Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 263

2090

(a) Calculate the magnitude of the Fermi wavevector for 4.2 x 1021

(b) Compute the Fermi energy (in eV) for this system.

(c) If the electrons are replaced by neutrons, compute the magnitude

(UC, Berkeley)

electrons confined in a box of volume 1 cm3.

of the Fermi wavevector and the Fermi energy.

Solution:

(a) The total number of particles is

The Fermi wavelength is

= 1.25 x 10-gm = 12.5A ,

h 8rV ‘I3

XF=-= (x)

PF

(b) The Fermi energy is

(c) If the electrons are replaced by neutrons, we find that

= XF = 12.5A ,

m

m‘

and E& = -EF = 5.2 x 10-4eV.

2091

Calculate the average energy per particle, E, for a Fermi gas at T = 0,

(UC, Berkeley)

given that EF is the Fermi energy.

Solution:

We consider two cases separately, non-relativistic and relativistic.

(a) For a non-relativistic particle, p << mc (p is the momentum and m

is the rest mass), it follows that

&= -. P2

2m