Planck's 1900 Quantum Hypothesis, Step by Step

Quantum physics did not begin with atoms or electrons. It began with a hot, blackened box and a formula that fitted the data too well to ignore. In the last three months of 1900 Max Planck went from a lucky guess to a derivation that required energy to come in packets of size hν, the idea behind our Max Planck sticker. Here is how it happened, step by step.

What problem was Planck trying to solve?

A black body absorbs all radiation that falls on it, and when hot it emits a spectrum that depends only on its temperature. In 1898 Otto Lummer and Ferdinand Kurlbaum described a practical version, a heated platinum cavity whose small opening radiates almost exactly like a black body Planck's law, Wikipedia^[1]. With sources like this, experimenters in Berlin measured the spectrum carefully.

Planck had backed Wilhelm Wien's 1896 formula, which fitted at short wavelengths. By September 1900 the experimentalists had shown beyond doubt that it failed at long wavelengths Planck's law, Wikipedia^[1]. On 7 October 1900 Heinrich Rubens told Planck that in the long-wavelength region, and only there, Rayleigh's new formula fitted well, with the energy roughly proportional to temperature Planck's law, Wikipedia^[1].

Step 1: the guess of 19 October 1900

Within days Planck produced a formula that joined the two limits, behaving like Rayleigh's at long wavelengths and like Wien's at short ones Planck's law, Wikipedia^[1]. He sent it to Rubens, who found that it fitted his and Kurlbaum's data at all wavelengths remarkably well. On 19 October 1900 Rubens and Kurlbaum briefly reported the fit and Planck added a short theoretical sketch Planck's law, Wikipedia^[1]. In modern notation, the spectral radiance is

B_ν(ν,T) = (2hν³/c²)·1/(exp(hν/k_BT) − 1)

where B_ν is the power emitted per unit area, per unit solid angle, per unit frequency (W·m⁻²·sr⁻¹·Hz⁻¹), ν is frequency in Hz, T is temperature in kelvin, c is the speed of light in m/s, h is the Planck constant in J·s and k_B is the Boltzmann constant in J/K Planck's law, Wikipedia^[1]. At low frequency it reduces to the Rayleigh–Jeans law and at high frequency to Wien's approximation Planck's law, Wikipedia^[1].

Step 2: the derivation of 14 December 1900

A formula that fits is not an explanation. Planck spent the next weeks looking for one, and presented his derivation to the German Physical Society on 14 December 1900 Max Planck, Wikipedia^[2]. Following a method Boltzmann had used for gases, he counted the ways of distributing energy among a large number of model oscillators in the cavity walls Planck's law, Wikipedia^[1]. To make the counting finite, he divided the total energy into P equal elements, each of size

ε = h·ν

where ε is the energy element in joules and ν is the oscillator frequency in hertz Max Planck, Wikipedia^[2]. The number of ways to share P identical elements among N oscillators is R = (N + P − 1)! / ((N − 1)! P!), a formula that appears in his 1901 paper in Annalen der Physik Planck 1901^[3]. For example, 4 elements among 3 oscillators can be arranged in 6!/(2!·4!) = 15 ways. Taking the logarithm of R gives the entropy, and requiring the result to match thermodynamics led him straight to the formula he had guessed Planck 1901^[3] Planck's law, Wikipedia^[1]. For the background on why counting gives entropy, see our post on Boltzmann entropy.

The decisive point is that ε must be proportional to frequency. If the elements are allowed to shrink to zero, the counting gives back the classical result, which fails badly at high frequency. The graph below shows the consequence: high-frequency oscillators, whose energy step hν is much larger than k_BT, are almost never excited.

Step 3: the numbers Planck extracted

The formula contains two constants, and Planck fitted both to the data. His value of h, 6.55×10⁻³⁴ J·s, is within 1.2% of today's value, and from the same data he made the first determination of the Boltzmann constant k_B Planck constant, Wikipedia^[4]. Since the 2019 redefinition of SI units, h is fixed exactly at 6.626 070 15×10⁻³⁴ J·s NIST CODATA: Planck constant^[5].

Worked example: where does the spectrum peak?

Differentiating Planck's law per unit wavelength gives Wien's displacement law,

λ_peak·T = b ≈ 2.898×10⁻³ m·K

where λ_peak is the wavelength of maximum emission in metres and b = 2.897 771 955×10⁻³ m·K NIST CODATA: Wien wavelength displacement law constant^[6]. Our arithmetic gives these peaks:

• The Sun's surface (about 5,772 K) peaks near 502 nm, in visible green light.
• An incandescent filament at 2,800 K peaks near 1.03 µm, in the infrared, which is why such bulbs are inefficient.
• Human skin at 310 K peaks near 9.3 µm.
• The cosmic microwave background at 2.7255 K peaks near 1.06 mm, which is why it is seen with radio telescopes; see our post on the CMB.

Did Planck solve the ultraviolet catastrophe?

Not in the way it is often told. The ultraviolet catastrophe, the classical prediction of infinite total power from a cavity, was pointed out independently by Einstein, Rayleigh and Jeans in 1905, and the name was first used by Paul Ehrenfest in 1911 Ultraviolet catastrophe, Wikipedia^[7]. Planck in 1900 was fitting data and deriving a formula, not escaping a known paradox. It was Einstein in 1905 who argued that light quanta are physically real Ultraviolet catastrophe, Wikipedia^[7].

What did Planck think the quanta were?

Less than the textbooks suggest. The historian Thomas Kuhn concluded that Planck's 1900 and 1901 papers and his 1906 book contain no clear statement that oscillator energy is restricted to multiples of hν, and that only around 1908 did Planck more or less accept part of Einstein's case for physical discreteness Planck's law, Wikipedia^[1]. Kuhn called these conclusions “heretical”, because they went against the common habit of reading Planck through later ideas, and they have since been accepted by other historians Planck's law, Wikipedia^[1]. Even in 1912 Planck still disputed Einstein's light quanta Planck's law, Wikipedia^[1]. The physics is settled; the history is subtler than the legend. The wider story of how the quantum idea spread is in our post on who started quantum mechanics.

Quick FAQ

What is the Planck constant in simple terms?

It sets the size of an energy step: a light quantum of frequency ν carries energy hν. At 500 nm that is about 4.0×10⁻¹⁹ J, or 2.5 eV.

Was Planck's derivation rigorous?

Wikipedia's account describes it as heuristic Planck's law, Wikipedia^[1]. Rigorous derivations came later with quantum statistics.

Where can I find more quantum designs?

See the Quantum Physics collection.

References

1. Wikipedia, “Planck's law”. https://en.wikipedia.org/wiki/Planck%27s_law
2. Wikipedia, “Max Planck”. https://en.wikipedia.org/wiki/Max_Planck
3. Max Planck (1901), “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Annalen der Physik 309, 553–563 (DOI 10.1002/andp.19013090310; metadata verified via Crossref). https://doi.org/10.1002/andp.19013090310
4. Wikipedia, “Planck constant”. https://en.wikipedia.org/wiki/Planck_constant
5. NIST Reference on Constants, Units and Uncertainty, “Planck constant h” (CODATA, exact value since the 2019 SI redefinition). https://physics.nist.gov/cgi-bin/cuu/Value?h
6. NIST Reference on Constants, Units and Uncertainty, “Wien wavelength displacement law constant b”. https://physics.nist.gov/cgi-bin/cuu/Value?bwien
7. Wikipedia, “Ultraviolet catastrophe”. https://en.wikipedia.org/wiki/Ultraviolet_catastrophe

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.