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Qm Autonomous Dynamics — Examples

1 · Prerequisites

2 · Summary

The companion page to qm-autonomous-dynamics collects finite worked computations of the framework's isolated evolution and one refutation of an informal claim about superpositions. Precession of a two-level preparation: exact phases, probabilities and a Bloch-vector rotation computes, on the two-level state space, the spectral measure and flow of the assigned energy observable $E_{1} T_{u_{1}, u_{1}} + E_{2} T_{u_{2}, u_{2}}$ with $E_{1} \neq E_{2}$ , the exact phases of the evolved preparation, the outcome law and expectations of the bounded observables $X, Y, Z$ , and the rotation of the expectation-value triple about the third axis with angular frequency $ω = (E_{2} - E_{1}) / ℏ$ . A finite superposition of energy eigenrays: exact phases, laws and return times computes the evolution of a finite superposition of orthogonal energy eigenrays, the trigonometric-polynomial form of the outcome laws of a bounded assigned sharp observable, the exact return-time criterion $(λ_{m} - λ_{n}) T / ℏ \in 2 π Z$ , the existence criterion for a positive return time, and the two-energy least period $2 π ℏ / ∣ λ_{1} - λ_{2} ∣$ . The counterexample A superposition of distinct energy eigenrays need not be stationary refutes the universal claim that every superposition of distinct energy eigenrays is stationary: in the equal-weight two-level superposition the outcome law of the flip observable takes the value $1$ at $t = 0$ and $0$ at $t = π / ∣ ω ∣$ , while the individual eigenrays remain stationary. The scope remark What the worked items of the companion page establish records that none of the items is an experiment: no counts, calibration, resolution, uncertainty or data comparison is asserted anywhere on the page.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passOpen item page →

Precession of a two-level preparation: exact phases, probabilities and a Bloch-vector rotation

Physical scope

A finite, exact model computation in the two-level state space of the framework: the assigned energy observable with two distinct energies in joules, its unitary flow, the exact evolution of a pure preparation, the outcome laws and expectations of three bounded self-adjoint observables, the resulting rotation of the expectation-value triple, and the explicit period of the laws. The energies are model parameters; the item asserts no apparatus realization, no detection model and no empirical value, and it identifies the two-level system with no specific physical species.

Example

Let $H = C^{2}$ carry the standard inner product $⟨ z, w ⟩ = z_{1} \overline{w_{1}} + z_{2} \overline{w_{2}}$ (Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space, The induced length is a norm), with orthonormal basis $u_{1} = (1, 0)$ , $u_{2} = (0, 1)$ and rank-one calculus $T_{u, v} x = ⟨ x, v ⟩ u$ (Rank-one operator calculus). Fix two distinct energies $E_{1} \neq E_{2}$ in joules (model parameters, not measured values), and consider the assigned energy observable of Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (a), $H_{o p} : = E_{1} T_{u_{1}, u_{1}} + E_{2} T_{u_{2}, u_{2}}, u_{H} = J,$ of an isolated system whose state space is the two-level state space of The state space of the framework: preparations, pure rays and density operators. Define the bounded self-adjoint observables $X : = T_{u_{1}, u_{2}} + T_{u_{2}, u_{1}}, Y : = i T_{u_{2}, u_{1}} - i T_{u_{1}, u_{2}}, Z : = T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}},$ with the same dimensionless unit, and let $ψ_{0} = α u_{1} + β u_{2}$ be a unit vector. Writing $ω : = \frac{E_{2} - E_{1}}{ℏ} (units s^{- 1}),$ the example establishes the following exact model statements.

(i) The spectral projection-valued measure of $H_{o p}$ is $E_{H} (B) = 1_{E_{1} \in B} T_{u_{1}, u_{1}} + 1_{E_{2} \in B} T_{u_{2}, u_{2}}$ , and the flow is $U (t) = e^{- i E_{1} t / ℏ} T_{u_{1}, u_{1}} + e^{- i E_{2} t / ℏ} T_{u_{2}, u_{2}}$ .

(ii) $X, Y, Z$ are bounded self-adjoint with $X^{2} = Y^{2} = Z^{2} = I$ and $X Y = i Z$ ; their spectra are ${\pm 1}$ , with $Y$ and $Z$ obtained from $X$ by the same pattern.

(iii) The evolved preparation is $ψ (t) = U (t) ψ_{0} = α e^{- i E_{1} t / ℏ} u_{1} + β e^{- i E_{2} t / ℏ} u_{2}$ , of unit norm, and the expectations are $⟨ X ⟩ (t) = 2 Re (α \overline{β} e^{i ω t}), ⟨ Y ⟩ (t) = - 2 Im (α \overline{β} e^{i ω t}), ⟨ Z ⟩ (t) = ∣ α ∣^{2} - ∣ β ∣^{2} .$

(iv) The triple $v (t) : = (⟨ X ⟩ (t), ⟨ Y ⟩ (t), ⟨ Z ⟩ (t)) \in R^{3}$ has constant norm one and is the rotation of $v (0)$ about the third axis by the angle $- ω t$ : $v (t) = R_{3} (- ω t) v (0)$ , where $R_{3} (θ) (x, y, z) = (x \cos θ - y \sin θ, x \sin θ + y \cos θ, z)$ . Thus the expectation-value triple precesses with angular frequency $ω$ .

(v) The outcome law of $X$ in the preparation $[ψ (t)]$ is the two-point law $μ_{ψ (t)}^{X} ({+ 1}) = (1 + ⟨ X ⟩ (t)) / 2$ , $μ_{ψ (t)}^{X} ({- 1}) = (1 - ⟨ X ⟩ (t)) / 2$ ; in particular for $α = β = 1 / \sqrt{2}$ one has $μ_{ψ (t)}^{X} ({+ 1}) = (1 + \cos ω t) / 2$ .

(vi) The energy law is time-independent, $μ_{ψ (t)}^{H_{o p}} ({E_{1}}) = ∣ α ∣^{2}$ and $μ_{ψ (t)}^{H_{o p}} ({E_{2}}) = ∣ β ∣^{2}$ for every $t$ , and every one of the displayed expectations is periodic in $t$ with period $2 π / ∣ ω ∣$ (when $ω \neq 0$ , which holds because $E_{1} \neq E_{2}$ ).

The computation is a model-internal one: the energies are parameters of the assigned observable, the probabilities are those of the framework's outcome law, and no laboratory measurement, apparatus or empirical value is asserted.

Facts & Assumptions

Given: The Hilbert space $H = C^{2}$ with the standard inner product and orthonormal basis $(u_{1}, u_{2})$ ; distinct real energies $E_{1} \neq E_{2}$ in joules, and the reduced Planck constant $ℏ$ with its value and unit, in the framework's quantity conventions (Dimensions, SI unit conventions and quantity types of the framework); the assigned energy observable $H_{o p} = E_{1} T_{u_{1}, u_{1}} + E_{2} T_{u_{2}, u_{2}}$ with unit $J$ ; the observables $X, Y, Z$ ; a unit vector $ψ_{0} = α u_{1} + β u_{2}$ ; the number $ω = (E_{2} - E_{1}) / ℏ$ ; the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), the choice hypotheses of the cited spectral suppliers.

[F1]

Rank-one calculus and one-dimensional projections. $T_{u, v} x = ⟨ x, v ⟩ u$ is bounded with $T_{u, v}^{*} = T_{v, u}$ , and the assignment is linear in $u$ and conjugate-linear in $v$ ; for every nonzero $ψ$ the operator $T_{ψ, ψ}$ is self-adjoint positive trace class, and for every unit $u$ the operator $T_{u, u}$ is the Hilbert orthogonal projection onto the one-dimensional subspace $C u$ , so that $T_{u, u}$ is idempotent and $T_{u, u} ψ = ⟨ ψ, u ⟩ u$ (Rank-one operator calculus, Rays, unit vectors and rank-one Hilbert projections clause (b)).

[F2]

Finite spectral structure. For the bounded self-adjoint $H_{o p}$ with orthonormal eigenvectors $u_{1}, u_{2}$ at the distinct eigenvalues $E_{1}, E_{2}$ , the assignment $E^{'} (B) : = 1_{E_{1} \in B} T_{u_{1}, u_{1}} + 1_{E_{2} \in B} T_{u_{2}, u_{2}}$ takes orthogonal-projection values by [F1], satisfies $E^{'} (R) = I$ by [F7], and is additive and multiplicative over the two-point eigenvalue set, so it is a projection-valued measure: for a disjoint sequence each of its two atoms occurs in at most one member, giving strong countable additivity. Its scalar measures are finite sums of point masses and hence regular; its integral domain is all of $H$ , since $\int λ^{2} d E_{x}^{'} (λ) = \sum_{k = 1}^{2} E_{k}^{2} ∣ ⟨ x, u_{k} ⟩ ∣^{2} < \infty$ . Thus $\int λ d E^{'} (λ) = E_{1} T_{u_{1}, u_{1}} + E_{2} T_{u_{2}, u_{2}} = H_{o p}$ ; by the uniqueness of the spectral measure it is $E_{H}$ (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely, Spectral theorem for unbounded self-adjoint operators (PVM form), Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (a)).

[F7]

Resolution of the identity. Since $(u_{1}, u_{2})$ is an orthonormal basis of the two-dimensional space $H$ , every $v \in H$ satisfies $v = ⟨ v, u_{1} ⟩ u_{1} + ⟨ v, u_{2} ⟩ u_{2}$ , and therefore $T_{u_{1}, u_{1}} + T_{u_{2}, u_{2}} = I$ (Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis).

[F3]

Bounded calculus for the two-point measure. For bounded Borel $f : R \to C$ , put $g = f (E_{1}) 1_{{E_{1}}} + f (E_{2}) 1_{{E_{2}}}$ . The simple integral is $Φ_{E_{H}} (g) = f (E_{1}) T_{u_{1}, u_{1}} + f (E_{2}) T_{u_{2}, u_{2}}$ . By [F2], $E_{H} (R ∖ {E_{1}, E_{2}}) = 0$ , so $g = f$ almost everywhere for every scalar measure $B \mapsto ⟨ E_{H} (B) x, x ⟩$ , $x \in H$ . The constant sequence $g_{n} = g$ is bounded uniformly by $∥ f ∥_{\infty}$ and converges to $f$ in that almost-everywhere sense; clause (4) of Pvm integral is a star homomorphism therefore gives $Φ_{E_{H}} (f) = Φ_{E_{H}} (g)$ , proving the formula without uniform approximation on $R$ .

[F4]

Flow and state evolution. The flow is $U (t) = \exp (- i t H_{o p} / ℏ) = Φ_{E_{H}} (e^{- i t λ / ℏ})$ and the preparation evolves as $ψ (t) = U (t) ψ_{0}$ , with $∥ ψ (t) ∥ = ∥ ψ_{0} ∥ = 1$ ; the energy law is time-independent (The Schroedinger-picture flow: unitarity, domain invariance and the domain-qualified equation clauses (a) and (d), Energy conservation, conserved sharp observables and the stationary eigenrays clause (a), Assigned energy observable, unitary flow and the two pictures of autonomous evolution clauses (b) and (c)).

[F5]

Outcome law. For an assigned sharp observable $(A_{0}, u_{A_{0}})$ the law of the pure preparation of a unit vector $ψ$ is $μ_{ψ}^{A_{0}} (B) = ∥ E_{A_{0}} (B) ψ ∥^{2}$ , and the mean for a unit vector in $D (A_{0})$ is $⟨ A_{0} ψ, ψ ⟩$ ; the law depends only on the ray (Assigned sharp observables and the outcome law of the framework clauses (b) and (e), State vectors, unit vectors, rays, normalization and phases of a Hilbert space).

[F6]

Complex exponential algebra. For real $θ, ϕ$ : $e^{i θ} = \cos θ + i \sin θ$ , $e^{i (θ + ϕ)} = e^{i θ} e^{i ϕ}$ , $∣ e^{i θ} ∣ = 1$ and $\overline{e^{i θ}} = e^{- i θ}$ ; for complex $z$ , $2 Re z = z + \overline{z}$ and $2 i Im z = z - \overline{z}$ (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ , $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive). Moreover $\ker (\exp) = 2 π i Z$ , so for real $θ$ one has $e^{i θ} = 1$ exactly for $θ \in 2 π Z$ ( $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ ).

Verification

technique · direct; compute the spectral measure, the flow, the algebra and the laws of the explicit two-dimensional model

1.1givenF1F2F3F7algebra

[Spectral measure and flow] $H_{o p}$ is bounded and self-adjoint because $T_{u_{1}, u_{1}}, T_{u_{2}, u_{2}}$ are self-adjoint projections by [F1] and $E_{1}, E_{2} \in R$ , and $H_{o p} u_{k} = E_{k} u_{k}$ ; on the two-point spectrum the assignment $E^{'}$ of [F2] has orthogonal-projection values with $E^{'} (R) = I$ by [F7] and is additive and multiplicative, so [F2] identifies $E_{H}$ , and [F3] with $f (λ) = e^{- i t λ / ℏ}$ gives $U (t) = \sum_{k} e^{- i E_{k} t / ℏ} T_{u_{k}, u_{k}}$ . This is (i).

1.2givenF1F7algebra

[The three observables] By [F1] each of $X = T_{u_{1}, u_{2}} + T_{u_{2}, u_{1}}$ , $Y = i T_{u_{2}, u_{1}} - i T_{u_{1}, u_{2}}$ , $Z = T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}}$ is bounded and self-adjoint. Composing rank-one operators from their definition, $T_{u, v} T_{w, z} x = ⟨ ⟨ x, z ⟩ w, v ⟩ u = ⟨ w, v ⟩ ⟨ x, z ⟩ u$ , so with orthonormality $T_{u_{1}, u_{2}} T_{u_{2}, u_{1}} = T_{u_{1}, u_{1}}$ , $T_{u_{2}, u_{1}} T_{u_{1}, u_{2}} = T_{u_{2}, u_{2}}$ and all products of two distinct rank-one terms $T_{u_{j}, u_{k}} T_{u_{l}, u_{m}}$ with $k \neq l$ vanish; hence $X^{2} = T_{u_{1}, u_{1}} + T_{u_{2}, u_{2}} = I$ by [F7], $Y^{2} = I$ , $Z^{2} = I$ and $X Y = i (T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}}) = i Z$ . The eigenvalues of $X$ are $\pm 1$ with eigenvectors $e_{\pm} = (u_{1} \pm u_{2}) / \sqrt{2}$ . For $Y$ , the vectors $f_{\pm} = (u_{1} \pm i u_{2}) / \sqrt{2}$ satisfy $Y f_{\pm} = \pm f_{\pm}$ ; for $Z$ , one has $Z u_{1} = u_{1}$ and $Z u_{2} = - u_{2}$ . Since each of $X, Y, Z$ is self-adjoint and squares to $I$ , its spectrum is contained in ${- 1, + 1}$ , and these explicit eigenvectors show that both values occur for each operator. This proves (ii).

2.1givenstep 1.1F4F6algebra

[Evolved preparation] By [F4] and step 1.1, $ψ (t) = \sum_{k} c_{k} e^{- i E_{k} t / ℏ} u_{k}$ with $c_{1} = α$ , $c_{2} = β$ ; all coefficients have modulus $∣ c_{k} ∣$ because $∣ e^{- i E_{k} t / ℏ} ∣ = 1$ by [F6], so $∥ ψ (t) ∥^{2} = ∣ α ∣^{2} + ∣ β ∣^{2} = 1$ . This is the first assertion of (iii).

3.1givenstep 1.2step 2.1F5F6algebra

[Expectations] With $θ_{k} : = E_{k} t / ℏ$ and $ψ (t) = α e^{- i θ_{1}} u_{1} + β e^{- i θ_{2}} u_{2}$ , the actions $X u_{1} = u_{2}$ , $X u_{2} = u_{1}$ , $Y u_{1} = i u_{2}$ , $Y u_{2} = - i u_{1}$ , $Z u_{k} = \pm u_{k}$ from step 1.2 give $⟨ X ψ (t), ψ (t) ⟩ = α \overline{β} e^{i (θ_{2} - θ_{1})} + \overline{α \overline{β} e^{i (θ_{2} - θ_{1})}} = 2 Re (α \overline{β} e^{i ω t})$ and $⟨ Y ψ (t), ψ (t) ⟩ = i α \overline{β} e^{i (θ_{2} - θ_{1})} - i \overline{α \overline{β} e^{i (θ_{2} - θ_{1})}} = - 2 Im (α \overline{β} e^{i ω t})$ and $⟨ Z ψ (t), ψ (t) ⟩ = ∣ α ∣^{2} - ∣ β ∣^{2}$ , using $i (w - \overline{w}) = - 2 Im w$ for $w = α \overline{β} e^{i ω t}$ and [F6]; by [F5] these are the means of $X, Y, Z$ in $[ψ (t)]$ , which is (iii).

4.1givenstep 3.1F6algebra

[Precession of the triple] Write $c : = α \overline{β} = a + i b$ with $a, b \in R$ and $z : = ∣ α ∣^{2} - ∣ β ∣^{2}$ . By [F6], $c e^{i ω t} = (a \cos ω t - b \sin ω t) + i (a \sin ω t + b \cos ω t)$ , so step 3.1 gives $v (t) = (2 a \cos ω t - 2 b \sin ω t, - 2 a \sin ω t - 2 b \cos ω t, z)$ . This equals $R_{3} (- ω t) v (0)$ for $v (0) = (2 a, - 2 b, z)$ by the displayed definition of $R_{3}$ . Its squared Euclidean norm is $∥ v (t) ∥^{2} = 4 ∣ c e^{i ω t} ∣^{2} + z^{2} = 4 ∣ α ∣^{2} ∣ β ∣^{2} + (∣ α ∣^{2} - ∣ β ∣^{2})^{2} = (∣ α ∣^{2} + ∣ β ∣^{2})^{2} = 1$ by [F6]; taking the nonnegative square root gives $∥ v (t) ∥ = 1$ . This proves (iv), including $c = 0$ .

4.2givenstep 1.2step 3.1F1F5F6algebra

[Law of the flip observable] By step 1.2, $X = Q_{+} - Q_{-}$ with the orthogonal projections $Q_{\pm} = T_{e_{\pm}, e_{\pm}}$ onto the eigenspaces of the eigenvalues $\pm 1$ , so $E_{X} ({+ 1}) = Q_{+}$ and, by [F1], $Q_{+} ψ = ⟨ ψ, e_{+} ⟩ e_{+} = \frac{1}{\sqrt{2}} (⟨ ψ, u_{1} ⟩ + ⟨ ψ, u_{2} ⟩) e_{+}$ . For $ψ = ψ (t)$ this gives $∥ Q_{+} ψ (t) ∥^{2} = \frac{1}{2} ∣ α e^{- i θ_{1}} + β e^{- i θ_{2}} ∣^{2} = \frac{1}{2} (1 + ⟨ X ⟩ (t))$ by [F6] and step 3.1, and by [F5] this is $μ_{ψ (t)}^{X} ({+ 1})$ ; the minus case is the complement, and for $α = β = 1 / \sqrt{2}$ the value is $(1 + \cos ω t) / 2$ . This is (v).

4.3givenstep 1.1step 3.1F4F6algebra

[Energy law and periodicity] By [F4], $μ_{ψ (t)}^{H_{o p}} ({E_{k}}) = ∥ E_{H} ({E_{k}}) ψ (t) ∥^{2} = ∥ c_{k} e^{- i E_{k} t / ℏ} u_{k} ∥^{2} = ∣ c_{k} ∣^{2}$ using step 1.1 and [F6], so the two-point energy law is time-independent; the displayed expectations and probabilities are functions of $e^{i ω t}$ , and $e^{i ω (t + 2 π / ∣ ω ∣)} = e^{i ω t}$ because $\pm 2 π i \in \ker (\exp)$ by [F6], so they are periodic with period $2 π / ∣ ω ∣$ ; here $ω \neq 0$ because $E_{1} \neq E_{2}$ . This is (vi).

5.1step 1.1step 1.2step 2.1step 3.1step 4.1step 4.2step 4.3∎

Steps 1.1, 1.2, 2.1, 3.1, 4.1, 4.2 and 4.3 establish (i)-(vi) for the explicit model data.

Remarks

"Precession" is a bookkeeping device. The rotation of $v (t)$ is a statement about three expectation values of three bounded observables of a two-level system; no rotating rigid body, magnetic moment or specific species is asserted. The frequency $ω = (E_{2} - E_{1}) / ℏ$ is fixed by the two energy parameters.
Relative versus global phase. The evolution multiplies the two components by different phases; the relative phase $e^{i ω t}$ is what moves the observables $X$ and $Y$ , while a common phase of the whole vector would change no outcome law. For $α = 0$ or $β = 0$ the preparation is an eigenray and the triple is constant, in agreement with the stationary-state theorem.
No detector is modeled. The probabilities displayed are model outcome laws for the declared observables; no measurement interaction, resolution, efficiency or record is introduced.

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passOpen item page →

A finite superposition of energy eigenrays: exact phases, laws and return times

Physical scope

A finite, exact model computation for a declared isolated system: the evolution of a nonzero finite superposition of orthogonal energy eigenrays, the resulting trigonometric-polynomial form of the outcome laws of bounded assigned sharp observables, the exact return-time criterion for the ray, the two-energy period, and the stationary case. The eigenvalues are model parameters; the item asserts no apparatus, no detection model and no empirical value, and no finite set of energies is identified with a physical spectrum.

Example

Let $H \neq {0}$ be the state space of a declared isolated system of the framework (The state space of the framework: preparations, pure rays and density operators), let $(H_{o p}, u_{H})$ be its assigned energy observable with 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 let $u_{1}, \dots, u_{N}$ be a finite orthonormal family of eigenvectors of $H_{o p}$ with real eigenvalues $λ_{1}, \dots, λ_{N}$ : $H_{o p} u_{n} = λ_{n} u_{n}$ for $n = 1, \dots, N$ . Fix coefficients $c_{n} \in C$ and the nonzero vector $ψ_{0} = \sum_{n = 1}^{N} c_{n} u_{n}, S : = {n : c_{n} \neq 0} .$ For a nonzero vector $ϕ$ , write $μ_{[ϕ]}^{A}$ for the outcome law of the pure preparation $[ϕ]$ , computed on the unit representative $ϕ / ∥ ϕ ∥$ as in Assigned sharp observables and the outcome law of the framework clause (b) and State vectors, unit vectors, rays, normalization and phases of a Hilbert space. Then, with $ψ (t) = U (t) ψ_{0}$ :

(i) $ψ (t) = \sum_{n \in S} c_{n} e^{- i λ_{n} t / ℏ} u_{n}$ for every $t$ , and $∥ ψ (t) ∥ = ∥ ψ_{0} ∥$ ; if $S$ has one element, or more generally if all $λ_{n}$ with $n \in S$ coincide, the ray $[ψ (t)]$ is constant.

(ii) For every bounded assigned sharp observable $(A, u_{A})$ and every Borel set $B \subseteq R$ , the outcome law in the preparation $[ψ (t)]$ is $μ_{[ψ (t)]}^{A} (B) = \frac{1}{∥ ψ_{0} ∥^{2}} \sum_{m, n \in S} c_{n} \overline{c_{m}} e^{- i (λ_{n} - λ_{m}) t / ℏ} ⟨ E_{A} (B) u_{n}, u_{m} ⟩ .$ Thus the probability is a finite trigonometric polynomial in $t$ with frequencies $(λ_{n} - λ_{m}) / ℏ$ .

(iii) For $T > 0$ the following are equivalent: (a) $[ψ (t + T)] = [ψ (t)]$ for every $t$ ; (b) $e^{- i λ_{n} T / ℏ}$ is independent of $n \in S$ ; (c) $(λ_{m} - λ_{n}) T / ℏ \in 2 π Z$ for all $m, n \in S$ .

(iv) Consequently a positive return time exists if and only if there are $ω_{0} > 0$ and integers $k_{n}$ ( $n \in S$ ) with $(λ_{n} - λ_{n_{0}}) / ℏ = ω_{0} k_{n}$ for all $n \in S$ , where $n_{0}$ is any fixed element of $S$ ; and then $T = 2 π / ω_{0}$ is a return time.

(v) Two-energy case $N = 2$ , $λ_{1} \neq λ_{2}$ , $c_{1} c_{2} \neq 0$ : the positive return times are exactly $T = 2 π ℏ k / ∣ λ_{1} - λ_{2} ∣$ , $k = 1, 2, 3, \dots$ ; the least return time is $2 π ℏ / ∣ λ_{1} - λ_{2} ∣$ , which is the period $2 π / ∣ ω ∣$ of the precession example with $ω = (λ_{2} - λ_{1}) / ℏ$ .

(vi) In the stationary case of (i), every outcome law of every assigned sharp observable is independent of $t$ : $μ_{[ψ (t)]}^{A} (B) = μ_{[ψ (0)]}^{A} (B)$ .

The computation is a model-internal one: the eigenvalues are parameters of the assigned observable, and no identification with a physical spectrum, apparatus or measured value is asserted.

Facts & Assumptions

Given: A nonzero complex Hilbert space $H$ ; the assigned energy observable $(H_{o p}, u_{H})$ with flow $U (t)$ ; a finite orthonormal family $u_{1}, \dots, u_{N}$ of eigenvectors with real eigenvalues $λ_{n}$ ; coefficients $c_{n}$ and the nonzero vector $ψ_{0}$ ; $S = {n : c_{n} \neq 0}$ ; the evolved vectors $ψ (t) = U (t) ψ_{0}$ ; a bounded assigned sharp observable $(A, u_{A})$ with spectral measure $E_{A}$ ; a Borel set $B$ ; the Axiom of Countable Choice and the Axiom of Choice, the choice hypotheses of the cited spectral suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice).

[F1]

Eigenrays evolve by phases. If $H_{o p} u = λ u$ with $λ \in R$ , then $U (t) u = e^{- i λ t / ℏ} u$ for every $t$ , and the flow is linear with $∥ ψ (t) ∥ = ∥ ψ_{0} ∥$ (Energy conservation, conserved sharp observables and the stationary eigenrays clause (c), The Schroedinger-picture flow: unitarity, domain invariance and the domain-qualified equation clause (a), Conserved spectral probability laws for commuting observables and stationary eigenrays clause (d)).

[F2]

Outcome law for a pure preparation. $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2} = ⟨ E_{A} (B) ψ, ψ ⟩$ for a unit vector, and the number depends only on the ray $[ψ]$ ; the inner product is linear in the first argument and conjugate-linear in the second (Assigned sharp observables and the outcome law of the framework clause (b), State vectors, unit vectors, rays, normalization and phases of a Hilbert space, Real and complex inner product spaces, with the inner product linear in the first argument).

[F3]

Complex exponential algebra. For real $θ, ϕ$ : $e^{i (θ + ϕ)} = e^{i θ} e^{i ϕ}$ , $∣ e^{i θ} ∣ = 1$ and $\overline{e^{i θ}} = e^{- i θ}$ ; for complex $z, w$ , $\overline{z w} = \overline{z} \overline{w}$ (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ , $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive). Moreover $\ker (\exp) = 2 π i Z$ , so for real $μ$ one has $e^{- i μ} = 1$ exactly for $μ \in 2 π Z$ ( $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ ).

[F4]

Orthonormality. $⟨ u_{n}, u_{m} ⟩ = δ_{n m}$ , and every $ψ_{0}$ as above is nonzero exactly when $S$ is nonempty (Real and complex inner product spaces, with the inner product linear in the first argument, The induced length is a norm).

Verification

technique · direct; expand the evolved vector along the eigenfamily, evaluate the pairing term by term and read the return-time condition off the phases

1.1givenF1F3algebra

[Evolution of the superposition] By [F1] applied to each eigenvector, $U (t) u_{n} = e^{- i λ_{n} t / ℏ} u_{n}$ ; linearity of $U (t)$ gives $ψ (t) = \sum_{n = 1}^{N} c_{n} e^{- i λ_{n} t / ℏ} u_{n} = \sum_{n \in S} c_{n} e^{- i λ_{n} t / ℏ} u_{n}$ , and $∥ ψ (t) ∥ = ∥ ψ_{0} ∥$ by the norm statement of [F1]. If all $λ_{n}$ with $n \in S$ equal a common $λ$ , then $ψ (t) = e^{- i λ t / ℏ} ψ_{0}$ , so $[ψ (t)] = [ψ_{0}]$ for every $t$ . This is (i).

2.1givenstep 1.1F2F3F4algebra

[Trigonometric-polynomial law] Assume first $∥ ψ_{0} ∥ = 1$ . By [F2] and step 1.1, using conjugate-linearity in the second argument, $μ_{[ψ (t)]}^{A} (B) = ⟨ E_{A} (B) \sum_{n \in S} c_{n} e^{- i λ_{n} t / ℏ} u_{n}, \sum_{m \in S} c_{m} e^{- i λ_{m} t / ℏ} u_{m} ⟩ = \sum_{m, n \in S} c_{n} \overline{c_{m}} e^{- i (λ_{n} - λ_{m}) t / ℏ} ⟨ E_{A} (B) u_{n}, u_{m} ⟩$ . For general nonzero $ψ_{0}$ , the unit representative $ψ (t) / ∥ ψ_{0} ∥$ has coefficients $c_{n} / ∥ ψ_{0} ∥$ by step 1.1; substituting these into the unit-vector calculation divides the displayed sum by $∥ ψ_{0} ∥^{2}$ , as asserted.

2.2givenstep 1.1F2F3F4algebra

[Return-time criterion] Let $T > 0$ . If (a) holds, then for every $t$ there is $ζ (t) \neq 0$ with $ψ (t + T) = ζ (t) ψ (t)$ ; pairing with $u_{m}$ for $m \in S$ and using orthonormality [F4] gives $c_{m} e^{- i λ_{m} (t + T) / ℏ} = ζ (t) c_{m} e^{- i λ_{m} t / ℏ}$ , hence $e^{- i λ_{m} T / ℏ} = ζ (t)$ for every $m \in S$ , which is (b). If (b) holds with common value $ζ$ , then $ψ (t + T) = ζ ψ (t)$ by step 1.1, which is (a). Finally (b) holds exactly when $e^{- i (λ_{m} - λ_{n}) T / ℏ} = 1$ for all $m, n \in S$ , and by the exponential kernel $\ker (\exp) = 2 π i Z$ of [F3] this is exactly the integrality condition (c). This proves (iii).

3.1givenstep 2.2F3algebra

[Common frequency and return time] Condition (iv) is the statement that the numbers $(λ_{n} - λ_{n_{0}}) / ℏ$ are integer multiples of one positive number $ω_{0}$ ; by step 2.2 with $T = 2 π / ω_{0}$ and [F3], $(λ_{m} - λ_{n}) T / ℏ = 2 π (k_{m} - k_{n}) \in 2 π Z$ for all $m, n \in S$ , so $T$ is a return time. Conversely, if $T > 0$ is a return time, then by step 2.2(c) applied with $m = n_{0}$ we have $(λ_{n} - λ_{n_{0}}) T / ℏ \in 2 π Z$ for all $n \in S$ , that is $(λ_{n} - λ_{n_{0}}) / ℏ = ω_{0} k_{n}$ with $ω_{0} : = 2 π / T > 0$ and integers $k_{n}$ ; this proves the equivalence of (iv).

3.2givenstep 2.2algebra

[Two energies] Let $N = 2$ , $λ_{1} \neq λ_{2}$ , $c_{1} c_{2} \neq 0$ and $T > 0$ . By step 2.2(c) with ${m, n} = {1, 2}$ the return times are exactly the solutions of $∣ λ_{1} - λ_{2} ∣ T / ℏ \in 2 π Z$ , that is $T = 2 π ℏ k / ∣ λ_{1} - λ_{2} ∣$ with $k \in Z$ , $k \geq 1$ ; the least such $T$ is $2 π ℏ / ∣ λ_{1} - λ_{2} ∣ = 2 π / ∣ ω ∣$ for $ω = (λ_{2} - λ_{1}) / ℏ$ . This is (v).

3.3givenstep 1.1step 2.1F1F2algebra

[Stationary case] If all $λ_{n}$ , $n \in S$ , equal a common $λ$ , then by step 1.1 $ψ (t) = e^{- i λ t / ℏ} ψ_{0}$ and $[ψ (t)] = [ψ_{0}]$ ; by [F2] the outcome law of every bounded assigned sharp observable depends only on the ray, so $μ_{[ψ (t)]}^{A} (B) = μ_{[ψ (0)]}^{A} (B)$ for every $t$ . This is (vi), and it agrees with the stationary-state characterization of Energy conservation, conserved sharp observables and the stationary eigenrays clause (c).

4.1step 1.1step 2.1step 2.2step 3.1step 3.2step 3.3∎

Steps 1.1, 2.1, 2.2, 3.1, 3.2 and 3.3 establish (i)-(vi) for the explicit model data.

Remarks

The relative phases are physical. Only the relative phases $e^{- i (λ_{n} - λ_{m}) t / ℏ}$ appear in the laws; a common phase of $ψ (t)$ would not change any outcome law, in agreement with the ray reading of pure preparations.
Existence of a common frequency is a number-theoretic condition on the model data. For two distinct energies it always holds, and the least return time is the inverse Bohr frequency of the precession example. For three or more energies it may fail, in which case the ray never returns to itself, although the laws remain finite sums of oscillations with the Bohr frequencies.
No detector is modeled. Exactly as on the precession example, the probabilities displayed are model outcome laws; no measurement interaction, finite resolution, efficiency or record is introduced.

CounterexampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passOpen item page →

A superposition of distinct energy eigenrays need not be stationary

Physical scope

A refutation, inside the framework's autonomous-dynamics reading, of the informal claim that every superposition of two or more distinct energy eigenrays of an isolated system is a stationary state with time-independent outcome laws: in the explicit two-level model with two distinct energies, an equal-weight superposition carries the bounded flip observable with an outcome probability oscillating between one and zero. The item refutes that specific universal claim; it asserts no apparatus, no measurement statistics from a laboratory and no empirical result, and it does not deny that individual eigenrays are stationary or that special superpositions can be stationary.

Statement refuted

Adopt the autonomous-dynamics reading of the framework (Assigned energy observable, unitary flow and the two pictures of autonomous evolution, P4). Then: every preparation of an isolated system that is a superposition of two or more energy eigenrays with distinct eigenvalues is a stationary state, in the sense that all of its outcome laws are independent of time; in particular a superposition of two eigenrays of the two-level model is stationary.

Facts & Assumptions

Given: The adopted autonomous-dynamics reading of the framework with the two-level model data of Precession of a two-level preparation: exact phases, probabilities and a Bloch-vector rotation: the Hilbert space $H = C^{2}$ , distinct energies $E_{1} \neq E_{2}$ in joules, the assigned energy observable $H_{o p} = E_{1} T_{u_{1}, u_{1}} + E_{2} T_{u_{2}, u_{2}}$ with flow $U (t)$ and frequency $ω = (E_{2} - E_{1}) / ℏ \neq 0$ , the bounded self-adjoint flip observable $X = T_{u_{1}, u_{2}} + T_{u_{2}, u_{1}}$ , and the unit vector $ψ_{0} = (u_{1} + u_{2}) / \sqrt{2}$ ; the Axiom of Countable Choice and the Axiom of Choice, the choice hypotheses of the cited spectral suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice).

[F1]

The model data and the law of $X$ . In the two-level model, $ψ (t) = U (t) ψ_{0} = α e^{- i E_{1} t / ℏ} u_{1} + β e^{- i E_{2} t / ℏ} u_{2}$ for unit $ψ_{0} = α u_{1} + β u_{2}$ , $⟨ X ⟩ (t) = 2 Re (α \overline{β} e^{i ω t})$ , and $μ_{ψ (t)}^{X} ({+ 1}) = (1 + ⟨ X ⟩ (t)) / 2$ (Precession of a two-level preparation: exact phases, probabilities and a Bloch-vector rotation clauses (i), (iii) and (v)).

[F2]

Stationary states are the eigenrays. A pure preparation $[ψ]$ is stationary, that is $[ψ (t)] = [ψ (0)]$ for every $t$ , exactly when $ψ$ is an eigenvector of $H_{o p}$ ; for a non-eigenvector the ray is not constant (Energy conservation, conserved sharp observables and the stationary eigenrays clause (c)).

[F3]

Trigonometric values. For real $x$ one has $\cos (x + π) = - \cos x$ and $\cos (- x) = \cos x$ ; in particular $\cos π = - 1$ and $\cos (- π) = - 1$ (Quarter-turn values and shifts by pi/2 and pi, Parity and the Pythagorean identity for sine and cosine).

[F4]

Euler form of the phase. $e^{i θ} = \cos θ + i \sin θ$ for real $θ$ , so $2 Re (e^{i θ}) = 2 \cos θ$ (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ ). Moreover $\ker (\exp) = 2 π i Z$ , so for real $θ$ one has $e^{i θ} = 1$ exactly for $θ \in 2 π Z$ ( $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ ).

Counterexample

technique · direct; evaluate the outcome law of the flip observable of the two-level model at two times and show that it takes different values

1.1givenF1F4algebra

[The preparation and its frequency] The vector $ψ_{0} = (u_{1} + u_{2}) / \sqrt{2}$ is a unit vector with $α = β = 1 / \sqrt{2}$ , so $α \overline{β} = 1 / 2$ and, by [F1], $⟨ X ⟩ (t) = 2 Re (\frac{1}{2} e^{i ω t}) = \cos ω t$ by [F4], with $ω = (E_{2} - E_{1}) / ℏ \neq 0$ because $E_{1} \neq E_{2}$ .

2.1givenstep 1.1F1F3F4algebra

[Two values of the law] At $t = 0$ one has $e^{i ω \cdot 0} = 1$ since $0 \in \ker (\exp)$ by [F4], so $⟨ X ⟩ (0) = 2 Re (1 / 2) = 1$ by step 1.1 and the law of $X$ in $[ψ_{0}]$ is $μ_{ψ_{0}}^{X} ({+ 1}) = (1 + 1) / 2 = 1$ . At $t_{0} : = π / ∣ ω ∣ > 0$ one has $ω t_{0} = \pm π$ , so $\cos (ω t_{0}) = \cos (\pm π) = - 1$ by [F3], and therefore $μ_{ψ (t_{0})}^{X} ({+ 1}) = (1 - 1) / 2 = 0$ .

3.1givenstep 1.1step 2.1F1F2algebra∎

[The claim fails] The outcome law of the bounded assigned sharp observable $X$ takes the value $1$ at $t = 0$ and the value $0$ at $t = t_{0} > 0$ in the same preparation $[ψ_{0}]$ ; hence this law is not independent of time and the superposition $[ψ_{0}]$ is not stationary. By [F2] this is exactly as it must be, since $ψ_{0}$ is not an eigenvector of $H_{o p}$ : $H_{o p} ψ_{0} = (E_{1} u_{1} + E_{2} u_{2}) / \sqrt{2}$ is a multiple of $ψ_{0} = (u_{1} + u_{2}) / \sqrt{2}$ only if $E_{1} = E_{2}$ . This refutes the displayed universal claim.

Remarks

What is not refuted. Each eigenray is stationary by [F2], and a superposition whose nonzero coefficients all carry the same eigenvalue is stationary too; the refuted assertion is the universal one about superpositions of distinct eigenvalues. The failure is produced by the relative phase $e^{i ω t}$ , not by any external disturbance.
The two-times computation is exact model arithmetic. Both probabilities are consequences of the framework's outcome law for the declared finite model; no detector, efficiency, resolution or recorded data is involved.

RemarkRemark: AI-adaptedProof: Not applicableOpen item page →

What the worked items of the companion page establish

Physical scope

Scope of the companion examples page: an inventory of the finite model computations it contains and of the physical questions it does not address. The remark asserts no dynamics and records no empirical data.

What the page establishes

Two-level precession. Precession of a two-level preparation: exact phases, probabilities and a Bloch-vector rotation computes, on the explicit two-dimensional state space, the spectral measure and flow of the assigned energy observable $E_{1} T_{u_{1}, u_{1}} + E_{2} T_{u_{2}, u_{2}}$ with $E_{1} \neq E_{2}$ , the exact phases of an evolved preparation, the outcome probabilities and expectations of three bounded observables, and the rotation of the expectation-value triple about the third axis with angular frequency $ω = (E_{2} - E_{1}) / ℏ$ . The computation verifies the flow and law of Assigned energy observable, unitary flow and the two pictures of autonomous evolution and Assigned sharp observables and the outcome law of the framework on finite data, and it confirms the energy conservation of Energy conservation, conserved sharp observables and the stationary eigenrays clause (a) in the model. No apparatus realization, detector resolution or measured frequency is asserted.

Finite superpositions. A finite superposition of energy eigenrays: exact phases, laws and return times computes the evolution of a nonzero finite superposition of orthogonal energy eigenrays, the trigonometric-polynomial form of the outcome laws of a bounded assigned sharp observable, the exact return-time criterion $(λ_{m} - λ_{n}) T / ℏ \in 2 π Z$ for $m, n$ in the support of the coefficients, the existence criterion for a positive return time via a common frequency, and the two-energy specialization with least return time $2 π ℏ / ∣ λ_{1} - λ_{2} ∣$ . These are exact model statements under the flow of The Schroedinger-picture flow: unitarity, domain invariance and the domain-qualified equation.

A failed universal claim. A superposition of distinct energy eigenrays need not be stationary refutes the informal universal statement that every superposition of energy eigenrays with distinct eigenvalues is stationary: on the two-level model with $ψ_{0} = (u_{1} + u_{2}) / \sqrt{2}$ the outcome law of the flip observable takes the values $1$ at $t = 0$ and $0$ at $t = π / ∣ ω ∣$ . The refutation isolates the relative phase as the mechanism and is consistent with the exact stationarity criterion of Energy conservation, conserved sharp observables and the stationary eigenrays clause (c): the individual eigenrays are stationary, while their equal-weight superposition is not when their eigenvalues differ. Distinct eigenrays with a common eigenvalue are not covered by this refutation: every nonzero superposition of their representatives is again an eigenvector with that eigenvalue and hence represents a stationary preparation, as in A finite superposition of energy eigenrays: exact phases, laws and return times clause (i).

What the page does not establish

None of the three items is an experiment: no counts, calibration, resolution, efficiency, background, uncertainty or data comparison appears anywhere on the page, and no energy value is measured. The two-level system is an abstract model and is not identified with a spin, a magnetic moment, a photon polarization or any other species; the coupling of spin to a magnetic field belongs to the corresponding model pages, and driven or resonant dynamics is deferred to the driven-dynamics development, which adopts its own time-dependent assumptions. The return-time and precession results are statements about the model's flow and outcome law, not about the stability, decoherence or control of any laboratory preparation.

Qm Autonomous Dynamics — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Precession of a two-level preparation: exact phases, probabilities and a Bloch-vector rotation

Example

Facts & Assumptions

Verification

Remarks

A finite superposition of energy eigenrays: exact phases, laws and return times

Example

Facts & Assumptions

Verification

Remarks

A superposition of distinct energy eigenrays need not be stationary

Statement refuted

Facts & Assumptions

Counterexample

Remarks

What the worked items of the companion page establish

What the page establishes

What the page does not establish

Sources