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| {{Quantum mechanics|cTopic=Fundamental concepts}}
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| The '''Pauli exclusion principle''' is the [[quantum mechanics|quantum mechanical]] principle that no two [[identical particles|identical]] [[fermions]] (particles with half-integer [[spin (physics)|spin]]) may occupy the same [[quantum state]] simultaneously. A more rigorous statement is that the total [[wave function]] for two identical fermions is [[Skew-symmetric matrix|anti-symmetric]] with respect to exchange of the particles. The principle was formulated by Austrian physicist [[Wolfgang Pauli]] in 1925.
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| For example, in an isolated atom no two electrons can have the same four [[quantum number]]s; if ''n'', ''{{ell}}'', and ''m<sub>{{ell}}</sub>'' are the same, ''m<sub>s</sub>'' must be different such that the electrons have opposite spins, and so on.
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| Integer spin particles, [[boson]]s, 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]] and [[Bose–Einstein condensate]].
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| ==Overview==
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| The Pauli exclusion principle governs the behavior of all [[fermion]]s (particles with "half-integer [[spin (physics)|spin]]"), while [[boson]]s (particles with "integer spin") are not subject to it. Fermions include [[elementary particle]]s such as [[quark]]s (the constituent particles of protons and neutrons), [[electron]]s and [[neutrino]]s. In addition, [[proton]]s and [[neutron]]s ([[subatomic particle]]s composed from three quarks) and some [[atom]]s are fermions, and are therefore subject to the Pauli exclusion principle as well. Atoms can have different overall "spin", which determines whether they are fermions or bosons — for example [[helium-3]] has spin 1/2 and is therefore a fermion, in contrast to [[helium-4]] which has spin 0 and is a boson. As such, the Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability, to the [[periodic table|chemical behavior of atoms]].
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| "Half-integer spin" means that the intrinsic [[angular momentum]] value of fermions is <math>\hbar = h/2\pi</math> (reduced [[Planck's constant]]) times a [[half-integer]] (1/2, 3/2, 5/2, etc.). In the theory of [[quantum mechanics]] fermions are described by [[identical particles|antisymmetric states]]. In contrast, particles with integer spin (called bosons) have symmetric wave functions; unlike fermions they may share the same quantum states. Bosons include the [[photon]], the [[Cooper pairs]] which are responsible for [[superconductivity]], and the [[W and Z bosons]]. (Fermions take their name from the [[Fermi–Dirac statistics|Fermi–Dirac statistical distribution]] that they obey, and bosons from their [[Bose–Einstein statistics|Bose–Einstein distribution]]).
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| ==History==
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| In the early 20th century it became evident that atoms and molecules with even numbers of electrons are more [[Chemical stability#Outside chemistry|chemically stable]] than those with odd numbers of electrons. In the famous 1916 article [http://osulibrary.oregonstate.edu/specialcollections/coll/pauling/bond/papers/corr216.3-lewispub-19160400.html "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 the shell and especially to hold eight electrons which are normally arranged symmetrically at the eight corners of a cube (see: [[cubical atom]]). 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 shell]]s about the nucleus.<ref>{{cite journal
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| | last=Langmuir | first=Irving
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| | title=The Arrangement of Electrons in Atoms and Molecules
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| | journal=Journal of the American Chemical Society
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| | year=1919 | volume=41 | issue=6 | pages=868–934
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| | url=http://www.physics.kku.ac.th/estructure/files/Langmuir_1919_AEA.pdf
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| | accessdate=2008-09-01 | |
| | doi=10.1021/ja02227a002
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| }}</ref> 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".
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| Pauli looked for an explanation for these numbers, which were at first only [[Empirical relationship|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 Clifton Stoner|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 level]]s are separated, is equal to the number of electrons in the closed shell of the [[noble gas]]es 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 two-valued quantum number, identified by [[Samuel Goudsmit]] and [[George Uhlenbeck]] as [[electron spin]].
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| == Connection to quantum state symmetry ==
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| The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric. An antisymmetric two-particle state is represented as a [[superposition principle|sum of states]] in which one particle is in state <math>\scriptstyle |x \rangle</math> and the other in state <math>\scriptstyle |y\rangle</math>:
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| :<math>
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| |\psi\rangle = \sum_{x,y} A(x,y) |x,y\rangle
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| </math>
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| and antisymmetry under exchange means that {{nowrap|1=''A''(''x'',''y'') = −''A''(''y'',''x'')}}. This implies that {{nowrap|1=''A''(''x'',''x'') = 0}}, which is Pauli exclusion. It is true in any basis, since unitary changes of basis keep antisymmetric matrices antisymmetric, although strictly speaking, the quantity {{nowrap|1=''A''(''x'',''y'')}} is not a matrix but an antisymmetric rank-two [[tensor]].
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| Conversely, if the diagonal quantities {{nowrap|1=''A''(''x'',''x'')}} are zero ''in every basis'', then the wavefunction component:
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| :<math>
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| A(x,y)=\langle \psi|x,y\rangle = \langle \psi | ( |x\rangle \otimes |y\rangle )
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| </math>
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| is necessarily antisymmetric. To prove it, consider the matrix element:
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| :<math>
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| \langle\psi| ((|x\rangle + |y\rangle)\otimes(|x\rangle + |y\rangle))
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| \,</math>
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| This is zero, because the two particles have zero probability to both be in the superposition state <math>\scriptstyle |x\rangle + |y\rangle</math>. But this is equal to
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| :<math>
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| \langle \psi |x,x\rangle + \langle \psi |x,y\rangle + \langle \psi |y,x\rangle + \langle \psi | y,y \rangle
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| \,</math>
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| The first and last terms on the right hand side are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:
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| :<math>
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| \langle \psi|x,y\rangle + \langle\psi |y,x\rangle = 0
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| \,</math>.
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| or
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| :<math>
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| A(x,y)=-A(y,x)
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| \,</math>
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| ===Pauli principle in advanced quantum theory===
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| According to the [[spin-statistics theorem]], particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics.
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| In relativistic [[quantum field theory]], the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin. Since, nonrelativistically, particles can have any statistics and any spin, there is no way to prove a spin-statistics theorem in nonrelativistic quantum mechanics.
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| In one dimension, bosons, as well as fermions, can obey the exclusion principle. A one-dimensional Bose gas with delta function repulsive interactions of infinite strength is equivalent to a gas of free fermions. The reason for this is that, in one dimension, 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 delta function interactions,<ref>[http://insti.physics.sunysb.edu/~korepin/pauli.pdf A. Izergin and V. Korepin, Letter in Mathematical Physics vol 6, page 283, 1982 ]</ref> as well as for [[Heisenberg model (quantum)|interacting spins]] and [[Hubbard model]] in one dimension, and for other models solvable by [[Bethe ansatz]]. The [[Stationary state|ground state]] in models solvable by Bethe ansatz is a [[Fermi energy|Fermi sphere]].
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| == Consequences ==<!-- This section is linked from [[Newton's laws of motion]] -->
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| ===Atoms and the Pauli principle===
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| 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 [[atom]]s and the way atoms share electrons, explaining the variety of chemical elements and their chemical combinations. An [[electric charge|electrically neutral]] atom contains bound [[electron]]s 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.
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| An example is the neutral [[helium]] atom, 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, with three bound electrons, the third electron cannot reside in a ''1s'' state, and must occupy one of the higher-energy ''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 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]].
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| ===Solid state properties and the Pauli principle===
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| In [[Electrical conductor|conductor]]s and [[semi-conductor]]s, there are very large numbers of [[molecular orbital]]s which effectively form a continuous [[band structure]] of [[energy level]]s. In strong conductors ([[metal]]s) electrons are so [[Degenerate energy level|degenerate]] that they can not even contribute much to the [[thermal capacity]] of a metal. Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion.
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| ===Stability of matter===
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| The stability of the electrons in an atom itself is not related to the exclusion principle, but is described by the quantum theory of the atom. The underlying idea is that close approach of an electron to the nucleus of the atom necessarily increases its kinetic energy, an application of the [[uncertainty principle]] of Heisenberg.<ref name=Lieb>[http://arxiv.org/abs/math-ph/0209034v1 Elliot J. Lieb] ''The Stability of Matter and Quantum Electrodynamics''</ref> However, stability of large systems with many electrons and many nuclei is a different matter, and requires the Pauli exclusion principle.<ref name=Lieb2>This realization is attributed by [http://arxiv.org/abs/math-ph/0209034v1 Lieb] and by {{cite book |author=GL Sewell |title=Quantum Mechanics and Its Emergent Macrophysics |isbn=0-691-05832-6 |year=2002|publisher=Princeton University Press}} to FJ 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>
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| 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 FJ 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>
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| A more rigorous proof was provided in 1967 by [[Freeman Dyson]] and Andrew Lenard, 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>FJ 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) ); FJ Dyson: ''Ground-State Energy of a Finite System of Charged Particles (''J.Math.Phys. '''8''', 1538–1545 (1967) )</ref> 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.
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| ===Astrophysics and the Pauli principle===
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| Dyson and Lenard did not consider the extreme magnetic or gravitational forces which 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.<ref>E.H. Lieb, M. Loss and J.P. Solovej, Phys. Rev. Letters, 75, 985–9 (1995) "Stability of Matter in Magnetic Fields"</ref> It is a consequence of [[general relativity]] that, in sufficiently intense gravitational fields, matter collapses to form a [[black hole]].
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| Astronomy provides a spectacular demonstration of the effect of the Pauli principle, in the form of [[white dwarf]] and [[neutron star]]s. In both types of body, atomic structure is disrupted by large [[gravitation]]al forces, leaving the constituents supported by "degeneracy pressure" alone. This exotic form of matter is known as [[degenerate matter]]. In white dwarfs atoms are held apart by [[electron degeneracy pressure]]. In neutron stars, subject to even stronger gravitational forces, electrons have merged with [[proton]]s to form [[neutron]]s. Neutrons are capable of producing an even higher 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. Neutrons 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 massive star or by the pressure of a [[supernova]], leading to the formation of a [[black hole]].
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| == See also ==
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| * [[Exchange force]]
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| * [[Exchange interaction]]
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| * [[Exchange symmetry]]
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| * [[Hund's rule]]
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| * [[Fermi hole]]
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| ==References==
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| {{Reflist}}
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| {{Refbegin}}
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| *{{Cite book | author=Dill, Dan | title=Notes on General Chemistry (2nd ed.) | chapter = Chapter 3.5, Many-electron atoms: Fermi holes and Fermi heaps | publisher=W. H. Freeman | year=2006 | isbn=1-4292-0068-5 }}
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| *{{Cite book | author=Griffiths, David J.|title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |isbn=0-13-805326-X}}
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| *{{Cite book | author=[[Liboff, Richard L.]] | title=Introductory Quantum Mechanics | publisher=Addison-Wesley | year=2002 | isbn=0-8053-8714-5}}
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| *{{Cite book | author=Massimi, Michela | title=Pauli's Exclusion Principle | publisher=Cambridge University Press | year=2005 | isbn=0-521-83911-4}}
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| *{{Cite book | author=Tipler, Paul; Llewellyn, Ralph | title=Modern Physics (4th ed.) | publisher=W. H. Freeman | year=2002 | isbn=0-7167-4345-0}}
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| {{Refend}}
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| ==External links==
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| *[http://nobelprize.org/nobel_prizes/physics/laureates/1945/pauli-lecture.html Nobel Lecture: Exclusion Principle and Quantum Mechanics] Pauli's own account of the development of the Exclusion Principle.
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| {{DEFAULTSORT:Pauli Exclusion Principle}}
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| [[Category:Concepts in physics]]
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| [[Category:Pauli exclusion principle| ]]
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| [[Category:Spintronics]]
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| [[Category:Chemical bonding]]
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| [[Category:Lorentz Medal winners]]
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