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# Angular momentum (3)

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

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## Angular momentum (3)

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1. 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

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

3. Ladder operators So is eigenstates of J2 and J:: Other important relations:

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

5. Values of j and m and matrices For a given m value m0, m0-n, m0-n+1,…,m0, m0+1, are all possible values. So max(m)=j, min (m) = -j to truncate the sequence Matrix of J2, J+, J-, Jx, Jy, Jz J2 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

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

7. 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

8. Qualitative results So we can denote Other partners for J=j1+j2 can be generated using the action of J- and J+

9. Qualitative results Assume j1j2 So J=j1+j2, j1+j2-1, …, j1-j2 once and once only! The two states of M= j1+j2-1, In general:

10. Clebsch-Gordan coefficients Projection of the above to and using orthornormal of basis • Properties: • CGC can be chosen to be real; • CGC vanishes unless M=m1+m2, |j1-j2|J j1+j2 • j1+j2+J is integer • Sum of square moduli of CGCs is 1 http://personal.ph.surrey.ac.uk/~phs3ps/cgjava.html

11. 3-j symbols Wigner 3-j symbols, also called 3j or 3-jm symbols, are related to Clebsch–Gordan coefficients through Properties: • Even permutations: (1 2 3) = (2 3 1) = (3 1 2) • Old permutation: (3 2 1) = (2 1 3) = (1 3 2） = (-1)j1+j2+j3 (1 2 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

12. Irreducible Tensor Operators • A set of operators Tqk with integer k and q=-k,-k+1,…,k: • Then Tqk’s are called a set of irreducible spherical tensors • Wigner-Echart theorem: Example of irreducible tensors with k=1, and q=-1,0,1: (J0=Jz, J1=-(Jx+iJy)/2, J-1= =(Jx-iJy)/2 Similar for r, p

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

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