Note: this question is sort of at the boundary of chemistry and physics, but seems a bit more physics-y (details of wave functions and so on), so I'm posting it here
This LibreTexts subsection presents what seems to be a pretty typical treatment of electron orbitals in basic undergraduate chemistry. According to this "standard story":
- Each electron "occupies" a particular orbital, the orbitals being indexed by the quantum numbers $n, l, m$ (or the equivalent "chemistry-style" notations like $1s, 2s, 2p_x, 2p_y, 2p_z$, etc.)
- An electron "occupying" a given orbital has the wave function $\Psi_{n, l, m}(\mathbf{r})$, and a graphical illustration of $|\Psi|^2$ for various such wave functions is given in Figure 1 of the aforementioned LibreTexts passage.
However, it seems sort of implausible that this "standard story" is a complete representation of what's really going on, as it implies wave functions always having "preferred" directions in space:
For example, the three $2p$ orbitals each have a sort of "dumbell" shape along a particular axis. So if we have a single (ground-state) Boron atom, the "standard story" tells us the 5th electron "occupies" one of the $2p$ orbitals (either $2p_x$, $2p_y$, or $2p_z$), meaning that its "dumbell-shaped" wave function is oriented in one particular direction (this will be true regardless of which particular directions in space our $\hat{x}$, $\hat{y}$, and $\hat{z}$ happen to point in). In other words, the "standard story" implies that our Boron atom's 5th electron is always in a state of having "chosen" a particular direction in space (the axis of its "dumbell-shaped" wave function).
This seems a little suspect, given that this is a quantum-mechanical system with radial symmetry. It seems way more plausible that the 5th electron is, in general, in some superposition state like $\sum_{m=-1}^{1} c_m \cdot \Psi_{2,1,m}(\mathbf{r})$.
So, my specific questions are:
- Is the "standard story" indeed over-simplistic in its implication that the wave function of each $2p$ electron must always be oriented in one of the three "preferred" directions ($2p_x$ or $2p_y$ or $2p_z$, i.e. the "dumbell" shape along either the $\hat{x}$, $\hat{y}$, or $\hat{z}$ direction)? Can a $2p$ electron's wave function in reality be some superposition state (e.g. some linear superposition of the $2p_x$, $2p_y$ and $2p_z$ waveforms, or something similar to that)?
- In what special-case scenarios might the "standard story's" implication of "preferred directions" actually be (mostly) true?
Notes Related To Questions:
- I'm assuming here that the state of the $n$-electron system can reasonably be approximated as the product of $n$ individual-electron wave functions. This seems to be deemed sufficient for many chemistry purposes, so hopefully(?) it's adequate for the purposes of this question, and thus it makes sense to speak of "the wave function" of an individual electron.
- Point #1 above is framed as a couple of essentially "yes or no" questions. Of course knowing the reasons behind the yes or no answers would be helpful. However I certainly wouldn't expect anyone to write a dissertation (a few paragraphs with perhaps references to further reading would be quite sufficient for me).
- My background is confined to non-relativistic quantum mechanics, so an answer which delves too deep into details of relativistic effects and/or QFT will present some challenges (hopefully at least the gist of what's going on can be explained without delving too deep into such details?).
- I've seen mention of "hybridized orbitals", which seem sort of like the superposition states I'm conjecturing, except that discussion of "hybridized orbitals" seems to come up in the context of molecular bonds (not the single-atom scenario I'm contemplating here), so I'm not sure it's relevant to this question.
Update: I just noticed that, by pure coincidence, someone posted a somewhat similar question just a few minutes before I hit the submit button on this one. However the questions are not the same: That other question contemplates an electron in a superposition of different-energy eigenstates, but what I’m asking about here is a superposition of states that have the same energy and total angular momentum ($n$ and $l$ numbers) but different $m$-index
Update 2: This question has been flagged as a possible duplicate, but it is not entirely duplicative. Aside from the question being kind of vague, there are a couple ways in which the answers provided do not fully address what I am asking here:
- One of the answers talks only about "hydrogen-like" (single-electron) atoms/ions. What I am asking about is more general.
- The other answer (most upvoted) implies that the "requirement" of a "preferred direction" is eliminated only by considering a mixed state. But it's not at all clear, physically, why this would be so. Why can't I have a pure state where a $2p$ electron's wave function is a simple superposition of the $2p_x$, $2p_y$ and $2p_z$ wave functions? That's kind of the whole point of my question.
