Why did Max Planck assume discrete energy values?

[Mikeal]

I've read several articles discussing how Max Planck decided to assume that the energy radiated by oscillators in a black-body came only in discrete increments of En = nhf, where n is an integer. Using this concept, he determined that the average energy of an oscillator was given by:
∞
<E> = ∑ Enp(En)
n=0

Where p(En) is the probability of its occurrence = e(-En/kT)
∞
∑ e-En/kT)
n=0
This led to the correct equation for the black-body radiation curve.

The texts claim the use of discrete values was a "mathematical expedient" to reduce the computational load. This reason doesn't make sense to me, as the continuous solution could have been computed by the fairly simple integral:
∞
<E> = ∫Enp(En)
n=0

My question is, what was the real reason/underlying science that drove Planck to use discrete energy values?

 

[Heinera]

Because it gave an excellent fit to the empirical black-body radiation curve. It was a combination of trial and error, and a stroke of genius.

See also his autobiography:
https://www.amazon.com/dp/0806530758/?tag=pfamazon01-20

 

[Mikeal]

This is what I'm having a problem with. The guy was a gifted and well respected scientist and we're supposed to believe that he floundered around until he had a epiphany (i.e. discrete energy values) which even he admitted, had no scientific basis at that time. That's not the way that scientists operate. Something must have driven him in that direction. If it wasn't science, it must have been a mathematical insight that showed the way forward. In fact, if you go to Wikipedia - Planck's Constant, there is some insight that suggests that non-discrete oscillator energy levels produced multiple solutions, whereas discrete energy levels produced the correct/unique solution. I'd like to understand that part a little more.

 

[phyzguy]

This is what I'm having a problem with. The guy was a gifted and well respected scientist and we're supposed to believe that he floundered around until he had a epiphany (i.e. discrete energy values) which even he admitted, had no scientific basis at that time. That's not the way that scientists operate.

Says who? In my experience, this is exactly the way scientists operate when faced with a difficult problem. You basically say to yourself, "Well, our accepted picture obviously isn't working, so what else could we assume that might give the correct result?" Then you cast around with different ideas, discarding the ones that don't work until you find one that does. There are many examples.

 

[stevendaryl]

I sort of thought of it as Planck using discrete energies as calculational device, in the same way that you might approximate an integral by a sum. He thought that in the end, he would take the limit as $h \rightarrow 0$, but he found that he got better agreement by sticking to a non-infinitesimal value for $h$.

I can't remember whether I actually read that, or made it up.

 

[bhobba]

This is what I'm having a problem with. The guy was a gifted and well respected scientist and we're supposed to believe that he floundered around until he had a epiphany (i.e. discrete energy values) which even he admitted, had no scientific basis at that time.

Yea - it's basically how things are creatively done sometimes.

I hate to tell you of the number of computer programs I 'fixed' that way.

Here is one. We released a new system I worked on with my team leader. It performed like a dog. I went to meetings saying we need more powerful hardware - letters went around saying I was full of the proverbial just sprouting computer mumbo jumbo - I told some guys the results of queuing theory how you pass a certain knee and things go to pieces - I now know I should have shut up - but what can I say - I was young foolish and full of it. Maybe I have gotten better since then - it was a LONG time ago - I suppose it depends on who you ask.

Anyway we were under pressure and had all sorts of meetings with finger pointing going all over the place, but nobody, including me really had any idea.

But I sat down and took a punt. To cut a long story short I thought our name searching algorithm was the culprit - it read stuff into database memory then scanned through the memory. Despite the fact I wrote it, I just aesthetically didn't like how crude it was, took the punt and drastically improved the algorithms efficiency. Conventional wisdom was it shouldn't matter - the disk I/O should have swamped the memory scan. But to everyone's amazement it worked - as one guy said it was like pulling some kind of stopper out. I got a lot of (undeserved) kudos because of that - my bosses unfairly were told how come I solved it and they didn't - of course it was just for a while - things like that are soon forgotten. But why did it work? Well we found out later after talking to the Database administrator. Sure it was in memory - but you had to do what was called a SVC (supervisor call) every-time you accessed it and guess what a SVC is - it wrote to disk before the call and returned the result to disk. It was SVC bound even though it was in memory. I hit on the answer by pure dumb luck while those who knew a lot more than me floundered.

Its just the way it is sometimes. Nothing to do with physics - but just an example of dumb guesses sometimes working.

Thanks
Bill

 

[bhobba]

I can't remember whether I actually read that, or made it up.

That's about right - here is the source:
https://www.amazon.com/dp/B00FN6JEJO/?tag=pfamazon01-20

Thanks
Bill

 

[fresh_42]

I'd like to understand that part a little more.

This article might be of interest for you (kudos to @DrClaude):
https://pdfs.semanticscholar.org/7aeb/37159b682b0e5ce54cdeb2834ccb4dbcd5be.pdf

 

[friend]

Or was it the case that he first found a curve that matched the data and worked it backwards to discover the discrete nature?

 

[stevendaryl]

Or was it the case that he first found a curve that matched the data and worked it backwards to discover the discrete nature?

@fresh_42 already posted a link to an article discussing the historical development of Planck's law. There were several steps:

1. An approximate formula by Wien, which didn't have a rigorous derivation
2. A derivation by Planck that did not use quantized energy.
3. The discovery that Wien's formula isn't accurate for low energies.
4. A revised formula by Planck that correctly described the low energy spectrum.
5. A derivation of the revised formula by Planck using discrete energy quanta.

The derivation in step 5 was indeed after-the-fact---he already had the correct formula, obtained in a more ad-hoc way.

 

[Paul Colby]

Only vague memories but, Planck's area of expertise was thermodynamics and statistical mechanics. Perhaps Planck recognized that it was the proliferation of states at high frequency that gave rise to the ultraviolet catastrophe. Limiting to finite energy quanta was a natural way to limit the number of high energy states making it finite. As mentioned above, he could then look at the $\hbar \rightarrow 0$ limit. The black body data he had access to was very good making the let's stop at $\hbar$ the obvious choice of interest.

 

[stevendaryl]

Only vague memories but, Planck's area of expertise was thermodynamics and statistical mechanics. Perhaps Planck recognized that it was the proliferation of states at high frequency that gave rise to the ultraviolet catastrophe. Limiting to finite energy quanta was a natural way to limit the number of high energy states making it finite. As mentioned above, he could then look at the $\hbar \rightarrow 0$ limit. The black body data he had access to was very good making the let's stop at $\hbar$ the obvious choice of interest.

Well, according to the paper mentioned (https://pdfs.semanticscholar.org/7aeb/37159b682b0e5ce54cdeb2834ccb4dbcd5be.pdf) Wein's formula got the right high-energy behavior, but Planck had to resort to quantizing energy in order to derive the correct low-energy behavior.

 

[friend]

@fresh_42

4. A revised formula by Planck that correctly described the low energy spectrum.
5. A derivation of the revised formula by Planck using discrete energy quanta.
The derivation in step 5 was indeed after-the-fact---he already had the correct formula, obtained in a more ad-hoc way.

Right. So is it possible to derive the quantum energies from the curve that fit the data? And is it possible that Planck derived quantum theory this way?

 

[Paul Colby]

Wien's formula got the right high-energy behavior, but Planck had to resort to quantizing energy in order to derive the correct low-energy behavior.

Wien's derivation of the high frequency behavior was not a direct and simple application of statistical mechanics to the problem but rather a hand wavy one based on how things generally aught to go. While Wien's expression is correct, the argument he used to obtain it tweaked Planck's sensibilities. The ultraviolet catastrophe is a divergence of the classical statistical mechanics (Rayleigh-Jeans analysis) at high (ultraviolet) frequencies, exactly where the Wien result applies. This is the point at which the classical measure of the "number of states" needed for a straight forward statistical mechanics calculation diverges.

 

[fresh_42]

Wein's formula got the right high-energy behavior ...

Wein's derivation of the high frequency behavior ...

Wien.

 

[Paul Colby]

Wien.

I was just copying the spelling in #12.

 

[stevendaryl]

I was just copying the spelling in #12.

Somehow, I got it right in #10, and then wrong in #12.

 

[Charles Link]

One thing that is quite important about the Planck blackbody function, besides solving the problem of the ultraviolet catastrophe, is that it can be integrated in closed form: $\int\limits_{0}^{+\infty} L(\lambda,T) \, d \lambda=\frac{\sigma T^4}{\pi}$ and it gives a theoretical value for the Stefan-Boltzmann constant $\sigma=\frac{\pi^2}{60} \frac{k_b^4}{\hbar^3 c^2}$, where $\sigma=5.67 \, E-8$ watts/(m^2 K^4) was previously only known experimentally, and wasn't known in terms of the fundamental constants. $\\$ (The integral is a non-trivial one that requires special integration techniques, but in any case it does have a closed form which, in fact, is precisely the Stefan-Boltzmann law). $\\$ In looking over the paper that @fresh_42 gave a "link" to in post #8, it is likely that the derivation of the Planck function has evolved considerably from what Planck originally presented around the year 1900. The article of the "link" in post #8 seems to suggest that Planck didn't even use a periodic boundary condition approach, which is now used in the standard derivation for the Planck function in the present day Statistical Physics textbooks.

 

[vanhees71]

As far as I know, the history of the black-body spectrum was as follows: Planck has been interested always in the fundamental questions, and one of the most pressing fundamental questions was that of the black-body spectrum since it was known from quite some time (~mid 19th century) that it is a very fundamental spectrum, independent of the material of the cavity that can only depend on fundamental constants. Planck made his name with the foundations of phenomenological thermodynamics, particularly a clear theoretical definition of entropy. He was quite sceptical against statistical methods a la Boltzmann first. In any case it was natural for him to study the black-body-radiation problem, and so he did from the 1880ies on. His trick was to use the independence of the spectrum from any specifics of the medium the cavity walls may consist and just using a single harmonic oscillator coupled to the electromagnetic field and asking for the thermal-equilibrium conditions between radiation and the oscillator (coupled to a thermal bath).

The breakthrough came, as almost always in physics, with high-precision experimental data by Rubens, Kurlbaum, et al from the Physikalisch Technische Reichsanstalt, who wanted to measure the black-body spectrum over a wide range of wave lengths in order to create a "normal spectrum" to compare light sources with it to have an industrial standard for various such light sources (particularly the upcoming electric ones).

First Planck investigated completely phenomenologically, how the spectrum (i.e., energy density per frequency or wave-length interval) should look, and he interpolated between the Rayleigh-Jeans and Wien's law, introducing the fudnamental constant, now named after him, the Planck quantum of action. He got what's also named after him, the Planck spectrum. Of course, as a theoretician that was not a very satisfactory state of affairs, and he wanted to derive his radiation law from the first principles of electrodynamics and thermodynamics. To that end he finally adopted the statistical methods from Boltzmann (on the way also writing down the famous relation between entropy and number of microstates, making up a macrostate, $S=k \ln W$, which is now engraved on Boltzmann's tombstone). The trick was to introduce an equidistant grid in (radiation) energy and counting the ways to distribute such portions of energy of radiation over the frequencies to get the given total (average) energy due to the temperature of the cavity walls, equilibrium being defined by the maximum entropy in accordance with the corresponding constraints. As it turned out, to get his formula he had to assume that the "energy quanta" are proportional to the frequency of the em. radiation with the proportionalit constant given by his quantum of action. Of course, the original plan was to let the energy quanta go to 0 at the very end of the calculation somehow to get into accordance with good old classical electrodynamics, for which the exchange of energy between the em. field and charged particles should be continuous and not in portions dependent on the frequency of the radiation. He tried so till the end of his life, but as is well known, what he discovered was the end of classical physics and the beginning of quantum theory, with Einstein the next player in this story.

Nowadays it's clear that the black-body radiation formula is indeed a direct hint at the quantization of the electromagnetic field, but in another way than Einstein thought. The discreteness of the absorption and induced emission is due to the quantization of the charges bound to the medium, as became clear with the elementary treatment of the photoeffect with modern quantum theory using first-order time-dependent perturbation theory. For this part the classical treatment of the em. field is sufficient, but as Einstein has found out in 1917, to get Planck's radiation law right from a kinetic approach one has to assume that there is also spontaneous emission, and that's only possible to derive from first principles when the quantization of the electromagnetic field is taken into account (Dirac 1927).

So indeed first it was just a calculational technique used by Planck to count the distribution of energies to the fundamental modes of the electromagnetic field to be able to use Boltzmann's statistical approach to thermal-equilibrium distributions (maximization of entropy), but then it turned out to be the key to the discovery of quantum theory, i.e., that one must quantize the energy of the radiation field in the way Planck did as a calculational tool, i.e., it's what's realized in Nature rather than a calculational tool, and to make the energy quanta arbitrarily small, i.e., letting Plancks constant formally going to 0 to get the "classical limit" doesn't work out. It's only a valid approximation of the Bose distribution for low frequencies:
$$\frac{1}{\exp(\hbar \omega/(k_B T))-1} \simeq \frac{k_B T}{\hbar \omega} \quad \text{for} \quad \hbar \omega \ll k_B T.$$
In fact it's of course the quantum limit rather than the classical one. The latter you get for the other limit, i.e., $\hbar \omega \gg k_B T$, for which you can neglect the 1 in the Bose distribution. The two limits are the Rayleigh-Jeans and the Wien radiation law, respectively:

https://en.wikipedia.org/wiki/Rayleigh–Jeans_law
https://en.wikipedia.org/wiki/Wien_approximation

 

[stevendaryl]

To that end he finally adopted the statistical methods from Boltzmann (on the way also writing down the famous relation between entropy and number of microstates, making up a macrostate, $S=k \ln W$, which is now engraved on Boltzmann's tombstone).

Planck first wrote that equation? Did Boltzmann have a different statistical definition of entropy?

 

[vanhees71]

That's a good question. I only read in some history-of-science books that it was Planck who wrote this equation for the first time explicitly. On the other hand he attributed it right away to Boltzmann. I'll try to find the original paper(s), where Boltzmann derives his H-theorem. There he must have given, at least implicitly, that statistical definition of entropy.

 

[vanhees71]

So, here it is:

L. Boltzmann, "Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen." Sitzungsberichte Akademie der Wissenschaften 66 (1872): 275-370.
English translation: Boltzmann, L. (2003). "Further Studies on the Thermal Equilibrium of Gas Molecules". The Kinetic Theory of Gases. History of Modern Physical Sciences. 1. pp. 262–349. Bibcode:2003HMPS...1..262B. ISBN 978-1-86094-347-8. doi:10.1142/9781848161337_0015.

from

https://en.wikipedia.org/wiki/H-theorem

The upshot is, as far as I can see from just flipping through the paper, that Boltzmann indeed gave the very general off-equilibrium expression for the entropy (or rather the negative of the modern definition):
$$E=\int_0^{\infty} \mathrm{d} x f(x) [\log[f(x)/\sqrt{x}]-1],$$
where $x$ is the energy and $f$ the one-particle distribution function.

 

[Robert M]

After-the-fact presentations of scientific discovery tend to gloss over a lot of details. Planck's understanding of the state of the art was apparently encyclopedic, and he apparently really covered the territory in getting to his famous result which was, in my estimation, one of the top ten (or something) of all time for combining theoretical insight with experimental fact.

http://bado-shanai.net/Map of Physics/mopPlancksderivBRL.htm