Quantum Physics For Dummies, Revised Edition

Chapter 8

Solving Problems in Three Dimensions: Spherical Coordinates

In This Chapter

• Problems in spherical coordinates
• Free particles in spherical coordinates
• Square well potentials
• Isotropic harmonic oscillators

In your other life as a sea captain-slash-pilot, you're probably pretty familiar with latitude and longitude — coordinates that basically name a couple of angles as measured from the center of the Earth. Put together the angle east or west, the angle north or south, and the all-important distance from the center of the Earth, and you have a vector that gives a good description of location in three dimensions. That vector is part of a spherical coordinate system.

Navigators talk more about the pair of angles than the distance (“Earth's surface” is generally specific enough for them), but quantum physicists find both angles and radius length important. Some 3D quantum physics problems even allow you to break down a wave function into two parts: an angular part and a radial part.

In this chapter, I discuss three-dimensional problems that are best handled using spherical coordinates. (For 3D problems that work better in rectangular coordinate systems, see Chapter 7.)

A New Angle: Choosing Spherical Coordinates Instead of Rectangular

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:

Because you can easily break this potential down in the x, y, and ...