Find the Eigenfunctions of Lz in Spherical Coordinates

At some point, your quantum physics instructor may ask you to find the eigenfunctions of L_z in spherical coordinates. In spherical coordinates, the L_z operator looks like this:

which is the following:

And because

this equation can be written in this version:

Cancelling out terms from the two sides of this equation gives you this differential equation:

This looks easy to solve, and the solution is just

where C is a constant of integration.

You can determine C by insisting that

be normalized — that is, that the following hold true:

(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.)

So this gives you

You are now able to determine the form of

which equals