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When to Use Spherical Coordinates Instead of Rectangular Coordinates

In quantum physics, you sometimes need to use spherical coordinates instead of rectangular coordinates. For example, say you have a 3D box potential, and suppose that the potential well that the particle is trapped in looks like this, which is suited to working with rectangular coordinates:

image0.png

Because you can easily break this potential down in the x, y, and z directions, you can break the wave function down that way, too, as you see here:

image1.png

Solving for the wave function gives you the following normalized result in rectangular coordinates:

image2.png

The energy levels also break down into separate contributions from all three rectangular axes:

E = Ex + Ey + Ez

And solving for E gives you this equation:

image3.png

But what if the potential well a particle is trapped in has spherical symmetry, not rectangular? For example, what if the potential well were to look like this, where r is the radius of the particle's location with respect to the origin and where a is a constant?

image4.png

Clearly, trying to stuff this kind of problem into a rectangular-coordinates kind of solution is only asking for trouble, because although you can do it, it involves lots of sines and cosines and results in a pretty complex solution. A much better tactic is to solve this kind of a problem in the natural coordinate system in which the potential is expressed: spherical coordinates.

The following figure shows the spherical coordinate system along with the corresponding rectangular coordinates, x, y, and z.

The spherical coordinate system.
The spherical coordinate system.

In the spherical coordinate system, you locate points with a radius vector named r, which has three components:

image6.png
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