The Spherical Harmonics can be generalized to vector spherical harmonics by looking for
a Scalar Function and a constant Vector such that
(1) |
(2) |
(3) | |||
(4) |
(5) |
(6) |
Construct another vector function
(7) |
(8) |
(9) |
(10) |
In this formalism, is called the generating function and 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 is taken as
(11) |
(12) |
(13) |
(14) |
A number of conventions are in use. Hill (1954) defines
(15) | |||
(16) | |||
(17) |
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.
© 1996-9 Eric W. Weisstein