Equation 39.14 defines the wave function for a particle in terms of the probability of finding the particle at different positions . But our interests go beyond merely characterizing experimental data. We would like to develop a new theory of matter. But just what is a theory? Although this is not a book on scientific methodology, we can loosely say that a physical theory needs two basic ingredients:
A descriptor, a mathematical quantity used to describe our knowledge of a physical object.
One or more laws that govern the behavior of the descriptor.
For example, Newtonian mechanics is a theory of motion. The primary descriptor in Newtonian mechanics is a particle’s position as a function of time. This describes our knowledge of the particle at all times. The position is governed by Newton’s laws. These laws, especially the second law, are mathematical statements of how the descriptor changes in response to forces. If we predict for a known set of forces, we feel confident that an experiment carried out at time t will find the particle right where predicted.
Newton’s theory of motion assumes that a particle’s position is well defined at every instant of time. The difficulty facing physicists early in the 20th century was the astounding discovery that the position of an atomic-size particle is not well defined. An electron in a double-slit experiment must, in some sense, go through both slits to produce an electron interference pattern. It simply does not have a well-defined position as it interacts with the slits. But if the position function is not a valid descriptor for matter at the atomic level, what is?
We will assert that the wave function is the descriptor of a particle in quantum mechanics. In other words, the wave function tells us everything we can know about the particle. The wave function plays the same leading role in quantum mechanics that the position function plays in classical mechanics.
Whether this hypothesis has any merit will not be known until we see if it leads to predictions that can be verified. And before we can do that, we need to learn what new law of physics determines the wave function in a given situation. We will answer this question in the next chapter.
It may seem to you, as we go along, that we are simply “making up” ideas. Indeed, that is at least partially true. The inventors of entirely new theories use their existing knowledge as a guide, but ultimately they have to make an inspired guess as to what a new theory should look like. Newton and Einstein both made such leaps, and the inventors of quantum mechanics had to make such a leap. We can attempt to make the new ideas plausible, but ultimately a new theory is simply a bold assertion that must be tested against reality via controlled experiments. The wave-function theory of quantum mechanics passed the only test that really matters in science—it works!