This year I’ll be teaching a new version of the same course on quantum mechanics aimed at mathematicians that I taught during the 2012-3 academic year (there’s a web-page here). During the last course I started writing up notes, and have spent a lot of the last academic year working on these, the current version will always be here. At this point I have a few of the last chapters to finish writing, as well as a long list of improvements to be made in the earlier ones. I’ll be teaching the course based on these notes, and hope to improve them as I go along, partly based on seeing what topics students have trouble with, and what they would like to hear more about.

I’ve learned a lot while doing this, and it now seems like a good idea to write something on the blog discussing topics that don’t seem to me to be dealt with well in the standard textbook treatments. Johan de Jong’s Stacks Project has recently added a sloganerator (written by Pieter Belmans and Johan Commelin), which has inspired me to try to organize things as slogans. Slogan 0 to appear soon….

It would be great if all of the problem sets be posted on the web page of the course so one can follow things better.

Thank You!

Being a physicist myself I always enjoy a clear and not to technical math description of fields in physics. I’ve read the first 15 pages or so and first impression is very good. Thanks for posting.

Cheers, Per

Per,

Thanks. Feedback about how to make this document more readable by physicists is definitely welcome.

QGravity,

The problem sets from the previous course are at

http://www.math.columbia.edu/~woit/QM/oldprobsets.pdf

As I get problem sets for the new course written, they’ll appear on the course page. Solutions won’t ever appear there, since I want to be able to reuse the problems…

Exciting stuff! Thanks for making it available. Would be looking out for notes and exercises.

Prof. Woit,

As someone who doesn’t know a lot about physics, I’ve started reading the linked notes, but I’ve hit a snag a few pages in. In particular, at the bottom of page 4 I read

“Given an observable O and states psi_1 and psi_2 that are eigenvectors of O with eigenvalues lambda_1 and lambda_2, the state

c_1 psi_1 + c_2 psi_2

may not have a well-defined value for the observable O. If one attempts to measure this observable, one will get either lambda_1 or lambda_2, with probabilities

|c_1^2|/(|c_1^2| + |c_2^2|)

and

|c_2^2|/(|c_1^2| + |c_2^2|)”

However, these numbers don’t appear to be well-defined. For instance, if psi_1 is a lambda_1-eigenvector for O then 2 psi_1 is also a lambda_1-eigenvector for O, so we could regard the state

2 psi_1 + psi_2

as either

2 (psi_1) + 1 (psi_2),

giving us a 4/5 chance of observing lambda_1 and a 1/5 chance of observing lambda_2, or we could regard it as

1 (2 psi_1) + 1 (psi_2),

giving us a 50/50 chance of observing either lambda_1 or lambda_2. Are there perhaps some constraints on the psi_j here, say that they should be unit vectors with respect to the Hermitian form on H? (That interpretation seems to match up with something you wrote in the blog post above, assuming that this is the same thing as Born’s rule.) Or have I misinterpreted something?

And of course thanks for writing this up, making it publicly available, and taking the time to engage with your readers!

Thanks Daniel,

You’re right, that’s ambiguous. I was trying to be clever and write things in a normalization-independent way, but for what I wrote you need psi_1 and psi_2 to be normalized the same way, e.g. by having unit norm. I’ll fix this right now.

After scanning the first four chapters, I’m really excited about these notes. Many thanks for making them available!

Could you record a version number and/or date last updated on the cover page?

Charlie,

There should be a date on the first page of the pdf.