The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions

(a) Framework data. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4, the kinematics and massive-particle model postulate P5, the identical-particle and spin postulate P6, and the preparation and apparatus discipline P7 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Fix a mass parameter $m>0$ and a spin label $s=0$, so that the framework's one-particle model The massive spin-s particle model of the framework: state space, operators, units and frame readings reads its state space as $H_0=L^2({\mathbb{R}}^3;\mathbb{C})$ with the pairing linear in the first argument, its componentwise position family $Q_1,Q_2,Q_3$, its componentwise momentum family $P_1,P_2,P_3$ and the action constant $\mathrm{\hslash }>0$; the spin-zero internal factor carries the trivial representation of P6. Units are those of Dimensions, SI unit conventions and quantity types of the framework: position is read in metres, momentum in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, energy and potential energy in joules, and $\mathrm{\hslash }$ in $\mathrm{J}\mathrm{s}$. Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) The declared effective one-dimensional model. As an additional model assumption of the pair, adopt the line model. Its declared state space is $H_{\mathrm{l}\mathrm{i}\mathrm{n}}:=L^2(\mathbb{R};\mathbb{C})=\{f:\mathbb{R}\to \mathbb{C}\text{ measurable}:{\int }_{\mathbb{R}}\mathrm{\mid }f(x){\mathrm{\mid }}^{2}dx<\mathrm{\infty }\}/\text{\hspace{0.17em}⁣}\sim ,$ the complex Hilbert space of almost-everywhere classes with the pairing $⟨f,g⟩={\int }_{\mathbb{R}}f\bar{g}dx$ linear in the first argument (The space $L^p(\mu )$ as the quotient by null functions). Its declared position assignment is the multiplication assignment $Q$, defined on the classes with $xf\in H_{\mathrm{l}\mathrm{i}\mathrm{n}}$, with the declared outcome law ${\mu }_{\psi }^Q(B):={\int }_B\mathrm{\mid }\psi (x){\mathrm{\mid }}^{2}dx(B\subseteq \mathbb{R}\text{ Borel})$ for a unit class $\psi$. Indeed, the positive sigma-finite Lebesgue measure space $\mathbb{R}$ and the finite measurable multiplier $x\mapsto x$ satisfy Spectral measure of a real L2 multiplication operator clauses (a)-(c) under the choice assumptions of (a): $Q$ is self-adjoint on the displayed domain, with spectral PVM $E_Q(B)=M_{{\mathbf{1}}_B}$ and $\mathrm{\parallel }{E}_Q(B)\psi {\mathrm{\parallel }}^2={\int }_B\mathrm{\mid }\psi {\mathrm{\mid }}^2$. The probability reading of this mathematical law is adopted under P2. Its declared momentum assignment is $P:=-i\mathrm{\hslash }D$ on the classes whose weak derivative $D\psi$ lies in $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$ (Weak derivative of a locally integrable function). The pair asserts no derivation of $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$, $Q$ or $P$ from the three-dimensional model $H_0$ of P5: the reduction to one spatial dimension, for instance by an idealized strong transverse confinement or by a declared one-dimensional sector, is an assumption of the model. An assignment of a concrete material system to the line model is an identification in the sense of clause (iii) of P7 and carries the corresponding uncertainty.

(c) The assigned energy operator. Let $V:\mathbb{R}\to \mathbb{R}$ be real, bounded and measurable; write $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}:={\sup }_{x\in \mathbb{R}}\mathrm{\mid }V(x)\mathrm{\mid }$. The assigned energy operator of the line model with potential $V$ is $H_V:=H_0^{\mathrm{l}\mathrm{i}\mathrm{n}}+M_V,H_0^{\mathrm{l}\mathrm{i}\mathrm{n}}:={\mathcal{F}}_2^{\ast }{M}_{c\mathrm{\mid }\xi {\mathrm{\mid }}^2}{\mathcal{F}}_2,c:=\frac{2{\pi }^2{\mathrm{\hslash }}^2}{m},$ with $M_V$ the multiplication operator of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (a), on the common domain $D(H_V):=D(H_0^{\mathrm{l}\mathrm{i}\mathrm{n}})=\{f\in H_{\mathrm{l}\mathrm{i}\mathrm{n}}:\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f\in H_{\mathrm{l}\mathrm{i}\mathrm{n}}\}$, where ${\mathcal{F}}_2$ is the unitary Plancherel transform of that theorem (Real multiplication operators and free plus bounded-potential Schrödinger operators clauses (a)-(c)). The declared energy unit is the joule. By The one-dimensional free Schr"odinger operator as the weak second-derivative operator the domain $D(H_V)$ is the Sobolev space $W^{2,2}(\mathbb{R};\mathbb{C})$ and $H_{V}f=-\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f+Vf(f\in D(H_V)),$ $D^2$ the second weak derivative, so the model's time-independent equation inside any open interval on which $V$ is (say) constant is the ordinary differential equation $-\frac{{\mathrm{\hslash }}^2}{2m}{\psi }^{\mathrm{\prime }\mathrm{\prime }}+V\psi =E\psi$. The model adopts $(H_V,\mathrm{J})$ as its assigned energy observable in the sense of Assigned sharp observables and the outcome law of the framework clause (a); its self-adjointness is established by the line-model energy theorem of this page and is not presupposed here. A declared potential is compactly supported when $V$ vanishes almost everywhere outside some bounded interval.

(d) Bound states and stationary vocabulary. A stationary solution of the model with potential $V$ and energy $E\in \mathbb{R}$ is a nonzero class $\psi \in D(H_V)$ with $H_V\psi =E\psi$; the pair $(\psi ,E)$ is an eigenpair of the model. A bound state of the model is a stationary solution with energy $E<0$ whose representative is normalized, $\mathrm{\parallel }\psi \mathrm{\parallel }=1$; its eigenvalue $E$ is its signed binding energy, read in joules, and its positive binding-energy magnitude is $-E$. The value $0$ is the reference energy in this negative-eigenvalue definition of a bound state. For a general bounded measurable $V$, this convention makes no assertion about the location of any spectral threshold. The model reads the energy outcome law of the pure preparation of a unit eigenstate $\psi$ with eigenvalue $E$ through clause (c) of Assigned sharp observables and the outcome law of the framework and Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a)-(b): for self-adjoint $H_V$ on the nonzero Hilbert space $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$, its spectral projection at $E$ fixes the unit eigenvector, so the outcome in $\{E\}$ has probability one and the law is the point mass ${\delta }_E$. The position density of the preparation is $\mathrm{\mid }\psi {\mathrm{\mid }}^2$ by the one-dimensional multiplication-PVM construction in clause (b) above. These readings are declarations of the model under P2 and construct no sample, detector or measured value.

(e) Declared generalized stationary solutions. Let $-\mathrm{\infty }=x_0<x_1<\cdots <x_n=\mathrm{\infty }$ be a finite partition of $\mathbb{R}$, let $E\in \mathbb{R}$, and let $\psi :\mathbb{R}\to \mathbb{C}$ be bounded and classically $C^1$ on $\mathbb{R}$ with a continuous derivative, and twice continuously differentiable on each open interval $(x_j,x_{j+1})$ restricted to which $V$ is continuous, satisfying $-\frac{{\mathrm{\hslash }}^2}{2m}{\psi }^{\mathrm{\prime }\mathrm{\prime }}(x)+V(x)\psi (x)=E\psi (x)$ there. In addition, require the stationary equation on the whole line in the weak sense: $-\frac{{\mathrm{\hslash }}^2}{2m}{\int }_{\mathbb{R}}\psi (x){\phi }^{\mathrm{\prime }\mathrm{\prime }}(x)dx+{\int }_{\mathbb{R}}V(x)\psi (x)\phi (x)dx=E{\int }_{\mathbb{R}}\psi (x)\phi (x)dx$ for every smooth compactly supported real test function $\phi \in C_c^{\mathrm{\infty }}(\mathbb{R};\mathbb{R})$; the equality is complex, equivalently componentwise. All integrals are finite because $V$ and $\psi$ are bounded and the tests have compact support. This is exactly the assertion $D^2\psi =(2m\mathrm{/}{\mathrm{\hslash }}^2)(V-E)\psi$ with the weak derivative convention of Weak derivative of a locally integrable function; it is required even when $V$ is continuous on no open interval. For a piecewise constant potential and a $C^1$ piecewise $C^2$ solution of the displayed classical equations, integration by parts twice on the pieces gives this identity: the internal boundary terms cancel by continuity of $\psi$ and ${\psi }^{\mathrm{\prime }}$, and the external terms vanish by compact support of the test. Such a $\psi$ is a declared generalized stationary solution of the model at energy $E$. It need not be an element of $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$ (for example, a function $x\mapsto A{e}^{ikx}$ with $A\in \mathbb{C}\setminus \{0\}$ and $k\in \mathbb{R}$ has constant density $\mathrm{\mid }A{\mathrm{\mid }}^2>0$, which is not integrable on $\mathbb{R}$). When it is not in $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$, it does not represent a preparation of the model and clause (d) does not apply to it; its scattering reading uses only the declared flux quantities computed from it, and every such use states the idealization. Square-integrable solutions are not excluded: if the nonzero class of $\psi$ belongs to $D(H_V)$, the weak equation above is the eigenvalue equation of clause (d), and its unit normalization has the preparation and outcome-law readings of that clause. Merely declaring a generalized stationary solution imposes no such normalization. For dimensional bookkeeping, assign $\psi$ and its amplitudes the one-dimensional position-wavefunction dimension ${\mathrm{\ell }}^{-1\mathrm{/}2}$ of Dimensions, SI unit conventions and quantity types of the framework clause (d)1 (the SI reference unit is ${\mathrm{m}}^{-1\mathrm{/}2}$); this convention does not normalize $\psi$ or make it an element of $H_{\mathrm{l}\mathrm{i}\mathrm{n}}$. The pair's scattering reading assigns to $\psi$ the probability current $j(x)=\frac{\mathrm{\hslash }}{m}\mathrm{Im}(\overline{\psi (x)}{\psi }^{\mathrm{\prime }}(x))$, read in ${\mathrm{s}}^{-1}$ by clause (d)5 of Dimensions, SI unit conventions and quantity types of the framework, and reads a ratio of currents at infinity as a transmitted or reflected fraction of the incident current, conditional on the declared solution and on the asymptotics displayed in each statement; no such fraction is ascribed to a single particle and no count is predicted.

(f) Status and scope. Clauses (a)-(e) fix the data of the pair: a declared effective line model with a declared potential, energy operator, bound-state vocabulary and generalized-solution vocabulary. In addition to P1, P2, P4, P5, P6, P7 and the cited mathematics, they adopt the effective line-model assignments of clauses (b)-(e); these assignments are additional model assumptions and are not consequences of the three-dimensional model. No numerical value of $m$, $V$ or $\mathrm{\hslash }$ and no measured quantity is asserted.