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.](https://www.dummies.com/wp-content/uploads/395023.image0.jpg)
A rotating diatomic molecule.
The Hamiltonian is
![image1.png](https://www.dummies.com/wp-content/uploads/395024.image1.png)
I is the rotational moment of inertia, which is
![image2.png](https://www.dummies.com/wp-content/uploads/395025.image2.png)
where r = |r1 – r2| and
![image3.png](https://www.dummies.com/wp-content/uploads/395026.image3.png)
Because
![image4.png](https://www.dummies.com/wp-content/uploads/395027.image4.png)
Therefore, the Hamiltonian becomes
![image5.png](https://www.dummies.com/wp-content/uploads/395028.image5.png)
So applying the Hamiltonian to the eigenstates, | l, m >, gives you the following:
![image6.png](https://www.dummies.com/wp-content/uploads/395029.image6.png)
And as you know,
![image7.png](https://www.dummies.com/wp-content/uploads/395030.image7.png)
so this equation becomes
![image8.png](https://www.dummies.com/wp-content/uploads/395031.image8.png)
And because H | l, m > = E | l, m >, you can see that
![image9.png](https://www.dummies.com/wp-content/uploads/395032.image9.png)
And that’s the energy as a function of l, the angular momentum quantum number.