Quantum asymmetric top ($H=a_x{L}_x^2+a_y{L}_y^2+a_z{L}_z^2$)

The quantum asymmetric top has different rotational-energy coefficients on its principal axes. Its Hamiltonian commutes with $L^2$ but generally not with an individual component, so diagonalization proceeds within fixed total-angular-momentum multiplets. For a mechanical rigid rotor the positive coefficients are reciprocals of twice its principal moments of inertia. The component sign convention for body-fixed generators does not affect this squared-component energy expression.

Within the $l=1$ multiplet, $|1,0⟩$ has energy ${\hslash }^2(a_x+a_y)$. The symmetric and antisymmetric combinations of $|1,1⟩$ and $|1,-1⟩$ have the other two pairwise coefficient sums. Principal-axis permutation relates these energies; equality requires corresponding coefficient degeneracies rather than angular-momentum conservation alone.