Planck's hypothesis

In December 1900 Max Planck presented to the German Physical Society a hypothesis that he himself described as “an act of desperation”: he had tried every classical route without success, and in the end accepted an idea that seemed to him devoid of physical sense, just to make the numbers work. The hypothesis is this: the oscillators making up the walls of the cavity cannot possess any (continuous) energy, but only integer multiples of an elementary quantity $h\,f$, where $f$ is the oscillator’s frequency and $h$ a new constant of nature. An oscillator of frequency $f$ can therefore only have the energies

$E_n = n\,h\,f, \qquad n = 0,1,2,\dots$

The constant $h$, Planck’s constant, has the value

$h \approx 6{,}63\cdot 10^{-34}\;\text{J}\cdot\text{s}$

and is so tiny that it explains why quantisation remains invisible in the macroscopic world: the “step” of energy $hf$ is enormously smaller than the energies involved in ordinary objects, so the scale appears continuous.

Principle — Quantum of energy (Planck, 1900)

The energy of a microscopic oscillator of frequency $f$ is quantised: it can take only values that are multiples of the elementary quantum $h\,f$.

Why does this hypothesis cure the catastrophe? The physical idea is that a high-frequency mode (small $\lambda$) requires a large quantum $hf$. But thermal agitation has on average only an energy of order $k_B T$ available: if $hf \gg k_B T$, that mode almost never manages to gather enough energy even for its first step, and remains effectively “switched off”. Classical equipartition, which gave $k_B T$ to every mode, is thus suppressed exactly where it was exploding.

Applying Boltzmann statistics to the modes of the cavity with this single hypothesis, Planck obtained the spectral formula

$u_P(\lambda,T) = \frac{8\pi\,h\,c}{\lambda^5}\cdot\frac{1}{\exp\!\bigl(hc/(\lambda\,k_B T)\bigr) - 1}$

which reproduces the experimental curve exactly, at every temperature. Its correctness is immediately clear by examining the two limits:

$\begin{aligned} &\lambda \text{ large:}\quad \frac{hc}{\lambda k_B T}\ll 1 \;\Rightarrow\; \exp\!\Bigl(\tfrac{hc}{\lambda k_B T}\Bigr)-1 \approx \frac{hc}{\lambda k_B T} \;\Rightarrow\; u_P \approx \frac{8\pi k_B T}{\lambda^4} = u_{RJ} \\ &\lambda \text{ small:}\quad \frac{hc}{\lambda k_B T}\gg 1 \;\Rightarrow\; \exp\!\Bigl(\tfrac{hc}{\lambda k_B T}\Bigr)\to\infty \;\Rightarrow\; u_P \to 0 \end{aligned}$

In the limit of long wavelengths the classical Rayleigh-Jeans formula is recovered (where it worked); in the limit of short wavelengths the exponential crushes $u_P$ to zero, killing the ultraviolet catastrophe. Moreover, the two experimental laws also follow from Planck’s formula as consequences: Wien’s law by maximising $u_P$ with respect to $\lambda$, and the Stefan-Boltzmann law by integrating $u_P$ over all wavelengths. A single hypothesis explains everything that classical physics could not explain.

Historical context

Planck himself did not interpret his hypothesis as a quantisation of energy: he thought that the “finite” nature of the quanta was a mathematical property of the energy exchanges with the walls, not a real property of the modes of the electromagnetic field. It was Einstein, in 1905, who took the radical step — light itself is quantised, and the photoelectric effect is direct proof of it. Planck received the Nobel Prize in 1918, Einstein in 1921 (for his work on the photoelectric effect, not for relativity). (Simonyi 2012)

The birth of a new physics

In 1900 nobody, not even Planck, suspected that the formula $E = hf$ would open a new century. Five years later Einstein applied the same idea to photons; eight years later Bohr used it for atoms; twenty-five years later Heisenberg, Schrödinger and Dirac wrote the equations of quantum mechanics; sixty years later lasers, transistors and tunnelling-effect microscopes were born. It all starts here, from the catastrophe that never actually occurred: an iconic example of how a small discrepancy between theory and experiment can open an enormous crack. (Gamow 1968)

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Topics: Quantum physics Concepts: Black body and Planck’s hypothesis · Photon

Related exercises: Worked exercise — Compton scattering with a 0.8 MeV photon · Connecting Malus’s law and a single photon · Hydrogen levels, Balmer and Lyman