December 2017: Let us reconsider Planck

Later in the same section of the book, Planck says, “These statements define completely the way in which the radiation processes considered take place, as time goes on, and the properties of the stationary state.  …  It is true that we shall not thereby prove that this hypothesis represents the only possible or even the most adequate expression of the elementary dynamical law of the vibrations of the oscillators.  On the contrary I think it very probable that it may be greatly improved as regards form and contents.”

In the years since Planck’s book was published in 1912, the form of the quantum harmonic oscillator problem has been greatly formalized, especially by Dirac in the 1920s, but I don’t think the “contents” or basic ideas have been improved upon yet.  We still don’t have the answer to Einstein’s concern about the “mechanism of energy transfer” between matter and radiation. In quantum mechanics, we really don’t deal with trying to find mechanisms, which are physically sensible mental pictures.  We deal instead with finding mathematical formalisms.

Planck’s idea of a stationary state of an electromagnetic oscillator could be called the first quantum mechanical idea, but it's probably also the last quantum mechanical idea that attempts to give a physical picture of the energy transfer between matter and radiation.  As quoted in my journal entry above, he says, “While the oscillator is absorbing it must also be emitting, for otherwise a stationary state would be impossible.”

Compare this with Bohr’s model of the hydrogen atom, published only a few months after Planck’s book:  the electron undergoing centripetal acceleration as it orbits the nucleus in the ground state doesn’t radiate, so the ground state is stable, but when the electron has been kicked to an excited state and then falls back to the ground state, there is no consideration of the proton-electron attraction as being involved in this event.  Thus, there's no causality associated with the emission process: it is spontaneous, and this is the name Einstein gave it in his 1916-17 derivation of the Planck spectrum.

The success of quantum theory depends on the acceptance of the idea that electrons can be in stationary states without radiating away their energy.  But there is no physical mechanism in quantum theory to account for this lack of radiation in a stationary state.  At least Planck had an idea that could physically account for the existence of an electromagnetic stationary state.  (See last paragraph below.)

This model of continuous absorption and quantized emission is from Planck's "second theory" put forth in a paper in 1912 as well as in his book.  He was trying to make his theory less of a radical departure from classical physics, or in any case he wanted to make it more intuitively understandable.  One result of his 1912 work, however, was that he discovered the zero-point energy term in the electromagnetic spectral density formula, a result that later was found unavoidable  as a result of the uncertainty principle.  So he found an even more radical departure from  classical physics rather than his hoped-for less radical approach. 

But, on the other hand, zero-point energy can also be understood in the context of classical electromagnetism, as pointed out in this 2015 article by Timothy H. Boyer, among other places.  And zero-point fluctuations of the e.m. field itself (around its zero average value in the vacuum) are by many physicists nowadays thought to be the cause for spontaneous emission.  But that's not all there is to the story.

December 1900: Planck unveils h and k

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.

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 E_t (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.]

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) / dE₀

The 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)

Planck's other constant, k, named for Boltzmann

Time to get back to the Planck.  To re-start the discussion, here's a relevant statement by another famous German physicist:

Boltzmann’s principle interprets entropy in terms of the probability of states and expresses it in the terse formula

S = k log W.

So it stands carved out on Boltzmann’s memorial in the Central Cemetery in Vienna, floating in the clouds over his majestic bust.

It is immaterial that Boltzmann himself never wrote down the equation in this form.  This was first done by Planck … .  The constant, k, was also introduced by Planck and not by Boltzmann.  Boltzmann only referred to the proportionality between S and the logarithm of the probability of a state.  The designation ‘Boltzmann’s principle’ was advocated by Einstein for …

W = exp(S/k)

in which S was considered to be known empirically, the quantity W being the unknown for which an expression was sought.

       --Arnold Sommerfeld, Thermodaynamics and Statistical Mechanics, English translation, Academic Press © 1956, fifth printing 1967, page 213.   (But, you'd need to know k to calculate W... .  Never mind!)

 And below, a couple of relevant drawings from the time of Planck's discovery.

Taken from the book Statistical Physics by Bernard Lavenda, here's a figure showing the actual experimental results Planck used when was trying to fit his theory to the data, and eventually of course succeeded:







And here's the device, not exactly box-like but more cylinder-like (see bottom part of drawing), used to experimentally produce the radiation whose energy spectrum is plotted above.  You know how you have to swing a hand-held spectrometer back and forth to view the spectrum of a visible light source, or at least you've seen how a prism disperses visible light.  This black-body cylindrical cavity is on wheels so the stationary spectrum analyzer equipment can get measurements from the full spectrum, which in this case is in the infrared region.  From a September 2016 American Journal of Physics article about Planck.