Development of Quantum Physics II – The Quantum Hypothesis

In the first part of this series of articles on the topic of quantum physics, I tried to explain the black body radiation and the fact why a tiny hole in the wall of a cavity can be seen as a black body.
In this second article, we want to find a law for this black body radiation. But we will quickly debunk this law and replace it with a new one. As already announced, this new and extraordinary principle messes the old ideas up.

(Be undeterred by the few equations! I only write them down for the sake of completeness – you really don’t have to understand them, if you don’t want to. 😉 )

(You can find the other articles of this series here.)

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The Rayleigh-Jeans law

The wave equation for the electromagnetic field only allows certain (stationary) natural oscillations (= eigen oscillations). Or, to say it more understandably: There are given boundary conditions which apply to the waves in a cavity (= radiation, light, …). To put it simply, the electromagnetic field cannot oscillate at the walls of the cavity and therefore the waves must obey (boundary) conditions. Standing (or stationary) waves form in the cavity as a consequence. These stationary waves can only oscillate in certain, defined frequencies (natural frequencies) which are called “normal modes” of a cavity. (If the dimensions of the cavity are much bigger than the wavelength of the radiation, the “mode density” will become independent of the cavity’s shape…just in case someone’s asking about that.)

To determine the mean energy per eigen oscillation (which is temperature-dependent), Rayleigh and Jeans made use of the classical model of the harmonic oscillator and found a law (the Rayleigh-Jeans law, surprisingly) which connects the spatial energy density wν and the frequency ν as follows:

$w_\nu(\nu)\,\text{d}\nu = \frac{8\pi\nu^2}{c^3}kT\,\text{d}\nu$

For our purposes, the only important thing is that the energy density $w_\nu$ in the (infinitesimal) interval $\text{d}\nu$ increases as the square of the frequency $\nu$.

The Rayleigh-Jeans law fits the experimental data quite well – but only for sufficient small frequencies (e.g., infrared light). For frequencies in the visible spectrum, it differs from experiment dramatically – and even more so for ultraviolet light. This is a problem indeed! (If the frequency $\nu$ approaches infinity, so does the spectral energy density. This is known as the “ultraviolet catastrophe”.)

John William Strutt, 3rd Baron Rayleigh

Sir James Hopwood Jeans

What is wrong with the Rayleigh-Jeans law?

This question also bothered the famous physicist Max Planck in the year 1900.
Trying to avoid the ultraviolet catastrophe, he constructed a novel hypothesis and called it the quantum hypothesis.

Planck’s law

Planck considered the normal modes of the black-body radiation as oscillators, as did Rayleigh and Jeans. But unlike his colleagues, he postulated that there exists a smallest energy the oscillators can absorb. (This means that the normal modes cannot soak up arbitrarily tiny energies.) He called the smallest possible energy packages “quanta“.
These quanta only depend on the frequency $\nu$ of the normal mode and they always are multiples of the smallest energy quantum $h\cdot\nu$.
The lowercase letter $h$ is called “Planck’s constant” and has the approximate (tiny) value $h=6.626\cdot 10^{-34}~\text{Js}$.

The smallest possible energy quanta of the electromagnetic field’s normal modes are called “photons“.

Max Planck

The energy quanta in Planck’s just introduced concept are so small that, normally, we don’t notice them. Let’s take light as an example: It is naturally found in “big portions” only – in rays of light, consisting of countless photons. Thus, we don’t usually think of the possible existence of minimal “light packages” (= photons = energy quanta), because were are simply “too big” to recognize them.

Furthermore, it turned out that the presence of these quanta h·ν offers a solution for the ultraviolet catastrophe.
The prominent Planck’s law

$w_\nu(\nu)\,\text{d}\nu = \frac{8\pi h\nu^3}{c^3}\frac{\text{d}\nu}{e^{h\nu / (kT)}-1}$

of the spectral energy density distribution $w_\nu(\nu)$ of the black-body radiation is able to describe nature very accurately and flawlessly agrees with experiments. (For very small energies, Planck’s law turns in the Rayleigh-Jeans law again.)

This is Planck’s law for various temperatures as a plot:

Black body spectrum according to Planck’s law for temperatures between 300 K and 1000 K in a linear diagram.

As you can see, the energy doesn’t approach infinity as the temperatur rises arbitrarily (= as the wavelengths approach zero) and no ultraviolet catastrophe arises.

Planck’s law is consistent with all experimental data, no exception has been found until today.

Conclusion

I hope you got the basic statement of this article: By assuming that the electromagnetic radiation field is quantized, the experimental results can be interpreted correctly while classical physics fail in this respect. So, Max Planck discretizes nature with his quantum hypothesis. The consequences of this will be sweeping!

Next time, I’ll try to explain the photoelectric effect – Einstein got the Nobel Prize for its theoretical explanation. Using the example of this effect, we will see that only the model of energy quanta is able to describe certain fundamental properties of nature.

The story about the development of quantum physics has just begun…

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