Find the Eigenfunctions of Lz in Spherical Coordinates
At some point, your quantum physics instructor may ask you to find the eigenfunctions of Lz in spherical coordinates. In spherical coordinates, the Lz 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
