The last term in (6.4) is the “interference term.” So much for water waves. The intensity can have any value, and it shows interference.
An experiment with electrons
Now we imagine a similar experiment with electrons. It is shown diagrammatically in Fig. 6–3. We make an electron gun which consists of a tungsten wire heated by an electric current and surrounded by a metal box with a hole in it. If the wire is at a negative voltage with respect to the box, electrons emitted by the wire will be accelerated toward the walls and some will pass through the hole. All the electrons which come out of the gun will have (nearly) the same energy. In front of the gun is again a wall (just a thin metal plate) with two holes in it. Beyond the wall is another plate which will serve as a “backstop.” In front of the backstop we place a movable detector. The detector might be a geiger counter or, perhaps better, an electron multiplier, which is connected to a loudspeaker.
We should say right away that you should not try to set up this experiment (as you could have done with the two we have already described). This experiment has never been done in just this way. The trouble is that the apparatus would have to be made on an impossibly small scale to show the effects we are interested in. We are doing a “thought experiment,” which we have chosen because it is easy to think about. We know the results that would be obtained because there are many experiments that have been done, in which the scale and the proportions have been chosen to show the effects we shall describe.
The first thing we notice with our electron experiment is that we hear sharp “clicks” from the detector (that is, from the loudspeaker). And all “clicks” are the same. There are no “half-clicks.”
We would also notice that the “clicks” come very erratically. Something like: click….. click-click… click….. click…. click-click…… click…, etc., just as you have, no doubt, heard a geiger counter operating. If we count the clicks which arrive in a sufficiently long time — say, for many minutes — and then count again for another equal period, we find that the two numbers are very nearly the same. So we can speak of the average rate at which the clicks are heard (so-and-so-many clicks per minute on the average).
As we move the detector around, the rate at which the clicks appear is faster or slower, but the size (loudness) of each click is always the same. If we lower the temperature of the wire in the gun the rate of clicking slows down, but still each click sounds the same. We would notice also that if we put two separate detectors at the backstop, one or the other would click, but never both at once. (Except that once in a while, if there were two clicks very close together in time, our ear might not sense the separation.) We conclude, therefore, that whatever arrives at the backstop arrives in “lumps.” All the “lumps” are the same size: only whole “lumps” arrive, and they arrive one at a time at the backstop. We shall say: “Electrons always arrive in identical lumps.”
Just as for our experiment with bullets, we can now proceed to find experimentally the answer to the question: “What is the relative probability that an electron ‘lump’ will arrive at the backstop at various distances x from the center?” As before, we obtain the relative probability by observing the rate of clicks, holding the operation of the gun constant. The probability that lumps will arrive at a particular x is proportional to the average rate of clicks at that x.
The result of our experiment is the interesting curve marked P12 in part (c) of Fig. 6–3. Yes! That is the way electrons go.
The interference of electron waves
Now let us try to analyze the curve of Fig. 6–3 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 P1. The result of the measurement is shown by the curve marked P1 in part (b) of Fig. 6–3. The result seems quite reasonable. In a similar way, we measure P2, the probability distribution for the electrons that come through hole 2. The result of this measurement is also drawn in the figure.
The result P12 obtained with both holes open is clearly not the sum of P1 and P2, the probabilities for each hole alone. In analogy with our water-wave experiment, we say: “There is interference.”
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, P12 is more than twice as large as P1 + P2. 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 P12 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 P12 in terms of P1 and P2.
Yet, surprisingly enough, the mathematics for relating P1 and P2 to P12 is extremely simple. For P12 is just like the curve I12 of Fig. 6–2, and that was simple. What is going on at the backstop can be described by two complex numbers that we can call φ̂1 and φ̂2 (they are functions of x, of course). The absolute square of φ̂1 gives the effect with only hole 1 open. That is, P1 = |φ̂1|2. The effect with only hole 2 open is given by φ̂2 in the same way. That is, P2 = |φ̂1|2. And the combined effect of the two holes is just P12 = |φ̂1 + φ̂2|. The mathematics is the same as what 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.)