Particles-to-waves detour: Einstein vs. Planck part 3a

To continue the discussion of the Planck-versus-Einstein time averages for the energies of Planck’s electromagnetic resonators, let’s start with an elementary example of a time average from the David J. Griffiths and Darrell F. Schroeter book Introduction to Quantum Mechanics, 3^rd edition, 2018.

Griffiths & Schroeter (G&S) give an example of a discrete variable average and an example of a continuous variable average. We’re interested in the continuous average, but the discrete data example is a good way to prepare for the continuous case.

Discrete probability example

You probably know the drill on this one. I’m purposely choosing this simplest example because I want to embellish it into a more complicated example—related to the Boltzmann-Planck complexion-counting versus the Einstein time-average  counting—later.

There are  N = 14 people in a room with ages ranging from 14 to 25.  Let N(j) represent the number of people who are j years old. The given dataset is

 

N(14) = 1

N(15) = 1

N(16) = 3

N(22) = 2

N(24) = 2

N(25) = 5

 

I won’t belabor this example too much by repeating it here. I’ll just note that G&S use it to introduce a few statistical concepts: the most probable age, the average (mean) age, and the median age. They then show how to calculate the variance and standard deviation from the average.

In making these calculations, the concept of probability isn’t even needed. The “most probable age,” for instance, is based on looking at the data and finding which age is represented by the largest N(j) value. In this example, it’s N(25) = 5. And you know how to calculate the average age, right?  No concept of probability required for that either, but it’s a good place to “invent” the concept. Writing out the sum of the weighted ages, the weight being N(j), and dividing by the number of people in the room gives the average age,

 

$$\frac{1(14)+1(15)+3(16)+2(22)+2(24)+5(25)}{14} = 21,$$

 

from which we can extract a notion of probability by observing that, when the divisor 14 is distributed to each term separately, each term has a weighting factor equal to  (number of people that age) divided by (total number of people), i.e., N(j)/N. This is what G&S call the probability,

$$P( j ) = \frac{N(j)}{N},$$

But we know, for instance, that there are two people in the room who are 24, so what’s probability got to do with it? The probability that a person in the room is 24 is unity, and the same is true for the probability of two people in the room being 24. The probability that three or more people in the room are 24 is zero. These numbers are known from the data set.

To have any meaning, a probability must describe something that is uncertain or has a random chance of having different values when we measure it. Therefore, the randomness in this example is introduced when we ask: what is the probability that a randomly chosen person in this room is 24? Since there are two 24-year-olds, the probability is 2/14 or 1/7.

Something G&S don’t discuss is how things would change if the ages of the people in this room were given in months, or even in weeks or days.  If everyone in the room knew the exact time of day they were born, their ages could even be calculated in hours. But with every passing hour their numerical hourly age increases by one, so we’d need to say something like “everyone write down their age in hours today at 3:10 p.m.”

I mention this because we’re about to make the transition from discrete variables to continuous variables. G&S just sort of jump to the continuous case in one fell swoop (which I’ll get to in a minute) but I want to make a slower transition.

So here’s a question to think about: is it less likely for two or more people in the room to be the same age in months compared with the same age in years? If so, why is that the case, and how would you calculate how much less likely it is? Now we’re getting into some nontrivial probability calculations. For instance, we’re given that there are three 16-year-olds in the room, so how would we calculate the probability that all three were born in the same month? But I’m getting off the immediate subject, so I’ll just put a footnote here.*

The point I’m making is when you ask for the probability that someone selected at random from this small group is a certain number of months old, you’re more likely to get zero, and you’re also more likely get a zero probability that anyone in the room is the same age in months. Change the unit of measurement to days or hours and these probabilities become even smaller.

This is where G&S jump all the way into the existential void—the continuum of numbers on the number line—when they make the transition to the continuous case. They say the probability of picking someone whose age “is exactly 16 years, 4 hours, 27 minutes, and 3.33… seconds is zero.” They mean exactly zero, whereas in my above discussion you can see that as the time interval becomes smaller, the probability of someone being a certain age measured in the smaller time intervals gets smaller.

Thus, choosing the feminine pronoun,  G&S say:

The only sensible thing to speak about is the probability that her age lies in some interval—say, between 16 and 17. If the interval is sufficiently short, this probability is proportional to the length of the time interval. For example, the chance that her age is between 16 and 16 plus two days is presumably twice the probability that it is between 16 and 16 plus one day. (Unless, I suppose, there was some extraordinary baby boom 16 years ago, on exactly that day—in which case we have simply chosen an interval too long for the rule to apply. If the baby boom lasted six hours, we’ll take intervals of a second or less, to be on the safe side. Technically, we’re talking about infinitesimal intervals.)

Continuous probability example 

The fact that G&S mention this fluctuation in birth rates is the reason I chose to use this example. Fluctuations are the deciding factor in considering Planck’s shorter time average versus Einstein’s longer time average. Planck uses the idea of fluctuations, without saying that word, to define the entropy of a resonator:

Entropy means disorder, and I thought that one should find this disorder in the irregularity with which even in a completely stationary radiation field the vibrations of the resonator change their amplitude and phase, as long as one considers time intervals long compared to the period of one vibration, but short compared to the duration of a measurement.

He famously found in this same paper, based on the 14 Dec 1900 talk he gave, that the resonators could only "change their amplitude" (amplitude-squared actually) by very small discrete amounts \(\epsilon = h\nu\) so that nothing smaller is allowed, although this seems to be what Planck implies by the "irregularity" of a change in amplitude. To be discussed in a later post! (The h here is Planck's constant, of course, and the h below is the height of a cliff.)

For now, we’ll just look at what G&S give as the continuum definition of probability, starting with:  "ρ(x)dx = probability of a randomly chosen individual’s age being between x and x + dx.  The proportionality factor, ρ(x), is often loosely called ‘the probability of getting x,’ but this is sloppy language; a better term is probability density. The probability that x lies between a and b (a finite interval) is given by the integral of ρ(x)"

 $$P(x)=\int_{b}^{a} \rho (x)dx. $$

Now for the example of a time average using continuous random variables. 

“Suppose someone drops a rock off a cliff of height h. As it falls, I snap a million photographs, at random intervals. On each picture I measure the distance the rock has fallen. Question: What is the average of all these distances? That is to say, what is the time average of the distance traveled?”

Neglecting air resistance, the distance x the rock has fallen in time t is 

$$x(t) = ½ gt^2,$$

the speed variable is dx/dt = gt, and the total time of the fall is 

$$T=\sqrt \frac{2h}{g}.$$ 

Thus, say G&S, “the probability that a particular photograph was taken between t and t + dt is dt/T, so the probability that it shows a distance in the corresponding range x to x + dx is"

$$\frac{dt}{T} =\frac{dx}{gt}\sqrt \frac{g}{2h}=\frac{1}{2\sqrt hx}dx.$$

The probability density is thus

$$\rho (x)=\frac{1}{2\sqrt hx}.$$

I leave it to you to write and solve the time integral based on the above relations, and to use the total time in terms of total distance to get the average distance, the answer being \(\frac{h}{3}\). (In comparison, what is the average speed?) 

Could we also, or instead, calculate average distance by adding up the measured distances on the photos (measured from the top of the cliff to the end point on each photo) and dividing by the number of photos? I'll be applying these ideas to the Planck and Einstein time averages in the next post, i.e., part 3b. And then, after that, back to the future: the final particles-to-waves post! We hope.

 

* A question you may be contemplating is what happens to these probabilities for a much larger population, say the approximately eight billion people on Earth right now? Then you will have nonzero probabilities of people being the same age in months, even in minutes. There is an age continuum. (Okay, we'd need a statistically significant sample from this overall population to calculate anything from actual numbers. Let's assume we have that.) You can ask for the probability that two randomly selected people were born in the same minute and get a nonzero answer. But you still will get an answer of zero when you pick a person at random and ask for the probability that he or she is “exactly 16 years, 4 hours, 27 minutes, and 3.33… seconds” old. And, not forgetting the question leading to this footnote, the probability that the three 16-year-olds were born in the same month is 

$$P = \frac{1}{1} \frac{1}{12}\frac{1}{12}$$

 

Binding Energy of Classical Electron now online

Just learned by searching under my name and the word "physics" that my master's thesis is available for viewing online now. That's good news for modern man! ("Download PDF" will just open the PDF in a new window, not actually download it to your computer.) It has a few problems, conceptual and otherwise, and is mainly historical, like the current Boltzmann-Planck-Einstein wave-versus-particle blog posts of mine, but I'm still working on some of the ideas in it. Send me an email with your comments if you have any (see my profile).

26 Oct 2025: After finding my thesis online, I was going to contact Victor Michalk, my thesis advisor. It's been about 20 years since I exchanged emails with him. I looked for his email address on the Texas State University physics department website, but didn't find it. When I did a general search, I found his obituary from last year at about this time. Thanks for your good teaching from the pretty bad, but commonly used, Herbert Goldstein graduate text in mechanics, Professor Michalk! And for your guidance on my thesis.

19 August 2026, a few corrections to the thesis:

1. A slight comma misplacement in the "trial by fire," quote on the acknowledgements page (the comma should come after the parentheses). And in that same sentence I say I learned that I like teaching physics, but I have to admit, liking it didn't last long, although I still did it occasionally when I needed the money.

2. On page 45, I say: "In common with other references consulted for this thesis, Feynman and Panofsky and Phillips do not explicitly say that the 'mass of unknown origin' can be considered to be responsible for the binding energy of the classical electron." 

This sentence was a bit of a ruse on my part. The exact two words "binding energy" aren't used by Panofsky and Phillips, but they explicitly say that such a mass of unknown origin is responsible for making the charge system stable: "An additional mass, \(-U_0/3c^2\), whose origin is not electromagnetic, is needed  to account for the observed mass, which obeys the relativistically correct equations. This extra mass (or energy) presumably represents the nonelectromagnetic binding which must be present to make the charge system of the electron stable; its need was pointed out by Poincaré." 

3. On pages 48 and 49, the radiation damping equations for the electric dipole moment \(p(x,t)\) are functions of x and t, position and time, but my \(x=Acos\omega t\) simple harmonic approximation is just a function of time. So I overlooked the need to mention the electric dipole moment approximation that gets rid of the x-dependence of the equations, firstly, then I didn't show that the amplitude A in the "simple harmonic approximation" isn't just a constant, it's a complex function of the incoming electromagnetic wave's changing frequency \(\omega\) and the dipole oscillator's constant frequency \(\omega_0\), and of the damping constant itself. 

The dipole approximation to my equation on page 49 means the incoming electromagnetic wave and the dipole oscillator itself are treated as time-dependent only,  The characteristic damping equation modifying the simple harmonic oscillator of natural frequency \(\omega _{0}\) becomes \(\ddot{x} + \gamma \dot{x} + \omega_0^2 x = \frac{e}{m}E_0cos\omega t\), with the damping constant \(\gamma = \frac{2e^2 \omega_0^2}{3mc^3}\). 

A trial solution is \(x(t)=Acos(\omega t + \phi)\), which works well as long as we have the right initial conditions. Try it and see what you get for the amplitude A. The usual trial solution is \(x(t)=Aexp(i\omega t)\), which gives what I originally had in mind for A when I realized The Corrections needed to be done:

$$A = \frac{eE_0}{m(\omega_0^2 - \omega^2 + i\gamma\omega)}$$

So you see the "simple harmonic approximation" isn't as simple(minded) as I made it appear in my thesis. Indeed, to give you (and me) more to study before I move on to the next post, here's something Francis Low says in his book Classical Field Theory (1997), at the beginning of the section on "Scattering by a Damped Harmonic Oscillator and Radiation Reaction," page 212: that the damping constant in this model of the light-matter interaction is something "that we will adjust to give overall energy conservation via the optical theorem." The result of Low's linear adjustment is the value for \(\gamma\) shown above, but it's worth looking into the meaning (beyond just energy conservation) of the optical theorem and how it's used, in case you're not already familiar with it. I recommend starting with the History section of the Wikipedia article.

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). Note added 26 September: probably should have dυ on each side, so that the equation applies to region between υ and υ + dυ. See note added New Years Day 2026 below in paragraph just after the UNPAUSE below the chalkboard photo.

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). Jan 1st 2026: The "per unit frequency" nature of this expression is the reason a frequency range dυ should be included in the equation in the first paragraph above. 

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

September 2025: Sorry to say, the links below to Einstein's papers don't get you there anymore because a paywall is being constructed. 

(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.

---

[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! 

From particles to waves, Part 3a: Eigen Spiel

“. . . when Werner Heisenberg discovered ‘matrix’ mechanics in 1925, he didn’t know what a matrix was (Max Born had to tell him), and neither Heisenberg nor Born knew what to make of the appearance of matrices in the context of the atom. David Hilbert is reported to have told them to go look for a differential equation with the same eigenvalues, if that would make them happier. They did not follow Hilbert’s well-meant advice and thereby may have missed discovering the Schrödinger wave equation.”

                                — Manfred Schroeder, in the Forward to his book Number Theory in Science                                                 and  Communication, 2^nd edition, corrected printing, 1990.

Keeping Up with the Eigens

In Quantum Concepts in Physics (Cambridge University Press, 2013), Malcolm Longair says (page 267), “In seeking a wave equation to describe de Broglie’s matter waves, Schrödinger began by attempting to find an appropriate relativistic wave equation. … These first attempts at the derivation of the relativistic wave equation were never published, but the argument can be traced in Schrödinger’s notebooks and a three-page memorandum he wrote on the eigenvibrations of the hydrogen atom.”

Professor Longair then says on page 268 that “de Broglie’s waves were propagating waves whereas Schrödinger had converted the problem into one of standing waves, like the vibrations of a violin string under tension.”

However, we know from my previous post that the production of standing waves on a string can be done without the string being fixed at both ends. Traveling sine waves of any frequency can be sent in from –∞ on a semi-infinite string and their reflection at the x= 0 end of the string, where the string is tied, will produce a standing wave.

This is where we come to the subject of all things eigen. The standing waves on a semi-infinite string are an example of what “eigenvibrations” are NOT, simply because they can have any frequency. Eigenvibrations occur only at eigenfrequencies, and these are the frequencies that are characteristic of the length of a string tied at both ends and the given boundary conditions at both ends. Indeed, eigenfriends, “characteristic” and “proper” are most often used in math and physics books as the English translation of eigen.

Eigen as “Own”

But let’s see what A Brief Course in German by Peter Hagboldt and F. W. Kauffmann, published in 1946, has to say on the subject. In the back of the book is a section called Vocabulary, which gives the English translation of various German words, including Wien and Wiener, which, just in case you didn’t already know, translate respectively as “Vienna” and “Viennese.” I only recently thought of looking up eigen in the book.

Besides “characteristic” and “proper,” the physics and math books sometimes also translate eigen as “special.” But none of those are what A Brief Course in German says. There, the Vocabulary section says Eigen translates as “own.” That’s it, no foolin’ around with “characteristic” or “proper” or “special” by Hagboldt and Kauffmann.

 

Eigen as “Self”

And “own” itself has a near-synonym in English. Among the Math Stack Exchange answers to a question dated 11 February 2013 and titled “What exactly are eigen-things?” there was one answer I especially liked, written by Alex Chaffee. He (or she) says “eigen means proper only insofar as ‘proper’ means ‘for oneself,’ as in ‘proprietary’ or French propre. Mostly eigen means self-oriented.”

This reference to propre connects with what seems to be a mistranslation used in relativity, where we have something called “proper” time, which some writers on the subject say comes from the French word "propre." It does seem that proper time really should be called “own time,” because “own time” is indeed what you read on your own watch, which never moves relative to you and thus never changes its rate of ticking relative to you. This mistranslation of French may also be why eigen gets mistranslated as “proper” instead of “own.”

Before moving on to discussing eigenwaves (sorry about that) on a finite length of string, I’ll mention one other reference that says eigen refers to self. Last year in the Arkansas Democrat-Gazette, a columnist named Philip Martin wrote about his German grandmother and mentioned that one of the German phrases she sometimes used was eigenlob stinkt. A few readers of this blog, such as Tom Mellett, are no doubt aware of how this phrase translates, but for those who aren’t, here is the English translation:  self-praise stinks.

I encourage you to think in terms of “own” and “self” when you see eigen in the future. To help you with that, here’s the History section from the Wikipedia article on Eigenvalues and Eigenvectors (it helped me) . . . 

 

Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.

In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes. Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.

In the early 19th century, Augustin-Louis Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions. Cauchy also coined the term racine caractéristique (characteristic root), for what is now called eigenvalue; his term survives in characteristic equation.

Later, Joseph Fourier used the work of Lagrange and Pierre-Simon Laplace to solve the heat equation by separation of variables in his famous 1822 book Théorie analytique de la chaleur. Charles-François Sturm developed Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues. This was extended by Charles Hermite in 1855 to what are now called Hermitian matrices.

Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle, and Alfred Clebsch found the corresponding result for skew-symmetric matrices. Finally, Karl Weierstrass clarified an important aspect in the stability theory started by Laplace, by realizing that defective matrices can cause instability.

In the meantime, Joseph Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory. Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.

At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen, which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Hermann von Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.

The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Richard von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G. F. Francis and Vera Kublanovskaya in 1961.

[References aren't cited above, but they are in the Wikipedia article. Also, the applications section of the Wikipedia article on Eigenfunction looks at the 1D wave equation and the Schrö eqn.]

(September 2025: Sorry to say, a paywall is under construction at the Einstein Papers Project website, so the links below won't take you there anymore.)

In some cases, there might be confusion over whether "proper" in an English translation of "eigen" means proper in the relativistic sense or in the sense of the eigenvalues, etc. Here's a page from Einstein's 1909 paper "On the present status of the radiation problem," in English (Princeton Einstein Papers Project). In the first paragraph, you'll find the words " at the proper frequency." In the German version , not surprisingly, you'll find "bei der Eigenfrequenz." This usage is the one from relativity. If it hadn't been, the English translation of Eigenfrequenz would be eigenfrequency. Maybe! Translations depend on the translators' knowledge and preferences. 

On page 358 of the English translation of this paper, you'll find our well-known acquaintance, the wave equation (in 3-D), along with some functions in its superposition solution that have t - r/c and t + r/c in their arguments: the "retarded" and "advanced" potentials, respectively. This paper of Einstein's is important because he note's Planck's mistake of assuming equal (a priori) probabilities for Boltzmann's "complexions" and 

Pause.

15 January 2025: Incorrect statement there, sorry! The problem Einstein is pointing out is that in order to use The Big W as the number of complexions in the entropy formula S = k*log(W), the complexions must be equally weighted, i.e., each must "be equally probable on the basis of statistical considerations." This is something Planck didn't address, but he did accomplish it without being aware of it.  In following Boltzmann's statistical method as presented in Boltz's 1877 paper relating the entropy and the probability of all the "state  distributions" of imaginary discrete kinetic energy elements distributed among gas molecules, Planck, like Boltzmann, actually used Bose-Einstein statistics, long before its significance in physics was discovered. Einstein didn't realize that Plank's complexions are equally probable under the "statistical considerations" used by Boltz and Planck, which were (are) Bose-Einstein considerations.  A bit of irony in that, and also in Boltzmann being the physicist who first used B-E statistics but being known eponymously for Maxwell-Boltzmann statistics.

Unpause.

In the 1909 paper, Einstein finds (equation 36) a wave and a particle term in the mean square fluctuations of thermal (blackbody) radiation--early evidence for the wave-particle duality.

To be continued in Part 3b.