Angular momentum addition in quantum mechanics

[omoplata]

Homework Statement

There is a combined system with angular momenta $$j_{1}=\frac{1}{2}$$, $$j_{2}=\frac{3}{2}$$ and $$j_{3}=1$$. The Hamiltonian of the composite system depends only on the total angular momentum. What are the states of the combined system? What are the degeneracies of the eigenenergies? (neglect quantum numbers not related to angular momentum.

Homework Equations

Don't know.

The Attempt at a Solution

I have the solution.
$$\mathbf{1/2 \otimes 3/2 \otimes 1} = \mathbf{(1 \oplus 2) \otimes 1} = \mathbf{1 \otimes 1 \oplus 2 \otimes 1} = \mathbf{(0 \oplus 1 \oplus 2) \oplus (1 \oplus 2 \oplus 3)} = \mathbf{0 \oplus (} 2 \mathbf{\cdot 1 ) \oplus ( } 2 \mathbf{\cdot 2 ) \oplus 3}$$
There are $$1 + 6 + 10 + 7 = 24$$ states
There are only four different eigenenergies corresponding to $$j=0,1,2,3$$ with degeneracies $$1,6,10,7$$ respectively.

But I don't understand it. What is this algebra? The expression at the left of the second equal sign seems to be expanded to the one on the right using the distributive law. So distributive law is valid. That's pretty much all I understand about how to arrive at the final solution.

I skimmed through the addition of angular momenta of Sakurai, but can't find the relevant part where this is described. Can someone please refer me to a book where I can look this up?

Thanks.

 

[Mindscrape]

The notation is for the "tensor product" (the adding is the "direct sum"), and it functions quite a bit like the cartesian product that you've possibly seen in set builder notation. It's not used all that much that I know of, possibly quite a bit in particle physics (those people are goofy :p), but it's good to know I suppose. You could, of course, use your own method to count up the states, though you may eventually come to something a bit similar to the tensor product.

This article looks like it will let you in on the rules
http://socrates.berkeley.edu/~jemoore/p137a/spincorbo.pdf

My method would be to just say
2 states from j=1/2
4 states from j=3/2
3 states from j=1

the 2 and 4 will mix to give 8 states, and the 8 states there will mix to create 24. 2*4*3, yipee!

 

[vela]

Do you remember the rules for adding angular momenta? If you have S=S₁+S₂, with corresponding quantum numbers s, s₁, and s₂, the allowed values for s are |s₁-s₂| to s₁+s₂. So if you have s₁=1/2 and s₂=3/2, the allowed values for s are 1 and 2. This corresponds to writing 1/2⊗3/2 = 1⊕2.

 

[omoplata]

Thanks for the answer. I think I figured it out.