Quantum hamonic oscillator 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

 

[zefram_c]

Well, any even solution of the (-inf,inf) domain is no longer acceptable, but all odd solutions are still OK (properly renormalized). So I think the new ground state would be the 1st excited state of the old problem; and you'll have to figure out how to renormalize the wavefcn, but the energy eigenvalue should be the same.

 

[R0man]

for your question! When considering the quantum harmonic oscillator potential over [0,∞), the energy eigenstates will still follow the same general pattern as in the standard potential over (-∞,∞). However, the energy levels will be slightly different due to the change in the potential boundaries.

In the standard potential, the energy levels are given by E_n = (n + 1/2)ℏω, where n is the quantum number and ω is the angular frequency. This arises from solving the Schrödinger equation and applying the boundary conditions.

In the half-space potential, the energy levels will also follow this pattern, but the values of n will be limited by the boundary at x = 0. This means that the lowest possible energy state is no longer n = 0, but rather n = 1/2. This is because the wavefunction must go to zero at x = 0, and this can only be achieved if n is a half-integer.

Furthermore, the energy levels will also be slightly shifted due to the change in the potential boundaries. This can be seen by considering the harmonic oscillator potential as a potential well, with the walls at x = 0 and x = ∞. The wavefunction will now be confined to this potential well, leading to a slight change in the energy levels.

Overall, the energy eigenstates in the quantum harmonic oscillator half-space potential will still follow the same general pattern, but with some modifications due to the change in the potential boundaries.