But what corresponds to the amplitude or displacement of a matter wave? The ampli- tude of an electromagnetic wave is represented by the electric and magnetic fields, E and B . In quantum mechanics, this role is played by the wave function , which is given the symbol (the Greek capital letter psi, pronounced “sigh”). Thus represents the wave displacement, as a function of time and position, of a new kind of field which we might call a “matter” field or a matter wave. To understand how to interpret the wave function we make an analogy with light using the wave – particle duality. We saw in Chapter 11 that the intensity I of any wave is proportional to the square of the amplitude. This holds true for light waves as well, as we saw in Chapter 22. That is, where E is the electric field strength. From the particle point of view, the intensity of a light beam (of given frequency) is proportional to the number of photons, N , that pass through a given area per unit time. The more photons there are, the greater the intensity. Thus This proportion can be turned around so that we have That is, the number of photons (striking a page of this book, say) is proportional to the square of the electric field strength. If the light beam is very weak, only a few photons will be involved. Indeed, it is possible to “build up” a photograph in a camera using very weak light so the effect of photons arriving can be seen. If we are dealing with only one photon, the relationship above can be interpreted in a slightly different way. At any point, the square of the electric field strength is a measure of the probability that a photon will be at that location. At points where is large, there is a high probability the photon will be there; where is small, the probability is low. We can interpret matter waves in the same way, as was first suggested by Max Born (1882 – 1970) in 1927. The wave function may vary in magnitude from point to point in space and time. If describes a collection of many electrons, then at any point will be proportional to the number of electrons expected to be found at that point. When dealing with small numbers of electrons we can’t make very exact predictions, so takes on the character of a probability. If which depends on time and position, represents a single electron (say, in an atom), then is interpreted like this: at a certain point in space and time represents the probability of finding the electron at the gi v en position and time. Thus is often referred to as the probability density or probability distribution . Double-Slit Interference Experiment for Electrons To understand this better, we take as a thought experiment the familiar double-slit experiment, and consider it both for light and for electrons. Consider two slits whose size and separation are on the order of the wave- length of whatever we direct at them, either light or electrons, Fig. 28 – 3. We know very well what would happen in this case for light, since this is just Young’s double-slit experiment (Section 24 –
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