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Derive the Formula for the Rotational Energy of a Diatomic Molecule

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.

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