| 释义 |
Vector Spherical HarmonicThe Spherical Harmonics can be generalized to vector spherical harmonics by looking fora Scalar Function  and a constant Vector  such that
so
 | (2) |
so
 | (5) |
 satisfies the vector Helmholtz Differential Equation if satisfies the scalar HelmholtzDifferential Equation
 | (6) |
Construct another vector function
 | (7) |
which gives
 | (9) |
In this formalism, is called the generating function and is called the Pilot Vector. The choiceof generating function is determined by the symmetry of the scalar equation, i.e., it is chosen to solve the desiredscalar differential equation. If is taken as
 | (11) |
 is the radius vector, then is a solution to the vector wave equation in spherical coordinates. Ifwe want vector solutions which are tangential to the radius vector,
 | (12) |
 | (13) |
 | (14) |
A number of conventions are in use. Hill (1954) defines
Morse and Feshbach (1953) define vector harmonics called  ,  , and  using rather complicatedexpressions.
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.
|
