The unsolvable problem that foreshadowed quantum mechanics

The boys (physicists, among them Planck and Einstein) are back in town (Solvay, 1911)

A challenge

In the 1890s, physicists studying thermodynamics were stuck in a deadlock. It was already known that the radiation emitted by a blackbody — an idealized surface that absorbs light of all frequencies —varies as a function of the light’s energy. For lower frequencies (that is, less energetic light) the Raleigh-Jeans law could tell you the amount of radiation expected with accuracy.

But for higher frequencies, like ultraviolet light, this same model predicted that the blackbody should emit an approachingly infinite amount of energy. It so happens that an oven or container with a pinhole is a pretty good approximation of a blackbody; clearly, no such explosion occurs. And yet physicists could not account for this observation anywhere in their theory.

The ultraviolet catastrophe refers to this lapse in the history of physics: a discrepancy between the observed radiance of a blackbody and the radiance predicted by the best models of the time

Here, the red curve represents the energy predicted by the Raleigh-Jeans law

Many problems in physics have melodramatic endings. In the case of these miscalculated galactic distances, systemic bias on the part of the satellite Hipparcos was the culprit. For the supposed faster-than-light neutrino (a finding which would immediately put all of modern physics into question), human error and a sensationalist media shared the blame.

But others are solved by means of so-called paradigm shifts: groundbreaking insights that provide us with a new framework to understand reality. These changes in perspective often challenge us to question a deeply-held assumption or intuition, especially when they match scientific observation very well.

Readers of my last article on physics know that Mercury’s orbit is one such anomaly: it anticipated our shift from classical Newtonian physics to Einstein’s theory of relativity. But in order do so, we had to grapple with some departures from our intuitions about reality: among them, the idea of simultaneity, and absolute space and time more generally.

As we’ll see, the solution to the ultraviolet catastrophe is another such breakthrough; the mind-bending departure in this case was the idea of a quantum universe.

Planck’s solution

Enter a 40-year old physicist, Max Planck. While working as an associate professor at the University of Kiels in Germany, he turned his attention to the elusive description of the blackbody’s radiance.

The legend himself, looking rather unenthused

First, Planck tried to solve the problem by elaborating on another preexisting model, Wien’s law, which correctly described the blackbody at high frequencies but not low (the opposite situation of the Rayleigh-Jeans law). This did not work, and the Wien-Planck law became a short-lived entry in the history of physics.

Inatead, Planck’s breakthrough came with a simple but effective mathematical ‘trick’. He expressed energy not as a continuous value, but as increments of a certain unit, the Planck constant h. In mathematical terms, he quantized the oscillation of the blackbody’s energy so that it can only take certain values: multiples of a specific amount of energy given by the Planck constant.

In other words, Planck solved the ultraviolet catastrophe by assuming that energy was not continuously divisible as we expect, but rather that it comes in discrete ‘packets’.

By treating energy as a discrete quantity, Planck was able to arrive at a model which perfectly describes the radiance of a blackbody.

In one elegant solution, both the Rayleigh-Jeans and Wien’s law had become obsolete. At low frequencies of light, Planck’s law tends towards Wien’s approximation: for high frequencies it tends towards Rayleigh and Jeans’. The observed radiance of a blackbody had finally been described perfectly.

Shortly after Planck did his derivation, Albert Einstein described this ‘packet of electromagnetic radiation’ as a photon, formally designating the particle nature of light for the first time. Understanding light as photons yielded new explanations for the photoelectric effect, an accomplishment which earned Einstein the Nobel Prize in physics in 1921.

Later on, Planck’s notion of quantized energy would influence Bohr’s description of the electron levels, leading him to conclude that electrons can only occupy certain discrete energy levels around the nucleus.

Like the energy radiated by a blackbody, electron shell positions come in discrete quantities

The philosophy of Planck’s postulate

The findings of 20th century physics were hugely consequential for understanding the reality in which we live. Einstein’s theories of relativity taught us that space and time are not absolute categories, but change depending on the observer. Quantum physics has continued to shatter our ideas of reality, probability, and so on.

This represents a unique development in human thought. More and more it seems, the questions which would have been considered within the domain of metaphysics or philosophy more generally are shifting over to physics. We no longer rely on the contemplative metaphysician to answer questions like, “can something come from nothing?” or “is the universe created or eternal?” We simply head over to our local physics department, who presents us with data and articulates their best theory.

Planck’s postulate — the idea that energy is quantized — is another concept in physics with many philosophical ramifications. We all know that although air and water seem continuous, they are actually discrete substances; their fluid-like properties are simply the result of the sheer number and interactions of these singular particles. But to claim that a physical concept something as abstract as energy came in quanta — discrete amounts — was not only new to physics, but completely opposed to our common-sense intuition of the world.

Take, for example, Zeno’s paradox, which was put forth in order to disprove the existence of motion. To arrive at point A from point B, you need first to transverse half the distance, and also half that remaining distance, and also half of that remaining distance, and so on infinitely.

As calculus teaches us, even if this task of ‘infinite division of space’ worked, this infinite sum of half still ‘converges’ into a whole number. In other words, an infinite set of numbers can sometimes sum to a finite number.

But the truly subversive aspect of quantum mechanics is that, in addition to energy, even space and time itself are not infinitely divisible, continuous: at a certain level, known as the Planck scale, we can speak of discrete ‘units’ of space and time.

In a departure from our most deeply held intuitions, the Planck scale tells us that the fundamental constituents of reality — space, time, energy, and so on — are not infinitely divisible

It is something of an irony that a character like Planck should be the bringer of such a revolutionary concept. He remained a conservative figure till the very end, denying the physical reality of his postulate and maintaining that it was simply a mathematic trick.

But advances in quantum mechanics have confirmed the bizarre nature of his discovery. It’s enough for us to ask what intuitions about reality will continue to be challenged, not only by physics but by psychology, computer science, and so on. We are living in revolutionary times—the brave, tireless gears of modernity—that demand meticulous attention and shifts in perspective.

Subscribe to my newsletter for updates on future articles:



Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store