Observables in Quantum Mechanics and the Importance of SelfAdjointness
Abstract
:1. Introduction
2. Motivation
2.1. Paradox 1
2.2. Paradox 2
2.3. Paradox 3
2.4. Paradox 4
2.5. Paradox 5
3. Basic Ideas and Results on SA Operators
3.1. The Momentum Operator on a Finite Interval
 Is the momentum operator $\widehat{p},\phantom{\rule{4pt}{0ex}}D\left(\widehat{p}\right)$ symmetric?
 What is its adjoint, more importantly what is $D({\widehat{p}}^{\u2020})$?
 Is the momentum operator $\widehat{p}$ selfadjoint, i.e., is $D\left(\widehat{p}\right)=D({\widehat{p}}^{\u2020})$ fulfilled?
3.2. The von Neumann’s Method: SelfAdjoint Extensions of a Symmetric Operator
 Is the symmetric operator T selfadjoint in $D\left(T\right)$?
 If it is not SA, can it be made selfadjoint?
 If it can be made SA, what is the suitable domain of selfadjointness?
 T is (essentially) SA iff $({n}_{+},{n}_{})=(0,0)$, or we say that it has a unique SA extension.
 If ${n}_{+}={n}_{}$, then the operator T is not SA but admits SA extensions.
 In case that ${n}_{+}\ne {n}_{}$, then the operator T is not SA and has no SA extensions.
3.3. Motion of a Free Particle on $\left[0,+\infty \right)$
 Is the Hamilton operator H symmetric in $D\left(H\right)$?
 What is the domain of its adjoint $D\left({H}^{\u2020}\right)$?
 Is the operator H SA in the domain $D\left(H\right)$?
 Operator H is symmetric in ${D}_{\alpha}\left(H\right)$.
 ${D}_{\alpha}\left({H}^{\u2020}\right)={D}_{\alpha}\left(H\right)$.
 Concluding that H is SA in ${D}_{\alpha}\left(H\right)$.
4. Some Physical Applications
4.1. New Bound States
4.2. Anomalies in Hamiltonian Formalism
4.3. Symmetry and Degeneracy of the Spectrum: Circle vs. Finite Interval
4.4. Pauli’s Theorem
5. Quantum Mechanics in Dimensions Higher Than 1
6. Final Remarks
 (i)
 hermitian if $(T\psi ,\phi )=(\psi ,T\phi )$$\forall \psi ,\phi \in D\left(T\right)$;
 (ii)
 symmetric if it is densely defined ($\overline{D\left(T\right)}=\mathcal{H}$) and hermitian40;
 (iii)
 selfadjoint if it is symmetric and $T={T}^{\u2020}$;
 (iv)
 essentially selfadjoint if it is symmetric and ${\left({T}^{\u2020}\right)}^{\u2020}={T}^{\u2020}$.
 (i)
 point spectrum ${\sigma}_{p}\left(T\right)$, where $(T\lambda I)$ is not injective so that ${\sigma}_{P}\left(T\right)$ is actually the set of all the eigenvalues of T;
 (ii)
 continuous spectrum ${\sigma}_{c}\left(T\right)$, where $(T\lambda I)$ is injective, $\overline{Im(T\lambda I)}=\mathcal{H}$ and ${(T\lambda I)}^{1}$ is not bounded;
 (iii)
 residual spectrum ${\sigma}_{r}\left(T\right)$, where $(T\lambda I)$ is injective, $\overline{Im(T\lambda I)}\ne \mathcal{H}$.
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Postulates of QM
 For a given physical system we assume the existence of a Hamiltonian formulation of classical mechanics. States of the system are points in an evendimensional phase space $\mathcal{P}={T}^{*}M$ and are labeled by canonical generalized coordinates ${q}_{a}$ and momenta ${p}_{a}$, $a=1,\dots ,n$, where n is the number of degrees of freedom. Hamilton equations of motion govern the time evolution of the system$${\dot{q}}_{a}=\left\{{q}_{a},H\right\},\phantom{\rule{1.em}{0ex}}{\dot{p}}_{a}=\left\{{p}_{a},H\right\},$$$$\left\{f,g\right\}=\sum _{a}\left(\frac{\partial f}{\partial {q}_{a}}\frac{\partial g}{\partial {p}_{a}}\frac{\partial f}{\partial {p}_{a}}\frac{\partial g}{\partial {q}_{a}}\right),$$
 A state in QM is defined as a vector $\psi $ in a suitable Hilbert space $\mathcal{H}$. A scalar product of such two vectors ${\psi}_{1}$ and ${\psi}_{2}$ is denoted by $({\psi}_{1},{\psi}_{2})$. It is assumed that any state $\psi \in \mathcal{H}$ can be realized physically (or at least to a first approximation) and that the superposition principle holds, that is if states ${\psi}_{1}$ and ${\psi}_{2}$ are realizable, then the state $\psi ={a}_{1}{\psi}_{1}+{a}_{2}{\psi}_{2}$ with any ${a}_{1},{a}_{2}\in \mathbb{C}$ is also realizable.
 To an each classical observable $f=f(x,p)$ we uniquely assigned a linear SA operator $\widehat{f}$ acting in Hilbert space $\mathcal{H}$. The operator $\widehat{f}$ is called a quantum observable. It is assumed that any operator $\widehat{f}$ is well defined46 on any state $\psi $, i.e., $\widehat{f}\psi \in \mathcal{H}$, $\forall \psi \in \mathcal{H}$. If so, the operator $\widehat{f}$ is uniquely determined by its matrix elements $({\psi}_{1},\widehat{f}{\psi}_{2})$, $\forall {\psi}_{1}{\psi}_{2}\in \mathcal{H}$, that is by its matrix ${f}_{mn}=({e}_{m},\widehat{f}{e}_{n})$ with respect to an orthonormal basis47${\left\{{e}_{n}\right\}}_{1}^{\infty}$ in $\mathcal{H}$. To any such operator $\widehat{f}$ we can assign its adjoint ${\widehat{f}}^{\u2020}$$$\left({\psi}_{1},{\widehat{f}}^{\u2020}{\psi}_{2}\right)=\left({\widehat{f}}^{\u2020}{\psi}_{1},{\psi}_{2}\right),\phantom{\rule{1.em}{0ex}}\forall {\psi}_{1},{\psi}_{2}\in \mathcal{H},$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\left({\widehat{f}}^{\u2020}\right)}^{\u2020}=\widehat{f},\phantom{\rule{1.em}{0ex}}{\left(a\widehat{f}\right)}^{\u2020}={a}^{*}{\widehat{f}}^{\u2020},\phantom{\rule{1.em}{0ex}}\forall a\in \mathbb{C}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\left(\widehat{f}+\widehat{g}\right)}^{\u2020}={\widehat{f}}^{\u2020}+{\widehat{g}}^{\u2020},\phantom{\rule{1.em}{0ex}}{\left(\widehat{f}\widehat{g}\right)}^{\u2020}={\widehat{g}}^{\u2020}{\widehat{f}}^{\u2020}.\hfill \end{array}$$The operator $\widehat{f}$ is SA48 if $\widehat{f}={\widehat{f}}^{\u2020}$, or$$\left({\psi}_{1},\widehat{f}{\psi}_{2}\right)=\left(\widehat{f}{\psi}_{1},{\psi}_{2}\right),\phantom{\rule{1.em}{0ex}}\forall {\psi}_{1},{\psi}_{2}\in \mathcal{H}$$The expectation value ${\u2329\widehat{f}\u232a}_{\psi}$ of any quantum observable $\widehat{f}$ in a state $\psi $ and the corresponding dispersion $\Delta f$ are defined as$${\u2329\widehat{f}\u232a}_{\psi}=\frac{\left(\psi ,\widehat{f}\psi \right)}{\left(\psi ,\psi \right)},\phantom{\rule{1.em}{0ex}}\Delta \widehat{f}=\sqrt{{\u2329{\widehat{f}}^{2}\u232a}_{\psi}{\u2329\widehat{f}\u232a}_{\psi}^{2}}.$$The observables are assumed to be SA operators because the corresponding eigenvalues are real and the eigenvectors form an orthonormal basis in $\mathcal{H}$. The spectrum (set of all the eigenvalues) represent all the possible measurements, while the complete orthonormalized set of the eigenstates of the observable provides a probabilistic interpretation of its measurements.
 The correspondence principle implies a connection between the Poisson bracket of classical observables and the commutator of the quantum observables. Namely,$$\left\{{f}_{1},{f}_{2}\right\}\to \frac{1}{i\hslash}\left[{\widehat{f}}_{1},{\widehat{f}}_{2}\right]+\widehat{O}(\hslash )$$The position operators ${\widehat{q}}_{a}$ and momentum operators ${\widehat{p}}_{a}$ are postulated to be SA and satisfy the canonical commutation relations$$\left[{\widehat{q}}_{a},{\widehat{q}}_{b}\right]=\left[{\widehat{p}}_{a},{\widehat{p}}_{b}\right]=0,\phantom{\rule{1.em}{0ex}}\left[{\widehat{q}}_{a},{p}_{b}\right]=i\hslash \left\{{q}_{a},{p}_{b}\right\}=i\hslash {\delta}_{ab}$$The correspondence principle imposes the form of the quantum observable as $\widehat{f}=f(\widehat{x},\widehat{p})+\widehat{O}(\hslash )$, where $\widehat{O}(\hslash )$ is chosen in such a way that it insures the selfadjointness. Since ${\widehat{q}}_{a}$ and ${\widehat{p}}_{a}$ don’t commute, due to the socalled ordering problem49, there is no unique construction of $f(\widehat{x},\widehat{p})$ via $f(x,p)$. Any two commuting observables ${\widehat{f}}_{1}$ and ${\widehat{f}}_{2}$ can be simultaneously measured because the commutativity implies the existence of common eigenvectors and joint spectrum. A minimum set of N commuting observables ${\widehat{f}}_{k}$, $k=1,\dots ,N,$$[{\widehat{f}}_{k},{\widehat{f}}_{l}]=0$, $\forall k,l$ whose joint spectrum is nondegenerate and whose common eigenvectors provide a unique specification of any vector in terms of the corresponding expansion with respect to these eigenvectors define a complete set of observables. Complete sets of observables completely specify the quantum description of a system under consideration.
 The time evolution of any quantum state $\psi \left(t\right)$ is governed by the Schrödinger equation,$$i\hslash \frac{\partial \psi}{\partial t}=\widehat{H}\psi ,$$
Appendix B. Bounded Linear Operators
Appendix C. Hilbert Spaces
 Hilbert space $\mathcal{H}$ is a vector space over the complex numbers. Usually, the elements (also called vectors or points) of $\mathcal{H}$ we denote by Greek letters, while the complex numbers with Latin letters. The vector space structure is encoded in $(\mathcal{H},\xb7,+)$ where $\xb7:\mathbb{C}\times \mathcal{H}\u27f6\mathcal{H}$ is the scalar multiplication and $+:\mathcal{H}\times \mathcal{H}\u27f6\mathcal{H}$ is the vector addition satisfying$$a\psi +b\eta \in \mathcal{H}\phantom{\rule{4pt}{0ex}}\forall a,b\in \mathbb{C}\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}\psi ,\eta \in \mathcal{H}$$
 The inner product $(\xb7,\xb7):\mathcal{H}\times \mathcal{H}\u27f6\mathbb{C}$ is a positive definite sesquilinear form on $\mathcal{H}$ satisfying$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& (\xi ,\eta )=\overline{(\eta ,\xi )};\phantom{\rule{1.em}{0ex}}(\xi ,\xi )\ge 0,\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}(\xi ,\xi )=0\iff \xi =0;\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& (\xi ,a\zeta +b\eta )=a(\xi ,\zeta )+b(\xi ,\eta )\Rightarrow (a\xi +b\zeta ,\eta )=\overline{a}(\xi ,\eta )+\overline{b}(\zeta ,\eta ).\hfill \end{array}$$$$\left\right\xi +\eta \left\right\le \left\right\xi \left\right+\left\right\eta \left\right$$$$\xi =\underset{n\to \infty}{lim}{\xi}_{n}$$$$\underset{n\to \infty}{lim}\left\right{\xi}_{m}{\xi}_{n}\left\right=0,$$$$\underset{n\to \infty}{lim}\left\right{\xi}_{m}{\xi}_{n}\left\right=0\Rightarrow \underset{n\to \infty}{lim}({\xi}_{n},\eta )=(\xi ,\eta ),\phantom{\rule{1.em}{0ex}}\forall \eta \in \mathcal{H}.$$
 $\mathcal{H}$ is a complete normed space. This means that every Cauchy sequence ${\left\{{\xi}_{n}\right\}}_{1}^{\infty}$ in $\mathcal{H}$ is convergent, i.e., it has a limit in $\mathcal{H}$:$$\underset{n\to \infty}{lim}\left\right{\xi}_{m}{\xi}_{n}\left\right=0\phantom{\rule{1.em}{0ex}}\phantom{\rule{4pt}{0ex}}\u27f9\phantom{\rule{1.em}{0ex}}\phantom{\rule{4pt}{0ex}}\exists \xi \in \mathcal{H}:\underset{n\to \infty}{lim}{\xi}_{n}=\xi .$$Notice that any convergent sequence ${\left\{{\xi}_{n}\right\}}_{1}^{\infty}$ is a Cauchy sequence and in a Hilbert space, the converse also holds54. A preHilbert space is a vector space with an inner product satisfying (A13). Physicists usually deal with just a preHilbert space, since there is no difference between preHilbert space and a Hilbert space of finite dimension. Fortunately, any preHilbert space can be made a complete Hilbert space by adding the “limits” of all Cauchy sequences. Note that the requirement of completeness is crucial, and not only technical, for applications of Hilbert spaces to QM.
 A Hilbert space $\mathcal{H}$ is called separable if it contains a countable dense set. Separable Hilbert spaces are sufficient for treating conventional QM. All infinitedimensional separable Hilbert spaces are isomorphic to the Hilbert space of complex square summable sequences ${l}^{2}\left(\mathbb{N}\right)$ defined by$${l}^{2}\left(\mathbb{N}\right):=\left(\left\{a:\mathbb{N}\u27f6\mathbb{C}\sum _{n=0}^{\infty}{\left{a}_{n}\right}^{2}<\infty \right\},{(a,b)}_{{l}^{2}}:=\sum _{n=0}^{\infty}{a}_{n}^{*}{b}_{n}\right)$$
 The vector space of all squareintegrable functions on a real interval $(a,b)$ is usually denoted by ${L}^{2}(a,b)$ and can be made into a Hilbert space$${L}^{2}(a,b)=\left\{\psi \left(x\right):{\int}_{a}^{b}\mathrm{d}x\phantom{\rule{4pt}{0ex}}{\left\psi \left(x\right)\right}^{2}<\infty \right\}.$$The scalar product in ${L}^{2}(a,b)$ is defined by$$({\psi}_{1},{\psi}_{2})={\int}_{a}^{b}\mathrm{d}x\phantom{\rule{4pt}{0ex}}{\psi}_{1}^{*}\left(x\right){\psi}_{2}\left(x\right).$$It is important to note that here the integrals are Lebesgue integrals, and strictly speaking, the elements of ${L}^{2}(a,b)$ are equivalence classes of functions that are equal almost everywhere55.
Appendix D. Adjoint of an Operator and Its Properties
 1.
 $D\left({A}^{\u2020}\right):=\left\{\psi \in \mathcal{H}\exists \eta \in \mathcal{H}:\forall \alpha \in D(A):(\psi ,A\alpha )=(\eta ,\alpha )\right\}$
 2.
 ${A}^{\u2020}\psi =\eta $
 1.
 $D\left(A\right)\subseteq D\left({A}^{\u2020}\right)$
 2.
 ${A}^{\u2020}\psi =A\psi ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \psi \in D\left(A\right)$
Appendix E. SelfAdjoint Operators and Their Properties
 For $A={A}^{\u2020}$ and $a\in \mathbb{R}\phantom{\rule{4pt}{0ex}}\Rightarrow \phantom{\rule{4pt}{0ex}}aA={\left(aA\right)}^{\u2020},{D}_{aA}={D}_{A}$.
 For $A={A}^{\u2020}$ and $B={B}^{\u2020}$, with $\overline{{D}_{A}\cap {D}_{B}}=\mathcal{H}\phantom{\rule{4pt}{0ex}}\Rightarrow \phantom{\rule{4pt}{0ex}}{(A+B)}^{\u2020}\supseteq A+B$.The sum of two SA operators $A+B$ is just a symmetric operator, in general, but if one of the operators, for example B, is bounded, then the sum is also an SA operator, i.e., $A+B={(A+B)}^{\u2020}$ with ${D}_{A+B}={D}_{A}$.
 For $A={A}^{\u2020}$ and $B={B}^{\u2020}$, with $\overline{{D}_{AB}}=\mathcal{H}\phantom{\rule{4pt}{0ex}}\Rightarrow {\left(AB\right)}^{\u2020}\supseteq BA$.The product $AB$ of two SA operators is not even symmetric. The product $AB$ is symmetric, i.e., ${\left(AB\right)}^{\u2020}\supseteq AB$, if A is bounded, and $BA\subseteq AB$, that is if A and B commute. The product $AB$ is SA, i.e., ${\left(AB\right)}^{\u2020}=AB$, if both operators are bounded, and they commute $[A,B]=0$.
Appendix F. The VonNeumann’s Theorems
Appendix F.1. The first von Neumann theorem
Appendix F.2. The second von Neumann theorem
Appendix F.3. The main theorem
Appendix G. Some Basics of Rigged Hilbert Space
Appendix H. Scale Symmetry in Classical Physics
Notes
1  See Appendix A for the usual postulates of QM. 
2  See Appendix B for the definition of bounded operators, Appendix C for the details on Hilbert space, Appendix D for the definition of the adjoint operator and Appendix E for the definition and properties of a SA operator. 
3  Here “state” is used in the physical jargon and actually means an element of the Hilbert space $\mathcal{H}$ (or an equivalence class of elements up to a phase). The state of a system is a positive linear map $\rho :\mathcal{H}\u27f6\mathcal{H}$ for which $Tr\left(\rho \right)=1$. States can be pure or mixed. A state ${\rho}_{\psi}$ is called pure if it maps $\psi \u27fc\frac{(\psi ,\phantom{\rule{4pt}{0ex}}\xb7\phantom{\rule{4pt}{0ex}})}{(\psi ,\psi )},\phantom{\rule{4pt}{0ex}}\forall \psi \in \mathcal{H}$. Thus, we can associate to each pure state ${\rho}_{\psi}$ an element in $\mathcal{H}$. However, this correspondence is not onetoone, and one should have this fact in mind. The physicist refer to the operators $\rho $ as density matrices and often in Dirac notation just write $\rho =\left\psi \right.\u232a\left.\u2329\psi \right$. 
4  Claims like “rigorous definition” or “rigorous proof” are also very often used in the physicist jargon. The thing is that objects are either well defined or not, or the claim is proven or not. There is no degree of “how much” something is or can be proven, but when physicist says that something is rigorously done, it just means that “all the necessary” assumptions that deal with questions of convergence, completeness, domains etc. are assumed to be valid (or better say omitted). Or simply, the nonrigouros proofs or definitions seems to work well in several cases and one just takes it for granted that the results apply for other situations also. 
5  As they are usually presented in conventional undergraduate or graduate physics courses. 
6  See Appendix A for postulates of QM. 
7  Here one should be aware of the full details of the construction of the Hilbert space ${L}^{2}$. Namely, one needs first a measurable space $(M,\Sigma ,\mu )$ (here M is a set, $\Sigma $ is the Borel$\sigma $algebra and $\mu $ is a measure) together with an equivalence relation $f\sim g\iff {\u2225f\u2225}_{2}={\u2225g\u2225}_{2}$, so that
$${L}^{2}\left(M\right):={\mathcal{L}}^{2}/\sim =\left\{\left[f\right]f\in {\mathcal{L}}^{2}\right\}$$
$${\mathcal{L}}^{2}(M,\Sigma ,\mu ):=\left\{f:M\to \mathbb{C}\phantom{\rule{4pt}{0ex}}\Re \left(f\right)\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}\Im \left(f\right)\phantom{\rule{4pt}{0ex}}\mathrm{are}\phantom{\rule{4.pt}{0ex}}\mathrm{measurable}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{\int}_{M}{\leftf\right}^{2}d\mu <\infty \right\}$$

8  In finite dimensional space the spectrum of an operator is the set of its eigenvalues. In general Hilbert space the spectrum of a SA operator is much more. It can have the point spectrum part (related to the set of eigenvalues) and the continuous spectrum part (usually related to generalized eigenfunctions that physicist sometimes call scattering states) [67,68,69,70]. 
9  This is actually just the symmetric condition, which in the finaldimensional case is hermiticity. 
10  ${\left\{{a}_{n}\right\}}_{1}^{\infty}$ denotes a set of members of the sequence ${a}_{n}$ starting from $n=1$ to $n=\infty $. 
11  This claim stands even rigorously, because if for a symmetric operator H its domain $D\left(H\right)\subset \mathcal{H}$ contains the orthonormal basis of $\mathcal{H}$, then H is SA (see [20]). 
12  Actually, this just a dense subspace and its completion is the Hilbert space ${L}^{2}[0,2\pi ]$. 
13  Physicist often thinks of the spectrum as the set of eigenvalues (point spectrum) which come from solving differential equations with certain boundary conditions and imposing square integrability of the solutions. However, to grasp the continuous part of the spectrum (or energy of the scattering states), physicist solves the same differential equation but “drops” the squareintegrability condition and looks for the solution outside the Hilbert space. This at first glance strange procedure is imposed by the Dirac notation and its justification can be found in the theory of rigged Hilbert spaces (or Gelfand triples). For more details see Appendix G and [72,76]. 
14  We also assume that $D\left(T\right)$ is dense in $\mathcal{H}$, that is $\overline{D\left(T\right)}=\mathcal{H}$, which means that the topological closure of $D\left(T\right)$ is the same as $\mathcal{H}$. $D\left(T\right)$ may not be a complete space, so dense means that the space together with limit vectors of all of its Cauchy sequences is exactly $\mathcal{H}$. 
15  For simplicity we take the unit mass and $\hslash =1$ system of units. 
16  From now on we understand that all derivatives are in the weak sense. 
17  A function $f:[a,b]\u27f6\mathbb{C}$ is absolutely continuous if it has a derivative ${f}^{\prime}$ almost everywhere, the derivative is Lebesgue integrable and
$$f\left(x\right)=f\left(a\right)+{\int}_{a}^{x}{f}^{\prime}\left(t\right)dt,\phantom{\rule{1.em}{0ex}}\forall x\in [a,b].$$

18  For more details see Appendix C. 
19  This reduces to the hermicity condition in finitedimensional Hilbert spaces. 
20  See Appendix D. 
21  
22  Actually we should consider ${T}^{\u2020}{\psi}_{\pm}=\pm i\kappa {\psi}_{\pm}$, where $\kappa \in \mathbb{R}$ due to dimensionality reasons (namely $\left[T\right]=\left[\kappa \right]$), but this we omit here and set $\kappa =1$ as our “natural” choice of units. However the dimensionality of $\kappa $ is very important and its physical meaning is that this is the scale of the anomaly, i.e., the breaking of the classical symmetry and it somewhat governs the physically allowed types of interaction on the boundary. 
23  
24  Here the domain is given a bit wage, since one might ask how a finite dimensional matrix U acts on ${\psi}_{}$. The details are given in Appendix F, and here our wage definition is good enough since in all of our examples the matrix U will be a pure phase. 
25  
26  For simplicity we set the mass to be $m=\frac{1}{2}$. 
27  And the parametrization $\alpha =\pm \frac{cos(\gamma /2+\pi /4)}{cos(\gamma /2)}$ is used. 
28  To see this more closely, act on (61) with $(\psi ,\xb7)$ to obtain
$$E=\frac{\alpha {\left\psi \left(0\right)\right}^{2}+{\u2225{\psi}^{\prime}\u2225}^{2}}{{\u2225\psi \u2225}^{2}}$$

29  There are no squareintegrable solutions of this equation for positive energy. Here one actually looks for a solution of a generalized eigenequation within the formalism of rigged Hilbert space (see Appendix G for further comments.) 
30  This is usually the case, for example the canonical commutation relation $[\widehat{x},\widehat{p}]=i\hslash I$, where the right hand side is algebraically obtained and, as an operator, is well defined on all of ${L}^{2}\left(\mathbb{R}\right)$. Algebraically obtained would here mean that we calculated $[H,A]$ evaluated on $\mathcal{T}$. 
31  Actually it is $SO(2,1)$ invariant and the symmetry is generated by the Hamiltonian H, dilatation D and conformal generator K. 
32  For more details on scale symmetry in classical physics see Appendix H. 
33  Which is equivalent to a situation where we consider the whole real line, but divide it in segments of length L and demand periodicity, i.e., identify all the points with respect to $\mathbb{R}$ mod L. 
34  Where we again take $\hslash =1$ and $m=1/2$ for simplicity. 
35  Here the maximal invariant subspace of the algebra generated by $\widehat{H}$ and $\widehat{p}$ is $\mathcal{T}={\bigcap}_{n,m=0}^{\infty}D({\widehat{H}}^{n}{\widehat{p}}^{m})=D(\widehat{H})=D\left(\widehat{p}\right)$. 
36  Similarly as for the momentum operator on a half plane. 
37  see Appendix G. 
38  For the inner product for $d=2$ we have $(\psi ,\phi )=\int dxdy\phantom{\rule{4pt}{0ex}}\overline{\psi}(x,y)\phi (x,y)=\int rdrd\varphi \phantom{\rule{4pt}{0ex}}\overline{\psi}(r,\varphi )\phi (r,\varphi )$. 
39  Or over some manifold M in general. 
40  This is equivalent to saying that the adjoint ${T}^{\u2020}$ is the extension of T, i.e., $T\subset {T}^{\u2020}$, meaning $D\left(T\right)\subset D\left({T}^{\u2020}\right)$ and ${T}^{\u2020}\psi =T\psi $$\forall \psi \in D\left(T\right)$. 
41  Essentially selfadjoint operators have a unique SA exstension. 
42  Notice that we assume that if $T:D\left(T\right)\u27f6\mathcal{H}$ is injective then ${T}^{1}$ is the inverse restricted to the image of T, i.e., ${T}^{1}:Im\left(T\right)\u27f6D\left(T\right)$. 
43  
44  There are mainly three well established formulation of QM: 1. Heisenberg’s matrix mechanics (historically first, but of less practicable use for a working physicist); Scrödinger’s wave mechanics (standard and most used formalism); Feynman’s path integral (wildly used in relativistic quantum field theories). Actually, in general, one can formulated consistent nonrelativistic QM in (at least) nine independent ways [118] 
45  In mathematics, quantization is a quantum deformation of classical structures; the deformation parameter is the Planck constant ℏ. 
46  To be precise this only holds for finitedimensional Hilbert spaces and for bounded operators in infinitedimensional Hilbert spaces. In physics we are usually dealing with unbounded operators, for which every operator has its domain. 
47  ${\left\{{\psi}_{n}\right\}}_{1}^{\infty}$ is an infinite sequence of vectors. 
48  More precisely see Appendix C. 
49  A substantial contribution to the resolution of this problem is due to Berezin [123] 
50  A Banach space is a normed and complete space (all Cauchy sequences are convergent in it). 
51  with respect to the topologies induced by the respective norms on $\mathcal{V}$ and $\mathcal{W}$ 
52  For all $\psi ,\eta ,\xi \in \mathcal{H}$ and $a,b\in \mathbb{C}$ we have
$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& (\mathrm{Commutativity}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{addition})\phantom{\rule{4pt}{0ex}}\psi +\eta =\eta +\psi \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& (\mathrm{Associativity}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{addition})\phantom{\rule{4pt}{0ex}}(\psi +\eta )+\xi =\psi +(\eta +\xi )\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& (\mathrm{Existence}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{a}\phantom{\rule{4.pt}{0ex}}\mathrm{zero}\phantom{\rule{4.pt}{0ex}}\mathrm{vector})\phantom{\rule{4pt}{0ex}}\mathrm{There}\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{a}\phantom{\rule{4.pt}{0ex}}\mathrm{vector}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}o\in \mathcal{H}\phantom{\rule{4pt}{0ex}}\mathrm{with}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}o+\psi =\psi +o=\psi \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& (\mathrm{Existence}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{additive}\phantom{\rule{4.pt}{0ex}}\mathrm{inverses})\phantom{\rule{4pt}{0ex}}\mathrm{For}\phantom{\rule{4.pt}{0ex}}\mathrm{each}\phantom{\rule{4pt}{0ex}}\psi ,\phantom{\rule{4pt}{0ex}}\mathrm{there}\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4pt}{0ex}}\psi \in \mathcal{H}\phantom{\rule{4pt}{0ex}}\mathrm{such}\phantom{\rule{4.pt}{0ex}}\mathrm{that}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\psi +(\psi )=(\psi )+\psi =o\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& (\mathrm{Distributivity}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{scalar}\phantom{\rule{4.pt}{0ex}}\mathrm{multiplication}\phantom{\rule{4.pt}{0ex}}\mathrm{over}\phantom{\rule{4.pt}{0ex}}\mathrm{vector}\phantom{\rule{4.pt}{0ex}}\mathrm{addition})\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}a(\psi +\eta )=a\psi +a\eta \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& (\mathrm{Distributivity}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{scalar}\phantom{\rule{4.pt}{0ex}}\mathrm{addition}\phantom{\rule{4.pt}{0ex}}\mathrm{over}\phantom{\rule{4.pt}{0ex}}\mathrm{scalar}\phantom{\rule{4.pt}{0ex}}\mathrm{multiplication})\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}(a+b)\psi =a\psi +b\psi \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& (\mathrm{Associativity}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{scalar}\phantom{\rule{4.pt}{0ex}}\mathrm{multiplication})\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(ab\right)\psi =a\left(b\psi \right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& (\mathrm{Scalar}\phantom{\rule{4.pt}{0ex}}\mathrm{multiplication}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}1\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{identity})\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}1\psi =\psi \hfill \end{array}$$

53  A Hilbert space is a particular case of a normed and metric space in which a norm and a metric (distance) satisfying standard requirements are generated by a scalar product; see [117] 
54  In short, a Hilbert space is complete with respect to a metric generated by a scalar product. 
55  When speaking about some function belonging to ${L}^{2}(a,b)$ and possessing some additional specific properties like absolute continuity, we actually mean the representative of the corresponding equivalence class. 
56  Densely defined means that the domain ${D}_{f}$ is dense in $\mathcal{H}$, i.e., $\overline{{D}_{f}}=\mathcal{H}$ 
57  Note that unitary operators are bounded and defined everywhere, and the notion of commutativity for such operators is unambiguous. 
58  Where ${\mathbb{C}}^{\prime}$ is a set of complex numbers with nonzero imaginary part, ${\mathbb{C}}^{\prime}=\left\{z=x+iy,y\ne 0\right\}={\mathbb{C}}_{+}\cup {\mathbb{C}}_{}$. If we define $A\left(z\right)=AzI$, then ${\Sigma}_{z}=\mathrm{ker}{A}^{\u2020}\left({z}^{*}\right)=\left\{{\xi}_{{z}^{*}}\in {D}_{{A}^{\u2020}}:{A}^{\u2020}{\xi}_{{z}^{*}}={z}^{*}{\xi}_{{z}^{*}}\right\}$ and ${\Sigma}_{{z}^{*}}=\mathrm{ker}{A}^{\u2020}\left(z\right)=\left\{{\xi}_{z}\in {D}_{{A}^{\u2020}}:{A}^{\u2020}{\xi}_{z}=z{\xi}_{z}\right\}$, and $\overline{A}$ is the closure of A 
59  Deficiency index is defined as $m\left(z\right)=\mathrm{dim}\phantom{\rule{4pt}{0ex}}\mathrm{ker}{A}^{\u2020}\left({z}^{*}\right)=\left\{{m}_{+},\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}z\in {\mathbb{C}}_{+}\phantom{\rule{4pt}{0ex}}\mathrm{or}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{m}_{}\mathrm{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}z\in {\mathbb{C}}_{}\right\}$ 
60  where we used $H(p,q)=\left(\frac{\partial L}{\partial \dot{q}}\dot{q}L\right)=\frac{{p}^{2}}{2m}+V\left(q\right)$, $L=\frac{m{\dot{q}}^{2}}{2}V\left(q\right)$ and $p=\frac{\partial L}{\partial \dot{q}}=m\dot{q}$. 
61  This equation can be generalized to higher dimensions by $q\u27f6{q}^{i}$ and $d=q\frac{\partial}{\partial q}\u27f6{q}^{i}{\nabla}_{i}$. 
62  Here we have assumed that the potential energy $V\left(q\right)$ is of class ${\mathcal{C}}^{1}\left(\mathbb{R}\right)$, but notice that there are also generalized functions or distributions that satisfy $V\left({\lambda}^{\frac{1}{2}}q\right)={\lambda}^{1}V\left(q\right)$. 
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Jurić, T. Observables in Quantum Mechanics and the Importance of SelfAdjointness. Universe 2022, 8, 129. https://doi.org/10.3390/universe8020129
Jurić T. Observables in Quantum Mechanics and the Importance of SelfAdjointness. Universe. 2022; 8(2):129. https://doi.org/10.3390/universe8020129
Chicago/Turabian StyleJurić, Tajron. 2022. "Observables in Quantum Mechanics and the Importance of SelfAdjointness" Universe 8, no. 2: 129. https://doi.org/10.3390/universe8020129