Advice on coursework - Is Statmech Important?

[Broccoli21]

Hey PF,
I'm in a bit of a tight spot with respect to classes next year, and essentially I have to choose between:
1) Taking Statistical mechanics in fall, then abstract algebra I in spring (then representation theory in junior year spring)
2) Taking abstract 1 in fall, representation theory in spring (which means statistical mechanics will be moved to junior year fall)

I am interested in theoretical physics, but also really like linear algebra and representation theory sounds really cool. I was just wondering - which choice would be the most useful for a physicist? Is stat mech really that important?

Feel free to ask me any questions to clarify, and thanks in advance!

 

[chiro]

Hey Broccoli21.

My advice would be to take the Statistical Mechanics. The reason for this is that you are doing a physics degree instead of a mathematics degree, even though it is theoretical physics.

In my view, it's more important that you have a grounded physics education to be a physicist (experimental, theoretical, whatever) as opposed to being a mathematician.

Chances are you can take the math courses required later on and if you end up working as a physicist, you'll probably end up teaching yourself the math required if you do require it by just reading papers and journals, getting book recommendations and so on: in other words you won't have to take classes to learn something, but rather just learn it by yourself more or less when you need to.

 

[nonequilibrium]

Statistical mechanics really is an important area of mathematical physics; it introduces a new way of thinking that is often used in a lot of other courses and other courses tend to plug in stat mech formulas when suitable, without giving explanations, so you'd be left mystified without a background in stat mech. At least that is my personal experience. That being said, I also simply love stat mech for its own sake (it's the most conceptually clear branch of theoretical physics if you ask me), just mentioning that since I understand the name "statistical mechanics" sounds quite boring.

Algebra and more specifically representation theory are also quite useful for a theoretical physicist, and the further you go, the more important they get, but stat mech is important from the get-go, so I wouldn't wait with that.

EDIT: well I wouldn't wait with algebra either, since it's probably the backbone for other math courses you're planning to take. Would it mess with your math class schedule (if existing!) if you take algebra in spring? (for physics classes, algebra isn't urgent at all) If not, then the order stat mech, algebra, representation seems ideal and the one I would prefer.

 

[lsaldana]

Broccoli21,

I heavily suggest that you take Statistical Mechanics as soon as possible, especially if you've completed thermodynamics and (at least) quantum mechanics 1. Abstract algebra really is a beautiful subject and will show up (especially group theory) in advanced physics courses although most likely in graduate school. However, if your interests are primarily in physics, statistical mechanics is a must. You will need the material in preparation for the Physics GRE and for other advanced topics, e.g. solid state physics. As for me, I opted to fill up on physics electives and leave some of the math for self-teaching. For instance, I'm doing research in theoretical nonlinear dynamics at the moment and have had to teach myself group theory. Unfortunately for me, I didn't find the time to take the formal course since I'm headed to a ph.d. physics program in the fall. But it sounds like you're a sophomore (?) so you'll have plenty of time.

 

[Fredrik]

Statistical mechanics is important. Without it, you don't even know what temperature is. But I don't think it's so urgent that you need to take it as soon as possible. Do it when it's convenient. Just don't try to skip it entirely.

Abstract algebra is one of those subjects I think you might just want to study on your own, and then only study the parts you need, instead of taking a whole course on it. What you need is usually just the definitions of rings, fields etc., and a little about homomorphisms and isomorphisms.

Representation theory is very useful in QM. You need it to understand things like spin, and what a particle is. (This point of view is explained in chapter 2 of Weinberg's QFT book).

 

[Broccoli21]

Thanks for the replies folks, and sorry for the late-ish response. I wasn't aware that statmech was really all that important apparently. Will statistical mechanics help me if I do research in quantum mechanics/be of use in further courses in physics?

As for Algebra, I don't think the math department will let me take rep theory until I take algebra I, so if I don't take it next year, then i'll be a year behind in algebra courses. I will also be taking a QFT course in junior year. Will representation theory help there?

Thanks again guys!

 

[Fredrik]

Will statistical mechanics help me if I do research in quantum mechanics/be of use in further courses in physics?

It might. It depends on what you will do research about. I'm sure you will encounter something that will touch on what you learned in your statistical mechanics course, but I can't guarantee that it will be absolutely necessary for you to have taken that course. It would however be pretty weird if you start doing research in QM without really knowing what temperature and entropy are, so you should make sure that you at least learn the basics.

It won't be used much in other courses. But it might be used a little. I don't think there's a course that you would be unlikely to pass just because you haven't taken statistical mechanics.

As for Algebra, I don't think the math department will let me take rep theory until I take algebra I, so if I don't take it next year, then i'll be a year behind in algebra courses. I will also be taking a QFT course in junior year. Will representation theory help there?

Many books on QFT don't even mention representations. I would however recommend that you study chapter 2 of Weinberg (skip chapter 1) no matter what book is used for your QFT course, and representation theory will definitely help you there. On the other hand, it's definitely possible to read that chapter without having taken a course in representation theory.

If you have taken a QM class or two, you may already have done some representation theory without knowing it. Denote the spin operators by $S_1,S_2,S_3$. The proof that $\vec S^2$ and $S_3$ has common eigenstates, that the eigenvalues of $\vec S^2$ are of the form j(j+1) where j is an integer, and that an eigenvalue m of $S_3$ satisfies $-j\leq m\leq j$, that's all representation theory. It's the problem of finding all irreducible representations of the Lie algebra of the rotation group SO(3).

Sounds a bit odd to require abstract algebra before representation theory. I think all you will need from abstract algebra is the definition of a group, and an understanding of homomorphisms and isomorphisms.