Einstein and the Quantum (12 page)

The source of the impossible answer was easy to trace. In a gas there is a very large but finite number of molecules; thus each molecule can have a fixed amount of energy,
kT
, and the total energy will be some large but finite quantity. In a box with trapped radiation, however, there are an
infinite
number of wavelengths of radiation that fit inside the box (remember that the wavelength of Maxwell radiation can be arbitrarily small). If each wavelength of radiation carries the same energy, then the total energy adds up to infinity, just as Einstein found. Is it possible that only some wavelengths actually get their share of the energy, leaving others below their thermal quota? No, it is not possible. Since the radiation and electrons are always interacting, if some wavelengths didn't get their “fair share” of the energy, then radiation would continually suck energy out of matter until all the matter cooled to absolute zero. Much later the physicist Paul Ehrenfest
²
came up with a catchy name for this phenomenon:
the ultraviolet catastrophe
(almost all the energy should flow into the shorter, ultraviolet wavelengths). Not to worry, it doesn't happen.

This is the brick wall that Einstein had run into in trying to explain blackbody radiation from statistical mechanics. The only answer you can get from Newtonian mechanics and Maxwell/Boltzmann theory is not just wrong (i.e., out of agreement with experiment); it is absurd. This was why Planck had to twist himself into knots and introduce “energy elements” to get his answer. Einstein had major reservations about this gimmick, and had devised a means to make his conjectures about quanta of light without committing himself to whether there was any truth in Planck's notion.

Despite his reservations about Planck's method for deriving his radiation law, Einstein treats Planck quite delicately in his paper. One might have expected that the angry young maverick who had confronted Drude and Boltzmann about their supposed errors would have liked nothing more than to emphasize the shortcomings of Planck's approach in print. Planck had tiptoed to the very edge of the ultraviolet catastrophe when he noted that his smallest energy constant,
h
, could not be taken arbitrarily small (which would have restored the continuous nature of energy). If it were so taken, one would immediately encounter Einstein's absurd result for the total radiation energy; however, Planck had not seen fit to mention the menace he had narrowly escaped. Einstein had discovered that the kaiser Planck had no clothes, and who better than him to tell the world?

However, a cryptic comment by Besso much later, in 1928, sheds light on his change of tone: “
On my side, I have been
your public during the years 1904, 1905; by helping you to edit your communications on the problem of quanta, I deprived you of some of your glory; but on the other hand, I procured you a friend, Planck.” It appears that Besso prevailed on Einstein to revise an earlier version of this paper in which more pointed comments were made about the correctness of Planck's derivation. Besso is on target: he probably did deprive Einstein of some of his glory; had Einstein been more direct in discussing Planck's work, his own role as the first person to propose seriously the quantization of energy, as a property of atomic mechanics, would have been much clearer.

At any rate, in the published version of the paper Einstein confines himself to the innocuous comment, “
Planck's formula for
ρ
(
ν
)
… has
been sufficient to account for all observations made so far.” Einstein is not trying to determine the correct radiation law here anyway, but instead his goal is interpreting the
meaning
of this law, which apparently
is
correct but, as he has just shown, conflicts with atomistic theory as then conceived. To do this he finds it sufficient to consider only the shorter wavelengths of thermal radiation, and to use for convenience the older Wien law, which fails for the longer wavelengths but actually fits the data very well in the short-wavelength, high-frequency range.
³
Assuming that Wien's law is
approximately
correct, he can work backward à la Planck and find the approximate entropy of blackbody radiation. He finds something very striking. The mathematical equation for this entropy is identical to that of a molecular gas, except that where the number,
N
, of gas molecules would appear in the formula for the molecular entropy, the expression
E
/
hν
appears for the blackbody entropy (where
E
is the total energy of the radiation of frequency
ν
). Einstein makes the immediate connection: if light consists of a train of particles (“quanta”) each of energy
hν
, then
E
/
hν
is the number of those quanta, and the blackbody entropy behaves exactly like a gas of independent molecules. In other words, the short-wavelength limit of the blackbody law suggests that light has particulate properties!

Of course this is not a proof. The only laws physicists believed at the time give the absurd answer for the total radiation energy (infinity) that he has already stated at the beginning of the paper. Thus Einstein doesn't know any more fundamental way of justifying his light-quanta hypothesis than through the mathematical analogy he has just given. However, this picture, even in the absence of the new and more accurate set of atomic laws that must underlie it, is enough to explain many puzzling experimental observations.

Here is how he was able to do this. First, he assumed (correctly) that whatever the new laws were, energy is still conserved. Energy conservation means that whenever some energy is given up by something (e.g.,
a molecule or a light wave), that energy is transferred to some other physical object or process so that energy is not created or destroyed, just redistributed.
⁴
His new picture implied that light transfers energy to matter in a different way than in the Maxwell wave picture. In classical physics the energy in a wave is proportional to its intensity, which is proportional to the square of the maximum height of the wave. This makes a lot of sense intuitively. Waves are disturbances in a medium that transport energy from one region in the medium to another. For example, water waves bring energy into the shore from winds in the oceans. We all can see that a gnarly thirty-footer brings in more energy than a five-foot swell; we instinctively measure the power of waves by their height. This is exactly how the energy of a light wave is measured in the classical wave theory of electromagnetism, their “height” being the strength of their electric field.

Einstein was proposing a radical reimagining of this on the atomic level. He proposed that the light wave really is a train of particles (of indefinite but presumably very small size), the quanta, which, because of their localized nature, can interact and exchange energy only with individual atoms or molecules. The intensity or height of the light wave tells one
how many
quanta the wave contains in a certain region of space, but it doesn't determine the energy of each quantum of light. Instead, the energy of the quanta depends only on the frequency (or wavelength) of the wave according to the relation
ε
_quant
=
hν
. This is just the Planck relation, but not for the energy of the molecular vibrations, rather for the quanta of light (photons).

Notice how strange this conclusion is from the point of view of the traditional wave picture. The frequency of a water wave just determines its wavelength (the spacing of the crests), which doesn't affect how much energy is transferred each time a crest crashes on the beach. Einstein agreed that higher-intensity light waves carry more
total
energy (because they consist of more quanta), but carrying more quanta matters little for how they interact with a
single
molecule or atom. That is because it is very improbable that two quanta will “meet” at the same point at the same time and thus be able to transfer twice as much energy to the same molecule.
⁵
So, in his new picture, what can happen in such a transfer is controlled by the frequency of the light and not its intensity.

Suddenly very puzzling observations made perfect sense. Einstein's first example was a phenomenon involving the absorption of light known as the Stokes rule. George Gabriel Stokes, born in the Irish county of Sligo in 1819, was the first of the three great Cambridge mathematical physicists
⁶
of the nineteenth century, along with Maxwell and William Thomson (later named Lord Kelvin). One of his most famous discoveries was that certain substances, when they absorbed light would “fluoresce”; that is, they would reemit the light, not as blackbody radiation, for which the wavelength is determined by temperature, but rather as visible light. However, Stokes noted that the reemitted light was always of a lower frequency (longer wavelength) than the absorbed light: this is the Stokes rule. Most dramatically, many substances could absorb ultraviolet (invisible) incident radiation and shift it down to lower frequencies in the visible range (this effect is now used as a key diagnostic probe in modern chemistry and biochemistry). Ironically, Stokes never published his famous rule;
⁷
instead Thomson announced it in 1883 but gave Stokes full credit for the discovery.

This rule makes no sense from the classical wave point of view. Since the energy in a classical light wave has nothing to do with its frequency, why can't a conversion of light to higher frequency sometimes happen? After all, the frequencies at which molecules emit light presumably have something to do with how tightly the atoms are bonded together, how the molecules vibrate, and so on. If the light wave is dumping energy in and it is being reemitted as a new light wave, then shouldn't it come out at whatever the natural frequencies of emission are for that molecule, independent of whatever frequency is used to dump the energy in? (Perhaps this obvious conundrum deterred Stokes from publishing, since there was no reasonable explanation for his rule.) Einstein's quantum picture handles this with ease.

If the best a molecule can do is absorb all the energy in
one
quantum of light, then the most energetic quantum it can reemit will have at most the same energy, and hence the same frequency; however, absorption almost always is accompanied by some amount of energy lost to the molecule (a sort of molecular friction), so in fact the highest energy and hence the highest frequency that can be reemitted is always lower. Einstein realizes that this conclusion is quite general: “
it makes no difference
by what kind of intermediate process this end result [the emission of a photon] is mediated. If the photoluminescent substance is not to be regarded as a permanent source of energy, then, according to [conservation of energy], the energy of the produced energy quantum cannot be greater than that of the producing energy quantum, hence we must have [the produced frequency less than or equal to the producing one]. This is the well-known Stokes rule.”

This was a very nice first flexing of Einstein's quantum muscles, but it was still just a qualitative explanation for something that was already known. His idea was so radical that it would take something more to get people to take it even a little bit seriously. Fortunately, he comes up with a more dramatic example and quantitative prediction in the next section of the paper. The great German physicist Heinrich Hertz, the first demonstrator of Maxwell radiation, had also discovered by
accident a phenomenon that would ultimately spell the downfall of pure Maxwellian electrodynamics. This phenomenon is called the “photoelectric effect.”

When light, particularly blue or ultraviolet light, is incident on and absorbed by a substance, yet another process can occur (besides the emission of blackbody radiation or the fluorescence of lower-frequency Stokes radiation). Sometimes the substance can eject an energetic “cathode ray,” which by that time had been identified as a speeding electron. These fast electrons can be collected and run through a circuit to generate a “photocurrent.” The same principle is operative in all modern solar cells, except that the electrons are not ejected from the material but are just promoted to a higher energy state, where they move much more freely and can be extracted as electrical energy in a circuit. By exactly the same kind of reasoning he had applied to the Stokes rule, Einstein explained the photoelectric effect. He analyzed the consequences of conservation of energy, combined with the principle that only
one
quantum of light can interact with
one
electron at a time.

First, it always takes some energy to knock electrons out of a solid. The electrons are trapped by their attraction to positively charged atoms and also by the surfaces of the material, so they need to absorb extra energy to be kicked out into space. This energy is pretty large on the atomic scale, comparable to the amount of energy in one quantum of blue or even ultraviolet light. So if one bathes the material in a beam of red light (no matter how intense), since each quantum of red light is individually too feeble,
no
photoelectrons are produced. Again, this is totally baffling from the classical point of view. From that perspective, the more intense the red beam is, the more energy it has, and the more likely it should be to produce photoelectrons. But increasing the red beam's intensity does nothing. In contrast, a rather weak beam of ultraviolet light does produce photoelectrons. A paradox for Maxwell, but not in the quantum picture—one solitary ultraviolet photon, with its higher frequency, has enough energy to do the job. So this observation is immediately explained by Einstein's light quanta.

But he easily goes further. The ejected photoelectrons are winging their way toward the collector, which is a metal contact absorbing
electrons and delivering them to the attached circuit. If this collector is charged with a large enough negative voltage, the electrons are repelled and the photocurrent ceases. A simple further application of conservation of energy allows Einstein to predict that this “stopping voltage” is precisely proportional to the frequency of the light. In other words, if you make a graph of stopping voltage versus frequency, it is a perfectly straight line with a specific value for its slope. And what is that value? It is exactly Planck's constant,
h
, for any material! Now that was a strong and precise prediction.