Extracting eigenvalues from wavefunction

[Ichimaru]

Homework Statement

The Hamiltonian for a rigid rotator which is confined to rotatei n the xy plane is

\begin{equation}
H=-\frac{\hbar}{2I}\frac{\delta^{2}}{\delta\phi^{2}}
\end{equation}

where the angle $\phi$ specifies the orientation of the body and $I$ is the moment of inertia. Interpret this experession and fint the energy levels and egenfunctions of H.

The unnormalised wavefunction of the rotator at time t is

\begin{equation}
\psi(x) = 1 + 4 sin^{2}(x)

\end{equation}

Determine the possible results of a measurement of its energy and their relative probabilities. What is the expectation value of the energy of this state.

Homework Equations

The Attempt at a Solution

I've found the eigenfunctions as

\begin{equation}

\psi_{m}(x)= Ae^{im\phi}

\end{equation}
\begin{equation}
E_{m}=\frac{ \hbar^{2} m^{2} }{2I}
\end{equation}
And m is any positive / negative integer.

But I don't really know how to analyse the wavefunction properly. We have always used the ket approach which is much simpler conceptually. Analagously I would apply the Hamiltonian to the wavefunction in the position representation, and try and factor out the original wavefunction times some number, which would be the relevant energy eigenvalue. But I can't seem to get anything understandable out.

Any help with how to understand getting measurements and their probabilities out of wavefunctions in general would be useful. Thanks.

 

[mfb]

We have always used the ket approach which is much simpler conceptually.

You can define |m> as your $\psi_m$ if you like, or just use $|\psi_m \rangle$.
Calculating $H \psi_m$ in the position basis is straightforward here and you directly get the eigenvalue as result you posted.