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Extracting eigenvalues from wavefunction

  1. Apr 11, 2013 #1
    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

    \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.

    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Apr 11, 2013 #2

    mfb

    User Avatar
    2016 Award

    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.
     
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