Physics:Quantum Postulates - Revision history

Maintenance script: Normalize quantum page header order

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Harold at 11:05, 22 May 2026

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	{{Short description\|Core assumptions of quantum mechanics}}~~Book I~~		{{Short description\|Core assumptions of quantum mechanics}}

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	'''Postulates''' ~~is a Book I topic in the Quantum Collection. Quantum~~ postulates are the core assumptions of quantum mechanics. They specify how physical systems are represented, how observables are defined, how measurements produce probabilities and state updates, and how states evolve in time. In the standard formulation, a system is described by a complex Hilbert space, observables are represented by self-adjoint operators, measurement statistics are given by the Born rule, and time evolution is governed by the Schrödinger equation. Quantum postulates are the core assumptions of quantum mechanics. They specify how physical systems are represented, how observables are defined, how measurements produce probabilities and state updates, and how states evolve in time. In the standard formulation, a system is described by a complex Hilbert space, observables are represented by self-adjoint operators, measurement statistics are given by the Born rule, and time evolution is governed by the Schrödinger equation.		'''Postulates''' quantum postulates are the core assumptions of quantum mechanics. They specify how physical systems are represented, how observables are defined, how measurements produce probabilities and state updates, and how states evolve in time. In the standard formulation, a system is described by a complex Hilbert space, observables are represented by self-adjoint operators, measurement statistics are given by the Born rule, and time evolution is governed by the Schrödinger equation. Quantum postulates are the core assumptions of quantum mechanics. They specify how physical systems are represented, how observables are defined, how measurements produce probabilities and state updates, and how states evolve in time. In the standard formulation, a system is described by a complex Hilbert space, observables are represented by self-adjoint operators, measurement statistics are given by the Born rule, and time evolution is governed by the Schrödinger equation. The postulates can be grouped into a small number of operational rules.
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Maintenance script: Clean Book I red links, intro, and image slots

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	'''~~Quantum postulates~~''' are the core assumptions of ~~[[Physics:Quantum mechanics\|~~quantum mechanics]]. They specify how physical systems are represented, how observables are defined, how measurements produce probabilities and state updates, and how states evolve in time. In the standard formulation, a system is described by a complex ~~[[Physics:Quantum~~ Hilbert space~~\|Hilbert space]]~~, observables are represented by self-adjoint operators, measurement statistics are given by the ~~[[Physics:Born rule\|~~Born rule]], and time evolution is governed by the ~~[[Physics:Quantum Schrödinger equation\|~~Schrödinger equation]].~~<ref>{{cite book \| last1=Cohen-Tannoudji \| first1=Claude \| last2=Diu \| first2=Bernard \| last3=Laloë \| first3=Franck \| title=~~Quantum mechanics. ~~Volume 2: Angular momentum~~, ~~spin~~, and ~~approximation methods \| publisher=Wiley~~-~~VCH Verlag GmbH & Co~~. ~~KGaA \| publication-place=Weinheim \| date=2020 \| isbn=978-3-527-82272-0}}</ref>~~		'''Postulates''' is a Book I topic in the Quantum Collection. Quantum postulates are the core assumptions of quantum mechanics. They specify how physical systems are represented, how observables are defined, how measurements produce probabilities and state updates, and how states evolve in time. In the standard formulation, a system is described by a complex Hilbert space, observables are represented by self-adjoint operators, measurement statistics are given by the Born rule, and time evolution is governed by the Schrödinger equation. Quantum postulates are the core assumptions of quantum mechanics. They specify how physical systems are represented, how observables are defined, how measurements produce probabilities and state updates, and how states evolve in time. In the standard formulation, a system is described by a complex Hilbert space, observables are represented by self-adjoint operators, measurement statistics are given by the Born rule, and time evolution is governed by the Schrödinger equation.
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	Several important consequences follow from the postulates.		Several important consequences follow from the postulates.

	* Physical symmetries act on the Hilbert space by unitary or antiunitary transformations, as stated by ~~[[Physics:Quantum~~ Wigner's theorem~~\|Wigner's theorem]]~~.<ref>{{cite book \| last=Weyl \| first=Hermann \| title=The Theory of Groups and Quantum Mechanics \| title-link=Gruppentheorie und Quantenmechanik \| publisher=Dover \| translator-first=H. P. \| translator-last=Robertson \| year=1950 \| orig-year=1931 \| bibcode=1950tgqm.book.....W }}</ref>		* Physical symmetries act on the Hilbert space by unitary or antiunitary transformations, as stated by Wigner's theorem.<ref>{{cite book \| last=Weyl \| first=Hermann \| title=The Theory of Groups and Quantum Mechanics \| title-link=Gruppentheorie und Quantenmechanik \| publisher=Dover \| translator-first=H. P. \| translator-last=Robertson \| year=1950 \| orig-year=1931 \| bibcode=1950tgqm.book.....W }}</ref>
	* Pure states correspond to one-dimensional orthogonal projectors, while general density operators describe mixed states.		* Pure states correspond to one-dimensional orthogonal projectors, while general density operators describe mixed states.
	* The [[Physics:Quantum Uncertainty principle\|uncertainty principle]] can be derived as a theorem of the operator formalism.		* The [[Physics:Quantum Uncertainty principle\|uncertainty principle]] can be derived as a theorem of the operator formalism.
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	== Spin ==		== Spin ==

	All particles possess intrinsic angular momentum called ~~[[Physics:Quantum~~ spin~~\|spin]]~~. Unlike classical rotation, quantum spin is an intrinsic property with no direct classical analogue. For a particle of spin <math>S</math>, the spin degree of freedom introduces the discrete values		All particles possess intrinsic angular momentum called spin. Unlike classical rotation, quantum spin is an intrinsic property with no direct classical analogue. For a particle of spin <math>S</math>, the spin degree of freedom introduces the discrete values

	<math display="block">\sigma=-S\hbar,\;-(S-1)\hbar,\;\dots,\;0,\;\dots,\;(S-1)\hbar,\;S\hbar.</math>		<math display="block">\sigma=-S\hbar,\;-(S-1)\hbar,\;\dots,\;0,\;\dots,\;(S-1)\hbar,\;S\hbar.</math>
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	}}		}}

	This requirement underlies the distinction between bosons and fermions and is closely related to the ~~[[Physics:Quantum~~ spin-statistics theorem~~\|spin-statistics theorem]]~~. In two spatial dimensions, more general exchange behavior can occur, leading to ~~[[Physics:Quantum anyon\|anyon]]s~~.<ref>{{cite book \| last1=Sakurai \| first1=Jun John \| last2=Napolitano \| first2=Jim \| title=Modern quantum mechanics \| publisher=Cambridge University Press \| isbn=978-1-108-47322-4 \| edition=3rd \| location=Cambridge \| year=2021}}</ref>		This requirement underlies the distinction between bosons and fermions and is closely related to the spin-statistics theorem. In two spatial dimensions, more general exchange behavior can occur, leading to anyons.<ref>{{cite book \| last1=Sakurai \| first1=Jun John \| last2=Napolitano \| first2=Jim \| title=Modern quantum mechanics \| publisher=Cambridge University Press \| isbn=978-1-108-47322-4 \| edition=3rd \| location=Cambridge \| year=2021}}</ref>

	=== Pauli exclusion principle ===		=== Pauli exclusion principle ===

Maintenance script: Normalize Book I Quantum page structure

2026-05-19T22:47:53Z

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Harold: Clean Quantum header BOM and legacy backlink

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2026-05-17T21:51:30Z

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	=Physical system=		== Physical system ==
	A physical system is described by states, observables, and dynamics. In classical mechanics, states are points in phase space and observables are real-valued functions on that space. In quantum mechanics, by contrast, states are rays or density operators on a Hilbert space, and observables are self-adjoint operators acting on that space.<ref>{{cite book \| last=Weyl \| first=Hermann \| title=The Theory of Groups and Quantum Mechanics \| title-link=Gruppentheorie und Quantenmechanik \| publisher=Dover \| translator-first=H. P. \| translator-last=Robertson \| year=1950 \| orig-year=1931 \| bibcode=1950tgqm.book.....W }}</ref>		A physical system is described by states, observables, and dynamics. In classical mechanics, states are points in phase space and observables are real-valued functions on that space. In quantum mechanics, by contrast, states are rays or density operators on a Hilbert space, and observables are self-adjoint operators acting on that space.<ref>{{cite book \| last=Weyl \| first=Hermann \| title=The Theory of Groups and Quantum Mechanics \| title-link=Gruppentheorie und Quantenmechanik \| publisher=Dover \| translator-first=H. P. \| translator-last=Robertson \| year=1950 \| orig-year=1931 \| bibcode=1950tgqm.book.....W }}</ref>

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