Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

What is the proportionality constant?

What is the proportionality constant that would make an actual equation out of (probability distribution) (amplitude)2?

The probability that the photon is in a certain small region of volume v should equal the fraction of the wave's energy that is within that volume:

We assume v is small enough so that the electric and magnetic fields are nearly constant throughout it. We then have

We can simplify this formidable looking expression by recognizing that in an electromagnetic wave, |E| and |B| are related by |E| = c|B|. With some algebra, it turns out that the electric and magnetic fields each contribute half the total energy, so we can simplify this to

As advertised, the probability is proportional to the square of the wave's amplitude.

Last Update: 2009-06-21