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Correct me if at any point I'm wrong. My understanding is that spins up and down are associated with left($L$) and right($R$) circular polarizations of a photon. Then any other polarization $P$ is a superposition of these states, i.e. $P =\alpha\left|L\right> + \beta\left|R\right>$. I can get any linear polarization by adding $L$ and $R$ and by adjusting their phase difference (check here). By adjusting amplitudes we can generalize to any elliptical polarization.

So do I understand correctly that the coefficients $\alpha$ and $\beta$ encode full information about amplitudes (or maybe their ratio) and relative phase between $L$ and $R$?

And next thing: is this reasoning true also for electrons? What happens when linear polarized electron gets near proton and they bound to make a hydrogen? Does the state of the electron collapses to up or down spin, even if there was not measurement made? Or if we have Li$^+$ ion why will vertically polarized electron populate shell $p$ and not the $s$ since vertical polarization is different than $L$ and $P$ polarizations? Why does not vertical polarization count as another quantum state?

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My understanding is that spins up and down are associated with left($L$) and right($R$) circular polarizations of a photon.

Your idea is not completely wrong, but it is technically rather annoying to get this correct, and it should be known by more people than it currently is.

There is no such thing as a usual spin for a photon. It does, however, have the projection of spin onto the momentum of a photon, and that is called helicity, and it is helicity that is corresponding to $\left|L\right>$ and $\left|R\right>$. The most satisfactory treatment of spins and helicity (and chirality) together, lies in Wigner's work on the topic, and this is usually only done in QFT, which is a shame, because it is so nice and elegant, not abhorrently difficult, even though, of course, it does have its difficulties.

As for the rest of how spins and photon polarisations work, and with a bonus introduction to Weyl's work on the topic, eigenchris has an entire playlist detailing how $(\left|\leftrightarrow\right>,\left|\updownarrow\right>),(\left|\nearrow\right>,\left|\searrow\right>),(\left|\circlearrowleft\right>,\left|\circlearrowright\right>)$ light polarisations match with $(\left|+x\right>,\left|-x\right>),(\left|+y\right>,\left|-y\right>),(\left|\uparrow\right>,\left|\downarrow\right>)$ spin states and it deserves much more attention.

What happens when linear polarized electron gets near proton and they bound to make a hydrogen? Does the state of the electron collapses to up or down spin, even if there was not measurement made?

That is just an electron with a spin in a direction different from the $z$-axis and it should be clear that it will not collapse to up or down spin alone. It will stay in its superposition, i.e. spin in its direction.

will vertically polarized electron populate shell $p$ and not the $s$

No, it will populate $s$ shell with whatever spin it started with. Orbital and spin angular momenta can be approximately taken as separate degrees of freedom.

Why does not vertical polarization count as another quantum state?

It counts. Just not new compared to spin up and down. If we pick $x$ direction, then it would be purely spin up or purely down in $x$ direction.

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