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 & many related eqs in Arfken are in error.
i.e., A is a rotational invariance. Addition Theorem Consider D l (R) unitary
Set R such that Addition Theorem
Example Angle Between Two Vectors l = 1 :
Spherical Wave Expansion Ex : Spherical Wave Expansion
Laplace Spherical Harmonic Expansion § 15.3 :
Example Spherical Green’s Function Ex : Set
§ 10.2 :
General Multipoles q i at r i : multipole moment (Caution : definition not unique) (r) = charge distribution
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.
Integrals of Three Y l m Lemma : Proof :Let QED Warning: all eqs derived from eq in Arfken are in error ( some R should be R 1 )
All Y l m evaluated at same point
ifTriangle 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 K j is related to the angular momentum L j by Einstein notation
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
Vector Coupling Vector spherical harmonics
i.e. Relation to Jackson’s vector harmonics (§16.2) : Ex
Partial Proof : Coef. of e 0 :
Useful Formulas Spatial inversion :
Ref: E.H.Hill, Am.J.Phys. 22,211 (54)