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  • $\begingroup$ I am perfectly in agreement with your reasoning but since I'm new on the quantum world, maybe I'm losting something. I'm confused on this point: if the orbitals are defined as the space region delimited by a surface where $\lvert \psi\rvert ^2$ is constant, why we consider the $|\Psi_{2,1,0}|^2$ as an orbital, instead this does not happen with $|\Psi_{2,1,1}|^2$ and $|\Psi_{2,1,-1}|^2$ (in fact in the last two cases we (or better as many books do) consider $px$ and $py$ as orbitals)? $\endgroup$ Commented Apr 27, 2017 at 12:30
  • $\begingroup$ I have read many times this sentence: "the quantum number $m$ defines the orientation of the orbitals, while $l$ their shape." This seems not to be true looking at the second image. $\endgroup$ Commented Apr 27, 2017 at 12:59
  • $\begingroup$ @SimonG Re "if the orbitals are defined as the space region delimited by a surface where $|\psi|^2$ is constant", that's not how you define orbitals. Orbitals are $\psi$, period. We often visualize orbitals using those contour plots, but that doesn't mean that's what orbitals are. Normally you require that $\psi$ be an energy eigenstate to call it an orbital, in which case all the wavefunctions you've mentioned are orbitals; some texts might only focus on a subset for didactic reasons but that's all that's going on. $\endgroup$ Commented Apr 27, 2017 at 13:16
  • $\begingroup$ Similarly, the sentence "the quantum number $m$ defines the orientation of the orbitals, while $l$ their shape" is pretty far from correct, and taken literally is dead wrong. If you loosen your definitions of "shape" and "orientation" enough, then it's true, but those terms would be unrecognizable to you in their 'true' forms. (More specifically, "shape" means irreducible representation of SO(3), while "orientation" means a specific member of that representation.) I can elaborate if you want but I don't think it's that important. $\endgroup$ Commented Apr 27, 2017 at 13:21
  • $\begingroup$ I was referring to my book which state that "orbitals are defined as the space region delimited by a surface where $|\Psi|^2$ is constant, into which the probability to find the electron is 0.9", maybe only for didactic purpose. Now for $n=2$ and $l=1$ I have three $2p$ orbitals. To be tightly practical, if the hydrogen atom is excited (2p) and I want to look to their orbitals, I expect to find something like the second image: instead I find always image that represent $px$, $py$, $pz$ as real atomic orbitals. Why this happen? If this represents the reality why I need to do a superposition? $\endgroup$ Commented Apr 27, 2017 at 13:57