Spherical harmonics play an important role in electrodynamics and in quantum mechanics. We shall show how they appear in the representation theory of the rotation group SO(3). Each irreducible representation of SO(3) can be realized in a finite-dimensional Hilbert space of functions on the sphere, the restrictions of harmonic homogeneous polynomials of a given degree, and this representation is unitary. We shall determine an orthonormal basis of this space that is transformed in a simple way under the action of the group of rotations: the elements of such a basis are simultaneous eigenfunctions for the operators J 3 and J 2, defined below. The functions thus defined on the sphere are called spherical harmonics.
KeywordsIrreducible Representation Orthonormal Basis Nonnegative Integer Spherical Harmonic Homogeneous Polynomial
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