Let us now consider an alternative explanation of the double-slit experiment, however, due to Richard Feynman (who, by the way, was born in Queens!). This explanation is published in his 1965 book, Quantum Mechanics and Path Integrals. Feynman's explanation is closer in spirit to a classical-like picture, yet it still represents a radical departure from classical mechanics. Feynman postulated that electrons could still behave as particle in a double-slit experiment. The twist is that the particles do not follow definite paths, as they would if they were classical particles. Rather, they can trace out a myriad of possible paths that might differ considerably from the path that would be predicted by classical mechanics. In fact, electrons initialized in the same manner, will follow different paths if they are allowed to wend their way through the double-slit apparatus! This is illustrated in the figure below:
Since the electron can follow any path and will follow different paths in different realizations of the experiment, physical quantities must be obtained by summing over all possible paths that the electron can follow. In order to carry out this sum, Feynman assigned a weight or importance to each path in the sum over paths. Since each path takes the electron from the source to a point on the screen (call this point , where varies over the legnth of the screen) in a time , the probability associated with each path can only depend on and . If is fixed, the can only depend on . Feynman proposed that each path could be assigned a quantity called the action , where has the following properties:
It is important to note that is a complex number.
An amplitude is related to an actual probability
by taking the absolute value squared of the amplitude.
Thus, if there were only one path with amplitude ,
the probability that the electron would follow that path is
Generally, a probability amplitude is a generalization of the square root of a probability that allows the amplitude to be a complex number. If is a probability, and is the associated probability amplitude, then if were restricted to be real, then there would be only two possible values of , i.e., and . If we let be complex, the relation between and is
Now, suppose we have two interfering paths with amplitudes and (they should depend on , but for notational simplicity, we will suppress the dependence). The total amplitude is , and the corresponding probability is , which gives
At this point, several comments are in order. It is tempting to try to impose either the wave-like picture or the many-paths picture on the experiment. Indeed, both of these pictures provide a useful physical picture that helps us understand the outcome of the experiment. In the wave-like picture, we can think of each electron that leaves the source as feeling the presence of both slits simultaneously, and therefore interfering with itself (rather than with other electrons). In the many-paths picture, each electron follows not one path in the path sum but all possible paths at once, and these paths interfere with each other. However, the infuriating thing about quantum mechanics is that we have no way of knowing what is taking place between the source and the detector. All we have is the observation that there is an interference pattern. Feynman's picture makes this rather manifest. The implications of his picture can be summarized as
The rationalizations of the three experiments we have examined, blackbody radiation, the photoelectric effect, and electron diffraction, leads us to conclude that classical mechanics, with its deterministic, predictable view of the universe, must be overthrown in favor of a much more radial theory, now known as quantum mechanics. It is interesting to note that the idea of probabilistic outcomes of experiments and the fact that we can ONLY predict the probabilities, lead Albert Einstein ultimately to reject quantum mechanics, saying: ``Gott spielt nicht Würfel'' (``God does not play dice'').