Editing Bound state (section)

==Definition==
{{see also|Decomposition of spectrum (functional analysis) #Quantum mechanics}}
Let [[Measure space#Important classes of measure spaces|{{math|''&sigma;''}}-finite measure space]] <math>(X, \mathcal A, \mu)</math> be a [[Measure space#Important classes of measure spaces|probability space]] associated with [[Separable space|separable]] [[complex number|complex]] [[Hilbert space]] <math>H</math>. Define a [[Stone's theorem on one-parameter unitary groups|one-parameter group of unitary operators]] <math> (U_t)_{t\in \mathbb{R}} </math>, a [[density operator]] <math>\rho = \rho(t_0) </math> and an [[observable]] <math>T</math> on <math>H</math>. Let <math>\mu(T,\rho)</math> be the induced probability distribution of <math>T</math> with respect to <math>\rho</math>. Then the evolution
:<math>\rho(t_0)\mapsto [U_t(\rho)](t_0) = \rho(t_0 +t)</math> 
is '''bound''' with respect to <math>T</math> if 
:<math>\lim_{R \rightarrow \infty}{\sup_{t \geq t_0}{\mu(T,\rho(t))(\mathbb{R}_{> R})}} = 0 </math>, 
where <math>\mathbb{R}_{>R} = \lbrace x \in \mathbb{R} \mid x > R \rbrace </math>.{{dubious|date=November 2016}}<ref>{{cite book | last1=Reed | first1=M. | last2=Simon | first2=B. | title=Methods of Modern Mathematical Physics: I: Functional analysis | publisher=Academic Press | year=1980 |page=303 | isbn=978-0-12-585050-6}}</ref>

A quantum particle is in a '''bound state''' if at no point in time it is found “too far away" from any finite region <math>R\subset X</math>. Using a [[wave function]] representation, for example, this means<ref>{{cite book | last=Gustafson | first=Stephen J. | last2=Sigal | first2=Israel Michael | title=Mathematical Concepts of Quantum Mechanics | publisher=Springer International Publishing | publication-place=Cham | year=2020 | isbn=978-3-030-59561-6 | issn=0172-5939 | doi=10.1007/978-3-030-59562-3|chapter=Bound and Decaying States}}</ref>
:<math>\begin{align}
0 &= \lim_{R\to\infty}{\mathbb{P}(\text{particle measured inside }X\setminus R)} \\
&= \lim_{R\to\infty}{\int_{X\setminus R}|\psi(x)|^2\,d\mu(x)},
\end{align}</math>
such that
:<math>\int_X{|\psi(x)|^{2}\,d\mu(x)} < \infty.</math>
In general, a quantum state is a bound state ''if and only if'' it is finitely [[Probability amplitude#Normalization|normalizable]] for all times <math>t\in\mathbb{R}</math> and remains spatially localized.<ref>{{cite journal | last=Ruelle | first=D. | title=A remark on bound states in potential-scattering theory | journal=Il Nuovo Cimento A | publisher=Springer Science and Business Media LLC | volume=61 | issue=4 | year=1969 | issn=0369-3546 | doi=10.1007/bf02819607 | url=https://www.ihes.fr/%7Eruelle/PUBLICATIONS/%5B25%5D.pdf}}</ref> Furthermore, a bound state lies within the [[Spectrum_(functional_analysis)#Classification_of_points_in_the_spectrum|pure point part]] of the spectrum of <math>T</math> ''if and only if'' it is an [[eigenvector]] of <math>T</math>.<ref>{{cite web | last=Simon | first=B. | title=An Overview of Rigorous Scattering Theory | date=1978 |page=3| url=https://api.semanticscholar.org/CorpusID:16913591}}</ref> 

More informally, "boundedness" results foremost from the choice of [[domain of definition]] and characteristics of the state rather than the observable.<ref group=nb>See [[Expectation_value_(quantum_mechanics)#Example_in_configuration_space|Expectation value (quantum mechanics)]] for an example.</ref> For a concrete example: let <math>H := L^2(\mathbb{R}) </math> and let <math>T</math> be the [[position operator]]. Given compactly supported <math>\rho = \rho(0) \in H</math> and <math>[-1,1] \subseteq \mathrm{Supp}(\rho)</math>.

*If the state evolution of <math>\rho</math> "moves this wave package to the right", e.g., if <math>[t-1,t+1] \in \mathrm{Supp}(\rho(t)) </math> for all <math>t \geq 0</math>, then <math>\rho</math> is not bound state with respect to position.
*If <math>\rho</math> does not change in time, i.e., <math>\rho(t) = \rho</math> for all <math>t \geq 0</math>, then <math>\rho</math> is bound with respect to position.
*More generally: If the state evolution of <math>\rho</math> "just moves <math>\rho</math> inside a bounded domain", then <math>\rho</math> is bound with respect to position.