Problems and Solutions on Thermodynamics and Statistical Mechanics

240 Problems d Sdutiom on Thermcdpam'ca d Statistical Mechanics

(b) Repeat the calculation, now using quantum ideas, to obtain an
expression that properly accounts for the observed spectral distribution
(Planck's Law).
(c) Find the temperature dependence of the total power emitted from

(CUSPEA)

Solution:
(a) For a set of three positive integers (nl,nz,ns), the electromagnetic
field at thermal equilibrium in the cavity has two modes of oscillation with

the frequency u(n1, n2, n3) = -(n: + nz + ni)1/2. Therefore, the number

of modes within the frequency interval Au is

the hole.

C

2L

Equipartition of energy then gives an energy density

47r

1 dE 1 8

kT. -u'Au.

• Au
  u, = -- L3 du - -. L3
  = 87ru2kT/c3.

When u is very large, this expression does not agree with experimental

observations since it implies u, o< u2.

(b) For oscillations of freqeuncy u, the average energy is

L-

n=O

which is to replace the classical quantity kT to give