In the following two problems, consider the fictional "spinless" hydrogen atom: a zero-spin electron
bound to a zero-spin proton.
The classical formula for the potential of a dipole $\vec{\mu}$ interacting with a magnetic field $\vec{B}$ is $U = -\vec{\mu} \cdot \vec{B}$.
For a point charge of mass $m$ and charge $-e$ in a closed orbit, the formula for the dipole moment
is $\vec{\mu} = -\frac{e}{2m}\vec{L}$, where $\vec{L}$ is its angular momentum.
Correspondingly, the quantum-mechanical expression for the perturbation of the energy of an or-
biting electron by a magnetic field is (ignoring spin)
$\hat{H}\prime = \frac{e}{2m}\vec{L} \cdot \vec{B}$
1 Stern-Gerlach / Zeeman effect: the hard way
Use first-order degenerate perturbation theory to calculate the energy level shifts of the first excited
state of the spinless hydrogen atom if $\vec{B} = B\hat{z}$. Unfortunately for you, you are required to use
the basis of $\psi_{nlm}$'s specified by 4.89, construct the matrix form of $\hat{H}\prime$ in this basis, and find its
eigenvalues.
Hint 1: I would strongly recommend against evaluating the integrals here (unless you
love integrals). Instead, I'd express $L_x$ in terms of raising and lowering operators (as
you've done before), or use a matrix form similar to what you developed for spin=1 in
a previous homework.
Hint 2: The first excited state of the spinless hydrogen atom is 4-fold degenerate, so I'd
expect a 4x4 matrix here.
