Particles-to-waves detour: Einstein vs. Planck part 2

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) U_υ

 

relating the cavity blackbody radiation density ρ(υ) to the average resonator energy U_υ 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 U_υ. 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 U_υ 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, 2^nd 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 U_υ. 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.)

Particles-to-waves detour: Einstein vs. Planck part 1

 (Revised 31 July 2025, mainly the footnote.)

Albert Einstein finished writing his paper “On a Heuristic Point of View Concerning the Production and Transformation of Light” on this date in 1905. I’ve recently been writing—trying to write—about Einstein’s criticisms of Planck’s papers of 1900 and 1901 introducing the quantum energy element into physics. This 120^th anniversary of Einstein’s completion of the above paper is a good time to post the first part of what I’ve been working on.

Yes, I’m detouring from my Particles to Waves series before posting Part 3b, the Final Segment. Who knows if it’ll ever get done? Who cares! Who gives a hoot? Well, Woodsy Owl, for one. He wants you to, too! ‘Cause this little detour series will segue nicely back into the Particles to Waves finale, friends, I promise. Einstein’s “Heuristic” paper, after all, introduced the light quantum, a particle-based description of electromagnetic waves. Like several other papers Einstein wrote during the first decade of the 20^th century, this paper was not easy for other physicists to understand and accept. It was different, however, in that it was the only one of these papers Einstein himself considered to be “revolutionary.”

On the other hand, there were the Max Planck papers published at the turn of the century that Einstein, like many other physicists, didn’t understand. Planck’s 1900 and 1901 papers on the blackbody radiation spectrum (the idealized thermal energy radiation spectrum) were themselves revolutionary and provided the basic idea for Einstein and his light quantum.

The questions I want to deal with here are “what did Einstein think Planck did wrong, in comparison to what Planck actually did do wrong, even though he got the right answer?” and “what did Planck actually do, in comparison to what he thought he did?”

 

In deriving what turned out to be the correct formula for the energy spectrum of blackbody radiation, 

                    ρ(υ,T)  =  (8πhυ³/c³) / [exp(hυ/kT) – 1] =  (8πυ²/c³)U_υ ,

Planck in his December 1900 and January 1901 papers postulated the existence of equally spaced discrete, integer-valued, energy levels for electrically charged “resonators” that absorbed and emitted electromagnetic waves—a very strange proposition indeed! In the equation, ρ(υ,T) is the spectral energy density of the radiation and U_υ is the average energy of a resonator. (As noted by Steven Weinberg, it was in the calculation of average oscillator energy that Planck made his "revolutionary suggestion" of discrete energy levels. See page 20 of Weinberg's 1977 Daedalus article, and see last equation below for explicit  U_υ expression.)

The electromagnetic (EM) waves, which Planck called a “stationary radiation field,” and the resonators together have a total energy E_t and are contained in “a diathermic medium with perfectly reflecting walls.” You can think of this system as an insulated box (with air or another gas in it, index of refraction approximately unity) and inside walls made of polished, mirror-like metal, with a few little black dots scattered around on the surface.[i] The black dots are the resonators, needed for thermalizing the enclosed radiation. (Whoa horsie, Rayleigh didn’t need em! But then after doing his reflecting-walls standing-wave analysis, he just used the "Maxwell-Boltzmann doctrine" of equipartition as the thermodynamic basis for his result. How to describe this detail? IN detail! Later!) 

Planck, in his December 1900 paper, says of the total energy, “The question is how in a stationary state this energy is distributed over the vibrations of the resonators and over the various frequencies of radiation present in the medium, and what will be the temperature of the total system.”

He first focuses on a single frequency υ and assigns N resonators with collective energy E to this frequency. Then, boom!, he introduces quantization by saying  this amount of energy E  is considered “to be composed of a very definite number of equal parts.” He introduces the constant h = 6.55 X 10^-27 erg-sec, and says, “This constant multiplied by the common frequency υ of the resonators gives us the energy element ε.” So there it is, the soon-to-be famous ε = hυ. This is for ONE frequency only, an arbitrary frequency, so the equation is not a proportionality in the usual sense. (It’s best to think of “ΔE = hυ,” i.e., the spacing of the energy levels equals hυ.) Every frequency has its own energy element, and Planck used “accents” to label the various frequencies, υ, υ’, υ’’, υ’’’, …, and their corresponding energies ε, ε’, ε’’, ε’’’, ….  

(Since the spectrum of energy vs. frequency of blackbody radiation is continuous, there’s an uncountable infinity of these frequencies. I think that may be why Planck used the accents rather than numerical subscripts 1, 2, 3, etc. for the different frequencies and energies. Subscripts imply discreteness. Later I’ll discuss the normal modes for radiation confined to a box--Dudley Towne's "waves confined to a limited region"-- where there are discrete frequencies, but the blackbody energy-versus-frequency spectrum, or energy versus wavelength spectrum, is continuous.)

Planck associated the energy elements only with the resonators, as if it were the resonators themselves that broke up the EM energy continuum into equal parts during absorption, then after emission the elements somehow melded together into the continuum of EM energies consisting of waves. Planck didn’t believe the EM field itself existed in the form of energy elements, and that’s the reason he didn’t think introducing his ε = hυ energy elements was a revolutionary idea. Here we have our first instance of what Planck thought he did versus what he actually did.

At first Einstein also believed Planck’s theory did not predict the existence of radiation quanta. In his “Heuristic Point of View” paper, Einstein assumed he was going beyond Planck’s work when he explicitly suggested the electromagnetic field was made of energy quanta:  “According to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of energy quanta that are localized in points in space, move without dividing, and can be absorbed or generated only as a whole.”

This idea of an electromagnetic field in free space consisting of light quanta is even stranger than Planck’s discrete energy levels for resonators. Planck’s resonators and their integer-valued energy levels nhυ were—and still are—abstractions. The energy of light waves consisting of “a finite number of energy quanta that are localized in points in space” gives a definite mental picture, and not a very believable mental picture, either, given the success of the wave theory of light propagation. Einstein was cautiously suggesting it was possible to think of light as particles or quanta, and he thus used the “heuristic point of view” terminology in his title.

 

Einstein in 1906 realized Planck’s published ideas about energy quanta applied to radiation as well as resonators. He said in his paper “On the theory of Light Production and Light Absorption” that “Planck’s theory makes implicit use of the aforementioned hypothesis of light quanta … the energy of a resonator changes by jumps of integral multiples of (R/N)βυ.” (At this time, instead of directly using Planck’s constant h, Einstein was still using (R/N)β, where R/N = k = Boltzmann’s constant, N = Avogadro's number, and β = h/k.) Thus belatedly realizing Planck has unwittingly gone Full Monty in uncovering the reality of quantization, Einstein then asks: how could Planck justify the use of the Maxwell-Lorentz electromagnetic wave theory  in deriving the very important relation (shown in the equation above)

ρ_{υ  =}  (8πυ²/c³)U_υ

for radiation density ρ_υ as a function of the average energy U_υ of the resonators?  Specifically, Planck derived the frequency dependent coupling constant 8πυ²/c³,  or electromagnetic mode density function, this way, not the average energy itself. He used his quantum energy element idea to calculate average resonator energy, which he found to be

U_υ  =  hυ/[exp(hυ/kT) – 1].

This mixed usage of both Maxwell-Lorentz ("classical" we say now) theory and the quantum-of-energy idea is the first of the three things Einstein believed Planck did wrong. I’ll finish discussing it and move on to the next two things in the next segment of this detour from Particles to Waves.

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[i] The mirrored walls, being totally reflective, are supposedly incapable of producing thermal radiation in equilibrium with the walls, as discussed by the great Sommerfeld on page 135 (page 142 in online PDF) of his Thermodynamics and Statistical Mechanics:   "when the inner walls of the cavity are made of a perfectly reflecting material and cannot, therefore, influence the rays falling on them, the radiation filling the cavity may become one which is not in equilibrium." However, earlier on the same page he says of the blackbody cavity, "The cavity constitutes a thermodynamic system which is independent of the particular physical and chemical processes of emission and absorption taking place in the walls."   Also, a simple harmonic oscillator, even an electrically charged one, isn’t a thermodynamic entity, but Planck gave his monochromatic electrical resonators randomized amplitudes and phases. The disorder of the random phases and amplitudes, he says, gives the resonators the thermodynamic property of entropy, and thus also the property of having a temperature. This issue of the thermalization of radiation in a box is related to my offhand parenthetic comment above on Rayleigh's use of energy equipartition in doing the classical thermal spectrum calculation in 1900. Blackened walls are used in real blackbody experiments. And Sommerfeld says, on the same page as his above comment on perfectly reflecting walls, "The introduction of a speck of soot into the cavity will turn the radiation into black body radiation. (The speck of dust performs the role of a catalyzer.)" We appreciate your thoughtful and thorough comments, Arnold, but you seem to be waffling a bit on whether it matters what the walls are made of!