# How did Planck come up with 'h'?

In my new found passion to understand the "inner meaning" of this thing called quantum theory, I've been trying to find Planck's derivation of -- or explanation for choosing -- the "constant of nature" that has become so popular.

I expected to be able to find the empirical basis for it very easily on google, but all I could find were explanations like, "a photon is a unit of radiation whose energy content can be found by E=hv." Well, this doesn't help me too much.

The closest thing I have been able to find to an "explanation" is from this article...

http://physicsworld.com/cws/article/print/373

To find W, Planck had to be able to count the number of ways a given energy can be distributed among a set of oscillators. It was in order to find this counting procedure that Planck, inspired by Boltzmann, introduced what he called "energy elements", namely the assumption that the total energy of the black-body oscillators, E, is divided into finite portions of energy, epsilon, via a process known as "quantization". In his seminal paper published in late 1900 and presented to the German Physical Society on 14 December - 100 years ago this month - Planck regarded the energy "as made up of a completely determinate number of finite equal parts, and for this purpose I use the constant of nature h = 6.55 x 10-27 (erg sec)". Moreover, he continued, "this constant, once multiplied by the common frequency of the resonators, gives the energy element epsilon in ergs, and by division of E by epsilon we get the number P of energy elements to be distributed over the N resonators".
What's the reason for using this precise value?

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mgb_phys
Homework Helper
You can measure it experimentally.
You use a vacuum tube and excite electrons from the cathode with a light source of known wavelength and measure the voltage accross the tube.
You can then get h/e from the slope - without having to know the work function of the cathode. If you know e you can get h.

epenguin
Homework Helper
Gold Member
It fitted an equation to experimental data, in his case of dependence of intensity of radiation on frequency for black body, that was the start of it all.

You can measure it experimentally.
You use a vacuum tube and excite electrons from the cathode with a light source of known wavelength and measure the voltage accross the tube.
You can then get h/e from the slope - without having to know the work function of the cathode. If you know e you can get h.
Thanks. Can you point me in a direction where someone has actually done this? I'm looking for experimental details...

mgb_phys
Homework Helper
Milikan did it originally in 1916 but you still have to pay for access to the paper !
http://focus.aps.org/story/v3/st23
There are lots of ugrad lab classes on the web describing it.

epenguin
Homework Helper
Gold Member
These are the first things in any textbook that goes beyond classical or elementary physics - as the fact of my knowing them proves. mgb_phys
Homework Helper
It's also quite an easy experiment to do nowadays with LEDs and vacuum tubes.
Unlike Milikan's other experiments - it actualy works!

Milikan did it originally in 1916 but you still have to pay for access to the paper !
http://focus.aps.org/story/v3/st23
There are lots of ugrad lab classes on the web describing it.
Thanks for the heads up on Milikan. However, the question still remains: What was Planck's justification for using it in 1900? I mean, how did the inventor of h derive it?

mgb_phys
Homework Helper
Sorry - Planck didn't measure it, and apparently didn't really believe in it!
He just realised mathemtically that if you made energy changes only available in discrete lumps it solved some problems in thermodynamics. Einstein used the idea to explain the photoelectric effect - so it's really as much Einstein's constant as Planck's.

Sometimes it takes centuries for the experiment to measure a constant proposed by a theory - it took a long time for the speed of light and the gravitational constant G to be measured accurately.

jtbell
Mentor
Planck came up with the idea of quantized radiation and the constant we know as h, in deriving a formula to fit the observed spectrum from a hot "black body." The details are covered in many introductory modern physics textbooks, e.g. Beiser, "Concepts of Modern Physics", section 2.2; Krane, "Modern Physics", section 3.3; Ohanian, "Modern Physics", section 3.1.

You might find these details on line by Googling for "black body radiation" or "blackbody radiation" or "Planck radiation formula".

Sorry - Planck didn't measure it, and apparently didn't really believe in it!
So, where in the heck did it come from? Did manna from heaven fall into his lap? He just realised mathemtically that if you made energy changes only available in discrete lumps it solved some problems in thermodynamics. Einstein used the idea to explain the photoelectric effect - so it's really as much Einstein's constant as Planck's.

Sometimes it takes centuries for the experiment to measure a constant proposed by a theory - it took a long time for the speed of light and the gravitational constant G to be measured accurately.

Why did he pick these precise discrete lumps that now seem to fit so perfectly with empirical observation? Are you saying that the mathematics just "manifested it"?

Planck came up with the idea of quantized radiation and the constant we know as h, in deriving a formula to fit the observed spectrum from a hot "black body."
I don't think that a constant is the reason why the Rayleigh-Jeans Law was a failure. Wein's displacement law is the thing that corrected it, and it had nothing to do with a simple constant.

mgb_phys
Homework Helper
Why did he pick these precise discrete lumps that now seem to fit so perfectly with empirical observation? Are you saying that the mathematics just "manifested it"?
No he came up with the idea that if you only allow energy to transfer in small lumps then some problems in thermodynamics are solved. He didn't need to know what value h had - just that it was very small.
It's the same as Newton's law of gravity - he knew that if the gravitational force was proportional the product of the masses and the distance squared then the orbits of planets worked out. It wasn't until a 100years later that Cavendish worked out what the value of 'G' was.

No he came up with the idea that if you only allow energy to transfer in small lumps then some problems in thermodynamics are solved. He didn't need to know what value h had - just that it was very small.
It's the same as Newton's law of gravity - he knew that if the gravitational force was proportional the product of the masses and the distance squared then the orbits of planets worked out. It wasn't until a 100years later that Cavendish worked out what the value of 'G' was.
Planck, in 1900 used: 6.55 x 10-34 J s
(source in first post)

The NIST standard, in 2008 says: 6.62606896 x 10-34 J s
(source:http://physics.nist.gov/cgi-bin/cuu/Value?h)

I'm assuming this is just a lucky guess?

mgb_phys
Homework Helper
You can derive it quite easily from Wien's law, which tells you frequency of the peak on the curve predicted by Planck's equation - of course this relies on you having a good value for Wien's constant!

Planck's original paper (translated!) http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html [Broken]

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You can derive it quite easily from Wien's law, which tells you frequency of the peak on the curve predicted by Planck's equation - of course this relies on you having a good value for Wien's constant!
Yes... this is all very interesting, indeed. I'm doing my best to trace this thing back in history. My guess that Boltzmann is going to end up being known as the "real" father of QM!

Post # 3 is as close as you'll get:

Wilhelm Wien....showed that under adiabatic expansion or contraction, the energy of light changes in the exact same way as the frequency. This means that the peak frequency should change with temperature as the energy goes. Wien did not interpret his constant b as a new fundamental constant of nature. This was done by Planck.
http://en.wikipedia.org/wiki/Wien_displacement_law

I'm sitting here reading Kuhn's "Black Body Theory and the Quantum Discontinuity 1894-1912". This is an extremely in depth look into the origins of QM.

The first mention of the "b" constant that eventually becomes Planck's "h" is mentioned here:
Planck then, with no preparatory argument whatsoever, simply "defines" [Kuhn's quotes] the entropy of a resonator of frequency v and energy U by the equation

S = (-U/av)log(U/ebv),

where e is the base of natural logarithms and "a and b are two universal positive constants the numerical values of which in the absolute c.g.s. system will be developed from thermodynamics in the next section".
The internal quote from Planck is footnoted as: "Funfte Mittheilung" (Planck, 1899), p. 465; I, 585.

Hello, thermodynamics! The plot thickens...

Post # 3 is as close as you'll get
Oh yeah? I'm reading a source of a source of a Wikipedia article!

In p. 104 of Kuhn's book, already cited:

To compute the entropy of an arbitrary distribution Planck must introduce combinatorials, and for this purpose he follows Boltzmann in subdividing the energy continuum into elements of finite size. It is at this point that he introduces the further novelty that was soon to prove the most consequential of all. For his purpose, unlike Boltzmann's, the size of the energy elements epsilon, epsilon-prime, epsilon-2prime, etc., must be fixed and proportional to frequency. Consideration of that vital step is the subject of the next chapter, but the passage in which Planck introduces it must be noted here, for it illustrates an aspect of his lecture that helped mislead readers about his intent.

"The distribution of energy over each type of resonator must now be considered, first the distribution of the energy E over the N resonators with frequency v. If E is regarded as infinitely divisible, an infinite number of different distributions is possible. We, however, consider -- and this is the essential point -- E to be coposed of a determinate number of equal parts and employ in their determination the natural constant h = 6.55 x 10-27 (erg x sec). This constant multiplied by the frequency, v, of the resonator yields the energy element epsilon in ergs, and, dividing E by epsilon, we obtain the number, P, of energy elements to be distributed over the N resonators." [Citation: "Zur Theorie des Gesetzes" (Planck, 1900e), pp. 239f.; I, 700f.]

Because Planck, here and for some time after, considers only the single set of resonators with frequency v and because he later omits the computation of a maximum, which would have demanded explicit recourse to resonators at other frequencies, the difference between his argument and that of Lorentz is obscured.
From what I understand of Kuhn's argument, Planck's use of Boltzmann's combinatorial method as applied to Wein's empirically correct law is the thing that got this whole "integer only energy quanta" thing off the ground...

Okay, this is what I've been looking for. h (Planck's) is nothing other that k (Boltzmann's). k is simply the gas constant (8.314472 J/K mol) divided by the Avogadro constant (6.02214179 entities per mol). There is apparently a controversy about who deserves credit for "discovering" this thing. According to http://en.wikipedia.org/wiki/Boltzmann_constant#Historical_note, Planck wrote in his Nobel Prize lecture:

"This constant is often referred to as Boltzmann's constant, although, to my knowledge, Boltzmann himself never introduced it — a peculiar state of affairs, which can be explained by the fact that Boltzmann, as appears from his occasional utterances, never gave thought to the possibility of carrying out an exact measurement of the constant."
It seems as though Planck was positively obsessed with proving the laws of thermodynamics on the "micro" rather than on the "statistical" level; the latter is all that concerned Boltzmann. According to Kuhn, Planck did quite a lot of "mathematical fudging," and his proofs did not impress too many of his peer reviewers in the time around December 1900.

But the "atomists" were ready and waiting to jump on anything that would bring Maxwell's continuous energy relations into a discrete form. So, when Einstein came along and popularized h in his own work, the physics world was primed for one heck of a party over the next few decades.

jtbell
Mentor
h (Planck's) is nothing other that k (Boltzmann's).
No. Planck's constant and Boltzmann's constant are two different things. They even have different units.

marcus
Gold Member
Dearly Missed
http://www.knowledgerush.com/kr/encyclopedia/Planck_time/
==quote==

Max Planck's creation of the natural units

Max Planck first listed his set of units (and gave values for them remarkably close to those used today) in May of 1899 in a paper presented to the Prussian Academy of Sciences. Max Planck: 'Über irreversible Strahlungsvorgänge'. Sitzungsberichte der Preußischen Akademie der Wissenschaften, vol. 5, p. 479 (1899). At the time he presented the units, quantum mechanics had not been invented. He himself had not yet discovered the theory of blackbody radiation (first published December 1900) in which the constant h made its first appearance and for which Planck was later awarded the Nobel prize. The relevant parts of Planck's 1899 paper leave some confusion as to how he managed to come up with the units of time, length, mass, temperature etc. which today we define using h-bar and motivate by references to quantum physics before things like h-bar and quantum physics were known. Here's a quote from the 1899 paper that gives an idea of how Planck thought about the set of units.

...ihre Bedeutung für alle Zeiten und für alle, auch ausserirdische und ausser menschliche Culturen nothwendig behalten und welche daher als 'natürliche Maasseinheiten' bezeichnet werden können...

...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as 'natural units'...
==endquote==

I've seen the German text of the 1899 article and I have an English translation somewhere. He gives a value for h that is very close to the right one, but it is hard to see how he came by it. He found it before the black body radiation law.

The title of the 1899 paper is "On irreversible radiation processes". If I remember right, he delivered it sometime late spring or early summer to the Prussian Academy of Sciences in Berlin.

Maybe googling the title would come up with an online copy.

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This 46 page article looks promising - I haven't read it yet:

Planck, the Quantum, and the Historians
Clayton A. Gearhart Phys. perspect. 4 (2002) 170–215
http://employees.csbsju.edu/cgearhart/pubs/PQH.pdf

Translations to accompany it:

http://employees.csbsju.edu/cgearhart/Planck/planck_1906.pdf

This contains detailed discussion of how Planck developed his blackbody formula, in particular concerning how he approached entropy.

I found this as one of a bunch of interesting looking papers at:

http://enjoy.phy.ntnu.edu.tw/course/view.php?id=120#section2

Also, perhaps (if one could get the text):

The Historians' Disagreements over the Meaning of Planck's Quantum
Darrigol, O Centaurus, Volume 43, Numbers 3-4, 1 December 2001 , pp. 219-239(21)
http://www.ingentaconnect.com/content/mksg/cnt/2001/00000043/F0020003/art00044

and

"Max Planck and black-body radiation," essay in Dieter Hoffmann, ed., Max Planck:
Annalen Papers (Wiley-VCH (2008), 395¬–414).
http://employees.csbsju.edu/cgearhart/pubs/sel_pubs.htm

- Robin