# Why do we use Planck's constant?

I have been trying to reason why energy packets (i.e. photons) are assumed to be quantized. I know this originated from Max Planck, but may someone explain why energy couldn't be emitted continuously rather than in packets?

Also:

Planck said he didn't understand the reason for this, but it happens. So, how did he arrive at the assumption?

Thanks for any clarification!

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The fundamental reason is that light waves at temperature T have a definite length scale, a typical wavelength, and classical electromagnetic theory has a scaling invariance which forbids such a scale from emerging. In the classical theory, the energy in electromagnetic waves just leaks into ever smaller distances.

The leaking can be understood by thermal arguments. If you have a certain amount of energy in a gas, you can distribute this energy in many ways--- you can put all the energy in one molecule, and have the other molecules sitting still, or you can distribute it roughly equally. The roughly equal distribution is vastly more likely. The average kinetic energy of a gas molecule is one definition of the thermodynamic temperature $T$, and in thermal equilibrium, all molecules have roughly the same kinetic energy, with a probability distribution determined completely just from the energy of each motion.

The same is true for fields. If you have a certain amount of energy in an electromagnetic field, you can distribute it over long-wavelength oscillations only, or over all oscillations equally. By probability considerations, you would prefer the maximally broad distribution. Since there are infinitely many modes at short distances, the classical energy will partition itself into little tiny increments, and leak to short distance electromagnetic jiggles, and this is called the ultraviolet catastrophe: all energy in the universe goes into the shortest wavelengths of light, and everything else stops and freezes.

This is not what is observed in nature. Rather, we see that at any temperature T there is a preferred wavlength $\lambda$ of emitted light. So that as a stove gets hotter, it gets red, then yellow, then white, then blue, then ultraviolet. It doesn't leak energy into short-wavelengths until T is at least a certain amount.

This means that there is something preventing energy from going into short wavelengths. What is it? Whatever it is, it must include a physical constant with dimensions mixing length and energy, which can convert from temperature (average kinetic energy) to wavelength. Since all light in thermal equilibrium with a gas looks the same, no matter what the gas or what the materials of the wall is, it must be a universal constant of nature.

Similarly, you have all sorts of parts inside an atom, like electrons, or nuclei. If these things could jiggle around, the energy would distribute into these too, and all atoms would be jiggling around with a lot of extra energy. This is not observed--- atoms and small molecules don't jiggle. This is the paradox of specific heats noted by Maxwell in the 1870s. These puzzles prevented full acceptance of Boltzmann's idea of statistical entropy.

The breakthrough came in 1898, when Wien noted that the distribution of thermal light in a container has to be consistent with the principle of adiabatic invariance. This is the principle that the thermal equilibrium state, if you change the parameters slowly, stays a thermal equilibrium state with the same entropy.

From this, Wien was led to study what happens to light in a box of mirrors when you move one of the mirrors inward slowly. A moving mirror doesn't reflect light like a stationary mirror, it does some work against the radiation pressure. Further, light reflected from a moving mirror is slightly bluer than the light that comes in. So the energy and the frequency of the reflected light are both changed.

If you want to check this today, you can see it is true by considering photon reflection. Wien showed this for classical EM waves, and the justification of photons comes from this reasoning, not the other way around.

Although the energy and the frequency are both changed when you move the mirror inward, their ratio stays the same. This ratio is an adiabatic invariant:

$$J= {E\over f}$$

This relation showed Wein that whatever the correct thermal equilibrium law, it must have this property: if you take the probability distribution of energy at any one frequency, shift the frequency by a certain amount, then the energy should change by the same amount. This means that, up to a scaling and normalization, the probability distribution is a function of the adiabatic invariant $E/f$

Wein then guessed that this probability distribution is exponential in E, since Boltzmann had shown that at temperature T, the probability of a classical state with energy E goes like $e^{-\beta E}$. This led Wien to guess the form $e^{-b{E\over f}}$ for the probability of having energy E at frequency f, where the constant $b$ is a new constant of nature. In units where Boltzmann's constant $k$ is set to 1, so that energy and temperature are measured with the same units, b is just the reciprocal of Planck's constant.

Wien's law explains the emergence of a length scale. The typical energy determines a typical frequency $f = bkT$, and determines a typical wavelength by $\lambda= {c\over f}$.

Wien's form is not correct except at high frequencies. At low frequencies, you find that the energy in each mode is not exponentially distributed, but on average equal to $kT$, like for any harmonic oscillation. Planck interpolated the two forms with a clever guess, and tried to justify the interpolation. His reasoning is not optimal, since he was motivated more by experimental data, and he wasn't sure of Boltzmann's hypothesis, nor did he appreciate the ultraviolet catastrophe fully (at least not according to Kuhn, who examined the original papers in detail--- I don't know if Kuhn is right, I just skimmed the original paper). I will follow Einstein's 1905 reasoning, which starts where Wien left off.

Einstein calculated the entropy of Wien's distribution, assuming it is correct, and noticed that the entropy at each frequency goes like

$$Nlog({V\over N})$$

Where $N= {bE\over f}$ will be seen to be the number of photons at frequency f.

The form of the entropy is counting the number of possible configurations of high-frequency light. The number of configurations is the exponential of the entropy, and it goes like

$${V^N\over N!}$$

it's the entropy of an ideal gas. This suggests strongly that the light is composed of photons, localizable particles each carrying an energy $hf$ where $h$ is Planck's constant, the reciprocal of Wein's constant $b$. The adiabatic invariance of the thermal photon gas means that the photons have to be able to bounce off slowly moving walls and stay single photons, and consistency with classical EM means that as the frequency of a photon changes, the energy changes proportionally, so that the photon must carry an energy proportional to the frequency, and the proportionality constant is a universal constant of nature.

From this hypothesis, Einstein went and solved the specific heat problem. He also was able to resolve the paradoxes of light emission in relativity, and deduce $E=mc^2$. It was Einstein who created the modern photon concept, and pretty much he alone believed in photons until 1919, when Milliken established experimentally that the photoelectric effect goes as Einstein says, and Compton showed that photons can individually collide with electrons in the famous Compton scattering experiments.

Once you know that there are bundles of energy of size $hf$, the ideas of Boltzmann allow you to rederive Planck's distribution law for thermal light, by assuming that the probability of having N photons at frequency f goes as $e^{-\beta Nhf}$ where $\beta=1/kT$ is the inverse temperature, and $Nhf$ is the total energy the $N$ photons have together.

While the deductive path is clearest through Einstein, it was Planck who came up with the distribution first. Planck's reasoning is based on interpolation, and I wouldn't be able to reproduce it logically, because it involves an unjustified assumption that the specific heat of the photon gas (or something like that, I don't remember what) would be the average of the Wien law and the Stephan-Boltzmann law.

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"This suggests strongly that the light is composed of photons, localizable particles each carrying an energy $hf$ where $h$ is Planck's constant" --- we take the localization to be in Fourier space, not in real space, right? Your argument, using only frequencies, doesn't justify localization in real space. We can construct states that are continuous superpositions of different number states, using arbitrary wave number modes, which somewhat takes the gloss off an argument that the energy isn't continuous. Analogously, a drum has continuous energy and has discrete modes. – Peter Morgan Apr 28 '12 at 19:30
That's the beauty of Einstein's argument--- the entropy is the same as particles localized in real space! The number of states goes as the volume to the n-th power, which only makes sense for localizable things. That this is a frequency picture is strange, but nowadays we understand that the entropy of a short wavelength gas is the same as the classical entropy of localizable particles, so there is no paradox. For Einstein, he was thinking for a while that there are particles riding on top of waves, like deBroglie Bohm. – Ron Maimon Apr 28 '12 at 23:27
and anyway, a drum has discrete energy in each discrete mode. (or at least a very cold drum, anyway). – Emilio Pisanty Apr 29 '12 at 9:43

Assuming continuous emission generated results at odds with experiment (the UV catastrophe): that is the theory was false to fact.

Those are the circumstances under which theorists start trying all kinds of crazy stuff just to see if it will work, and sometimes the justification comes after the math.

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I expanded on your answer below--- Planck's original work is sort of obscure on this, but Wein had the main idea right (but not photons), and Planck was following Wein. – Ron Maimon Apr 28 '12 at 5:22
@RonMaimon: Do you mean Wien? Wein is wine and Wien is Vindobona a.k.a. Vienna. – NikolajK Apr 28 '12 at 18:59
@NickKidman: Probably the least of my atrocious spelling sins. – Ron Maimon Apr 28 '12 at 23:28
@RonMaimon: I'd put names on people on the top of the not-to-spell-wrong list. – NikolajK Apr 29 '12 at 0:00
@RonMaimon: Also, I'm from Wien. ;) – NikolajK Apr 29 '12 at 0:00

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