Pauli exclusion principle

<h2 id="overview">Overview</h2>
The Pauli exclusion principle describes the behavior of all <a href="/facts/Fermion/LEckdDsR">fermions</a> (particles with half-integer <a href="/facts/Spin_(physics)/4Jvp7e8P">spin</a>), while <a href="/facts/Boson/XtPFcjb5">bosons</a> (particles with integer spin) are subject to other principles. Fermions include <a href="/facts/Elementary_particle/CTFDM97U">elementary particles</a> such as <a href="/facts/Quark/hwuuVkV5">quarks</a>, <a href="/facts/Electron/L0KBo5cW">electrons</a> and <a href="/facts/Neutrino/yGWVfSg2">neutrinos</a>. Additionally, <a href="/facts/Baryon/QFwAUWK2">baryons</a> such as <a href="/facts/Proton/clCjepXP">protons</a> and <a href="/facts/Neutron/oRCM1geb">neutrons</a> (<a href="/facts/Subatomic_particle/tMEzcTWQ">subatomic particles</a> composed from three quarks) and some <a href="/facts/Atom/69syB8dH">atoms</a> (such as <a href="/facts/Helium-3/SmPC5oT3">helium-3</a>) 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, <a href="/facts/Helium-3/SmPC5oT3">helium-3</a> has spin 1/2 and is therefore a fermion, whereas <a href="/facts/Helium-4/vNsL5qhC">helium-4</a> has spin 0 and is a boson.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>: 123–125  The Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability to the <a href="/facts/Periodic_table/umdGIfUb">chemical behavior of atoms</a>.
Half-integer spin means that the intrinsic <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a> value of fermions is 
 
 
 
 ℏ
 =
 h
 
 /
 
 2
 π
 
 
 {\displaystyle \hbar =h/2\pi }
 
 (<a href="/facts/Reduced_Planck_constant/qGHAe3qq">reduced Planck constant</a>) times a <a href="/facts/Half-integer/oJ29mf8G">half-integer</a> (1/2, 3/2, 5/2, etc.). In the theory of <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, fermions are described by <a href="/facts/Identical_particles/qg0mHeVR">antisymmetric states</a>. In contrast, particles with integer spin (bosons) have symmetric wave functions and may share the same quantum states. Bosons include the <a href="/facts/Photon/mFNhAEzA">photon</a>, the <a href="/facts/Cooper_pairs/uHILVyXu">Cooper pairs</a> which are responsible for <a href="/facts/Superconductivity/pqQAlSzJ">superconductivity</a>, and the <a href="/facts/W_and_Z_bosons/gAncY5XV">W and Z bosons</a>. Fermions take their name from the <a href="/facts/Fermi%25E2%2580%2593Dirac_statistics/aLo4bton">Fermi–Dirac statistical distribution</a>, which they obey, and bosons take theirs from the <a href="/facts/Bose%25E2%2580%2593Einstein_statistics/Utb2jN3n">Bose–Einstein distribution</a>.

<h2 id="history">History</h2>
In the early 20th century it became evident that atoms and molecules with even numbers of electrons are more <a href="/facts/Chemical_stability/j0YIgc6F">chemically stable</a> than those with odd numbers of electrons. In the 1916 article "The Atom and the Molecule" by <a href="/facts/Gilbert_N._Lewis/wWnV6QXL">Gilbert N. Lewis</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 <a href="/facts/Cubical_atom/ajNtecop">at the eight corners of a cube</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> In 1919 chemist <a href="/facts/Irving_Langmuir/mPt8xwUp">Irving Langmuir</a> suggested that the <a href="/facts/Periodic_table/umdGIfUb">periodic table</a> 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 <a href="/facts/Electron_shell/QxI30eh9">electron shells</a> around the nucleus.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> In 1922, <a href="/facts/Niels_Bohr/ydLPvQXT">Niels Bohr</a> updated <a href="/facts/Bohr_model/FmTHjaxy">his model of the atom</a> by assuming that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable "closed shells".<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>: 203 
Pauli looked for an explanation for these numbers, which were at first only <a href="/facts/Empirical_relationship/WSxeL4St">empirical</a>. At the same time he was trying to explain experimental results of the <a href="/facts/Zeeman_effect/A6NMlp2D">Zeeman effect</a> in atomic <a href="/facts/Spectroscopy/mj71FuNK">spectroscopy</a> and in <a href="/facts/Ferromagnetism/fH3HFnSl">ferromagnetism</a>. He found an essential clue in a 1924 paper by <a href="/facts/Edmund_Clifton_Stoner/30kenmeg">Edmund C. Stoner</a>, which pointed out that, for a given value of the <a href="/facts/Principal_quantum_number/8SzSWV6w">principal quantum number</a> (n), the number of energy levels of a single electron in the <a href="/facts/Alkali_metal/IkWbCLk2">alkali metal</a> spectra in an external magnetic field, where all <a href="/facts/Degenerate_energy_level/ulrnMXTR">degenerate energy levels</a> are separated, is equal to the number of electrons in the closed shell of the <a href="/facts/Noble_gas/j0yHMEAE">noble gases</a> 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 <a href="/facts/Samuel_Goudsmit/C5UszfUx">Samuel Goudsmit</a> and <a href="/facts/George_Uhlenbeck/IVKgKcVF">George Uhlenbeck</a> as <a href="/facts/Electron_spin/guAK2JJ2">electron spin</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="connection-to-quantum-state-symmetry">Connection to quantum state symmetry</h2>
In his Nobel lecture, Pauli clarified the importance of quantum state symmetry to the exclusion principle:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<blockquote>Among the different classes of symmetry, the most important ones (which moreover for two particles are the only ones) are the <a href="/facts/Boson/XtPFcjb5">symmetrical class</a>, in which the wave function does not change its value when the space and spin coordinates of two particles are permuted, and the <a href="/facts/Fermion/LEckdDsR">antisymmetrical class</a>, in which for such a permutation the wave function changes its sign...[The antisymmetrical class is] the correct and general wave mechanical formulation of the exclusion principle.</blockquote>
The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be <a href="/facts/Identical_particles/qg0mHeVR">antisymmetric with respect to exchange</a>. If 
 
 
 
 
 |
 
 x
 ⟩
 
 
 {\displaystyle |x\rangle }
 
 and 
 
 
 
 
 |
 
 y
 ⟩
 
 
 {\displaystyle |y\rangle }
 
 range over the basis vectors of the <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> describing a one-particle system, then the tensor product produces the basis vectors 
 
 
 
 
 |
 
 x
 ,
 y
 ⟩
 =
 
 |
 
 x
 ⟩
 ⊗
 
 |
 
 y
 ⟩
 
 
 {\displaystyle |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 <a href="/facts/Superposition_principle/GKGs9C92">superposition</a> (i.e. sum) of these basis vectors:

|
        
        ψ
        ⟩
        =
        
          ∑
          
            x
            ,
            y
          
        
        A
        (
        x
        ,
        y
        )
        
          |
        
        x
        ,
        y
        ⟩
        ,
      
    
    {\displaystyle |\psi \rangle =\sum _{x,y}A(x,y)|x,y\rangle ,}

where each A(x, y) is a (complex) scalar coefficient. Antisymmetry under exchange means that A(x, y) = −A(y, x). This implies A(x, y) = 0 when x = y, which is Pauli exclusion. It is true in any basis since local changes of basis keep antisymmetric matrices antisymmetric.
Conversely, if the diagonal quantities A(x, x) are zero in every basis, then the wavefunction component

A
        (
        x
        ,
        y
        )
        =
        ⟨
        ψ
        
          |
        
        x
        ,
        y
        ⟩
        =
        ⟨
        ψ
        
          |
        
        
          
            (
          
        
        
          |
        
        x
        ⟩
        ⊗
        
          |
        
        y
        ⟩
        
          
            )
          
        
      
    
    {\displaystyle 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

⟨
        ψ
        
          |
        
        
          
            (
          
        
        (
        
          |
        
        x
        ⟩
        +
        
          |
        
        y
        ⟩
        )
        ⊗
        (
        
          |
        
        x
        ⟩
        +
        
          |
        
        y
        ⟩
        )
        
          
            )
          
        
        .
      
    
    {\displaystyle \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
 ⟩
 +
 
 |
 
 y
 ⟩
 
 
 {\displaystyle |x\rangle +|y\rangle }
 
. But this is equal to

⟨
        ψ
        
          |
        
        x
        ,
        x
        ⟩
        +
        ⟨
        ψ
        
          |
        
        x
        ,
        y
        ⟩
        +
        ⟨
        ψ
        
          |
        
        y
        ,
        x
        ⟩
        +
        ⟨
        ψ
        
          |
        
        y
        ,
        y
        ⟩
        .
      
    
    {\displaystyle \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:

⟨
        ψ
        
          |
        
        x
        ,
        y
        ⟩
        +
        ⟨
        ψ
        
          |
        
        y
        ,
        x
        ⟩
        =
        0
        ,
      
    
    {\displaystyle \langle \psi |x,y\rangle +\langle \psi |y,x\rangle =0,}

or

A
        (
        x
        ,
        y
        )
        =
        −
        A
        (
        y
        ,
        x
        )
        .
      
    
    {\displaystyle A(x,y)=-A(y,x).}

For a system with n > 2 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
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 )
 
 
 {\displaystyle 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
 (
 …
 ,
 
 x
 
 i
 
 
 ,
 …
 ,
 
 x
 
 j
 
 
 ,
 …
 )
 =
 −
 A
 (
 …
 ,
 
 x
 
 j
 
 
 ,
 …
 ,
 
 x
 
 i
 
 
 ,
 …
 )
 
 
 {\displaystyle A(\ldots ,x_{i},\ldots ,x_{j},\ldots )=-A(\ldots ,x_{j},\ldots ,x_{i},\ldots )}
 
 for any 
 
 
 
 i
 ≠
 j
 
 
 {\displaystyle i\neq j}
 
. The exclusion principle is the consequence that, if 
 
 
 
 
 x
 
 i
 
 
 =
 
 x
 
 j
 
 
 
 
 {\displaystyle x_{i}=x_{j}}
 
 for any 
 
 
 
 i
 ≠
 j
 ,
 
 
 {\displaystyle i\neq j,}
 
 then 
 
 
 
 A
 (
 …
 ,
 
 x
 
 i
 
 
 ,
 …
 ,
 
 x
 
 j
 
 
 ,
 …
 )
 =
 0.
 
 
 {\displaystyle A(\ldots ,x_{i},\ldots ,x_{j},\ldots )=0.}
 
 This shows that none of the n particles may be in the same state.

<h3>Advanced quantum theory</h3>
According to the <a href="/facts/Spin%25E2%2580%2593statistics_theorem/j2qqVpez">spin–statistics theorem</a>, 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 <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>, the Pauli principle follows from applying a <a href="/facts/Rotation_operator_(quantum_mechanics)/vdUA900a">rotation operator</a> in <a href="/facts/Imaginary_time/UTTYSUw7">imaginary time</a> 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 <a href="/facts/Nonlinear_Schr%25C3%25B6dinger_equation/gfGkRApW">nonlinear Schrödinger equation</a>. In momentum space, the exclusion principle is valid also for finite repulsion in a Bose gas with delta-function interactions,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> as well as for <a href="/facts/Heisenberg_model_(quantum)/8RPu8n5n">interacting spins</a> and <a href="/facts/Hubbard_model/5pphLNde">Hubbard model</a> in one dimension, and for other models solvable by <a href="/facts/Bethe_ansatz/wSEGLbK6">Bethe ansatz</a>. The <a href="/facts/Stationary_state/HBQvbb6s">ground state</a> in models solvable by Bethe ansatz is a <a href="/facts/Fermi_energy/S4Q8ShGR">Fermi sphere</a>.

<h2 id="applications">Applications</h2>
<h3>Atoms</h3>
The Pauli exclusion principle helps explain a wide variety of physical phenomena. One particularly important consequence of the principle is the elaborate <a href="/facts/Electron_configuration/TQjRr22P">electron shell structure</a> of atoms and the way atoms share electrons, explaining the variety of chemical elements and their chemical combinations. An <a href="/facts/Electric_charge/xxsBfmlB">electrically neutral</a> atom contains bound electrons equal in number to the protons in the <a href="/facts/Atomic_nucleus/orpIm5Xl">nucleus</a>. 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 <a href="/facts/Helium_atom/ASrTuP3v">helium atom</a> (He), which has two bound electrons, both of which can occupy the lowest-energy (<a href="/facts/Electron_shell/QxI30eh9">1s</a>) 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 (<a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a>). In a <a href="/facts/Lithium/7vXX6BWA">lithium</a> 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 <a href="/facts/Ground_state/ML7cfCug">ground state</a> 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 <a href="/facts/Periodic_table/umdGIfUb">periodic table of the elements</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>: 214–218 
To test the Pauli exclusion principle for the helium atom, Gordon Drake<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> 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.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> 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 5×10−6. (The exclusion principle implies a weight of zero.)

<h3>Solid state properties</h3>
In <a href="/facts/Electrical_conductor/BSuyupLP">conductors</a> and <a href="/facts/Semiconductor/zFf3mGTE">semiconductors</a>, there are very large numbers of <a href="/facts/Molecular_orbital/0dP30jaa">molecular orbitals</a> which effectively form a continuous <a href="/facts/Electronic_band_structure/W5o7oE2R">band structure</a> of <a href="/facts/Energy_level/66x51Khc">energy levels</a>. In strong conductors (<a href="/facts/Metal/pqQVs8QU">metals</a>) electrons are so <a href="/facts/Degenerate_energy_level/ulrnMXTR">degenerate</a> that they cannot even contribute much to the <a href="/facts/Thermal_capacity/I63nzSyp">thermal capacity</a> of a metal.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>: 133–147  Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion.

<h3>Stability of matter</h3>
Further information: <a href="/facts/Stability_of_matter/TGWIkzKI">Stability of matter</a>
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 <a href="/facts/Uncertainty_principle/UUSoSp4T">uncertainty principle</a> of Heisenberg.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> However, stability of large systems with many electrons and many <a href="/facts/Nucleons/6fguU9Xk">nucleons</a> is a different question, and requires the Pauli exclusion principle.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
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 <a href="/facts/Paul_Ehrenfest/Q3Q7hXeJ">Paul Ehrenfest</a>, 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.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
The first rigorous proof was provided in 1967 by <a href="/facts/Freeman_Dyson/kSRxTh2r">Freeman Dyson</a> 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 class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
A much simpler proof was found later by <a href="/facts/Elliott_H._Lieb/Vr4mrmtb">Elliott H. Lieb</a> and <a href="/facts/Walter_Thirring/6QtorhHY">Walter Thirring</a> in 1975. They provided a lower bound on the quantum energy in terms of the <a href="/facts/Thomas-Fermi_model/d4fNornu">Thomas-Fermi model</a>, which is stable due to a <a href="/facts/Density_functional_theory/wwz4Gqdg">theorem of Teller</a>. The proof used a lower bound on the kinetic energy which is now called the <a href="/facts/Lieb%25E2%2580%2593Thirring_inequality/92kINJC4">Lieb–Thirring inequality</a>.
The consequence of the Pauli principle here is that electrons of the same spin are kept apart by a repulsive <a href="/facts/Exchange_interaction/orogl7qP">exchange interaction</a>, which is a short-range effect, acting simultaneously with the long-range electrostatic or <a href="/facts/Coulombic_force/V0lR5NKv">Coulombic force</a>. 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.

<h3>Astrophysics</h3>
Dyson and Lenard did not consider the extreme magnetic or gravitational forces that occur in some <a href="/facts/Astronomical/0ArJVM1p">astronomical</a> objects. In 1995 <a href="/facts/Elliott_Lieb/Vr4mrmtb">Elliott Lieb</a> and coworkers showed that the Pauli principle still leads to stability in intense magnetic fields such as in <a href="/facts/Neutron_star/4enfoxBA">neutron stars</a>, although at a much higher density than in ordinary matter.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> It is a consequence of <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> that, in sufficiently intense gravitational fields, matter collapses to form a <a href="/facts/Black_hole/1r2lmkcF">black hole</a>.
Astronomy provides a spectacular demonstration of the effect of the Pauli principle, in the form of <a href="/facts/White_dwarf/gIYlTtrC">white dwarf</a> and <a href="/facts/Neutron_star/4enfoxBA">neutron stars</a>. In both bodies, the atomic structure is disrupted by extreme pressure, but the stars are held in <a href="/facts/Hydrostatic_equilibrium/e75gVkh2">hydrostatic equilibrium</a> by <a href="/facts/Degeneracy_pressure/S5bc5MwK">degeneracy pressure</a>, also known as Fermi pressure. This exotic form of matter is known as <a href="/facts/Degenerate_matter/S5bc5MwK">degenerate matter</a>. The immense gravitational force of a star's mass is normally held in equilibrium by <a href="/facts/Ideal_gas_law/Pphokwhf">thermal pressure</a> caused by heat produced in <a href="/facts/Thermonuclear_fusion/fzQUW49h">thermonuclear fusion</a> in the star's core. In white dwarfs, which do not undergo nuclear fusion, an opposing force to gravity is provided by <a href="/facts/Electron_degeneracy_pressure/fupuyb5c">electron degeneracy pressure</a>. In <a href="/facts/Neutron_star/4enfoxBA">neutron stars</a>, subject to even stronger gravitational forces, electrons have merged with protons to form neutrons. Neutrons are capable of producing an even higher degeneracy pressure, <a href="/facts/Neutron_degeneracy_pressure/S5bc5MwK">neutron degeneracy pressure</a>, albeit over a shorter range. This can stabilize neutron stars from further collapse, but at a smaller size and higher <a href="/facts/Density/92p6hh9I">density</a> than a white dwarf. Neutron stars are the most "rigid" objects known; their <a href="/facts/Young_modulus/MV5JMCIM">Young modulus</a> (or more accurately, <a href="/facts/Bulk_modulus/yCRFQonn">bulk modulus</a>) is 20 orders of magnitude larger than that of <a href="/facts/Diamond/IBUdnd1y">diamond</a>. However, even this enormous rigidity can be overcome by the <a href="/facts/Gravitational_field/K1uTmJFy">gravitational field</a> of a neutron star mass exceeding the <a href="/facts/Tolman%25E2%2580%2593Oppenheimer%25E2%2580%2593Volkoff_limit/msQN8GkI">Tolman–Oppenheimer–Volkoff limit</a>, leading to the formation of a <a href="/facts/Black_hole/1r2lmkcF">black hole</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>: 286–287 

<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Spin-statistics_theorem/j2qqVpez">Spin-statistics theorem</a></li>
<li><a href="/facts/Exchange_interaction/orogl7qP">Exchange interaction</a></li>
<li><a href="/facts/Exchange_symmetry/qg0mHeVR">Exchange symmetry</a></li>
<li><a href="/facts/Fermi%25E2%2580%2593Dirac_statistics/aLo4bton">Fermi–Dirac statistics</a></li>
<li><a href="/facts/Fermi_hole/TQYGjjrs">Fermi hole</a></li>
<li><a href="/facts/Hund%2527s_rule/SqQFS8vR">Hund's rule</a></li>
<li><a href="/facts/Pauli_effect/lltQUTcS">Pauli effect</a></li></ul>

General

<ul><li>Dill, Dan (2006). "Chapter 3.5, Many-electron atoms: Fermi holes and Fermi heaps". Notes on General Chemistry (2nd ed.). W. H. Freeman. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4292-0068-5.</li>
<li><a href="/facts/Liboff%2c_Richard_L./eMKNaAUI">Liboff, Richard L.</a> (2002). Introductory Quantum Mechanics. Addison-Wesley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8053-8714-5.</li>
<li><a href="/facts/Michela_Massimi/sJDBa4s8">Massimi, Michela</a> (2005). Pauli's Exclusion Principle. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-83911-4.</li>
<li>Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7167-4345-0.</li>
<li>Scerri, Eric (2007). <a href="https://archive.org/details/periodictableits0000scer">The periodic table: Its story and its significance</a>. New York: Oxford University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-530573-9.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://nobelprize.org/nobel_prizes/physics/laureates/1945/pauli-lecture.html">Nobel Lecture: Exclusion Principle and Quantum Mechanics</a> Pauli's account of the development of the Exclusion Principle.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Kenneth S. Krane (5 November 1987). Introductory Nuclear Physics. Wiley. ISBN 978-0-471-80553-3. <a href="978-0-471-80553-3" target="_blank">978-0-471-80553-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Linus Pauling and The Nature of the Chemical Bond: A Documentary History". Special Collections & Archives Research Center - Oregon State University – via scarc.library.oregonstate.edu. <a href="http://scarc.library.oregonstate.edu/coll/pauling/bond/index.html" target="_blank">http://scarc.library.oregonstate.edu/coll/pauling/bond/index.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Langmuir, Irving (1919). "The Arrangement of Electrons in Atoms and Molecules" (PDF). Journal of the American Chemical Society. 41 (6): 868–934. doi:10.1021/ja02227a002. Archived from the original (PDF) on 2012-03-30. Retrieved 2008-09-01. <a href="https://www.webcitation.org/66YZ6UWkA?url=http://www.physics.kku.ac.th/estructure/files/Langmuir_1919_AEA.pdf" target="_blank">https://www.webcitation.org/66YZ6UWkA?url=http://www.physics.kku.ac.th/estructure/files/Langmuir_1919_AEA.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Shaviv, Glora (2010). The Life of Stars: The Controversial Inception and Emergence of the Theory of Stellar Structure. Springer. ISBN 978-3-642-02087-2. <a href="978-3-642-02087-2" target="_blank">978-3-642-02087-2</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Straumann, Norbert (2004). "The Role of the Exclusion Principle for Atoms to Stars: A Historical Account". Invited Talk at the 12th Workshop on Nuclear Astrophysics: 184–196. arXiv:quant-ph/0403199. Bibcode:2004quant.ph..3199S. CiteSeerX 10.1.1.251.9585. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Pauli, W. (1925). "Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren". Zeitschrift für Physik. 31 (1): 765–783. Bibcode:1925ZPhy...31..765P. doi:10.1007/BF02980631. S2CID 122941900. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"Wolfgang Pauli, Nobel lecture (December 13, 1946)" (PDF). <a href="https://www.nobelprize.org/uploads/2018/06/pauli-lecture.pdf" target="_blank">https://www.nobelprize.org/uploads/2018/06/pauli-lecture.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">A. G. Izergin; V. E. Korepin (July 1982). "Pauli principle for one-dimensional bosons and the algebraic Bethe ansatz" (PDF). Letters in Mathematical Physics. 6 (4): 283–288. Bibcode:1982LMaPh...6..283I. doi:10.1007/BF00400323. S2CID 121829553. Archived from the original (PDF) on 2018-11-25. Retrieved 2009-12-02. <a href="https://web.archive.org/web/20181125205409/http://insti.physics.sunysb.edu/~korepin/pauli.pdf" target="_blank">https://web.archive.org/web/20181125205409/http://insti.physics.sunysb.edu/~korepin/pauli.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 0-13-111892-7 <a href="0-13-111892-7" target="_blank">0-13-111892-7</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Drake, G.W.F. (1989). "Predicted energy shifts for "paronic" Helium". Phys. Rev. A. 39 (2): 897–899. Bibcode:1989PhRvA..39..897D. doi:10.1103/PhysRevA.39.897. PMID 9901315. S2CID 35775478. <a href="https://scholar.uwindsor.ca/physicspub/85" target="_blank">https://scholar.uwindsor.ca/physicspub/85</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Deilamian, K.; et al. (1995). "Search for small violations of the symmetrization postulate in an excited state of Helium". Phys. Rev. Lett. 74 (24): 4787–4790. Bibcode:1995PhRvL..74.4787D. doi:10.1103/PhysRevLett.74.4787. PMID 10058599. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Kittel, Charles (2005), Introduction to Solid State Physics (8th ed.), USA: John Wiley & Sons, Inc., ISBN 978-0-471-41526-8 <a href="978-0-471-41526-8" target="_blank">978-0-471-41526-8</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Lieb, Elliott H. (2002). "The Stability of Matter and Quantum Electrodynamics". arXiv:math-ph/0209034. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">This realization is attributed by Lieb, Elliott H. (2002). "The Stability of Matter and Quantum Electrodynamics". arXiv:math-ph/0209034. and by G. L. Sewell (2002). Quantum Mechanics and Its Emergent Macrophysics. Princeton University Press. ISBN 0-691-05832-6. 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) ). <a href="0-691-05832-6" target="_blank">0-691-05832-6</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">As described by F. J. Dyson (J.Math.Phys. 8, 1538–1545 (1967)), Ehrenfest made this suggestion in his address on the occasion of the award of the Lorentz Medal to Pauli. <a href="/wiki/Lorentz_Medal" target="_blank">/wiki/Lorentz_Medal</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">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) ) <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Dyson, Freeman (1967). "Ground-State Energy of a Finite System of Charged Particles". J. Math. Phys. 8 (8): 1538–1545. Bibcode:1967JMP.....8.1538D. doi:10.1063/1.1705389. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Lieb, E. H.; Loss, M.; Solovej, J. P. (1995). "Stability of Matter in Magnetic Fields". Physical Review Letters. 75 (6): 985–9. arXiv:cond-mat/9506047. Bibcode:1995PhRvL..75..985L. doi:10.1103/PhysRevLett.75.985. PMID 10060179. S2CID 2794188. <a href="/wiki/Physical_Review_Letters" target="_blank">/wiki/Physical_Review_Letters</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Martin Bojowald (5 November 2012). The Universe: A View from Classical and Quantum Gravity. John Wiley & Sons. ISBN 978-3-527-66769-7. <a href="978-3-527-66769-7" target="_blank">978-3-527-66769-7</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
</ol>

Pauli exclusion principle open-in-new

Pauli exclusion principle