Editing Pauli exclusion principle (section)

== Applications ==
<!-- This section is linked from [[Newton's laws of motion]] -->

=== Atoms ===
The Pauli exclusion principle helps explain a wide variety of physical phenomena. One particularly important consequence of the principle is the elaborate [[electron configuration|electron shell structure]] of atoms and the way atoms share electrons, explaining the variety of chemical elements and their chemical combinations. An [[electric charge|electrically neutral]] atom contains bound electrons equal in number to the protons in the [[atomic nucleus|nucleus]]. Electrons, being fermions, cannot occupy the same quantum state as other electrons, so electrons have to "stack" within an atom, i.e. have different spins while at the same electron orbital as described below.

An example is the neutral [[helium atom]] (He), which has two bound electrons, both of which can occupy the lowest-energy ([[Electron shell|1s]]) states by acquiring opposite spin; as spin is part of the quantum state of the electron, the two electrons are in different quantum states and do not violate the Pauli principle. However, the spin can take only two different values ([[eigenvalue]]s). In a [[lithium]] atom (Li), with three bound electrons, the third electron cannot reside in a 1s state and must occupy a higher-energy state instead. The lowest available state is 2s, so that the [[ground state]] of Li is 1s<sup>2</sup>2s. Similarly, successively larger elements must have shells of successively higher energy. The chemical properties of an element largely depend on the number of electrons in the outermost shell; atoms with different numbers of occupied electron shells but the same number of electrons in the outermost shell have similar properties, which gives rise to the [[periodic table|periodic table of the elements]].<ref name=Griffiths2004>{{citation| author=Griffiths, David J.|title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |isbn= 0-13-111892-7}}</ref>{{rp|214–218}}

To test the Pauli exclusion principle for the helium atom, Gordon Drake<ref>{{cite journal | last = Drake | first = G.W.F.| year = 1989| title = Predicted energy shifts for "paronic" Helium| url = https://scholar.uwindsor.ca/physicspub/85| journal = Phys. Rev. A| volume = 39 | issue = 2 
| pages = 897–899 | doi =10.1103/PhysRevA.39.897| pmid = 9901315| bibcode = 1989PhRvA..39..897D| s2cid = 35775478}}</ref> carried out very precise calculations for hypothetical states of the He atom that violate it, which are called '''paronic states'''. Later, K. Deilamian et al.<ref>{{cite journal | last = Deilamian | first = K.|display-authors=etal|year = 1995 | title = Search for small violations of the symmetrization postulate in an excited state of Helium| journal = Phys. Rev. Lett.| volume = 74 | issue = 24| pages = 4787–4790 | doi=10.1103/PhysRevLett.74.4787| pmid = 10058599| bibcode = 1995PhRvL..74.4787D}}</ref> used an atomic beam spectrometer to search for the paronic state 1s2s <sup>1</sup>S<sub>0</sub> calculated by Drake. The search was unsuccessful and showed that the statistical weight of this paronic state has an upper limit of {{val|5|e=-6}}. (The exclusion principle implies a weight of zero.)

=== Solid state properties ===
In [[Electrical conductor|conductor]]s and [[semiconductor]]s, there are very large numbers of [[molecular orbital]]s which effectively form a continuous [[electronic band structure|band structure]] of [[energy level]]s. In strong conductors ([[metal]]s) electrons are so [[Degenerate energy level|degenerate]] that they cannot even contribute much to the [[thermal capacity]] of a metal.<ref name=Kittel2005>{{citation|last=Kittel|first=Charles|title=[[Introduction to Solid State Physics]]|publisher=John Wiley & Sons, Inc.|year=2005|location=USA|edition=8th|isbn=978-0-471-41526-8}}</ref>{{rp|133–147}} Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion.

=== Stability of matter ===

{{further|Stability of matter}}

The stability of each electron state in an atom is described by the quantum theory of the atom, which shows that close approach of an electron to the nucleus necessarily increases the electron's kinetic energy, an application of the [[uncertainty principle]] of Heisenberg.<ref name=Lieb>{{Cite arXiv |eprint = math-ph/0209034|last1 = Lieb|first1 = Elliott H.|title = The Stability of Matter and Quantum Electrodynamics|year = 2002}}</ref> However, stability of large systems with many electrons and many [[nucleons]] is a different question, and requires the Pauli exclusion principle.<ref name=Lieb2>This realization is attributed by {{cite arXiv |eprint = math-ph/0209034|last1 = Lieb|first1 = Elliott H.|title = The Stability of Matter and Quantum Electrodynamics|year = 2002}} and by {{cite book |author=G. L. Sewell |title=Quantum Mechanics and Its Emergent Macrophysics |isbn=0-691-05832-6 |year=2002|publisher=Princeton University Press}} to F. J. Dyson and A. Lenard: ''Stability of Matter, Parts I and II'' (''J. Math. Phys.'', '''8''', 423–434 (1967); ''J. Math. Phys.'', '''9''', 698–711 (1968) ).</ref>

It has been shown that the Pauli exclusion principle is responsible for the fact that ordinary bulk matter is stable and occupies volume. This suggestion was first made in 1931 by [[Paul Ehrenfest]], who pointed out that the electrons of each atom cannot all fall into the lowest-energy orbital and must occupy successively larger shells. Atoms, therefore, occupy a volume and cannot be squeezed too closely together.<ref>As described by F. J. Dyson (J.Math.Phys. '''8''', 1538–1545 (1967)), Ehrenfest made this suggestion in his address on the occasion of the award of the [[Lorentz Medal]] to Pauli.</ref>

The first rigorous proof was provided in 1967 by [[Freeman Dyson]] and Andrew Lenard ([[:de:Andrew Lenard|de]]), who considered the balance of attractive (electron–nuclear) and repulsive (electron–electron and nuclear–nuclear) forces and showed that ordinary matter would collapse and occupy a much smaller volume without the Pauli principle.<ref>F. J. Dyson and A. Lenard: ''Stability of Matter, Parts I and II'' (''J. Math. Phys.'', '''8''', 423–434 (1967); ''J. Math. Phys.'', '''9''', 698–711 (1968) )</ref><ref name=Dyson1967a>
{{cite journal
 | last =Dyson
 | first =Freeman
 | title =Ground-State Energy of a Finite System of Charged Particles
 | journal =J. Math. Phys.
 | volume =8
 | issue =8
 | pages =1538–1545
 | year =1967
 | doi =10.1063/1.1705389
 |bibcode = 1967JMP.....8.1538D
}}</ref>
A much simpler proof was found later by [[Elliott H. Lieb]] and [[Walter Thirring]] in 1975. They provided a lower bound on the quantum energy in terms of  the [[Thomas-Fermi model]], which is stable due to a [[Density functional theory#Thomas–Fermi model|theorem of Teller]]. The proof used a lower bound on  the kinetic energy which is now called the [[Lieb–Thirring inequality]].

The consequence of the Pauli principle here is that electrons of the same spin are kept apart by a repulsive [[exchange interaction]], which is a short-range effect, acting simultaneously with the long-range electrostatic or [[Coulombic force]]. This effect is partly responsible for the everyday observation in the macroscopic world that two solid objects cannot be in the same place at the same time.

=== Astrophysics ===
Dyson and Lenard did not consider the extreme magnetic or gravitational forces that occur in some [[astronomical]] objects. In 1995 [[Elliott Lieb]] and coworkers showed that the Pauli principle still leads to stability in intense magnetic fields such as in [[neutron star]]s, although at a much higher density than in ordinary matter.<ref>{{cite journal |first1=E. H. |last1=Lieb |first2=M. |last2=Loss |first3=J. P. |last3=Solovej |journal=[[Physical Review Letters]] |volume=75 |issue=6 |pages=985–9 |year=1995 |title=Stability of Matter in Magnetic Fields |doi=10.1103/PhysRevLett.75.985 |pmid=10060179 |arxiv = cond-mat/9506047 |bibcode = 1995PhRvL..75..985L |s2cid=2794188 }}</ref> It is a consequence of [[general relativity]] that, in sufficiently intense gravitational fields, matter collapses to form a [[black hole]].

Astronomy provides a spectacular demonstration of the effect of the Pauli principle, in the form of [[white dwarf]] and [[neutron star]]s. In both bodies, the atomic structure is disrupted by extreme pressure, but the stars are held in [[hydrostatic equilibrium]] by ''[[degeneracy pressure]]'', also known as Fermi pressure. This exotic form of matter is known as [[degenerate matter]]. The immense gravitational force of a star's mass is normally held in equilibrium by [[Ideal gas law|thermal pressure]] caused by heat produced in [[thermonuclear fusion]] in the star's core. In white dwarfs, which do not undergo nuclear fusion, an opposing force to gravity is provided by [[electron degeneracy pressure]]. In [[neutron star]]s, subject to even stronger gravitational forces, electrons have merged with protons to form neutrons. Neutrons are capable of producing an even higher degeneracy pressure, [[neutron degeneracy pressure]], albeit over a shorter range. This can stabilize neutron stars from further collapse, but at a smaller size and higher [[density]] than a white dwarf. Neutron stars are the most "rigid" objects known; their [[Young modulus]] (or more accurately, [[bulk modulus]]) is 20 orders of magnitude larger than that of [[diamond]]. However, even this enormous rigidity can be overcome by the [[gravitational field]] of a neutron star mass exceeding the [[Tolman–Oppenheimer–Volkoff limit]], leading to the formation of a [[black hole]].<ref name="Bojowald2012">{{cite book|author=Martin Bojowald|title=The Universe: A View from Classical and Quantum Gravity|date=5 November 2012|publisher=John Wiley & Sons|isbn=978-3-527-66769-7}}</ref>{{rp|286–287}}