## Sunday, September 27, 2020

### How did Max Planck come up with the quantum theory of light?

Reposted from Quora

Back in 1900, Max Planck just wanted to find the right formula – which we know as Planck's law now – saying how much energy a black body (which absorbs everything and doesn’t reflect any light) emits in electromagnetic waves in any interval of frequencies (or wavelengths) when it’s heated up to a given temperature.

At low frequencies, it was known that the Rayleigh–Jeans law – a simple power law – did a good job to describe the density of energy per unit wavelength. A classical derivation from the first principles was known, too: it’s based on the simple Boltzmann distribution applied to the possible values of energy carried by the electromagnetic waves.

At high frequencies, the much newer Wien approximation – with an exponentially dropping factor – was relevant and phenomenologically matched the observations. That already included a new constant, now known as Planck’s constant, but wasn’t called in this way, and no deeper derivation of that other limit was known. That Wien approximation was extracted from pure experiments.

Planck knew that a better formula that works in both limits and in between should behave as something that interpolates between exponentials and power laws. So at high frequencies, it should be a decreasing exponential. But the exponential shouldn’t affect the behavior at low frequencies.

So he simply replaced the decreasing exponential by$\frac{1}{\exp(X)-1}=\dots$ When the (absolute value of the) exponent $$X=hf/kT$$ is large, this becomes $$\exp(-X)$$. When it’s small, the denominator may be Taylor expanded and the fraction becomes $$1/X$$ which can be completed to the power law.

This new Planck formula agreed with the data beautifully and just a few months later in Fall 1900, he also gave a better explanation why this formula is correct. The inverse of the “exponential minus one” may be written as a geometric series$\dots = e^{-X}+e^{-2X}+e^{-3X}+\dots = \dots$ and this geometric series looks like the “integral” (over the phase space – well, only the integration over the momentum space is nontrivial, the regular space is just a volume factor) that one uses in standard thermodynamics. But it looks like we are only summing over discrete levels:$\dots = \sum_{N=1}^\infty e^{-NX}$ So he figured out that if the electromagnetic field isn’t allowed to carry any energy but only the energy $$E = N\cdot hf$$ which is the product of a non-negative integer (now interpreted as the number of photons in a state – the sum above goes from $$N=1$$ rather than $$N=0$$ because it calculates the average energy, not the partition sum or probability), Planck’s constant, and frequency, one may derive the perfectly correct result that works at all frequencies, low ones, high ones, and those in between, using the “modified classical derivation” but with sums instead of integrals.

He had derived the “light quanta” or, using a modern word introduced by the chemist Gilbert Lewis (1926), “photons” – the energy of the electromagnetic field had to be a multiple of the photon’s energy, it was clear from his argumentation – but he didn’t take his derivation seriously. He thought it was just a trick and the right explanation would avoid the light quanta. But in 1905, Einstein took the photons much more seriously and explained the photoelectric effect as a mechanical collision of photons with matter that kicks electrons out (Einstein got his Nobel prize for this explanation in 1921; he didn’t get any Nobel prize for relativity for rather silly, partly political, reasons). Doubts about photons evaporated when the Compton scattering was explained as a “mechanical, elastic collision of photons with electrons” in 1923.

So Planck established the research of quantum theory – where quantities aren’t always continuous but may be quantized in steps – because he successfully explained the black body radiation using a “sum over allowed energies” instead of the “integral over allowed energies” which classical physics had previously suggested. The replacement of the continuous energies (and integrals) by discrete energies (and sums) is the magic that allows the energy carried by the electromagnetic waves to drop at high frequencies of the light (and the total to be finite or convergent; the total energy emitted by a black body was predicted to be divergent i.e. infinite according to the Rayleigh-Jones-Boltzmann classical derivation above and that’s clearly nonsense, no light bulb can emit an infinite amount of energy per second). That was a crucial contribution to the birth of quantum theory but as far as I know, it was also his last major contribution to quantum theory or quantum mechanics.

Planck became irrelevant in physics research when the true quantum mechanics was born in the 1920s and he was among the folks who basically expected quantum mechanics to go away – be replaced by some classical waves – but it couldn’t have happened and it never happened.