Eigenstates of rotor with magnetic field (Zeeman effect)

Question

The representation of angular momentum (of $L^2$ and $L_z$) in the space is given by

$$ \langle\vec{r}|L_z|\psi\rangle = \frac{\hbar}{i}\frac{\partial}{\partial\phi}\langle\vec{r}|\psi\rangle $$

and $$ \langle\vec{r}|L^2|\psi\rangle = -\hbar^2\left[\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2} + \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)\right]\langle\vec{r}|\psi\rangle $$

I want to solve the Zeeman's effect problem, consisting of a Hamiltonian of the form

$$ H = \frac{L^2}{2\mu R^2} - \frac{eBL_z}{2\mu}, $$ where $\mu,R,e,B$ are constants. I have to solve the equation

$$ H|\psi\rangle = E|\psi\rangle. $$

Clearly the eigenstates must be eigenstates of $L^2$ and $L_z$ simultaneously. I write this as $|\psi\rangle = |\ell, m \rangle$. Taking into account that we define

$$ L^2|\ell, m\rangle = \hbar^2\ell(\ell+1)|\ell, m\rangle, \quad\quad L_z|\ell, m\rangle = \hbar m |\ell, m\rangle. $$

Then we can find the eigenenergies: $$ E_{\ell, m} = \frac{\hbar^2\ell(\ell+1)}{r\mu R^2}-\frac{e\hbar B}{2\mu}m $$

I want to find the eigenstates, therefore I do the following: $$ \langle\theta, \phi | H |\ell,m\rangle = E_{\ell,m}\langle\theta, \phi |\ell,m\rangle. $$

This differential equation is difficult to solve. Is my approach correct to find the eigenstates?

Thank you for your help.

Answer 1

In the standard angular momentum (=spherical harmonic) basis, your Hamiltonian is already diagonal, so no further diagonalisation is needed. In other words, the usual basis is $|\ell,m\rangle$, defined as \begin{aligned} L^2|\ell,m\rangle&=\ell(\ell+1)|\ell,m\rangle\\ L_z|\ell,m\rangle&=m|\ell,m\rangle \end{aligned} where $\langle \theta,\phi|\ell,m\rangle\equiv Y^m_\ell(\theta,\phi)$ are the spherical harmonics (modulo, perhaps, the normalisation).

In this basis, $E_{\ell,m}$ is given by the formula you wrote, while the eigenvectors are just spherical harmonics. You don't need to solve any eigenvalue (or differential) equation, mathematicians solved it for you long ago!

(In other words, your approach is indeed correct, but you don't have to solve the partial differential equation yourself: its solutions are already worked out, and you can find the explicit functions in the linked wiki page).