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There cannot be two or more Fermions in one quantum state. A quantum state is defined by a full set of observables / operators. For example, in the case of an electron in the H atom it is $H, L^2, L_z$. The state is then $|n,l,m\rangle$.

However, where does the Pauli principle "stop"? E.g. assume we have another H atom closeby. Then there can be two electrons in e.g. the 1s state: $|n,l,m\rangle$, $|n',l',m'\rangle$.

So, is the limit to the Pauli principle a "different potential well" e.g. the different Coulomb field of the two protons? Or is it the distance between Fermions? Can we define the boundaries of the Pauli principle quantitatively?

Addendum: If it is in fact the distance: In special relativity, time and space should be treated symetrically in all regards (as seen for example in the Dirac equation.). On the other hand, taking the distance between the Fermions would impose a fundamental, non symetric (in space and time) constrain on Fermions.

Welcome_Green
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Your $|{n,l,m}\rangle$ is not the complete wavefunction, it's just the wavefunction relative to the center-of-mass part of the nuclear(proton)-electron part. Another atom has an independent COM part and therefore the two wavefunctions are already orthogonal.

RoderickLee
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The better way to think about this is to antisymmetrize your states w/r to $\vert n\ell m\rangle$, so that the two-electron state is \begin{align} \psi_{12}=\frac{1}{\sqrt{2}}\left(\vert n\ell m\rangle_1\vert n’\ell’m’\rangle_2- \vert n\ell m\rangle_2\vert n’\ell’m’\rangle_1\rangle\right)\, , \end{align} (assuming $\vert n\ell m\rangle$ and $\vert n’\ell’ m’\rangle$ are orthonormal.) Then quite clearly the principle is satisfied in the sense that if both electrons are “in the same state” then $(n’\ell’m’)=(n\ell m)$ and $\psi_{12}=0$.

ZeroTheHero
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Roughly speaking, the pairwise self-organization of electrons affects both inneratomic and molecular bonding as well as ionic and covalent bonding. The Pauli principle of pairwise anti-parallel alignment of electrons always applies. Covalent radii are in the range of 100 pm. However, no cross-section of action is assigned to the spin. This is exactly where your question comes in.

It is interesting that a mental equalization of the spin alignment with an alignment of the magnetic dipoles of the electron can lead to a solution approach. It is probably possible to calculate the torque between two electrons with oppositely directed dipole moments and compare it with the thermal collisions between the bound atoms.

HolgerFiedler
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