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 .

To find the dependence, we will use the fact that there are limits on . The state with maximum must give zero when raised.

This gives us a differential equation for that state.

The solution is

Check the solution.

Its correct.

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 can continue to lower to get all of the eigenfunctions.

We call these eigenstates the **Spherical Harmonics**.
The spherical harmonics are **normalized**.

Since they are eigenfunctions of Hermitian operators, they are

We will use the **actual function** in some problems.

The spherical harmonics with negative can be easily compute from those with positive .

Any function of
and
can be **expanded in the spherical harmonics**.

The spherical harmonics form a **complete set**.

When using bra-ket notation, is sufficient to identify the state.

The spherical harmonics are **related to the Legendre polynomials** which are functions of
.