A. LAPLACE OPERATOR

The operator ? ² can be written in spherical coordinates as

(A1)

(A2)

B. SPHERICAL HARMONICS

The spherical harmonics are eigenfunctions of the preceding operator L ² and of L _Z ,

(A3)

such that

(A4)

(A5)

They are related to the Legendre functions, for example,

(A6)

where , etc. They are orthonormal,

(A7)

Phases used here are the so-called Condon and Shortley phases,

(A8)

The spherical harmonic addition theorem states that

(A9)

where ? is the angle between vectors pointing, respectively, in the directions ( ?, ) and ( ? ?, ?). We often denote the angle ?, by , it being understood that ?, are polar coordinates of the vector r. Consequently,

(A10)

There is a very useful theorem for the product of two spherical harmonics of the same angles:

(A11a)

or

(A11b)

where the large parentheses ( ) are called 3- j symbols (described later) and . Note that the 3- j symbol with the zeros in the lower position requires that

(A12)

C. ANGULAR-MOMENTUM COUPLING; CLEBSCH-GORDAN COEFFICIENTS

Let and be eigenfunctions of angular momentum of, say, two quantum systems with the eigenvalues indicated. Sometimes they are denoted by

Denote by L, the vector sum

(A13)

Eigenfunctions of L ² and L _z with eigenvalues L( L+1) and M are

(A14a)

This is referred to as the vector coupling of