## Angular Momentum

For the common problem of central potentials , we use the obvious rotational symmetry to find that the angular momentum, , operators commute with ,

but they do not commute with each other.

We want to find two mutually commuting operators which commute with , so we turn to which does commute with each component of .

We chose our two operators to be and .

Some computation reveals that we can write

With this the kinetic energy part of our equation will only have derivatives in assuming that we have eigenstates of .

The Schrödinger equation thus separates into an angular part (the term) and a radial part (the rest). With this separation we expect (anticipating the angular solution a bit)

will be a solution. The will be eigenfunctions of

so the radial equation becomes

We must come back to this equation for each which we want to solve.

We solve the angular part of the problem in general using angular momentum operators. We find that angular momentum is quantized.

with and integers satisfying the condition . The operators that raise and lower the component of angular momentum are

We derive the functional form of the Spherical Harmonics using the differential form of the angular momentum operators.

Jim Branson 2013-04-22