I've now worked my way through the problems at the end of Chapter 1 of James Binney and David Skinner's 'The Physics of Quantum Mechanics'. And I found them to be manageable. Much more so than some of the problems in later chapters, by the looks of it. The ones in Chapter 1 are no doubt intended as gentle warm-up exercises.
Useful warm-up exercises. If nothing else, they forced me to remind myself of some complex number calculation rules.
The last problem, about an infinite series of potential wells, was the most interesting one. The first task is to find the probability of an electron being in the nth well. A quantum state vector for the electron's location is specified, with probability amplitudes provided as a function of n. So it's pretty straightforward to calculate the probability of the electron being in the nth well as a function of n.
The next task is to determine the probability of finding the electron in well 0 or anywhere to the right of it. I duly calculated the sum of all those probabilities from well 0 to infinity after looking up the appropriate summation formula on Wikipedia.
Only to realise that I could have spared myself the trouble: I could have found the answer more easily by employing a symmetry argument, since the probabilities are arranged symmetrically around 0 and must add up to 1.
But I don't regret my detour. It's given me a bit more practice in infinite series calculations. And that's of course what these problems are all about: practice, practice, practice.
Indeed, I find it very difficult to just read a physics textbook and understand it all without getting some practice in manipulating the mysterious mathematical symbols those texts tend to be peppered with. While I fear that at least some practice is necessary to achieve understanding in physics, I don't think it's sufficient.
Is the potential well probability amplitude distribution given in the problem discussed above realistic? If so, where do those probabilities come from? How and why do electrons move from one well to another? Is it meaningful to say that an electron is in a particular well before we've measured its location? Is it ever meaningful to say that an electron is in a particular well? I'm sure it's possible to be rather good at problem solving without understanding very well what the problem is about.
To adapt a quote I read somewhere many years ago: Physicists often expect to be regarded as profound thinkers even though you can find the most simple-minded people among them, unable to do anything that requires them to think unless it can be achieved by combining symbols in a way which is more a matter of practice than of deep thought.
