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Energy conservation, conserved sharp observables and the stationary eigenrays

Physical scope

A conditional consequence of the autonomous-dynamics postulate P4 and the observable and probability postulate P2 in the reading of this page for a declared isolated system: the outcome law of the assigned energy observable is time-independent in every preparation; for a pure preparation represented by a nonzero vector in the energy domain, its finite first and second energy moments, evaluated on a unit representative, are conserved; every bounded assigned sharp observable compatible with the energy has a time-independent outcome law; and a pure preparation is stationary exactly when it is an eigenray of the energy, in which case all its outcome laws are time-independent. No relaxation, equilibration or decay is asserted; no converse of the compatibility criterion is claimed for unbounded energy operators; no measurement, apparatus or empirical claim is made.

Statement

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 (P2), 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 ( ${A C}_{ω}$ ), The Axiom of Choice). Let $(H_{o p}, u_{H})$ be the assigned energy observable of a declared isolated system with state space $H$ (The state space of the framework: preparations, pure rays and density operators), spectral projection-valued measure $E_{H}$ and flow $U (t) = \exp (- i t H_{o p} / ℏ)$ (Assigned energy observable, unitary flow and the two pictures of autonomous evolution), and write $ρ (t) = U (t) ρ (0) U (t)^{*}$ . Then:

(a) Energy law and moments. For every preparation $ρ$ , every Borel set $B \subseteq R$ and every $t$ , $μ_{ρ (t)}^{H_{o p}} (B) = tr (ρ (t) E_{H} (B)) = tr (ρ (0) E_{H} (B)) = μ_{ρ (0)}^{H_{o p}} (B);$ for the pure preparation of a unit vector $ψ$ likewise $∥ E_{H} (B) U (t) ψ ∥^{2} = ∥ E_{H} (B) ψ ∥^{2}$ . If $ψ (0) \neq 0$ and $ψ (0) \in D (H_{o p})$ , the mean and second moment of the energy law of the pure preparation $[ψ (0)]$ are finite and time-independent. For any nonzero representative $ψ (0)$ their time-zero values are $\frac{⟨ H_{o p} ψ (0), ψ (0) ⟩}{∥ ψ (0) ∥^{2}} and \frac{∥ H_{o p} ψ (0) ∥^{2}}{∥ ψ (0) ∥^{2}},$ respectively. The displayed conservation of moments makes no assertion that these moments are finite for preparations outside this domain hypothesis.

(b) Conserved sharp observables. Let $(A, u_{A})$ be a bounded assigned sharp observable that is compatible with the energy (Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (f)). Then its outcome law is time-independent: for every preparation $ρ$ , every Borel set $B$ and every $t$ , $μ_{ρ (t)}^{A} (B) = μ_{ρ (0)}^{A} (B) .$

(c) Stationary states are exactly the eigenrays. Let $ψ \in H$ , $ψ \neq 0$ . The pure preparation $[ψ]$ is stationary, that is $[ψ (t)] = [ψ (0)]$ for every $t$ , if and only if $ψ$ is an eigenvector of $H_{o p}$ : $ψ \in D (H_{o p})$ and $H_{o p} ψ = λ ψ$ for some $λ \in R$ . In that case $ψ (t) = e^{- i t λ / ℏ} ψ (0)$ for every $t$ , and every assigned sharp observable has a time-independent outcome law in this preparation: $μ_{[ψ (t)]}^{A} (B) = μ_{[ψ (0)]}^{A} (B)$ for every $(A, u_{A})$ , every Borel $B$ and every $t$ .

(d) Status and exclusions. No equilibration, relaxation, decay or convergence of the law of an arbitrary preparation is asserted; no moment beyond those finite at time zero is claimed; the converse of (b) — constancy of the law of a bounded observable in every preparation implying strong spectral compatibility with the energy — is not asserted for a general (possibly unbounded) $H_{o p}$ ; and the item is conditional on the isolated-system idealization of P4, asserting no apparatus realization and no empirical result.

Facts & Assumptions

Given: The adopted postulates P1, P2 and P4; the Axiom of Countable Choice and the Axiom of Choice; the assigned energy observable $(H_{o p}, u_{H})$ with PVM $E_{H}$ and flow $U (t)$ ; a preparation $ρ$ ; a bounded assigned sharp observable $(A, u_{A})$ ; a nonzero vector $ψ$ ; a Borel set $B$ ; real $t$ .

[F1]

Flow and preparation evolution. $U (t) = \exp (- i t H_{o p} / ℏ)$ is the strongly continuous unitary flow of P4(ii), $ρ (t) = U (t) ρ (0) U (t)^{*}$ is a preparation for every $t$ , and $[ψ (t)]$ is a pure preparation whenever $ψ (0) \neq 0$ (The Schroedinger-picture flow: unitarity, domain invariance and the domain-qualified equation clauses (a) and (d), Assigned energy observable, unitary flow and the two pictures of autonomous evolution clauses (b) and (c)).

[F2]

Conserved spectral laws. Let $K$ be self-adjoint with PVM $E_{K}$ and $U (t) = e^{- i t K / ℏ}$ . Then: for every preparation $ρ$ the energy law $tr (ρ U (t)^{*} E_{K} (C) U (t)) = tr (ρ E_{K} (C))$ is independent of $t$ ; if $ψ \in D (K)$ the scalar moments $\int λ^{k} d E_{K, U (t) ψ}$ for $k = 1, 2$ equal those at time zero, with values $⟨ K ψ, ψ ⟩$ and $∥ K ψ ∥^{2}$ ; if a bounded operator $B_{0}$ commutes with every $E_{K} (C)$ then $B_{0} U (t) = U (t) B_{0}$ ; if a PVM $F$ commutes with $E_{K}$ in the strong spectral sense then $tr (U (t) ρ U (t)^{*} F (S)) = tr (ρ F (S))$ for all $t, S$ ; and a unit eigenvector $K ψ = λ ψ$ satisfies $U (t) ψ = e^{- i t λ / ℏ} ψ$ (Conserved spectral probability laws for commuting observables and stationary eigenrays clauses (a), (b)(ii), (c) and (d)).

[F3]

Outcome law and moments. For an assigned sharp observable $(A_{0}, u_{A_{0}})$ the outcome law of a preparation is $μ_{ρ}^{A_{0}} (B) = tr (ρ E_{A_{0}} (B))$ , for a unit pure preparation $∥ E_{A_{0}} (B) ψ ∥^{2}$ , and this number depends only on the ray $[ψ]$ ; for a unit vector in $D (A_{0})$ the mean and the second moment are $⟨ A_{0} ψ, ψ ⟩$ and $∥ A_{0} ψ ∥^{2}$ , and they exist exactly under the stated domain hypotheses (Assigned sharp observables and the outcome law of the framework clauses (b) and (e)).

[F4]

Stationary-ray characterization. For a self-adjoint $T$ with flow $U (t) = e^{i t T}$ and a nonzero $ψ$ , the invariance of the ray, $U (t) ψ \in C ψ$ for all $t$ , is equivalent to $ψ \in D (T)$ and $T ψ = λ ψ$ for some real $λ$ ; and then $U (t) ψ = e^{i t λ} ψ$ (Stationary rays of a self-adjoint flow are exactly the eigenrays).

[F5]

The energy convention. $U (t) = \exp (- i t H_{o p} / ℏ)$ equals $e^{i t T}$ for the self-adjoint $T = - H_{o p} / ℏ$ with $D (T) = D (H_{o p})$ , and the PVM of $K = H_{o p}$ is $E_{H}$ (Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b), Integral of a measurable function against a projection-valued measure, Spectral theorem for unbounded self-adjoint operators (PVM form)).

Proof

technique · direct; read the energy law, moments, compatible-observable law and eigenray evolution from the conserved-spectral-law theorem with $K=H_{\mathrm{op}}$, whose convention $U(t)=e^{-itK/\hbar}$ is exactly the framework's, and prove the converse of the stationarity characterization with the stationary-ray lemma

Given: The data of the statement and [F1]-[F5].

1.1givenF1F2F3F5algebra

[Energy law] Instantiate [F2] with $K = H_{o p}$ and its PVM $E_{K} = E_{H}$ ; this is the same flow as [F1] by [F5]. Its first statement gives $tr (ρ (0) E_{H} (B)) = tr (U (t) ρ (0) U (t)^{*} E_{H} (B)) = tr (ρ (t) E_{H} (B))$ for every preparation and every Borel $B$ , which is the first display of (a) after reading the outcome law through [F3]; the pure-vector form follows from [F3] with $[ψ (t)]$ and the identity $U (t)^{*} E_{H} (B) U (t) = E_{H} (B)$ , or directly from the second statement of [F2] with $F = E_{H}$ .

1.2givenF2F3algebra

[Eigenrays are stationary] Let $H_{o p} ψ = λ ψ$ with $λ \in R$ and $ψ \neq 0$ , and set $u : = ψ / ∥ ψ ∥$ . Then $u$ is a unit eigenvector with the same eigenvalue. The last statement of [F2] with $K = H_{o p}$ gives $U (t) u = e^{- i t λ / ℏ} u$ for every $t$ ; multiplying by $∥ ψ ∥$ gives $ψ (t) = U (t) ψ = e^{- i t λ / ℏ} ψ$ . Since the scalar is nonzero, $[ψ (t)] = [ψ (0)]$ and $[ψ]$ is stationary. Moreover, for every assigned sharp observable $(A_{0}, u_{A_{0}})$ , its law $μ_{[ψ (t)]}^{A_{0}} (B)$ depends only on the ray $[ψ (t)] = [ψ (0)]$ by [F3], so $μ_{[ψ (t)]}^{A_{0}} (B) = μ_{[ψ (0)]}^{A_{0}} (B)$ ; this is the forward direction and the final assertion of (c).

2.1givenstep 1.1F2F3F5algebra

[Moments] Suppose $ψ (0) \neq 0$ and $ψ (0) \in D (H_{o p}) = D (K)$ , and set $u_{0} : = ψ (0) / ∥ ψ (0) ∥$ . Then $u_{0}$ is a unit vector in $D (K)$ . Apply the moment statement of [F2] with $k = 1, 2$ to $u_{0}$ ; its scalar-measure moments are time-independent and their time-zero values are $⟨ K u_{0}, u_{0} ⟩ = ⟨ H_{o p} ψ (0), ψ (0) ⟩ / ∥ ψ (0) ∥^{2}$ and $∥ K u_{0} ∥^{2} = ∥ H_{o p} ψ (0) ∥^{2} / ∥ ψ (0) ∥^{2}$ . By [F3], these are the mean and second moment of the energy outcome law of the ray $[ψ (0)]$ ; by [F2] and [F3] the law is unchanged for every $t$ , so both moments are finite and time-independent. This proves the moment assertion of (a).

2.2givenstep 1.1F2F3F5algebra

[Compatible bounded observables] Let $(A, u_{A})$ be bounded and energy-compatible, so $F : = E_{A}$ satisfies $F (S) E_{H} (C) = E_{H} (C) F (S)$ for all Borel $S, C$ . The third statement of [F2], with $K = H_{o p}$ , gives $tr (ρ (t) F (S)) = tr (ρ (0) F (S))$ for every $t$ and every Borel $S$ ; reading $tr (ρ F (S))$ through [F3] as $μ_{ρ}^{A} (S)$ gives the display of (b).

2.3givenstep 1.2F4F5algebra

[Only eigenrays are stationary] Conversely, let $[ψ]$ be stationary, that is $U (t) ψ \in C ψ$ for every $t$ , with $ψ \neq 0$ . By [F5], $U (t) = e^{i t T}$ with the self-adjoint $T = - H_{o p} / ℏ$ , $D (T) = D (H_{o p})$ . Applying [F4] to $T$ and $ψ$ gives $ψ \in D (T) = D (H_{o p})$ and $T ψ = λ^{'} ψ$ with $λ^{'} \in R$ ; multiplying by $- ℏ$ gives $H_{o p} ψ = λ ψ$ with $λ = - ℏ λ^{'} \in R$ . Together with step 1.2 this proves (c).

3.1step 1.1step 2.1step 2.2step 1.2step 2.3∎

Steps 1.1, 2.1, 2.2, 1.2 and 2.3 prove (a)-(c); clause (d) records that the statement asserts neither equilibration nor the converse of (b) for unbounded $H_{o p}$ .

Remarks

What "conservation of energy" means here. The conserved object is the outcome law of the assigned energy observable — a probability measure on the Borel sets — and, when they exist, its first two moments. The statement does not say that a single numerical value of the energy is revealed at each time, and it does not identify $⟨ H_{o p} ⟩$ with a measured calorimetric energy.
Stationarity is a ray statement. Because the pure law depends only on the ray, the stationary property of an eigenray is stable under the phase $e^{- i t λ / ℏ}$ ; a superposed preparation of two eigenrays with distinct eigenvalues is generally not stationary, and that failure is exhibited on the companion page.

Energy conservation, conserved sharp observables and the stationary eigenrays

Statement

Facts & Assumptions

Proof

Remarks

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