ABSTRACT
Computing a rotation in 3D space requires specifying both a rotation axis and an angle of rotation. The most elegant approach is to use Euler angles, which allows the rotation operator to be represented by a simple 3 by 3 unitary matrix. The Euler angles provide the foundation for the Wigner matrix, which performs the rotation of the eigenvalues of the angular momentum operator including spherical harmonics. This chapter explains the derivation of the Wigner matrix for integer and half-integer values of the orbital index. While the derivation for integer values of the orbital index uses Cartesian 3D space, the derivation for half-integer values relies on the concept of spinors. These appear in the math with the introduction of an isotropic vector of length zero. A detailed introduction to the theory of spinors is given and used in deriving the general form of the Wigner matrix. This is done using a purely algebraic approach, completely avoiding group theory. Derivation of the addition theorem for spherical harmonics then explains the expansion of the Legendre polynomials into a product of two spherical harmonic functions.
