It seems that Planck's constant was made from fitting a curve for blackbody radiation, is it just experimental-further more his assumption that energy comes in quanta seems to have been a guess. Why would energy come in quanta? Wikipedia says that he didn't think about it much, but I don't know why harmonic oscillators would even be suggested to only have a quantized number of energy modes, moreover what an energy mode would be. With a set frequency, how does one vary the energy? The main question I have, is what the reasoning behind the quantization of oscillators?
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1$\begingroup$ Planck was actually driven the to the mathematical form the curve by a genius sequence of heuristic arguments from empirical laws of thermodynamics and Wien's law. He certainly didn't just guess it, or extract it from some curve-fitting procedure. $\endgroup$– David HJul 29, 2013 at 3:36
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$\begingroup$ Do you know a good link to those arguments? My textbook isn't the best. $\endgroup$– user24082Jul 29, 2013 at 3:51
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1$\begingroup$ Sure. Here's a pdf version of Planck's On the Law of Distribution of Energy in the Normal Spectrum. $\endgroup$– David HJul 29, 2013 at 3:57
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2$\begingroup$ I suppose you are just asking how Planck was inspired and are not puzzled by how really BB radiation happens . If the latter have a look at my answer physics.stackexchange.com/questions/70896/… . A harmonic oscillator is a good approximation to any symmetric potential as the first term in an expansion of a symmetric potential is x**2 . $\endgroup$– anna vJul 29, 2013 at 5:05
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2 Answers
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As I understand, you are asking for Planck's motivation for his law and thus quantization of energy. Well, there is a letter written in 1931 by Planck to Robert Williams Wood. I haven't found it anywhere on the web, so I'll quote part of it here. I'm taking it from Theoretical Concepts in Physics by Malcolm S. Longair (as Longair, I also find it rather moving):
Briefly summarised, what I did can be described as simply an act of desperation. (...) A theoretical interpretation therefore had to be found at any cost, no matter how high. (...) a new constant is required to assure that energy does not disintegrate. But the only way to recognize how this can be done is to start from a definite point of view. This approach was opened to me by maintaining the two laws of thermodynamics. The two laws, it seems to me, must be upheld under all circumstances. For the rest, I was ready to sacrifice every one of my previous convictions about physical laws. (...) one finds that the continuous loss of energy into radiation can be prevented by assuming that energy is forced at the outset to remain together in certain quanta. This was purely a formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result.
He later tells Wood that he is sending him an English version copy of his Nobel lecture, so I guess that's also worth reading. If you find the full letter on the web, please let me know.
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$\begingroup$ But is the quantization so that the series converges? As opposed to an integral? $\endgroup$– user24082Aug 1, 2013 at 2:58
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1$\begingroup$ Well, basically, yes. When he says "the continuous loss of energy into radiation..." he refers to the ultraviolet catastrophe, meaning that replacing the distribution of energy $f(E)$ in the expression $\langle{E}\rangle=\frac{\int_0^\infty{E}\,f(E)\,dE}{\int_0^\infty{f}(E)\,dE}$ with a discrete one, $\langle{E}\rangle=\frac{\sum_{n=0}^\infty{E}\,f(E)}{\sum_{n=0}^\infty{f}(E)}$ one gets rid of the divergence seen e.g. in Rayleigh-Jeans law. That can be read in likely any introductory quantum mechanics book, and it may be seen as the "formal" justification for what Planck did, $\endgroup$– user24999Aug 1, 2013 at 4:01
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$\begingroup$ but it's still kind of funny /odd /interesting that it was practically (judging by the letter), as you say, a guess. In the Wikipedia article about Planck, it is said: Subsequently, Planck tried to grasp the meaning of energy quanta, but to no avail: "My unavailing attempts to somehow reintegrate the action quantum into classical theory extended over several years and caused me much trouble." Still this same classical limit is of interest in branches like quantum chaos. That I find pretty fascinating. $\endgroup$– user24999Aug 1, 2013 at 4:08
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The answers posted so far are correct but miss the questioner's point. Planck originally (and incorrectly) attempted to derive the law of black-body radiation by assuming that energy was an infinitely divisible "fluid" which could be apportioned smoothly into a large collection of containers (i.e., the resonators - molecules - in the walls of the black-body cavity). This derivation (which he published) failed to reproduce the actual spectrum observed only a few months later. In order to rescue his derivation (and his reputation), Planck re-derived his formula using Boltzmann's statistical-based equation for entropy which involves counting the number of discrete ways indivisible "droplets of energy" could be used to fill the aforementioned containers. He assumed that at some point, he could arrive at a correct expression by allowing these hypothetical "droplets" to become arbitrarily small, but the mathematics of his derivation did not allow such a limit to be taken while still matching the experimental data. Planck was therefore stuck with his "droplets" (which he called quanta) distributed among the resonators.
It should be noted that Planck did not associate these quanta with black-body radiation itself (i.e., photons). That conceptual leap was made by Einstein 5 years after Planck's reputation-saving derivation using Boltzmann's $S=k_B\ln(W)$ counting formula for the states of the atoms in the cavity walls. Planck himself was never entirely comfortable with the idea of radiation (in the Maxwellian sense) in the form of discrete packets.
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1$\begingroup$ To elaborate, the "droplets of energy" would not be at odds with classical physics, provided the limit could be taken afterwards. So, a priori, I don't think he was throwing anything out the window. $\endgroup$ Nov 23, 2013 at 22:49
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$\begingroup$ @jim thanks for your answer, btw, you can typeset math nicely by enclosing it between dollar signs. $\endgroup$ Nov 23, 2013 at 22:51