quantum rules | large hole expt | small hole expt | probability
Two rules set quantum mechanical predictions apart from those of classical mechanics. First, quantum mechanics asserts that some quantities, such as an electron's position, can never be known with certainty (at least not over "short" distances). We can, at best, give only the probability of detecting an electron in a particular location. Second, we are told that the rules for combining probabilities is not what we would guess from everyday experience. The correct rules introduce a strange quantity called the "probability amplitude", and seem to imply all sorts of weird things, such as electrons being in two places simultaneously!
Of course, it is easier to say what the rules are then it is to explain or use them. Therefore, to build a case for these two rules, and to illustrate their use, I will describe some simple electron "diffraction" experiments whose results cannot be explained using classical physics.
The basic experimental apparatus is shown below. It consists of an evacuated bottle containing devices for emitting, blocking, and detecting electrons. Our apparatus is not too different, in fact, from the picture tube on your TV set or computer. These tubes are also evacuated, and contain an electron emitter and detector (the picture screen). In our experiment, however, we will use a single small detector which can be moved around and which will count the number of electrons striking it in a given time period.
In the "large hole" experiment, a screen with a large hole in it is placed between the emitter and detector ("large" means "much larger than an electron"). The screen blocks electrons perfectly, so the electrons must pass through the hole in order to reach the detector. The results of the experiment are shown on the blue graph on the right (the vertical axis corresponds to detector position, and the horizontal axis gives the number of electrons detected in a given time period). The locations where electrons are detected correspond to places where an electron can travel through the hole in the screen on a straight line connecting the emitter and the detector. This result is consistent with classical physics which says that particles should travel in a straight line unless they are acted on by some force (note: gravitational forces are much too weak to cause any observable effects).
The next version of this experiment uses a screen with a much smaller hole ("small" means "not much larger than an electron"). This simple change has a big impact on where electrons are detected. The largest number of electrons still appear to travel through the hole in a straight line, i.e., we detect them at points that line up with the hole. However, we also detect electrons at points that clearly do not line up with the hole. Since the electrons must go through the hole before they can arrive at the detector, we must conclude that these "odd" electrons change direction as they pass through the hole, and that this "change of direction" is a consequence of hole size.
The "change in direction" phenomenon is referred to as "electron diffraction" and was first observed in "small hole" experiments involving light beams. The diffraction phenomenon has many chemical applications, one of which is the measurement of interatomic distances in molecules. A variety of diffraction techniques can be used (electron, X-ray, neutron), and all rely on the existence of "small holes" to create diffraction.
The two previous experiments give very different views of electron behavior. The first experiment suggests that electron behavior can be predicted with certainty ("electrons travel in straight lines"). The second experiment, however, shows that is not true. We cannot say with certainty how an electron travels through a "small" hole. It might pass through the hole by traveling in a straight line, OR it might change direction in an unpredictable manner. Also, we cannot say which electron will go straight and which will be diffracted. These appear to be fundamental limitations on our ability to predict electron behavior.
Although we cannot predict where an electron will go with certainty, we can predict detection probabilities. For example, the following chart shows what might be observed if 1000 electrons were passed through a small hole and detected at 10 different locations. If it turned out that only detector positions 5, 6, and 7 lined up with the hole, and 555 electrons (175 + 195 + 185) or 55.5% were detected at these positions, we would predict that each electron has a 55.5% probability of passing straight through the hole. 445 electrons were detected at other positions, so we would predict that each electron has a 44.5% probability of being diffracted to one of these positions.
Note that we could make much more detailed probability predictions. For example, we could use the data in the chart to estimate the probability that an electron's path would change by a certain specified amount. What is important to remember, though, is that this is the BEST we can do. We cannot say with certainty where any particular electron will go.
(last updated 6/7/97)