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:

image0.png

which is the following:

image1.png

And because

image2.png

this equation can be written in this version:

image3.png

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

image4.png

This looks easy to solve, and the solution is just

image5.png

where C is a constant of integration.

You can determine C by insisting that

image6.png

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

image7.png

(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

image8.png

You are now able to determine the form of

image9.png

which equals

image10.png
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