Einstein and the Quantum (31 page)

Einstein had realized that making sense of the behavior of light requires that that we describe radiation by an extended field or disturbance, which would
determine
the measurement of energy transfer at specific points in space, while at the same time the energy transfer would itself be composed of localized, and quantized, units. He came up with the picturesque name of “ghost fields” (
Gespensterfeld
) for his guiding disturbance, and it seemed that he was inclined to regard the particulate quanta as the “real” things while the extended, wavelike entity was relegated to a secondary, spectral existence. Nonetheless he took the idea quite seriously, and from 1918 on discussed it widely, so that not just Einstein's inner circle of confidants knew of it but also young students, such as Eugene Wigner,
¹
who recalled that Einstein was “
quite fond of it
.” Moreover, according to Lorentz, Einstein also foresaw that the ghost field would be used to determine probabilities for the light quanta. Continuing his description of Einstein's idea, he states: “
It has to be assumed
that at each reflection and diffraction, whenever an incident light beam is split into two or more beams, the
probability
that a light quantum takes one or the other path is proportional to the intensities of the motions of light along these various paths,
calculated according to the classical laws
” (italics added). “Classical laws” here refers to Maxwell's equations, which determine the reflection and diffraction of electromagnetic waves, and hence determine the probability that the quanta will go one way or the other. For individual photons, it seems, there
are
no deterministic laws of motion; only the properties of the guiding field are set by deterministic laws. These two ideas—that quantum particles are guided by an extended field, which allows them to “interfere with themselves,” and that this extended field obeys definite laws but doesn't determine the fate of individual quanta—are key concepts in the modern quantum theory of radiation (and of matter). Usually they are attributed to Max Born,
who applied them first to electrons, but he always credited Einstein as their source, even on the Nobel podium in Stockholm.

In 1917, at the same time as Einstein was beginning to develop his concept of “ghost fields” for quanta of light and trying not to give up the ghost himself from his various ailments, he somehow mustered the energy to look at the other side of the quantum dilemma: the quantum theory of atoms and electrons. Einstein had begun his work on quantum theory treating light and thermal radiation, but by 1907 had realized that Planck's law required that atoms and molecules must also obey a non-Newtonian mechanics, since their vibrational energy was necessarily quantized. At that time he chose to pursue a revised theory of radiation that would contain quanta of light, only to set the project aside in frustration in 1911. Instead it was Bohr who took the next step in atomic mechanics, in 1913, with his quantized electron orbits, followed closely by the elaborations of his theory due to Arnold Sommerfeld in 1915 that so impressed Einstein. Until the spring of 1917, in twelve years of work, Einstein had never proposed an equation to describe the quantum behavior of electrons, the key to understanding the atom. Rather remarkably, he took up exactly this challenge during his convalescence in March and April of 1917. His work would uncover a surprising deficit in the Bohr-Sommerfeld version of quantum theory.

The early version of the quantum theory of matter, which Bohr and Sommerfeld pioneered (and which is now known as the “old quantum theory”), had overcome the first major hurdle in generalizing the Planck-Einstein restriction that energy is quantized on the molecular scale. As we saw earlier, Planck and Einstein had imposed this restriction only on the simplest type of periodic motion, linear harmonic oscillation, whereas Bohr had generalized it through rather laborious arguments to the periodic motion of an electron “orbiting” the nucleus of the atom. Sommerfeld had realized that Bohr's prescription amounted to a more general rule than Planck's
ε
=
nhυ
, but that the quantity actually quantized, both in electron orbits and in molecular vibrations, was the
action
.

Yes,
that
“action.” Recall how Herr Planck always referred to his constant,
h
, as the “quantum of action.” “Action” is a technical term in
classical mechanics; it refers to a mathematical quantity that was introduced in the nineteenth century as a convenient and powerful way to think about any particle trajectory that obeys Newton's laws. The “action of a trajectory” is a number that physicists can calculate, but its meaning is not as intuitive and familiar as the concepts of mass, momentum, and energy introduced by Newton. Roughly it corresponds to the minimum amount of
time
that a particle can take in going from point A to point B, given the forces it experiences. Somewhat miraculously, the
actual
trajectory a Newtonian particle follows according to
F
=
ma
manages to minimize this global measure, as if the particle had planned its path in advance. You can actually throw away Newton's laws and do all of classical mechanics just based on this “principle of least action.”

And just like other quantities in classical mechanics, such as energy, action is also a continuously varying quantity, so there was no obvious way to break it up into smallest units. No way, that is, until the discovery of Planck's constant. Planck's constant is the first known fundamental physical constant that has the same units as action, the units of length times momentum. If I want to cut a rope into one-, two-, and three-foot sections, I need a ruler to measure this out; I could use this ruler to “quantize” the lengths of rope. Similarly, Sommerfeld had realized that Planck's constant was the natural “ruler” for quantizing periodic trajectories; the action along each trajectory had to be a whole number times
h
. You could throw out all the classically allowed electron orbits in the atom that didn't have action divisible by
h
with no remainder. For some reason, they are not allowed. This rule didn't explain
why
microscopic motion was quantized in a more fundamental sense than Planck's original hypothesis, but it allowed the quantum theory to describe a much wider range of motions than just simple harmonic oscillation. In particular Sommerfeld had been able to add elliptical electron orbits to Bohr's simple circular orbits, and even include the effect of relativity on the orbital energy as the electron whizzed by the nucleus. In fact this approach would occupy the community for the next seven years as more and more elaborate models were made for quantized orbits in the atom.

Einstein was intrigued enough by the elegance of Sommerfeld's method that immediately after finishing his new radiation theory, he turned his attention to it. But soon he would realize that there was a problem with this new theory of quantized actions, one that only he recognized, and which he set out to address while he was still barely off life support in early spring 1917.

The first clear indication of his new perspective comes in a letter to Besso on April 29, 1917: “
yesterday I presented a little thing
on the Sommerfeld-Epstein
²
formulation of quantum theory before the thinned ranks of our Phys. Society. I want to write it up in the next few days.” By May 11 he was ready to present the finished product to the German Physical Society, a paper titled “On the Quantum Theorem of Sommerfeld and Epstein.”

Einstein was motivated to write this paper by a core principle of his relativity theory: the laws of physics should not depend on the reference system of the observer or, in the case of the atom, on the choice of a coordinate system for describing electron orbits. He begins his paper by praising the Sommerfeld approach but then says “
it still remains unsatisfying
that one has to depend on [a specific choice of the coordinate system], because this probably has nothing to do with the quantum problem,
per se
.”

Einstein was searching for a more general statement of the principle of quantizing action. And, armed with a much more advanced set of mathematical tools than he possessed before his development of general relativity theory, he found one: topological invariance. Topology is the branch of mathematics that studies the properties of objects that remain unchanged when the objects can be deformed continuously to another shape, but without breaking anything, so that the object retains the same number of “holes.” In topology a doughnut and a coffee cup are considered to be the same shape, because the handle
of the cup corresponds to the hole in the doughnut and you could imagine taking the rest of the cup (if it were malleable) and squeezing it smoothly around the handle without ever breaking off a piece until it molded into a doughnut shape. In modern physics topological invariants have been found to play critical roles in many fundamental theories, but none had been found prior to Einstein's 1917 work. In a tour de force, Einstein, was able to show in a few short pages that the Bohr-Sommerfeld rule can be written in a manner that depends only on topological properties of the electron trajectories, essentially on how these orbits “wrap back” on themselves after each period of the motion. Since it is a topological property, it doesn't change if you rotate or deform the system, and hence it doesn't depend on any particular coordinate system.

So it looks like Einstein is satisfied that he has firmed up the Bohr-Sommerfeld theory with a little more rigorous mathematics and that is going to be the end of it. But there is more to the story. In a formulation that surely sounded as odd then as it does today, Einstein continues, “
we now come to a very essential point
which I carefully avoided mentioning
during the … sketch of the basic idea.” Actually, he points out, if the electron orbits are sufficiently complicated or irregular, so that there
is
no single period in which the trajectory wraps back on itself, then his method goes out the window, along with the whole approach of Bohr and Sommerfeld. Such complicated, irregular motion is now familiar to modern physicists, where it is described by the term
dynamical chaos
.

“
One notices immediately
,” Einstein concludes, “that [motion of this type] excludes the quantum condition we have formulated.” So there is a looming problem for the Bohr-Sommerfeld approach, which has baffled Einstein himself, the problem of quantizing chaotic motion.

Einstein's work was so advanced that few people understood it at the time; chaotic motion was not really appreciated until the advent of modern computers, and topological quantum rules were beyond everyone's ken. Despite the fame of its author, the problem identified in his article was ignored for more than fifty years, before a subfield of
theoretical physics emerged known as “quantum chaos theory.”
³
But the paper he had written was not unimportant at the time; in 1926, when Erwin Schrödinger wrote his epochal series of papers defining the wave equation of quantum mechanics, he included the following footnote: “
The framing of the quantum conditions
[in Einstein's 1917 paper] is the most akin, out of all the earlier attempts, to the present one.”

 

¹
Wigner arrived in Berlin in 1921 at the age of nineteen and attended the famous physics colloquia of Berlin University. He witnessed firsthand the completion of quantum theory and would go on to win the Nobel Prize in 1963 for seminal applications of the theory, particularly for his identification and use of symmetry principles.

²
Paul Epstein was a young Jewish theoretical physicist from Poland who did his PhD with Sommerfeld and contributed to some further details of the Sommerfeld formulation. Einstein was rather impressed by him and helped him when he had difficulty finding a permanent job; he eventually immigrated to the United States with the help of Millikan and became a professor at Caltech.

³
It was Einstein's 1917 paper that inspired the author to look again at Einstein's contributions to quantum theory, as described in the introduction (see A. Douglas Stone, “Einstein's Unknown Insight and the Problem of Quantizing Chaos,”
Physics Today
, August 2005, pp. 37–43).

CHAPTER 23

FIFTEEN MILLION MINUTES OF FAME

But now I'm just about fed up
to the teeth with relativity! Even such a thing pales when one is too occupied with it.

—ALBERT EINSTEIN TO ELSA EINSTEIN, JAN. 8, 1921

While Einstein struggled to regain his vitality and refocus his efforts on understanding the atom during the years 1917 and 1918, a singular confluence of social forces was occurring that would change his life irrevocably. On November 9, 1918, the German Reich surrendered, the kaiser abdicated, and Germany was thrown into political turmoil. The Great War would leave a residue of hatred, resentment, and disillusionment, which would affect international scientific cooperation for a decade. Yet within less than a year a British scientific expedition would catapult Einstein to a level of global fame unprecedented in the history of science. Arthur Stanley Eddington, the young Plumian Professor of Astronomy at Cambridge, led the expedition; he had been a conscientious objector during the war and was a person of undiplomatic stubbornness in whom Einstein would have recognized a kindred soul. When the Cambridge dons obtained for him a deferment from military service, on the grounds of his superior value as a scientist, he insisted on signing it with the stipulation that, had he been deemed of greater military value, he would have refused service anyway. Eddington had championed Einstein's theory of general relativity in England and, through the offices of the astronomer
royal, Sir Frank Dyson, had been put in charge of the eclipse-observing expeditions that departed in February of 1919 to test Einstein's quantitative predictions of the bending of the path of starlight due to the gravitational field of the sun. With some eventfulness, in the end these expeditions confirmed Einstein's theory and reported their results at the historic meeting of the Royal Society on November 6, 1919.