Charge distribution in positronium Inspired by this: Electrical neutrality of atoms
If I have a wavefunction of the 'reduced mass coordinate' for a hydrogen like atom made from an electron and a positron, what is the spatial charge distribution?
When we solve the hydrogen atom, we change into coordinates of the center of mass, and the separation distance with the reduced mass.  Here, the masses of the constituent particles are the same.  So the center of mass is equidistant from the positron and electron, and so discussing r and -r is just swapping the particles.  Since the probability distribution for all the energy levels of the hydrogen atom are symmetric to inversion (images can be seen here http://panda.unm.edu/Courses/Finley/P262/Hydrogen/WaveFcns.html ), this seems to say no matter what energy level positronium is in, the charge distribution is neutral?  Since the energy level basis is complete, this seems to say we can't polarize a positronium atom without dissociating it!? This doesn't make sense to me, so I'm  probably making a big mistake here.
 A: Okay we have the center of mass coordinate $r_{cm} = (r_e + r_p)/2$, and the reduced mass coordinate $r = r_e - r_p$.  So given the wavefunction $\psi(r_{cm},r)$ what you are asking is just a change of basis from $|r_{cm},r\rangle$ to $|r_e,r_p\rangle$.  So you just need to consider
$$\langle r_e,r_p|r_{cm},r\rangle = \delta\left(r_{cm}-(r_e+r_p)/2\right) \ \delta\left(r-(r_e-r_p)\right)$$

To get more at what appears to be confusing you, let's focus on how positronium can be polarized.  The spherical harmonics will have a defined even or odd parity to inversion, and so yes the square of the wavefunction (probability density) is invariant to inversion.  But this does not mean all superpositions of these harmonics will have this property.
Consider for instance just s + p_z orbital.  On one side the wavefunction amplitude will add constructively, while on the other it will add deconstructively.  The density is no longer symmetric to inversion.  This is the standard chemistry description of orbitals, so probably a good way to get follow up reading is searching for hybrid orbitals.
Here's a link I found with some images for you:
http://www.uwosh.edu/faculty_staff/gutow/Orbitals/N/What_are_hybrid_orbitals.shtml
A: Normally we speak of charge distribution in terms of atomic form-factors. It means Born approximation which means in turn fast charged projectiles or photons. 
Para- and orto-positroniums have different magnetic fields due to different spin orientation.
For slow charged projectiles its field strongly polarizes the atom and the charge distribution is not the same as in a "free" atom.
