# Eigenvalues of total angular momentum

1. Aug 25, 2004

### stefano

Who knows the formula to calculate the eigenvalues of total angular momentum between two different states? In particular, what is the matrix element of

<S, L, J, M_J | J^2 | S', L', J', M'_J> ?

Thank's...

2. Aug 25, 2004

### zefram_c

Eh? You've given the states here with j being one of the quantum numbers, which makes the problem trivial. Can you clarify?

3. Aug 25, 2004

### stefano

And so? What is the trivial answer?

4. Aug 25, 2004

### vanesch

Staff Emeritus
J^2 delta_(J,J') delta(s,s') delta(l,l') delta(m,m')

cheers,
Patrick

5. Aug 25, 2004

### Fredrik

Staff Emeritus

$$|j_1 j_2; m_1 m_2\rangle$$

or

$$|j_1 j_2;j m\rangle$$

If you're using the second option, the problem is trivial, as Vanesch said, but the eigenvalue of J² is

$$\hbar^2 j(j+1)$$

not j².

If you're using the first option, it gets much more complicated. In principle, you can calculate your matrix element if you first expand one of the kets in eigenkets of the second kind. The coefficients of the expansion are called Clebsch-Gordan coefficents.

6. Aug 26, 2004

### stefano

I am agree with you. In fact I made bad the question, because my original problem is that I need to write eigenstate of J for three particles. Then J=j_1+j_2+j_3; to do this I have to fix the projection (for example M=1/2) and the three sets of single particles for which M=1/2 are (5/2,1/2, -5/2), (3/2,1/2, -3/2) and (5/2, -1/2, -3/2) named A, B, C. Now I have to build the matrix U of elements of J, that I arrive at the original question: for example U_12 is <A|J^2|B> where |A> and |B> are
the states written before, but they aren't eigenstates of J^2.
I found a formula that may be mine answer:

<A| J |B> = delta(j_a, j_b)*delta(m_a+-1,m_b)*sqrt ((j_b+-m_b)*(j_b-+j_b+1))

with |a> = !j_a, m_a>
I need to perform all this matrix elements and then I have to diagonalize this matrix to have the three coefficients to combine |A>, |B> and |C>, in order to have eigenstates of J.

Sorry for my first question that was badly formed!