![Wolfgang Pauli young](https://upload.wikimedia.org/wikipedia/commons/4/43/Wolfgang_Pauli_young.jpg)
In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the Pauli exclusion principle states that two or more
identical particles
In quantum mechanics, identical particles (also called indistinguishable 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
In mathematics, a half-integer is a number of the form
:n + \tfrac,
where n is an whole number. For example,
:, , , 8.5
are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
spins
The spins (as in having "the spins")Diane Marie Leiva. ''The Florida State University College of Education''Women's Voices on College Drinking: The First-Year College Experience"/ref> is an adverse reaction of intoxication that causes a state of v ...
(i.e.
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s) cannot occupy the same
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
within a
quantum system
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
simultaneously. This principle was formulated by Austrian physicist
Wolfgang Pauli
Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics fo ...
in 1925 for
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s, and later extended to all fermions with his
spin–statistics theorem
In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles that ...
of 1940.
In the case of electrons in atoms, it can be stated as follows: it is impossible for two electrons of a poly-electron atom to have the same values of the four
quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
s: ''n'', the
principal quantum number
In quantum mechanics, the principal quantum number (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable.
A ...
; ', the
azimuthal quantum number
The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe t ...
; ''m
'', the
magnetic quantum number
In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes th ...
; and ''m
s'', the
spin quantum number
In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe th ...
. For example, if two electrons reside in the same
orbital, then their ''n'', ', and ''m
'' values are the same; therefore their ''m
s'' must be different, and thus the electrons must have opposite half-integer spin projections of 1/2 and −1/2.
Particles with an integer spin, or
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 odd half-integer s ...
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
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" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
or atoms 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 densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...
.
A more rigorous statement is that, concerning the
exchange
Exchange may refer to:
Physics
*Gas exchange is the movement of oxygen and carbon dioxide molecules from a region of higher concentration to a region of lower concentration. Places United States
* Exchange, Indiana, an unincorporated community
* ...
of two identical particles, the total (many-particle)
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
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.
Overview
The Pauli exclusion principle describes the behavior of all
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s (particles with "half-integer
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
"), 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 odd half-integer s ...
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. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, an ...
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 nuclei. All commonly o ...
s,
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s and
neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s. Additionally,
baryon
In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classified ...
s such as
proton
A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
s and
neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
s (
subatomic particle
In physical sciences, a subatomic particle is a particle that composes 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 pr ...
s composed from three quarks) and some
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
s (such as
helium-3
Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron (the most common isotope, helium-4, having two protons and two neutrons in contrast). Other than protium (ordinary hydrogen), helium-3 is the ...
) 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 (the most common isotope, helium-4, having two protons and two neutrons in contrast). Other than protium (ordinary hydrogen), helium-3 is the ...
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 consis ...
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
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
value of fermions is
(reduced
Planck's constant) times a
half-integer
In mathematics, a half-integer is a number of the form
:n + \tfrac,
where n is an whole number. For example,
:, , , 8.5
are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
(1/2, 3/2, 5/2, etc.). In the theory of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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, so they always ...
, the
Cooper pairs
In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper ...
which are responsible for
superconductivity
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
, 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 , , and ...
. Fermions take their name from the
Fermi–Dirac statistical distribution, which they obey, and bosons take theirs from the
Bose–Einstein distribution Bose–Einstein may refer to:
* Bose–Einstein condensate
** Bose–Einstein condensation (network theory)
* Bose–Einstein correlations
* Bose–Einstein statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describe ...
.
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 bond a ...
, 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 engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface chemistry.
Langmuir's most famous publication is the 1919 art ...
suggested that the
periodic table
The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of ch ...
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 followed by electrons 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 physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. B ...
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 for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
. At the same time he was trying to explain experimental results of the
Zeeman effect
The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel prize ...
in atomic
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
and in
ferromagnetism
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
. 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 (symbolized ''n'') is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable.
A ...
(''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
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descri ...
s are separated, is equal to the number of electrons in the closed shell of the
noble gas
The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under standard conditions, they are all odorless, colorless, monatomic gases with very low chemi ...
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
Samuel Abraham Goudsmit (July 11, 1902 – December 4, 1978) was a Dutch-American physicist famous for jointly proposing the concept of electron spin with George Eugene Uhlenbeck in 1925.
Life and career
Goudsmit was born in The Hague, Neth ...
and
George Uhlenbeck
George Eugene Uhlenbeck (December 6, 1900 – October 31, 1988) was a Dutch-American theoretical physicist.
Background and education
George Uhlenbeck was the son of Eugenius and Anne Beeger Uhlenbeck. He attended the Hogere Burgerschool (High S ...
as
electron spin
In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin (physics), spin and electric charge. The value of the ...
.
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
and
range over the basis vectors of the
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
describing a one-particle system, then the tensor product produces the basis vectors
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:
:
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
:
is necessarily antisymmetric. To prove it, consider the matrix element
:
This is zero, because the two particles have zero probability to both be in the superposition state
. But this is equal to
:
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:
:
or
:
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
are identified by ''n'' one-particle states. The condition of antisymmetry states that the coefficients must flip sign whenever any two states are exchanged:
for any
. The exclusion principle is the consequence that, if
for any
then
This shows that none of the ''n'' particles may be in the same state.
Advanced quantum theory
According to the
spin–statistics theorem
In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles that ...
, 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 classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, the Pauli principle follows from applying a
rotation operator in
imaginary time
Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories.
...
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 nonlin ...
. 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 approximate model used to describe the transition between conducting and insulating systems.
It is particularly useful in solid-state physics. The model is named for John Hubbard.
The Hubbard model states that each el ...
in one dimension, and for other models solvable by
Bethe ansatz. 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 is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
atom contains bound electrons equal in number to the protons in the
nucleus
Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
*Atomic nucleus, the very dense central region of an atom
*Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA
Nucle ...
. 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 either one or two neutrons, depending on the isotope, held together by ...
, 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 of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s). In a
lithium
Lithium (from el, λίθος, lithos, lit=stone) is a chemical element with the symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard conditions, it is the least dense metal and the least dense solid el ...
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 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
The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of c ...
.
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
1S
0 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 which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
s, there are very large numbers of
molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding ...
s which effectively form a continuous
band structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
of
energy level
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The te ...
s. In strong conductors (
metal
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
s) electrons are so
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
that they cannot even contribute much to the
thermal capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
of a metal.
Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion.
Stability of matter
''(for more information, read the
Stability of matter page)''
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
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
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 (nucleon number).
Until the 1960s, nucleons we ...
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 physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition an ...
, 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 an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum m ...
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
Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physics#Mathematically rigorous physics, mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, Cond ...
and
Walter Thirring
Walter Thirring (29 April 1927 – 19 August 2014) was an Austrian physicist after whom the Thirring model in quantum field theory is named. He was the son of the physicist Hans Thirring.Thirring, H. Über die Wirkung rotierender ferner Massen ...
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 (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical ...
, 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 phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxies ...
objects. In 1995
Elliott Lieb
Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.
Lieb is ...
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 collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. white ...
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 Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
that, in sufficiently intense gravitational fields, matter collapses to form a
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
.
Astronomy provides a spectacular demonstration of the effect of the Pauli principle, in the form of
white dwarf
A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
and
neutron star
A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. white ...
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 (hydrostatic balance, 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. In the planetary ...
by ''
degeneracy pressure
Degenerate matter is a highly dense state of fermionic matter in which the Pauli exclusion principle exerts significant pressure in addition to, or in lieu of, thermal pressure. The description applies to matter composed of electrons, protons, neu ...
'', also known as Fermi pressure. This exotic form of matter is known as
degenerate matter
Degenerate matter is a highly dense state of fermionic matter in which the Pauli exclusion principle exerts significant pressure in addition to, or in lieu of, thermal pressure. The description applies to matter composed of electrons, protons, neu ...
. The immense gravitational force of a star's mass is normally held in equilibrium by
thermal pressure
In thermodynamics, thermal pressure (also known as the thermal pressure coefficient) is a measure of the relative pressure change of a fluid or a solid as a response to a temperature change at constant volume. The concept is related to the Pressur ...
caused by heat produced in
thermonuclear fusion
Thermonuclear fusion is the process of atomic nuclei combining or “fusing” using high temperatures to drive them close enough together for this to become possible. There are two forms of thermonuclear fusion: ''uncontrolled'', in which the re ...
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
Electron degeneracy pressure is a particular manifestation of the more general phenomenon of quantum degeneracy pressure. The Pauli exclusion principle disallows two identical half-integer spin (physics), spin particles (electrons and all other fe ...
. In
neutron star
A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. white ...
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
Degenerate matter is a highly dense state of fermionic matter in which the Pauli exclusion principle exerts significant pressure in addition to, or in lieu of, thermal pressure. The description applies to matter composed of electrons, protons, neu ...
, 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 substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
than a white dwarf. Neutron stars are the most "rigid" objects known; their
Young modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
(or more accurately,
bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume.
Other moduli describe ...
) 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. Another solid form of carbon known as graphite is the Chemical stability, chemically stable form of car ...
. However, even this enormous rigidity can be overcome by the
gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
of a neutron star mass exceeding the
Tolman–Oppenheimer–Volkoff limit The Tolman–Oppenheimer–Volkoff limit (or TOV limit) is an upper bound to the mass of cold, nonrotating neutron stars, analogous to the Chandrasekhar limit for white dwarf stars. If the mass of the said star reaches the limit it will collapse to ...
, leading to the formation of a
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
.
See also
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Spin-statistics theorem
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Exchange force In physics the term exchange force has been used to describe two distinct concepts which should not be confused.
Exchange of force carriers in particle physics
The preferred meaning of exchange force is in particle physics, where it denotes a forc ...
*
Exchange interaction
In chemistry and physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical ...
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Exchange symmetry
In quantum mechanics, identical particles (also called indistinguishable 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, ...
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Fermi–Dirac statistics
Fermi–Dirac statistics (F–D 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 di ...
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Fermi hole
Fermi heap and Fermi hole refer to two closely related quantum physics, quantum phenomenon, phenomena that occur in many-electron atoms. They arise due to the Pauli exclusion principle, according to which no two electrons can be in the same quantum ...
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Hund's rule
Hund's rule of maximum multiplicity is a rule based on observation of atomic spectra, which is used to predict the ground state of an atom or molecule with one or more open electronic shells. The rule states that for a given electron configuration ...
*
Pauli effect
The Pauli effect or Pauli's device corollary is the supposed tendency of technical equipment to encounter critical failure in the presence of certain people. The term was coined after mysterious anecdotal stories involving Austrian theoretical phy ...
References
;General
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External links
Nobel Lecture: Exclusion Principle and Quantum MechanicsPauli's account of the development of the Exclusion Principle.
{{DEFAULTSORT:Pauli Exclusion Principle
Concepts in physics
Spintronics
Chemical bonding
Lorentz Medal winners