How to Change Rectangular Coordinates to Spherical Coordinates

In quantum physics, to find the actual eigenfunctions (not just the eigenstates) of angular momentum operators like L² and L_z, you turn from rectangular coordinates, x, y, and z, to spherical coordinates because it’ll make the math much simpler (after all, angular momentum is about things going around in circles). The following figure shows the spherical coordinate system.

The spherical coordinate system.

In the rectangular (Cartesian) coordinate system, you use x, y, and z to orient yourself. In the spherical coordinate system, you also use three quantities:

as the figure shows. You can translate between the spherical coordinate system and the rectangular one this way: The r vector is length of the vector to the particle that has angular momentum,

is the angle of r from the z axis, and

is the angle of r from the x axis.

Consider the equations for angular momentum:

When you take the angular momentum equations with the spherical-coordinate-system conversion equations, you can derive the following:

Okay, these equations look pretty involved. But there’s one thing to notice: They depend only on

which means their eigenstates depend only on

not on r. So the eigenfunctions of the operators in the preceding list can be denoted like this:

Traditionally, you give the name

to the eigenfunctions of angular momentum in spherical coordinates, so you have the following:

All right, time to work on finding the actual form of

You know that when you use the L² and L_z operators on angular momentum eigenstates, you get this:

So the following must be true:

In fact, you can go further. Note that L_z depends only on

which suggests that you can split

up into a part that depends on

and a part that depends on

Splitting

up into parts looks like this:

That’s what makes working with spherical coordinates so helpful — you can split the eigenfunctions up into two parts, one that depends only on

and one part that depends only on