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Quantum Physics For Dummies, Revised Edition by Steven Holzner

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Chapter 5

Working with Angular Momentum on the Quantum Level

In This Chapter

  • Angular momentum
  • Angular momentum and the Hamiltonian
  • Matrix representation of angular momentum
  • Eigenfunctions of angular momentum

In classical mechanics, you may measure angular momentum by attaching a golf ball to a string and whirling it over your head. In quantum mechanics, think in terms of a single molecule made up of two bound atoms rotating around each other. That's the level at which quantum mechanical effects become noticeable. And at that level, it turns out that angular momentum is quantized. And since that has tangible results in many cases, such as the spectrum of excited atoms, it's an important topic.

Besides having kinetic and potential energy, particles can also have rotational energy. Here's what the Hamiltonian (total energy — see Chapter 4) looks like:

images

Here, L is the angular momentum operator and I is the rotation moment of inertia. What are the eigenstates of angular momentum? If L is the angular momentum operator, and l is an eigenvalue of L, then you could write the following:

images

But that turns out to be incomplete because angular momentum is a vector in three-dimensional space — and it can be pointing any direction. Angular momentum is typically given by a magnitude and a component ...

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