Why the hydrogen radial wave function is real?
Is it a coincidence?
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up.Sign up to join this community
Since quantum states that differ by multiplication by a complex number of length $1$ are all equivalent, you can multiply any wavefunction of the Hydrogen atom by such a complex number, and you'll get a vector in Hilbert space that is an equivalently valid description of the corresponding physical state.
Perhaps Another way to say this is that for a matter wave the oscillations are the presence and absence of matter, mediated by the resistance above the speed of light, whereas for an electromagnetic wave the loss of electric field gives rise to the magnetic field. Here, we have no significance for the absence of matter, just empty space? That is my understanding.