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Section 2.3 Ultraviolet Catastrophe

Electromagnetism and thermodynamics were well-developed and successful theories by the end of the 19\(^\text{ th }\) century. The theory of electromagnetism (Maxwell's equations) demonstrated the unification of electricity and magnetism, and led to the realization that light is an electromagnetic wave. This, in turn, led to the technology of generating and receiving radio waves, which was the second great step in the information technology revolution. 1  Similarly, our theories of thermodynamics explained the states of matter and provided an understanding of the engines that powered the industrial revolution.

There was just one big problem: classical E&M and thermodynamics are incompatible. Here is the basic issue. Picture some substance at a temperature \(T\text{.}\) The particles in that substance are vibrating around with thermal motion. This amounts to accelerating charges, and accelerating charges emit EM waves. Conclusion: we expect matter at a temperature \(T\) to be radiating away some of its energy into electric and magnetic fields.

But the energy exchange goes both ways. These EM fields exert forces on the charged particles, giving them back some energy. So all together we see that energy sloshes between the moving particles and the EM fields, much like it sloshes from particle to particle within the substance. Then we should expect the EM fields, like the particle motion, to have some thermal equilibrium values determined by the temperature \(T\text{.}\)

So far, so good. Let us now try to calculate what the thermal equilibrium EM fields should be for the simplest case we can construct. Imagine a cavity bounded by a pair of walls separated by a distance \(L\) (i.e., a one-dimensional box), and to keep things simple we will only have one spatial dimension. The situation is illustrated in Figure 2.1. The charges in the walls are in thermal equilibrium at temperature \(T\text{,}\) so they are moving and creating electric and magnetic fields. What form can these EM fields take? Basically, we get all the possible standing wave modes, just like waves on a string or sound waves in a tube. As we found in the waves unit, the longest wavelength is \(\lambda=2L\text{,}\) the second longest \(\lambda=L\text{,}\) the third longest is \(\lambda=2L/3\text{,}\) and so on. The actual EM fields contain all those modes and can be expressed as some superposition of these waves.

Figure 2.1. EM waves in cavity of length \(L\text{.}\) These EM waves are created by the thermal motion of the charges in the walls.

Now let's try to calculate the amplitude of these waves. First, recall that the energy of a wave is proportional to the amplitude squared. For a superposition of wave modes the energy turns out to be the superposition of the energies of each individual wave:

\begin{equation} E = \alpha A_1^2 + \alpha A_2^2 + \alpha A_3^2 + \dots\tag{2.1} \end{equation}

where \(\alpha\) is some constant and \(A_n\) is the amplitude of the \(n^\text{ th }\) longest wavelength mode of the EM wave. Now comes the thermodynamics: the equipartition theorem says that in thermal equilibrium any quadratic term in the energy has an average value of \(\textstyle\frac{1}{2}k_BT\text{.}\) Therefore we can conclude

\begin{equation} \langle \alpha A_1^2\rangle = \langle \alpha A_2^2\rangle = \langle \alpha A_3^2\rangle = \dots = \frac{1}{2}k_BT\text{.}\label{eq_EM_wave_equipartition}\tag{2.2} \end{equation}

The equipartition theorem implies that each of those modes has to have the same average value, and they are all related to the temperature \(T\text{.}\) It's a powerful theorem!

And now we are ready to compute the total energy in the EM fields. All we need to do is count the number of modes in the cavity (\(n=1\text{,}\) \(2\text{,}\) \(3\text{,}\) …) and then multiply by \(\textstyle\frac{1}{2}k_BT\text{.}\) But here is the problem: There are an infinite number of modes!

We can keep drawing waves with shorter and shorter wavelength, and we never run out of modes. Each new one we draw brings another \(\textstyle\frac{1}{2}k_BT\) to the energy. So we are led to conclude that there is an infinite amount of energy in the EM fields. This is clearly wrong — thankfully, or we would all be blasted by the infinite radiation all around us. And now we can see why this is called the ultraviolet catastrophe: it is the infinite piling up of shorter and shorter wavelengths, or higher and higher frequencies, that is where something in the theories are breaking down.

So where did it go wrong?

The printing press was the first.