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After learning some rudimentary Quantum Mechanics, I have found that the wavefunctions of harmonic Oscillators and particles in potential well are all real valued. The ground state of harmonic oscillators, for example, has a wavefunction similar to a Normal distribution.

I wonder whether or not there are interesting examples complex valued wavefunctions. Since QM's formulation needs a lot of complex numbers, I think some systems must have complex valued wavefunctions.

Or can we say that all systems can be described with real valued wavefunctions?

EDIT: Really sorry for giving an unclear question. What I am going to find is a wavefunction that never become real valued after evolutioning according to the time dependent Schrodinger equation $i\hbar \frac{d|\phi>}{dt}=H |\phi>$

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    $\begingroup$ Duplicate? physics.stackexchange.com/q/77894 Note that they talk about eigenstates. You can trivially obtain a complex wavefunction by a linear combination of eigenstates. $\endgroup$ – jinawee Aug 8 '18 at 6:24
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First and foremost if you take a Harmonic oscillator and try to find the time dependent wavefunction you necessarily get a complex phase factor $\lvert n\rangle \to e^{-i\hbar^{-1}E_nt}\lvert n\rangle$.

Secondly having the wavefunctions that are solutions be real values is just a convenient choice - the wave equation being real this choice is always possible. You can multiply the wavefunction by a constant phase factor $e^{i\phi}$ and it changes nothing.

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  • $\begingroup$ For the case of hydrogen atom, can we still always choose eigenstates to be real-valued functions? I remember that the eigenstates consist of spherical harmonics $Y_{l}^{m_{l}}(\theta, \phi) \sim e^{i m_{l} \phi}$ which is complex for $l > 0$. $\endgroup$ – K_inverse Aug 8 '18 at 10:05
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  1. The wavefunction solution to the TISE can be chosen real, cf. this & this Phys.SE posts.

  2. The wavefunction solution to the TDSE is oscillatory and therefore generically complex.

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