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 (P1 + P2), 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.
Watching the electrons
We shall now try the following experiment. To our electron apparatus we add a very strong light source, placed behind the wall and between the two holes, as shown in Fig. 6–4. We know that electric charges scatter light. So when an electron passes, however it does pass, on its way to the detector, it will scatter some light to our eye, and we can see where the electron goes. If, for instance, an electron were to take the path via hole 2 that is sketched in Fig. 6–4, we should see a flash of light coming from the vicinity of the place marked A in the figure. If an electron passes through hole 1 we would expect to see a flash from the vicinity of the upper hole. If it should happen that we get light from both places at the same time, because the electron divides in half… Let us just do the experiment!
Here is what we see: every time that we hear a “click” from our electron detector (at the backstop), we also see a flash of light either near hole 1 or near hole 2, but never both at once! And we observe the same result no matter where we put the detector. From this observation we conclude that when we look at the electrons we find that the electrons go either through one hole or the other. Experimentally, Proposition A is necessarily true.
What, then, is wrong with our argument against Proposition A? Why isn’t P12 just equal to P1 + P2? Back to experiment! Let us keep track of the electrons and find out what they are doing. For each position (x-location) of the detector we will count the electrons that arrive and also keep track of which hole they went through, by watching for the flashes. We can keep track of things this way: whenever we hear a “click” we will put a count in Column 1 if we see the flash near hole 1, and if we see the flash near hole 2, we will record a count in Column 2. Every electron which arrives is recorded in one of two classes: those which come through 1 and those which come through 2. From the number recorded in Column 1 we get the probability P‘1 that an electron will arrive at the detector via hole 1; and from the number recorded in Column 2 we get P’2, the probability that an electron will arrive at the detector via hole 2. If we now repeat such a measurement for many values of x, we get the curves for P‘1 and P’2 shown in part (b) of Fig. 6–4.
Well, that is not too surprising! We get for P‘1 something quite similar to what we got before for P, by blocking off hole 2; and P’2 is similar to what we got by blocking hole 1. So there is not any complicated business like going through both holes. When we watch them, the electrons come through just as we would expect them to come through. Whether the holes are closed or open, those which we see come through hole 1 are distributed in the same way whether hole 2 is open or closed.
But wait! What do we have now for the total probability, the probability that an electron will arrive at the detector by any route? We already have that information. We just pretend that we never looked at the light flashes, and we lump together the detector clicks which we have separated into the two columns. We must just add the numbers. For the probability that an electron will arrive at the backstop by passing through either hole, we do find P‘12 = P’1 + P‘2. That is, although we succeeded in watching which hole our electrons come through, we no longer get the old interference curve P12, but a new one, P’12, showing no interference! If we turn out the light P12 is restored.
We must conclude that when we look at the electrons the distribution of them on the screen is different than when we do not look. Perhaps it is turning on our light source that disturbs things? It must be that the electrons are very delicate, and the light, when it scatters off the electrons, gives them a jolt that changes their motion. We know that the electric field of the light acting on a charge will exert a force on it. So perhaps we should expect the motion to be changed. Anyway, the light exerts a big influence on the electrons. By trying to “watch” the electrons we have changed their motions. That is, the jolt given to the electron when the photon is scattered by it is such as to change the electron’s motion enough so that if it might have gone to where P12 was at a maximum, it will instead land where P12 was a minimum; that is why we no longer see the wavy interference effects.
You may be thinking: “Don’t use such a bright source! Turn the brightness down! The light waves will then be weaker and will not disturb the electrons so much. Surely, by making the light dimmer and dimmer, eventually the wave will be weak enough that it will have a negligible effect.” OK. Let’s try it. The first thing we observe is that the flash of light scattered from the electrons as they pass by does not get weaker. It is always the same-sized flash. The only thing that happens as the light is made dimmer is that sometimes we hear a “click” from the detector but see no flash at all. The electron has gone by without being “seen.” What we are observing is that light also acts like electrons; we knew that it was “wavy,” but now we find that it is also “lumpy.” It always arrives — or is scattered — in lumps that we call “photons.” As we turn down the intensity of the light source we do not change the size of the photons, only the rate at which they are emitted. That explains why, when our source is dim, some electrons get by without being seen. There did not happen to be a photon around at the time the electron went through.