Einstein and the Quantum (13 page)

Moreover, it was an interesting and surprising prediction because it moved Planck's constant,
h
, from the arena of thermal radiation to the seemingly unrelated area of photoelectric phenomena. It strongly suggested that this constant was of general significance for physics. This is just the kind of thing experimental physicists love: an important new idea connected to a clear and easily falsified prediction. The statement that stopping voltage must have a linear dependence on the light beam's frequency, independent of its intensity, was bizarre enough. But also the graph of this line must have a universal, material-independent slope? It was too good to be true. Surely someone would jump on this.

Except … the photoelectric experiments required were very difficult, and Einstein, a virtual unknown who was contradicting the wave theory of light, had hardly more credibility than a crackpot, whose writings were to be thrown in the wastebasket. And unfortunately the existing data did not help very much. They did appear to show the strange dependence on frequency and independence of intensity that Einstein's picture explained, but they were nowhere near accurate enough to verify or contradict his universal-slope prediction. He was only able to make a rather hopeful statement: “
As far as I can see
, our conception does not conflict with the properties of the photoelectric effect observed by [the German physicist Philip Lenard].” Einstein would have to wait several more years for his quantum of fame.

 

¹
This relation is that the total radiation energy flowing out of the surface of a black-body is
J
=
σT
⁴
. It was derived by Boltzmann's teacher, Josef Stefan, and generalized by Boltzmann;
σ
is called the Stefan-Boltzmann constant and its value was known by 1905.

²
Paul Ehrenfest was an Austrian-born Jewish physicist whom Einstein would meet in Prague in 1912. As Einstein recalled, “
within a few hours
we were friends, as if made for one another by our strivings and longings.” Later that year Ehrenfest became a professor at Leiden and arranged regular visits there by Einstein for the next two decades.

³
In this context, high frequency means that energy element,
hν
, is larger than the classical thermal energy,
E
_mol
=
kT
, and low frequency refers to the opposite situation.

⁴
Later that year Einstein would generalize this rule via the famous equation
E
=
mc
²
. Sometimes energy can be transformed into mass and is “lost,” although in principle it can always be transformed back, so that mass-energy is still conserved. Such effects are typically negligible with absorption and emission of light, although essential in nuclear reactions.

⁵
While this is improbable for normal beams of light, with the particularly intense beams generated by modern lasers it becomes much more likely, leading to so-called nonlinear optical effects. An important new form of microscopy, known as two-photon microscopy, which gives superior imaging of living tissue, is based on this new possibility. Einstein is aware of and mentions the possibility of nonlinear processes involving several photons in his article but assumes, correctly, that they are rare.

⁶
None of the three were English: Thomson was born in Belfast in 1824 and Maxwell in Edinburgh in 1831. Both Stokes and Thomson lived into the twentieth century, whereas Maxwell, to the great detriment of science, passed away in 1879. Maxwell, now regarded as the greatest of the three, was the only one not to be knighted.

⁷
Lord Rayleigh, the successor to these three leaders, commented that Stokes may have suffered from a “
Morbid dread of mistakes
,” which inhibited publication.

CHAPTER 10

ENTERTAINING THE CONTRADICTION

“
I do not seek the meaning
of the quantum of action (light quantum) in the vacuum but at the sites of absorption and emission, and assume that processes in vacuum are described exactly by Maxwell's equations.” This was Max Planck's first known response to Einstein's heuristic theory of light quanta, sent to Einstein in a letter of July 6, 1907, more than two years after the publication of the “revolutionary” paper of 1905. Planck must surely have known of Einstein's ideas much earlier, since he was the theory editor of the journal in which they were published,
Annalen der Physik
. Unfortunately none of the referee or editorial comments on Einstein's papers of 1905 has survived, so we don't know how direct a role Planck had in approving them for publication. Planck was known for being open to publishing scientific contributions with which he disagreed, as long as they did not contain clear errors, and this tolerance likely came into play with Einstein's paper. Planck consistently opposed the idea of the particulate nature of light in vacuum, in a respectful but firm manner, for at least a decade after its publication. And he was not alone in this attitude; among the eminent physicists of the time, only Johannes Stark, whose outstanding work on the photoelectric effect Einstein had mentioned, openly supported Einstein's radical new view of light. In May of 1909, in response to a letter from Einstein, Lorentz himself sent a lengthy, technical reply to Einstein, detailing what he saw as the insuperable problems with his hypothesis. He concludes, “
It is a real pity
that the light quantum hypothesis encounters such serious difficulties
because otherwise the hypothesis is very pretty, and many of the applications that you and Stark have made of it are very enticing.”

What were the “serious difficulties” of which Lorentz spoke? They arose because of the conflict between two basic categories in the physics of the time: waves and particles. Newton's laws had introduced the idea of mass, the quality of a material that resists change of motion and that responds to and generates gravitational attraction. Although atomic theory was in its infancy, the idea that everyday massive objects were made of smaller, more fundamental building blocks (i.e., atoms and molecules) had been widely used by physicists since the time of Maxwell and was winning the day by 1905. Atoms were the fundamental particles of physics, although soon to be understood as made up of protons, neutrons, and electrons. (Modern physics has added many other particles to this classification and further subdivided the atomic nucleus into quarks). So the idea that macroscopic “stuff” (solids, liquids, and gases) is an aggregate of atomic-scale particles was commonplace at the time of Einstein's early work.

Moreover, it was clear that when these particles aggregate in large quantities, such as in an ocean or in the atmosphere, they create media in which disturbances can propagate, disturbances that in these cases are called water waves or sound waves. It is important to realize that the particles of water or air are the
substrate
in which the waves propagate; it not possible to generate such “classical” waves without their substrate (in space no one can hear you scream). In the classical view, the fundamental objects are the particles that make up the medium, and the waves are derivative objects. Moreover, it is critical to understand that these waves are collective phenomena; waves move through the medium, but the particles don't. Waves are not a whole bunch of particles moving in the same direction.

This is illustrated nicely by a new kind of wave, discovered sometime in the 1980s, which we will refer to as “fan waves.” In the classical fan wave the particles are the sports fans in a stadium, who, because of boredom or some other stimulus, spontaneously generate collective motion. In an ideal, fully developed, clockwise fan wave all the fans from the first to last row in the upper deck stand up and then sit down
in a brief period (about two seconds), causing, by poorly understood interactions, the fans immediately next to them on their left to do the same thing immediately thereafter. This disturbance in the crowd then propagates around the stadium, creating a nice visual effect, until it is damped out either by loss of synchronization or loss of interest. Anyone who has contributed to such a wave realizes that the particles (the fans) do not move in the direction of the wave; they just bob up and down. It is the disturbance, the wave, that propagates, not the “particles” of the medium.

To this extent fan waves are typical classical waves in a medium. To complete the analogy, however, we will have to embellish a bit on the conventional fan wave to allow for a further critical feature, interference of waves. Imagine that all the fans are standing already (it is a particularly exciting moment in the game) and can make two different kinds of waves, by either raising their arms above their heads or lowering them down to their knees, as in a revival meeting. Also allow for the possibility that waves can go either clockwise or counterclockwise. You look to your left or right, and if the fan next to you raises his hands, you do the same; if he lowers them you do the same as well. Now some wise guy starts a clockwise wave by raising his hands, and his friend behind him starts simultaneously a counterclockwise wave by lowering his hands. These two waves propagate around the stadium in opposite directions at the same speed, and so halfway around they meet. At the column where they meet, the fans on the right raise their hands just as the fans on the left lower theirs: the fans in the middle don't know what to do. So they do nothing. The two waves have met up, “out of phase” as the physicists say, and they cancel each other out.

This is a somewhat fanciful illustration of the interference of waves, which are extended disturbances in a medium, having both an amplitude (how big the wave is at any given point) and a phase (how close the wave is to a peak or a trough at any given point). Depending on their phases, waves interfere and are larger where the crests coincide and smaller (or zero) where a crest and peak coincide. This is the sine qua non of a wave. But note that when we have destructive interference and two waves cancel out, the particles of the medium are still there
(i.e., the fans in our example); they are just undisturbed. Waves are disturbances, so they can be positive or negative and can cancel each other out; you can add one and one and get zero. Particles cannot. (Two fans claiming a single seat
will
create destructive interference, but of a different kind.)

This was how all waves were conceived of until 1905. However, a major challenge to this understanding was implicit in Maxwell's discovery of electromagnetic waves. Here the propagating disturbance was an electric and magnetic field, but there was no obvious medium in which it could propagate. Physicists since the time of Newton had hypothesized that heat and possibly even light propagated through a transparent medium known as the “ether.” Maxwell's discovery now confirmed its existence and that it was the substrate through which all EM waves propagated.

The absolute necessity for such a medium was so evident that Heinrich Hertz, the first to demonstrate reception and transmission of radio waves, expressed it thus: “
Take electricity out
of our world and light vanishes; take the luminiferous ether out of our world and electric and magnetic fields can no longer travel through space.”

This medium was, however, highly problematic. Despite its stubborn invisibility, it had to be all-pervasive, since apparently EM waves could propagate everywhere. It couldn't have much (if any) mass, because it would then have gravitational effects, which were not in evidence. And since the earth moves in different directions at different times of year, the velocity of light on earth should vary in some manner, just as a water wave appears to move more slowly to a boat moving in the same direction. However, experiments testing the speed of light showed no hint of this effect.

But what choice did one have other than postulating an ether? Try creating a fan wave in an empty stadium. You don't have to be an Einstein to see that you can't have the wave without the medium. It turns out, however, that you
do
have to be Einstein to suggest that you
can
have the wave without the medium.

As noted above, the familiar waves of classical physics, which propagate as a disturbance in a medium, look different to an observer
moving with respect to that medium. The surfer on the crest of a water wave sees an almost stationary wall of water roiling around him. By the same logic the young Einstein, in his school days at Aarau, had imagined moving along next to a light wave at the speed
c
and seeing a stationary electric field that no longer oscillated. This apparently conceivable physical situation made no sense to him: “
But such a thing does not seem to exist
, either on the grounds of experience or according to the Maxwellian equations.” The leading theorists of the time, Lorentz and the French mathematical physicist Henri Poincaré, grappled with this conundrum and, while making major mathematical advances, kept the physical picture of electromagnetic waves tied to the ether. Einstein also pondered this puzzle on and off during his student years and afterward, and it was finally in May of 1905, two months after he had submitted his paper on light quanta, that the answer came to him: if time itself were not absolute but “flowed” differently for observers in uniform relative motion, then all the apparent contradictions could be resolved!

This was the key idea of Einstein's “rough draft” on the “electrodynamics of moving bodies which employs a modification of the theory of space and time” that he spoke of in his vivacious letter to Habicht in May 1905. Within two months this idea has been developed into his famous paper on what became known as the special theory of relativity. This work has received much attention in the literature, and we will not review it here except to quote part of one critical sentence: “
The introduction of a ‘light ether'
will prove superfluous, inasmuch as … no ‘space at absolute rest' endowed with special properties will be introduced.”