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An exact two-level energy readout and its moments

Physical scope

A finite, exact model computation in the two-level state space of the framework: the energy observable E_1 P_1 + E_2 P_2 for two distinct energies in joules, its spectral measure, the outcome probabilities of a pure and of a statistical mixture (mixed for interior weights and pure at the endpoints), and the mean, second moment and variance of the readout. The energies are model parameters; the item asserts no apparatus realization, no measurement statistics from a laboratory and no empirical value.

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, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive), with orthonormal basis $e_{1} = (1, 0)$ , $e_{2} = (0, 1)$ . Fix two distinct energies $E_{1} \neq E_{2}$ in joules (model parameters, not measured values), and consider the assigned sharp observable $A : = E_{1} T_{e_{1}, e_{1}} + E_{2} T_{e_{2}, e_{2}}, u_{A} = J,$ with $T_{u, v} x = ⟨ x, v ⟩ u$ the rank-one calculus of Rank-one operator calculus (Assigned sharp observables and the outcome law of the framework). Then:

(i) $A$ is bounded and self-adjoint with $D (A) = H$ ; its spectral projection-valued measure is $E_{A} (B) = 1_{E_{1} \in B} T_{e_{1}, e_{1}} + 1_{E_{2} \in B} T_{e_{2}, e_{2}} (B \subseteq R Borel),$ so the spectrum is $σ (A) = {E_{1}, E_{2}}$ and the outcome lies in it almost surely: $μ_{ρ}^{A} ({E_{1}, E_{2}}) = 1$ for every preparation $ρ$ . This probability-one statement does not logically exclude realizations in a null event.

(ii) For the pure preparation of the unit vector $ψ = c_{1} e_{1} + c_{2} e_{2}$ , $∣ c_{1} ∣^{2} + ∣ c_{2} ∣^{2} = 1$ , the outcome law is $μ_{ψ}^{A} ({E_{1}}) = ∣ c_{1} ∣^{2}, μ_{ψ}^{A} ({E_{2}}) = ∣ c_{2} ∣^{2}, μ_{ψ}^{A} (B) = \sum_{i : E_{i} \in B} ∣ c_{i} ∣^{2},$ and for the statistical mixture $ρ = w T_{e_{1}, e_{1}} + (1 - w) T_{e_{2}, e_{2}}$ with $0 \leq w \leq 1$ the law is $μ_{ρ}^{A} ({E_{1}}) = w$ , $μ_{ρ}^{A} ({E_{2}}) = 1 - w$ . This preparation is mixed exactly when $0 < w < 1$ ; at $w = 0$ it is the pure state $T_{e_{2}, e_{2}}$ , and at $w = 1$ it is the pure state $T_{e_{1}, e_{1}}$ .

(iii) With $p : = ∣ c_{1} ∣^{2}$ and $q : = ∣ c_{2} ∣^{2} = 1 - p$ , the mean, second moment and variance of the pure preparation are $⟨ E ⟩_{ψ} = E_{1} p + E_{2} q, ⟨ E^{2} ⟩_{ψ} = E_{1}^{2} p + E_{2}^{2} q, {Var}_{ψ} = p q (E_{1} - E_{2})^{2},$ and for the mixture replace $p$ by $w$ and $q$ by $1 - w$ ; all three numbers are finite and exact because $A$ is bounded.

The computation is a model-internal readout: the $E_{i}$ 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 $(e_{1}, e_{2})$ ; distinct real energies $E_{1} \neq E_{2}$ in joules; the assigned observable $A = E_{1} T_{e_{1}, e_{1}} + E_{2} T_{e_{2}, e_{2}}$ with unit $J$ ; a unit vector $ψ = c_{1} e_{1} + c_{2} e_{2}$ ; a weight $w \in [0, 1]$ ; 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 linear in $x$ , with $T_{u, v}^{*} = T_{v, u}$ and $T_{u, u}$ self-adjoint and positive; for unit $u$ the operator $T_{u, u}$ is the orthogonal projection onto $C u$ with $T_{u, u}^{2} = T_{u, u}$ , its trace is $1$ , and for orthonormal $u, v$ one has $T_{u, u} T_{v, v} = 0$ (Rank-one operator calculus, Rays, unit vectors and rank-one Hilbert projections,).

[F2]

Finite-dimensional spectral structure. The eigenvectors $e_{1}, e_{2}$ of the normal operator $A$ form an orthonormal basis of $H$ , so $T_{e_{1}, e_{1}} + T_{e_{2}, e_{2}} = I$ (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F3]

Spectral PVM uniqueness. A self-adjoint operator has a unique regular PVM on the Borel sets of $R$ representing it, and the spectral PVM is that measure (Spectral theorem for unbounded self-adjoint operators (PVM form)).

[F4]

Outcome law. For an assigned observable the law is $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ , equal to $∥ E_{A} (B) ψ ∥^{2}$ for a pure preparation, and it is affine under mixtures (Assigned sharp observables and the outcome law of the framework clauses (b) and (c), The Born outcome law: probability measure, mixtures, atoms and spectral support clauses (a) and (b)).

[F5]

Moments. For a unit vector in $D (A)$ the mean and second moment are $⟨ A ψ, ψ ⟩$ and $∥ A ψ ∥^{2}$ , and the variance is the second moment minus the square of the mean (clauses (a)-(c), Mean, second moment and variance of an assigned observable in a preparation clauses (a) and (b)).

[F6]

States of the model. A state is a self-adjoint positive trace-class operator of trace one. A pure state is $T_{ψ, ψ}$ for a unit vector $ψ$ , equivalently a rank-one orthogonal projection; a state is mixed when it is not pure (The state space: density operators, pure and mixed states, and purity, Density operators and Pure states). The statistical mixture here is a state by positivity and the rank-one trace formulas in [F1].

Verification

technique · direct; compute the spectral measure, the outcome probabilities and the moments on the explicit two-dimensional data

1.1givenF1F2F3F4F6algebra

[The observable and its spectral measure] $A = E_{1} T_{e_{1}, e_{1}} + E_{2} T_{e_{2}, e_{2}}$ is bounded with $∥ A ∥ \leq \max {∣ E_{1} ∣, ∣ E_{2} ∣}$ and self-adjoint because each $T_{e_{i}, e_{i}}$ is self-adjoint by [F1] and $E_{i} \in R$ ; $A e_{1} = E_{1} e_{1}$ and $A e_{2} = E_{2} e_{2}$ . Define $E^{'} (B) : = 1_{E_{1} \in B} T_{e_{1}, e_{1}} + 1_{E_{2} \in B} T_{e_{2}, e_{2}}$ ; [F1] gives that each value is an orthogonal projection and that $T_{e_{1}, e_{1}} T_{e_{2}, e_{2}} = T_{e_{2}, e_{2}} T_{e_{1}, e_{1}} = 0$ , while [F2] gives $E^{'} (R) = I$ ; multiplicativity and countable additivity hold because the two-point space ${E_{1}, E_{2}}$ is finite, and the integral of the identity against $E^{'}$ is $E_{1} T_{e_{1}, e_{1}} + E_{2} T_{e_{2}, e_{2}} = A$ . By uniqueness [F3], $E^{'}$ is the spectral PVM of $A$ , so $σ (A) = {E_{1}, E_{2}}$ . Its displayed PVM has $E_{A} ({E_{1}, E_{2}}) = I$ , so [F4] gives $μ_{ρ}^{A} ({E_{1}, E_{2}}) = tr (ρ) = 1$ for every preparation, proving the almost-sure support claim of (i).

2.1givenstep 1.1F1F2F4F6algebra

[Outcome probabilities] For the unit vector $ψ = c_{1} e_{1} + c_{2} e_{2}$ , the identity $T_{e_{i}, e_{i}} ψ = ⟨ ψ, e_{i} ⟩ e_{i} = c_{i} e_{i}$ from [F1] gives $E_{A} (B) ψ = \sum_{i : E_{i} \in B} c_{i} e_{i}$ , whose norm squared is $\sum_{i : E_{i} \in B} ∣ c_{i} ∣^{2}$ ; by [F4] this is $μ_{ψ}^{A} (B)$ , so in particular $μ_{ψ}^{A} ({E_{i}}) = ∣ c_{i} ∣^{2}$ and the total mass is $∣ c_{1} ∣^{2} + ∣ c_{2} ∣^{2} = 1$ . For the mixture, $ρ$ is self-adjoint, positive and trace class by [F1] and $w, 1 - w \geq 0$ , with trace $w + (1 - w) = 1$ , so it is a state by [F6]. If $0 < w < 1$ , then $ρ e_{1} = w e_{1}$ and $ρ e_{2} = (1 - w) e_{2}$ give range $H$ , of dimension two; hence $ρ$ is not a rank-one projection and is mixed. At $w = 0$ or $w = 1$ it equals the respective rank-one projection $T_{e_{2}, e_{2}}$ or $T_{e_{1}, e_{1}}$ and is pure. Now [F4] and linearity of the trace give $μ_{ρ}^{A} ({E_{1}}) = tr (ρ T_{e_{1}, e_{1}}) = w$ because $tr (T_{e_{1}, e_{1}}^{2}) = 1$ and $tr (T_{e_{2}, e_{2}} T_{e_{1}, e_{1}}) = 0$ by [F1], and similarly $μ_{ρ}^{A} ({E_{2}}) = 1 - w$ ; this is clause (ii).

3.1givenstep 2.1F1F5F6algebra∎

[Moments of the readout] With $p = ∣ c_{1} ∣^{2}$ , $q = ∣ c_{2} ∣^{2}$ , the vector $ψ$ lies in $D (A) = H$ , so [F5] applies: the mean is $⟨ A ψ, ψ ⟩ = E_{1} p + E_{2} q$ , the second moment is $∥ A ψ ∥^{2} = E_{1}^{2} p + E_{2}^{2} q$ , and the variance is their difference, computed as $E_{1}^{2} p + E_{2}^{2} q - (E_{1} p + E_{2} q)^{2} = p (1 - p) E_{1}^{2} + q (1 - q) E_{2}^{2} - 2 p q E_{1} E_{2} = p q (E_{1} - E_{2})^{2}$ , using $p + q = 1$ and $[F 6]$ . For the mixture $ρ = w T_{e_{1}, e_{1}} + (1 - w) T_{e_{2}, e_{2}}$ , which is a state by [F6], the same identities give mean $w E_{1} + (1 - w) E_{2}$ and variance $w (1 - w) (E_{1} - E_{2})^{2}$ , since the law is the two-point law of (ii) and the moments of a finite two-point law are the displayed sums. All quantities are finite because $A$ is bounded. This is clause (iii).

Remarks

The two eigenvalues $E_{1} \neq E_{2}$ are distinct, so the eigenspaces are one-dimensional and the spectral projections are exactly the rank-one projections $T_{e_{i}, e_{i}}$ ; a degenerate energy would replace them by the projection onto the eigenspace and the exact-value probabilities by its trace weight, in agreement with Atoms of the spectral measure and eigenvalues of a self-adjoint operator.
The example is a finite model computation, not a reported experiment: nothing here identifies $E_{1}, E_{2}$ with the levels of a specific physical system, and no counts, uncertainty or apparatus behaviour is asserted.

An exact two-level energy readout and its moments

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

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