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Vector Spherical Harmonic

The spherical harmonics can be generalized to vector spherical harmonics by looking for a scalar function psi and a constant vector c such that

M=del x(cpsi)
(1)
=psi(del xc)+(del psi)xc
(2)
=(del psi)xc
(3)
=-cxdel psi,
(4)

so

del ·M=0.
(5)

Now interchange the order of differentiation and use the fact that multiplicative constants can be moving inside and outside the derivatives to obtain

del ^2M=del ^2(cxpsi)
(6)
=del xdel ^2(cpsi)
(7)
=del x(cdel ^2psi)
(8)

and

k^2M=k^2del x(cpsi)
(9)
=del x(ck^2psi).
(10)

Putting these together gives

del ^2M+k^2M=del x[c(del ^2psi+k^2psi)],
(11)

so M satisfies the vector Helmholtz differential equation if psi satisfies the scalar Helmholtz differential equation

del ^2psi+k^2psi=0.
(12)

Construct another vector function

N=(del xM)/k,
(13)

which also satisfies the vector Helmholtz differential equation since

del ^2N=1/kdel ^2(del xM)
(14)
=1/kdel x(del ^2M)
(15)
=1/kdel x(-k^2M)
(16)
=-kdel xM
(17)
=-k^2N,
(18)

which gives

del ^2N+k^2N=0.
(19)

We have the additional identity

del xN=1/kdel x(del xM)
(20)
=1/kdel (del ·M)-1/kdel ·(del M)
(21)
=-1/kdel ·(del M)
(22)
=-(del ^2M)/k
(23)
=kM.
(24)

In this formalism, psi is called the generating function and 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 M is taken as

M=del x(rpsi),
(25)

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

M·r=r·(del psixc)
(26)
=(del psi)·(cxr)
(27)
=0,
(28)

so

cxr=0
(29)

and we may take

c=r
(30)

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

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

V_l^m=-sqrt((l+1)/(2l+1))Y_l^mr^^+1/(sqrt((l+1)(2l+1)))(partialY_l^m)/(partialtheta)theta^^+(iMsintheta)/(sqrt((l+1)(2l+1)))Y_l^mphi^^
(31)
W_l^m=sqrt(l/(2l+1))Y_l^mr^^+1/(sqrt(l(2l+1)))(partialY_l^m)/(partialtheta)theta^^+(iM)/(sqrt(l(2l+1))sintheta)Y_l^mphi^^
(32)
X_l^m=-M/(sqrt(l(l+1))sintheta)Y_l^mtheta^^-i/(sqrt(l(l+1)))(partialY_l^m)/(partialtheta)phi^^.
(33)

Morse and Feshbach (1953) define vector harmonics called B, C, and P using rather complicated expressions.

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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.

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Vector Spherical Harmonic

Cite this as:

Weisstein, Eric W. "Vector Spherical Harmonic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VectorSphericalHarmonic.html

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