Qm Autonomous Dynamics

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the choice hypotheses of the cited spectral and functional-calculus suppliers. Let $H\mathrm{\ne }\{0\}$ be a complex Hilbert space (Hilbert space), let $T$ be self-adjoint on $H$ (Symmetric, self-adjoint and essentially self-adjoint operators) with spectral projection-valued measure $E$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure), and put $U(t):=e^{itT}={\int }_{\mathbb{R}}{e}^{it\lambda }dE(\lambda )(t\in \mathbb{R})$ in the sense of the integral of a measurable function against a projection-valued measure (Integral of a measurable function against a projection-valued measure). Then:

(a) $U$ is a strongly continuous one-parameter unitary group (Strongly continuous one-parameter unitary group) with $U(0)=I$, $U(t+s)=U(t)U(s)$ and $U(t{)}^{\ast }=U(t{)}^{-1}=U(-t)$ for all $s,t$; moreover $U(t)D(T)=D(T)$, $TU(t)=U(t)T$ on $D(T)$, and the difference quotient $\frac{1}{t}(U(t)x-x)$ has a norm limit as $t\to 0$ exactly for $x\in D(T)$, where that limit is $iTx$ (A self-adjoint operator generates a strongly continuous unitary group, Infinitesimal generator of a unitary group).

(b) For every bounded Borel function $h:\mathbb{R}\to \mathbb{C}$ and every $t\in \mathbb{R}$ one has $h(T)U(t)=U(t)h(T)$; in particular $E(B)U(t)=U(t)E(B)$ for every Borel set $B\subseteq \mathbb{R}$.

(c) For $x\in D(T)$ the map $t\mapsto U(t)x$ is differentiable at every $t\in \mathbb{R}$, with $\frac{d}{dt}U(t)x=iU(t)Tx=iTU(t)x;$ for $x\in H\setminus D(T)$ the map $t\mapsto U(t)x$ is differentiable at no point of $\mathbb{R}$.

• Sign and scaling conventions. The physical reading of this lemma uses $T=-K\mathrm{/}\mathrm{\hslash }$ for a self-adjoint energy operator $K$ and a positive constant $\mathrm{\hslash }$; then $D(T)=D(K)$ and $U(t)=e^{-itK\mathrm{/}\mathrm{\hslash }}$ in the spectral calculus, while the derivative statement of clause (c) becomes $i\mathrm{\hslash }\frac{d}{dt}U(t)x=KU(t)x$ for $x\in D(K)$. The identification of the group generated by $-iK\mathrm{/}\mathrm{\hslash }$ with the spectral-calculus exponential is the uniqueness statement of Stone's theorem; it is used, not reproved, on the physics pages.
• Only the domain carries the equation. Clause (c) shows that the norm-differentiability of the flow is equivalent to membership in $D(T)$; the flow itself, clause (a), is defined on all of $H$ and is continuous there.

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H\mathrm{\ne }\{0\}$ be a complex Hilbert space (Hilbert space), let $T$ be self-adjoint (Symmetric, self-adjoint and essentially self-adjoint operators) and $U(t)=e^{itT}$ as in Flow calculus of a self-adjoint generator: commutation, invariance and differentiability; for $A\in \mathcal{B}(H)$ (A bounded linear operator between normed spaces) put $A(t):=U(t{)}^{\ast }AU(t)=U(-t)AU(t)(t\in \mathbb{R})\mathrm{.}$ Then:

(a) $A(t)\in \mathcal{B}(H)$ with $A(0)=A$, $\mathrm{\parallel }A(t)\mathrm{\parallel }=\mathrm{\parallel }A\mathrm{\parallel }$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum) and $A(t{)}^{\ast }=A^{\ast }(t)$; in particular $A$ self-adjoint implies every $A(t)$ self-adjoint (Self-adjoint, positive, unitary and normal operators). Moreover $A(t+s)=U(s{)}^{\ast }A(t)U(s)$ for all $s,t$.

(b) If $AD(T)\subseteq D(T)$, then $A(t)D(T)\subseteq D(T)$ for every $t$, and for every $x\in D(T)$ the map $t\mapsto A(t)x$ is differentiable at every $t$ with $\frac{d}{dt}A(t)x=-i[T,A(t)]x=-i(TA(t)-A(t)T)x\mathrm{.}$

(c) For every $x\in H$ the map $t\mapsto A(t)x$ is continuous, and the uniform operator norm bound $\mathrm{\parallel }A(t)\mathrm{\parallel }\le \mathrm{\parallel }A\mathrm{\parallel }$ holds.

(d) If $A$ is self-adjoint with spectral projection-valued measure $E_A$ (Spectral theorem for unbounded self-adjoint operators (PVM form)), then every $A(t)$ is self-adjoint with spectral projection-valued measure $E_{A(t)}(B)=U(-t)E_A(B)U(t)(B\subseteq \mathbb{R}\text{ Borel}),$ and the spectra agree, $\sigma (A(t))=\sigma (A)$.

• Where the hypothesis is needed. Clauses (a) and (c) require nothing beyond boundedness of $A$; clause (b), and with it the Heisenberg equation, uses the product-domain hypothesis $AD(T)\subseteq D(T)$, which is exactly the condition under which the differentiated products are defined. A bounded observable whose spectral PVM commutes with that of $T$ satisfies the hypothesis trivially, because then $A$ maps every $D(h(T))$ into itself.
• Transported measurable functional calculus. Step 2.3 shows more than the statement of (d): for every bounded Borel $g$ the operator $g(A(t))$ equals $U(-t)g(A)U(t)$; this is the operator form of the statement that the unitary flow is an automorphism of the algebra of bounded Borel functions of the observable.

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H\mathrm{\ne }\{0\}$ be a complex Hilbert space (Hilbert space, Real and complex inner product spaces, with the inner product linear in the first argument), let $T$ be self-adjoint (Symmetric, self-adjoint and essentially self-adjoint operators) with spectral projection-valued measure $E$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure), let $U(t)=e^{itT}=\int e^{it\lambda }dE(\lambda )$ be the generated strongly continuous unitary group (Strongly continuous one-parameter unitary group, Integral of a measurable function against a projection-valued measure), and let $\psi \in H$ with $\psi \mathrm{\ne }0$. Then the following are equivalent:

(i) for every $t\in \mathbb{R}$ the vector $U(t)\psi$ is a complex multiple of $\psi$ (the ray of $\psi$ is invariant under the flow);

(ii) $\psi \in D(T)$ and $T\psi =\lambda \psi$ for some $\lambda \in \mathbb{R}$ (the ray of $\psi$ is an eigenray of $T$).

If these hold for the eigenvalue $\lambda$, then $U(t)\psi =e^{it\lambda }\psi$ for every $t\in \mathbb{R}$.

• Normalization is bookkeeping. The proof normalizes $\psi$ only to use the unit vector in the inner-product computations of [F6] and step 1.1; the statement itself does not require $\mathrm{\parallel }\psi \mathrm{\parallel }=1$.
• No converse beyond rays. Clause (i) concerns the invariance of the ray; it does not claim that $U(t)\psi =\psi$ for an eigenvector, because the eigenvalue phase $e^{it\lambda }$ is present. The statement that the two vectors lie in one ray, hence determine the same pure preparation, is the content of the phase-invariance of states, not an equality of vectors.

(a) Assigned energy observable. Adopt the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4) for a declared isolated system whose state space is the model's The state space of the framework: preparations, pure rays and density operators. An assigned energy observable of the system is an assigned sharp observable $(H_{\mathrm{o}\mathrm{p}},u_H)$ in the sense of Assigned sharp observables and the outcome law of the framework clause (a), whose declared unit $u_H$ is an energy unit in the sense of Dimensions, SI unit conventions and quantity types of the framework (joules in the SI conventions fixed there). Thus $H_{\mathrm{o}\mathrm{p}}:D(H_{\mathrm{o}\mathrm{p}})\subset H\to H$ is densely defined and self-adjoint (Symmetric, self-adjoint and essentially self-adjoint operators, Hilbert space), with spectral projection-valued measure $E_H$ on the Borel sets of $\mathbb{R}$ under the choice hypotheses of Spectral theorem for unbounded self-adjoint operators (PVM form) (Projection valued measure), and its outcome law in a preparation is the law of Assigned sharp observables and the outcome law of the framework clause (b). The assignment of $H_{\mathrm{o}\mathrm{p}}$ is part of the model specification, as in clause (i) of P4; this definition asserts nothing about which laboratory quantity of a concrete system realizes it.

(b) The unitary flow. The time evolution or flow of the isolated system is the family $U(t):=\exp \text{\hspace{0.17em}⁣}(-\frac{i}{\mathrm{\hslash }}t{H}_{\mathrm{o}\mathrm{p}}):={\int }_{\mathbb{R}}{e}^{-it\lambda \mathrm{/}\mathrm{\hslash }}d{E}_H(\lambda )(t\in \mathbb{R}),$ the reading of clause (ii) of P4 in the spectral calculus of $H_{\mathrm{o}\mathrm{p}}$ (Integral of a measurable function against a projection-valued measure); here $\mathrm{\hslash }>0$ is the reduced Planck constant with its value and unit fixed by Dimensions, SI unit conventions and quantity types of the framework, so that $t{H}_{\mathrm{o}\mathrm{p}}\mathrm{/}\mathrm{\hslash }$ is dimensionless. Equivalently, $U(t)=e^{itT}$ for the self-adjoint generator $T:=-H_{\mathrm{o}\mathrm{p}}\mathrm{/}\mathrm{\hslash }$ with $D(T)=D(H_{\mathrm{o}\mathrm{p}})$; the two readings agree by the uniqueness of the generator in Stone's theorem, and by A self-adjoint operator generates a strongly continuous unitary group (Flow calculus of a self-adjoint generator: commutation, invariance and differentiability clause (a)) the family $U$ is a strongly continuous one-parameter unitary group: $U(0)=I$, $U(t+s)=U(t)U(s)$, $U(t{)}^{\ast }=U(t{)}^{-1}=U(-t)$ (Strongly continuous one-parameter unitary group).

(c) Schroedinger picture. The Schroedinger-picture evolution of a preparation is fixed by clause (iii) of P4. A state vector evolves by $\psi (t):=U(t)\psi (0)$ for every $\psi (0)\in H$, and the state at time $t$ of the pure preparation $[\psi (0)]$ is the pure preparation $[\psi (t)]$, which is well defined because $U(t)$ is injective; a general preparation $\rho$, that is an element of the state space $D(H)$ of The state space of the framework: preparations, pure rays and density operators, evolves by $\rho (t):=U(t)\rho (0)U(t{)}^{\ast },$ the reading of clause (iii) of P4 in the density-operator vocabulary of The state space: density operators, pure and mixed states, and purity (Self-adjoint, positive, unitary and normal operators, Density operators and scalar spectral probability). In this picture the assigned observables are held fixed at their assigned values and the preparation moves.

(d) Heisenberg picture. For a bounded assigned sharp observable $(A,u_A)$ — that is, an assigned sharp observable with $A\in \mathcal{B}(H)$ (A bounded linear operator between normed spaces), so that $A$ is self-adjoint and everywhere defined — the Heisenberg representative of $(A,u_A)$ at time $t$ is the pair $(A_H(t),u_A)$ with the same declared unit and $A_H(t):=U(t{)}^{\ast }AU(t)=U(-t)AU(t)\mathrm{.}$ By Heisenberg flow of a bounded operator: conjugates, covariance and the equation of motion clauses (a) and (d), $A_H(t)$ is again bounded and self-adjoint with spectral projection-valued measure $E_{A_H(t)}(B)=U(-t)E_A(B)U(t)$, so $(A_H(t),u_A)$ is again an assigned sharp observable of the framework; in this picture the preparation is held fixed at its value at time $0$ and the observables move. The two pictures are two bookkeeping devices for the same model; the statement that they assign the same outcome laws is proved in the Heisenberg-picture theorem of this page, not assumed here.

(e) Stationary preparations. A preparation $\rho \in D(H)$ is stationary for the assigned energy observable when $\rho (t)=\rho$ for every $t\in \mathbb{R}$. A pure preparation $[\psi ]$ is stationary when $[\psi (t)]=[\psi (0)]$ for every $t$, that is, when the evolved vector remains in the ray of $\psi$; the characterization of this condition by the eigenrays of $H_{\mathrm{o}\mathrm{p}}$ is proved in the stationary-states theorem of this page, not assumed here. An energy eigenstate is a pure preparation of a unit eigenvector of $H_{\mathrm{o}\mathrm{p}}$, and a stationary state in the model is a stationary pure preparation; by that theorem the two notions coincide.

(f) Energy-compatible observables. An assigned sharp observable $(A,u_A)$ is compatible with the energy, or energy-compatible, when its spectral projection-valued measure and that of the energy commute strongly, $E_A(B)E_H(C)=E_H(C)E_A(B)\text{for all Borel }B,C\subseteq \mathbb{R}\mathrm{.}$ This is an equality of bounded operators on $H$, not a relation between formal commutators on a small domain; the phrase conserved quantity for an energy-compatible assigned observable is a claim about the time invariance of its outcome law and is established in the energy-conservation theorem of this page for bounded $A$.

(g) Status. Clauses (a)-(f) record the readings of P4 together with P1 and P2 in the vocabulary of this page and fix the terminology used by the later pages of the category. They adopt no physical assumption beyond P4, P1 and P2 and the choice hypotheses of the cited mathematics, assert no dynamics for a prescribed time-dependent family or for an open system (clauses (iv) and (v) of P4 are separate model assumptions treated on their own pages), assert no measurement model, no apparatus realization and no empirical result, and make no claim that any laboratory system is exactly isolated.

• Units and dimensionless phases. The energy unit $u_H$ and $\mathrm{\hslash }$ are part of the model's quantity conventions: $H_{\mathrm{o}\mathrm{p}}t\mathrm{/}\mathrm{\hslash }$ is a dimensionless spectral variable, and the argument of the exponential is read in those units. No numerical value of $H_{\mathrm{o}\mathrm{p}}$ is asserted.
• Domain matters. The flow $U(t)$ is defined on all of $H$, while the energy operator and every unbounded assigned observable are defined only on their domains; the Schroedinger equation and the Heisenberg equation are domain-qualified statements and are proved as such on this page.
• Rays and global phases. Multiplying $\psi (t)$ by a common unit-modulus scalar leaves its ray unchanged. For nonzero coefficients $c_j,c_k$ in a fixed orthonormal basis, their relative phase is $(c_j\mathrm{/}\mathrm{\mid }{c}_j\mathrm{\mid })\overline{(c_k\mathrm{/}\mathrm{\mid }{c}_k\mathrm{\mid })}$; it is unchanged by a common phase and can affect outcome laws. Autonomous evolution need not change relative phases: for $H_{\mathrm{o}\mathrm{p}}=EI$ with $E\in \mathbb{R}$, clause (b) gives $U(t)=e^{-iEt\mathrm{/}\mathrm{\hslash }}I$, so every ray is stationary.

Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1), the observable and probability postulate in the reading of Assigned sharp observables and the outcome law of the framework, and the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4), and 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), the choice hypotheses of the cited spectral suppliers. Let $H$ be the state space assigned to a declared isolated system of the massive-matter sector (The state space of the framework: preparations, pure rays and density operators), let $(H_{\mathrm{o}\mathrm{p}},u_H)$ be its assigned energy observable with spectral projection-valued measure $E_H$ (Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (a)), and let $U(t)=\exp \text{\hspace{0.17em}⁣}(-\frac{i}{\mathrm{\hslash }}t{H}_{\mathrm{o}\mathrm{p}})={\int }_{\mathbb{R}}{e}^{-it\lambda \mathrm{/}\mathrm{\hslash }}d{E}_H(\lambda ),\psi (t):=U(t)\psi (0),\rho (t):=U(t)\rho (0)U(t{)}^{\ast }$ be the flow and the state evolutions of clauses (b) and (c) of Assigned energy observable, unitary flow and the two pictures of autonomous evolution, with $\mathrm{\hslash }>0$ and the energy unit in the sense of Dimensions, SI unit conventions and quantity types of the framework. Then:

(a) Unitary flow. $U$ is a strongly continuous one-parameter unitary group (Strongly continuous one-parameter unitary group) with $U(0)=I$, $U(t+s)=U(t)U(s)$ and $U(t{)}^{\ast }=U(t{)}^{-1}=U(-t)$; every orbit has constant norm, $\mathrm{\parallel }\psi (t)\mathrm{\parallel }=\mathrm{\parallel }\psi (0)\mathrm{\parallel }$, so for $\psi (0)\mathrm{\ne }0$ the ray $[\psi (t)]$ is a pure preparation of the model for every $t$ (The state space: density operators, pure and mixed states, and purity).

(b) Domain invariance. $U(t)D(H_{\mathrm{o}\mathrm{p}})=D(H_{\mathrm{o}\mathrm{p}})$ for every $t$ and $H_{\mathrm{o}\mathrm{p}}U(t)=U(t)H_{\mathrm{o}\mathrm{p}}$ on $D(H_{\mathrm{o}\mathrm{p}})$: the flow preserves the energy domain and commutes with the energy there.

(c) Schroedinger equation on the domain. For $\psi (0)\in D(H_{\mathrm{o}\mathrm{p}})$ the curve $t\mapsto \psi (t)$ is norm-differentiable at every $t\in \mathbb{R}$, it stays in $D(H_{\mathrm{o}\mathrm{p}})$, and $i\mathrm{\hslash }\frac{d}{dt}\psi (t)=H_{\mathrm{o}\mathrm{p}}\psi (t)=H_{\mathrm{o}\mathrm{p}}U(t)\psi (0);$ for $\psi (0)\in H\setminus D(H_{\mathrm{o}\mathrm{p}})$ the curve is norm-differentiable at no $t\in \mathbb{R}$, so the equation is not asserted there, although the evolution $t\mapsto \psi (t)$ remains continuous.

(d) Density-operator evolution. For every preparation $\rho$ (The state space of the framework: preparations, pure rays and density operators) and every $t$ the operator $\rho (t)=U(t)\rho (0)U(t{)}^{\ast }$ is again a preparation: self-adjoint, positive, trace class with $\mathrm{tr}\rho (t)=1$ (Trace class operator, Self-adjoint, positive, unitary and normal operators). Moreover for every bounded operator $B$ the identity $\mathrm{tr}(\rho (t)B)=\mathrm{tr}(\rho (0)U(t{)}^{\ast }BU(t))$ holds.