You often see in atomic and molecular physics texts that bonding occurs between two atomic orbitals when their wave functions are in phase. These pictures often depict the 'phase' as whether or not the wave function is positive of negative. This is obviously not the same as the complex phase component of the wave function. The $ p_z $ orbital is completely real: $$ \psi (r, \theta)\propto \frac{r}{a_o}e^{-r/2a_0}\cos\theta $$ While the $p_x$ orbital is actually complex: $$ \psi (r, \theta, \phi)\propto \frac{r}{a_o}e^{-r/2a_0}\sin \theta e^{i\phi} $$

But both these p orbitals have the same behaviour when bonding, as if their complex phases are the same. What is the difference between this atomic orbital 'phase' and the actual complex phase of the wave function?

  • $\begingroup$ As a sidenote, a complex valued function also includes the purely real values. In your px, you can set phi to 0 or pi, for example. $\endgroup$
    – BjornW
    Commented Jun 9, 2015 at 16:55

1 Answer 1


If we write:

$$ \psi_+ \propto \frac{r}{a_o}e^{-r/2a_0}\sin \theta e^{i\phi} $$


$$ \psi_- \propto \frac{r}{a_o}e^{-r/2a_0}\sin \theta e^{-i\phi} $$

then because any superposition of solutions is also a solution we can write the solutions:

$$ \psi_{px} = \psi_+ + \psi_- $$

and likewise:

$$ \psi_{py} = \psi_+ - \psi_- $$

which gives us:

$$ \psi_{px} \propto \frac{r}{a_o}e^{-r/2a_0}\sin \theta \cos \phi $$


$$ \psi_{py} \propto \frac{r}{a_o}e^{-r/2a_0}\sin \theta \sin \phi $$

These are the more usual forms of the $p_x$ and $p_y$ orbitals that you'll find in textbooks, and both are real. Not that it makes any difference whether the wavefunction is real or complex because all observables are given by Hermitian operators that are guaranteed to be real anyway.

  • 1
    $\begingroup$ +1: Thanks for providing the link, sir. I just want to know why the books don't use orbitals having definite $m$ i.e. either $\psi_+$ or $\psi_-$; why only the superposition is the main interest; cannot electron have the either wavefunction?? $\endgroup$
    – user36790
    Commented Jun 22, 2015 at 19:51

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