Problems and Solutions on Thermodynamics and Statistical Mechanics

256 Problems 8 Solution3 on Thermdpmics 8 Statistical Mechanics

2085

Consider an ideal quantum gas of Fermi particles at a temperature T.

(a) Write the probability p(n) that there are n particles in a given

(b) Find the root-mean-square fluctuation ((n - (n))z))'/z in the occu-
pation number of a single particle state as a function of the mean occupation
number (n). Sketch the result.

single particle state as a function of the mean occupation number, (n).

(MIT)

Solution:

potential. The partition function is

(a) Let E be the energy of a single particle state, p be the chemcial

z = exp[n(p - &)/kT] = 1 + exp[(p - &)/kT].

n

The mean occupation number is

The probability is

(b) ((n - (n))2) = kT- a(n) - - (n)(l- (n))

aP

So we have ((n - (n))z))'/z = J(n)(l - (n))

The result is shown in Fig. 2.17.

1

<(n- <n>,2>'

Fig. 2.17.