Left to
right: James Peebles (Princeton), George Abell
(UCLA), Malcolm Longair (Cambridge), and Jaan Einasto (Tartu Observatory, Tõravere,
Estonia), at an International Astronomical Union symposium in 1977. From Five
Decades of Missing Matter, by Jaco de Swart, Physics Today, August 2024.
The
first of three things Einstein questioned about Planck’s blackbody radiation
formula derivations in 1900 and 1901 was Planck’s use of electromagnetic wave
theory instead of his own new quantized energy level theory to derive the
equation
ρ(υ)
= (8πυ2/c3) Uυ
relating the cavity blackbody radiation density ρ(υ) to the average resonator energy Uυ at any given arbitrary frequency υ (and arbitrary temperature T, not shown).
Plank’s overall goal was to find ρ(υ), or actually ρ(υ,T), so his next step after deriving this equation would be to find Uυ. As Einstein said in his paper “On the Theory of Light Production and Light Absorption,” published in 1906, “This [equation] reduced the problem of black-body radiation to the problem of determining Uυ as a function of temperature.”
And here’s what our main physics man in
England, Malcolm Longair, pictured above, says about the equation after he shows Planck's derivation of it in Theoretical
Concepts in Physics: An Alternative View of Theoretical Reasoning in
Physics, 2nd edition published in 2003:
This
is the result which Planck derived in a paper published in June 1899. It is a
remarkable formula. All information about the nature of the oscillator has
completely disappeared from the problem. There is no mention of its charge or
mass. All that remains is its average energy Uυ. The meaning behind the formula is
obviously very profound and fundamental in a thermodynamic sense. I find this
an intriguing calculation: the whole analysis has proceeded through a study of
the electrodynamics of oscillators, and yet the final result contains no trace
of the means by which we arrived at the answer. One can imagine how excited
Planck must have been when he discovered this basic result.
Well,
it’s hard for me to imagine Planck in an excited state, even though he invented
the concept, but nevermind that. What did Albert Einstein see as a problem with
this equation? It’s what Longair says in the next to last sentence above: “the
whole analysis has proceeded through a study of the electrodynamics of
oscillators . . .”. How, Einstein asked in his 1906 paper quoted above, could Planck’s discovery of the discrete nature of the energy of an oscillator make
use of the wave-based logic of Maxwell-Lorentz oscillator theory that "does not allow distinguished energy values of a resonator"?
Like
a physics bloodhound on the right scent, Einstein was barking up the right
tree. But it turned out this time there was just a Cheshire Cat in the tree,
and its smile was saying, “Actually, Albert, classical physics is okay for this
calculation.”
Almost
everybody (and their dog) uses classical physics nowadays in textbook derivations
of the factor 8πυ2/c3 in the above equation, and
this factor is what Planck found in his “study of the electrodynamics of
oscillators.” We now call it a density of states or a mode density, and textbooks obtain it most often by doing a “study” of electromagnetic waves confined to a
cubical box or "cavity".
For
instance, referring to electromagnetic waves in a cavity as “field
excitations,” Rodney Loudon on page 1 in The Quantum Theory of Light
says “The field excitations are then limited to an infinite discrete set of
spatial modes determined by the boundary conditions at the cavity walls. The
allowed standing wave spatial variations of the electromagnetic field in the
cavity are identical in the classical and quantum theories, but the time
dependencies of each mode are governed by classical and quantum
harmonic-oscillator equations, respectively.”
PAUSE:
Here you may well ask “What ARE the ‘classical and quantum harmonic-oscillator
equations’ anyway, Rodney?” Good
question! Loudon is kind of skimming over the issue because the wavefunction of
a quantum harmonic oscillator in one of its stationary states (yep, an eigenstate,
or self-state) has no time dependence. (And neither does an atom in a stationary state, thus the reason it's called a stationary state, although physically it makes no sense.) As pointed out by John Townsend in Quantum
Mechanics: A Modern Approach, time dependence only comes from a
superposition of quantum harmonic oscillator energy states. Here’s my chalkboard solution to Townsend’s
Problem 7-9, which asks for the position expectation value, <x>, of the superposition of the n
and n+1 energy states of a quantum harmonic oscillator:
This
time dependence is what you expect for a classical harmonic oscillator, so I’m
not sure what Loudon is saying in regard to “time dependencies” but I’ll come
back to that later, maybe. If he’d said temperature dependence, I could
understand that. Or energy dependence, i.e., amplitude squared for classical
and nhυ for quantum. But anyway! Our subject
of the moment is the nature of the calculation of the spatial dependence of cavity radiation
in equilibrium with the cavity walls, so I’ll get back to that now. UNPAUSE.
The
units of 8πυ2/c3 are inverse volume times inverse
frequency, so this factor multiplied by the oscillator average energy gives
units of energy per unit volume per unit frequency for ρ(υ).
Since 8π/c3
≈ 10-24,
the order of magnitude of the mode density at any given frequency is υ2
x 10-24,
or (υ
x 10-12)2. For
microwave frequencies around 1012 Hz, this average-oscillator-energy
to radiation-density conversion factor is unity. Going up or down the frequency
scale by a factor of 10 causes the conversion of average oscillator energy to
radiation energy to go up or down by a factor of 100.
The higher the frequency of the oscillator (resonator), the more efficiently its energy is converted into radiation. Turn the equation around, however, and you see that the conversion of radiation energy into oscillator energy is more difficult at higher frequencies. Not your great-great-grandfather’s equipartion theorem!
Well, since we haven’t considered temperature dependence here yet, we’re not in the equipartition-or-not-equipartition realm. I'll get to that in my next post when I look at Rayleigh's 1900 calculation versus Planck's 1899 calculation. We are in the equilibrium realm, I remind you, where the resonators are absorbing and emitting radiation at the same rate on average. Which doesn't mean resonators at different frequencies are emitting and absorbing energy at the same rate--that would give a white noise or uniform spectral energy density function instead of the Planck function. (What kind of system would have a white noise spectrum?) It means absorption and emission rates are equal for each resonator, individually. The Planck function ρ(υ) shows what the constant equilibrium absorption and emission rates are at different frequencies.
The conclusion I’m passing along here is one of the answers to my question from my previous post: “what did Einstein think Planck did wrong, in comparison to what Planck actually did wrong?” Answer: Planck was wrong to use Maxwell-Lorentz oscillator theory to find the conversion factor 8πυ2/c3 (he should have used Rayleigh's "mode count" method), and Einstein was right to be concerned about that. But Einstein’s concern was a Cheshire Cat illusion, as illustrated by Professor Longair’s comments above: “All information about the nature of the oscillator has completely disappeared from the problem.” Planck's result for the radiation density versus oscillator average energy was and is correct.
Next I'll look at Einstein's opinion that the reason it was okay for Planck to use Maxwell-Lorentz theory was because Planck's calculation involves average oscillator energy. The problem with Einstein's opinion in this matter is that average oscillator energy is precisely where Planck made his energy quantization hypothesis. (See the quote from Steven Weinberg in my previous post.) Is there a difference in the average Planck was calculating and the one Einstein had in mind? Perhaps Planck's was a shorter time average?
(I didn't plan to post this today, and even thought of not posting it today, since this is the 80th anniversary of Hiroshima, the first "application" of physicists' unleashing of uncontrolled nuclear fission energy. "Oh, vey!" said Einstein when he heard about it, which translates from Yiddish as "Woe is me!" or just "Oh, woe!" I haven't found a reference to what Planck, who died in 1947, said about it.)
(I revised this post on August 9, 2025, Nagasaki Remembrance Day number 80. To remember and not to repeat the application of nuclear weapons, that is the goal.)
