I Why does an electron orbital have a preferred z axis?

[PeterDonis]

It means the potential energy function is spherically symmetrical.

Ok, but the potential energy is not an "object".

Object here means the hydrogen atom

Which is not the same as the potential energy. Nobody disputes that the potential energy is spherically symmetrical (a function of $r$ only in spherical coordinates) for the cases under discussion, and that's the only interpretation of "spherically symmetric" that you've given a precise mathematical definition for. Yet you keep talking about "objects", and evidently have in mind some other definition of "spherically symmetric" for objects, which can't possibly be "the potential energy function is spherically symmetrical" since the potential energy function is not an object. So what are you talking about?

 

[Happiness]

Ok, but the potential energy is not an "object".
Which is not the same as the potential energy. Nobody disputes that the potential energy is spherically symmetrical (a function of $r$ only in spherical coordinates) for the cases under discussion, and that's the only interpretation of "spherically symmetric" that you've given a precise mathematical definition for. Yet you keep talking about "objects", and evidently have in mind some other definition of "spherically symmetric" for objects, which can't possibly be "the potential energy function is spherically symmetrical" since the potential energy function is not an object. So what are you talking about?

An electron has spherical symmetry when it experiences a potential energy function that is spherically symmetrical.

 

[PeterDonis]

An electron has spherical symmetry when it experiences a potential energy function that is spherically symmetrical.

If this is your definition, it's quite perverse (not to mention different from any definition used in the literature), since we have already seen that there are plenty of solutions to the Schrodinger equation with a spherically symmetric potential that are not spherically symmetric (the wave functions are not just functions of $r$). I see no good reason to call those solutions "spherically symmetric" simply because the potential is (and neither, as I mentioned, does any of the literature).

 

[PeterDonis]

The OP question has been more than sufficiently addressed. Thread closed.