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/PP NN
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/PP NN]
=
C {log [(N+P)N+P] — log PP — log NN}
= 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, Uυ
= E/N. If we rearrange, we have E = NUυ , and can
write P in terms of the average energy: P = NUυ /
ε. So, voilá, the total entropy of the N oscillators in terms
of the average energy of a single oscillator:
S
= C{ (N + NUυ / ε)log (N + NUυ / ε) — (NUυ /
ε) log( NUυ / 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 + Uυ / ε)log (N + NUυ / ε) — (Uυ / ε)
log( NUυ / ε) — log N },
S/N
= C { (1
+ Uυ / ε)log (N + NUυ / ε) — (Uυ / ε) log( NUυ
/ ε) — log N }.
This
is still expressed in terms of N and NUυ , 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 Sυ instead
of S/N for average entropy, to match it with average energy Uυ
:
Sυ
= C {
(1 + Uυ / ε)log N(1 + Uυ / ε) — (Uυ / ε)
logN(Uυ / ε) — log N }
= C { (1
+ Uυ / ε)logN + (1 + Uυ / ε)log(1 + Uυ / ε) —
(Uυ / ε) logN + (Uυ / ε)log(Uυ / ε) — log N}
— (Uυ / ε) log N — (Uυ / ε) log (Uυ / ε) — log N }.
Sυ
= C {(1 + Uυ / ε) log(1 + Uυ / ε)) — (Uυ
/ ε) log(Uυ / 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.
