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

 

[quantumdude]

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.

 

[wais]

for the question! When the quantum harmonic oscillator potential is over [0..infinity], instead of [-infinity..infinity], the energy eigenstates will still follow the same pattern of hw(n + 1/2), where h is Planck's constant and w is the frequency of the oscillator. However, the main difference is that the energy eigenstates will now be confined to a half-space, rather than being able to exist in both positive and negative regions. This is because the potential in the region from 0 to infinity acts as a barrier, not allowing the oscillator to exist in the negative region. This will result in a change in the energy levels and wavefunctions of the oscillator, as the boundary conditions are now different. The energy levels will be shifted and the wavefunctions will be modified to reflect the confined nature of the oscillator. Overall, the behavior of the oscillator will be affected by the presence of the half-space potential, but the general formula for the energy eigenstates will remain the same.