quantum harmonic oscl. half space potential

[YZer]

I'm trying to figure out what happens to the energy eigenstates when the quantum harmonic oscillator potential is over [0..infinity] rather then
[-infinity..infinity]. Originaly its hw(n +1/2)...

Thanks

[Tom Mattson]

The half space SHO is what you get when you superpose the SHO with an infinite wall. Since a wavefunction cannot penetrate an infinite wall, you have to have a node there. So of all the regular SHO solutions, only the ones with a node at the origin survive. Their corresponding energies are the only allowed energies.