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](https://www.dummies.com/wp-content/uploads/395073.image0.png)
which is the following:
![image1.png](https://www.dummies.com/wp-content/uploads/395074.image1.png)
And because
![image2.png](https://www.dummies.com/wp-content/uploads/395075.image2.png)
this equation can be written in this version:
![image3.png](https://www.dummies.com/wp-content/uploads/395076.image3.png)
Cancelling out terms from the two sides of this equation gives you this differential equation:
![image4.png](https://www.dummies.com/wp-content/uploads/395077.image4.png)
This looks easy to solve, and the solution is just
![image5.png](https://www.dummies.com/wp-content/uploads/395078.image5.png)
where C is a constant of integration.
You can determine C by insisting that
![image6.png](https://www.dummies.com/wp-content/uploads/395079.image6.png)
be normalized — that is, that the following hold true:
![image7.png](https://www.dummies.com/wp-content/uploads/395080.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](https://www.dummies.com/wp-content/uploads/395081.image8.png)
You are now able to determine the form of
![image9.png](https://www.dummies.com/wp-content/uploads/395082.image9.png)
which equals
![image10.png](https://www.dummies.com/wp-content/uploads/395083.image10.png)