Harmonic oscillator derivation of wave functions

[jtaa]

here is a link to the pdf file with my question and answershttp://dl.dropbox.com/u/2399196/harmonic%20osc.pdf

i'm not sure where to start, because i don't want to assume anything that i haven't been given.
i'm stuck on part (iv) where i have to derive explicit expressions for 2 wave functions.

thanks!

 

[gabbagabbahey]

Well, from your 2 expressions for $H$ you have

$$\left( \frac{1}{2}\hat{p}^2+\frac{1}{2}\hat{x}^2\right) \Psi_0(x)=\hbar\left(\hat{N}+\frac{1}{2} \right)\Psi_0(x) = \frac{\hbar}{2}\Psi_0(x)$$

So, if you express $\hat{p}^2$ and $\hat{x}^2$ in the $x$-basis, you will have a differential equation you can solve for $\Psi_0(x)$

 

[jtaa]

ok i get a second order diff equation that looks like this:

http://dl.dropbox.com/u/2399196/2orderdiff.png

but how do i solve that?

 

[vela]

This is a topic covered in many quantum mechanics and mathematical methods texts. I'd suggest you start there.