Qm Driven Dynamics And Adiabatic Limits

(a) Framework data. Adopt the state and composition postulate P1, the observable and probability postulate P2 and the autonomous-dynamics postulate P4 (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) for a declared system, in the state, sharp-observable and evolution readings of The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity, Assigned sharp observables and the outcome law of the framework and Assigned energy observable, unitary flow and the two pictures of autonomous evolution; units are those of Dimensions, SI unit conventions and quantity types of the framework. 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). The readings below are the reading of clause (iv) of P4 for the page's models; clause (iv) records that a prescribed time-dependent family is a different model assumption from the autonomous one, and clause (b) below adopts it.

(b) Prescribed energy family. As an additional model assumption of the pair, fix a nonempty compact interval $I=[a,b]\subset \mathbb{R}$ of physical times (seconds), the declared state space $\mathfrak{H}$ of the system, a value $\mathrm{\hslash }>0$ of the reduced Planck constant as in Dimensions, SI unit conventions and quantity types of the framework, and a prescribed family $t\mapsto H(t)$ of self-adjoint operators on $\mathfrak{H}$ (Symmetric, self-adjoint and essentially self-adjoint operators) in one of the following three regularity classes, whose hypotheses are part of the model declaration:

• (i) Bounded norm-continuous class. $D(H(t))=\mathfrak{H}$ for every $t$; each $H(t)\in \mathcal{B}(\mathfrak{H})$ is self-adjoint (A bounded linear operator between normed spaces); $t\mapsto H(t)$ is norm-continuous on $I$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum); and $K:={\sup }_{t\in I}\mathrm{\parallel }H(t)\mathrm{\parallel }\mathrm{/}\mathrm{\hslash }<\mathrm{\infty }$.
• (ii) Bounded-control class. $H(t)=H_0+V(t)$, where $H_0$ is self-adjoint with domain $D(H_0)\subseteq \mathfrak{H}$ and $V:I\to \mathcal{B}(\mathfrak{H})$ is strongly continuous with self-adjoint values and $\mathrm{\parallel }V(t)\mathrm{\parallel }=v(t)$, $v:={\sup }_{t\in I}v(t)<\mathrm{\infty }$.
• (iii) Common-domain class. There is a dense linear subspace $Y\subseteq \mathfrak{H}$ that is itself a Hilbert space continuously embedded in $\mathfrak{H}$ (Hilbert space, for instance $Y=D(H_0)$ with the graph norm of a self-adjoint $H_0$), with $D(H(t))=Y$ for every $t$, each $H(t)$ self-adjoint as an operator on $\mathfrak{H}$, and $t\mapsto H(t)$ of class $C^1(I;\mathcal{B}(Y,\mathfrak{H}))$.

The family, its class and its data are prescribed: no field equation, source model, back-reaction or feedback of the system on the source is assumed, and no claim is made that any laboratory arrangement realizes the family. The family is coupled to the system through the material hypotheses of the page or of the item that declares a specific model (for instance a magnetic-moment or electric-dipole coupling, or the external-source Coulomb family of the page's adiabatic models); the reading of $H(t)$ as the energy of the declared system is a model assumption.

(c) The model's evolution. The driven model declares that the time evolution of the system between times $s,t\in I$ is carried by a family $U(t,s)$ of unitary operators on $\mathfrak{H}$ with $U(s,s)=I,U(t,r)=U(t,s)U(s,r)(r,s,t\in I),U(t,s{)}^{\ast }=U(s,t),$ jointly strongly continuous in $(t,s)$, solving the time-dependent Schroedinger equation on $I$ in the sense of the class: in class (i) $U$ is the unique norm-continuous solution of $i\mathrm{\hslash }{\mathrm{\partial }}_{t}U(t,s)=H(t)U(t,s)$ with $U(s,s)=I$ constructed by the Dyson series of Norm Dyson series and uniqueness (whose hypothesis is exactly the norm-continuity of the bounded self-adjoint family); in class (ii) $U$ is the mild propagator constructed in Strong bounded interaction picture, satisfying $U(t,s)\psi =E(t-s)\psi -\frac{i}{\mathrm{\hslash }}{\int }_s^{t}E(t-r)V(r)U(r,s)\psi dr$ for every $\psi \in \mathfrak{H}$, with $E(u)=e^{-iu{H}_0\mathrm{/}\mathrm{\hslash }}$ the autonomous flow of clause (b) of Assigned energy observable, unitary flow and the two pictures of autonomous evolution; in class (iii) $U$ is the strong propagator of Common-domain nonautonomous evolution with physical generator $A=H\mathrm{/}\mathrm{\hslash }$, preserving $Y$, for which $t\mapsto U(t,s)y$ is $C^1$ in $\mathfrak{H}$ with $i\mathrm{\hslash }{\mathrm{\partial }}_{t}U(t,s)y=H(t)U(t,s)y$ for every $y\in Y$. In each class the cited item is what makes the declaration well posed: it constructs the family and proves uniqueness under the class hypotheses. The model asserts nothing beyond those hypotheses; in particular it asserts no invariant domain and no strong equation for $H_0$-domain vectors in class (ii), and no common domain in classes (i) and (ii).

On a singleton interval $I=\{a\}$, the evolution declaration consists only of $U(a,a)=I$; all differential assertions in these class readings apply only when $a<b$, with one-sided derivatives at the endpoints.

(d) State and outcome readings. The evolved preparation at time $t$ of a preparation $\rho$ specified at time $s$ is $\rho (t):=U(t,s)\rho (s)U(t,s{)}^{\ast }$; it is a preparation of the model in the sense of The state space: density operators, pure and mixed states, and purity for every $t$, by unitarity of $U(t,s)$ and Density operators and scalar spectral probability. A pure preparation $[\psi (s)]$ evolves to the pure preparation $[\psi (t)]$ with $\psi (t):=U(t,s)\psi (s)\mathrm{\ne }0$. The outcome law at time $t$ of an assigned sharp observable $(A,u_A)$ (Assigned sharp observables and the outcome law of the framework) in that preparation is ${\mu }_{\rho (t)}^A$, the reading of P2; the definition records the timing convention once and asserts nothing about how or when a record is produced.

(e) Energy-basis amplitudes and level populations. In class (ii), let $({\phi }_j{)}_{j\in J}$ be an orthonormal family in $D(H_0)$ with $H_0{\phi }_j=E_j{\phi }_j$ and $E_j\in \mathbb{R}$ (a complete orthonormal family of eigenvectors is not assumed here). For $s\le t$ and labels $i,j$ the transition amplitude from $i$ at time $s$ to $j$ at time $t$ is the scalar $A_{ji}(t,s):=e^{i{E}_j(t-s)\mathrm{/}\mathrm{\hslash }}⟨U(t,s){\phi }_i,{\phi }_j⟩,$ and the transition probability is $p_{ji}(t,s):=\mathrm{\mid }{A}_{ji}(t,s){\mathrm{\mid }}^2$. If additionally $E_j$ is a simple eigenvalue of $H_0$ with spectral projection $P_j=E_{H_0}(\{E_j\})$ onto $\mathbb{C}{\phi }_j$, then the definition states, as the reading of P2, the identity $p_{ji}(t,s)={\mu }_{\psi (t)}^{H_0}(\{E_j\})=\mathrm{\parallel }{E}_{H_0}(\{E_j\})\psi (t){\mathrm{\parallel }}^2,\psi (t)=U(t,s){\phi }_i,$ which follows from the rank-one form of $P_j$ and the normalization $\mathrm{\parallel }{\phi }_j\mathrm{\parallel }=1$; $p_{ji}$ is the reading of the outcome probability of the value $E_j$ of the assigned energy observable $(H_0,\mathrm{J})$ in the evolved preparation, and it is a property of the declared model, not a measured frequency.

(f) Declared two-level drive models. Let $({\sigma }_x,{\sigma }_y,{\sigma }_z)$ be the Pauli matrices of Finite spin matrices clause (e) acting on ${\mathbb{C}}^2$ in the orthonormal basis $(e_{\uparrow },e_{\downarrow })$ (Real and complex inner product spaces, with the inner product linear in the first argument), let ${\omega }_0,{\mathrm{\Omega }}_1,\omega \in \mathbb{R}$, and put $H_0:=\frac{\mathrm{\hslash }}{2}{\omega }_0{\sigma }_z$ (class (i) with $I$ any compact interval). The circular two-level drive model is the class-(i) family $H_{\mathrm{c}}(t):=\frac{\mathrm{\hslash }}{2}({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1(\cos (\omega t){\sigma }_x+\sin (\omega t){\sigma }_y\left)\right),$ and the linear two-level drive model is the class-(i) family $H_{\mathrm{l}}(t):=\frac{\mathrm{\hslash }}{2}({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1\cos (\omega t){\sigma }_x)\mathrm{.}$ The real parameters ${\omega }_0,{\mathrm{\Omega }}_1,\omega$ represent angular frequencies of dimension duration${}^{-1}$, read in ${\mathrm{s}}^{-1}$; their numerical values are expressed in that unit, so $\omega t$ is dimensionless and $\mathrm{\hslash }{\omega }_0$ and $\mathrm{\hslash }{\mathrm{\Omega }}_1$ have energy units; ${\omega }_0$ is the splitting frequency of the model (the level gap is $\mathrm{\hslash }\mathrm{\mid }{\omega }_0\mathrm{\mid }$), $\omega$ the drive frequency, ${\mathrm{\Omega }}_1$ the transverse coupling, and $\mathrm{\Delta }:={\omega }_0-\omega$ the detuning. Populations of the level pair are the diagonal entries $⟨P_{\bullet }\psi ,\psi ⟩$ of the orthogonal projections $P_{\uparrow },P_{\downarrow }$ onto $\mathbb{C}{e}_{\uparrow },\mathbb{C}{e}_{\downarrow }$ in the evolved vector $\psi (t)$. The two families are model data: no identification with a laboratory spin, molecule or field, and no relation of ${\mathrm{\Omega }}_1$ to a field amplitude, is asserted; such an identification is a tested identification in the sense of P7.

(g) Status and scope. Clauses (b), (c), (f) are adopted model assumptions and readings; the existence and uniqueness of the propagator in each class is cited mathematics, not a consequence of P1, P2 or P4. The model asserts no open-system or environment coupling, no non-unitary reduced dynamics, no measurement or apparatus model, no thermalization or equilibration, no long-time limit and no empirical record; the identification of the family's parameters with laboratory fields and of the system with a laboratory sample is made, if at all, only in separately declared models with their own uncertainty.

• Why classes (i)-(iii) instead of one general theorem. A general unbounded time-dependent family requires a domain condition; classes (ii) and (iii) are the two standard sufficient regimes used on this page, and the bounded class (i) covers the finite-level models. Clause (iv) of P4 explicitly does not adopt a general unbounded family, so each class is declared with its own mathematics.
• Amplitudes are not outcomes. The scalar $A_{ji}$ is a complex bookkeeping amplitude; the model's outcome probability is the Born number of clause (e). Global phases of the eigenvectors change $A_{ji}$ by a phase and leave $p_{ji}$ unchanged.

(a) Setting. Adopt the framework data and the declared driven model of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models in the bounded-control class of its clause (b)(ii): $\mathfrak{H}$ is the model's state space (Hilbert space), $H_0$ is self-adjoint on its exact dense domain $D(H_0)$ (Symmetric, self-adjoint and essentially self-adjoint operators), $V:I\to \mathcal{B}(\mathfrak{H})$ is strongly continuous with self-adjoint values and $v={\sup }_{t\in I}\mathrm{\parallel }V(t)\mathrm{\parallel }<\mathrm{\infty }$ (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum), $H(t)=H_0+V(t)$ on $D(H_0)$, and $U(t,s)$ is the mild propagator of clause (c)(ii) constructed in Strong bounded interaction picture. The readings of P1 and P2 are those of 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, The state space: density operators, pure and mixed states, and purity and Assigned sharp observables and the outcome law of the framework; all times lie in $I$ and are read in seconds as in Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) The free-flow factor. The free-flow factor of the model is the autonomous flow of the unperturbed part, $E(t):=e^{-it{H}_0\mathrm{/}\mathrm{\hslash }}(t\in \mathbb{R}),$ the reading of clause (b) of Assigned energy observable, unitary flow and the two pictures of autonomous evolution applied to the assigned energy observable $(H_0,\mathrm{J})$; it is a strongly continuous one-parameter unitary group on all real arguments. This auxiliary group supplies $E(-t)$ and $E(t-s)$ even when those arguments lie outside $I$; the physical driven evaluation times $s,t$ remain in $I$. The interaction-picture control is the family $V_I:I\to \mathcal{B}(\mathfrak{H})$, $V_I(t):=E(-t)V(t)E(t),$ which is strongly continuous with self-adjoint values and $\mathrm{\parallel }{V}_I(t)\mathrm{\parallel }=\mathrm{\parallel }V(t)\mathrm{\parallel }\le v$. The interaction propagator is $U_I(t,s):=E(-t)U(t,s)E(s)(s,t\in I),$ and the model's clause (c)(ii) supplies, as the cited theorem proves, the decomposition $U(t,s)=E(t)U_I(t,s)E(-s)\mathrm{.}$ The interaction propagator is unitary, jointly strongly continuous, satisfies $U_I(s,s)=I$ and the cocycle $U_I(t,r)=U_I(t,s)U_I(s,r)$, and its first-endpoint map is strongly differentiable with $i\mathrm{\hslash }{\mathrm{\partial }}_t{U}_I(t,s)\psi =V_I(t)U_I(t,s)\psi (\psi \in \mathfrak{H}),$ all as supplied by Strong bounded interaction picture; these are readings of that theorem in the model's vocabulary, not new assumptions.

(c) Interaction states and preparations. For a Schroedinger-picture state vector $\psi (t)=U(t,s)\psi (s)$ the interaction-picture state is ${\psi }_I(t):=E(-t)\psi (t),\text{equivalently}{\psi }_I(t)=U_I(t,s){\psi }_I(s),$ and for a preparation $\rho$ with $\rho (t)=U(t,s)\rho (s)U(t,s{)}^{\ast }$ the interaction-picture preparation is ${\rho }_I(t):=E(-t)\rho (t)E(t),$ which is again a preparation of the model by Density operators and scalar spectral probability; for a pure preparation $[\psi (s)]$, $[{\psi }_I(t)]$ is pure. The map $\psi (t)\mapsto {\psi }_I(t)$ is unitary at every $t$, so $\mathrm{\parallel }{\psi }_I(t)\mathrm{\parallel }=\mathrm{\parallel }\psi (t)\mathrm{\parallel }$.

(d) The two pictures assign the same outcome laws. Let $(A,u_A)$ be an assigned sharp observable with $A\in \mathcal{B}(\mathfrak{H})$ bounded and let $A_I(t):=E(-t)AE(t)$ be its interaction-picture representative, which is again a bounded self-adjoint observable with $E_{A_I(t)}(B)=E(-t)E_A(B)E(t)$ for every Borel set $B$ by clause (d) of Heisenberg flow of a bounded operator: conjugates, covariance and the equation of motion applied to the flow $E$ of the self-adjoint $T=-H_0\mathrm{/}\mathrm{\hslash }$. Then for every preparation $\rho$ specified at $s$ and every $t\in I$, ${\mu }_{\rho (t)}^A(B)={\mu }_{{\rho }_I(t)}^{A_I(t)}(B)(B\subseteq \mathbb{R}\text{ Borel}),$ because $\rho (t)=E(t){\rho }_I(t)E(-t)$ and $\mathrm{tr}(E(t){\rho }_{I}E(-t)E_A(B))=\mathrm{tr}({\rho }_{I}E(-t)E_A(B)E(t))$ by cyclicity of the trace; equivalently, for a pure preparation, $\mathrm{\parallel }{E}_A(B)\psi (t)\mathrm{\parallel }=\mathrm{\parallel }{E}_{A_I(t)}(B){\psi }_I(t)\mathrm{\parallel }$. This identity is the precise sense in which the two pictures are two bookkeeping devices for the same model; it is a reading of P2 together with the cited mathematics, not a claim about apparatuses.

(e) Energy-basis transition amplitudes. Let ${\phi }_a,{\phi }_b\in D(H_0)$ be unit eigenvectors, $H_0{\phi }_a=E_a{\phi }_a$, $H_0{\phi }_b=E_b{\phi }_b$, and let $A_{ba}(t,s)$ and $p_{ba}(t,s)$ be the transition amplitude and probability of clause (e) of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models. Then $A_{ba}(t,s)=e^{is(E_a-E_b)\mathrm{/}\mathrm{\hslash }}⟨U_I(t,s){\phi }_a,{\phi }_b⟩,\text{hence}{p}_{ba}(t,s)=\mid ⟨U_I(t,s){\phi }_a,{\phi }_b⟩{\mid }^2,$ by the decomposition of clause (b) and $E(-s){\phi }_a=e^{is{E}_a\mathrm{/}\mathrm{\hslash }}{\phi }_a$, $⟨E(t)\chi ,{\phi }_b⟩=e^{-it{E}_b\mathrm{/}\mathrm{\hslash }}⟨\chi ,{\phi }_b⟩$; the phases are of modulus one (Real and complex inner product spaces, with the inner product linear in the first argument). The transition probability is therefore computable in either picture, and the definition records this as the reading used by the page's amplitude theorem; no measurement frequency or rate is defined.

(f) Status and scope. Clauses (b)-(e) are readings of the declared driven model, of clauses (b)-(e) of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models, and of P1 and P2, in the vocabulary of the cited mathematics. The interaction picture is a change of description of the same model: it adds no assumption, no physical process, no measurement and no empirical claim, and it asserts nothing about detectors, records or the laboratory realization of the model.

Let $({\sigma }_x,{\sigma }_y,{\sigma }_z)$ be the Pauli matrices of Finite spin matrices clause (e), acting on ${\mathbb{C}}^2$ in the orthonormal basis $(e_{\uparrow },e_{\downarrow })$ of that clause, so that ${\sigma }_z=\mathrm{diag}(1,-1)$, ${\sigma }_x{\sigma }_y=i{\sigma }_z$, ${\sigma }_y{\sigma }_z=i{\sigma }_x$ and ${\sigma }_z{\sigma }_x=i{\sigma }_y$; write $I$ for the identity and let $P_{\uparrow },P_{\downarrow }$ be the orthogonal projections onto $\mathbb{C}{e}_{\uparrow }$ and $\mathbb{C}{e}_{\downarrow }$. Let $\mathrm{\hslash }>0$, $\omega ,{\omega }_0,{\mathrm{\Omega }}_1\in \mathbb{R}$, put $\mathrm{\Delta }:={\omega }_0-\omega$ and, for $t\in \mathbb{R}$, $R(t):=\cos \text{\hspace{0.17em}⁣}\left(\frac{\omega t}{2}\right)I+i\sin \text{\hspace{0.17em}⁣}\left(\frac{\omega t}{2}\right){\sigma }_z\mathrm{.}$ Then:

(a) Frame. For every $t$ the matrix $R(t)$ is unitary with $R(0)=I$, the map $R$ is of class $C^{\mathrm{\infty }}$ with $R^{\mathrm{\prime }}(t)=\frac{i\omega }{2}{\sigma }_{z}R(t),$ and $R(t)P_{\uparrow }R(t{)}^{\ast }=P_{\uparrow }$, $R(t)P_{\downarrow }R(t{)}^{\ast }=P_{\downarrow }$.

(b) Conjugation. For every $t$, $R(t){\sigma }_{x}R(t{)}^{\ast }=\cos (\omega t){\sigma }_x-\sin (\omega t){\sigma }_y,R(t){\sigma }_{y}R(t{)}^{\ast }=\sin (\omega t){\sigma }_x+\cos (\omega t){\sigma }_y,R(t){\sigma }_{z}R(t{)}^{\ast }={\sigma }_z\mathrm{.}$

(c) Circular drive. For the family $H_{\mathrm{c}}(t):=\frac{\mathrm{\hslash }}{2}({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1(\cos (\omega t){\sigma }_x+\sin (\omega t){\sigma }_y))$ one has, for every $t$, $R(t)H_{\mathrm{c}}(t)R(t{)}^{\ast }+i\mathrm{\hslash }{R}^{\mathrm{\prime }}(t)R(t{)}^{\ast }=\frac{\mathrm{\hslash }}{2}(\mathrm{\Delta }{\sigma }_z+{\mathrm{\Omega }}_1{\sigma }_x),$ which is a constant Hermitian matrix independent of $t$.

(d) Linear drive. For the family $H_{\mathrm{l}}(t):=\frac{\mathrm{\hslash }}{2}({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1\cos (\omega t){\sigma }_x)$ put $A:=\frac{\mathrm{\hslash }}{2}(\mathrm{\Delta }{\sigma }_z+\frac{{\mathrm{\Omega }}_1}{2}{\sigma }_x),B:=\frac{\mathrm{\hslash }{\mathrm{\Omega }}_1}{8}({\sigma }_x+i{\sigma }_y)\mathrm{.}$ Then for every $t$ $R(t)H_{\mathrm{l}}(t)R(t{)}^{\ast }+i\mathrm{\hslash }{R}^{\mathrm{\prime }}(t)R(t{)}^{\ast }=A+e^{2i\omega t}B+e^{-2i\omega t}{B}^{\ast },$ and $\mathrm{\parallel }A\mathrm{\parallel }\le \frac{\mathrm{\hslash }}{2}(\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }+\frac{\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{2})$, $\mathrm{\parallel }B\mathrm{\parallel }=\frac{\mathrm{\hslash }\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{4}$.

(e) Transformation of the dynamics. Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H:\mathbb{R}\to \mathrm{End}({\mathbb{C}}^2)$ be continuous with Hermitian values, let $\psi :\mathbb{R}\to {\mathbb{C}}^2$ be of class $C^1$ with $i\mathrm{\hslash }{\psi }^{\mathrm{\prime }}(t)=H(t)\psi (t)(t\in \mathbb{R}),$ and put ${\psi }_R:=R\psi$ and $H^R(t):=R(t)H(t)R(t{)}^{\ast }+i\mathrm{\hslash }{R}^{\mathrm{\prime }}(t)R(t{)}^{\ast }$. Then ${\psi }_R$ is of class $C^1$ and satisfies $i\mathrm{\hslash }{\psi }_R^{\mathrm{\prime }}(t)=H^R(t){\psi }_R(t)$ for every $t$; the family $H^R$ is continuous with Hermitian values. Consequently, if $U$ is the unitary propagator of the norm-continuous family $H$ supplied by Norm Dyson series and uniqueness, then $U^R(t,s):=R(t)U(t,s)R(s{)}^{\ast }(s,t\in \mathbb{R})$ is the unitary propagator of $H^R$: it satisfies $i\mathrm{\hslash }{\mathrm{\partial }}_t{U}^R(t,s)=H^R(t)U^R(t,s)$, $U^R(s,s)=I$, the cocycle, and $U^R(t,s{)}^{\ast }=U^R(s,t)$. Moreover for every solution $\psi$ of the $H$-equation, every $t\in \mathbb{R}$ and both signs $\bullet \in \{\uparrow ,\downarrow \}$, $⟨P_{\bullet }\psi (t),\psi (t)⟩=⟨P_{\bullet }{\psi }_R(t),{\psi }_R(t)⟩\mathrm{.}$

The lemma is a statement of finite-dimensional linear algebra and calculus for the displayed matrices; no physical identification, no measurement and no asymptotic claim is made.