Following on my notes about Euler’s formula, I’ve finally finished some work on another piece of elementary exposition, a discussion of the free quantum particle, which can be found as chapters 10, 11 and 12 of the book I’m working on.
These chapters are a complete rewrite and major expansion of what used to be there, a rather slap-dash single chapter on the subject. The excuse for this in my mind had been that it’s a topic treated in detail in every quantum mechanics textbook, so best if I passed over it quickly and moved on to things that weren’t so well treated elsewhere. Another reason for this was that my understanding of analysis has never been what it should be, and it seemed best if I not make that too obvious by how I handled the mathematics of this subject.
This summer I started rewriting the book from the beginning, and once I hit the chapter on the free particle it became very clear that it needed improvement, both for its own sake and for how the material was needed in later chapters. I spent some time doing some remedial study in analysis, and after a while got to a point such that I felt capable of writing something that captured more of the relevant mathematics. Finally, today I got to the point where these three chapters are in decent shape, and soon I’ll move on to later ones.
One thing that I’d never thought much about before, but that struck me while rewriting these chapters, is the quite peculiar nature of a position eigenstate in quantum mechanics. Normally one only thinks about this in relativistic quantum field theory, where the problems associated with localizing a relativistic particle motivate the move to a quantum field theory. Of course a position eigenstate is just a delta-function, but what is peculiar is the dynamics, what happens if you take that as an initial condition. See the end of chapter 12 for what I’m talking about (be sure you have the latest version, today’s date on the front), I won’t try and reproduce that here. Part of this story is the tricky nature of the free-particle propagator in real time, as opposed to its much better behavior in imaginary time. The issue of analytic continuation in time continues to fascinate me, including the quite non-trivial nature of what happens even for the supposedly trivial case of a free particle in one spatial dimension.