Simple Nature - Light and Matter

We assume v is small enough so that the electric and magneticfields are nearly constant throughout it. We then haveP =(18πk |E|2 + c28πk |B|2 )vhf.We can simplify this formidable looking expression by recognizingthat in a plane wave, |E| and |B| are related by |E| = c|B|. Thisimplies (problem 40, p. 725), that the electric and magnetic fieldseach contribute half the total energy, so we can simplify the resultto( 18πkP = 2|E|2) vhfv=4πkhf |E|2 .The probability is proportional to the square of the wave’s amplitude,as advertised. 33 But note that along the way, we had to make two crucial assumptions: thatthe wave was sinusoidal, and that it was a plane wave. These assumptions willnot prevent us from describing examples such as double-slit diffraction, in whichthe wave is approximately sinusoidal within some sufficiently small region suchas one pixel of a camera’s imaging chip. Nevertheless, these issues turn out tobe symptoms of deeper problems, beyond the scope of this book, involving theway in which relativity and quantum mechanics should be combined. As a tasteof the ideas involved, consider what happens when a photon is reflected froma conducting surface, as in example 23 on p. 699, so that the electric field atthe surface is zero, but the magnetic field isn’t. The superposition is a standingwave, not a plane wave, so |E| = c|B| need not hold, and doesn’t. A detector’sprobability of detecting a photon near the surface could be zero if the detectorsensed electric fields, but nonzero if it sensed magnetism. It doesn’t make senseto say that either of these is the probability that the photon “was really there.”848 Chapter 13 Quantum Physics