Find eigenfunctions and eigenvalues of an operator

[fdbjruitoirew]

Homework Statement

$\hat T = \frac{{\hat L_z^2}}{{2I}} = - \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}}$

Homework Equations

Find eigenfunctions and eigenvalues of this operator

The Attempt at a Solution

It leads to the differential eqn
$- \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}}f = \lambda f$
it has the characteristic eqn
$\lambda + \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}} = 0$
then I don't know how to do the next step

 

[Jilang]

If I said its a wave equation would that help?

 

[fdbjruitoirew]

then just follow the steps for solving Schrödinger eqn that was written in textbook, is it your idea?

 

[Jilang]

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?

 

[vela]

It leads to the differential eqn
$- \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}}f = \lambda f$
it has the characteristic eqn
$\lambda + \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}} = 0$
then I don't know how to do the next step

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.