Planck's 1st Derivation of Planck's Law?

[tade]

My information comes from: http://www.cems.uvm.edu/~tlakoba/AppliedUGMath/notes/lecture_13.pdf

Quote A:

In 1899, a German physicist Max Planck rederived Wien’s formula (i.e., (13.4) with γ = 5) from phenomenological thermodynamical considerations.

Quote B:

His first derivation of this formula, done in October 1900, was based solely on phenomenological Thermodynamics and required no assumptions about microscopic properties of radiation.

Quote C:

Next, Planck attempted to rederive this “good” formula using microscopic considerations of Statistical Mechanics developed by Boltzmann.

Quote B and Quote C concern Planck's Law, not Wien's Law.

I know how Planck derived it as mentioned in Quote C.


However, what are the "phenomenological Thermodynamics" derivations as mentioned in Quote A and Quote B?

 

[tade]

Because from what is written here, Planck's Law is not so significant to QM if it can be derived by other means?

 

[mpresic]

Best reference I know of that describes the "phenomological thermodynamics" in detail iby Oliver Darrigol, From C-numbers to q- numbers. Another shorter book is Malcolm Longair, Concepts in theoretical physics.

 

[bhobba]

Because from what is written here, Planck's Law is not so significant to QM if it can be derived by other means?

Forget this early stuff - its of historical interest - its not really the foundation of QM.

The following from the modern viewpoint is much much better:
http://www.scottaaronson.com/democritus/lec9.html

Thanks
Bill

 

[vanhees71]

Yes, the black-body spectrum is most clearly derived directly from QED.

 

[dextercioby]

My information comes from: http://www.cems.uvm.edu/~tlakoba/AppliedUGMath/notes/lecture_13.pdf

Quote A:

Quote B:

Quote C:

Quote B and Quote C concern Planck's Law, not Wien's Law.

I know how Planck derived it as mentioned in Quote C.

However, what are the "phenomenological Thermodynamics" derivations as mentioned in Quote A and Quote B?

You can find the two original papers of Max Planck to the German Society of Physics (DPG) from October and Dec. 1900 in this old book: https://www.amazon.com/dp/0470691050/?tag=pfamazon01-20 - both in English translation and German original. You can also check Gallica (the digital Library of France) for the mentioned volumes of Annalen der Physik from 1900 and 1901.

For some unkown reason the Amazon direct link is not working.

 

[mpresic]

I am puzzled by tade's first two posts. The second post seems to suggest that if Planck's law can be understood by other means, why are the thermodynamic considerations that led Planck to quantization important. Darrigol's book, (I have only read parts) shows that Planck deeply analyzed all disciplines; statistical mechanics, electrodynamics, mechanics, and the early quantum mechanics and understood all of these ideas as a whole.

Practically, I suppose if you can use the most powerful concepts available in the 21 century, you need not appreciate the limitations Planck reckoned with in the late 19th century except from a historical perspective, But I do think it is exciting to understand just how ingenious Planck was. Moreover, learning the "old" physics can be useful, when encountering new physical ideas. Why else do we learn the old classical mechanics when it is just a special case of quantum mechanics with h-bar = 0, or a special case of relativity with c = infinity.

 

[vanhees71]

Indeed, the most simple way in the 21st century to derive Planck's Law is to use QED.

 

[tade]

I am puzzled by tade's first two posts. The second post seems to suggest that if Planck's law can be understood by other means, why are the thermodynamic considerations that led Planck to quantization important. Darrigol's book, (I have only read parts) shows that Planck deeply analyzed all disciplines; statistical mechanics, electrodynamics, mechanics, and the early quantum mechanics and understood all of these ideas as a whole.

Practically, I suppose if you can use the most powerful concepts available in the 21 century, you need not appreciate the limitations Planck reckoned with in the late 19th century except from a historical perspective, But I do think it is exciting to understand just how ingenious Planck was. Moreover, learning the "old" physics can be useful, when encountering new physical ideas. Why else do we learn the old classical mechanics when it is just a special case of quantum mechanics with h-bar = 0, or a special case of relativity with c = infinity.

anyway do you know what the "phenomenological Thermodynamics" was?

 

[mpresic]

This is my interpretation, and I may be wrong, and I will definitely be incomplete. The whole argument from Darrigol is scattered over perhaps 30 or 40 pages, but here goes. First, the phenomenological is an adjective. It is not like you can refer to irreversible thermodynamics and investigate the texts on it, (here the irreversible is more than an adjective, and guides us to a discipline). It seems that the thermodynamics in Planck's time predated statistical considerations. At this time the idea of atoms was by no means universally accepted. Planck's thermodynamics involved what he could measure macroscopically. Planck considered macroscopic measurable quantities, temperature, volume, energy, entropy (but not the statistical definition of entropy), pressure, heat, and possibly other thermodynamic properties. These are phenomenological, you measure all them directly and you need not model them with things you cannot see or directly sense (with tools available in the late 19th century)
Planck was a much better thermodynamicist than I am, or any of his contemporaries. He even admitted to being relieved that he thought his contemporaries were on the wrong track. He said, this allowed him to investigate the problem of black body radiation at his leisure, without worrying about being "scooped". I cannot see all of his argument but it involves stability considerations of states of d2S/dE2 (I think, I do not know). A briefer account instead of Darrigol, is given in a chapter of Malcolm Longair's book, " Theoretical Concepts in Physics" (case study 5).. I recommend this shorter threatment but if want a true derivation of Longair's equation 9.21, you need the original papers of Planck or Darrigol.
Longair and Darrigol also share the ideas in the correspondence where Einstein is unhappy with Planck's use of (later to be called Bose-Einstein) statistics in place of Maxwell-Boltzman statistics. Planck was also puzzled by this. I may be wrong in some of this, but I hope it has whetted your appetite. I think a reading of the short case study by Longair, and (perhaps) later the exposition by Darrigol will be rewarding. Few people are bored by the progress of revolutionary scientific ideas

 

[tade]

Yes, the black-body spectrum is most clearly derived directly from QED.

true, but I am curious about the historical development of the formula.

 

[DrClaude]

anyway do you know what the "phenomenological Thermodynamics" was?

phenomenological:

Of or relating to phenomenology; dealing with the description and classification of phenomena, rather than with their explanation or cause.

Planck wanted to find a way to reproduce the blackbody spectrum. Even after having found the law that bears his name, he thought of the quantization of the light as a mathematical artifact, not something corresponding to underlying physics. See Max Planck: the reluctant revolutionary.

 

[tade]

phenomenological:

Planck wanted to find a way to reproduce the blackbody spectrum. Even after having found the law that bears his name, he thought of the quantization of the light as a mathematical artifact, not something corresponding to underlying physics. See Max Planck: the reluctant revolutionary.

sorry, I wish to know what sort of "phenomenological thermodynamics" Planck used as mentioned in the article.

 

[tade]

This is my interpretation, and I may be wrong, and I will definitely be incomplete. The whole argument from Darrigol is scattered over perhaps 30 or 40 pages, but here goes. ... Few people are bored by the progress of revolutionary scientific ideas

Thanks, I'll check it out. Few or many?

 

[tade]

I did some research on this. It turns out that Planck's phenomenological thermodynamics should have led him to the Rayleigh-Jeans law, but he made a mistake and ended up deriving his famous Planck's law by mistake!
He already had access to black body spectra experimental data by this time though.

 

[PeterDonis]

It turns out that Planck's phenomenological thermodynamics should have led him to the Rayleigh-Jeans law, but he made a mistake and ended up deriving his famous Planck's law by mistake!

Reference, please?

 

[vanhees71]

If you read the original paper by Planck, there's no "mistake", but an ingenious discovery of Bose statistics (at least that's how I read the paper with the knowledge we have today about quantum statistics) without stating explicitly that he gets the Planck Law by applying statistics in a different way than for classical Particles a la Boltzmann. Planck counts indeed no particles but energy quanta of the electromagnetic field.

 

[tade]

Reference, please?

The book Black-Body Theory and the Quantum Discontinuity, 1894-1912 by Thomas S. Kuhn covers the historical development of Planck's law. The almost 400-page-long book is almost entirely about nothing but the development of Planck's law.

In the afterword of the book, Revisiting Planck, at the end of section 3. The Counter-evidence: Planck's Derivation, Kuhn wrote:

...there is one another putative difficulty with Planck's derivation, his apparent failure to maximize his count of the number of ways of realizing a particular distribution. That difficulty, too, proves to be only apparent. ...Planck's result is incompatible with the continuity of resonator energy. He ought to have arrived at the Rayleigh-Jeans law. There is a mistake in his derivation, the one that Einstein pointed out in 1910.
Boltzmann's method of counting ways of realizing a distribution is valid only if the distribution function is effectively constant within each cell. In Planck's case this requires hν << kT, an approximation that does not generally hold. This difficulty remained invisible to all but Einstein, however, and even he did not recognize it until about 1910.

In the link in my OP, it states that the derivation of Planck's law that most students are taught was derived by Lorentz in 1910. So it appears that Einstein and Lorentz fixed the issue with Boltzmann's 'way-counting' method in 1910.