Vector Spherical Harmonic

The Spherical Harmonics can be generalized to vector spherical harmonics by looking for a Scalar Function $\psi$ and a constant Vector ${\bf c}$ such that

$${\bf M} \equiv \nabla\times({\bf c}\psi)=\psi(\nabla\times{\bf c})+(\nabla\psi)\times {\bf c}$$

$$= (\nabla\psi)\times{\bf c} = -{\bf c}\times\nabla\psi,$$ (1)

so

$$\nabla\cdot{\bf M}=0.$$ (2)

Now use the vector identities

$$\nabla^2{\bf M} = \nabla^2(\nabla\times{\bf M}) = \nabla\times(\nabla^2{\bf M})$$

$$= \nabla\times(\nabla^2{\bf c} \psi) = \nabla\times({\bf c}\nabla^2\psi)$$ (3)

$$k^2{\bf M} = k^2\nabla\times({\bf c}\psi) = \nabla\times({\bf c}\nabla^2\psi),$$ (4)

so

$$\nabla^2{\bf M}+k^2{\bf M}=\nabla\times[{\bf c}(\nabla^2\psi+k^2\psi)],$$ (5)

and ${\bf M}$ satisfies the vector Helmholtz Differential Equation if $\psi$ satisfies the scalar Helmholtz Differential Equation

$$\nabla^2\psi+k^2\psi=0.$$ (6)

Construct another vector function

$${\bf N}\equiv {\nabla\times{\bf M}\over k},$$ (7)

which also satisfies the vector Helmholtz Differential Equation since

$$\nabla^2{\bf N} = {1\over k}\nabla^2(\nabla\times{\bf M})={1\over k}\nabla\times(\nabla^2{\bf M})$$

$$= {1\over k}\nabla\times(-k^2{\bf M}) = -k\nabla\times{\bf M}=-k^2{\bf N},$$ (8)

which gives

$$\nabla^2{\bf N}+k^2{\bf N}=0.$$ (9)

We have the additional identity

$$\nabla\times{\bf N} = {1\over k}\nabla\times(\nabla\times{\bf M}) = {1\over k}\nabla(\nabla\cdot {\bf M})$$

$$= {1\over k}\nabla^2 {\bf M} -{1\over k}\nabla^2{\bf M}= {-\nabla^2{\bf M}\over k} = k{\bf M}.$$ (10)

In this formalism, $\psi$ is called the generating function and ${\bf c}$ is called the Pilot Vector. The choice of generating function is determined by the symmetry of the scalar equation, i.e., it is chosen to solve the desired scalar differential equation. If ${\bf M}$ is taken as

$${\bf M}=\nabla\times({\bf r}\psi),$$ (11)

where ${\bf r}$ is the radius vector, then ${\bf M}$ is a solution to the vector wave equation in spherical coordinates. If we want vector solutions which are tangential to the radius vector,

$${\bf M}\cdot {\bf r}={\bf r}\cdot(\nabla\psi\times{\bf c}) = (\nabla\psi)({\bf c}\times{\bf r})=0,$$ (12)

so

$${\bf c}\times{\bf r}={\bf0}$$ (13)

and we may take

$${\bf c}={\bf r}$$ (14)

(Arfken 1985, pp. 707-711; Bohren and Huffman 1983, p. 88).

A number of conventions are in use. Hill (1954) defines

$${\bf V}_l^m \equiv -\sqrt{l+1\over 2l+1} Y_l^m \hat{\bf r} + {1\over\sqrt{(l+1)(2l+1)}} {\partial Y_l^m\over\partial\theta}\hat{\boldsymbol{\theta}}$$

$$\phantom{=} \mathop{+} {iM\sqrt{(l+1)(2l+1)}\sin\theta} Y_l^m \hat{\boldsymbol{\phi}}$$ (15)

$${\bf W}_l^m = \sqrt{l\over 2l+1} Y_l^m \hat{\bf r} + {1\over\sqrt{l(2l+1)}} {\p... ...ymbol{\theta}}+ {iM\over\sqrt{l(2l+1)}\sin\theta} Y_l^m \hat{\boldsymbol{\phi}}$$ (16)

$${\bf X}_l^m = - {M\over\sqrt{l(l+1)}\sin\theta} Y_l^m \hat{\boldsymbol{\theta}}... ...\over\sqrt{l(l+1)}} {\partial Y_l^m\over\partial\theta}\hat{\boldsymbol{\phi}}.$$ (17)

Morse and Feshbach (1953) define vector harmonics called ${\bf B}$, ${\bf C}$, and ${\bf P}$ using rather complicated expressions.

References

Arfken, G. ``Vector Spherical Harmonics.'' §12.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 707-711, 1985.

Blatt, J. M. and Weisskopf, V. ``Vector Spherical Harmonics.'' Appendix B, §1 in Theoretical Nuclear Physics. New York: Wiley, pp. 796-799, 1952.

Bohren, C. F. and Huffman, D. R. Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983.

Hill, E. H. ``The Theory of Vector Spherical Harmonics.'' Amer. J. Phys. 22, 211-214, 1954.

Jackson, J. D. Classical Electrodynamics, 2nd ed. New York: Wiley, pp. 744-755, 1975.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part II. New York: McGraw-Hill, pp. 1898-1901, 1953.