The Radiation Laws and the Birth of Quantum Mechanics

Quantum mechanics was born on the 14th December 1900 when Max Planck explained the derivation of his radiation law at a meeting of the German Physical Society in Berlin. It was a desperate attempt to explain findings by Lummer, Pringsheim, Kurlbaum, Paschen, and Rubens who had performed precise experiments on thermal radiation of certain objects called `black body radiators'.

The black body radiator was a concept from the middle of the 19th century, introduced by Kirchhoff in 1859. Kirchhoff discussed the thermal equilibrium between the radiation (heat) within an arbitrarily shaped container and the walls of the container (black bodies) that completely absorb all incident radiation. The walls also emit radiation since otherwise there would be no radiation within the container at all (which contradicts experiments). The radiation inside such a cavity is called black body radiation.

From thermodynamics (second law) one can show that the spectral energy density $u$ of black body radiation, i.e. the radiation energy per volume and per frequency interval, is only a function of the frequency $\nu$ and the temperature $T$ of the walls, and does not depend, e.g., on the shape of the container:

$$u=u(\nu,T)$$ (1)

Note: $\int_{\nu_1}^{\nu_2}d \nu u=u(\nu,T)$ is the radiation energy within the frequency interval $[\nu_1,\nu_2]$.

Kirchhoff had already pointed out the importance of determining the explicit form function $u(\nu,T)$. Its importance should lie in the fact that it was universal, i.e. independent of any details of the geometry of the container. Such a universal function could be expected to contain deep physical insights about thermodynamics and radiation.

In fact, it took 40 years and the efforts of many physicists to finally find the explicit form of $u(\nu,T)$. After Kirchhoff's introduction of the `black body', people where wondering how to realise this theoretical concept experimentally. They first tried to blacken metallic (platinum) plates without much success. The success came by Otto Lummer and Wilhelm Wien (`Physikalisch-Technische-Reichsanstalt', the PTR in Berlin, a precessor of the nowadays PTB, the German national Bureau of standards). They went back to the original definition (thermal equilibrium with the walls of the container) of the black body and argued that one should use a cavity with a small hole inside to get the black body radiation out of it, without disturbing it too much.

Figure: Lummer and Kurlbaum's black-body experiment from 1898: platinum cylinder sheet within a ceramic tube [Phys. Bl. 12/2000, p. 43].

At that time, there was already the theoretical prediction by Wien who had found in 1893 a scaling law for $u(\nu,T)$, stating

$$u(\nu,T)=\nu^3f(\nu/T)$$ (2)

with an (unknown) `scaling function' $f$ of only one variable, i.e. the ratio $\nu/T$. In particular, this scaling law immediately explained the Stefan-Boltzmann-law

$$U(T):=\int_0^{\infty}d \nu u(\nu,T)=\sigma T^4,\quad \sigma = const.$$ (3)

Wien even made a suggestion for the explicit form of $f$ in analogy to Maxwell's velocity distribution in a gas (Wien's law),

$$u(\nu,T)=\frac{4\nu^3}{c^3}b\exp\left(-\frac{a\nu}{T}\right),\quad a,b = const,$$ (4) (valid for large frequencies and small temperatures: quantum limit !)

where $c$ is the speed of light. Wien's law was compatibel with the experimental results until the middle of the year 1900. Lummer and his coworker Kurlbaum had developed a very precise bolometer, based on the bolometer by Samuel P. Langley used in astrophysics from 1880. Furthermore, Lummer and his coworker Pringsheim developed black body radiators that could operate in a very large temperature range between -188$^o$C and 1200$^o$C, later up to temperatures of 1600$^o$C.

It turned out that Wien's law (1.4) was quite a good description of the experimental data but there were small deviations at large temperatures. Lummer and Pringsheim again improved their experiment into the range of up to wavelengths $\lambda=c/\nu$ of $\lambda=8.3 \mu$m and $T=1650$K, and the deviations became even stronger. The story became even more confusing in the autumn of 1900 when Friedrich Paschen in Hannover claimed good agreement of his data with (1.4), and Max Planck also had `proven' it by thermodynamic considerations.

The bomb came with new measurements by a guest scientist at the PTR (Heinrich Rubens) which extended up to $\lambda=50\mu$m. The deviations from Wien's law could not discussed away any longer. Rather, in the extreme long wave-length limit, Rubens found good agreement with another radiation law that had previously set up by Lord Rayleigh (Rayleigh-Jeans-law),

$$u(\nu,T)=\rho(\nu)\bar{E}(\nu)=\frac{8\pi\nu^2}{c^3}k_BT,$$ (5) (valid for small frequencies and large temperatures: classical limit !)

where $k_B$ is the Boltzmann constant. Rayleigh's law followed from the density of states $\rho(\nu)=8\pi\nu^2/c^3$ (density of electromagnetic eigenmodes per volume, polarization direction and frequency) of the electromagnetic field in a cavity, and the theorem of thermodynamics that gives each degree of freedom of an oscillation in thermal equilibrium an average energy $\bar{E}(\nu)=k_BT$ ($1/2k_BT$ for kinetic and potential energy each), independent of the frequency $\nu$.

Rubens told Planck of his observations over afternoon tea, and the same evening Planck, in a desperate attempt to `improve' Wien's law, suggested an interpolation formula between (1.4) and (1.5), Planck's law

$$u(\nu,T)=\frac{8\pi\nu^2}{c^3} \frac{h\nu}{\exp\left(\frac{h\nu}{k_BT}\right)-1},$$ (6)

where the new constant $h$ is the Planck constant

$$h= 6.626 \times 10^{-34}Js.$$ (7)

Planck's law turned out to give excellent agreement with all the experimental data. He solved this puzzle by the hypothesis Planck's hypothesis, that oscillators change their energies $E(\nu)$ only in integer multiples of a fundamental energy unit $\varepsilon$. He didn't explicitely assume $E(\nu)=n\varepsilon$ at that time, but he showed that $\varepsilon$ must be proportional to the frequency $\nu$, i.e. $\varepsilon=h\nu$.