A B C D E F
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 19th-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)
A B C D E F