2. Angular momentum (3) Summary of orbit and spin angular momentum Matrix elements Combination of angular momentum Clebsch-Gordan coefficients and 3-j symbols Irreducible Tensor Operators

3. Summary of orbit and spin angular momentum In General: Eigenvalues j=0,1/2,1,3/2,…; m j =-j, -j+1,…,j Eigenvector |j,m>

4. Ladder operators So is eigenstates of J 2 and J  : : Other important relations:

5. Matrix elements Denote the normalization factor as C: Similarly, we can calculate the norm for J -

6. Values of j and m and matrices For a given m value m 0, m 0 -n, m 0 -n+1,…,m 0, m 0 +1, are all possible values. So max(m)=j, min (m) = -j to truncate the sequence Matrix of J2, J+, J-, Jx, Jy, Jz J 2 diagonal, j(j+1) for each block Jz diagonal, j,j-1,…,-j for each block J +, J - upper or lower sub diagonal for each block Jx=(J + +J - )/2, Jy = =(J + -+J - )/2i also block diagonal

7. Submatrix for j=1/2, spin Pauli matrices:

8. Combination of angular momentum Angular momenta of two particles (  =x,y,z): Angular momentum is additive: It can be verified that obeys the commutation rules for angular momentum Construction of eigenstates of

9. Qualitative results So we can denote Other partners for J=j 1 +j 2 can be generated using the action of J - and J +

10. Qualitative results Mj 1 +j 2 j 1 +j 2 -1j 1 +j 2 -2…j 1 -j 2 …-j 1 …-j 1 -j 2 N(M)123…2j 2 +1 …1 So J=j 1 +j 2, j 1 +j 2 -1, …, j 1 -j 2 once and once only! Assume j 1  j 2 The two states of M= j 1 +j 2 -1, In general:

11. Clebsch-Gordan coefficients Projection of the above to and using orthornormal of basis Properties: 1.CGC can be chosen to be real; 2.CGC vanishes unless M=m 1 +m 2, |j 1 -j 2 |  J  j 1 +j 2 3.j 1 +j 2 +J is integer 4.Sum of square moduli of CGCs is 1 http://personal.ph.surrey.ac.uk/~phs3ps/cgjava.html

12. 3-j symbols Wigner 3-j symbols, also called 3j or 3-jm symbols, are related to Clebsch– Gordan coefficients through Properties: (1)Even permutations: (1 2 3) = (2 3 1) = (3 1 2) (2)Old permutation: (3 2 1) = (2 1 3) = (1 3 2 ） = (-1) j1+j2+j3 (1 2 3) (3)Chainging the sign of all Ms also gives the phase (-1) j1+j2+j3 http://plasma-gate.weizmann.ac.il/369j.html http://personal.ph.surrey.ac.uk/~phs3ps/tjjava.html

13. Irreducible Tensor Operators A set of operators T q k with integer k and q=-k,-k+1,…,k: Then T q k ’s are called a set of irreducible spherical tensors Wigner-Echart theorem: Example of irreducible tensors with k=1, and q=-1,0,1: (J 0 =J z, J 1 =-(J x +iJ y )/  2, J -1 = =(J x -iJ y )/  2 Similar for r, p

14. Products of tensors Tensors transform just like |j,m> basis, so Two tensors can be coupled just like basis to give new tensors: