Quantum Mechanics a' la Feynman
Interference is a fundamental feature of quantum mechanics. Richard Feynman
alludes to it in his celebrated textbook as the ``only mystery'' of quantum
mechanics. There he starts his discussion of quantum mechanics by considering a
succession of thought experiments in which first bullets, then water waves, and
then electrons pass through a double-slit apparatus and strike a screen some
distance away. In the case of bullets, each bullet goes through one slit or the
other, and the bullet hits on the screen are smoothly distributed. In contrast,
the disturbances of the water waves passing through the slits result in a
regular pattern of successively large and then small disturbances on the screen,
the familiar ``interference'' pattern. When electrons are incident, we are
surprised to find that, as long as our apparatus is not arranged to reveal
through which slit an electron passes, the electron hits recorded on the screen
by electron detectors show the interference pattern. If we modify the apparatus
so that it can reveal which slit the electrons take, we observe the classical
bullet pattern rather than the interference pattern.
Feynman has summarized the quantum mechanics of amplitudes in three simple rules:1 (1) the probability P of a particular outcome from the interaction of a particle with an apparatus is given by the square of the absolute value of a complex number f which is called the probability amplitude: P = | f |2; (2) when the same outcome can occur in indistinguishable alternative ways, its probability amplitude is the sum of the probability amplitudes for each way considered separately: P = | f1 + f2 |2; but (3) when an experiment is performed that is capable of determining which way the outcome occurred, the probability of the outcome is the sum of the probabilities of each alternative: P = P1 + P2 = |f1|2+|f2|2.
The study of interference of light with single photons offers us a chance to study experimentally these fundamentals of quantum superposition and interference. Through the process of parametric fluorescence we produce pairs of photons that are correlated in energy and momentum. In one experiment we send one photon through an interferometer to a detector. The other photon goes directly to a detector. By detecting the two in coincidence is as if we were “tagging” the photon going through the interferometer, allowing us to measure the interference of that photon with itself. When we cannot tell which path the photon takes in the interferometer we see interference, following Feynman's rule #2 above. If the apparatus is set up so that we can determine which path the photon takes, then we do not see interference, and the probability follows rule #3.
1R. P. Feynman, R. B. Leighton and M. Sands, The Feynman
Lectures on Physics (Addison-Wesley, Reading, 1965) V. 3, p. 1-1.
E.J. Galvez/C.H. Holbrow/Colgate U.