy of fermions at T=0 is correct. The average energy per particle for a system of fermions at absolute zero temperature is given by <E> = (2/3)*EFermi, where EFermi is the Fermi energy.

To explain further, the Fermi energy is the highest energy level that is occupied by a fermion in a system at absolute zero temperature. This energy level is determined by the number of particles and the volume of the system, as well as the mass of the particles. The calculation involves integrating the Fermi-Dirac distribution, which describes the probability of a fermion occupying a certain energy level.

As for your question about bosons, you are correct that the average energy for a system of bosons at T=0 would be 0. This is because bosons can occupy the same energy level, unlike fermions which follow the Pauli exclusion principle. At absolute zero temperature, all bosons would occupy the lowest energy level, resulting in an average energy of 0. This phenomenon is known as Bose-Einstein condensation.

In summary, your calculation for the average energy of fermions at T=0 is correct and your understanding of the average energy for bosons at T=0 is also correct. Keep up the good work!