Grad Student Newsletter UT-Austin 20 Feb 1967

Sorry, the copy I have of this newsletter was cut off at this point. I found it in a paperback book of Enrico Fermi's Lecture Notes on Quantum Mechanics that I got from David Potter of Austin, who was a physics grad student at UT-Austin in 1967.  I arrived as a grad physics student there a little over 20 years later.  See my journal entry from 30 August 1987 about grad student orientation, posted on 30 August 2017.

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.