Physics:Quantum Fermi–Dirac statistics - Revision history

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	'''Fermi–Dirac statistics''' describe the occupation of quantum states by identical [[Physics:Quantum fermion\|fermions]]. They apply to particles with half-integer spin, including [[Physics:Quantum electron\|electrons]], [[Physics:Quantum neutrino\|neutrinos]], [[Physics:Quantum quark\|quarks]], and composite fermions such as many atoms and nuclei.<ref>{{Cite journal \|last=Dirac \|first=Paul A. M. \|title=On the Theory of Quantum Mechanics \|journal=Proceedings of the Royal Society A \|year=1926 \|volume=112 \|issue=762 \|pages=661-677 \|doi=10.1098/rspa.1926.0133 \|jstor=94692}}</ref>		'''Fermi–Dirac statistics''' is a Book II topic in the Quantum Collection. '''Fermi–Dirac statistics''' describe the occupation of quantum states by identical [[Physics:Quantum fermion\|fermions]]. They apply to particles with half-integer spin, including [[Physics:Quantum electron\|electrons]], [[Physics:Quantum neutrino\|neutrinos]], [[Physics:Quantum quark\|quarks]], and composite fermions such as many atoms and nuclei.<ref>{{Cite journal \|last=Dirac \|first=Paul A. M. \|title=On the Theory of Quantum Mechanics \|journal=Proceedings of the Royal Society A \|year=1926 \|volume=112 \|issue=762 \|pages=661-677 \|doi=10.1098/rspa.1926.0133 \|jstor=94692}}</ref> Fermi-Dirac statistics limit each available one-particle quantum state to one identical fermion. This rule explains electron shells, the structure of metals and semiconductors, Fermi surfaces, and degeneracy pressure in compact stars. The distribution becomes especially important at low temperature or high density, where quantum indistinguishability dominates thermal randomness.
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	'''Fermi–Dirac statistics''' describe the occupation of quantum states by identical [[Physics:Quantum fermion\|fermions]]. They apply to particles with half-integer spin, including [[Physics:Quantum electron\|electrons]], [[Physics:Quantum neutrino\|neutrinos]], [[Physics:Quantum quark\|quarks]], and composite fermions such as many atoms and nuclei.		'''Fermi–Dirac statistics''' describe the occupation of quantum states by identical [[Physics:Quantum fermion\|fermions]]. They apply to particles with half-integer spin, including [[Physics:Quantum electron\|electrons]], [[Physics:Quantum neutrino\|neutrinos]], [[Physics:Quantum quark\|quarks]], and composite fermions such as many atoms and nuclei.<ref>{{Cite journal \|last=Dirac \|first=Paul A. M. \|title=On the Theory of Quantum Mechanics \|journal=Proceedings of the Royal Society A \|year=1926 \|volume=112 \|issue=762 \|pages=661-677 \|doi=10.1098/rspa.1926.0133 \|jstor=94692}}</ref>
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	== Description ==		== Description ==
	Fermions obey the [[Physics:Quantum Pauli exclusion principle\|Pauli exclusion principle]]: no two identical fermions may occupy the same quantum state. For a system in thermal equilibrium, the average occupation of a one-particle state with energy <math>E</math> is given by the Fermi-Dirac distribution:		Fermions obey the [[Physics:Quantum Pauli exclusion principle\|Pauli exclusion principle]]: no two identical fermions may occupy the same quantum state. For a system in thermal equilibrium, the average occupation of a one-particle state with energy <math>E</math> is given by the Fermi-Dirac distribution:<ref>{{Cite journal \|last=Fermi \|first=Enrico \|title=Sulla quantizzazione del gas perfetto monoatomico \|journal=Rendiconti Lincei \|year=1926 \|volume=3 \|pages=145-149}}</ref>

	<math>f(E)=\frac{1}{\exp\left((E-\mu)/k_{\mathrm B}T\right)+1}</math>		<math>f(E)=\frac{1}{\exp\left((E-\mu)/k_{\mathrm B}T\right)+1}</math>
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	At absolute zero, states below the Fermi energy are filled and states above it are empty. At nonzero temperature the boundary is smoothed, but the exclusion principle still limits each available state to one fermion of a given quantum state.		At absolute zero, states below the Fermi energy are filled and states above it are empty. At nonzero temperature the boundary is smoothed, but the exclusion principle still limits each available state to one fermion of a given quantum state.

	This statistical rule explains the structure of electron shells in atoms, the behavior of electrons in metals and semiconductors, degeneracy pressure in white dwarfs and neutron stars, and the existence of [[Physics:Quantum Fermi surfaces\|Fermi surfaces]] in condensed matter systems.		This statistical rule explains the structure of electron shells in atoms<ref>{{Cite journal \|last=Sommerfeld \|first=Arnold \|title=Zur Elektronentheorie der Metalle \|journal=Naturwissenschaften \|year=1927 \|volume=15 \|issue=41 \|pages=824-832 \|doi=10.1007/BF01505083}}</ref>, the behavior of electrons in metals and semiconductors, degeneracy pressure in white dwarfs and neutron stars, and the existence of [[Physics:Quantum Fermi surfaces\|Fermi surfaces]] in condensed matter systems.

	== Historical names ==		== Historical names ==
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	== References ==		== References ==
	{{reflist\|3}}		{{reflist\|3}}
	* {{Cite journal \|last=Fermi \|first=Enrico \|title=Sulla quantizzazione del gas perfetto monoatomico \|journal=Rendiconti Lincei \|year=1926 \|volume=3 \|pages=145-149}}
	* {{Cite journal \|last=Dirac \|first=Paul A. M. \|title=On the Theory of Quantum Mechanics \|journal=Proceedings of the Royal Society A \|year=1926 \|volume=112 \|issue=762 \|pages=661-677 \|doi=10.1098/rspa.1926.0133 \|jstor=94692}}
	* {{Cite book \|last=Dirac \|first=Paul A. M. \|title=Principles of Quantum Mechanics \|edition=4th revised \|location=London \|publisher=Oxford University Press \|year=1967 \|isbn=978-0-19-852011-5}}
	* {{Cite journal \|last=Sommerfeld \|first=Arnold \|title=Zur Elektronentheorie der Metalle \|journal=Naturwissenschaften \|year=1927 \|volume=15 \|issue=41 \|pages=824-832 \|doi=10.1007/BF01505083}}

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

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	== References ==		== References ==
	{{reflist\|3}}		{{reflist\|3}}
	* {{Cite journal \|last=Fermi \|first=E. \|title=Sulla quantizzazione del gas perfetto monoatomico \|journal=Rendiconti Lincei \|year=1926 \|volume=3 \|pages=145-149}}		* {{Cite journal \|last=Fermi \|first=Enrico \|title=Sulla quantizzazione del gas perfetto monoatomico \|journal=Rendiconti Lincei \|year=1926 \|volume=3 \|pages=145-149}}
	* {{Cite journal \|last=Dirac \|first=P. A. M. \|title=On the Theory of Quantum Mechanics \|journal=Proceedings of the Royal Society A \|year=1926 \|volume=112 \|issue=762 \|pages=661-677 \|doi=10.1098/rspa.1926.0133}}		* {{Cite journal \|last=Dirac \|first=Paul A. M. \|title=On the Theory of Quantum Mechanics \|journal=Proceedings of the Royal Society A \|year=1926 \|volume=112 \|issue=762 \|pages=661-677 \|doi=10.1098/rspa.1926.0133 \|jstor=94692}}
	* {{Cite ~~web~~ \|~~title~~=~~Fermi-~~Dirac ~~statistics~~ \|~~url~~=~~https://www~~.~~britannica~~.~~com/science/Fermi~~-~~Dirac~~-~~statistics~~ \|~~website~~=~~Encyclopaedia Britannica~~ \|~~access~~-~~date~~=~~2026-05-23~~}}		* {{Cite book \|last=Dirac \|first=Paul A. M. \|title=Principles of Quantum Mechanics \|edition=4th revised \|location=London \|publisher=Oxford University Press \|year=1967 \|isbn=978-0-19-852011-5}}
			* {{Cite journal \|last=Sommerfeld \|first=Arnold \|title=Zur Elektronentheorie der Metalle \|journal=Naturwissenschaften \|year=1927 \|volume=15 \|issue=41 \|pages=824-832 \|doi=10.1007/BF01505083}}

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

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	== See also ==		== See also ==
	* [[Physics:Quantum fermion]]		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also/Matter}}
	* [[Physics:~~Quantum Pauli exclusion principle]]~~
	* [[Physics:Quantum electron]]
	* [[Physics:Quantum Fermi surfaces]]
	* [[Physics:Quantum ~~Bose–Einstein statistics]]~~

	== References ==		== References ==

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	* [[Biography:Wolfgang Pauli\|Wolfgang Pauli]] formulated the exclusion principle on which the statistics depends.		* [[Biography:Wolfgang Pauli\|Wolfgang Pauli]] formulated the exclusion principle on which the statistics depends.

	== ~~Related concepts~~ ==		== See also ==
	* [[Physics:Quantum fermion]]		* [[Physics:Quantum fermion]]
	* [[Physics:Quantum Pauli exclusion principle]]		* [[Physics:Quantum Pauli exclusion principle]]

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	~~[[File~~:~~Quantum_Fermi_Dirac_statistics_educational_yellow.png\|thumb\|right\|Fermi~~-~~Dirac statistics fill one~~-~~particle states subject to the Pauli exclusion principle.]]~~		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

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	'''Fermi–Dirac statistics''' describe the occupation of quantum states by identical [[Physics:Quantum fermion\|fermions]]. They apply to particles with half-integer spin, including [[Physics:Quantum electron\|electrons]], [[Physics:Quantum neutrino\|neutrinos]], [[Physics:Quantum quark\|quarks]], and composite fermions such as many atoms and nuclei.		'''Fermi–Dirac statistics''' describe the occupation of quantum states by identical [[Physics:Quantum fermion\|fermions]]. They apply to particles with half-integer spin, including [[Physics:Quantum electron\|electrons]], [[Physics:Quantum neutrino\|neutrinos]], [[Physics:Quantum quark\|quarks]], and composite fermions such as many atoms and nuclei.
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			[[File:Quantum_Fermi_Dirac_statistics_educational_yellow.png\|thumb\|280px\|Fermi-Dirac statistics fill one-particle states subject to the Pauli exclusion principle.]]
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	== Description ==		== Description ==

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+[[File:Quantum_Fermi_Dirac_statistics_educational_yellow.png|thumb|right|Fermi-Dirac statistics fill one-particle states subject to the Pauli exclusion principle.]]
-== See also ==
+'''Fermi–Dirac statistics''' describe the occupation of quantum states by identical [[Physics:Quantum fermion|fermions]]. They apply to particles with half-integer spin, including [[Physics:Quantum electron|electrons]], [[Physics:Quantum neutrino|neutrinos]], [[Physics:Quantum quark|quarks]], and composite fermions such as many atoms and nuclei.
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 == References ==
 {{reflist|3}}
 {{Author|Harold Foppele}}

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{{Short description|Fermi–Dirac statistics in quantum matter}}
{{Quantum matter backlink|Atoms}}

'''Fermi–Dirac statistics''' describe the occupation of quantum states by fermions. They are central to electrons in atoms, metals, semiconductors, neutron matter, and many-body systems.

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also/Matter}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Fermi–Dirac statistics|1}}