Einstein and the Quantum (29 page)

Following Bohr, Einstein can now be more precise about the nature of this dynamical equilibrium state. He assumes the matter in contact with radiation is a gas of molecules with discrete, quantized energy levels, Bohr's stationary states. His argument is, however, so general that he never needs to assign specific values to the allowed energies: it is enough that they exist. Thus his reasoning becomes independent of the details of Bohr's method for calculating atomic energy levels. Energy is then exchanged between the molecules and the radiation field according to the Bohr prescription. If a molecule has any energy greater than its minimum energy state (called the “ground state”), then it can either emit radiation of the appropriate energy,
hυ
₁
, so as to drop down to a lower
energy state, or absorb radiation energy of another specific frequency,
hυ
₂
, and jump up to a higher energy state. In order for equilibrium to be maintained, these processes of emission and absorption of energy from the radiation must balance out on average; the same amount of energy must flow into the molecule as out. In fact, Einstein points out, not only must the energy flow balance out overall; the exchange of energy must balance out independently for
each pair
of molecular levels. For each such pair “upward transitions” (absorption) must equal “downward transitions” (emission) or the system could not maintain its equilibrium.

So far so good; the logic and math are so simple he can hardly have made a mistake, but he also can hardly have gained much deductive power. Then, a crucial insight: “
We shall distinguish here
two types of transitions.” When the molecule is in any of its higher-energy, “excited” states,
even if there is no radiation at all present
, he assumes that there is still some chance that it will emit radiation “without external influence. One can hardly imagine it to be other than similar to radioactive reactions.” What a leap! Radioactive decay was a mysterious
nuclear
phenomenon that appeared to be completely random. A radioactive substance has a certain half-life, that is, the period of time during which on average half of the nuclei emit radioactive particles. But for any specific nucleus all one could say was that the probability was one-half that it would not decay in one half-life and one-fourth that it wouldn't decay in two, etcetera. The actual time and direction of decay was (and is still) unpredictable. Now Einstein is claiming that at least
some
of the events in which atoms emit light are just like this, are what we now call “spontaneous emission” events, and he writes down exactly the same mathematical rule for the number of spontaneous emission events per unit time as for radioactive decay.

He then considers other emission events of a more conventional sort, which we now call “stimulated emission”; the number of
these
events depends on the amount of radiation that is already present, which means that in equilibrium this number will be proportional to the blackbody radiation density. These events would have been more familiar to his readers because in classical electrodynamics one pictures the radiation field as “driving” or “being driven by” the electron charges in the atom, increasing or decreasing the amount of energy
contained in the electron's orbital motion. When the radiation field subtracts energy from the electrons, we have “stimulated emission”; when it adds energy to the electrons, we have absorption. Whew, at least some of this sounds familiar, but … he completely throws out the classical method of calculating absorption and emission.
Instead he treats
these
processes as random too
.

Now all he has to do is balance out the various processes; the energy lost by the molecules in spontaneous and stimulated emission must on average equal that gained by absorption. By one last sleight of hand he relates the rate of stimulated emission to that of absorption, simplifying the equation of balance. Two more lines of algebra and,
Mein Gott
, he has derived Planck's radiation law! Einstein has ingeniously bypassed all the complicated counting and gone straight to the answer. He can't resist giving himself a little pat on the back: “
The simplicity of the hypotheses
, the generality with which the analysis can be carried out so effortlessly, and the natural connection to Planck's linear oscillator … seem to make it highly probable that these are the basic traits of a future theoretical representation.” In his jubilant letter to Besso a few weeks later he is even more effusive: “
A brilliant idea has dawned
on me about radiation absorption and emission…. An astonishingly simple derivation, I should say
the
derivation of Planck's formula. A thoroughly quantized affair.”

Einstein was correct; his approach has become
the
derivation of Planck's formula. It is completely valid within the modern theory of quantum mechanics and electrodynamics and is in fact the reasoning still used in the majority of textbooks. Einstein introduced two unknown constants of proportionality (one for the rate of spontaneous emission, denoted by
A
, and one for the rate of stimulated emission, denoted by
B
), and then used additional arguments to replace them with known constants. These new fundamental quantities can be calculated directly in the modern theory, but in homage to the master are still labeled the “Einstein
A
and
B
coefficients.”

Einstein did not, however, rest on his laurels. Rederiving Planck's law in a purely quantum framework was progress, but it did not in itself clarify the basic question of the existence of light quanta. He was still hunting for the resolution of the “spherical wave paradox,”
that a classical point source emits a uniformly expanding spherical wave front, like the ripples a rock makes when dropped into a pond. Such waves seemed to rule out conceptualizing atomic emission as the release of a localized particle of light, which flies out in a particular direction. It was hard to visualize dropping a rock in a pond and having a single bump of water move out in a specific direction. However, it wasn't obvious that single atoms really behaved like classical point sources; maybe, he thought, the classical viewpoint was just wrong. Perhaps real atomic emission was a directed process, emitting “bumps” of light in specific directions; only when there were many atoms randomly emitting in all directions (or one emitting repeatedly) would it
appear
spherical.

A few weeks after his first 1916 paper Einstein realized, to his delight, that his new hypotheses of quantum emission allowed him to prove just this fact, and he eagerly drafted a second paper containing this demonstration. When he wrote to Besso on August 11, crowing about having found “
the
derivation,” he added, “I am writing the paper
⁴
right now.” In a follow-up letter to Besso two weeks later he added, “
it can be demonstrated convincingly
that the elementary processes of emission and absorption are directed processes.”

So what was the new insight that so excited him? Einstein had realized that he could return yet again to the well that had yielded the theory of Brownian motion of particles in suspension and of radiation energy fluctuations. In this famous second quantum paper of 1916, titled “On the Quantum Theory of Radiation,” he reviews his elegant new derivation of Planck's law, stating, “
this derivation deserves attention
not only because of its simplicity, but especially because it seems to clarify somewhat the still unclear process of emission and absorption of radiation by matter.” Now, he says, we must go beyond just considerations of energy exchange. “
The question arises: does the molecule
receive an
impulse
(i.e. a push in a specific direction) when it absorbs or emits the energy,
hυ
? …
It turns out that we arrive at a theory which is free of contradictions, only if we interpret those elementary processes as completely directed processes
[italics in the original]. Herein lies the main result of the following considerations.”

We have already seen that radiation exerts pressure even in classical electrodynamics and thus can push on matter, that is, transfer momentum to matter. Einstein had used this fact in his Salzburg “sliding mirror” thought experiment. The effect is similar to the recoil that occurs if you fire a gun; a light wave emitted in a specific direction causes the emitting atom to recoil in the opposite direction. Similarly, the analogue of absorption of light waves by an atom is the unfortunate process of “absorbing” an incoming bullet, which among the less problematic of its effects causes the “absorber” to be pushed in the direction of motion of the bullet. However, if the atom emitted a spherical light wave, the recoil pressure would be equal in all directions and no net momentum would be transferred. One can picture this by imagining a platoon of soldiers on a raft. If they all line up and fire their rifles in the same direction, then the recoil pushes the raft strongly in the opposite direction, but if they form a circle and fire outward at the same time, the raft will not move. (They could also form a circle and all fire inward, and again the raft would not move, but that would have other effects.) So the consequences of directed as opposed to undirected (spherical) emission are different, and in the process of energy exchange between molecules and radiation that Einstein is discussing, different motions of the molecules occur, depending on whether one assumes the emission and absorption is directed or undirected (“isotropic”).

To prove that each molecular interaction with radiation is a directed process, he makes the following argument. Imagine a gas molecule moving around in an enclosure filled with radiation, both of which are at the same temperature,
T
(this is the condition for thermal equilibrium). For the radiation this means that its energy density will be described by a universal radiation law, which depends on
T
and on the frequency,
υ
. Einstein is not here assuming that this law is given
by the (now) well-known Planck formula; his goal here is to derive the Planck law in a new way, based on Bohr's quantum atom and general considerations from statistical physics. In this effort he assumes that the atoms in the gas have a kinetic energy given by our old friend the equipartition theorem, which he knew failed for the vibrational energy of molecules but was well confirmed for the energy associated with the free motion of atoms in a gas.
⁵

So our equally partitioned gas molecules are gliding along with the average kinetic energy 3
kT
/2, but their motion is not free of all forces. The presence of all that radiation generates a kind of frictional force, due to the Doppler effect. The Doppler effect, which is familiar for sound waves, is the observed change in frequency (pitch) that occurs when a sound source is moving toward or away from the receiver: when moving toward, its pitch is measured to be higher than when at rest, and similarly it is found to be lower when the sound source is moving away. The same effect occurs for light waves: when you move toward them their frequency increases, and when away from them it decreases. Actually the details are a bit different for light because, unlike sound, it always is measured to move at
c
, but Einstein had worked out the formulas for this while banishing the ether way back in the miracle year of 1905.

Why does this effect cause friction? Imagine a situation where water waves of equal frequency are being generated in opposite directions at two ends of a swimming pool and you are standing in the middle, being buffeted forward and backward by those waves from each side. If you stand still, then on average you get an equal number of forward and backward shoves as the waves hit you, and you are not, on average, pushed toward either end of the pool. However, if you move at some reasonable speed (compared with the speed of the waves) toward one end, then you hit the incoming waves from that end more frequently than the waves generated behind you at the opposite end. This is the
Doppler effect in a very concrete form: you are moving in a medium in which waves are also moving, such that you encounter more crests (at a higher frequency) when you move against the direction of wave propagation, and at a lower frequency when you move in the same direction as the waves. In this case the visceral effect is that you get knocked backward more than you get knocked forward, and you feel an effective force impeding your motion toward the end of the pool you are approaching. But if you turn around and start walking toward the other end, exactly the same thing happens, except now the force is pointing in the opposite direction; that is, it behaves like friction, slowing you down no matter which direction you go. It takes Einstein a dense three pages of algebra to work out the exact mathematical formula for this frictional force, but this is the essential idea.
⁶

But this is not the only force acting on the molecule; it cannot be, because if it were, over time the radiation field would extract all the kinetic energy from the molecules, leaving them at absolute zero temperature. (In our pool analogy, the walker gets too tired to walk against the current and just stands still.) Again we would have a version of the ultraviolet catastrophe. But Einstein knows how nature avoids this. The previous reasoning assumed that the absorption events occur in a perfectly regular sequence, whereas in actuality the molecule is being randomly buffeted by photons at irregular intervals, so that in any short interval it gets a net kick from the radiation that can push it in either direction, forward or backward. Einstein calculates the magnitude of this
fluctuating
force. And then he assumes that these two forces, the frictional one and the fluctuating one, must on average balance, precisely to avert the unobserved cooling of matter by radiation. But this balance equation depends on the mathematical form of the radiation distribution law, the infamous universal function
ρ
(
υ
,
T
). With great relish, Einstein shows that Planck's law, and only Planck's law, will make the two forces cancel each other on average.