Ultraviolet Catastrophe

This is a follow-up to my previous post where I walked through deriving the speed of light using Maxwell’s third and fourth equations. Honestly, once I got to this point things started getting harder to wrap my head around, especially after seeing where the Rayleigh-Jeans law actually ends up.

After Maxwell showed that the light we see is nothing more than an Electromagnetic Wave, a few decades passed and the world stepped into the age of electrification. Companies in Germany were in a full-on race to build the most efficient street light bulbs. The problem is, when you heat a carbon filament with electricity, it glows (yellow or white). But push it too hot and the filament burns out. Keep it too cool and it only gives off a dim red glow, throwing away most of the energy as invisible infrared heat.

So the German government set up a dedicated research institute and basically told physicists: “Figure out a proper mathematical formula that tells us exactly how much energy and what color of light comes out when you heat something to temperature T.”

Before anyone could write a formula, physicists needed a clean ideal model to work with. They came up with the idea of a perfect cubic cavity, a box with length , width , and height , so the total volume is .

The inner walls are assumed to be a perfect conductor that reflects electromagnetic waves with zero loss (100% reflection). Heat the box to some absolute temperature , and the walls start emitting radiation that bounces around endlessly, forming what are called Standing Waves. The whole point of this thought experiment is to count how many of those wave modes can exist.

Boundary Conditions in 1 Dimension

Rayleigh started in 1900 by thinking about this in just one dimension. Picture the radiation like a guitar string running from the left wall ( ) to the right wall ( ). Since the walls are perfect reflectors, no wave can get through, which means the electric field has to be exactly zero right at the walls.

For a wave to actually fit inside the box, the room’s length has to hold an integer number of half-wavelengths ( ):

(where is the number of antinodes)

This is where things start going the formal physics route. Rather than sticking with wavelength ( ), physicists prefer to work in terms of Frequency ( ). From the basic wave relation , we get .

Plug that into the boundary condition and the mode index ( ) becomes:

Expanding to 3 Dimensions

Unlike the earlier Maxwell derivation where we went from three dimensions down to one, here we do the reverse. The box is 3D, so waves can travel diagonally and bounce off all three pairs of walls at once, giving each mode components along all three axes ( ).

Since the three axes are all perpendicular to each other, Rayleigh combined them into a single mode vector using the 3D Pythagorean Theorem:

Which makes the mode sphere radius:

Counting Modes with a Spherical Shell

The equation tells us that all possible wave modes sit on a sphere in this coordinate space. Instead of computing the full solid sphere volume, we want the differential version: the Volume of a Spherical Shell.

Think of it this way: we want to count modes only within a very thin slice of frequency (from to ). That is exactly the same as asking for the volume of an infinitesimally thin shell of thickness .

First, differentiate with respect to :

Shell volume is just surface area times thickness:

Substituting and :

The Positive Octant and Valid Modes ( )

Here is the catch: , , represent physical bounce counts, so none of them can be negative. In 3D space, that means we can only use the chunk of the shell where all three coordinates are positive, which is exactly of the full sphere.

So the actual number of valid modes ( ) in our frequency slice is:

James Jeans’ Fix (The Polarization Factor)

In 1905, James Jeans pointed out that everything above was treating waves as scalar, which light is not. Maxwell’s equations say light is a transverse electromagnetic wave, and that means for every direction a wave travels, the electric field has 2 independent perpendicular polarization directions.

So we need to multiply our mode count by two:

Assigning Energy via the Equipartition Theorem

Now we have the seat count ( ), time to fill the seats with energy. Classical thermodynamics tells us through the Equipartition Theorem that each electromagnetic mode behaves like a harmonic oscillator with two degrees of freedom (kinetic and potential) each contributing . So the average energy per mode is:

(where is the Boltzmann Constant)

Total radiation energy ( ) in our frequency range is just modes times energy per mode:

To make the result independent of box size, divide everything by the cavity volume ( ). This gives us the Energy Density per Unit Frequency ( ):

Converting from Frequency to Wavelength

The formula above is correct, but labs at the time (like Lummer and Pringsheim’s experiments) measured spectra using prisms and diffraction gratings, which output data in Wavelength ( ), not frequency. So we need to convert.

Starting from , differentiate with respect to :

(The minus sign just reflects the fact that shorter wavelength means higher frequency. For converting a density we just take the absolute value: ).

Now substitute and :

And here is where the algebra works out nicely. The numerator gives , which cancels the in the denominator entirely. The denominator picks up .

Everything collapses down to:

This is the moment classical physics started to crack. Rayleigh himself had actually spotted the issue back in 1900 and tried patching it by adding a suppressing exponential ( ) to stop the numbers from exploding. But Jeans came in 1905, cleaned up the math, and showed that the exponential had no grounding in classical physics whatsoever. Note: that exponential turns out to correspond to what Planck later called the second radiation constant, . Rayleigh had just thrown it in as an arbitrary constant.

Once the equation was left naked as , the consequences were pretty alarming: as shrinks toward zero (ultraviolet, X-rays), the in the denominator sends total energy flying toward Infinity ( ). That breaks Conservation of Energy, and it directly contradicts experimental data where radiation actually falls off in the ultraviolet range. This failure got a name that stuck: the Ultraviolet Catastrophe.

At this point, classical physics was genuinely stuck. All three of its pillars, Newton’s mechanics, Maxwell’s electromagnetism, and Boltzmann’s thermodynamics and statistical mechanics, had nothing left to offer. Until Planck eventually did what he himself called an “Act of Desperation” (which I am currently trying to work through because it is genuinely rough :p). This is where quantum physics is born, out of heated arguments about what light actually is, and eventually the realization that we had just been staring at the macroscopic world for too long (a remarkable comment someone left in my notes about the Laplace Demon genuinely cracked my worldview open). Well, that’s science for you, endlessly complicated, endlessly interesting.

“What I did can be described as simply an act of desperation… it was purely a formal assumption and I really did not give it much thought.” — Max Planck, in a letter to a colleague, 1931