Introduction to quantum mechanics

This chapter gives a brief introduction to quantum mechanics. Quantum mechanics can be thought of roughly as the study of physics on very small length scales, although there are also certain macroscopic systems it directly applies to. The descriptor “quantum” arises because in contrast with classical mechanics, certain quantities take on only discrete values. However, some quantities still take on continuous values, as we’ll see. In quantum mechanics, particles have wavelike properties, and a particular wave equation, the Schrodinger equation, governs how these waves behave. The Schrodinger equation is different in a few ways from the other wave equations we’ve seen in this book. But these differences won’t keep us from applying all of our usual strategies for solving a wave equation and dealing with the resulting solutions. In some respect, quantum mechanics is just another example of a system governed by a wave equation. In fact, we will find below that some quantum mechanical systems have exact analogies to systems we’ve already studied in this book. So the results can be carried over, with no modifications whatsoever needed. However, although it is fairly straightforward to deal with the actual waves, there are many things about quantum mechanics that are a combination of subtle, perplexing, and bizarre. To name a few: the measurement problem, hidden variables along with Bell’s theorem, and wave-particle duality. You’ll learn all about these in an actual course on quantum mechanics. Even though there are many things that are highly confusing about quantum mechanics, the nice thing is that it’s relatively easy to apply quantum mechanics to a physical system to figure out how it behaves. There is fortunately no need to understand all of the subtleties about quantum mechanics in order to use it. Of course, in most cases this isn’t the best strategy to take; it’s usually not a good idea to blindly forge ahead with something if you don’t understand what you’re actually working with. But this lack of understanding can be forgiven in the case of quantum mechanics, because no one really understands it. (Well, maybe a couple people do, but they’re few and far between.) If the world waited to use quantum mechanics until it understood it, then we’d be stuck back in the 1920’s. The bottom line is that quantum mechanics can be used to make predictions that are consistent with experiment. It hasn’t failed us yet. So it would be foolish not to use it. The main purpose of this chapter is to demonstrate how similar certain results in quantum mechanics are to earlier results we’ve derived in the book. You actually know a good