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I understand that the orbitals are a 3D contour inside of which there is a certain probability ( for example 90% ) to find the electron ( this is my understanding of it, so correct me if I am wrong ).
Now, I want to know the meaning of shapes like that of p-orbital. They are dumbbell-shaped ( sort of ) and are located on the x-, y- and z-axis. But aren't the axis completely arbitrary. Since the axis are arbitrary, we should expect only shapes that remain invariant upon any kind of rotations of the axis. The only thing that fits in that criterion is the spherical shape (or s-orbital).
So, what is the meaning of the non-spherical shapes of various orbitals, in the context I have mentioned above?