# Angular momentum coupling - worked out examples?

## Main Question or Discussion Point

3j, 6j, 9j -symbols, Wigner-Eckart theorem, reducible matrix element, irreducible (spherical) tensor. I just cannot understand the notation. I just cannot fully understand the physics behind those concepts. Where can I find fully worked out example calculations?

Especially, the notation is difficult for me. Please, give some help. I am a Bachelor of Science in Physics. :uhh:

## Answers and Replies

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quasar987
Homework Helper
Gold Member

Things like |a>|b> actually represent the tensor product of the vectors |a> and |b>. If you know that, and understand the mathematics of tensor products, and understand what a tensor product of kets represent physically, then you should, in principle, not have any more difficulty solving problems.

I personally prefer this approach to looking at a lot of examples and trying to osmose in a problem solving algorithm. I cannot redirect you to solved problems but if you want to try and tame the beast the way I suggested, then have a look at Cohen-Tanoudji's Quantum Mechanics Vol. I Chapter II §F (p.153 in my edition): Tensor product of State Space.

I used Cohen-Tanoudji in the past. At the time I hated it because I could never find anything I was looking for. Ive used it to study on my own a bit more recently and some things I was fuzzy about involving tensor products came clear. I also didn't like their over use of notation for EVERYTHING, the first time around, but it proved to be helpfull when i was reviewing angular momentum. They have a nice worked example involving the fine and hyper-fine structure of hydrogen in vol II somewhere, but the Wigner-Eckart theorem is relegated to one of the post chapter appendices. I also like their treatment of identical particles.

Another angular momentum gold mine is Sakurai's book Modern Quantum Mechanics. He treats this along side symmetries and does a very nice job.

Tom Mattson
Staff Emeritus
Gold Member
If it's sample problems you want then get the Schaum's Outline called Quantum Mechanics. It's at a level similar to Sakurai.

Dr Transport
Gold Member
3j, 6j, 9j -symbols, Wigner-Eckart theorem, reducible matrix element, irreducible (spherical) tensor. I just cannot understand the notation. I just cannot fully understand the physics behind those concepts. Where can I find fully worked out example calculations?

Especially, the notation is difficult for me. Please, give some help. I am a Bachelor of Science in Physics. :uhh:
Normally these topics are taught in a graduate course, Cohen-Tanoudji has been mentioned, Liboff has some but not much. At a more advanced level, try Tinkham's Group Theory and Quantum Mechanics book (Dover). The problem I have found in the past is that you do not see anything worked out in detail. An oldie is Condon and Shortley or even Wigner's book, they are highly mathematical but if you can wade through them you will come out ahead. I'd start with Condon and Shortley because they actually work out some of the numerics you want to know.

dextercioby