# Find eigenfunctions and eigenvalues of an operator

1. Jan 22, 2014

### fdbjruitoirew

1. The problem statement, all variables and given/known data
$\hat T = \frac{{\hat L_z^2}}{{2I}} = - \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}}$

2. Relevant equations
Find eigenfunctions and eigenvalues of this operator

3. 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 dont know how to do the next step

2. Jan 22, 2014

### Jilang

If I said its a wave equation would that help?

3. Jan 22, 2014

### fdbjruitoirew

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

4. Jan 22, 2014

### 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?

5. Apr 5, 2015

### vela

Staff Emeritus
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.