Blackbody Radiation
We have a cavity, carved out of a conducting material, with only a small entrance/exit
hole so that we can see inside. The material is at some temperature T. The hole in the
cavity is small compared to the cavity volume so that radiation going into the hole has a
small probability of escaping from the hole before being absorbed by the cavity walls.
The hole is a good approximation of what is known as a blackbody. Intuitively, we know
that a body radiates light at many different frequencies, and that the power and the most
distinct frequency radiated are related to the temperature. We define the spectral radiancy
RT(ν)dν as the energy emitted per unit time in radiation with a frequency in the interval ν
to ν + d ν from a unit area of the emitter. Experimentally, the spectral radiancy is shown
below to have several distinct features: 1) The radiancy drops to zero at low frequencies,
2) it increases up to a maximum value that appears to depend linearly on temperature
(vmax  T is called Wien’s Displacement Law) and then 3) decreases again at higher
frequencies to zero.

The integral of the radiancy, RΤ   RΤ (v) dv appears to increase more rapidly than
0
4
-8
2
4
linearly. In fact, by experiment, RT = σT (Stefan’s Law), where σ = 5.67x10 W/(m K )
is the Stefan-Boltzmann constant. Let us try to understand this from a Classical
perspective.
Classical statistical mechanics tells us that for an ensemble of entities at a particular
temperature T, the available energy will be shared among the entities equally such that
the average energy of each entity (averaged over their energy distribution) will be
-23
E  kT , where k = 1.38x10 J/K is the Boltzmann’s constant. (One can derive this from
e  E / kT
). What are the
kT
entities involved in our case? Well, first let us deal with the radiation inside the cavity,
rather than the radiation emitted by the hole so that we will try to determine the energy
density inside the cavity instead of the power emitted from the hole. Since the hole
samples the radiant energy inside the cavity, it is clear that these two quantities should be
proportional to each other: RT (v)  T (v) . But still, what are the entities involved? Well,
what is inside the cavity? Don’t say photons, because for now we are classical physicists,
and therefore only know about electromagnetic waves. The entities are the modes of
oscillation of these waves. For simplicity let’s look at the waves in a one-dimensional
cavity of length a:
the Boltzmann’s distribution of energy for the entities: P( E ) 
Classically, each of these modes of oscillation of the radiation field can carry energy, and
on average will carry kT amount of energy. So, in order to determine the energy density
function ρT(ν), we need to be able to count how many of these modes are available in a
certain frequency range ν + dν. We must first relate the mode number n to the frequency.
We know that for electromagnet waves, with velocity c, the frequency is related to the
wavelength by ν = c/λ. Examining the picture above, the wavelength for allowed modes
is λ = 2a/n so that the allowed frequencies are ν = cn/2a or that the number of allowed
states dn = 2a/c dν. One can go through a similar reasoning to find that in 3 dimensions,
the number of allowed states is given by:
8 a 3
N (v)dv  3 v 2 dv
c
and thus, that the energy density in the cavity is given by:
8 v 2 kT
 T (v)dv 
dv
c3
The result of this calculation is plotted below with the experimental result. Something is
obviously wrong and this failure of classical physics became known as the “ultraviolet
catastrophe”!
Did we do something wrong in our classical calculation? No! Classically, each mode of
oscillation can have an energy distribution with average energy kT – remember that
classically, each oscillation mode can have any energy – visualize this by just thinking
that the amplitude of oscillation can be a continuous function with any value. Now let’s
examine these energy distributions. I mentioned above that the average energy (kT) can
be derived from the Boltzmann distribution. The figure below shows this graphically.
The bottom plot again assumes that the energy can take on any value, which it of course
CAN in classical physics. So, if we have followed the classical description, and we get
th
the wrong answer, what do we, as 19 century physicists, do? Well, in ???? Plank made a
supposition that he felt made the smallest deviation from the well-tested theory and that
agreed with classical physics at low frequencies where classical physics agreed with
experiment. He relaxed the assumption that the energy must be continuous. [Noticed how
I phrased that – that continuous energies was in itself an assumption that was not tested at
very small energies.] If there are only certain allowed energy steps, say ∆E, and we still
use the very-well founded Boltzmann distribution (see Appendix A of Eisberg and
Resnick for a nice discussion of the Boltzmann distribution), we no longer get kT for the
average energy of a certain mode. In fact, if we make the energy step size a function of
the mode by saying for instance that ∆E = constant*frequency, then the average energy of
the modes decreases with increasing frequency as shown in the figure below. Plank
evaluated the average energy by taking the sum of the probability to have an energy (still
from the Boltmann distribution) times the energy from the step size, and then used this
average energy (instead of kT) to find the energy density distribution inside the cavity,
and then fit to the experimental data to evaluate the constant for his energy step size
above. He called this constant h, so that the energy step size ∆E = hν. [Don’t get ahead of
ourselves here! Plank did NOT hypothesize the photon, but only that the energies of
oscillation of the electromagnetic field were quantized.] The average energy he calculated
was then:
hv
E 
hv / kT
e
1
Using this average energy per mode his energy density becomes (again using our same
calculation for the density of modes):
8 v 2
hv
dv
3
hv / kT
c e
1
Which given the extracted value of the constant h, fit the data precisely.
Plank’s postulate was that entities whose “coordinate” oscillates in time can possess only
T (v) dv 
st
(Condensed from Eisberg Resnick’s Quantum Physics 1 Ed.)