Energy values of a half harmonic oscillator

[siifuthun]

energy values of a "half" harmonic oscillator

This is the full question:
Find the solution to the "half" harmonic oscillator:
http://img241.imageshack.us/img241/9181/02bt7.jpg
Compare the energy values and wave functions with those of the full harmonic oscillator. Why are some of the full solutions present and some missing in the "half" problem?

I'm having some trouble figuring out how to find a solution for the wave function of a "half" harmonic oscillator. I can picture what this would look like, since if we "halved" a harmonic oscillator the left end point for energy levels E_1, E_3, E_5 and so on wouldn't be any good since the left end point would no longer be zero at the boundary.

I'm having some trouble figuring out how I should adapt the time-indepedent Schrödinger equation to these new boundaries because the book we use only mentions it briefly in the appendix. I don't even know what equation to look at because when I searched it online I got an equation that looked different from my book.

 

[kuruman]

You have to assume an infinite wall potential at $x=0$. That's what makes the potential "half" harmonic. This means the eigenfunctions must vanish at $x=0$ and be appropriate for the full harmonic oscillator for $x>0$. Can you figure out what these are? Hint: The eigenstates of the full harmonic oscillator are even and odd functions in $x$.