What's up with Electron Probability Density?

[CollectiveRocker]

I realize that electron probability density is the probability of finding an electron in a given volume, but as I was working on some homework, I wasn't sure how this fact would apply. Under what circumstances is an atomic electron's probability-density distribution spherically symmetric? Why? The book says that an electron's probability density is symmetrical about the z-axis regardless of the quantum state it is in. But I'm still confused.

 

[theFuture]

The density is largely dependent on potential, in this case the coulomb potential. Because the potential is symmetric about the z, so should the wavefunction (and corresponding probability density function). Correct me, please, if this isn't correct.

 

[CollectiveRocker]

Does anyone know if this is right?