Is the Bound State Wave Function Always Real or Imaginary?

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SUMMARY

The discussion centers on the nature of bound state wave functions in quantum mechanics, specifically whether they are always real or can be imaginary. It is established that for bound states with a degeneracy of level one, the wave function can be expressed as ψ=ψ_{r}+iψ_{i}, where ψ_{r} and ψ_{i} are linearly dependent. The ordinary Schrödinger equation confirms that bound state wave functions are typically real, while the implications of the non-linear Schrödinger equation on wave function degeneracy remain uncertain. The conclusion emphasizes that this reasoning holds true primarily in one-dimensional systems.

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Nemanja989
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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 :)
 
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This is correct, but only in one dimension.
 

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