Black-Body Radiation
Maxwell's wave theory of light looked complete. Then physicists tried to predict, from that same classical theory, the simple spectrum of light glowing from a hot object — and the prediction diverged to infinity. Max Planck's fix, meant only as a mathematical trick, was the first crack in the classical picture.
What a black body is
A black body is an idealised physical object that absorbs all electromagnetic radiation falling on it, at every wavelength, reflecting none of it — hence "black." Because it is also a perfect absorber, a black body in thermal equilibrium is equally a perfect emitter, radiating energy across a continuous range of wavelengths purely as a result of its own temperature, following a spectrum (a graph of emitted intensity against wavelength) that depends on that temperature alone, and on nothing else about the object's composition, shape or surface finish. A heated metal filament, molten metal in a furnace, and even a star like the Sun are all, to a good approximation, physical realisations of this idealised black body.
Raising a black body's temperature has two well-established, experimentally observed effects on its emission spectrum: the total power radiated at all wavelengths increases sharply, and the wavelength at which emission is most intense — the peak of the spectrum — shifts to shorter wavelengths. This is why an object glows dull red when moderately hot, then orange, then white, then blue-white as its temperature climbs: the peak of its spectrum is sliding from the infrared, through the red end of the visible spectrum, and on toward the blue end as temperature rises.
The classical prediction versus reality
By the end of the nineteenth century, physicists tried to derive the shape of the black-body spectrum from first principles, treating the radiation inside a hot cavity as a continuous electromagnetic wave phenomenon — exactly the picture Maxwell's theory had so successfully established. The calculation modelled the radiation as arising from a very large number of possible oscillation modes inside the cavity, with classical physics assigning an equal average share of energy to every possible mode, regardless of its frequency.
This classical approach predicted that the intensity of radiation should keep rising, without any limit, as wavelength gets shorter and shorter — because higher-frequency (shorter-wavelength) modes become more and more numerous, and classical theory gave each of them an equal share of energy. Taken to its conclusion, this implied a hot object should radiate an infinite amount of energy at very short (ultraviolet and beyond) wavelengths — a result so obviously unphysical, and so starkly at odds with any real measurement, that it became known as the ultraviolet catastrophe.
Real, carefully measured black-body spectra look nothing like this. Instead of climbing without limit, the observed intensity rises smoothly from very long wavelengths, reaches a single peak at some particular wavelength that depends on temperature, and then falls back down again at shorter wavelengths, tailing off toward zero. The classical wave theory, for all its success elsewhere, was flatly wrong about this one specific measurement.
The ultraviolet catastrophe was not a small discrepancy or a rounding error — it was a qualitative failure. Classical theory predicted intensity rising forever at short wavelengths; every real measurement showed intensity rising to a peak and then falling. No adjustment of classical assumptions about the cavity, the material, or the temperature could close this gap. Something fundamental about the classical treatment of energy itself had to be wrong.
Planck's quantum hypothesis
In 1900, the German physicist Max Planck found a mathematical formula that reproduced the observed black-body spectrum exactly, across its whole range of wavelengths — but only by making one very specific, very unclassical assumption. Planck proposed that the oscillating charges responsible for emitting and absorbing the radiation inside the cavity could not exchange energy with the radiation field continuously, in any amount whatsoever, as classical physics assumed. Instead, they could only gain or lose energy in discrete packets, or quanta, with the energy of a single quantum fixed by the oscillation frequency $f$:
where $h$ is a new fundamental constant (now called the Planck constant, $h = 6.63\times10^{-34}\,\text{J s}$). Restricting energy exchange to whole multiples of $hf$, rather than allowing any continuous value, has the effect of suppressing high-frequency modes: at high frequency, a single quantum $hf$ becomes so large that very little energy is available to excite that mode at all, which is exactly the cut-off needed to stop the predicted intensity climbing without limit at short wavelengths.
It's worth being precise about what Planck did and didn't claim in 1900. He introduced quantisation of energy exchange purely as a device to make the mathematics fit the observed spectrum — he was not proposing that radiation itself was made of discrete particles, and he did not initially believe his own assumption reflected some deep new truth about the nature of light. Planck's quantum hypothesis fixed the black-body problem, and nothing more, as far as he was concerned at the time.
That distinction matters for what comes next in this module. It was Einstein, five years later in 1905, who took the radical extra step of proposing that light itself — not just the process of energy exchange between matter and radiation — is made of discrete quanta, or photons, each carrying energy $E=hf$ whether or not it happens to be interacting with matter at that moment. Einstein's photon hypothesis, covered in the next lesson, used Planck's formula but went well beyond Planck's own, more conservative interpretation of it.
Planck's 1900 result predates, and is logically independent of, Einstein's photoelectric-effect explanation. It is usually regarded as the very first moment classical physics was shown to break down at a fundamental level — the starting point of what would later become quantum theory — even though Planck himself did not fully grasp, or intend, the scale of the revolution his equation would eventually help set off.
Key takeaways
- A black body is an idealised perfect absorber and perfect emitter of radiation, with an emission spectrum depending only on its temperature.
- Raising temperature increases total radiated power and shifts the spectrum's peak to shorter wavelengths.
- Classical wave theory predicted intensity should rise without limit at short wavelengths (the "ultraviolet catastrophe") — flatly contradicted by every real measurement, which shows a single peak followed by a fall-off.
- Planck (1900) fixed this by assuming oscillating charges exchange energy with radiation only in discrete quanta, $E=hf$, rather than continuously.
- Restricting energy exchange to multiples of $hf$ suppresses high-frequency modes and reproduces the observed spectrum exactly.
- Planck intended this purely as a mathematical device to fit the data, not as a claim that light itself is made of particles.
- This was the first crack in the classical wave picture of light — distinct from, and earlier than, Einstein's 1905 photon hypothesis, which is covered next.