What is the range of Pauli's exclusion principle？

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.