Suppose we have a rigid cube whose centre is at rest but it can rotate in three dimensions. One might propose for the Hamiltonian and energy eigenstates the following argument:
$$
\hat{H} =
\frac{\hat{L}_x^2}{2 I_x} +
\frac{\hat{L}_y^2}{2 I_y} +
\frac{\hat{L}_z^2}{2 I_z}
= \frac{\hat{L}^2}{2 I}
$$
using $I \equiv I_x = I_y = I_z$ for coordinate axes aligned with the principle axes of the cube. One then gets energy eigenvalues $L(L+1) \hbar^2/2I$ and each energy eigenstate has degeneracy $2L+1$.
But the above is wrong. It is wrong because it understates the degeneracy of the eigenspace of $\hat{L}^2$ for this case. My question is: how can we see that?
Essentially the same question was already asked in
Quantum rotator and equipartition theorem
and there is an excellent answer there, but it simply states that the eigenspace degeneracy is not $2L+1$ without going into further details on that. I am trying to elucidate how the humble student can see "oh yes, clearly this is wrong because we forgot …"
Best Answer
The "naive" approach is wrong because it forgets to spell out what the actual space of states is $H$ acts on - sure, you can write down $H = L^2 = L_x^2 + L_y^2 + L_z^2$, but what is the state space of the system we're describing?
That all energy eigenvalues are of the form $\ell(\ell +1)$ is true - but just having the Hamiltonian in hand doesn't tell you what the allowed values of $\ell$ are or how often they are repeated. There is just no basis for the assumption that the space of states should contain the eigenspace of $L^2$ for each $\ell$ only once.
The correct quantization of the rigid rotor can be somewhat subtle because the Euler angles often used as generalized coordinates are $S^1$-valued (or "$2\pi$-periodic", if you prefer), not $\mathbb{R}$-valued, but in any case the $L_i$ are not the canonical momenta (as you can already see from them not commuting with each other!). Regardless of what quantization procedure exactly we implement, our state space however should be wavefunctions of either the coordinates (Euler angles) or the canonical momenta.
Now, if you note that "every integer $\ell$ once" is just the spherical harmonics $Y(\theta,\phi)$ which are functions of two angles, it should seem somewhat intuitive that the idea of each $\ell$-eigenspace occuring only once underestimates the actual state space, which should be functions of three angles.