Angular momentum (3)

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 j0,1/2,1,3/2, mj-j,
-j1,,j Eigenvector j,mgt
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-n1,,m0, m01,
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(j1) for each block Jz diagonal, j,j-1,,-j for
each block J, J- upper or lower sub diagonal for
each block Jx(JJ-)/2, Jy (J-J-)/2i also
block diagonal
6
Submatrix for j1/2, 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 Jj1j2 can be generated using
the action of J- and J
9
Qualitative results
Assume j1?j2
M j1j2 j1j2-1 j1j2-2 j1-j2 -j1-j1 -j1-j2
N(M) 1 2 3 2j21 2j21 2j21 1
So Jj1j2, j1j2-1, , j1-j2 once and once only!
The two states of M j1j2-1,
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 Mm1m2, j1-j2?J? j1j2
• j1j2J is integer
• Sum of square moduli of CGCs is 1

http//personal.ph.surrey.ac.uk/phs3ps/cgjava.htm
l
11
3-j symbols

• Wigner 3-j symbols, also called 3j or 3-jm
  symbols, are related to ClebschGordan
  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)j1j2j3 (1 2 3)
• Chainging the sign of all Ms also gives the phase
  (-1)j1j2j3

http//plasma-gate.weizmann.ac.il/369j.html
http//personal.ph.surrey.ac.uk/phs3ps/tjjava.htm
l
12
Irreducible Tensor Operators

• A set of operators Tqk with integer k and
  q-k,-k1,,k
• Then Tqks are called a set of irreducible
  spherical tensors
• Wigner-Echart theorem

Example of irreducible tensors with k1, and
q-1,0,1 (J0Jz, J1-(JxiJy)/?2, J-1
(Jx-iJy)/?2 Similar for r, p
13
Products of tensors

• Tensors transform just like j,mgt basis, so Two
  tensors can be coupled just like basis to give
  new tensors