How did Planck arrive at the idea that energy is quantized? [duplicate]

Max Planck, arguably the leading theoretical Physicist of his day, apparently used of experimental data to formulate his equation for black body radiation, which was a major step in the birth of quantum mechanics. For this seminal event of the 20th century, I would expect a detailed explanation of how Planck arrived at his conclusion that energy MUST be quantized. I've even read his Nobel lecture but was non the wiser.

Can some kind person please explain it, or give me even just a hint?

marked as duplicate by AccidentalFourierTransform, probably_someone, Kyle Kanos, Emilio Pisanty quantum-mechanics StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jun 6 '18 at 11:45

• I think Planck does have a claim to be the leading theoretical physicist of his day, but Lorentz and certainly Boltzmann might be contenders for the title. Probably others. Interesting... – Philip Wood Jun 5 '18 at 23:01
• I read that Planck knew the experimentally measured distribution of black body radiation power and was trying to recover that experimental data theoretically. He tried a lot of things that didn't work. Then, out of "despiration", he tried converting an integral over energies to a sum over discrete energies, and it worked. In other words, it was literally a random guess-and-check math idea with no physical basis whatsoever. I don't remember where I read that. It might have been The Making of the Atomic Bomb, but I'm not sure. – DanielSank Jun 6 '18 at 4:32
• Possible duplicate of How did Planck derive his formula $E=hf$? – probably_someone Jun 6 '18 at 4:36
• @DavidSank Yes, that's exactly my understanding of how Planck discovered quantisation, though he would have understood even as he was doing the maths, the quantisation implications of his trial sum of discrete energies. He was quite bright! – Philip Wood Jun 8 '18 at 21:48

Warning for purists: this is going to be very hand-wavy. Planck's work was esoteric, cutting-edge Physics at the time (c 1900) and is still highly technical second or third year undergraduate Physics, requiring a knowledge of statistical mechanics – glossed over in what follows.

Black body radiation is the radiation inside a cavity with walls at a fixed temperature. The radiation is in the form of 'standing' electromagnetic waves. Many frequencies of standing wave will co-exist. There will be more possible modes of standing wave with frequencies between (say) 80 THz and 90 THz than between (say) 10 THz and 20 THz and the number of modes per 10 THz interval goes up continuously with frequency. So if the energy of the radiation is distributed evenly among different modes of standing wave, one would expect the high frequency modes (ultraviolet and beyond) to hog the lot – simply because there are more and more modes (without bound!). This was later called 'the ultraviolet catastrophe'.

The word 'catastrophe' was used because this prediction seemed like a catastrophe for theoretical physics. The high frequency modes don't hog all the energy – far from it. Lummer and Pringsheim had conducted careful experiments which showed that energy density peaked at some intermediate frequency. Planck built on the work of Wien, Boltzmann and others and succeeded in finding an equation that fitted these experimental results.

Planck then discovered that he could derive this equation from pure theory, but only by making what he considered to be an outrageous hypothesis: that for a mode of frequency f, the energy in the cavity can only exist in 'lumps' or quanta of size E = hf. How does this avoid the ultraviolet catastrophe? Even a single quantum for a very high frequency mode would be huge. The chances of great chunks of the available energy going to more than a few (high frequency) modes were remote; these greedy modes were priced out of the market!

This attempt at explaining Planck's brilliant idea has made several oversimplifications. For one, Planck stressed quantisation of the energies emitted and absorbed by oscillators in the cavity walls, rather than quantisation of the waves themselves. Above all, I've totally omitted the mathematical treatment – which is what makes Planck's work so convincing for physicists.

• Would the downvoter be kind enough to give a reason? It could be instructive. – Philip Wood Jun 9 '18 at 11:00
• This explanation is physically correct but historically incorrect. Planck was not trying to explain away the ultraviolet catastrophe, which in fact was not understood until Ehrenfest's work in 1911. It wasn't until well after the understanding of the quantization of EM modes that people realized the problems with the classical approach, giving an after-the-fact justification for QM. Planck was simply trying to theoretically understand a phenomenological result, not to resolve any deep problem with (what we now call) classical physics. – tparker Nov 18 '18 at 18:05
• Moreover, Planck didn't take his quantization assumption very seriously, and certainly wouldn't have thought of it as "outrageous". He just thought it was a convenient math trick that simplified the calculation of the partition function, which approximated some continuous exact calculation. It wasn't until Einstein's explanation of the photoelectric effect five years later that people realized that EM radiation really is physically quantized. – tparker Nov 18 '18 at 18:06
• (a) I don't think I claimed that Planck was "trying to explain away the ultraviolet catastrophe", but I've inserted the word 'later' near the end of my second paragraph to remove what could be interpreted as an anachronism. (b) I know that Planck regarded quantisation as a mathematical trick, but wasn't he well aware of what it implied physically (at least as regards emission and absorption of radiation)? – Philip Wood Nov 18 '18 at 19:47
• My understanding is that until Einstein's explanation of the photoelectric effect, Planck did not believe that anything was physically quantized, but instead that the discrete sum in the partition function was just a discretization of what he believed to be fundamentally a continuous integral. And I still think it's misleading to discuss the ultraviolet catastrophe at all in the context of a question about Planck's historical development - it's not just the name, but the very idea that had not yet been developed, so it did not in any way motivate his discovery. – tparker Nov 18 '18 at 20:32