By Chad Orzel, Union College
Through the late 1800s, a lot of really smart people tried to explain the light we see from hot objects, but all of them failed. The guy who finally succeeded, in 1900, was Max Planck. But to do it, he had to resort to a weird trick, and in the process, he kicked off what became the quantum revolution.
Max Planck’s Quantum Hypothesis
Planck introduced an idea, now called the quantum hypothesis, about the way that light gets emitted. He said that each frequency of light that exists in a box must be produced by what he called an oscillator in the walls emitting light at that particular frequency.
He didn’t have a detailed physical model for these—this was more than a decade before the modern picture of atoms was developed—but he reasoned that something in there must be vibrating at the frequency associated with the light.
He assigned each of these oscillators a characteristic energy equal to its frequency multiplied by some small number. Then he added the really critical rule that a given oscillator can emit energy only in integer multiples of this characteristic energy: one unit, two units, three units, but never one-and-a-half units or pi units.
He was doing this to set up a particular calculus trick and expected to take the small number to zero at the end of the problem, which would go back to a smooth and continuous distribution of energy. To his surprise, though, this method worked out only if that small number was not zero, in which case the quantum hypothesis fixed the ultraviolet catastrophe.
How the Quantum Hypothesis Works
To see how it works, think of a simple set of modes, each getting an equal share of the energy available from heat, but add in Planck’s quantum hypothesis. Let’s say that the energy share is low enough that it’s just six times the characteristic energy of the fundamental mode. In that case, the blackbody would emit six units of light at the fundamental frequency.
The first harmonic, mode number 2, has twice the frequency of the fundamental, so its characteristic energy is twice as big. Thus, we get three units of light at this frequency. Mode 3 has a frequency three times the fundamental, so it emits two units of light.
When we get to mode four, though, Planck’s rule kicks in. This has a frequency four times that of the fundamental mode, so it should emit one-fourth as many units of light. But that would be one-and-a-half units, and only integer values are allowed. So thanks to the quantum hypothesis, we only get one unit of light at this frequency, which is less than what we would’ve expected.
As the frequency keeps going up, the characteristic energy gets larger and larger, and eventually is bigger than the share of the energy that would be allotted to that frequency—in our model, that happens at mode 7, which ‘should’ emit six-sevenths of a unit of light. It’s only allowed to emit light in integer units, though, so in fact, it can’t emit any light, and neither can any of the higher-frequency modes.
This is a transcript from the video series Einstein’s Legacy: Modern Physics All around You. Watch it now, on Wondrium.
Fixing the Ultraviolet Catastrophe
Planck’s quantum rule thus does exactly the thing needed to fix the ultraviolet catastrophe: It squashes the emission of light in short-wavelength, high-frequency modes. At some particular frequency, the characteristic energy needed to emit one unit of light becomes greater than the energy available from heat for emitting light at that frequency, and the emission gets shut off.
Combining the idea of equipartition with the quantum hypothesis gives us a very nice explanation of the shape of the blackbody spectrum. Starting at long wavelengths and moving down, the characteristic energies are very low, so we just see an increase in the total amount of light in a particular band of wavelengths due to the increasing number of modes in that wavelength range.
At short wavelengths, though, the characteristic energy gets too big, so while the number of modes keeps going up, for many of them we get no light emitted at all. This cutoff happens very rapidly, which is why the short-wavelength side of the peak drops off faster than the long-wavelength tail.
Planck’s Constant
The amount of energy available due to heat increases as the temperature increases, which means that the frequency where the quantum cutoff happens goes up with temperature, and that’s why the peak moves to shorter wavelengths for hotter objects.
When we carry through all the mathematical details, what we end up with is exactly the formula that Planck had found empirically in 1899, but now with a grounding in the fundamental physics of equipartition. It also gives us a very particular value for the small number that multiplies the frequency to get the characteristic energy, what we now call Planck’s constant.
In modern units, it’s 6.626 × 10−34 joules per hertz, or 33 zeroes to the right of the decimal point before we get to the 6626. That’s an extremely small number to be sure, but it’s very definitely not zero.
Success of Max Planck’s Formula
Planck resorted to the quantum hypothesis out of desperation, but it was a smashing success. His blackbody formula has been tested over and over, at temperatures from the hottest stars to the cold of interstellar space. The cosmic microwave background radiation, light leftover from just a few hundred thousand years after the big bang, fits perfectly to a blackbody spectrum with a temperature of only 2.71° above absolute zero.
Planck himself was never entirely happy with his trick, though, despite its success. He held out hope that some other physicist would find a clever and elegant trick that would explain the blackbody formula without resorting to this ugly business of oscillators that can only emit integer multiples of some characteristic energy.
Common Questions about How Max Planck Solved the Blackbody Problem
Max Planck introduced an idea, now called the quantum hypothesis, about the way that light gets emitted. He said that each frequency of light that exists in a box must be produced by what he called an oscillator in the walls emitting light at that particular frequency.
Planck’s quantum rule squashes the emission of light in short-wavelength, high-frequency modes. At some particular frequency, the characteristic energy needed to emit one unit of light becomes greater than the energy available from heat for emitting light at that frequency, and the emission gets shut off, thus fixing the ultraviolet catastrophe.
In modern units, the value of Planck’s constant is 6.626 × 10−34 joules per hertz, or 33 zeroes to the right of the decimal point before we get to the 6626.
