Physics:Quantum Dirac equation - Revision history

WikiHarold: Restore missing Quantum reference definitions

2026-05-24T00:31:04Z

Restore missing Quantum reference definitions

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	* In the low-energy limit it reduces to the Pauli equation, and then further to the Schrödinger equation when spin effects are neglected.<ref name="BjorkenDrell1964" />		* In the low-energy limit it reduces to the Pauli equation, and then further to the Schrödinger equation when spin effects are neglected.<ref name="BjorkenDrell1964" />
	* Applying another Dirac operator shows that each spinor component satisfies the relativistic Klein–Gordon equation.<ref name="Thaller1992" />		* Applying another Dirac operator shows that each spinor component satisfies the relativistic Klein–Gordon equation.<ref name="Thaller1992">{{cite book \|last=Thaller \|first=B. \|title=The Dirac Equation \|series=Texts and Monographs in Physics \|publisher=Springer \|year=1992}}</ref>
	* In the massless case the equation reduces to the Weyl equation, relevant for chiral fermions.<ref name="Ohlsson2011">{{cite book \|last=Ohlsson \|first=Tommy \|title=Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory \|publisher=Cambridge University Press \|year=2011 \|page=86 \|isbn=978-1-139-50432-4}}</ref>		* In the massless case the equation reduces to the Weyl equation, relevant for chiral fermions.<ref name="Ohlsson2011">{{cite book \|last=Ohlsson \|first=Tommy \|title=Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory \|publisher=Cambridge University Press \|year=2011 \|page=86 \|isbn=978-1-139-50432-4}}</ref>

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

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

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	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}
	{{Quantum article nav\|previous=Physics:Quantum Matter Elements and Particles\|previous label=Matter Elements and Particles\|next=Physics:Quantum Klein–Gordon equation\|next label=Klein–Gordon equation}}		{{Quantum article nav\|previous=Physics:Quantum Matter Elements and Particles\|previous label=Matter Elements and Particles\|next=Physics:Quantum Klein–Gordon equation\|next label=Klein–Gordon equation}}

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	{{Short description\|Relativistic quantum mechanical wave equation}}		{{Short description\|Relativistic quantum mechanical wave equation}}

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}
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	{{Short description\|Relativistic quantum mechanical wave equation}}~~Book I~~		{{Short description\|Relativistic quantum mechanical wave equation}}

	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}

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	{{Quantum book backlink\|Mathematical structure and systems}}		{{Quantum book backlink\|Mathematical structure and systems}}

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	'''Dirac equation''' is the relativistic wave equation for spin-<math>\tfrac{1}{2}</math> particles. Introduced by [[Biography:Paul Dirac\|Paul Dirac]] in 1928, it ~~provided the first~~ quantum~~-mechanical description fully consistent~~ with special relativity and ~~successfully accounted for~~ electron spin ~~and the fine structure~~ of the ~~hydrogen spectrum~~.<ref name="Dirac1928">{{cite journal \|last=Dirac \|first=P. A. M. \|title=The Quantum Theory of the Electron \|journal=Proceedings of the Royal Society A \|volume=117 \|issue=778 \|pages=~~610–624~~ \|year=1928 \|doi=10.1098/rspa.1928.0023 \|bibcode=1928RSPSA.117..610D \|jstor=94981 \|doi-access=free}}</ref><ref name="Atkins1974">{{cite book \|last=Atkins \|first=P. W. \|title=Quanta: A Handbook of Concepts \|publisher=Oxford University Press \|year=1974 \|page=52 \|isbn=978-0-19-855493-6}}</ref> It also implied the existence of antimatter, later confirmed experimentally through the discovery of the positron.<ref name="Anderson1933">{{cite journal \|last=Anderson \|first=Carl D. \|title=The Positive Electron \|journal=Physical Review \|volume=43 \|issue=6 \|page=491 \|year=1933 \|doi=10.1103/PhysRev.43.491 \|bibcode=1933PhRv...43..491A \|doi-access=free}}</ref>		'''Dirac equation''' is the relativistic wave equation for spin-<math>\tfrac{1}{2}</math> particles such as the electron. Introduced by [[Biography:Paul Dirac\|Paul Dirac]] in 1928, it joined quantum mechanics with special relativity in a first-order wave equation and explained electron spin as a natural part of the theory rather than as an added rule.<ref name="Dirac1928">{{cite journal \|last=Dirac \|first=P. A. M. \|title=The Quantum Theory of the Electron \|journal=Proceedings of the Royal Society A \|volume=117 \|issue=778 \|pages=610-624 \|year=1928 \|doi=10.1098/rspa.1928.0023 \|bibcode=1928RSPSA.117..610D \|jstor=94981 \|doi-access=free}}</ref>

	The equation acts on a four-component ~~spinor field, or~~ Dirac spinor~~, rather than on a single complex wavefunction~~. ~~In this way it naturally incorporates spin,~~ positive- and negative-energy ~~solutions~~, and the ~~correct relativistic dispersion relation~~.<ref name="~~Thaller1992~~">{{cite ~~book~~ \|last=~~Thaller~~ \|first=B. \|title=The ~~Dirac Equation~~ \|~~series~~=~~Texts and Monographs in Physics~~ \|~~publisher~~=~~Springer~~ \|year=~~1992~~}}</ref>		The equation acts on a four-component Dirac spinor and uses gamma matrices to preserve Lorentz covariance. Its solutions include positive- and negative-energy sectors, leading historically to the prediction of antimatter and the later discovery of the positron.<ref name="Anderson1933">{{cite journal \|last=Anderson \|first=Carl D. \|title=The Positive Electron \|journal=Physical Review \|volume=43 \|issue=6 \|page=491 \|year=1933 \|doi=10.1103/PhysRev.43.491 \|bibcode=1933PhRv...43..491A \|doi-access=free}}</ref> In the Quantum Collection it connects nonrelativistic wave mechanics, spin, the Pauli equation, the Klein-Gordon equation, and the transition toward quantum field theory.

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	{{Short description\|Relativistic quantum mechanical wave equation}}		{{Short description\|Relativistic quantum mechanical wave equation}}

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	'''Dirac equation''' is the relativistic wave equation for spin-<math>\tfrac{1}{2}</math> particles. Introduced by [[Biography:Paul Dirac\|Paul Dirac]] in 1928, it provided the first quantum-mechanical description fully consistent with [[Physics:Special relativity\|special relativity]] and successfully accounted for electron spin and the fine structure of the hydrogen spectrum.<ref name="Dirac1928">{{cite journal \|last=Dirac \|first=P. A. M. \|title=The Quantum Theory of the Electron \|journal=Proceedings of the Royal Society A \|volume=117 \|issue=778 \|pages=610–624 \|year=1928 \|doi=10.1098/rspa.1928.0023 \|bibcode=1928RSPSA.117..610D \|jstor=94981 \|doi-access=free}}</ref><ref name="Atkins1974">{{cite book \|last=Atkins \|first=P. W. \|title=Quanta: A Handbook of Concepts \|publisher=Oxford University Press \|year=1974 \|page=52 \|isbn=978-0-19-855493-6}}</ref> It also implied the existence of antimatter, later confirmed experimentally through the discovery of the positron.<ref name="Anderson1933">{{cite journal \|last=Anderson \|first=Carl D. \|title=The Positive Electron \|journal=Physical Review \|volume=43 \|issue=6 \|page=491 \|year=1933 \|doi=10.1103/PhysRev.43.491 \|bibcode=1933PhRv...43..491A \|doi-access=free}}</ref>		'''Dirac equation''' is the relativistic wave equation for spin-<math>\tfrac{1}{2}</math> particles. Introduced by [[Biography:Paul Dirac\|Paul Dirac]] in 1928, it provided the first quantum-mechanical description fully consistent with [[Physics:Special relativity\|special relativity]] and successfully accounted for electron spin and the fine structure of the hydrogen spectrum.<ref name="Dirac1928">{{cite journal \|last=Dirac \|first=P. A. M. \|title=The Quantum Theory of the Electron \|journal=Proceedings of the Royal Society A \|volume=117 \|issue=778 \|pages=610–624 \|year=1928 \|doi=10.1098/rspa.1928.0023 \|bibcode=1928RSPSA.117..610D \|jstor=94981 \|doi-access=free}}</ref><ref name="Atkins1974">{{cite book \|last=Atkins \|first=P. W. \|title=Quanta: A Handbook of Concepts \|publisher=Oxford University Press \|year=1974 \|page=52 \|isbn=978-0-19-855493-6}}</ref> It also implied the existence of antimatter, later confirmed experimentally through the discovery of the positron.<ref name="Anderson1933">{{cite journal \|last=Anderson \|first=Carl D. \|title=The Positive Electron \|journal=Physical Review \|volume=43 \|issue=6 \|page=491 \|year=1933 \|doi=10.1103/PhysRev.43.491 \|bibcode=1933PhRv...43..491A \|doi-access=free}}</ref>

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	[[File:Quantum_Dirac_equation.jpg\|~~450px~~\|~~Conceptual visualization of the~~ Dirac equation~~, showing the union of quantum mechanics, special relativity, spinor structure, and antimatter~~.]]		</div>
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			[[File:Quantum_Dirac_equation.jpg\|thumb\|280px\|Quantum Dirac equation.]]
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imported>WikiHarold: Replace raw Quantum Collection backlink with B backlink template

2026-05-08T19:02:12Z

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	{{Quantum article nav\|previous=Physics:Quantum Matter Elements and Particles\|previous label=Matter Elements and Particles\|next=Physics:Quantum Klein–Gordon equation\|next label=Klein–Gordon equation}}		{{Quantum article nav\|previous=Physics:Quantum Matter Elements and Particles\|previous label=Matter Elements and Particles\|next=Physics:Quantum Klein–Gordon equation\|next label=Klein–Gordon equation}}

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	'''Dirac equation''' is the relativistic wave equation for spin-<math>\tfrac{1}{2}</math> particles. Introduced by [[Biography:Paul Dirac\|Paul Dirac]] in 1928, it provided the first quantum-mechanical description fully consistent with ~~[[Physics:Special relativity\|~~special relativity]] and successfully accounted for electron spin and the fine structure of the hydrogen spectrum.<ref name="Dirac1928">{{cite journal \|last=Dirac \|first=P. A. M. \|title=The Quantum Theory of the Electron \|journal=Proceedings of the Royal Society A \|volume=117 \|issue=778 \|pages=610–624 \|year=1928 \|doi=10.1098/rspa.1928.0023 \|bibcode=1928RSPSA.117..610D \|jstor=94981 \|doi-access=free}}</ref><ref name="Atkins1974">{{cite book \|last=Atkins \|first=P. W. \|title=Quanta: A Handbook of Concepts \|publisher=Oxford University Press \|year=1974 \|page=52 \|isbn=978-0-19-855493-6}}</ref> It also implied the existence of antimatter, later confirmed experimentally through the discovery of the positron.<ref name="Anderson1933">{{cite journal \|last=Anderson \|first=Carl D. \|title=The Positive Electron \|journal=Physical Review \|volume=43 \|issue=6 \|page=491 \|year=1933 \|doi=10.1103/PhysRev.43.491 \|bibcode=1933PhRv...43..491A \|doi-access=free}}</ref>		'''Dirac equation''' is the relativistic wave equation for spin-<math>\tfrac{1}{2}</math> particles. Introduced by [[Biography:Paul Dirac\|Paul Dirac]] in 1928, it provided the first quantum-mechanical description fully consistent with special relativity and successfully accounted for electron spin and the fine structure of the hydrogen spectrum.<ref name="Dirac1928">{{cite journal \|last=Dirac \|first=P. A. M. \|title=The Quantum Theory of the Electron \|journal=Proceedings of the Royal Society A \|volume=117 \|issue=778 \|pages=610–624 \|year=1928 \|doi=10.1098/rspa.1928.0023 \|bibcode=1928RSPSA.117..610D \|jstor=94981 \|doi-access=free}}</ref><ref name="Atkins1974">{{cite book \|last=Atkins \|first=P. W. \|title=Quanta: A Handbook of Concepts \|publisher=Oxford University Press \|year=1974 \|page=52 \|isbn=978-0-19-855493-6}}</ref> It also implied the existence of antimatter, later confirmed experimentally through the discovery of the positron.<ref name="Anderson1933">{{cite journal \|last=Anderson \|first=Carl D. \|title=The Positive Electron \|journal=Physical Review \|volume=43 \|issue=6 \|page=491 \|year=1933 \|doi=10.1103/PhysRev.43.491 \|bibcode=1933PhRv...43..491A \|doi-access=free}}</ref>

	The equation acts on a four-component spinor field, or ~~[[Physics:~~Dirac spinor~~\|Dirac spinor]]~~, rather than on a single complex wavefunction. In this way it naturally incorporates spin, positive- and negative-energy solutions, and the correct relativistic dispersion relation.<ref name="Thaller1992">{{cite book \|last=Thaller \|first=B. \|title=The Dirac Equation \|series=Texts and Monographs in Physics \|publisher=Springer \|year=1992}}</ref>		The equation acts on a four-component spinor field, or Dirac spinor, rather than on a single complex wavefunction. In this way it naturally incorporates spin, positive- and negative-energy solutions, and the correct relativistic dispersion relation.<ref name="Thaller1992">{{cite book \|last=Thaller \|first=B. \|title=The Dirac Equation \|series=Texts and Monographs in Physics \|publisher=Springer \|year=1992}}</ref>

	<div style="float:right; border:1px solid #ccc; padding:4px; background:#f9f9f9; margin:0 0 1em 1em; width:470px;">		<div style="float:right; border:1px solid #ccc; padding:4px; background:#f9f9f9; margin:0 0 1em 1em; width:470px;">
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	== Why the Dirac equation was needed ==		== Why the Dirac equation was needed ==

	The nonrelativistic ~~[[Physics:~~Schrödinger equation~~\|Schrödinger equation]]~~ works well at low velocities, but it does not incorporate special relativity. A naive relativistic replacement leads to the ~~[[Physics:Klein–Gordon equation\|~~Klein–Gordon equation]], which is second order in time and does not naturally describe spin-<math>\tfrac{1}{2}</math> electrons.<ref name="RaeNapolitano2015">{{cite book \|last1=Rae \|first1=Alastair I. M. \|last2=Napolitano \|first2=Jim \|title=Quantum Mechanics \|edition=6th \|publisher=Routledge \|year=2015 \|isbn=978-1482299182}}</ref>		The nonrelativistic Schrödinger equation works well at low velocities, but it does not incorporate special relativity. A naive relativistic replacement leads to the Klein–Gordon equation, which is second order in time and does not naturally describe spin-<math>\tfrac{1}{2}</math> electrons.<ref name="RaeNapolitano2015">{{cite book \|last1=Rae \|first1=Alastair I. M. \|last2=Napolitano \|first2=Jim \|title=Quantum Mechanics \|edition=6th \|publisher=Routledge \|year=2015 \|isbn=978-1482299182}}</ref>

	Dirac’s key insight was to seek an equation linear in both the time and spatial derivatives. This required introducing matrix coefficients acting on a multi-component wavefunction. The resulting formalism explained electron spin from first principles rather than inserting it phenomenologically.<ref name="Dirac1928" /><ref name="Shankar1994">{{cite book \|last=Shankar \|first=R. \|title=Principles of Quantum Mechanics \|edition=2nd \|publisher=Plenum \|year=1994}}</ref>		Dirac’s key insight was to seek an equation linear in both the time and spatial derivatives. This required introducing matrix coefficients acting on a multi-component wavefunction. The resulting formalism explained electron spin from first principles rather than inserting it phenomenologically.<ref name="Dirac1928" /><ref name="Shankar1994">{{cite book \|last=Shankar \|first=R. \|title=Principles of Quantum Mechanics \|edition=2nd \|publisher=Plenum \|year=1994}}</ref>
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	The Dirac equation contains several important limiting cases and connections:		The Dirac equation contains several important limiting cases and connections:

	* In the low-energy limit it reduces to the ~~[[Physics:~~Pauli equation~~\|Pauli equation]]~~, and then further to the Schrödinger equation when spin effects are neglected.<ref name="BjorkenDrell1964" />		* In the low-energy limit it reduces to the Pauli equation, and then further to the Schrödinger equation when spin effects are neglected.<ref name="BjorkenDrell1964" />
	* Applying another Dirac operator shows that each spinor component satisfies the relativistic Klein–Gordon equation.<ref name="Thaller1992" />		* Applying another Dirac operator shows that each spinor component satisfies the relativistic Klein–Gordon equation.<ref name="Thaller1992" />
	* In the massless case the equation reduces to the ~~[[Physics:~~Weyl equation~~\|Weyl equation]]~~, relevant for chiral fermions.<ref name="Ohlsson2011">{{cite book \|last=Ohlsson \|first=Tommy \|title=Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory \|publisher=Cambridge University Press \|year=2011 \|page=86 \|isbn=978-1-139-50432-4}}</ref>		* In the massless case the equation reduces to the Weyl equation, relevant for chiral fermions.<ref name="Ohlsson2011">{{cite book \|last=Ohlsson \|first=Tommy \|title=Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory \|publisher=Cambridge University Press \|year=2011 \|page=86 \|isbn=978-1-139-50432-4}}</ref>

	These links make the Dirac equation a central bridge between nonrelativistic quantum mechanics and modern quantum field theory.		These links make the Dirac equation a central bridge between nonrelativistic quantum mechanics and modern quantum field theory.