Why the hydrogen radial wave function is real?
Is it a coincidence?
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2$\begingroup$ Related: The book of Griffiths, Intro to QM, Problem 2.1b, p.24; and this and this Phys.SE post. $\endgroup$– Qmechanic ♦Apr 9, 2013 at 17:01
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$\begingroup$ Thank you, but how can I conclude? $\endgroup$– ArnaudApr 9, 2013 at 17:04
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$\begingroup$ The main point is that a wave function solution to the TISE is not necessarily real, but it can be chosen so. $\endgroup$– Qmechanic ♦Apr 13, 2013 at 15:09
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3 Answers
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Wave functions which are Eigenfunctions of the stationary Schrödinger equation can always be chosen to be real. That's because the equation itself is real. Depending on the boundary conditions, the solution can also be complex (e.g. for scattering BC they are complex).
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3$\begingroup$ It can be chosen to be real. See e.g. $p_x$, $p_y$ and $p_z$ wave function which are equivalent to the complex counterparts for l=1. $\endgroup$– zonksoftApr 9, 2013 at 16:57
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1$\begingroup$ See en.wikipedia.org/wiki/… . For the spherical harmonics of one angular momentum number, there is an equivalent linear combination which is real. $\endgroup$– zonksoftApr 9, 2013 at 17:03
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$\begingroup$ Oh, thank you very much ! Have you a link that prove the fact that wave functions of bound systems can be chosen to be real ? $\endgroup$– ArnaudApr 9, 2013 at 17:05
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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.
answered Apr 9, 2013 at 16:57
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$\begingroup$ Yes, but it doesn't answer my question. $e^{ix}$ is never a real function, even when you multiply it by a constant. $\endgroup$– ArnaudApr 9, 2013 at 16:59
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$\begingroup$ Yes, but it doesn't answer my question. $e^{ix}/(x^2+1)$ is never a real function, even when you multiply it by a constant. $\endgroup$– ArnaudApr 9, 2013 at 17:02
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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.
