1 / 13

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

jess
Download Presentation

Angular momentum (3)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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:

More Related