Angular momentum eigenfunctions

[IHateMayonnaise]

1. The problem statement, all variables and given/known data

This is problem 18.1 from Merzbacher.

"The hamiltonian of a rigid rotator in a magnetic field perpendicular to the x axis is of the form H=AL^2+BL_z+CL_y, if the term \that is quadratic in the field is neglected. Obtain the exact energy eigenvalues and eigenfunctions of the hamiltonian."


2. Relevant equations

L^2Y_l^m(\theta,\phi)=l(l+1)\hbar^2 Y_l^m(\theta,\phi)

L_ZY_l^m(\theta,\phi)=m\hbarY_l^m(\theta,\phi)

L_Y Y_l^m(\theta,\phi)=\frac{1}{2\imath}(L_+-L_-)Y_l^m(\theta,\phi)

L_+ Y_l^m(\theta,\phi)=\sqrt{l(l+1)-m(m+1)}\hbar Y_l^{m+1}(\theta,\phi)

L_- Y_l^m(\theta,\phi)=\sqrt{l(l+1)-m(m-1)}\hbar Y_l^{m-1}(\theta,\phi)

3. The attempt at a solution

For the first two parts of the hamiltonian the answer is easy:

E=Al(l+1)\hbar^2+Bm\hbar

But what is the eigenvalue for the last part (of L_Y)? Is it just



C\left[\frac{\sqrt{l(l+1)-m(m+1)}\hbar-\sqrt{l(l+1)-m(m-1)}\hbar}{2\imath}
\right]


?

And for the eigenfunctions: for the first two parts of H it is just the spherical harmonics, but for the last part is it both

Y_l^{m+1}(\theta,\phi)

and

Y_l^{m-1}(\theta,\phi)

?

Many days without sleep. THis is an easy question I know, please don't judge me I am sleep deprived :)

IHateMayonnaise

[gabbagabbahey]

When you act on the state Y_{\ell}{}^{m}(\theta,\phi) with the Hamiltonian, do you get a constant multiple of Y_{\ell}{}^{m}(\theta,\phi) ? If not, it isn't an eigenfunction of the Hamiltonian.

[IHateMayonnaise]

When you act on the state Y_{\ell}{}^{m}(\theta,\phi) with the Hamiltonian, do you get a constant multiple of Y_{\ell}{}^{m}(\theta,\phi) ? If not, it isn't an eigenfunction of the Hamiltonian.

Right! This is what confused me. So I guess it cannot be an eigenfunction, and the only one is the regular old spherical harmonic.

And the eigenvalue is just


C\left[\frac{\sqrt{l(l+1)-m(m+1)}\hbar-\sqrt{l(l+1)-m(m-1)}\hbar}{2\imath}
\right]

?

Of course this could be simplified..