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B.  The solution, with a puzzle of its own

Planck had discovered the first accurate description of a certain property of light for which the previously known laws of light waves didn't account.  Those laws, formulated by James Clerk Maxwell, suggested that the energy of light waves was smoothly distributed in space.  But Planck recognized that his own discovery suggested something else:  objects that absorb light waves have to be absorbing the energy in lumps (or quanta) rather than in continuous streams.  In 1905, Einstein had considered light quanta further, and had shown that even in the space between objects, the waves' energy appeared to behave like tiny particles.

Einstein went further still.  By late 1906, he had reached a conclusion about the behavior of objects that emit and absorb light that was as radical as the idea of light quanta itself.  When applied to objects that interact with light, Einstein's idea helped make better sense of Planck's law.  But when the idea was applied to all objects, it also resolved the difficulty with the theory that thermal energy is the random motion of particles.

Maxwell's equations account for light waves as ripples in electromagnetic force fields.  According to these equations, such waves are stirred up whenever an object with an electric charge vibrates, with some of the object's energy of motion going into the wave.  Because electric charges move in response to the electromagnetic fields around them, the reverse also happens:  a light wave striking a charged object can set it to vibrating, with energy leaving the wave and going into the vibrating object.  If the charged object was already vibrating when the wave reached it, the wave may alter the object's vibration.

Just as Maxwell's equations suggest that the energy in these waves is smoothly distributed in space, nothing in the equations or in Newton's laws of motion directly implies that a charged object can't vibrate with any old motion, or any arbitrary energy, as long as there are forces on the object that could produce that motion.

Despite Maxwellian expectations, light energy appears to come in quanta anyway.  Einstein took the idea of energy quanta further by exploring the idea that a vibrating electric charge can only vibrate with certain energies.  If that were so, one consequence would be a more direct way to derive Planck's law.

Planck had arrived at the idea of light quanta by considering the entropy of vibrating charges that could emit and absorb light waves.  The entropy of a set of objects characterizes how much of its energy can be extracted to do useful work.  This, in turn, is related to how randomly the objects' energy is distributed among them, and it was this aspect of entropy on which Planck focused.  From these considerations, plus some experimental data, Planck derived a relationship between the frequency of a light wave and its intensity under certain conditions, which then led to the idea of light being emitted and absorbed in quanta.

Planck's idea of light quanta had come from his study of how energy would be distributed in a furnace among the light waves it contained and the vibrating electric charges in the atoms of whatever glowing hot objects produced the light.  These considerations led to a law relating the frequency of light in the furnace to its intensity, which in turn led to the idea that light energy was produced in units instead of a continuous flow.

In deducing his law, Planck made no special assumptions about what ways the electric charges can vibrate.  On the other hand, Einstein showed that if the vibrating particles could only vibrate with certain energies, with each possible energy differing from the next by the same amount, then light quanta with energies equal to the differences must exist.  Planck's law relating light waves' frequencies and intensities immediately follows from this.

As mentioned in the previous section, Einstein's idea also resolves the disagreement between the 19^th-century theory of heat as random atomic motion and how real materials respond to heat flow.  If vibrating particles, such as the atoms of which solids are made, are limited to having certain energies but not others, then it's quite natural that colder objects would require less heat than hotter objects to raise their temperatures by one degree.

Stated the way we've just put it, this limitation may not seem very remarkable.  But the assumption was, in fact, quite drastic.  It wasn't just that earlier theory didn't consider such a limitation; no obvious physical mechanism existed to impose it.

To better understand why, imagine a simple type of vibrating system:  a weight on the end of a spring.  If you put some energy into the spring by stretching it, and then let the spring go, the weight will vibrate up and down between two points.  If you add a little more energy, the weight's high and low points will get further apart; if the weight-and-spring system loses energy, its high and low points will get closer together.  One way to add more energy to the system would be to catch the weight at its low point and stretch the spring some more.  Apparently, there's nothing about the system that would keep you from stretching the spring by any amount you might want to, and thus adding any amount of energy that you could manage (at least up to the spring's breaking point).

If a vibrating atom could only have certain amounts of energy, it would be like a weight on a spring that could only reverse direction at certain maximum and minimum spring lengths but not at any intermediate lengths.  If you caught and stretched such an "atomic" spring at the atom's "low point", the spring would presumably have to stretch to its next maximum length all at once instead of moving gradually through all the intermediate points.

Here we find a behavior of atoms that does not make sense when we try to understand it in terms of larger objects whose behavior is more familiar to us.  As it turns out, the reason atoms can vibrate with some energies and not others is much easier to understand once we take into account some other features of atoms, which were still unknown when Einstein first studied the heat absorption of solids.  Einstein was thus stuck with the problem of reasoning about a behavior that was apparently real, but incomprehensible.  About all he could say at the time was that atoms had fewer possible states of motion than objects within our experience and consider the consequences, however puzzling the reasons behind them were.

One possible objection remains to Einstein's idea.  If larger objects are made of atoms, shouldn't those larger objects share the same strange behavior?  If they do, then why don't we see it?  These questions, at least, do have a sensible answer which we can understand more easily once we see how Einstein solved his immediate problem:  how should the heat absorption of a solid vary with temperature if energy quanta exist?  It is at this point that we will look at how temperature and energy are related.     (.....continued)

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