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