1967 Calendar

I was in the seventh and eighth grades in 1967.  I got a set of drums, a trap set, from my parents, after having received a snare drum for Christmas in 1966.  The condition under which I received the trap set, which was a used Slingerland set (black), was that I would join the Dial Jr. High band in the eighth grade.  Playing in the band would have been an embarrassment for me.  Not cool. I fortunately didn't have to play with the band in public, due to the fact that I was a beginner.  I was in the "training band," and recall spending quite a bit of the class period alone with the two other drummers in a practice room, shooting the breeze and not practicing. 

Also I remember getting sent to the principal's office for putting up a photo from Playboy in the bandroom window.  The older guy among us three training drummers brought it to school.  Not the only time I got sent to the principal's office that year, but the only band-period principal's office trip I can remember.

The credits at the end of A Serious Man say copyrighted material from Playboy was used with permission of the magazine, but I've looked for and haven't seen any Playboy material in the movie. It's most likely shown on the wall in the blurry background of the scene where Danny is practicing for "the Torah portion" (singing) of his bar mitzvah.  Nope it’s in the scenes where Danny and his profane friend pick the lock on Rabbi Turchek's desk drawer: the particular shot shows the drawer being closed, and lasts only about half a second. Instead of the contents of the drawer that Danny and his friend are shown rummaging through looking for Danny's transistor radio (and the $20 bill stuck in the faux leather case), the quick-shot of the closing drawer's contents shows different stuff, including a Playboy magazine.

I rediscovered a few years ago an International Paper Co. 1967 desk calendar that was kept on the desk in my parents bedroom that year: 

I wrote comical comments in it ("Get Smart" was one of my favorite shows)...

... more of which will be on display later.  In Rabbi Scott's office in the movie ("Things aren't so bad.  Look at the parking lot, Larry! Just look at that parking lot."), a calendar for May and June 1967 is shown, so I'm starting with May.

Planck oscillator average energy Part II

See the last part of my previous post for equations leading up to here. Leaving out the constant C for the moment, and using the total rather than the partial derivative, since U_υ  is the only independent variable,

dS_υ / dU_υ  =   (1/ε)log [(1 + U_υ / ε)] + ( 1/ε)(1+ U_υ / ε)/ (1+ U_υ / ε)

                                                           - (1/ε)log (U_υ / ε) - (U_υ /ε)(1/ε)(1/U_υ /ε)

The red factors cancel, giving

dS_υ / dU_υ = (1/ε)log [(1 + U_υ / ε)] + (1/ε) — (1/ε)log (U_υ / ε) — (1/ε),

where the red terms now cancel. With C reinserted, and dS_υ / dU_υ replaced with 1/T, the result is

1/T = C{(1/ε)log [(1 + U_υ / ε)] — (1/ε)log (U_υ / ε)} 

             = (C/ε){(log [(1 + U_υ / ε)] — log (U_υ / ε)} = (C/ε){(log [(1 + U_υ / ε) / (U_υ / ε)]}

or                                        ε /CT = log[(1 + U_υ / ε) / (U_υ / ε)].

Taking exponentials of both sides

exp(ε /CT ) = (1 + U_υ / ε) / (U_υ / ε)

= 1 / (U_υ / ε)  +  (U_υ / ε) / (U_υ / ε)

=  ε / U_υ   +  1

We want an expression for the average oscillator energy U_υ , so some more algebra is required, namely,

ε/U_υ = exp(ε /CT )  –  1,

 or                                                      U_υ = ε /[ exp(ε /CT ) – 1].

This was Planck's new expression for the average energy of a resonator in equilibrium with black-body radiation—new as of 1900. There's a lot more to discuss here: 1. why Planck's constant h and Boltzmann's constant k come out of this calculation,  2. the Rayleigh-Jeans and Wien limits at the low end and high end, respectively, of the black-body frequency spectrum, 3. the Wien displacement law, 4. the "wrong" assumption of indistinguishable energy elements Planck made when he chose the permutation calculation that goes into finding the entropy, and 5. the choice of perfectly reflecting (mirror) walls versus perfectly absorbing (black) walls of the cavity. For mirror walls there is absorption with immediate re-emission, while black walls would absorb and only re-emit after affecting the motions of the atoms of the material, right? Sure! But just in case, check with Born & Wolf or Hecht &Zajac if you want to be further enlightened.

On this last question, Sommerfeld, in his Thermodynamics & Statistical Mechanics book, Section 20, says that when the walls are perfectly reflecting, they cannot come into temperature equilibrium with the radiation, and a “speck of soot” must be introduced into the cavity as a “catalyzer” to achieve equilibrium!  And there's also the question of where the oscillators (resonators) are, exactly. Are they supposed to be in the cavity walls or are they imaginary particles, like soot, floating about in the air in the cavity? Apparently, with black walls, the resonators are in the walls, and with mirror walls, the resonators need to be free-floating in the air of the cavity. In calculations, we sort of assume a vacuum cavity, so that the speed of light = c = γυ can be used in the equations rather than c/n, where n is the index of refraction of the (heated) air in the cavity. For air (at 300 K) n ≈ 1.0003, so we won’t worry about that right now.

Right now we will look at Einstein's 1907 calculation using geometric series to find Planck's expression for the average energy of a simple harmonic oscillator system. Einstein says, "To arrive at Planck's theory of black-body radiation ... one assumes Maxwell's theory of electricity yields the correct relationship between radiation density and Ē.”  Yes, Einstein uses Ē instead of U_υ for the average resonator energy. The “correct relationship” he’s referring to is Planck's "simple relation," u_υdυ = (8πυ²/c³)U_υ dυ, where U_υ is average resonator energy, as discussed in the last part of my Walking further with Planck post.

“On the other hand,” Einstein continues, “one abandons equation (4),

Ē =  ʃE exp [(-N/RT)E]dE / ʃ exp [(-N/RT)E]dE  = RT/N = kT,

i.e., one assumes that it is the application of the molecular-kinetic theory which causes a conflict with experience. ... this stipulation involves the assumption that the energy of the elementary structure under consideration assumes only values infinitesimally close to 0, ε, 2ε, etc."

Einstein doesn't bother to show the actual equation (4) calculation, but I guess it was a routine calculation by that time. One way it can be done is by separately finding the average potential energy and the average kinetic energy of a one-dimensional harmonic oscillator. Then the E in the integrals in the expression above is, respectively, a constant times x-squared or a different constant times p-squared. The result of the integration in each case is kT/2.  Try it!  Adding the potential and kinetic contributions together gives kT.  This is the classical result, which Rayleigh (with a numerical correction made later by Jeans) calculated in 1900.

Page 216 of the Einstein paper shows this result. Page 217 shows the series representation that gives Planck's result. Einstein was apparently the first to obtain Planck's result using the infinite series calculation. Nowadays this is how most introductory textbooks do it, although they use discrete sums instead of the integral notation used by Einstein.

I will show the calculation as done by Koichi Shimoda on page 70 of Introduction to Laser Physics.  We now have E = nhυ, and use U_υ again instead of Ē (if the Greek letter nu comes out as u, I'll try to fix that later...)

U_υ =  Σ nhυ·exp(-nhυ/kT) / Σ exp(-nhυ/kT)

where the sums are over n = 0 to n = ∞. This is Boltzmann’s method for calculating a statistical average, but with discrete sums instead of integrals because we have discrete or quantized energy levels.  Shimoda lets exp(-hυ/kT) = r in order to make it clear that the denominator is the geometric series

Σ exp(-nhυ/kT)  =  Σ rⁿ  = 1/(1-r) =1/[1- exp(-hυ/kT)].

And he writes the numerator as 

Σ nhυ·exp(-nhυ/kT) = hυ Σn rⁿ

then use the derivative of rⁿ

drⁿ/dr  =  nr^n-1

and compensates for the n-1 exponent by using an extra factor of r out front when writing the sum:

Σn rⁿ  = rΣ drⁿ/dr  = r (d/dr )Σ rⁿ ,

since a sum of derivatives is the derivative of the sum.  Now we put in 1/(1-r) for the geometric series sum, and do the derivative

r (d /dr) (1-r)^-1 =  r(-1)(-1)(1-r)^-2  =  r/(1-r)².

Putting the closed-form expressions for numerator and denominator into the equation for average energy gives

U_υ = hυ[r/(1-r)²] / [1/(1-r)]  = hυr(1-r) / (1-r)² = hυr / (1-r),

and using the trick of pulling out a factor of r in the denominator gives

hυr / (1-r)  = hυr / [r( 1/r – 1 )]  =  hυ ( 1/r – 1 )^-1.

Now 1/r is replaced by exp(hυ/kT), and the Planck expression for average energy is the result:

U_υ  =  hυ / [exp(hυ/kT) – 1]

One overall final point about the average energy:  It is also given by the expression we used in calculating the entropy of an oscillator.  Remember that?  It’s just the arithmetic mean, E/N, where E is the total energy of N oscillators.  Planck specified that the energy E is partitioned among P elements, each with energy ε, so the average energy is

U_υ  =  E/N = Pε/N.

This simple average must equal the thermodynamic expression Planck found for average energy:

Pε/N  = ε /[ exp(ε /CT ) – 1],

or                                                       P/N   =  1/[ exp(ε /CT ) – 1].

This is sort of self-explanatory, but not quite. Planck has already said that these N resonators have a common frequency υ. (Whatever happened to the other N’, N’’, … resonators with frequencies υ’, υ’’, … and energies E’, E’’, … ? I don’t know!  I’m hoping to get back to that. November 2017:  Planck does discuss the other resonators, see my 31 Oct 2017 post.)  If we go ahead and use ε = hυ, then P/N could be called the occupation number associated with the frequency υ.   In modern terminology, it is the Planck thermal excitation function,

<n>  =  1/[ exp(hυ/ k_BT) – 1],

giving the  mean  number of photons excited in the field mode at temperature T.

To be discussed later:  Why did Planck choose a direct proportionality between energy and frequency? At what point does the quantization of energy occur? (The proportionality between energy and frequency is not in itself quantization.)  How did Planck find the constants h and k? And the biggest question, how can all this be explained—that a continuous oscillator can have discrete energy levels—or as Einstein put it, what is the "mechanism of energy transfer"?  In classical physics the mechanism is acceleration, caused by a net force acting on an electric charge. So far, there is no mechanism in quantum theory to explain how charges radiate, in spite of the potential energy appearing in the Schrödinger equation. There are only acausal calculational tools, such as transition probabilities and the use of time-dependent perturbation theory, as in the Feynman diagrammatic perturbation approach (well, okay, only for unbound energy states, so we’ll just come back to all that later).  

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.