wang
(Wang)
#1
Figure 3: Experiment with two birefringent plates
found after the analyzer will beNcos^2 (θ−α). Translated into probabilities, the original
beam has probability 1, and the analyzer will find polarization angleαwith probability
ptot= cos^2 (θ−α) (2.5)
But a different way in which to evaluate this same probability is to combine the probabilities
as the light traverses the two birefringent plates.
If probabilities combined according to classical rules, we would find the following se-
quence. After the first, but before the second birefringent plate, the probability that the
photon has polarizationsxoryis given by (2.4). After the second plate, the probabilities
that thexandypolarized beam yield polarization angleαin the analyzer is
p′x = cos^2 α
p′y = sin^2 α (2.6)
According to the classical rules, the combined probability would be
p′tot=pxp′x+pyp′y= cos^2 θcos^2 α+ sin^2 θsin^2 α (2.7)
which is in disagreement withptot = cos^2 (θ−α) for general θ andα. In fact, we have
ptot−p′tot = 2 cosθsinθcosαsinα. This is an interference term: it arises from the fact
that when the two beams from birefringent plate 1 recombine, theyinterfere, just as the
electric fields of electro-magnetic waves did. Although photons behave like particles in that
they form discrete quanta, their probabilities combine like waves. This is an aspect of the
particle/wave duality of quantum mechanics.
(4)The correct rule for the combination of probabilities is given in terms of theprobability
amplitudesax,ay,a′x,a′y, related to the previously defined probabilities by
px=|ax|^2 p′x=|a′x|^2
py=|ay|^2 p′y=|a′y|^2 (2.8)
