Celebrating Einstein

"The Momentum of Light"

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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.

  • 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.

  • 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.

  • 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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