P.S. According to an article in the March 2018 AJP by Timothy Boyer, Planck first introduced the constant h in 1899 when he numerically fit Wien's formula to experimental data. As the data on the black-body radiation curve improved during the next year, Wien's formula was found to be inadequate to fit the spectrum in the lower frequency region. Planck's constant was not to blame, and in fact found its fullest usage in the correct black-body spectral formula derived by Planck in late 1900.
Let us consider mimicking Planck by starting off this blog post with the same three words he uses (in English translation) to start the paragraph that describes “the question” he wants to answer in his December 1900 talk to the German Physical Society.
Let us consider mimicking Planck by starting off this blog post with the same three words he uses (in English translation) to start the paragraph that describes “the question” he wants to answer in his December 1900 talk to the German Physical Society.
His talk
is titled “On the theory of the energy distribution law of the normal spectrum.” Here’s the paragraph
stating the question(s) to be answered:
Let us consider a large number of monochromatically vibrating resonators – N of frequency υ (per second), N’ of frequency υ’, N’’ of frequency υ’’, …, with all N large numbers – which are at large distances apart and are enclosed in a diathermic medium with light velocity c and bounded by reflecting walls. Let the system contain a certain amount of energy, the total energy Et (erg), which is present partly in the medium as travelling radiation and partly in the resonators as vibrational energy. The question is how in a stationary state this energy is distributed over the vibrations of the resonators and over the various frequencies of radiation present in the medium, and what will be the temperature of the total system.
So “the
question” is really two questions, but they can’t be separated: In equilibrium, how is the total energy distributed,
and what is the expression that relates this distribution to the temperature of
the system? In answering the first
question, Planck introduced the quantum of action h and found its numerical value.
In answering the second question, he found the numerical value for what
has come to be called Boltzmann’s constant, k.
Planck presumed in a rather arbitrary way that
the resonators could absorb, hold, and release energy in discrete amounts. It’s arbitrary because he says that if “the
number P of energy elements which
must be divided over the N
resonators” is not an integer, “we take for P
an integer in the neighborhood.” He
is talking here about the first set of monochromatic resonators, of which there
are N, and is considering “the
distribution of the energy E over the
N resonators of frequency υ”.
At this
point, Planck wants to derive an entropy expression he already knows will lead
to the correct formula for the energy spectrum.
He found the entropy expression in his October 1900 paper that uses a
two-term formula—one term in the formula fits the high-frequency
part of the spectrum, the other fits the low end, and added together they fit
the whole spectrum. Indeed, this turned out to be the first hint of the particle-wave duality of quantum mechanics, although it wasn't and still isn't given much significance in that respect, usually just being referred to by Planck as well as most other physicists as an "interpolation" formula based on guesswork.
To get the right entropy expression, and to not make it seem like guesswork, Planck resorted to energy quantization (but not frequency quantization—keep that in mind), and he starts by saying why he doesn't use energy as a continuous quantity distributed over the N resonators: “If E is considered to be a continuously divisible quantity, this distribution is possible in infinitely many ways.” [compare "momentum is shared out in infinite number of ways"—Oppenheimer 1930, see Weinberg’s Daedalus article 1977.]
To get the right entropy expression, and to not make it seem like guesswork, Planck resorted to energy quantization (but not frequency quantization—keep that in mind), and he starts by saying why he doesn't use energy as a continuous quantity distributed over the N resonators: “If E is considered to be a continuously divisible quantity, this distribution is possible in infinitely many ways.” [compare "momentum is shared out in infinite number of ways"—Oppenheimer 1930, see Weinberg’s Daedalus article 1977.]
Let’s
pause for a moment before getting to what Planck calls “the most essential
point of the whole calculation.” Nowhere
in his discussion does Planck mention the tedious slide-rule calculations he
had to do to find the numbers to make his expression for average energy fit the
black-body energy spectrum curves. But it seems to me the curve fitting to find numerical values for
both h and k would have been what he later
described as “a few weeks of the most strenuous work” of his life.
All
right, h first, and for the sake of
completeness and continuity, I’ll quote again two of Planck’s phrases I quoted
above:
We consider, however—this is the most essential point of the whole calculation—E to be composed of a very definite number of equal parts and use thereto the constant of nature h = 6.55 X 10-27 erg-sec. This constant multiplied by the common frequency υ of the resonators gives us the energy element ε in ergs, and dividing E by ε we get the number P of the energy elements which must be divided over the N resonators. If the ratio is not an integer, we take for P an integer in the neighborhood.
Planck
now uses a combinatorial calculation for the number of ways the P energy elements can be distributed
over the N resonators, and says “Each
of these ways of distribution we call a ‘complexion,’ using an expression
introduced by Mr. Boltzmann for a similar quantity.” To be filled in here later… “Similar quantity” maybe, but
not that similar, everybody says. See my previous post Planck oscillator average energy Part I. for the combinatorial caclulation. Planck continues:
We perform the same calculations for the resonators of the other groups, by determining for each group of resonators the number of possible complexions for the energy given to the group. The multiplication of all numbers obtained in this way gives us then the total number R of all possible complexions for the arbitrary assigned energy distribution over all resonators.
In the same way, any other arbitrarily chosen energy distribution E, E’, E’’… will correspond to a definite number R of all possible complexions evaluated in the above manner. Among all energy distributions which are possible for a constant E0 = E + E’ + E’’ + … there is one well-defined one for which the number of possible complexions R0 is larger than for any other distribution. We look for this distribution, if necessary by trial, since this will just be the distribution taken up by the resonators in the stationary radiation field, if they together possess the energy E0
...
After the stationary energy distribution is thus determined using a constant h, we can find the corresponding temperature ... using a second constant of nature k = 1.346 X 10-6 erg per kelvin through the equation
1/T = d(k lnR0) / dE0The product k lnR0 is the entropy of the system of resonators; it is the sum of the entropy of all separate resonators.
(continued in More Walking with Max Karl Ernst Ludwig Planck)
