Absolutely Small - Michael D. Fayer [28]
The fact that the wavefunction oscillates positive and negative doesn’t matter. In explaining photon interference quantum mechanically following Figure 5.1, the Born interpretation of the wavefunction was introduced. In the Born interpretation, the absolute value squared of the wavefunction in a certain region of space gives the probability of finding a particle in that region of space. When the wavefunction is squared, it becomes only positive in the same way that 22 = 4 and (-2)2 = 4 because minus times a minus is a plus. Note that in Figure 6.1, whenever one wave is zero, the other wave is either at a positive or negative maximum. Where one wave is small, the other wave is big. When the wavefunction is analyzed mathematically, and as can be seen from the graph, at all locations the absolute value squared of the wavefunction is uniform along the x axis.
The absolute value squared of the wavefunction for a free particle is uniform along the x axis from +∞ to -∞. Therefore, the probability of finding the particle anywhere in space is uniform. The particle has equal probability of being found at x = 10, or x = -1,000,000, or anywhere else. Imagine that you are a tiny creature, frequently referred to as Maxwell’s Demon. You are standing next to the particle-wave shown in Figure 6.1. You make a grab for the particle. There is some probability that you will find it in your hand. If you start over and do this again and again, depending on the size of your hand, you may eventually come up with the particle. Each time you need to start fresh in your attempt to grab the particle. If you move somewhere else along the wave and do the same thing, the chance that you will come up with the particle is no different. This is what it means to say that there is equal probability for finding the particle anywhere. There is no best spot for Maxwell’s Demon to stand to try to grab the particle. All locations are equally good.
This picture of a free particle described by a wavefunction that represents equal probability of finding the particle anywhere doesn’t go along very well with our classical concept of a particle. In Figure 2.5, we described a classical particle as having a particular momentum and position at a given time. In discussing the photoelectric effect (Figure 4.3), Einstein described light as photons, which are quanta of light. One photon “hits” one electron, and the electron flies out of the piece of metal. This description almost sounds like both the photon and the electron are particles in the classical mechanics sense of particles. However, in discussing the interference of photons in conjunction with Figure 5.1, it was necessary to use the Born interpretation and describe photons as probability amplitude waves, with half of the probability going into each leg of the interferometer. In Figure 6.1, the plot of a free particle wavefunction is completely delocalized, spread out over all space. The description is the same for a photon or an electron.
INTERFERENCE OF WAVES WITH DIFFERENT WAVELENGTHS
So what are photons and electrons and rocks and anything else? Are they particles or waves? To see that there is no contradiction in the quantum mechanical description of the nature of things, we need to discuss waves and the interference of waves further. In connection with Figures 3.2 and 3.3, we discussed that waves could interfere constructively to give a bigger wave or destructively to give a smaller wave or no wave at all. In the examples in Figures 3.2 and 3.3, the waves have the same wavelengths. When they added constructively (Figure 3.2), all of the positive peaks lined up with the positive peaks and the negative peaks lined up with the negative peaks to give increased amplitude. When the waves added destructively (Figure 3.3), the positive peaks lined up with the negative peaks and vice versa, to give cancellation. However, waves