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.

The Hamiltonian is

I is the rotational moment of inertia, which is

where r = |r₁ – r₂| and

Because

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.