February 24, 2006

The Quantum

Max Planck On the Theory of the Energy Distribution Law of the Normal Spectrum 1900

The story of this paper begins, as do many great discoveries, with a puzzling observation.

By the end of the 19th century people had figured out how to make colour filters for light, and measure the brightness of light by comparing it against light sources of known strength. With these tools, people could measure the intensity (brightness) of any source of light at any frequency (colour), and, as scientists are wont to do, make graphs of intensity vs. frequency. Such a graph is called the spectrum of a light source.

Any light that strikes an object is either reflected back, passes through the object, or is absorbed by the object. Light that reflects or passes through is unchanged, and therefore relatively uninteresting, but something very interesting happens to the light that is absorbed. Atoms aren't any good at keeping absorbed energy, so they quickly re-emit it, usually at a different frequency. This changes the spectrum in interesting ways.

To study this effect, scientists used to measure black-box radiation; that is, they would cut a small hole in a box and see what light came out. If the box has been left closed up for long enough, you can be quite sure that any light that comes out has been absorbed and radiated from the interior walls of the box many times, so there can be no reflected light from an exterior source left inside. (At reasonable temperatures, of course, all the light inside a box is at frequencies below those we can see. You'll just have to take my word that there is, in fact, light inside a dark box.)

The puzzling observation was this: black-box radiation has a spectrum which can be predicted solely from the temperature of the box. At first glance this seems reasonable... until you start thinking about it. It doesn't matter what shape or size the box is, or even what it's made of. As long as they're the same temperature, boxes of any size or material produce exactly the same spectrum, called the black-body spectrum or the normal spectrum.

Since scientists had already observed that atoms of different materials re-radiate absorbed light in different ways, this was very upsetting. Even more upsetting was that then-current theories produced impossibly wrong answers when used to calculate what should be observed from the interior of a black box.

To understand black-box radiation, Planck needed statistics. In 1900 it was known that if every configuration of a system is equally probable, then one can calculate the number of possible configurations of some known system, and calculate its probability. A very simple example is rolling two dice: every possible roll is equally likely, but there's only 1 way to roll a 12 (6 and 6) and 6 ways to roll a 7, so a 7 is six times more probable than a 12.

Small changes to a system (like re-rolling one of the dice) are more likely to move the system toward a probable state than an improbable one. Once a system reaches the most highly probable state, any small changes are more likely to keep it there than move it towards an unprobable state. In other words, a system being subjected to small changes will reach an equilibrium, a steady state, and that state is exactly the most probable one.

The temperature of a black box is, in effect, a measurement of its total energy. Planck wanted to calculate the most probable distribution of energy among the atoms and colours of light in the interior of a black box. To do this, he was forced to assume that atoms cannot absorb or radiate arbitrary amounts of energy. In other words, there must be a smallest possible amount of energy that atoms can absorb or radiate, or else there would be an infinite number of ways to distribute the energy in the box, which causes the probability calculations to break. This is the fundamental idea, what Planck called "the most essential point of the whole calculation". Later, this smallest possible amount of energy would be named the quantum.

Planck showed that if the size of the quantum is determined by the frequency of the light being absorbed or radiated, then the normal spectrum is the equilibrium state inside a black box. In doing this, he provided the first valid explanation for the universality of the normal spectrum.

The relationship between the size of the quantum and the frequency of light that Planck discovered is simple: multiply the frequency of light in question by a constant. Planck called the constant h, and calculated it (based on observations of black-box light) to be 6.55 x 10-27 erg seconds. An erg is a very small amount of energy, so h, when multiplied by some frequency of light, gives you an astonishingly small amount of energy: the quantum of that frequency.


At November 16, 2007, Anonymous Anonymous said...



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