The Quantum Mechanics Experiment

I’ve decided to try an experiment: To use this blog as a platform to teach undergrad quantum mechanics. Each post will be the equivalent of a “lecture;” the comments section is a natural place for the back-and-forth of questions.

Why? Mostly because it’s fun. I love teaching, I love quantum mechanics, and I haven’t had students in far too long.

It’s also an interesting experiment — I want to see if a blog platform is really a good place to teach a course. It sounds good in theory; the “lectures” are persistent, so people can come to it whenever they find it; there’s a natural place to ask and answer questions, and even to see what other people have asked; and there is one key technical innovation now in place, namely that WordPress allows you to enter equations.

(For reference, and for use when commenting: You can type in equations as “$latex …. $“. It’s a bit finicky, so preview your comments before you post to make sure it worked)

So I’m going to do this, posting as time permits, and I intend to work my way through the entire standard undergrad QM syllabus. Come join!

What I’m Going To Cover

The key things which I’m absolutely going to cover are:

  • Quick math / notation overview
  • Deriving the Schrödinger equation, the Uncertainty Principle, etc.
  • Solving the Schrödinger equation for various important potentials: potential wells, tunneling, the simple harmonic oscillator, the Hydrogen atom, etc.
  • Spherical symmetry and angular momentum. (A far more interesting subject than it sounds, because it’s the lead-in to the general theory of symmetries in QM)
  • Identical particles and exchange symmetry; multi-electron atoms.
  • Perturbation theory, both time-independent and time-dependent.
  • Scattering.

Beyond this, I’m planning on going in to depth on any number of things of particular physical or mathematical interest. Some of them have a natural place in the syllabus, and others don’t; sometimes I’ll simply not feel like doing the “next official thing,” and will instead branch off and talk about some interesting bit of physics which there’s already enough background established to explain. There are also some subjects which I might go into in more depth if people are interested and/or I feel like it: mixed states, solid state physics, group theory, some basic physical chemistry, some basic quantum statistical mechanics, relativistic quantum theory. We’ll see.

I’m going to make an active effort to alternate between very physically concrete things and more mathematical topics. This means that some things won’t be presented in the most formally natural order; e.g., I’ll probably talk about the Hydrogen atom at some length before talking about spherically symmetric problems, using the special case to motivate the general discussion rather than the other way round. Mostly this is because it’s far too easy to branch off into some fairly heavy pure math if one is me and is not being particularly careful about it.

Background Material

Because this is meant to cover the material of a junior-level course, a certain amount of background knowledge will be required. On the math side, you should be fully fluent in calculus — I will treat the various integrals, derivatives, Taylor expansions and so on as obvious. A good deal of solving of differential equations will happen during this course, but most physicists tend to learn diff. eq. by doing quantum mechanics, so only a basic knowledge is required to begin with. (e.g., it should be clear to you that a 2nd-order linear ODE has two independent solutions, and why; and you should be able to figure out those solutions for an equation like y''=-a^2 y by simple inspection) Quantum mechanics also relies on a great deal of linear algebra; while I do intend to give a review of the relevant material (QM tends to use a slightly different notation than mathematics), it will be far easier to follow if you already know what inner product spaces, changes of bases, and eigenvectors are and how to find them. On the physics side, the required background is essentially freshman physics; the equation “E=\frac{p^2}{2m} + V” should have an obvious meaning to you, and a knowledge of electrodynamics including concepts like a dielectric constant and a vector potential will be useful later on.

Some books which may be useful:

  • Classical mechanics: Almost any text will work; I don’t have many of the standard freshman-level books on my shelf, but Taylor’s Classical Mechanics and Fowles and Cassiday’s Analytical Mechanics are there, and they’re both quite fine. Browsing Wikipedia entries may even be enough. For a more advanced treatment, try José and Saletan’s Classical Dynamics, which is very readable.
  • Electrodynamics: Probably the best-written textbook in all of physics is D. J. Griffiths’ Introduction to Electrodynamics. If you ever need to know anything about the subject, that’s the first place to go. (The standard grad text, Jackson’s Classical Electrodynamics, is basically a more thorough covering of the same material)
  • Linear algebra: AFAICT, the books on this are pretty interchangeable. I don’t seem to have any on my shelf at the moment.
  • Integrals: Even in the age of Mathematica, it’s surprisingly helpful, when doing any advanced mathematics or physics, to have a good table of integrals handy. I personally keep two, a little easy-to-carry one (Dwight’s Table of Integrals and Other Mathematical Data) and the great standard one, Gradshteyn and Ryzhik’s Table of Integrals, Series and Products. The latter is invaluable to anyone doing serious scientific work. Another handy book of data is the Particle Data Book, available online.
  • Quantum Mechanics: There are a lot of textbooks on the subject. Some that I’ll refer to are: Griffiths’ Introduction to Quantum Mechanics; Liboff’s Introductory Quantum Mechanics; Gasiorowicz’s Quantum Physics; Powell and Craseman’s Quantum Mechanics; Sakurai’s Modern Quantum Mechanics; and Cohen-Tannoudji’s Quantum Mechanics. The last book is particularly famous as an enormous repository of exercises, both worked and unworked.

This last bit brings up one more important point: The only way to learn quantum mechanics properly is to do it. Exercises, working computations, and so on, are the only way to get a real feeling both for how the math works and for how the physics works. I will try to post exercises of my own, but I realize that this is the thing most likely to be left by the wayside. A serious student should therefore get their hands on one (or more!) good QM book and do as many exercises as possible. (That’s also useful because, while I’m certain that everything I write will be absolutely scintillating and crystal-clear, it may be helpful to hear the same ideas explained in multiple ways.)

So: An experiment begins. Let’s see how it works out. 🙂

Published in: on August 2, 2010 at 10:00  Comments (1)  

One Comment

  1. I don’t see what any of this has to do with psychic powers. 😉

Comments are closed.

%d bloggers like this: