Blundell Thermal Physics · Problem 23.2

Photons

Physics — Statistical Mechanics · wien-displacement-law-maxima

Problem

Show that the maxima in the functions

u_\nu

and

u_\lambda

can be computed by maximizing the function

x^\alpha/(\mathrm{e}^x - 1)

for

\alpha = 3

and

\alpha = 5

respectively. Show that this implies that

x = \alpha(1 - \mathrm{e}^{-x}).\tag{23.66}

This equation can be solved by iterating

x_n = \alpha(1 - \mathrm{e}^{-x_{n-1}});\tag{23.67}

now show that (using an initial guess of

x_1 = 1

) this leads to the values given in eqns 23.53 and 23.54.