Pauli exclusion principle
Quantum mechanical principle that two identical fermions cannot occupy the same quantum state simultaneously
Top 10 Pauli exclusion principle related articles

Contents
Part of a series on 
Quantum mechanics 

$i\hbar {\frac {\partial }{\partial t}}\psi (t)\rangle ={\hat {H}}\psi (t)\rangle$

The Pauli exclusion principle is the quantum mechanical principle which states that two or more identical fermions (particles with halfinteger spin) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940.
In the case of electrons in atoms, it can be stated as follows: it is impossible for two electrons of a polyelectron atom to have the same values of the four quantum numbers: n, the principal quantum number, ℓ, the azimuthal quantum number, m_{ℓ}, the magnetic quantum number, and m_{s}, the spin quantum number. For example, if two electrons reside in the same orbital, then their n, ℓ, and m_{ℓ} values are the same, therefore their m_{s} must be different, and thus the electrons must have opposite halfinteger spin projections of 1/2 and −1/2.
Particles with an integer spin, or bosons, are not subject to the Pauli exclusion principle: any number of identical bosons can occupy the same quantum state, as with, for instance, photons produced by a laser or atoms in a Bose–Einstein condensate.
A more rigorous statement is that concerning the exchange of two identical particles: the total (manyparticle) wave function is antisymmetric for fermions, and symmetric for bosons. This means that if the space and spin coordinates of two identical particles are interchanged, then the total wave function changes its sign for fermions and does not change for bosons.
If two fermions were in the same state (for example the same orbital with the same spin in the same atom), interchanging them would change nothing and the total wave function would be unchanged. The only way the total wave function can both change sign as required for fermions and also remain unchanged is that this function must be zero everywhere, which means that the state cannot exist. This reasoning does not apply to bosons because the sign does not change.
Pauli exclusion principle Intro articles: 9
Overview
The Pauli exclusion principle describes the behavior of all fermions (particles with "halfinteger spin"), while bosons (particles with "integer spin") are subject to other principles. Fermions include elementary particles such as quarks, electrons and neutrinos. Additionally, baryons such as protons and neutrons (subatomic particles composed from three quarks) and some atoms (such as helium3) are fermions, and are therefore described by the Pauli exclusion principle as well. Atoms can have different overall "spin", which determines whether they are fermions or bosons — for example helium3 has spin 1/2 and is therefore a fermion, in contrast to helium4 which has spin 0 and is a boson.^{[1]}^{:123–125} As such, the Pauli exclusion principle underpins many properties of everyday matter, from its largescale stability, to the chemical behavior of atoms.
"Halfinteger spin" means that the intrinsic angular momentum value of fermions is $\hbar =h/2\pi$
Pauli exclusion principle Overview articles: 20
History
In the early 20th century it became evident that atoms and molecules with even numbers of electrons are more chemically stable than those with odd numbers of electrons. In the 1916 article "The Atom and the Molecule" by Gilbert N. Lewis, for example, the third of his six postulates of chemical behavior states that the atom tends to hold an even number of electrons in any given shell, and especially to hold eight electrons, thought to be typically arranged symmetrically at the eight corners of a cube.^{[2]} In 1919 chemist Irving Langmuir suggested that the periodic table could be explained if the electrons in an atom were connected or clustered in some manner. Groups of electrons were thought to occupy a set of electron shells around the nucleus.^{[3]} In 1922, Niels Bohr updated his model of the atom by assuming that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable "closed shells".^{[4]}^{:203}
Pauli looked for an explanation for these numbers, which were at first only empirical. At the same time he was trying to explain experimental results of the Zeeman effect in atomic spectroscopy and in ferromagnetism. He found an essential clue in a 1924 paper by Edmund C. Stoner, which pointed out that, for a given value of the principal quantum number (n), the number of energy levels of a single electron in the alkali metal spectra in an external magnetic field, where all degenerate energy levels are separated, is equal to the number of electrons in the closed shell of the noble gases for the same value of n. This led Pauli to realize that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state if the electron states are defined using four quantum numbers. For this purpose he introduced a new twovalued quantum number, identified by Samuel Goudsmit and George Uhlenbeck as electron spin.^{[5]}^{[6]}
Pauli exclusion principle History articles: 17
Connection to quantum state symmetry
The Pauli exclusion principle with a singlevalued manyparticle wavefunction is equivalent to requiring the wavefunction to be antisymmetric with respect to exchange. If $x\rangle$
 $\psi \rangle =\sum _{x,y}A(x,y)x,y\rangle ,$
where each A(x,y) is a (complex) scalar coefficient. Antisymmetry under exchange means that A(x,y) = −A(y,x). This implies A(x,y) = 0 when x = y, which is Pauli exclusion. It is true in any basis since local changes of basis keep antisymmetric matrices antisymmetric.
Conversely, if the diagonal quantities A(x,x) are zero in every basis, then the wavefunction component
 $A(x,y)=\langle \psi x,y\rangle =\langle \psi {\Big (}x\rangle \otimes y\rangle {\Big )}$
is necessarily antisymmetric. To prove it, consider the matrix element
 $\langle \psi {\Big (}(x\rangle +y\rangle )\otimes (x\rangle +y\rangle ){\Big )}.$
This is zero, because the two particles have zero probability to both be in the superposition state $x\rangle +y\rangle$
 $\langle \psi x,x\rangle +\langle \psi x,y\rangle +\langle \psi y,x\rangle +\langle \psi y,y\rangle .$
The first and last terms are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:
 $\langle \psi x,y\rangle +\langle \psi y,x\rangle =0,$
or
 $A(x,y)=A(y,x).$
For a system with n > 2 particles, the multiparticle basis states become nfold tensor products of oneparticle basis states, and the coefficients of the wavefunction $A(x_{1},x_{2},\ldots ,x_{n})$
Advanced quantum theory
According to the spin–statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with halfinteger spin occupy antisymmetric states; furthermore, only integer or halfinteger values of spin are allowed by the principles of quantum mechanics. In relativistic quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of halfinteger spin.
In one dimension, bosons, as well as fermions, can obey the exclusion principle. A onedimensional Bose gas with deltafunction repulsive interactions of infinite strength is equivalent to a gas of free fermions. The reason for this is that, in one dimension, the exchange of particles requires that they pass through each other; for infinitely strong repulsion this cannot happen. This model is described by a quantum nonlinear Schrödinger equation. In momentum space, the exclusion principle is valid also for finite repulsion in a Bose gas with deltafunction interactions,^{[7]} as well as for interacting spins and Hubbard model in one dimension, and for other models solvable by Bethe ansatz. The ground state in models solvable by Bethe ansatz is a Fermi sphere.
Pauli exclusion principle Connection to quantum state symmetry articles: 11
Applications
Atoms
The Pauli exclusion principle helps explain a wide variety of physical phenomena. One particularly important consequence of the principle is the elaborate electron shell structure of atoms and the way atoms share electrons, explaining the variety of chemical elements and their chemical combinations. An electrically neutral atom contains bound electrons equal in number to the protons in the 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, which has two bound electrons, both of which can occupy the lowestenergy (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 (eigenvalues). In a lithium atom, with three bound electrons, the third electron cannot reside in a 1s state and must occupy one of the higherenergy 2s states instead. 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 of the elements.^{[8]}^{:214–218}
To test the Pauli exclusion principle for the He atom, Gordon Drake^{[9]} 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.^{[10]} used an atomic beam spectrometer to search for the paronic state 1s2s ^{1}S_{0} calculated by Drake. The search was unsuccessful and showed that the statistical weight of this paronic state has an upper limit of 5x10^{−6}. (The exclusion principle implies a weight of zero.)
Solid state properties
In conductors and semiconductors, there are very large numbers of molecular orbitals which effectively form a continuous band structure of energy levels. In strong conductors (metals) electrons are so degenerate that they cannot even contribute much to the thermal capacity of a metal.^{[11]}^{:133–147} Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion.
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.^{[12]} However, stability of large systems with many electrons and many nucleons is a different question, and requires the Pauli exclusion principle.^{[13]}
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 lowestenergy orbital and must occupy successively larger shells. Atoms, therefore, occupy a volume and cannot be squeezed too closely together.^{[14]}
A more rigorous proof was provided in 1967 by Freeman Dyson and 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.^{[15]}^{[16]}
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 shortrange effect, acting simultaneously with the longrange 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 stars, although at a much higher density than in ordinary matter.^{[17]} 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 stars. 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 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 stars, 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.^{[18]}^{:286–287}
Pauli exclusion principle Applications articles: 38
See also
 Spinstatistics theorem
 Exchange force
 Exchange interaction
 Exchange symmetry
 Fermi–Dirac statistics
 Fermi hole
 Hund's rule
 Pauli effect
References
 ^ Kenneth S. Krane (5 November 1987). Introductory Nuclear Physics. Wiley. ISBN 9780471805533.
 ^ "Linus Pauling and The Nature of the Chemical Bond: A Documentary History". Special Collections & Archives Research Center  Oregon State University – via scarc.library.oregonstate.edu.
 ^ Langmuir, Irving (1919). "The Arrangement of Electrons in Atoms and Molecules" (PDF). Journal of the American Chemical Society. 41 (6): 868–934. doi:10.1021/ja02227a002. Archived from the original (PDF) on 20120330. Retrieved 20080901.
 ^ Shaviv, Glora (2010). The Life of Stars: The Controversial Inception and Emergence of the Theory of Stellar Structure. Springer. ISBN 9783642020872.
 ^ Straumann, Norbert (2004). "The Role of the Exclusion Principle for Atoms to Stars: A Historical Account". Invited Talk at the 12th Workshop on Nuclear Astrophysics. arXiv:quantph/0403199. Bibcode:2004quant.ph..3199S. CiteSeerX 10.1.1.251.9585.
 ^ Pauli, W. (1925). "Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren". Zeitschrift für Physik. 31 (1): 765–783. Bibcode:1925ZPhy...31..765P. doi:10.1007/BF02980631. S2CID 122941900.
 ^ A. G. Izergin; V. E. Korepin (July 1982). "Pauli principle for onedimensional bosons and the algebraic bethe ansatz" (PDF). Letters in Mathematical Physics. 6 (4): 283–288. Bibcode:1982LMaPh...6..283I. doi:10.1007/BF00400323. S2CID 121829553.
 ^ Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 0131118927
 ^ Drake, G.W.F. (1989). "Predicted energy shifts for "paronic" Helium". Phys. Rev. A. 39 (2): 897–899. Bibcode:1989PhRvA..39..897D. doi:10.1103/PhysRevA.39.897. PMID 9901315.
 ^ Deilamian, K.; et al. (1995). "Search for small violations of the symmetrization postulate in an excited state of Helium". Phys. Rev. Lett. 74 (24): 4787–4790. Bibcode:1995PhRvL..74.4787D. doi:10.1103/PhysRevLett.74.4787. PMID 10058599.
 ^ Kittel, Charles (2005), Introduction to Solid State Physics (8th ed.), USA: John Wiley & Sons, Inc., ISBN 9780471415268
 ^ Lieb, Elliott H. (2002). "The Stability of Matter and Quantum Electrodynamics". arXiv:mathph/0209034. Bibcode:2002math.ph...9034L. Cite journal requires
journal=
(help)  ^ This realization is attributed by Lieb, Elliott H. (2002). "The Stability of Matter and Quantum Electrodynamics". arXiv:mathph/0209034. and by G. L. Sewell (2002). Quantum Mechanics and Its Emergent Macrophysics. Princeton University Press. ISBN 0691058326. 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) ).
 ^ 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.
 ^ 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) )
 ^ Dyson, Freeman (1967). "Ground‐State Energy of a Finite System of Charged Particles". J. Math. Phys. 8 (8): 1538–1545. Bibcode:1967JMP.....8.1538D. doi:10.1063/1.1705389.
 ^ Lieb, E. H.; Loss, M.; Solovej, J. P. (1995). "Stability of Matter in Magnetic Fields". Physical Review Letters. 75 (6): 985–9. arXiv:condmat/9506047. Bibcode:1995PhRvL..75..985L. doi:10.1103/PhysRevLett.75.985. PMID 10060179. S2CID 2794188.
 ^ Martin Bojowald (5 November 2012). The Universe: A View from Classical and Quantum Gravity. John Wiley & Sons. ISBN 9783527667697.
 General
 Dill, Dan (2006). "Chapter 3.5, Manyelectron atoms: Fermi holes and Fermi heaps". Notes on General Chemistry (2nd ed.). W. H. Freeman. ISBN 1429200685.
 Liboff, Richard L. (2002). Introductory Quantum Mechanics. AddisonWesley. ISBN 0805387145.
 Massimi, Michela (2005). Pauli's Exclusion Principle. Cambridge University Press. ISBN 0521839114.
 Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0716743450.
 Scerri, Eric (2007). The periodic table: Its story and its significance. New York: Oxford University Press. ISBN 9780195305739.
External links
 Nobel Lecture: Exclusion Principle and Quantum Mechanics Pauli's account of the development of the Exclusion Principle.