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The electron cloud for the single electron orbital $1s^1$ is radially symmetric in 3 dimensions. H atom cloud

This is perhaps because of the probability distribution function (p.d.f.) $|\psi|^2 = ce^{-\frac{2r}{a_0}}$ depending on a single variable $r$. On the other hand, the p.d.f. of He atom is the bivariate function depending on the positions of both electrons $|\psi_{\text{He}}|^2 = ce^{-\frac{r_1 + r_2}{a_0}}$. Moreover, both electrons have opposite spins. Will this modify the distribution of dots in the electron cloud? If yes then how? More exactly, what is $\psi^2 (x,y,z)$ for each electron in the actual He atom (not the unperturbed He atom in which both electrons don't interact)? Do the electrons lose their individuality so that it is impossible to give separate wavefunctions for each interacting electron?

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    $\begingroup$ That turns out to be quite a tricky problem to solve, but fortunately Bethe and Salpeter did in 65 years ago. See amazon.com/Quantum-Mechanics-One-Two-Electron-Atoms/dp/… (no endorsement of the shop, just a pointer to the book, likely in your nearby physics library). $\endgroup$
    – Jon Custer
    Commented Nov 15, 2021 at 14:05
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    $\begingroup$ I am surprised how difficult is finding data in the standard web to this very simple question. Some plots I saw suggest radial symmetry, however it is hard to know if e-e interaction were taken in account. You can also try Physics SE, eventually. $\endgroup$
    – Alchimista
    Commented Nov 16, 2021 at 8:19
  • $\begingroup$ As you surmised, the two electrons do not have separate wave functions because of their interaction and interchangeability. The single wave function for helium which accounts for both electrons is unknown because we can't solve the Schrodinger equation for that case. See websites.umich.edu/~chem461/QMChap8.pdf for more detail. $\endgroup$
    – Andrew
    Commented Nov 16, 2021 at 14:56

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