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B. Are energy quanta also momentum quanta?
Einstein's review was similar to his 1905 analysis of
energy quanta. In that work, he had considered a furnace containing hot,
luminous objects at a given temperature T. The results indicate
what fraction of the light quanta we would find within any range of energy.
In his 1916 work, Einstein also considered a furnace.
In this analysis, the hot, luminous objects were molecules of a gas filling the
furnace. Using individual molecules offered some advantages. Whereas
a piece of wood, or a lump of coal, would scarcely be disturbed by exchanging
momentum with a single light quantum, the motion of a gas molecule would be
affected significantly. Furthermore, earlier physicists had already
figured out enough about the motion of gas molecules at temperature T to
tell what fraction of the molecules would have momenta in a given range, and
what the molecules' average energy would be. Remarkably, since these
features of a gas depend only on temperature, they should be unaffected by the
molecules' interactions with light quanta. While the effects on the
individual molecules' momenta would be considerable, these effects should cancel
each other out.
To proceed with his analysis, Einstein considered exactly
what ways a light quantum could interact with a single gas molecule. He
saw three basic possibilities.
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In the first type of process, a molecule could emit
some of its energy in the form of a light quantum. The light quantum's
momentum would be proportional in size to its energy and would be directed
whichever way the quantum was emitted. As it emitted the quantum, the
molecule would recoil in the opposite direction.
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In the second type of process, a molecule exposed to
light waves could absorb one of the light quanta. The molecule would
absorb not only the quantum's energy, but also its momentum. If the
molecule were originally stationary, absorbing the quantum would propel it
in the direction the quantum had been moving; if the molecule had been
moving, its initial momentum would change by the same amount in the
direction of the quantum's motion.
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The third possibility is that a quantum of the
surrounding light waves would stimulate a molecule to emit a
duplicate quantum. As with the first, spontaneous type of process, the
molecule would recoil away from the direction of the emitted quantum's
momentum.
Each of these processes is analogous to one that should
occur according to the classical wave theory of light. The only difference
from classical theory that we've noted so far is that the light's energy and
momentum are supposed to come in quanta instead of continuous streams.
To see how these processes would affect the momenta of the
molecules in the furnace, Einstein also had to consider how often each process
would occur. The chance that each type of process would occur over a given
time should depend on how long that given time is-wait twice as long, and the
process has twice the opportunity to happen. The probabilities might also
depend on how much energy the molecule has before and after interacting with the
light quantum. For the second and third processes listed above, the
probabilities should also depend on the intensity of the light surrounding the
molecule-doubling the intensity doubles the opportunity for the molecule to
react to a light quantum.
Each of the three processes will change the molecules'
energies and momenta. To complete his analysis, Einstein had to account
for how many molecules were in what states between the times these processes
occur. Other things being equal, the greater the change in energy between
two states of the same molecule, and the smaller the temperature, the more
likely it is that the molecule will be in the lower-energy state; if a molecule
is x times move likely to be in a lower-energy state than a higher-energy
state, then doubling the energy difference between the two states, or halving
the temperature, would make the lower-energy state x2 times more likely than the
higher-energy state.
Taking into account the proportion of molecules in each
state, the effect of light intensity on how often the molecules change their
state, and some other known facts about the way light and matter interact, the
first thing Einstein determined was how the light's intensity should vary with
its frequency and temperature. As with Einstein's 1905 analysis, the
result was simply Planck's law, which Planck had arrived at in 1900 from a
different starting point. This disposed of the "energy" part of
the problem.
Solving the "momentum" part of the problem
required accounting for one additional complication. A molecule at rest in
the midst of the furnace would be exposed to a rain of light quanta equally
intense in all directions. Because the intensity is equal, all directions
are equally likely to be the one from which the molecule will absorb a quantum
from this rain, or to be the direction in which the molecule will emit a quantum
to add to the rain. The molecule is therefore equally likely to recoil in
any direction as a result. But a moving molecule would be heading into the
rain of quanta, making that rain, from the molecule's own perspective, more
intense in the molecule's forward direction. It's therefore most likely
that the light would oppose the molecule's motion.
In making his 1916 analysis, Einstein had the advantage of
some earlier results from his relativity theory, namely how the frequency and
the intensity of light both differ as you examine the same light in different
frames of reference. Without these findings, analyzing the effect of the
light-quantum rain on the moving molecules might have been more difficult than
it was. With them, instead of trying to calculate the effect of the light
quanta on a molecule with a nonzero velocity, Einstein could simplify the math
by shifting to the molecule's own reference frame. (Einstein had used a
similar shift in perspective to arrive at his "E=mc2" equation.)
In this frame from the molecule's own perspective the molecule was stationary, and the
light's intensity and frequency distribution were easily shown to form a more
intense rain of quanta in "front" of the molecule. The size of
the intensity difference could be determined, thus showing exactly how likely it
was that the molecule would be pushed "back".
It turns out that either of the two effects the tendency of light to resist the motion of
a molecule moving through the furnace, or to move a molecule that's initially at
rest in the furnace-would, by itself, alter the average momentum of the gas
molecules. But when Einstein compared these two effects, he found that
they cancelled out. This is what should have happened, since Einstein's
earlier analysis had agreed with experiment even though it had completely
ignored momentum.
(.....continued)
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