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Quantum Physics For Dummies, Revised Edition by Steven Holzner

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Chapter 7

Rectangular Coordinates: Solving Problems in Three Dimensions

In This Chapter

  • Exploring the Schrödinger equation in the x, y, and z dimensions
  • Working with free particles in 3D
  • Getting into rectangular potentials
  • Seeing harmonic oscillators in 3D space

One-dimensional problems are all very well and good, but the real world has three dimensions. This chapter is all about leaving one-dimensional potentials behind and starting to take a look at spinless quantum mechanical particles in three dimensions.

Here, you work with three dimensions in rectangular coordinates, starting with a look at the Schrödinger equation in glorious, real-life 3D. You then delve into free particles, box potentials, and harmonic oscillators. Note: By the way, the next chapter uses spherical coordinates because some problems are better in one system than the other. Problems with spherical symmetry are best handled in spherical coordinates, for example.

The Schrödinger Equation: Now in 3D!

In one dimension, the time-dependent Schrödinger equation (of the type in Chapters 3 and 4 that let you find the wave function) looks like this:

image

And you can generalize that into three dimensions like this:

image

Using the Laplacian operator, you can recast this into a more compact form. Here's what the Laplacian looks ...

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