# Extracting eigenvalues from wavefunction

1. Apr 11, 2013

### Ichimaru

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

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

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

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

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

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.

2. Relevant equations

3. The attempt at a solution

I've found the eigenfunctions as

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

E_{m}=\frac{ \hbar^{2} m^{2} }{2I}

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.

2. Apr 11, 2013

### Staff: Mentor

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.