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In many introductions to the pauli's exclusion principle, it only said that two identical fermions cannot be in the same quantum state, but it seems that there is no explanation of the range of those two fermions. What is the scope of application of the principle of exclusion? Can it be all electrons in an atom, or can it be electrons in a whole conductor, or can it be a larger range?

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  • $\begingroup$ I quote: The quantum system is the atom or a molecule and the fermions are the electrons in the shells. Call it the electrons spin or the electrons magnetic dipole moment, they are the reason for how electrons behave like they behave in atoms. Pauli’s exclusion principle states this phenomenon, but not explain it. Just to get a better idea, put the spotlight on the electrons magnetic dipole moment (they are correlated one by one with the spin). physics.stackexchange.com/a/456830 $\endgroup$ – HolgerFiedler May 3 '20 at 14:25
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    $\begingroup$ This is a question that I have often had trouble with, but this answer has helped me the most in trying to understand it. physics.stackexchange.com/a/288766/23756 $\endgroup$ – RBarryYoung May 3 '20 at 18:46
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All electrons (and all elementary particles) in the universe are supposed to have exactly identical properties according to the standard model. This means that for electrons, the Pauli exclusion principle reads "No 2 electrons in the universe can occupy the same state".

But due to the phrasing of your question, I think you might also have a wrong idea of what exactly constitutes a "same state". For instance, if you have two atoms of hydrogen 1 km apart, both could have an electron in the "same" $1s$ state. This is simply because these two states are different. While they are both $1s$ states, they are associated with different atoms.

In a crystal, the picture is slightly different because strictly speaking the eigenstates are Bloch states which are delocalized over the while crystal. But for the deepest levels (the ones well below the conduction level), the picture of localized states localized around each atom is not so off. In that case, all atoms in the crystal will typically have these states occupied, but again this is not in opposition with Pauli's principle because the states are distinguishable due to being associated with different atoms.

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    $\begingroup$ your answer begs the question what "being associated with different atoms" distinguishes those atoms one from another? $\endgroup$ – hyportnex May 2 '20 at 18:01
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    $\begingroup$ That would be the natural next question, yes. I believe that as long as the interaction term between one orbital located on one atom, and the other located on the other atom, is small, the eigenstates of the two atom system can almost be seen as the product of the two independent states (one on each atom), and thus that these states are "different". $\endgroup$ – QuantumApple May 2 '20 at 19:12
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    $\begingroup$ You could distinguish spatially separated states (such as those on two different atoms) by noting that the expectation value of their positions are different, which means they are necessarily described by different wave-functions. $\endgroup$ – BBeast May 4 '20 at 4:44
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In principle it covers all Fermions in the Universe. Not two Fermions share the same quantum numbers. In a material with many moles of electrons each one of them has different values of energy level, etc. Of course, you have to consider, for example, that two electrons with the same n, l, m and spin numbers orbit two identical nuclei. They have, however, different quantum numbers since given a reference frame and the description of the system by some rather complicated quantum state vector, they would differ in their quantum numbers. The same applies for more complicated systems. So, final example, fermions in a collapsing star resist the collapse due to Pauli's exclusion principle even though they are in a huge system with not a very nicely defined quantum state vector.

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    $\begingroup$ What specifically are the quantum numbers that differ for the electrons of two different hydrogen (or helium) atoms on different sides of the universe? This is the part that confuses many people (including me), so it would be very helpful to explain what the different quantum numbers actually. Because to my understanding, the only differences between two otherwise identical hydrogen atoms, is entirely non-quantum. I.E., position and orientation, neither of which are quantized, AFAIK. $\endgroup$ – RBarryYoung May 3 '20 at 17:54
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    $\begingroup$ two different atoms could be described as a tensor product (if one disregards a lot of other things) of the basic states of each one, leaving a set of $n_1, n_2, l_1, l_2, \ldots$ quantum numbers. Each atom is sort of independent of the other and therefore provides a full set of quantum numbers. Again, not having in mind position and so on. Also, consider that if one tries to describe two atoms as one system it would be even more complex and again the quantum numbers for that system would definitely be different for each electron. $\endgroup$ – Nelson Vanegas A. May 3 '20 at 17:57
  • $\begingroup$ @RBarryYoung If the interactions between atoms are negligible, as in your example, then the atom which an electron is occupying is a good quantum number (with a characteristic position expectation value). If the atoms are in a periodic crystal, then your quantum number is the crystal momentum (with the corresponding states being Bloch states. If your system is a molecule, then your quantum number would enumerate the molecular orbitals. $\endgroup$ – BBeast May 4 '20 at 4:51
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The most common way to visualize the range of the exclusion principle comes to us from the study of ultra-dense objects like white dwarf stars and neutron stars. In a white dwarf, gravity squeezes the matter in it so hard that the wave functions of the electrons in it begin to overlap- and that's where the exclusion principle kicks in, and fights back against gravity to support the white dwarf and prevent it from being squeezed down more. This effect is called degeneracy pressure and a complete description of it would be the length of several chapters in an astrophysics text.

Degeneracy pressure only kicks in when the atoms are being squeezed together so hard that most of the empty space within the atoms has been compressed away. In effect, this means that the distance range over which degeneracy pressure becomes important is far smaller than the dimensions of a typical atom in its unsqueezed state.

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  • $\begingroup$ Does application of Pauli exclusion principle on 2 electrons anyhow relate to the degree of overlapping of their wave functions ? So as for 2 relatively distant hydrogen atoms they ( mathematicians forgive ) do not overlap, therefore "they do not know" there is another such electron in the same state. $\endgroup$ – Poutnik May 3 '20 at 12:16
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    $\begingroup$ @Poutnik if they are in atomic orbitals around different nuclei, they are different states even if they have the same quantum numbers. $\endgroup$ – John Dvorak May 3 '20 at 12:19
  • $\begingroup$ ...and when those nuclei get squeezed together enough, at some point the exclusion principle kicks in. . $\endgroup$ – niels nielsen May 3 '20 at 17:52
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It depends on the system to which the fermions belong. The exclusion principle says that no two fermions can have the same quantum state. The quantum state includes the system to which the fermion belongs. If you are looking at electrons in atoms, for example, the atom is the system, and the exclusion principle applies only to electrons within a particular atom. If you are looking at a fermi gas, then the range is the volume of the gas. If you are looking at a white dwarf, then it is the size of the white dwarf.

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In quantum mechanics, particle interactions can be of two types, scattering interactions and bound states.

What is the scope of application of the principle of exclusion?

The Pauli exclusion principle applies to bound states of electrons in the solutions of potential equations for atoms/molecules/lattices. It will apply to fermions in general , for example no two muons can occupy the same muonic hydrogen energy level.

Can it be all electrons in an atom,

All electrons of an atom have to occupy different energy levels. Energy levels might be degenerate, but they must be different in a quantum number( example spin orientation for example)

or can it be electrons in a whole conductor,

The electrons in a whole conductor are very lightly bound, which means the energy levels they occupy are very close to continuum, i.e. there will always be an available energy level with different quantum numbers to occupy, this is what allows to have more general quantum mechanical models for solids as the band theory of solids.

or can it be a larger range?

So range has meaning for the Pauli exclusion principle only when one is talking of bound states that have energy levels labeled by quantum numbers available for occupation.

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    $\begingroup$ All electrons of an atom must occupy different quantum states. Energy levels can be degenerate. $\endgroup$ – ProfRob May 3 '20 at 8:47
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    $\begingroup$ @RobJeffries thenks, I will qualify $\endgroup$ – anna v May 3 '20 at 9:07
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As you mentioned, the Pauli exclusion principle states that:

two identical fermions cannot be in the same quantum state

From your question, it is difficult to know how much quantum mechanics you know, but a state is basically everything you know to understand a system. In one representation of quantum mechanics, a state is represented as a complex number function of position in space, often denoted $\psi(x)$, with $x$ having as many dimensions as needed to represent your system. $x$ can therefore be a scalar or a vector. So, why do we have quantum numbers in atoms? The trick is that bound particles can only be in certain states, or linear combinations of these states. That, is, $\psi(x)$ cannot be arbitrary for bound particles, it needs to have a very specific form. This is analogous to stating in classical mechanics that a particle is bound to rotate about a point in a plane. From a 3D problem, you are now back to a 1D problem. The difference is that you now start from an uncountable set (all the $\psi(x)$) to a countable or even finite set. So, instead of writing $\psi(x)$, we write it as a linear combination of the fundamental, or pure states, the ones corresponding to quantum numbers, and we denote these states by the way we count them, with quantum numbers, instead of carrying the whole functions with us. Note that $\psi(x)$ can be in more than 3 dimensions if you have more than 1 particle, as you need more than 3 numbers to represent your system then. It's just like in classical mechanics: two particles in 1 dimension are represented by their respective positions, $x_1$ and $x_2$.

Given all this, the other answers give a very good idea of what the range of the exclusion principle is: it is in principle infinite. Note that if two particles are not in the same potential well, then the wave function is defined by the quantum numbers of the first potential well and the quantum numbers of the second potential well. There are the same "numbers" with the same values, but mathematically, they correspond to different wave functions as the functions are centered around a different origin, so you can have two helium atoms in the ground state side by side.

A more precise formulation of the Pauli exclusion principle is that the wave function representing a system of more than one particles must be antisymmetric with respect to the exchange of the two particles. That is, if you switch the location of the two particles, the wave function changes sign. Since fermions of a certain type, such as electrons, are all indistinguishable from each other if they have the same spin, the only way this can happen for electrons in the same potential well is if two electrons have different spin. With the same spin, you need $\psi(x) = -\psi(x)$, so 0: no electrons.

As a final note, in practice, when particles interact in wide potential wells, which relates to your "range", the energy levels get very close to each other at energies corresponding to large well width. Then, you can have many particles have the "same" energy at high enough energies, but the energy still differs between two electrons if they have the same spin. It just differs by a little. Of course, the electrons that have lower energy (bound closer to the center of the potential well) have energies that are spaced apart by a larger steps. To see the influence of the Pauli exclusion principle at higher energies in such systems, you need to have a lot of electrons at these energies.

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