The line model's energy observable, flow and stationary states

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the kinematics and massive-particle model postulate P5 (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), all read through the declared effective one-dimensional line model The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions with mass parameter $m>0$ and spin label $s=0$. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Let $V:\mathbb{R}\to \mathbb{R}$ be real, bounded and measurable, let $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}=\sup \mathrm{\mid }V\mathrm{\mid }$, and let $H_V$ be the assigned energy operator of clause (c) of the model, with flow $U(t):=\exp (-it{H}_V\mathrm{/}\mathrm{\hslash })$ and evolved preparations $\rho (t):=U(t)\rho (0)U(t{)}^{\ast }$ as in Assigned energy observable, unitary flow and the two pictures of autonomous evolution. Then:

(a) $(H_V,\mathrm{J})$ is an assigned sharp observable of the model in the sense of clause (a) of Assigned sharp observables and the outcome law of the framework: $H_V$ is densely defined and self-adjoint on $D(H_V)=W^{2,2}(\mathbb{R};\mathbb{C})$, bounded below by $-\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$, and $H_{V}f=-\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f+Vf$ for every $f\in D(H_V)$.

(b) Flow. The family $t\mapsto U(t)$ is the model's strongly continuous one-parameter unitary group of the assigned energy observable; the Schroedinger equation $i\mathrm{\hslash }\frac{d}{dt}\psi (t)=H_V\psi (t)$ holds in the norm-differentiable sense exactly for initial vectors $\psi (0)\in D(H_V)$, and for other vectors the evolution is continuous but the equation is not asserted; density preparations evolve as $\rho (t)=U(t)\rho (0)U(t{)}^{\ast }$ with $\rho (t)$ a density operator for every $t$.

(c) Stationary states. For $\psi \in H_{\mathrm{l}\mathrm{i}\mathrm{n}}$, $\psi \mathrm{\ne }0$, the pure preparation $[\psi ]$ is stationary, $[\psi (t)]=[\psi (0)]$ for all $t$, if and only if $\psi$ is an eigenvector of $H_V$; in that case, with eigenvalue $E$, one has $\psi (t)=e^{-itE\mathrm{/}\mathrm{\hslash }}\psi (0)$, the energy outcome law is time-independent, and if $\mathrm{\parallel }\psi \mathrm{\parallel }=1$ the law assigns probability one to the singleton $\{E\}$, so the energy mean exists and equals $E$, with second moment $E^2$. Moreover every bounded assigned sharp observable compatible with the energy has a time-independent outcome law in every preparation.

(d) Position reading. In a unit pure preparation $\psi$ the declared position assignment of the model has outcome law ${\mu }_{\psi }^Q(B)={\int }_B\mathrm{\mid }\psi {\mathrm{\mid }}^{2}dx$ for every Borel $B\subseteq \mathbb{R}$; and if $\psi$ is a unit eigenvector with eigenvalue $E$, the position density $\mathrm{\mid }\psi {\mathrm{\mid }}^2$ and the energy point mass at $E$ coexist as readings of the same preparation, of the action of the declared observables, without any joint measurement claim.

(e) Scope. The statement is conditional on P1, P2, P4, P5 and the declared line model; it asserts no discrete spectrum or eigenbasis for a general bounded measurable $V$, no completeness or scattering statement, no identification of $H_V$ with any laboratory Hamiltonian beyond the tested identification allowed by P7, and no empirical result.

• What the theorem does not say. For a general bounded measurable potential only self-adjointness, the flow and the stationary-state equivalence are claimed. Discreteness of the spectrum, completeness of eigenfunctions and the explicit negative spectrum are proved in the interval and finite-well items of this page under their own hypotheses, not here.
• Units. The lower bound $-\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$ and the eigenvalues $E$ are read in joules when $V$ and $\mathrm{\hslash }$ are read in the units of Dimensions, SI unit conventions and quantity types of the framework; $D^2$ acts on the coordinate $x$ in metres, so $-\frac{{\mathrm{\hslash }}^2}{2m}{D}^2$ has the dimension of energy.