Quantum Physics For Dummies, Revised Edition
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Here’s an example that involves finding the rotational energy spectrum of a diatomic molecule. The figure shows the setup: A rotating diatomic molecule is composed of two atoms with masses m1 and m2. The first atom rotates at r = r1, and the second atom rotates at r = r2. What’s the molecule’s rotational energy?

A rotating diatomic molecule.
A rotating diatomic molecule.

The Hamiltonian is

image1.png

I is the rotational moment of inertia, which is

image2.png

where r = |r1r2| and

image3.png

Because

image4.png

Therefore, the Hamiltonian becomes

image5.png

So applying the Hamiltonian to the eigenstates, | l, m >, gives you the following:

image6.png

And as you know,

image7.png

so this equation becomes

image8.png

And because H | l, m > = E | l, m >, you can see that

image9.png

And that’s the energy as a function of l, the angular momentum quantum number.

About This Article

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Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.

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