Anisotropic electron orbitals in hydrogen

Question

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? :

1. 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.

2. 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"?

Answer 1

The correct interpretation is the first one. A hydrogen atom in a pure $d$ state, like, say, the $l=2$, $m_{L_z}=0$ state, really does point in a specific direction (i.e. the quantization axis).

There is nothing wrong with anisotropy, and particularly, there is nothing wrong with the existence of anisotropic objects. Not everything in life is a sphere - if you want an anisotropic object, grab the nearest pen.

What you do have, with the hydrogen atom, is an example of isotropic dynamics. This does not preclude the existence of anisotropic solutions of those dynamics: what it does is require that, for every anisotropic solution $S$ that points in direction $\hat{\mathbf n}$ and every arbitrary direction $\hat{\mathbf n}'$, there must exist an equivalent solution $S'$ that points in direction $\hat{\mathbf n}'$. For the specific case of hydrogenic $d$ states, this requires the existence of $d$ states that point in any arbitrary direction, which is of course true.

In terms of the real world, saying that you have "a sample of hydrogen atoms in $d$ states" is not really enough information. In the typical case, you will have excited them into this state using laser radiation, using linear polarization to climb from the $s$ state to the $p_z$ state and from there to the $d_{m_{L_z}}=0$ state. In this case, you'll have a sample of hydrogen atoms all pointing in the same direction; this is of course OK with the isotropy requirement because the initial state was isotropic but the exciting radiation was not.

In some other cases, however, you can think of having a bunch of hydrogen atoms in $d$ states that point in a bunch of different directions. This is a bit of a more contrived experiment, but you could achieve it with e.g. multiple lasers with different polarizations. In this case, however, you do not use a linear superposition to describe the experiment; instead, you use something called a mixed state. Among other things, this is because an even linear superposition of $d_{m_{L_{\hat{\mathbf n}}}}$ states that point in all directions $\hat{\mathbf n}$ actually gives you a wavefunction which is identically zero - it's an interesting calculation, you should try it.