The interference of electron waves
Now let us try to analyze the curve of Fig. 37–3 to see whether we can understand the behavior of the electrons. The first thing we would say is that since they come in lumps, each lump, which we may as well call an electron, has come either through hole $1$ or through hole $2$. Let us write this in the form of a “Proposition”: Proposition A: Each electron either goes through hole $1$ or it goes through hole $2$.  Assuming Proposition A, all electrons that arrive at the backstop can be divided into two classes: (1) those that come through hole $1$, and (2) those that come through hole $2$. So our observed curve must be the sum of the effects of the electrons which come through hole $1$ and the electrons which come through hole $2$. Let us check this idea by experiment. First, we will make a measurement for those electrons that come through hole $1$. We block off hole $2$ and make our counts of the clicks from the detector. From the clicking rate, we get $P_1$. The result of the measurement is shown by the curve marked $P_1$ in part (b) of Fig. 37–3. The result seems quite reasonable. In a similar way, we measure $P_2$, the probability distribution for the electrons that come through hole $2$. The result of this measurement is also drawn in the figure. The result $P_{12}$ obtained with both holes open is clearly not the sum of $P_1$ and $P_2$, the probabilities for each hole alone. In analogy with our water-wave experiment, we say: “There is interference.” \begin{equation} \label{Eq:I:37:5} \text{For electrons:}\quad P_{12}\neq P_1+P_2. \end{equation} How can such an interference come about? Perhaps we should say: “Well, that means, presumably, that it is not true that the lumps go either through hole $1$ or hole $2$, because if they did, the probabilities should add. Perhaps they go in a more complicated way. They split in half and …” But no! They cannot, they always arrive in lumps … “Well, perhaps some of them go through $1$, and then they go around through $2$, and then around a few more times, or by some other complicated path … then by closing hole $2$, we changed the chance that an electron that started out through hole $1$ would finally get to the backstop …” But notice! There are some points at which very few electrons arrive when both holes are open, but which receive many electrons if we close one hole, so closing one hole increased the number from the other. Notice, however, that at the center of the pattern, $P_{12}$ is more than twice as large as $P_1 + P_2$. It is as though closing one hole decreased the number of electrons which come through the other hole. It seems hard to explain both effects by proposing that the electrons travel in complicated paths. It is all quite mysterious. And the more you look at it the more mysterious it seems. Many ideas have been concocted to try to explain the curve for $P_{12}$ in terms of individual electrons going around in complicated ways through the holes. None of them has succeeded. None of them can get the right curve for $P_{12}$ in terms of $P_1$ and $P_2$. Yet, surprisingly enough, the mathematics for relating $P_1$ and $P_2$ to $P_{12}$ is extremely simple. For $P_{12}$ is just like the curve $I_{12}$ of Fig. 37–2, and that was simple. What is going on at the backstop can be described by two complex numbers that we can call $\hat{\phi}_1$ and $\hat{\phi}_2$ (they are functions of $x$, of course). The absolute square of $\hat{\phi}_1$ gives the effect with only hole $1$ open. That is, $P_1 = \abs{\hat{\phi}_1}^2$. The effect with only hole $2$ open is given by $\hat{\phi}_2$ in the same way. That is, $P_2 = \abs{\hat{\phi}_2}^2$. And the combined effect of the two holes is just $P_{12} = \abs{\hat{\phi}_1 + \hat{\phi}_2}^2$. The mathematics is the same as that we had for the water waves! (It is hard to see how one could get such a simple result from a complicated game of electrons going back and forth through the plate on some strange trajectory.) We conclude the following: The electrons arrive in lumps, like particles, and the probability of arrival of these lumps is distributed like the distribution of intensity of a wave. It is in this sense that an electron behaves “sometimes like a particle and sometimes like a wave.” Incidentally, when we were dealing with classical waves we defined the intensity as the mean over time of the square of the wave amplitude, and we used complex numbers as a mathematical trick to simplify the analysis. But in quantum mechanics it turns out that the amplitudes must be represented by complex numbers. The real parts alone will not do. That is a technical point, for the moment, because the formulas look just the same. Since the probability of arrival through both holes is given so simply, although it is not equal to $(P_1 + P_2)$, that is really all there is to say. But there are a large number of subtleties involved in the fact that nature does work this way. We would like to illustrate some of these subtleties for you now. First, since the number that arrives at a particular point is not equal to the number that arrives through $1$ plus the number that arrives through $2$, as we would have concluded from Proposition A, undoubtedly we should conclude that Proposition A is false. It is not true that the electrons go either through hole $1$ or hole $2$. But that conclusion can be tested by another experiment.