It is well established that:
'In quantum mechanics, the behavior of an electron in an atom is described by an orbital, which is a probability distribution rather than an orbit.
There are also many graphs describing this fact: http://en.wikipedia.org/wiki/Electron: (hydrogen atomic orbital - one electron) In the figure, the shading indicates the relative probability to "find" the electron, having the energy corresponding to the given quantum numbers, at that point.
My question is: How do these symmetries shown in the above article occur? What about the 'preferable' axis of symmetries? Why these?
Answer
The hydrogen atom is spherically symmetric, so for any solution of the Schrödinger equation for the hydrogen atom, any rotation of that solution must also be a solution. If you do the math on how to rotate a solution, it turns out that the solutions with a particular energy $E_n$ fall into groups labeled by an integer $l < n$. The integer $l$ is physical: $\hbar^2 l(l+1)$ is the magnitude squared of the angular momentum. Within each group, rotating the solution gives you a new solution in the same group. These two facts are of course connected: a rotation can't change the length of a vector.
One can show that each group contains $2l+1$ independent solutions, in that any solution $|n,l\rangle$ where the energy is $E_n$ and the angular momentum $\hbar^2 l(l+1)$ can be written as a sum $$|n,l\rangle = \sum_{m=-l}^l c_m |n,l,m\rangle$$ (I apologize for the somewhat poor notation.)
This decomposition is based on choosing a particular axis, and taking each state to depend on the angle $\varphi$ around this axis as $e^{im\varphi}$. The appearance of axes of symmetry in these plots is due to this choice of axis and particular decomposition. With another choice of axis, which is the same as a rotation, the states will be mixed.
The bottom line is that it's not each solution -- wavefunction -- that needs to be spherically symmetric, but the total set of solutions.
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