Planck oscillator average energy Part I

 

This is the first page of my notes from Steven Weinberg’s introductory quantum mechanics course (Fall 1998, UT-Austin).  I will later post the other two pages from this day, and have previously (September 2016) posted my class notes from several other days. Robert Oppenheimer's colleague Robert Serber is one of the characters—he's quite a character, really—in Jon Else's Oscar-nominated documentary The Day After Trinity.  Weinberg's voice sounded to me almost exactly like Serber's, thus my note in the top left corner.

I'm posting the above page of notes as a prelude to showing Planck's method of calculating the average energy of a particular frequency of black-body radiation, which is also the average energy of a particular oscillator or resonator in his theory, as discussed at the end of my previous post.

You can see from my class notes that Weinberg didn't show Planck's calculations, and didn’t seem to think very highly of Planck’s line of thought, calling it a thermodynamic Mickey Mouse derivation. Others, including Einstein way back when, have noted Planck did not make correct use of statistical mechanics, but I won’t go into that here, since I don’t see why it was (is) wrong. Different people give different reasons for why it was wrong, at least according to the various books I have in my collection (Born, Sommerfeld, Milonni, Lavenda, and Longair, among others).

Going back to my Weinberg notes, it is interesting to compare his negative comments on Planck's derivation there with his laudatory comments from 21 years earlier in the journal Daedalus.  Weinberg's article in the Fall 1977 issue is titled "The Search for Unity:  Notes for a History of Quantum Field Theory."  Here are the paragraphs related to Planck:

It will be worthwhile for us to concentrate on Planck’s proposal for a moment, not only because it led to modern quantum mechanics, but also because an understanding of this idea is needed in order to understand what quantum field theory is about.

 

 

Planck supposed that the electrons in a heated body are capable of oscillating back and forth at all possible frequencies, like a violin with a huge number of strings of all possible lengths.  Emission or absorption of radiation at a given frequency occurs when the electron oscillations at that frequency give up energy or receive energy from the electromagnetic field.  The amount of energy being radiated per second by an opaque body at any frequency therefore depends on the average amount of energy in electron oscillations at that particular frequency.

It was in calculating this average energy that Planck made his revolutionary suggestion. He proposed that the energy of any mode of oscillation is quantized—that is, that it is not possible to set oscillation going with any desired energy, as in classical mechanics, but only with certain distinct allowed values of the energy.  More specifically, Planck assumed that the difference between any two successive allowed values for the energy is always the same for a given mode of oscillation, and is equal to the frequency of the mode times a new constant of nature which has come to be called Planck’s constant.

It follows that the allowed states of the modes of oscillation of very high frequency are widely separated in energy, so that it takes a great deal of energy to excite such a mode at all.  But the rules of statistical mechanics tell us that the probability of finding a great deal of energy in any one mode of oscillation falls off rapidly with increasing energy; hence the average energy in oscillations of very high frequency must fall off rapidly with the frequency of the radiation, thus avoiding the catastrophe of an infinite total rate of radiation.

You're not alone if you're wondering what "the energy in any mode of oscillation" means. One of the section titles in Peter Milonni's book The Quantum Vacuum says it simply:  "A Field Mode is a Harmonic Oscillator."  Thus a mode represents a single frequency of oscillation, but "mode" really is an abbreviation for "normal mode".  A single Planck oscillator, however, can have from zero to any integer number of quanta of energy stored in it (if stored is the right word).  And so can the electromagnetic field--there are so-and-so many "photons in the mode," is how physicists say it.

In the case of a physical cavity, which is what we're considering, normal modes are called "cavity modes" and are three dimensional, meaning they require three integers for a complete specification. See my post from September 2015 for a sample calculation.  For an electromagnetic wave in general (traveling in “free space”), a mode is specified by the direction of oscillation of the electric field (the polarization direction) and the frequency and direction of travel (the wave vector). Radiation is said to be unpolarized or isotropic when its component waves have random directions of polarization, and black-body or thermal radiation fits this description.

Now back to our program. Recall from last time that Planck’s expression for the number of complexions (also called microstates or probability, or as Sommerfeld in his Thermo. & Stat. Mech. book says, permutability) is given by

W = (N+P - 1)!/P!(N-1)! 

Ignoring the 1 as compared to the large numbers N and P, and using a truncated form of Stirling’s  approximation, where the factorial of a really large number is approximated by raising it to the power of itself, this becomes

W ≈ (N+P)^N+P/P^P N^N

The entropy, Planck’s favorite thing to calculate—but now he’s doing it using statistical mechanics rather than Clausius’ classical thermodynamics—is the natural logarithm of W and also includes a multiplicative constant Planck discovered by doing this calculation, although it is called Boltzmann’s constant.  Letting the constant be C, the entropy is

S = C log W = C log [(N+P)^N+P/P^P N^N] 

= C {log [(N+P)^N+P] — log P^P — log N^N}

   = C {(N+P)log (N+P) — P log P —N log N}.

The two variables in the equation are N, the number of oscillators (see previous post, where oscillators are separated into single-frequency groups N, N’, N’’ and so on), and P =E/ε, the combined energy of the N oscillators divided by a unit of energy ε, with ε presumed to be indivisible.

Question: Why is ε presumed to be an indivisible unit of energy?  Answer: To make the resulting equation for the energy spectrum fit the observed experimental energy spectrum. In fact, in order for the energy spectrum formula to fit the experimental results, ε has to be proportional to the frequency υ, and the proportionality constant has to be a particular value—this will be Planck’s constant. At the moment we just want to see how the average energy of an oscillator is related to the combined energy of N oscillators.  Then we can use that to find an expression for the entropy of an oscillator (the above expression is for the entropy of N oscillators).  Then, finally, we will find an expression that relates average energy to temperature—the expression we’re looking for.

In terms of the total energy of the N oscillators, the average energy of a single oscillator is just the usual arithmetical average, U_υ = E/N.  If we rearrange, we have E = NU_υ , and can write P in terms of the average energy: P = NU_υ / ε.  So, voilá, the total entropy of the N oscillators in terms of the average energy of a single oscillator: 

S = C{ (N + NU_υ / ε)log (N + NU_υ / ε) — (NU_υ / ε) log( NU_υ / ε) —N log N }.

There’s a common factor of N multiplying each log term, and a common factor of N inside each log.  Pulling out the common factor multiplying the log terms gives the entropy per oscillator, 

S  =  CN { (1 + U_υ / ε)log (N + NU_υ / ε) — (U_υ / ε) log( NU_υ / ε) — log N }, 

S/N  = C { (1 + U_υ / ε)log (N + NU_υ / ε) — (U_υ / ε) log( NU_υ / ε) — log N }.

This is still expressed in terms of N and NU_υ , so it’s not really a usable expression for average entropy.  The miracle is that the N’s inside the logarithms also cancel out.  Let’s use S_υ instead of S/N for average entropy, to match it with average energy U_υ :

S_υ  = C { (1 + U_υ / ε)log N(1 + U_υ / ε) — (U_υ / ε) logN(U_υ / ε) — log N }

     = C { (1 + U_υ / ε)logN + (1 + U_υ / ε)log(1 + U_υ / ε) — (U_υ / ε) logN + (U_υ / ε)log(U_υ / ε) — log N}

      = C { log N  +  (U_υ / ε) log N  +  (1 + U_υ / ε) log (1 + U_υ / ε)) 
                                                                       — (U_υ / ε) log N — (U_υ / ε) log (U_υ / ε) — log N }.

The terms in red cancel, giving

S_υ = C {(1 + U_υ / ε) log(1 + U_υ / ε)) — (U_υ / ε) log(U_υ / ε) }.

Now, by making use of both sides of the thermodynamic identity



                                                                ∂S/∂U = 1/T

the average energy can be calculated the way Planck did it.  The intrepid reader is encouraged to try it for himself or herself.   Next time, I’ll go through it and also show how Einstein used a different method to find the average energy, by relying on the Boltzmann method for finding a statistical average, which is also the method most often shown in introductory textbooks.  It involves use of the geometric series.

Walking further with Planck

Max Planck, in his 14 December 1900 presentation to the German Physical Society, reported he was able to “derive deductively an expression for the entropy of a monochromatically vibrating resonator and thus for the energy distribution in a stationary radiation state, that is, in the normal spectrum.”

 

The title of his talk was “On the Theory of the Energy Distribution Law of the Normal Spectrum.”  In more modern language, he was talking about electromagnetic standing waves or normal modes of radiation in what we now call a cavity, which contained abstract electrically charged oscillators or resonators We may imagine the cavity to be a heated metallic cube insulated from the outside but not sealed. It’s a sort of oven, actually, but instead of having a big door to open to see what’s going on inside, it has a small hole through which light to be spectrally analyzed is emitted. 

The small hole absorbs light which is then mixed in with the light already in the cavity. No light is reflected  from the hole, so it acts as a near-perfect absorber and emitter of radiation, and its spectrum is that of a black-body.

Planck informed the assembled GPS members that his theoretical analysis of black-body radiation was based on “the laws of electromagnetic radiation, thermodynamics and probability calculus.” Precise measurements were being made in 1900 of the spectrum of cavity radiation, so Planck had experimental results to compare to his theory. Fitting his calculated energy distribution to the experimental results played an important role  in his discovery.

Inside Planck’s imaginary cavity there are “monochromatically vibrating resonators,” which he imagines divided up into different groups according to their frequency of radiation, with N resonators in the group having frequency υ, N’ resonators in the group having frequency υ’, N’’ in the group having frequency υ’’, and so on.  Planck says these resonators are “at large distances apart,” and are “enclosed in a diathermic medium with light velocity c and bounded by reflecting walls.” 

He assigns a total energy E_t to the radiation and the resonators and says: “The question is how in a stationary state this energy is distributed over the vibrations of the resonator and over the various frequencies of the radiation present in the medium, and what will be the temperature of the total system.”

Planck’s prescription for the energy of the resonators is that they have “arbitrary definite energies.”  He labels the different definite energies just as he labeled the resonators and their frequencies, with an increasing number of apostrophes: E, E’, E’’, and so on.   Then he says:

 “The sum E + E’ + E’’ + E’’’ + … = E₀  must, of course, be less than E_t. The remainder E_t - E₀ pertains then to the radiation present in the medium. We must now give the distribution of the energy over the separate resonators of each group, first of all the distribution of the energy E over the N resonators of frequency υ.  If E is considered to be a continuously divisible quantity, this distribution is possible in infinitely many ways.  We consider, however—and 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, and dividing E by ε we get the number P of 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.”

What does this inexactness mean? If we take ε to be the unit of energy then E must consist of a certain, exact number of these units.  I mean, this is supposed to be where quantization occurs!  Contained in this question is the question of discrete energy levels versus continuous frequency values.  We’ll hold off on this question for the moment.

 

Planck then calculates the number of ways to distribute P energy elements over the N resonators in order to calculate the entropy of these resonators.  He is doing his “probability calculus” first, then his thermodynamics calculations. His calculations for electromagnetic radiation were done in earlier papers. The purpose of his December 19 presentation, he says, is “to explain as clearly as possible the real core of the theory.”

Regarding the number of ways to distribute P elements of energy over N resonators of frequency υ, Planck continues: “Each of these ways of distribution we call a ‘complexion,’ using an expression introduced by Mr. Boltzmann for a similar quantity.”  Then Planck gives a simple numerical example of a complexion:  He chooses P = 100 units of energy to be distributed in one particular way over N = 10 resonators.  His table of numbers is:

1	2	3	4	5	6	7	8	9	10
--------------------------------------
7	38	11	0	9	2	20	4	4	5

Regarding this table he says, “The number of all possible complexions is clearly equal to the number of all possible sets of numbers which one can obtain for the lower sequence, for given N and P.  To avoid all misunderstandings, we remark that two complexions must be considered to be different if the corresponding sequences contain the same numbers, but in different order.”

Now for the calculation. We have 10 “boxes” and 100 “balls,” and want to know how many ways there are to distribute the 100 balls in the 10 boxes. One way to do this calculation is to show Planck’s table of numbers in a pictorial fashion like so

|°°°°°°°|°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°|°°°°°°°°°°°||°°°°°°°°°|°°|°°°°°°°°°°°°°°°°°°°°|°°°°|°°°°|°°°°°|

We then imagine the ways that the different symbols shown here can be redistributed.  According to Bernard Lavenda (Statistical Physics: A Probablistic Approach, 1991, p. 100) this combinatorial idea was first expressed as a "joke" in an October 1914 letter from Kamerlingh Onnes and Paul Ehrenfest to H. A. Lorentz.  But it works quite well, which is one reason it can be considered funny.  Archimedes was probably laughing when he ran through the streets naked shouting "Eureka!"  In contrast, Planck in 1900 merely wrote down the general formula (see below), and said it came from “the theory of permutations.”

We take the symbols ° and | as items to be redistributed. Thus we have balls and “walls” rather than balls and boxes.  The key idea is that the walls on the ends don’t get redistributed, since we then would have some balls outside the boxes.  So we leave those two walls out of the calculation, and count the number of redistributions we can make of the items inside those two walls.  What we have is one long box with 100 balls and 9 walls (count ‘em yourself) inside it. So we have 9 identical items of one type and 100 identical items of another type. In the general case, we have (N-1) walls and P balls. Keep this in mind when you look at the general formula below!

To get to the general formula, we need to start with a simpler situation.  If we had 109 items and none were identical to each other—let’s say you typed in 109 unique keyboard characters—how many ways to redistribute these would there be? There would be 109! ways.

(The exclamation mark is factorial notation. Look it up, or try it for yourself with these four words: Bridge Ices Before Road.  There are 4! = 4·3·2·1 = 24 ways to distribute these four non-identical words. To show this, start by writing the 4 words in a column, then put the combinations of the remaining 3 words in a column to the right of each of these, and so on for the 2 remaining words and then the one remaining word.)

Now we go to the next level of complexity.  If 9 of these 109 are identical we would have fewer distinguishable ways of redistributing the 109 items. There are 9! indistinguishable ways of redistributing 9 identical items, and although it seems pointless to consider these indistinguishable configurations, we have to divide 109! by 9! to get the number of distinguishable combinations when we have 9 identical items among the 109.

In our case we also have a group of 100 items out of the 109 that are identical. So we have to divide 109! by 100! and by 9!,  giving a total of 

109!/(100!)(9!)  = 109·108·107·106·105·104·103·102·101·100!/(100!)(9!) 

= 109·108·107·106·105·104·103·102·101/9! = 4263421511271, a very big number on the order of 10¹², or a terabyte of data--well, okay, not the number itself. If you had to store every number between 0 and 4263421511271 that would be about a terabyte.

The general expression for the number of ways P identical energy elements can be divided up among N resonators is

[P + (N-1)]!/[P! (N-1)!], 

which Planck writes as

(N+P-1)!

------------

(N-1)! P!

 But remember we have N’, N’’, etc. and P’, P’’, etc.  Planck says:

We perform the same calculation for the resonators of the other groups, by determining for each group of resonators the number of possible complexions for the energy given to that group.  The multiplication of all numbers obtained 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, and R is evaluated in the above manner.  Among all energy distributions which are possible for a constant E₀ = E + 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 the resonators together possess the energy E₀.  These quantities E, E’, E’’,… can then be expressed in terms of E₀.  Dividing E by N, E’ by N’, and so on, we obtain the stationary value [the average value] of the energy U_υ , U’_υ’ , U’’_υ’’ , … of a single resonator of each group, and thus the spatial density of the corresponding radiation energy in a diathermic medium in the spectral range υ to υ+dυ,

u_υdυ = (8πυ²/c³) U_υ dυ,

so that the energy of the medium is also determined.

At this point we need to pause and be reminded of what Max Karl Ernst Ludwig (Planck) means when he says “normal spectrum” or “normal energy distribution.”  What he says is, “The normal energy distribution is then the one in which the radiation densities of all different frequencies have the same temperature.” 

The best way to understand this statement is to quote what he says just before it.  He talks about the hypothesis of “natural radiation,” and even puts those words in quotes himself, then says: “the law of energy distribution in the normal spectrum is completely determined when one succeeds in calculating the entropy S of an irradiated, monochromatic, vibrating resonator as a function of its vibrational energy U.  Since one then obtains, from the relationship dS/dU = 1/T, the dependence of the energy U on the temperature T, and since the energy is also related to the density of radiation at the corresponding frequency by a simple relation, one also obtains the dependence of this density of radiation on the temperature.” 

The “simple relation” is the one above for u_υ  in the frequency range υ to υ+dυ.  Planck had earlier derived the factor in parenthesis (Lord Rayleigh had done it even earlier that same year). This factor is what makes the makes the spectrum "normal," and it represents the number of resonators or oscillators (or modes, in modern language) in the given frequency range.  Planck’s big deal was that he found the expression for the average energy U_υ that fit the data for black-body cavity radiation.  So we’ll be coming back to that derivation next. 

In the meantime, here’s a quote from Peter Milonni’s book The Quantum Vacuum concerning how Planck used the permutation formula above:  “Planck counted the number of ways, or ‘complexions,’ over which P energy elements could be distributed among N radiators. His counting procedure was totally at odds with classical statistical methods in its treatment of the energy elements as fundamentally indistinguishable.  In one sense Planck was following Boltzmann in regarding all complexions equally likely, but of course his way of counting the number of complexions was radically different.  His ‘energy elements’ obeyed what would much later be recognized as Bose-Einstein statistics.”