MARGINALIA
Storied Theory
Science and stories are not only compatible, they're inseparable, as shown by Einstein's classic 1905 paper on the photoelectric effect
Roald Hoffmann
Science seems to be afraid of storytelling, perhaps because it
associates narrative with long, untestable yarns. Stories are
perceived as "just" literature. Worse, stories are not
reducible to mathematics, so they are unlikely to impress our peers.
This fear is misplaced for two reasons. First, in paradigmatic
science, hypotheses have to be crafted. What are alternative
hypotheses but competing narratives? Invent them as fancifully as
you can. Sure, they ought to avoid explicit violations of reality
(such as light acting like a particle when everyone knows it's a
wave?), but censor those stories lightly. There is time for
experiment—by you or others—to discover which story
holds up better.
The second reason not to fear a story is that human beings do
science. A person must decide what molecule is made, what instrument
built to measure what property. Yes, there are facts to begin with,
facts to build on. But facts are mute. They generate neither the
desire to understand, nor appeals for the patronage that science
requires, nor the judgment to do A instead of B, nor the will to
overcome a seemingly insuperable failure. Actions, small or large,
are taken at a certain time by human beings—who are living out
a story.
Better Theory Through Stories
One might think that experiments are more sympathetic than theories
to storytelling, because an experiment has a natural chronology and
an overcoming of obstacles (see my article, "Narrative,"
in the July-August 2000 American Scientist). However, I
think that narrative is indivisibly fused with the theoretical
enterprise, for several reasons.
One, scientific theories are inherently explanatory. In mathematics
it's fine to trace the consequences of changing assumptions just for
the fun of it. In physics or chemistry, by contrast, one often
constructs a theoretical framework to explain a strange experimental
finding. In the act of explaining something, we shape a story. So C
exists because A leads to B leads to C—and not D.
Two, theory is inventive. This statement is certainly true for
chemistry, which today is more about synthesis than analysis and
more about creation than discovery. As Anne Poduska, a graduate
student in my group, pointed out to me, "theory has a greater
opportunity to be fanciful, because you can make up molecules that
don't (yet) exist."
Three, theory often provides a single account of how the world
works—which is what a story is. In general, theoretical papers
do not lay out several hypotheses. They take one and, using a set of
mathematical mappings and proof techniques, trace out the
consequences. Theories are world-making.
Finally, comparing theory with experiment provides a natural ending.
There is a beginning to any theory—some facts, some
hypotheses. After setting the stage, developing the readers'
interest, engaging them in the fundamental conflict, there is the
moment of (often experimental) truth: Will it work? And if that test
of truth is not at hand, perhaps the future holds it.
The theorist who restates a problem without touching on an
experimental result of some consequence, or who throws out too many
unverifiable predictions, will lose credibility and, like a
long-winded raconteur, the attention of his or her audience. Coming
back to real ground after soaring on mathematical wings gives theory
a narrative flow.
Let me analyze a theoretical paper to show how this storytelling
imperative works. Not just any paper, but a classic appropriate to
the centennial of Albert Einstein's great 1905 papers.
The Puzzle of Dwarvish Work
Einstein's paper on the photoelectric effect, published that fecund
year, was singled out by the 1921 Nobel Committee (late as usual,
and perhaps still afraid of relativity) as the basis for their
award. It is also the only one of the 1905 papers that Einstein
himself deemed revolutionary. But when one reads the article, the
photoelectric effect appears late, as a denouement; the paper begins elsewhere.
The unwritten prologue is the contemporary interest in black-body
radiation—the tendency of any object, no matter what its
composition, to radiate light when it is heated. We see it in iron
nestled in the forge, glowing red, then yellow, then white.
The intensity of this emitted light varies with the color
(wavelength). At low temperatures, bodies radiate in the infrared.
As the temperature rises, the maximum intensity of the radiated
light moves into the red, then extends through the spectrum to the
ultraviolet. At high temperatures, objects radiate intense light
across the visible spectrum—that's white heat. The intensity
of radiated light diminishes in the extreme ultraviolet and far
infrared (see right). Astronomers estimate the temperatures
of stars from just such curves.
The standard (and eminently successful) understanding of light in
Einstein's day came from James Maxwell's electromagnetic theory.
Coupled with thermodynamics and the kinetic theory of gases—a
high expression of Newtonian mechanics—electromagnetic theory
led to a "radiation law" that described how the intensity
of light varied with wavelength at each temperature. The law fit the
data—at long wavelengths. At short wavelengths, the equation
derived from electromagnetic theory failed, in what became known as
"the ultraviolet catastrophe."
In 1900, Max Planck found an expression that fit over the entire
range of observations. Planck further perceived that his accurate
radiation law could be obtained only if the energies of the little
bits of oscillating charge that caused the light (he called them
"resonators") assumed discontinuous values. So the quantum
was born.
Planck had trouble believing that physics was, deep down,
discontinuous. He spent many years searching for a way around what
he discovered. But that is another story.
How Einstein Tells It
The photoelectric paper is modestly entitled, "On a Heuristic
Point of View Concerning the Production and Transformation of
Light." Einstein begins by stating the problem posed by the
quantum hypothesis: He defines the resonators as bound electrons and
takes us, with characteristic clarity, made possible by five years
of experience with quanta, through Planck's derivation. He develops
the characters in his tale—the radiation, Planck, his
resonators, classical electromagnetic theory.
Then Einstein does something new. He sets out to derive Planck's
radiation law without any assumptions about how light is generated.
How does he do that? By assigning an entropy (the measure of
randomness, a concept already in wide use by then) to the light and
relating that entropy to the density of the radiation. Einstein
proves that the entropy of the light in the black body varies with
volume just the way that entropy varies with volume for that standby
of freshman chemistry, the ideal gas.
This demonstration is direct. It's not Hemingway, but for scientific
prose, really exciting. Einstein is taking us somewhere—we
don't know where yet, but by the way he sets the scene, by his pace
and conviction, we know something is going to happen.
Pretty incredible. No resonators, just a functional analogy of atoms
or molecules to light. Playing out the analogy, light of a given
wavelength could be described as if its energy came in dollops
of what Einstein called Rβv/N, and today we
would call hv, a constant (h) times the light's
frequency (v). But that's just a way of looking at
things—it's not for nothing that Einstein put the word
heuristic in the title. Or is it? When do stories become real?
Back to the paper: Einstein has just rederived Planck's radiation
law without resonators. Yet the discreteness of the light's energy,
its quantization, is newly manifest in Einstein's work. There is no
mistaking it. From this climax the paper cruises along another
plateau, then swoops into a breathtaking shift of scene. Philipp
Lenard had three years earlier observed "cathode rays," or
beams of electrons, by shining light onto a metal. The phenomenon
happened only when the frequency of that light exceeded a certain
minimum; below that frequency (or above that
wavelength)—nothing. After seeing the electrons, Lenard
observed that their kinetic energy depended on the color of the
light, their number on the intensity of the light.
This phenomenon we now call the photoelectric effect. Aside from
being today a primary source of information on molecules and
surfaces, the effect is behind photoelectric cells opening elevator
doors, and is used in solar cells and light-sensitive diodes.
Back to 1905. Einstein just says: Let's assume light is quantized in
units of hv, and that a "light quantum" (we would
call it a photon today) gives up all its energy to a single
electron. The electron needs a certain energy to leave the surface;
if it has some left over, the extra contributes to its motion.
Einstein calculates, in a couple of terse sentences, the energies
involved and finds reasonable agreement with Lenard's measurements.
With this and another calculation on the ionization of gases, he
brings us down to experimental reality.
Except reality is not down, it is evidence. Evidence that this story
of light being quantized is not just any story. This one is worth
telling to our great-grandchildren.
Einstein's theory leaves us soaring, thinking what else this
strange, discontinuous view of light might explain. Soon Bohr will
use it to give us the first theory of an atom. This story is as
exciting as Thomas Mann's 1902 Buddenbrooks, which Einstein
might have been reading at the time.
The photoelectric paper was submitted to Annalen der Physik
(Annals of Physics) in March 1905. But Planck's quantum theory, and
the nature of light, had been on Einstein's mind for quite a while.
On April 30, 1901 he wrote to his future wife, Mileva Maric, "I
came recently on the idea that when light is generated, perhaps
there occurs a direct conversion of kinetic energy to light. Because
of the parallelism: motional energy of the molecules—absolute
temperature—spectrum (energy of radiation in equilibrium). Who
knows when a tunnel will be dug through these hard mountains!"
The Story Is in the Theory
All theories tell a story. They have a beginning, in which people
and ideas, models, molecules and governing equations take the stage.
Their roles are defined; there is a puzzle to solve. Einstein sets
his characters into motion so ingeniously, using entropy to tease
out the parallels between moving molecules and the energy of light.
The story develops; there are consequences of Einstein's approach.
And at the end, his view of light as quantized and particular
confronts the reality of the heretofore unexplained photoelectric
effect. The postscripted future, of all else that can be understood
and all new things that can be made, is implicit.
Perceptive reader Anne Poduska notes that the photoelectric paper
"is particularly interesting because of the layering of
perspectives (similar to legends being passed from one generation to
the next, with each storyteller adding their own
flair/details)." Indeed, Einstein uses Planck's development of
the radiation law even as the younger physicist claims he will do it
differently. He parlays belief in the discreteness of molecules
(some of his contemporaries still doubted their existence) into an
argument, first cautious, then growing in strength, of the
discreteness of light.
A young man of 25, Einstein had mastered the old stories. In this
paper he combined the ways others looked at the world, and trusting
analogy as much as mathematics, made something new. Science is an
inspired account of the struggle by human beings to understand the
world. Changing it in the process. How could this be anything but a story?
Acknowledgment
Thanks to Anne Poduska for her careful reading and suggestions.
© Roald Hoffmann