Atomic Orbitals: Superpositions and m-Eigenstates

Source: https://grok.com/c/4fff08f0-15a6-4899-b2c3-af45e4ed7a13?rid=a1a0e256-7c53-46f4-b12b-613e4b18c3c6

You're exactly right—that's a very common point of initial confusion, and your intuition about "superposition" is spot on in the broader quantum sense.
The key here is the specific context in which the term "superposition" is being used in that Wikipedia figure caption (and in most quantum chemistry/ atomic physics discussions of real vs. complex orbitals):

The complex m-eigenstates (right side: the ones with toroidal/doughnut shapes, phase-winding rainbows, definite m quantum number) are the "pure," fundamental solutions — direct eigenstates of the angular momentum operator L_z. They have a well-defined projection of orbital angular momentum along the chosen z-axis.
The real orbitals (left side: p_x, d_{x²-y²}, d_xy, the familiar Cartesian-aligned lobes in yellow/blue) are constructed as specific linear combinations (i.e., superpositions) of those complex m-eigenstates — usually pairs like ψm + ψ{-m} and ψm - ψ{-m} (with appropriate normalization and phase factors, often following the Condon–Shortley convention).

So yes, the real ones really are superpositions: each p_x (for example) is literally
p_x ∝ (Y_{1,+1} + Y_{1,-1})
(or a similar combination depending on phase convention), where Y are the complex spherical harmonics.
The complex terms don't "cancel out" in the sense of disappearing entirely — rather, the imaginary parts cancel between the +m and -m partners, leaving a purely real function overall. That's why we call them "real orbitals" or "real linear combinations." The superposition is deliberately chosen to eliminate the complex phase winding and produce shapes that point along x, y, z (or other preferred molecular directions), which is hugely convenient for describing chemical bonding, hybridization, crystal field splitting, etc.
In a deeper quantum sense, any valid wavefunction (including the complex m-states themselves) can be part of a superposition — an electron in a real atom isn't forced to be in just one of these basis states forever. But when we label something an "orbital" like 2p_x, we're saying it's that particular superposition of the underlying m-eigenstates.
So your original thought was correct: "superposition" does mean combining individual (here, the complex m) states. The figure just uses "superpositions" as shorthand for those chemically useful real combinations on the left, contrasting them with the pure m-eigenstates on the right.
Glad it clicked — it's one of those details that trips up almost everyone at first! If you'd like, I can walk through the exact math for how p_x / p_y emerge from p_{+1} and p_{-1}, or why the toroidal ones appear in physics-heavy contexts.

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