Is the Bound State Wave Function Always Real or Imaginary?

[Nemanja989]

Hi :),

recently I was thinking whether every bound state is only real or imaginary, not mixture? Since all bound states have degeneracy of level one, if we suppose that $ψ=ψ_{r}+iψ_{i}$, then $ψ_{r}$ and $ψ_{i}$ must be linearly dependent as in opposite case there would be a bound state with degeneracy of level two.

Is this type reasoning good or not?

In every single example that I have seen or was doing numerically bound state wave function was always real.

Also, degeneracy level is derived through ordinary Schrödinger equation. What about non-linear Schrödinger equation? Does in this case every bound state correspond to only one wave function, or there is degeneracy of higher level?

Thanks in advance :)

 

[Bill_K]

This is correct, but only in one dimension.