I would like to clarify my understanding of anisotropic electrons orbitals in the atom of hydrogen - I feel uncomfortable by the mere fact of asymmetry (anisotropy) existing. Clearly, many orbitals ("d" orbitals) point in a specific direction (often called "z" axis). Let me for the moment make a philosophical assumption, that one can think or wave function as of a real object. What is the correct interpretation? :
One should think of a "d-excited" atom (flying now somewhere in my room) as truly pointing to a specific direction: that atom points to window, this one to doors, the other one to the upper corner of the room. The justification may be that the process of formation of a "d-excited" atom is always asymmetric (anisotropic)(is it??) and the atom inherits the asymmetry.
The Schrodinger equation (and it special time-independent form) is linear! Therefore I can make a summation of the same d-orbital over all spatial directions, getting so a spherical symmetry:
$$d_\mathrm{symmetric} = \sum_{i : \text{all directions}} d_{\text{direction }i}$$
Am I missing something in this argument? Such state is time independent (isn't it?) and has well defined energy (the one of the "d" orbital). I must admit I am not sure now about prediction concerning projection on a given axis (well, for a completely symmetric state it has to be $1/2$).
So let me repeat the question: how should I think or "real" hydrogen atoms excited to a $d$ state? Symmetric or asymmetric or "it depends"?
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