Quantum Physics For Dummies
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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


I is the rotational moment of inertia, which is


where r = |r1r2| and




Therefore, the Hamiltonian becomes


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


And as you know,


so this equation becomes


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


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

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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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