Yonatan Zunger’s Blog

Some of you may have noticed a dearth of quantum mechanics posts in the past month. The project isn’t dead, it’s simply been delayed due to a great deal of other work piling up in the time which would normally go into it.

However, I’ve also been looking through the past posts in the sequence, and I’ve come to the conclusion that the approach needs to be overhauled. Teaching with and without a room full of students is a pair of very different propositions; I may have underestimated how much of what I do is normally based on the back-and-forth with people, and being able to see in real-time what is and isn’t being communicated well. As it stands, I can’t follow all of my own arguments in some of those posts (the derivation of the Uncertainty Principle being one of the worst offenders). Clearly some rethinking is in order to improve coherence.

So, the project will continue, but in a somewhat modified form and at a slightly later date. Stay tuned.

Last time, we looked at a particle with spin 1/2 in a constant magnetic field. We saw that, like with a classical magnetic dipole, this field caused the magnetic moment to rotate around the field axis. Whenever we encounter a rotating body, a natural thing to try to do is to apply a rotating force to it and see if we can make it resonate.¹ Today we’ll do that and see what happens.

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Alright, enough with the mathematical abstraction! This post, and the next quite a few posts, will all be looking at very concrete, physical examples, applying the methods of quantum mechanics and seeing what we can see. Today, we’re going to start with the simplest possible system that has nontrivial dynamics: a system with only two states in it. In a sign of great creativity, this is normally referred to as “the two-state system.”

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Last time, we walked through some of the history of quantum mechanics and came out with the Schrödinger Equation, the master equation of nonrelativistic quantum mechanics. Much of what we’ll do in this course will involve solving this equation in a variety of interesting cases; but before we begin, it’s worth plunging a bit more deeply into the equation itself and seeing what we can learn just from its structure. Among other things, we’ll see the relationship of the abstract vectors we get from the linear algebra approach to the functions we use in the differential-equation approach; see the (rather simple) way that real systems evolve over time; and encounter the fundamental limitations on measurement in quantum mechanics.

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Today I’d like to start doing some physics. I wish I could say that we were going to derive the Schrödinger equation — which is basically the master equation of quantum mechanics — but it doesn’t follow from a simple examination of mathematical principles. Its justification comes from the fact that it seems to accurately describe the physical world. So I’m going to walk through some of its history, and the experiments and physical facts leading up to it, and will end up with an equation and, more importantly, an explanation of what the quantities being solved for actually mean.

A bit of history

Our story begins in 1900. At this time, our understanding of physics wasn’t quite complete — there was still some argument over whether “atoms” had any physical reality or were simply a useful calculational tool for chemistry, and we were still trying to sort out just how we moved relative to the æther — but we were confident enough in our understanding that Lord Kelvin could comfortably say that “There is nothing new to be discovered in physics now; all that remains is more and more precise measurement.”

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And now for the first contentful post about quantum mechanics. It’s wild, wonderful, exciting… okay, not really. It’s the mathematical prelude, an overview of some key physical constants and some key bits of linear algebra which we’ll use rather extensively going forwards. This isn’t meant to be deep, or to be an introduction to linear algebra; it’s a way to establish the notations we’ll be using, and to point out the facts which will happen to be important to us later on. (more…)

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!

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