Stone’s theorem on one-parameter unitary groups

E443147

mathematical theorem result in operator theory result in quantum mechanics theorem in functional analysis

Stone’s theorem on one-parameter unitary groups is a fundamental result in functional analysis and quantum mechanics that characterizes strongly continuous one-parameter unitary groups as being generated by unique self-adjoint operators.

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Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in operator theory ⓘ result in quantum mechanics ⓘ theorem in functional analysis ⓘ
appliesTo	self-adjoint operators on a Hilbert space ⓘ strongly continuous one-parameter unitary groups on a Hilbert space ⓘ
assumes	real parameter t in R for the one-parameter group ⓘ
characterizes	self-adjoint operators as generators of unitary groups ⓘ strongly continuous one-parameter unitary groups ⓘ
classificationRole	classifies strongly continuous one-parameter unitary groups by their self-adjoint generators ⓘ
concerns	densely defined self-adjoint operator A ⓘ one-parameter unitary group {U(t)}_{t in R} ⓘ
converseStatement	If A is a self-adjoint operator on a Hilbert space H, then U(t)=exp(itA) defines a strongly continuous one-parameter unitary group ⓘ
domain	Hilbert spaces NERFINISHED ⓘ
ensuresProperty	the generator of a strongly continuous one-parameter unitary group is closed ⓘ the generator of a strongly continuous one-parameter unitary group is densely defined ⓘ
field	functional analysis ⓘ mathematical physics ⓘ operator theory ⓘ quantum mechanics ⓘ
formalStatement	If {U(t)}_{t in R} is a strongly continuous one-parameter unitary group on a Hilbert space H, then there exists a unique self-adjoint operator A such that U(t)=exp(itA) for all real t ⓘ
generalizationOf	finite-dimensional diagonalization of normal matrices to continuous unitary flows ⓘ
generatorDefinition	the generator A is defined by Aψ = i lim_{t→0} (U(t)ψ − ψ)/t on its domain ⓘ
historicalPeriod	20th century ⓘ
implies	the generator of a strongly continuous one-parameter unitary group is self-adjoint ⓘ the time evolution in quantum mechanics is governed by a self-adjoint Hamiltonian ⓘ
importance	links self-adjointness with physical observables in quantum mechanics ⓘ provides rigorous foundation for Schrödinger time evolution ⓘ
mathematicalArea	C*-algebras and unitary representations of groups ⓘ
namedAfter	Marshall Harvey Stone NERFINISHED ⓘ
relatedTo	Hille–Yosida theorem NERFINISHED ⓘ Stone–von Neumann theorem NERFINISHED ⓘ spectral theorem NERFINISHED ⓘ
relatesConcept	one-parameter group ⓘ self-adjoint operator ⓘ strong continuity ⓘ unitary group ⓘ
requiresCondition	U(0)=I, the identity operator ⓘ group property U(t+s)=U(t)U(s) ⓘ strong continuity of the unitary group ⓘ
statesThat	every self-adjoint operator generates a strongly continuous one-parameter unitary group ⓘ every strongly continuous one-parameter unitary group is generated by a unique self-adjoint operator ⓘ
typicalNotation	U(t)=e^{itA} ⓘ
usedIn	construction of unitary groups from self-adjoint Hamiltonians ⓘ description of time evolution of quantum states ⓘ mathematical formulation of quantum mechanics ⓘ
usesConcept	spectral theorem for self-adjoint operators ⓘ strong operator topology ⓘ

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Wigner’s theorem on symmetry transformations → relatedConcept → Stone’s theorem on one-parameter unitary groups ⓘ