The angular momentum eigenstates are eigenstates of two operators.
The differential operators take some work to derive.
Its easy to find functions that give the eigenvalue of .
Here we should note that only the integer value of work for these solutions. If we were to use half-integers, the wave functions would not be single valued, for example at and . Even though the probability may be single valued, discontinuities in the amplitude would lead to infinities in the Schrödinger equation. We will find later that the half-integer angular momentum states are used for internal angular momentum (spin), for which no or coordinates exist.
Therefore, the eigenstate is.
We call these eigenstates the Spherical Harmonics. The spherical harmonics are normalized.
We will use the actual function in some problems.
Any function of
can be expanded in the spherical harmonics.
The spherical harmonics form a complete set.
The spherical harmonics are related to the Legendre polynomials which are functions of .