Imaginary part of the wave function

[RandomUsernameToSave]

In the time independent harmonic oscillator from example, the eigen states doesn't have any imaginary part contrary to what could be expected by the analytical resolution of the harmonic oscillator. (exp(imphi))

Do you have any idea why, and how I could fix it ?

[rafael-fuente]

Hi,

The harmonic oscillator and any other system subjected to a real value potential V(x) only have real-valued eigenstates up to an arbitrary complex constant with a module equal to 1.

Eigenstates of the 1D harmonic oscillator are:

Which they are normalized in such a way that they only have a real part.

[RandomUsernameToSave]

Yes but the 3D harmonic oscillator has the spherical harmonics function :

[rafael-fuente]

The complex spherical harmonic basis is not the unique basis of the 3D harmonic oscillator due to energy degeneracy. This solver will normally converge separately to the real $\frac{Y^{m}_l + Y^{-m}_l}{2}$ and imaginary part $\frac{Y^{m}_l - Y^{-m}_l}{2i}$ of the wavefunction you cite, which form a basis with the same span as the m = +-1 spherical harmonics $Y^{m}_l$ and $Y^{-m}_l$

For the hydrogen atom, it will happen exactly the same: instead of converging to the complex spherical harmonics, it will converge to the basis it's usually used in chemistry books to represent p orbitals. However, if you turn on a magnetic field, as illustrated in the Zeeman effect example, the degeneracy breaks and the eigenstates will converge to wavefunctions with the angular part approaching the complex spherical harmonics.