Physics:Quantum Mathematical Foundations of Quantum Theory - Revision history

Maintenance script: Normalize quantum page header order

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	{{Short description\|Hilbert spaces, operators, and mathematical structures underlying quantum theory}}~~Book I~~		{{Short description\|Hilbert spaces, operators, and mathematical structures underlying quantum theory}}

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	'''Mathematical Foundations of Quantum Theory''' is a ~~Book I topic in the Quantum Collection. A~~ Hilbert space is a vector space equipped with an inner product and complete with respect to the norm induced by that inner product. It provides the natural mathematical setting for quantum mechanics, where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space. The geometry of Hilbert space generalizes the familiar Euclidean notions of length, angle, orthogonality, and projection to spaces of finite or infinite dimension. A Hilbert space is a vector space equipped with an inner product and complete with respect to the norm induced by that inner product.~~{{cite book \|title=Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis }}~~ It provides the natural mathematical setting for quantum mechanics, where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space.		'''Mathematical Foundations of Quantum Theory''' a Hilbert space is a vector space equipped with an inner product and complete with respect to the norm induced by that inner product. It provides the natural mathematical setting for quantum mechanics, where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space. The geometry of Hilbert space generalizes the familiar Euclidean notions of length, angle, orthogonality, and projection to spaces of finite or infinite dimension. A Hilbert space is a vector space equipped with an inner product and complete with respect to the norm induced by that inner product. It provides the natural mathematical setting for quantum mechanics, where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space. In quantum theory, Hilbert spaces are usually taken to be complex.
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	</math>		</math>

	leads to the ~~[[Wikipedia:Uncertainty principle\|~~uncertainty principle]].<ref>{{cite book \|last=Griffiths \|first=David J. \|title=Introduction to Quantum Mechanics \|publisher=Pearson \|year=2018}}</ref>		leads to the uncertainty principle.<ref>{{cite book \|last=Griffiths \|first=David J. \|title=Introduction to Quantum Mechanics \|publisher=Pearson \|year=2018}}</ref>

	== Spectral theorem ==		== Spectral theorem ==

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	A '''~~Hilbert space~~''' is a ~~[[Wikipedia:Vector~~ space\|vector space]] equipped with an ~~[[Wikipedia:Inner product space\|~~inner product]] and complete with respect to the norm induced by that inner product.~~<ref>{{cite book~~		'''Mathematical Foundations of Quantum Theory''' is a Book I topic in the Quantum Collection. A Hilbert space is a vector space equipped with an inner product and complete with respect to the norm induced by that inner product. It provides the natural mathematical setting for quantum mechanics, where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space. The geometry of Hilbert space generalizes the familiar Euclidean notions of length, angle, orthogonality, and projection to spaces of finite or infinite dimension. A Hilbert space is a vector space equipped with an inner product and complete with respect to the norm induced by that inner product.{{cite book \|title=Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis }} It provides the natural mathematical setting for quantum mechanics, where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space.
	~~\|last=Reed~~
	~~\|first=Michael~~
	~~\|last2=Simon~~
	~~\|first2=Barry~~
	~~\|title=Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis~~
	~~\|publisher=Academic Press~~
	~~\|year=1980~~
	~~\|isbn=9780125850506~~
	~~}}</ref>~~ It provides the natural mathematical setting for ~~[[Wikipedia:Quantum mechanics\|~~quantum mechanics]], where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space.~~<ref>{{cite book \|last=Dirac \|first=P. A. M. \|title=The Principles of Quantum Mechanics \|publisher=Oxford University Press \|year=1930 \|url=https://archive.org/details/principlesofquan0000dira}}</ref>~~ The geometry of Hilbert space generalizes the familiar Euclidean notions of length, angle, orthogonality, and projection to spaces of finite or infinite dimension.
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2026-05-19T22:48:13Z

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

	{{Sourceattribution\|Quantum Mathematical Foundations of Quantum Theory\|1}}		{{Sourceattribution\|Physics:Quantum Mathematical Foundations of Quantum Theory\|1}}

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	{{Quantum article nav\|previous=Physics:Quantum probability\|previous label=Probability\|next=Physics:Quantum Interpretations of quantum mechanics\|next label=Interpretations of quantum mechanics}}		{{Quantum article nav\|previous=Physics:Quantum probability\|previous label=Probability\|next=Physics:Quantum Interpretations of quantum mechanics\|next label=Interpretations of quantum mechanics}}

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	A '''~~Hilbert space~~''' is a ~~[[Wikipedia:Vector~~ space\|vector space]] equipped with an ~~[[Wikipedia:Inner product space\|~~inner product]] and complete with respect to the norm induced by that inner product.~~<ref>{{cite book~~		'''Mathematical Foundations of Quantum Theory''' is a Book I topic in the Quantum Collection. A Hilbert space is a vector space equipped with an inner product and complete with respect to the norm induced by that inner product. It provides the natural mathematical setting for quantum mechanics, where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space. The geometry of Hilbert space generalizes the familiar Euclidean notions of length, angle, orthogonality, and projection to spaces of finite or infinite dimension. A Hilbert space is a vector space equipped with an inner product and complete with respect to the norm induced by that inner product.{{cite book \|title=Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis }} It provides the natural mathematical setting for quantum mechanics, where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space.
	~~\|last=Reed~~
	~~\|first=Michael~~
	~~\|last2=Simon~~
	~~\|first2=Barry~~
	~~\|title=Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis~~
	~~\|publisher=Academic Press~~
	~~\|year=1980~~
	~~\|isbn=9780125850506~~
	~~}}</ref>~~ It provides the natural mathematical setting for ~~[[Wikipedia:Quantum mechanics\|~~quantum mechanics]], where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space.~~<ref>{{cite book \|last=Dirac \|first=P. A. M. \|title=The Principles of Quantum Mechanics \|publisher=Oxford University Press \|year=1930 \|url=https://archive.org/details/principlesofquan0000dira}}</ref>~~ The geometry of Hilbert space generalizes the familiar Euclidean notions of length, angle, orthogonality, and projection to spaces of finite or infinite dimension.
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