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Find eigenfunctions and eigenvalues of an operator

  1. Jan 22, 2014 #1
    1. The problem statement, all variables and given/known data
    [itex]\hat T = \frac{{\hat L_z^2}}{{2I}} = - \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}}[/itex]

    2. Relevant equations
    Find eigenfunctions and eigenvalues of this operator


    3. The attempt at a solution
    It leads to the differential eqn
    [itex]- \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}}f = \lambda f[/itex]
    it has the characteristic eqn
    [itex]\lambda + \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}} = 0[/itex]
    then I dont know how to do the next step
     
  2. jcsd
  3. Jan 22, 2014 #2
    If I said its a wave equation would that help?
     
  4. Jan 22, 2014 #3
    then just follow the steps for solving Schrodinger eqn that was written in textbook, is it your idea?
     
  5. Jan 22, 2014 #4
    No just solve it as a regular maths equation. What sort of function when differentiated twice gives you the same function multiplied by a negative constant?
     
  6. Apr 5, 2015 #5

    vela

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    That's not the characteristic equation for the differential equation. For one thing, derivatives shouldn't appear in it.

    If you rearrange the original differential equation slightly, you get
    $$\frac{\partial ^2}{\partial \varphi^2} f = -k^2 f$$ where ##k^2 = \frac{2I\lambda}{\hbar^2}## is a constant. Surely, you've seen that kind of differential equation before.
     
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