Pauli exclusion principle
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quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, the Pauli exclusion principle (German: Pauli-Ausschlussprinzip) states that two or more
identical particles In quantum mechanics, indistinguishable particles (also called identical or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
with half-integer spins (i.e.
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s) cannot simultaneously occupy the same
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
within a system that obeys the laws of
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. This principle was formulated by Austrian physicist
Wolfgang Pauli Wolfgang Ernst Pauli ( ; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and a pioneer of quantum mechanics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics "for the ...
in 1925 for
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s, and later extended to all fermions with his spin–statistics theorem of 1940. In the case of electrons in atoms, the exclusion principle can be stated as follows: in a poly-electron atom it is impossible for any two electrons to have the same two values of ''all'' four of their
quantum number In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantu ...
s, which are: ''n'', the
principal quantum number In quantum mechanics, the principal quantum number (''n'') of an electron in an atom indicates which electron shell or energy level it is in. Its values are natural numbers (1, 2, 3, ...). Hydrogen and Helium, at their lowest energies, have just ...
; ', the
azimuthal quantum number In quantum mechanics, the azimuthal quantum number is a quantum number for an atomic orbital that determines its angular momentum operator, orbital angular momentum and describes aspects of the angular shape of the orbital. The azimuthal quantum ...
; ''m'', the
magnetic quantum number In atomic physics, a magnetic quantum number is a quantum number used to distinguish quantum states of an electron or other particle according to its angular momentum along a given axis in space. The orbital magnetic quantum number ( or ) disting ...
; and ''ms'', the
spin quantum number In physics and chemistry, the spin quantum number is a quantum number (designated ) that describes the intrinsic angular momentum (or spin angular momentum, or simply ''spin'') of an electron or other particle. It has the same value for all ...
. For example, if two electrons reside in the same orbital, then their values of ''n'', ', and ''m'' are equal. In that case, the two values of ''m''s (spin) pair must be different. Since the only two possible values for the spin projection ''m''s are +1/2 and −1/2, it follows that one electron must have ''m''s = +1/2 and one ''m''s = −1/2. Particles with an integer spin (
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
s) are not subject to the Pauli exclusion principle. Any number of identical bosons can occupy the same quantum state, such as photons produced by a
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
, or atoms found in a
Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low Density, densities is cooled to temperatures very close to absolute zero#Relation with Bose–Einste ...
. A more rigorous statement is: under the exchange of two identical particles, the total (many-particle)
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
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 sign for fermions, but does not change sign for bosons. So, if hypothetically two fermions were in the same statefor example, in the same atom in the same orbital with the same spinthen interchanging them would change nothing and the total wave function would be unchanged. However, the only way a total wave function can both change sign (required for fermions), and also remain unchanged is that such a function must be zero everywhere, which means such a state cannot exist. This reasoning does not apply to bosons because the sign does not change.


Overview

The Pauli exclusion principle describes the behavior of all
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s (particles with half-integer spin), while
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
s (particles with integer spin) are subject to other principles. Fermions include
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a c ...
s such as
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nucleus, atomic nuclei ...
s,
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s and
neutrino A neutrino ( ; denoted by the Greek letter ) is an elementary particle that interacts via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is so small ('' -ino'') that i ...
s. Additionally,
baryon In particle physics, a baryon is a type of composite particle, composite subatomic particle that contains an odd number of valence quarks, conventionally three. proton, Protons and neutron, neutrons are examples of baryons; because baryons are ...
s such as
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
s and
neutron The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
s (
subatomic particle In physics, a subatomic particle is a particle smaller than an atom. According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a baryon, lik ...
s composed from three quarks) and some
atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s (such as
helium-3 Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron. (In contrast, the most common isotope, helium-4, has two protons and two neutrons.) Helium-3 and hydrogen-1 are the only stable nuclides with ...
) 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,
helium-3 Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron. (In contrast, the most common isotope, helium-4, has two protons and two neutrons.) Helium-3 and hydrogen-1 are the only stable nuclides with ...
has spin 1/2 and is therefore a fermion, whereas
helium-4 Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consi ...
has spin 0 and is a boson. The Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability to the chemical behavior of atoms. Half-integer spin means that the intrinsic
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
value of fermions is \hbar = h/2\pi (
reduced Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
) times a half-integer (1/2, 3/2, 5/2, etc.). In the theory of
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, fermions are described by antisymmetric states. In contrast, particles with integer spin (bosons) have symmetric wave functions and may share the same quantum states. Bosons include the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
, the Cooper pairs which are responsible for
superconductivity Superconductivity is a set of physical properties observed in superconductors: materials where Electrical resistance and conductance, electrical resistance vanishes and Magnetic field, magnetic fields are expelled from the material. Unlike an ord ...
, and the
W and Z bosons In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
. Fermions take their name from the Fermi–Dirac statistical distribution, which they obey, and bosons take theirs from the Bose–Einstein distribution.


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 Gilbert Newton Lewis (October 23 or October 25, 1875 – March 23, 1946) was an American physical chemist and a dean of the college of chemistry at University of California, Berkeley. Lewis was best known for his discovery of the covalent bon ...
, 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, which he assumed to be typically arranged symmetrically at the eight corners of a cube. In 1919 chemist
Irving Langmuir Irving Langmuir (; January 31, 1881 – August 16, 1957) was an American chemist, physicist, and metallurgical engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface chemistry. Langmuir's most famous publicatio ...
suggested that the
periodic table The periodic table, also known as the periodic table of the elements, is an ordered arrangement of the chemical elements into rows (" periods") and columns (" groups"). It is an icon of chemistry and is widely used in physics and other s ...
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 In chemistry and atomic physics, an electron shell may be thought of as an orbit that electrons follow around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called the "K shell"), followed by the "2 shell" (o ...
s around the nucleus. In 1922,
Niels Bohr Niels Henrik David Bohr (, ; ; 7 October 1885 – 18 November 1962) was a Danish theoretical physicist who made foundational contributions to understanding atomic structure and old quantum theory, quantum theory, for which he received the No ...
updated his model of the atom by assuming that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable "closed shells". Pauli looked for an explanation for these numbers, which were at first only
empirical Empirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. There is no general agreement on how t ...
. At the same time he was trying to explain experimental results of the
Zeeman effect The Zeeman effect () is the splitting of a spectral line into several components in the presence of a static magnetic field. It is caused by the interaction of the magnetic field with the magnetic moment of the atomic electron associated with ...
in atomic
spectroscopy Spectroscopy is the field of study that measures and interprets electromagnetic spectra. In narrower contexts, spectroscopy is the precise study of color as generalized from visible light to all bands of the electromagnetic spectrum. Spectro ...
and in
ferromagnetism Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromagne ...
. 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 In quantum mechanics, the principal quantum number (''n'') of an electron in an atom indicates which electron shell or energy level it is in. Its values are natural numbers (1, 2, 3, ...). Hydrogen and Helium, at their lowest energies, have just ...
(''n''), the number of energy levels of a single electron in the
alkali metal The alkali metals consist of the chemical elements lithium (Li), sodium (Na), potassium (K),The symbols Na and K for sodium and potassium are derived from their Latin names, ''natrium'' and ''kalium''; these are still the origins of the names ...
spectra in an external magnetic field, where all
degenerate energy level In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give th ...
s are separated, is equal to the number of electrons in the closed shell of the
noble gas The noble gases (historically the inert gases, sometimes referred to as aerogens) are the members of Group (periodic table), group 18 of the periodic table: helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), radon (Rn) and, in some ...
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 George Eugene Uhlenbeck (December 6, 1900 – October 31, 1988) was a Dutch-American theoretical physicist, known for his significant contributions to quantum mechanics and statistical mechanics. He co-developed the concept of electron spin, alo ...
as electron spin.


Connection to quantum state symmetry

In his Nobel lecture, Pauli clarified the importance of quantum state symmetry to the exclusion principle:
Among the different classes of symmetry, the most important ones (which moreover for two particles are the only ones) are the symmetrical class, in which the wave function does not change its value when the space and spin coordinates of two particles are permuted, and the antisymmetrical class, in which for such a permutation the wave function changes its sign... he antisymmetrical class isthe correct and general wave mechanical formulation of the exclusion principle.
The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric with respect to exchange. If , x\rangle and , y\rangle range over the basis vectors of the
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
describing a one-particle system, then the tensor product produces the basis vectors , x,y\rangle=, x\rangle\otimes, y\rangle of the Hilbert space describing a system of two such particles. Any two-particle state can be represented as a superposition (i.e. sum) of these basis vectors: : , \psi\rangle = \sum_ A(x,y) , x,y\rangle, where each is a (complex) scalar coefficient. Antisymmetry under exchange means that . This implies when , which is Pauli exclusion. It is true in any basis since local changes of basis keep antisymmetric matrices antisymmetric. Conversely, if the diagonal quantities 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. But this is equal to : \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 particles, the multi-particle basis states become ''n''-fold tensor products of one-particle basis states, and the coefficients of the wavefunction A(x_1,x_2,\ldots,x_n) are identified by ''n'' one-particle states. The condition of antisymmetry states that the coefficients must flip sign whenever any two states are exchanged: A(\ldots,x_i,\ldots,x_j,\ldots)=-A(\ldots,x_j,\ldots,x_i,\ldots) for any i\ne j. The exclusion principle is the consequence that, if x_i=x_j for any i\ne j, then A(\ldots,x_i,\ldots,x_j,\ldots)=0. This shows that none of the ''n'' particles may be in the same state.


Advanced quantum theory

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. In relativistic
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin. 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, 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 theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonli ...
. In momentum space, the exclusion principle is valid also for finite repulsion in a Bose gas with delta-function interactions, as well as for interacting spins and
Hubbard model The Hubbard model is an Approximation, approximate model used to describe the transition between Conductor (material), conducting and Electrical insulation, insulating systems. It is particularly useful in solid-state physics. The model is named ...
in one dimension, and for other models solvable by
Bethe ansatz In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly for one-dimensional lattice models. It was first used by Hans Bethe in 1931 to find the exact eigenvalues and eigenv ...
. The
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
in models solvable by Bethe ansatz is a Fermi sphere.


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 Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other an ...
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 A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with two neutrons, depending on the isotope, held together by the strong ...
(He), which has two bound electrons, both of which can occupy the lowest-energy ( 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 In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s). In a
lithium Lithium (from , , ) is a chemical element; it has chemical symbol, symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard temperature and pressure, standard conditions, it is the least dense metal and the ...
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 The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
of Li is 1s22s. 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. To test the Pauli exclusion principle for the helium atom, Gordon Drake 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. used an atomic beam spectrometer to search for the paronic state 1s2s 1S0 calculated by Drake. The search was unsuccessful and showed that the statistical weight of this paronic state has an upper limit of . (The exclusion principle implies a weight of zero.)


Solid state properties

In conductors and
semiconductor A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
s, there are very large numbers of molecular orbitals which effectively form a continuous band structure of
energy level A quantum mechanics, quantum mechanical system or particle that is bound state, bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical mechanics, classical pa ...
s. In strong conductors (
metal A metal () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, electricity and thermal conductivity, heat relatively well. These properties are all associated wit ...
s) electrons are so degenerate that they cannot 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.


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 The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
of Heisenberg. However, stability of large systems with many electrons and many
nucleons In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number. Until the 1960s, nucleons were thought to be ele ...
is a different question, and requires the Pauli exclusion principle.This realization is attributed by and by 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) ). 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 Paul Ehrenfest (; 18 January 1880 – 25 September 1933) was an Austrian Theoretical physics, theoretical physicist who made major contributions to statistical mechanics and its relation to quantum physics, quantum mechanics, including the theory ...
, 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. The first rigorous proof was provided in 1967 by
Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrix, random matrices, math ...
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. 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 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 In chemistry and physics, the exchange interaction is a quantum mechanical constraint on the states of indistinguishable particles. While sometimes called an exchange force, or, in the case of fermions, Pauli repulsion, its consequences cannot alw ...
, 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 Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest include ...
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 A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s, although at a much higher density than in ordinary matter. It is a consequence of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
that, in sufficiently intense gravitational fields, matter collapses to form a
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
. Astronomy provides a spectacular demonstration of the effect of the Pauli principle, in the form of
white dwarf A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
and
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s. In both bodies, the atomic structure is disrupted by extreme pressure, but the stars are held in
hydrostatic equilibrium In fluid mechanics, hydrostatic equilibrium, also called hydrostatic balance and hydrostasy, is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. I ...
by ''
degeneracy pressure In astrophysics and condensed matter physics, electron degeneracy pressure is a quantum mechanical effect critical to understanding the stability of white dwarf stars and metal solids. It is a manifestation of the more general phenomenon of quan ...
'', 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 Nuclear fusion is a reaction in which two or more atomic nuclei combine to form a larger nuclei, nuclei/neutron by-products. The difference in mass between the reactants and products is manifested as either the release or absorption of ener ...
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 astrophysics and condensed matter physics, electron degeneracy pressure is a quantum mechanical effect critical to understanding the stability of white dwarf stars and metal solids. It is a manifestation of the more general phenomenon of quan ...
. In
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
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 Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
than a white dwarf. Neutron stars are the most "rigid" objects known; their Young modulus (or more accurately,
bulk modulus The bulk modulus (K or B or k) of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other mo ...
) is 20 orders of magnitude larger than that of
diamond Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Diamond is tasteless, odourless, strong, brittle solid, colourless in pure form, a poor conductor of e ...
. However, even this enormous rigidity can be overcome by the
gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
of a neutron star mass exceeding the Tolman–Oppenheimer–Volkoff limit, leading to the formation of a
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
.


See also

* Spin-statistics theorem *
Exchange interaction In chemistry and physics, the exchange interaction is a quantum mechanical constraint on the states of indistinguishable particles. While sometimes called an exchange force, or, in the case of fermions, Pauli repulsion, its consequences cannot alw ...
* Exchange symmetry *
Fermi–Dirac statistics Fermi–Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac distribution of part ...
* Fermi hole * Hund's rule * Pauli effect


References

; General : * * * * *


External links


Nobel Lecture: Exclusion Principle and Quantum Mechanics
Pauli's account of the development of the Exclusion Principle. {{DEFAULTSORT:Pauli Exclusion Principle Concepts in physics Spintronics Chemical bonding