Albert Einstein finished writing his
revolutionary 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 120th 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 20th 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) = (8πυ2/c3)Uυ = (8πhυ3/c3) / [exp(hυ/kT) – 1],
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 Uυ is the average energy of a resonator.
The electromagnetic waves, which
Planck called a “stationary radiation field,” and the resonators together have
a total energy Et and are contained in “a diathermic medium
with perfectly reflecting walls.” You can think of this system as an insulated box
with air in it 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, he got the
classical equipartition 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 frequencies 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, 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, 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)
ρυ = (8πυ2/c3)Uυ
for radiation density ρυ as a function of the average energy Uυ of a single resonator? Planck used his quantum energy element idea to calculate this average energy, which he found to be
Uυ = hυ/[exp(hυ/kT) – 1],
but used
Maxwell-Lorentz wave-based oscillator theory to calculate the relation of
average oscillator energy to spectral radiation energy density. This 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,
but I can’t agree with that supposition, since any material is going to absorb
and emit thermal radiation (Kirchhoff’s radiation law). Also, a simple harmonic
oscillator, even an electrically charged one, isn’t a thermodynamic entity, but
Planck’s electrical resonators are considered to have randomized amplitudes and phases,
while maintaining a constant frequency. The disorder of the random phases and
amplitudes, says Planck, gives the resonators the thermodynamic property of
entropy. This thermalization of radiation question is related to my offhand parenthetic comment above on Rayleigh's use of energy equipartition in doing the thermal spectrum calculation in 1900.
