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So we're learning about the Hydrogen atom in QM and I'm having trouble reconciling something in my head. We're looking at potentials that are only radius dependent, like the Coulomb potential.

Now, I know the math. I see that we assume the wave function can be separated into the product of three single variable function, one for each spherical coordinate, then we apply the Laplacian to it, we do the whole rigmarole for the radial one. I've done it all out.

But for the θ,[itex]\phi[/itex] one, we get the Spherical Harmonics. And while I'm not disagreeing with the actual math, from a physical standpoint, I'm confused as to why the probability distribution could have different values at the same radius (but different values of [itex]\theta[/itex] or [itex]\phi[/itex]) if the potential is totally spherically symmetric.

The way I'm thinking about it is, nature has no concept of the z axis which the angles [itex]\theta[/itex] and [itex]\phi[/itex] are defined with respect to, so where would spherical harmonics for an atom actually be pointed? The closest classical analogy I can think of is a planet orbiting in the gravitational potential (same as the coulomb, really), but the planet only goes on its specific orbit because of the initial conditions that sent it spinning in the plane it currently orbits in.

Can someone help me out?

Thanks!

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# How do we get angular dependent wave function from only radially dependent potentials

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