3 Spherical Tensors Spherical tensors Objects that transform
3. Spherical Tensors Spherical tensors : Objects that transform like 2 nd tensors under rotations. § 15. 5 : {Ylm ; m = l, …, l } is a (2 l+1)-D basis for (irreducible) spherical tensors. Let Caution: we’ve used R to denote rotation in Euclidean, Hilbert & function spaces. Ylm orthonormal Dl (R) unitary : see Tung, § 7. 6 Eq. 16. 52 Warning: & many related eqs in Arfken are in error. Wigner matrices
Addition Theorem Consider Dl (R) unitary i. e. , A is a rotational invariance.
Set R such that Addition Theorem
Example 16. 3. 1. Angle Between Two Vectors l=1:
Spherical Wave Expansion Ex. 15. 2. 26 : Spherical Wave Expansion
Laplace Spherical Harmonic Expansion § 15. 3 :
Example 16. 3. 2. Ex. 16. 3. 9 : Set Spherical Green’s Function
Set § 10. 2 :
General Multipoles qi at ri : multipole moment (Caution : definition not unique) (r) = charge distribution
Multipole moment of unit charge placed at (x, y, z) : Mathematica Caution: Arfken’s table on p. 802 used Mlm differs from the conventional definition of multipoles by a scale factor. For given l, has 2 l+1 components but the Cartesian multipole has 3 l. Cartesian tensors are reducible.
Integrals of Three Ylm Warning: all eqs derived from eq. 16. 52 in Arfken are in error ( some R should be R 1 ) Lemma : Proof : Let QED
All Ylm evaluated at same point
if Triangle rule only if &
4. Vector Spherical Harmonics Vector Helmholtz eq. : Consider a complex 3 -D vector u is a spherical tensor of rank 1. Set Kj is related to the angular momentum Lj by Einstein notation
any vector & ( j ) exempts j from implicit summation is an eigenvector of K with eigenvalue Eigenvectors k of eigenvalue for K 3 are : Mathematica Condon-Shortley convention
Vector Coupling Vector spherical harmonics
i. e. Relation to Jackson’s vector harmonics (§ 16. 2) : Ex. 16. 4. 4
Partial Proof : Coef. of e 0 :
Useful Formulas Spatial inversion :
Ref: E. H. Hill, Am. J. Phys. 22, 211 (54)
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