A note on the Fermi energy of an ideal Fermi gas trapped under a generic power law potential in d -dimension

The average energy per fermion in the case of a Fermi gas with any kinematic characteristic, trapped under the most general power law potential in d-dimension has been calculated at zero temperature. In a previous paper (Acharyya M 2010 Eur. J Phys. 31 L89) it was shown, in the case of a free ideal Fermi gas, as the dimension increases the average energy approaches the Fermi energy and in infinite dimension the average energy becomes equal to the Fermi energy at T = 0. In this letter it is shown that, for a trapped system at finite dimension the average energy depends on a power law exponent, but as the dimension tends to infinity the average energy coincides with the Fermi energy for any power law exponent. The result obtained in this manuscript is more general, as we can describe the free system as well as any trapped system with an appropriate choice of power law exponent, and is true for any kinematic parameter.