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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 Lz, 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.
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:

image1.png

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,

image2.png

is the angle of r from the z axis, and

image3.png

is the angle of r from the x axis.

image4.png

Consider the equations for angular momentum:

image5.png

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

image6.png

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

image7.png

which means their eigenstates depend only on

image8.png

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

image9.png

Traditionally, you give the name

image10.png

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

image11.png

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

image12.png

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

image13.png

So the following must be true:

image14.png

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

image15.png

which suggests that you can split

image16.png

up into a part that depends on

image17.png

and a part that depends on

image18.png

Splitting

image19.png

up into parts looks like this:

image20.png

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

image21.png

and one part that depends only on

image22.png
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