1D bound state for a real potential

Question

The prof says: "for 1Dimensional bound states with a real potential, the wave function is real, up to a phase". The proof goes like this:

1D bound states are never degenerated. So $\Psi_{real}$ and $\Psi_{imaginary}$ are linearly dependent. So $\Psi \equiv \Psi_{real} +i\Psi_{imaginary}=\Psi_{real} (1+ic)=(1+c^2)e^{iArg(1+ic)}\Psi_{real}$

Whatever the proof, I don't understand the statement since any complex number (the wavefunction is one complex number) is in some way real up to a phase. So I don't really understand what this theorem is trying to teach us.

PS: I cannot ask directly the professor because I study from a video recorded 6 years ago

Answer 1

No, the wavefunction $\psi(\vec{r})$ is not just 1 complex number: it is infinitely many complex numbers, 1 for each value of position $\vec{r}$. In contrast, the professor is making the non-trivial statement that there exists a global (i.e. $\vec{r}$-independent) complex constant $c$.

For more details, see also this & this related Phys.SE posts.