Statistical Physics, Second Revised and Enlarged Edition

Phoneybosons 105

density,wealso needtoknow, (i)how manystates there areinthefrequencyrangeof
interest, and (ii) how the frequencyνof a photon relates to its energyε.The second
questionisimmediately answeredbyε=hν.Thefirstis yet another straightforward
exampleoffittingwavesintoboxes.
The density of photon states inkis given by (4. 5 ), with the polarization factor
G=2. Photons, since they are spin-1, masslessbosons,have two polarization states;
inclassicalterms electromagnetic waves are transverse,givingleft- or right-hand
polarizations, but there is no longitudinal wave. Hence

g(k)δk=V/( 2 π)^3 · 4 πk^2 δk·2( 9 .10)

Wewishto translate (9.10) to adensityofstatesinfrequencyν,correspondingto the
required spectral energy density. This is readily and accurately achieved for photons
in vacuum, sinceν=ck/ 2 π,wherecisthe speedoflight. Making thechange of
variables,(9.10)becomes

g(ν)δν=V· 8 πν^2 δν/c^3 (9.11)

The answer nowfollows at once. The energy in a rangeisthe number ofstatesinthat
range×the number ofphotonsper state×the energyper photon. That is

u(ν)δν=g(ν)δν×f(ν)×ε(ν)

=V· 8 πν^2 δν/c^3 × 1 /[exp(hν/kkkBT)− 1 ]×hν

=V·

8 πhν^3 δv

c^3

·

1

[exp(hν/kkkBT)− 1 ]

(9.12)

Equation (9.12)isthecelebratedPlanckradiationformula. ItisdrawninFig. 9.7for
three different temperatures. We can make several comments.

1. 
   

   This is not how Planck derived it! Photons had not been invented in 1 9 00. His
   argument was based on a localized oscillator model, in which each of theg(ν)δν
   oscillator modeshadan averagethermalenergynotofkkkBT,theclassicalincorrect
   result,but ofhν/[exp(hν/kkkBT)− 1 ]as derived in Chapter 3 (essentially (3.13)
   ignoring zero-point energy). Themodernderivationis muchtobe preferred.

   
   


2. 
   

   ThePlancklawisin excellent agreement withexperiment. One ofthefeatures
   concernsthemaximuminu(ν).Experiment (effectively Wien’s law) shows that
   νmaxis proportionaltoT.Thisisevidentfrom (9.12), since the maximum will
   occur at afixedvalue ofthedimensionlessvariabley=hν/kkkBT.Infactymaxis a
   little less than 3(see Exercise 9.5).

   
   


3. 
   

   Another experimentalpropertyisthat the totalenergyintheradiationis propor-
   tional toT^4 .ThisT^4 lawfollowsfrom anintegration ofthePlancklaw (9.12) over