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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 [itex]ψ=ψ_{r}+iψ_{i}[/itex], then [itex]ψ_{r}[/itex] and [itex]ψ_{i}[/itex] 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 Schrodinger equation. What about non-linear Schrodinger equation? Does in this case every bound state correspond to only one wave function, or there is degeneracy of higher level?

Thanks in advance :)