- #1
Ichimaru
- 9
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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.