1. 3.Spherical Tensors Spherical tensors : Objects that transform like 2 nd tensors under rotations.  {Y l m ; m =  l, …, l } is a (2l+1)-D basis for (irreducible) spherical tensors. § 15.5 : Let   see Tung, §7.6 Y l m orthonormal  D l (R) unitary : Wigner matrices Caution: we’ve used R to denote rotation in Euclidean, Hilbert & function spaces. Warning: Eq.16.52 & many related eqs in Arfken are in error.

2.  i.e., A is a rotational invariance. Addition Theorem Consider  D l (R) unitary

3. Set R such that    Addition Theorem

4. Example 16.3.1. Angle Between Two Vectors l = 1 : 

5. Spherical Wave Expansion  Ex.15.2.26 :   Spherical Wave Expansion

6. Laplace Spherical Harmonic Expansion § 15.3 : 

7. Example 16.3.2.Spherical Green’s Function Ex.16.3.9 :  Set 

8.  § 10.2 : 

9. General Multipoles q i at r i : multipole moment (Caution : definition not unique)  (r) = charge distribution

10. Multipole moment of unit charge placed at (x, y, z) : Caution: Arfken’s table on p.802 used Mathematica M l m differs from the conventional definition of multipoles by a scale factor. For given l, has 2l+1 components but the Cartesian multipole has 3 l.  Cartesian tensors are reducible.

11. Integrals of Three Y l m Lemma : Proof :Let    QED Warning: all eqs derived from eq.16.52 in Arfken are in error ( some R should be R  1 )

12.   All Y l m evaluated at same point

13.   

14. 

15.  ifTriangle rule  only if &

16. 4.Vector Spherical Harmonics Vector Helmholtz eq. : Consider a complex 3-D vector  u is a spherical tensor of rank 1. Set  K j is related to the angular momentum L j by  Einstein notation  

17.  any vectoris an eigenvector of K with eigenvalue  & ( j ) exempts j from implicit summation Eigenvectors k of eigenvalue for K 3 are : Mathematica Condon-Shortley convention

18. Vector Coupling Vector spherical harmonics 

19. i.e. Relation to Jackson’s vector harmonics (§16.2) : Ex.16.4.4

20.  

21.   

22. Partial Proof : Coef. of e 0 :

23. Useful Formulas Spatial inversion :

24. Ref: E.H.Hill, Am.J.Phys. 22,211 (54)