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Early Experiments in Quantum Physics 219

2
It is easy to see that as the frequency increases as n in the ultraviolet and the x-ray range, the
n

‘‘density of states’’ ¼ r will increase to 1 contrary to experimental data! We also need to alert
V
cdl
. Since the convention
l
the reader to an interesting dilemma here. Note that since c ¼ ln, dn ¼ 2
in plotting graphs is to show the x-axis increasing to the right we will have to agree to suppress the
2
8pk B Tn dn
sign of one or the other if we plot the graph in terms of n or l and we have r(n)dn ¼ 3
c

8pk B T c 2 c 8pk B Tdl

when using wavelength. How-
using frequency or r(l)dl ¼ 3 2 dl ¼ 4
c l l l
ever, as one goes to shorter wavelengths (l ! 0) the density of states still diverges to 1 so the
Rayleigh–Jeans formula diverges in either frequency or wavelength form! We did the tricky
counting scheme in detail so that you know it is correct and that is not the problem. Rayleigh
ð
e
1
ee k B T de 2
0 (k B T)
and Jeans depended on the average energy per mode as e ¼ ð ¼ ¼ k B T where
e (k B T)
1
e k B T de
0
ð
1 n!
n ax
we have used 0! ¼ 1 and our old friend from Introduction as x e dx ¼ nþ1 . Surely we do not
0 a
think Boltzmann was incorrect?
In 1901, Max Planck pondered this dilemma and tried a revolutionary assumption. Maybe the
ð
1
problem is in using de that assumes the energy is a smooth continuous variable? That is, maybe
0
ð
the energy is not a smooth continuous variable? Thus Planck tried using S instead of de.An
integral sign is a form of continuous addition but if energy exists as small chunks, quanta, then
maybe you have to use a discrete summation over the states. One can pose the question as to
whether smooth peanut butter is really an oil or just some finer version of chunky peanut butter.
Close examination shows that smooth peanut butter is just smaller chunks of peanut in an oil base!
Thus Planck considered the average energy of discrete ‘‘quantum’’ chunks (Figure 10.6).
ne
P P n
(ne)e (ne)x
1 k B T 1
n¼0 n¼0
e ¼ e ¼ P where we have used the previous analogy of the average
P n
1 e k B T 1 x
n¼0 n¼0
1.25
Spica
(23,000°K)
Intensity (normalized) 0.75 The sun (3,400°K)
1.0
Antares
0.50
0.25 (5,800°K)
0
0 5,000 10,000 15,000 20,000
Wavelength (Å)
FIGURE 10.6 Normalized blackbody radiation for three stars using the Stefan–Boltzman equation to estimate
the temperature of the stars. (From Prof. Mike Guidry of the University of Tennessee Knoxville Department of
Physics as used in their Astronomy 162 course. http:==csep10.phys.utk.edu=astr162=lect=light=radiation.html
With permission.)