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Qm Observables And Born Probabilities

1 · Prerequisites

2 · Summary

This page develops the outcome law of the framework's observable and probability postulate. The physical vocabulary is fixed in Assigned sharp observables and the outcome law of the framework: an assigned sharp observable is a densely defined self-adjoint operator together with a declared unit for its real spectral coordinate, its spectral projection-valued measure $E_{A}$ exists under the stated choice hypotheses, and the outcome law of a preparation $ρ$ is $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ , equal to $∥ E_{A} (B) ψ ∥^{2}$ for the pure preparation of a unit vector. The definition also fixes the readings of exact values, of the spectral support, of the mean and variance, and of the affinity of the law under statistical mixing; it adopts no assumption beyond P1 and P2 and defers the finite unsharp-effect extension to the measurement-and-instruments page.

The mathematical layer supplies the properties of the law. The spectral measure of a real $L^{2}$ multiplication operator is computed from the spectral theorem as multiplication by the pulled-back indicators, with its vanishing sets and essential-range spectrum (Spectral measure of a real L2 multiplication operator); this is the tool that later identifies the position observables. The atoms of the spectral measure are the eigenspace projections, so an exact value carries zero probability unless it is an eigenvalue (Atoms of the spectral measure and eigenvalues of a self-adjoint operator); the support of the spectral measure is exactly the spectrum, so outcomes outside the spectrum have probability zero (Support of the spectral measure, the spectrum and the values of the scalar law); the law is real linear in the self-adjoint trace-class variable, affine under statistical mixtures and uniformly continuous in the trace norm (Affinity and trace-norm continuity of the scalar spectral law); and the mean, second moment and variance have the spectral formulas with their exact domain criteria (Mean, second moment and variance of the scalar spectral law). Distinct density operators are separated by the law of a bounded rank-one projection (Distinct density operators are separated by the law of a bounded rank-one projection).

The physics theorems read these results as conditional statements about the model. The Born outcome law: probability measure, mixtures, atoms and spectral support assembles the probability-measure, mixture, atom, support and no-density clauses; Mean, second moment and variance of an assigned observable in a preparation states the moment formulas with their hypotheses and the possibility of non-existence; Position and momentum outcome laws and the Born density of the massive particle model computes the position and momentum laws of the spinless massive model - the Born densities $∣ ψ ∣^{2}$ and $∣ \hat{ψ} ∣^{2}$ , the joint position measure and the atomlessness of exact coordinate values - and Distinct preparations are separated by the statistics of a bounded sharp observable draws the separation statement for preparations. No item asserts a complete discrete eigenbasis for a general self-adjoint operator, a density for an arbitrary outcome law, a moment without its domain hypothesis, a joint measurement, an apparatus realization or an empirical result; the scope remark What the observable and Born-probability page adopts, proves and does not claim records these limits.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Atoms of the spectral measure and eigenvalues of a self-adjoint operator

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H \neq {0}$ be a complex Hilbert space (Hilbert space), let $A$ be a self-adjoint operator on $H$ with spectral projection-valued measure $E_{A}$ on the Borel $σ$ -algebra of $R$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure), and let $E_{A, x} (B) : = ⟨ E_{A} (B) x, x ⟩$ be its scalar measures (Scalar and complex measures from a pvm). For $λ \in R$ say that $λ$ is an eigenvalue of $A$ when $A x = λ x$ for some nonzero $x \in D (A)$ , the domain-restricted reading of Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism. Then:

(a) The orthogonal projection $E_{A} ({λ})$ is the projection of $H$ onto the eigenspace $\ker (A - λ) = {x \in D (A) : A x = λ x}$ ; in particular $E_{A} ({λ}) \neq 0$ if and only if $λ$ is an eigenvalue of $A$ .

(b) For every density operator $ρ$ on $H$ , that is every self-adjoint positive trace-class $ρ$ with $tr ρ = 1$ (Density operators and scalar spectral probability clause (a)), the number $μ_{ρ}^{A} ({λ}) = tr (ρ E_{A} ({λ}))$ is the weight that the scalar law $μ_{ρ}^{A}$ assigns to the singleton ${λ}$ : it vanishes for every $ρ$ whenever $λ$ is not an eigenvalue of $A$ , and it equals $1$ for the pure state $T_{u, u}$ of every unit eigenvector $u \in \ker (A - λ)$ .

(c) If $A$ has no eigenvalues at all, then $μ_{ρ}^{A} ({λ}) = 0$ for every density operator $ρ$ and every $λ \in R$ : the scalar law of every state carries no point mass, while its values on general Borel sets remain $tr (ρ E_{A} (B))$ .

Clauses (a)-(c) are mathematical statements about $A$ , $E_{A}$ and the density operators; no physical interpretation of $μ_{ρ}^{A}$ is asserted here.

Facts & Assumptions

Given: The Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )); a nonzero complex Hilbert space $H$ ; a self-adjoint operator $A$ on $H$ with spectral projection-valued measure $E_{A}$ and scalar measures $E_{A, x} (B) = ⟨ E_{A} (B) x, x ⟩$ ; and a real number $λ$ .

[F1]

Spectral representation. $E_{A}$ is the unique regular projection-valued measure on the Borel $σ$ -algebra of $R$ with $D (A) = {x \in H : \int_{R} λ^{2} d E_{A, x} (λ) < \infty}$ and $A x = \int_{R} λ d E_{A} (λ) x$ for every $x \in D (A)$ (Spectral theorem for unbounded self-adjoint operators (PVM form),).

[F2]

Functional calculus. For every Borel function $f : R \to C$ the operator $f (A) = f (E_{A})$ has domain $D (f (A)) = {x \in H : \int_{R} ∣ f ∣^{2} d E_{A, x} < \infty}$ , is closed and densely defined, satisfies $∥ f (A) x ∥^{2} = \int_{R} ∣ f ∣^{2} d E_{A, x}$ and $f (A)^{*} = \overline{f} (A)$ , and is the norm limit of its bounded truncations $f 1_{{∣ f ∣ \leq n}} (A)$ on that domain; on $D (A)$ the functions $f (λ) = λ$ and $f (λ) = λ - μ$ give $A$ and $A - μ I$ (Unbounded Borel functional calculus: domains, products, spectral mapping,).

[F3]

Projection-valued measure. $E_{A}$ takes values in the orthogonal projections, with $E_{A} (\emptyset) = 0$ , $E_{A} (R) = I$ and $E_{A} (B \cap C) = E_{A} (B) E_{A} (C)$ for Borel $B, C$ ; for pairwise disjoint Borel sets $B_{n}$ with union $B$ one has $\sum_{n} E_{A} (B_{n}) x = E_{A} (B) x$ in norm for every $x \in H$ ; and $B \subseteq C$ implies $E_{A} (B) = E_{A} (B) E_{A} (C) = E_{A} (C) E_{A} (B)$ and $E_{A} (C) - E_{A} (B) = E_{A} (C ∖ B)$ (Projection valued measure, Scalar and complex measures from a pvm).

[F4]

Density operators and their scalar law. A density operator $ρ$ admits an expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ with $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ and $(u_{n})$ a finite or countable orthonormal family; for every PVM $E$ on a measurable space the assignment $B \mapsto tr (ρ E (B))$ is a probability measure with $tr (ρ E (B)) = \sum_{n} w_{n} ∥ E (B) u_{n} ∥^{2}$ , and for a unit vector $u$ one has $μ_{T_{u, u}} (B) = ∥ E (B) u ∥^{2}$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F5]

Orthogonal projections. An orthogonal projection $P$ satisfies $P^{2} = P = P^{*}$ and $∥ P x ∥^{2} = ⟨ P x, x ⟩ \leq ∥ x ∥^{2}$ for every $x \in H$ , and has zero range exactly when $P = 0$ ; moreover $∥ x - P x ∥^{2} = ∥ x ∥^{2} - ∥ P x ∥^{2}$ (The Hilbert orthogonal projection onto a closed subspace, Hilbert projections are linear, self-adjoint and contractive).

Proof

1.1givenF1F2F3algebra

[ $E_{A} ({λ}) H$ lies in the eigenspace] Let $x \in H$ satisfy $E_{A} ({λ}) x = x$ . For every Borel set $B$ the projection identity gives $E_{A, x} (B) = ⟨ E_{A} (B) E_{A} ({λ}) x, x ⟩ = ⟨ E_{A} (B \cap {λ}) x, x ⟩$ , so the finite measure $E_{A, x}$ is concentrated on ${λ}$ with total mass $∥ x ∥^{2}$ ; hence $\int λ^{2} d E_{A, x} = λ^{2} ∥ x ∥^{2} < \infty$ and $x \in D (A)$ by [F1]. By [F2] applied to $f (μ) = μ - λ$ one has $∥ (A - λ) x ∥^{2} = \int (μ - λ)^{2} d E_{A, x} (μ)$ , and the integrand vanishes $E_{A, x}$ -almost everywhere because $E_{A, x}$ is concentrated on ${λ}$ ; hence $A x = λ x$ .

1.2givenF2F3F5algebra

[Every eigenvector is fixed by $E_{A} ({λ})$ ] Conversely let $x \in D (A)$ satisfy $A x = λ x$ . By [F2] with $f (μ) = μ - λ$ , $0 = ∥ (A - λ) x ∥^{2} = \int (μ - λ)^{2} d E_{A, x} (μ)$ . For $n \geq 1$ put $C_{n} : = {μ \in R : ∣ μ - λ ∣ \geq 1 / n}$ ; on $C_{n}$ the integrand is at least $1 / n^{2}$ , so $E_{A, x} (C_{n}) \leq n^{2} \int (μ - λ)^{2} d E_{A, x} = 0$ . Set $D_{1} : = C_{1}$ and $D_{n} : = C_{n} ∖ C_{n - 1}$ for $n \geq 2$ . The sets $D_{n}$ are pairwise disjoint, their union is $R ∖ {λ}$ , and $E_{A, x} (D_{n}) = 0$ because $D_{n} \subseteq C_{n}$ ; hence countable additivity [F3] gives $E_{A, x} (R ∖ {λ}) = 0$ . Therefore $E_{A, x} ({λ}) = E_{A, x} (R) = ∥ x ∥^{2}$ , and since $(I - E_{A} ({λ}))$ is an orthogonal projection, [F5] gives $∥ x - E_{A} ({λ}) x ∥^{2} = ∥ x ∥^{2} - ∥ E_{A} ({λ}) x ∥^{2} = ∥ x ∥^{2} - E_{A, x} ({λ}) = 0$ , so $x = E_{A} ({λ}) x$ .

2.1step 1.1step 1.2F5

[Clause (a)] Steps 1.1 and 1.2 show that the range of the orthogonal projection $E_{A} ({λ})$ is exactly $\ker (A - λ)$ . A nonzero eigenspace contains a nonzero vector, and a projection with nonzero range is nonzero by [F5]; conversely $E_{A} ({λ}) \neq 0$ supplies a nonzero element of its range, which is an eigenvector by step 1.1. Hence $E_{A} ({λ}) \neq 0$ exactly when $λ$ is an eigenvalue of $A$ .

3.1givenstep 1.2step 2.1F4algebra

[Clause (b)] Let $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ be the expansion of [F4]. Then $μ_{ρ}^{A} ({λ}) = tr (ρ E_{A} ({λ})) = \sum_{n} w_{n} ∥ E_{A} ({λ}) u_{n} ∥^{2} \geq 0$ by [F4], and the sum is zero if and only if $∥ E_{A} ({λ}) u_{n} ∥ = 0$ for every $n$ of positive weight. If $λ$ is not an eigenvalue then $E_{A} ({λ}) = 0$ by step 2.1, so every term vanishes and $μ_{ρ}^{A} ({λ}) = 0$ . If $u$ is a unit eigenvector at $λ$ , then $u = E_{A} ({λ}) u$ by step 1.2 and [F4] gives $μ_{T_{u, u}}^{A} ({λ}) = ∥ E_{A} ({λ}) u ∥^{2} = ∥ u ∥^{2} = 1$ .

4.1step 2.1F4algebra∎

[Clause (c)] If $A$ has no eigenvalues then $E_{A} ({λ}) = 0$ for every $λ$ by step 2.1, so $μ_{ρ}^{A} ({λ}) = tr (ρ 0) = 0$ for every density operator and every $λ$ : the scalar law assigns no mass to any singleton. The values of $μ_{ρ}^{A}$ on general Borel sets are unchanged and remain those of [F4].

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Affinity and trace-norm continuity of the scalar spectral law

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and let $H$ be a nonzero complex Hilbert space (Hilbert space). Let $A$ be a self-adjoint operator on $H$ with spectral projection-valued measure $E_{A}$ (Projection valued measure). For a self-adjoint trace-class operator $T$ on $H$ (Trace class operator, Trace of a trace class operator) and a Borel set $B \subseteq R$ put $Φ_{T} (B) : = tr (T E_{A} (B)) \in C .$ Then:

(a) $∣ Φ_{T} (B) ∣ \leq ∥ T ∥_{1}$ for every Borel set $B$ ; the map $T \mapsto Φ_{T} (B)$ is real-linear on the self-adjoint trace-class operators; and for every density operator $ρ$ one has $0 \leq Φ_{ρ} (B) \leq 1$ , so that $μ_{ρ}^{A} : = Φ_{ρ}$ is a probability measure on the Borel sets with $μ_{ρ}^{A} (R) = 1$ (Density operators and scalar spectral probability clause (b)).

(b) If $ρ = \sum_{n} w_{n} ρ_{n}$ is a statistical mixture of density operators $ρ_{n}$ with weights $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , the trace-norm convergent sum of The state space is convex, closed under countable mixtures, and complete in the trace norm, then for every Borel set $B$ $μ_{ρ}^{A} (B) = \sum_{n} w_{n} μ_{ρ_{n}}^{A} (B),$ the series converging absolutely.

(c) For density operators $ρ, σ$ one has $\sup {∣ μ_{ρ}^{A} (B) - μ_{σ}^{A} (B) ∣ : B \subseteq R Borel} \leq ∥ ρ - σ ∥_{1},$ and if density operators $ρ_{k}$ converge to $ρ$ in trace norm, then $μ_{ρ_{k}}^{A} (B) \to μ_{ρ}^{A} (B)$ uniformly over all Borel sets $B$ .

Facts & Assumptions

Given: The Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )); a nonzero complex Hilbert space $H$ ; a self-adjoint operator $A$ with spectral PVM $E_{A}$ ; self-adjoint trace-class operators $T$ and density operators $ρ, ρ_{n}, σ$ on $H$ ; Borel sets $B \subseteq R$ .

[F1]

Projections and the trace ideal. Every $E_{A} (B)$ is an orthogonal projection, hence $∥ E_{A} (B) ∥ \leq 1$ ; for trace-class $T$ and bounded $S$ the product $T S$ is trace class with $∥ T S ∥_{1} \leq ∥ T ∥_{1} ∥ S ∥$ (Projection valued measure, Trace class is a two sided Banach operator ideal clause 2).

[F2]

Trace linearity and bound. The trace is linear on the trace-class operators and $∣ tr S ∣ \leq ∥ S ∥_{1}$ for every trace-class $S$ (Trace is absolutely convergent and basis independent clauses 1 and 4).

[F3]

Scalar law. For a density operator $ρ$ with spectral expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ , $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , $(u_{n})$ orthonormal, and every PVM $E$ , the assignment $B \mapsto tr (ρ E (B))$ is a probability measure and $tr (ρ E (B)) = \sum_{n} w_{n} ∥ E (B) u_{n} ∥^{2}$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F4]

Convex combinations of states. If $ρ_{n}$ are density operators and $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ over a finite or countable index set, then $ρ : = \sum_{n} w_{n} ρ_{n}$ converges in trace norm to a density operator (The state space is convex, closed under countable mixtures, and complete in the trace norm).

Proof

1.1givenF1F2algebra

[Contraction and trace bound] By [F1] the operator $T E_{A} (B)$ is trace class with $∥ T E_{A} (B) ∥_{1} \leq ∥ T ∥_{1} ∥ E_{A} (B) ∥ \leq ∥ T ∥_{1}$ , since $E_{A} (B)$ is an orthogonal projection of norm at most one; [F2] gives $∣ Φ_{T} (B) ∣ = ∣ tr (T E_{A} (B)) ∣ \leq ∥ T E_{A} (B) ∥_{1} \leq ∥ T ∥_{1}$ .

1.2givenF1F2F3algebra

[Real-linearity and positivity on states] For self-adjoint trace-class $S, T$ and real $a, b$ the operator $a S + b T$ is self-adjoint and trace class, and [F2] gives $Φ_{a S + b T} (B) = tr ((a S + b T) E_{A} (B)) = a tr (S E_{A} (B)) + b tr (T E_{A} (B)) = a Φ_{S} (B) + b Φ_{T} (B)$ ; this proves real-linearity. If $ρ$ is a density operator with expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ , then [F3] gives $Φ_{ρ} (B) = \sum_{n} w_{n} ∥ E_{A} (B) u_{n} ∥^{2} \geq 0$ and $Φ_{ρ} (B) \leq \sum_{n} w_{n} ∥ u_{n} ∥^{2} = 1$ ; together with [F3] this makes $μ_{ρ}^{A} = Φ_{ρ}$ a probability measure.

2.1givenstep 1.2F1F2F4algebra

[Affinity under mixtures] Let $ρ_{n}$ be density operators and $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , with trace-norm convergent sum $ρ = \sum_{n} w_{n} ρ_{n}$ , a density operator by [F4]. Writing $S_{N} : = \sum_{n \leq N} w_{n} ρ_{n}$ , [F1] gives $∥ S_{N} E_{A} (B) - ρ E_{A} (B) ∥_{1} \leq ∥ S_{N} - ρ ∥_{1} ∥ E_{A} (B) ∥ \leq ∥ S_{N} - ρ ∥_{1} \to 0$ ; by [F2] and continuity of the scalar limit, $tr (ρ E_{A} (B)) = \lim_{N} tr (S_{N} E_{A} (B)) = \lim_{N} \sum_{n \leq N} w_{n} tr (ρ_{n} E_{A} (B)) = \sum_{n} w_{n} μ_{ρ_{n}}^{A} (B)$ , the series converging absolutely because the finite partial sums of the nonnegative terms are bounded by the limit $μ_{ρ}^{A} (B) \leq 1$ .

2.2givenstep 1.1algebra

[Uniform trace-norm continuity] For Borel $B$ , $ρ - σ$ is self-adjoint and trace class, so step 1.1 gives $∣ μ_{ρ}^{A} (B) - μ_{σ}^{A} (B) ∣ = ∣ Φ_{ρ - σ} (B) ∣ \leq ∥ ρ - σ ∥_{1}$ ; taking the supremum over Borel sets yields the displayed bound, and the convergence assertion follows by applying it to $ρ_{k} - ρ$ .

3.1step 1.1step 1.2step 2.1step 2.2∎

[Conclusion] Step 1.2 proves the bound, real-linearity, positivity and probability-measure clauses of (a); step 2.1 proves the mixture identity (b); step 2.2 proves the uniform bound and continuity (c).

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Mean, second moment and variance of the scalar spectral law

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a nonzero complex Hilbert space (Hilbert space) and let $A$ be a self-adjoint operator on $H$ with spectral projection-valued measure $E_{A}$ and scalar measures $E_{A, ψ} (B) = ⟨ E_{A} (B) ψ, ψ ⟩$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, Scalar and complex measures from a pvm). For a unit vector $ψ \in H$ write $μ_{ψ}^{A} : = μ_{T_{ψ, ψ}}^{A}$ for the scalar law of the pure state $T_{ψ, ψ}$ of Density operators and scalar spectral probability clause (b). Then:

(a) Second moment. $\int_{R} λ^{2} d μ_{ψ}^{A} (λ) = ∥ A ψ ∥^{2}$ if $ψ \in D (A)$ , and the integral equals $+ \infty$ if $ψ \notin D (A)$ ; thus the second moment of the scalar law is finite exactly on $D (A)$ .

(b) Mean. $\int_{R} ∣ λ ∣ d μ_{ψ}^{A} (λ) < \infty$ if and only if $ψ \in D (∣ A ∣^{1 / 2})$ ; in that case the mean $m (ψ) : = \int_{R} λ d μ_{ψ}^{A} (λ)$ is finite and real, and if $ψ \in D (A)$ then $m (ψ) = ⟨ A ψ, ψ ⟩$ .

(c) Variance. If $ψ \in D (A)$ and $m : = ⟨ A ψ, ψ ⟩$ , then $\int_{R} (λ - m)^{2} d μ_{ψ}^{A} (λ) = ∥ A ψ ∥^{2} - m^{2} = ∥ (A - m) ψ ∥^{2} \geq 0;$ if in addition $ψ \in D (A^{2})$ , that is $ψ \in D (A)$ and $A ψ \in D (A)$ , then $∥ (A - m) ψ ∥^{2} = ⟨ A^{2} ψ, ψ ⟩ - ⟨ A ψ, ψ ⟩^{2}$ .

(d) States. Let $ρ$ be a density operator with expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ , $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , $(u_{n})$ orthonormal. In every weighted sum below, a term with $w_{n} = 0$ is defined to be zero before any component integral is evaluated; equivalently, sum only over $n$ with $w_{n} > 0$ . This convention also applies to signed component means, which need not exist at zero-weight indices. Then for every Borel function $g \geq 0$ $\int_{R} g d μ_{ρ}^{A} = \sum_{n} w_{n} \int_{R} g d E_{A, u_{n}} \in [0, \infty],$ so the mean of $μ_{ρ}^{A}$ is finite if and only if $\sum_{n} w_{n} \int ∣ λ ∣ d E_{A, u_{n}} < \infty$ , and then it equals $\sum_{n} w_{n} \int λ d E_{A, u_{n}}$ . For the second moment, put $M_{n} : = \int λ^{2} d E_{A, u_{n}}$ , so $M_{n} = ∥ A u_{n} ∥^{2}$ when $u_{n} \in D (A)$ and $M_{n} = + \infty$ otherwise; a zero weight contributes zero. Then $\int λ^{2} d μ_{ρ}^{A} = \sum_{n} w_{n} M_{n} \in [0, \infty]$ . It is finite exactly when every $u_{n}$ with $w_{n} > 0$ lies in $D (A)$ and $\sum_{n : w_{n} > 0} w_{n} ∥ A u_{n} ∥^{2} < \infty$ ; componentwise domain membership alone does not ensure a finite second moment.

Facts & Assumptions

Given: The Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )); a nonzero complex Hilbert space $H$ ; a self-adjoint operator $A$ with PVM $E_{A}$ ; a unit vector $ψ \in H$ ; a density operator $ρ$ with expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ .

[F1]

Spectral domain and calculus. $D (A) = {ψ \in H : \int λ^{2} d E_{A, ψ} < \infty}$ and $A ψ = \int λ d E_{A} (λ) ψ$ on $D (A)$ ; for every Borel $f$ the operator $f (A)$ has domain $D (f (A)) = {ψ : \int ∣ f ∣^{2} d E_{A, ψ} < \infty}$ and satisfies $∥ f (A) ψ ∥^{2} = \int ∣ f ∣^{2} d E_{A, ψ}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Unbounded Borel functional calculus: domains, products, spectral mapping, Integral of a measurable function against a projection-valued measure).

[F2]

Mean and second moment of a pure state. For a unit vector $ψ \in D (A)$ one has $\int λ d E_{A, ψ} = ⟨ A ψ, ψ ⟩$ and $\int λ^{2} d E_{A, ψ} = ∥ A ψ ∥^{2}$ ; moreover $\int ∣ λ ∣ d E_{A, ψ} < \infty$ exactly when $ψ \in D (∣ A ∣^{1 / 2})$ , and then the mean $\int λ d E_{A, ψ}$ is finite (Density operators and scalar spectral probability clause (c)).

[F3]

Scalar law and its series form. For the density operator $ρ$ with the displayed expansion and every PVM $E$ , $tr (ρ E (B)) = \sum_{n} w_{n} ∥ E (B) u_{n} ∥^{2}$ for every Borel $B$ , and for a unit vector $u$ the law of $T_{u, u}$ is $∥ E (B) u ∥^{2} = E_{u} (B)$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F4]

Monotone convergence and approximation. If $0 \leq f_{m} ↑ f$ pointwise then $\int f_{m} d ν ↑ \int f d ν$ for every measure $ν$ (Monotone convergence for the integral); every nonnegative measurable $g$ is the pointwise increasing limit of nonnegative simple functions ; and if $f_{m} \to f$ pointwise with $∣ f_{m} ∣ \leq G$ for an integrable $G$ then $\int f_{m} d ν \to \int f d ν$ (Dominated convergence).

[F5]

Self-adjoint quadratic forms are real. For a self-adjoint $A$ and $ψ \in D (A)$ one has $⟨ A ψ, ψ ⟩ = \overline{⟨ ψ, A ψ ⟩} = \overline{⟨ A ψ, ψ ⟩} \in R$ , and $∥ (A - m) ψ ∥^{2} = ∥ A ψ ∥^{2} - 2 m ⟨ A ψ, ψ ⟩ + m^{2} ∥ ψ ∥^{2}$ for real $m$ (Symmetric, self-adjoint and essentially self-adjoint operators).

Proof

1.1givenF1F2F5algebra

[Second moment and mean for a state vector] Let $ψ \in H$ be a unit vector and identify $μ_{ψ}^{A}$ with the finite measure $E_{A, ψ}$ . If $ψ \in D (A)$ , [F1] and [F2] give $\int λ^{2} d μ_{ψ}^{A} = ∥ A ψ ∥^{2} < \infty$ and $\int λ d μ_{ψ}^{A} = ⟨ A ψ, ψ ⟩$ ; if $ψ \notin D (A)$ , then [F1] gives $\int λ^{2} d μ_{ψ}^{A} = \infty$ , so the second moment is infinite. In the first case [F5] makes $⟨ A ψ, ψ ⟩$ real. This proves (a) and the mean formula of (b).

1.2givenF3F4algebra

[Reduction of state moments to pure moments] By [F3], $μ_{ρ}^{A} (B) = \sum_{n} w_{n} E_{A, u_{n}} (B)$ for every Borel $B$ . Let $g \geq 0$ be Borel, and omit all zero-weight indices as stipulated in (d). For indicators $g = 1_{B}$ the identity $\int g d μ_{ρ}^{A} = \sum_{n} w_{n} \int g d E_{A, u_{n}}$ is exactly [F3]; it extends to nonnegative simple $g$ by linearity, and to all nonnegative Borel $g$ by writing $g$ as an increasing pointwise limit of nonnegative simple functions [F4] and applying monotone convergence twice, to the measure $μ_{ρ}^{A}$ and to the partial sums of the series. This proves the displayed identity of (d).

2.1givenstep 1.1F1F3algebra

[The first moment criterion] For any unit $ψ \in H$ , define $∣ A ∣^{1 / 2} : = f (A)$ with the Borel function $f (λ) = ∣ λ ∣^{1 / 2}$ . By [F1], $D (∣ A ∣^{1 / 2}) = {ψ \in H : \int ∣ λ ∣ d E_{A, ψ} < \infty}$ . By [F3], $E_{A, ψ} = μ_{ψ}^{A}$ , so this gives the claimed equivalence without assuming $ψ \in D (A)$ . When the absolute first moment is finite, the positive and negative parts of $λ$ have finite integrals, and their difference defines a finite real mean; otherwise the mean is not a finite Lebesgue integral. Together with step 1.1 this proves (b).

2.2givenstep 1.1F1F2F5algebra

[Variance of a state vector] Let $ψ \in D (A)$ be a unit vector and $m : = ⟨ A ψ, ψ ⟩$ , a real number by [F5]. Then $A ψ - m ψ \in H$ and by [F1] applied to $f (λ) = λ - m$ , $∥ (A - m) ψ ∥^{2} = \int (λ - m)^{2} d E_{A, ψ}$ . Expanding the square and using [F2] for the integrals of $λ^{2}$ , $λ$ and $1$ gives $\int (λ - m)^{2} d μ_{ψ}^{A} = \int λ^{2} d μ_{ψ}^{A} - 2 m \int λ d μ_{ψ}^{A} + m^{2} = ∥ A ψ ∥^{2} - 2 m^{2} + m^{2} = ∥ A ψ ∥^{2} - m^{2}$ .

3.1step 2.2F1F5algebra

[The second identity for the variance] If moreover $ψ \in D (A^{2})$ , then $A ψ \in D (A)$ and the inner product $⟨ A^{2} ψ, ψ ⟩$ is defined and equals $⟨ A (A ψ), ψ ⟩ = ⟨ A ψ, A ψ ⟩ = ∥ A ψ ∥^{2}$ because $A$ is self-adjoint; substituting this into step 2.2 gives $\int (λ - m)^{2} d μ_{ψ}^{A} = ⟨ A^{2} ψ, ψ ⟩ - m^{2} = ⟨ A^{2} ψ, ψ ⟩ - ⟨ A ψ, ψ ⟩^{2}$ , which together with step 2.2 proves (c).

4.1step 1.1step 1.2F2F4algebra∎

[Moments of a state] All sums in this step run over positive-weight indices, with the zero-term convention of (d). With $g = ∣ λ ∣$ in step 1.2, the mean of $μ_{ρ}^{A}$ is finite exactly when $\sum_{n} w_{n} \int ∣ λ ∣ d E_{A, u_{n}} < \infty$ ; in that case every positive-weight component has finite absolute first moment, and applying the signed version of the identity to the positive and negative parts and subtracting gives $\int λ d μ_{ρ}^{A} = \sum_{n} w_{n} \int λ d E_{A, u_{n}}$ , using dominated convergence [F4] for the tail sums. With $g = λ^{2}$ , step 1.2 and [F2] give $\int λ^{2} d μ_{ρ}^{A} = \sum_{n} w_{n} M_{n}$ , where $M_{n} = \int λ^{2} d E_{A, u_{n}}$ is $∥ A u_{n} ∥^{2}$ for $u_{n} \in D (A)$ and $+ \infty$ otherwise, and terms of zero weight contribute zero. Since this is a nonnegative series, it is finite exactly when every component of positive weight lies in $D (A)$ and the weighted sum of the squared norms is finite; having each positive-weight component in $D (A)$ by itself does not imply convergence of that series.

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Distinct density operators are separated by the law of a bounded rank-one projection

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the Axiom of Choice is used only where the spectral theorem for unbounded self-adjoint operators is invoked (The Axiom of Choice). Let $H$ be a nonzero complex Hilbert space (Hilbert space) and let $ρ, σ$ be density operators on $H$ with $ρ \neq σ$ (Density operators and scalar spectral probability clause (a)). Then:

(a) There is a unit vector $w \in H$ with $⟨ ρ w, w ⟩ \neq ⟨ σ w, w ⟩$ .

(b) For the rank-one orthogonal projection $P : = T_{w, w}$ of Rank-one operator calculus (The Hilbert orthogonal projection onto a closed subspace), whose spectral projection-valued measure assigns $P$ to the Borel set ${1}$ , the two scalar laws of Density operators and scalar spectral probability clause (b) satisfy $μ_{ρ}^{P} ({1}) = tr (ρ P) = ⟨ ρ w, w ⟩, μ_{σ}^{P} ({1}) = tr (σ P) = ⟨ σ w, w ⟩,$ so the bounded sharp observable $P$ assigns different outcome probabilities to $ρ$ and $σ$ .

(c) For every unit vector $w$ and every self-adjoint trace-class $T$ one has $∣ tr (T T_{w, w}) ∣ \leq ∥ T ∥_{1}$ ; in particular $∣ ⟨ ρ w, w ⟩ - ⟨ σ w, w ⟩ ∣ \leq ∥ ρ - σ ∥_{1}$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and, for the spectral PVM of clause (b), the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice); a nonzero complex Hilbert space $H$ ; density operators $ρ, σ$ on $H$ ; unit vectors $w \in H$ .

[F1]

Rank-one calculus. $T_{u, v} x = ⟨ x, v ⟩ u$ defines a trace-class operator with $∥ T_{u, v} ∥_{1} = ∥ u ∥ ∥ v ∥$ and $tr (T_{u, v}) = ⟨ u, v ⟩$ ; $T_{u, u}$ is self-adjoint and positive with $⟨ T_{u, u} x, x ⟩ = ∣ ⟨ x, u ⟩ ∣^{2}$ (Rank-one operator calculus).

[F2]

One-dimensional projections. For $u \in S (H)$ the operator $T_{u, u}$ is the Hilbert orthogonal projection onto the one-dimensional subspace $C u$ ; in particular $T_{u, u}^{2} = T_{u, u} = T_{u, u}^{*}$ (Rays, unit vectors and rank-one Hilbert projections, The Hilbert orthogonal projection onto a closed subspace).

[F3]

Density operators and their law. A density operator $τ$ is self-adjoint, positive and trace class with $tr τ = 1$ , and admits an expansion $τ = \sum_{n} t_{n} T_{v_{n}, v_{n}}$ with $t_{n} \geq 0$ , $\sum_{n} t_{n} = 1$ and $(v_{n})$ orthonormal; for every PVM $E$ , $B \mapsto tr (τ E (B))$ is a probability measure with $tr (τ E (B)) = \sum_{n} t_{n} ∥ E (B) v_{n} ∥^{2}$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F4]

Trace ideal and trace bounds. Products of a trace-class operator with a bounded operator are trace class and $∥ A T B ∥_{1} \leq ∥ A ∥ ∥ T ∥_{1} ∥ B ∥$ ; the trace is linear on trace-class operators with $∣ tr S ∣ \leq ∥ S ∥_{1}$ (Trace class is a two sided Banach operator ideal clause 2, Trace is absolutely convergent and basis independent clause 4).

[F5]

Quadratic form and norm. For a bounded self-adjoint $T$ one has $∥ T ∥ = \sup {∣ ⟨ T x, x ⟩ ∣ : ∥ x ∥ = 1}$ , and the value is $0$ exactly when $T = 0$ (Norm of a self adjoint operator from its quadratic form).

[F6]

Spectral PVM of a projection. Every orthogonal projection $P$ is a self-adjoint operator with $0 \leq P \leq I$ ; by the spectral theorem it has a unique regular PVM $E_{P}$ on the Borel sets of $R$ with $P = \int λ d E_{P} (λ)$ (Spectral theorem for unbounded self-adjoint operators (PVM form)).

Proof

1.1givenF3F4F5algebra

[Quadratic forms separate trace-class states] Put $T : = ρ - σ$ , a nonzero self-adjoint trace-class operator by [F3] and [F4]. Suppose $⟨ T w, w ⟩ = 0$ for every unit vector $w$ ; then the supremum of [F5] is $0$ , so $∥ T ∥ = 0$ and $T = 0$ , contradicting $ρ \neq σ$ . Hence there is a unit $w$ with $0 \neq ⟨ (ρ - σ) w, w ⟩ = ⟨ ρ w, w ⟩ - ⟨ σ w, w ⟩$ , proving (a).

1.2givenF1F3F4algebra

[The trace pairing of a rank-one projection] Let $τ = \sum_{n} t_{n} T_{v_{n}, v_{n}}$ be the expansion of a density operator and let $w$ be a unit vector. For each $n$ , the product rule $T_{v_{n}, v_{n}} T_{w, w} = T_{v_{n}, ⟨ v_{n}, w ⟩ w}$ holds by direct evaluation on any $x$ : $T_{v_{n}, v_{n}} T_{w, w} x = ⟨ x, w ⟩ ⟨ w, v_{n} ⟩ v_{n} = T_{v_{n}, ⟨ v_{n}, w ⟩ w} x$ ; hence by [F1] its trace is $⟨ v_{n}, ⟨ v_{n}, w ⟩ w ⟩ = ∣ ⟨ v_{n}, w ⟩ ∣^{2}$ . Multiplying by $t_{n}$ , summing and using the trace ideal and trace bound [F4] together with $\sum_{n} t_{n} ∥ v_{n} ∥^{2} ∥ w ∥^{2} = 1$ gives the absolutely convergent identity $tr (τ T_{w, w}) = \sum_{n} t_{n} ∣ ⟨ v_{n}, w ⟩ ∣^{2}$ . On the other hand $⟨ τ w, w ⟩ = \sum_{n} t_{n} ⟨ T_{v_{n}, v_{n}} w, w ⟩ = \sum_{n} t_{n} ∣ ⟨ w, v_{n} ⟩ ∣^{2}$ by [F1] and continuity of the pairing against the norm convergent series. Since $∣ ⟨ w, v_{n} ⟩ ∣ = ∣ ⟨ v_{n}, w ⟩ ∣$ , the two series agree: $tr (τ T_{w, w}) = ⟨ τ w, w ⟩$ for every density operator $τ$ .

2.1step 1.1step 1.2F2F3F6algebra

[The projection separates the two states] Fix a unit $w$ with $⟨ ρ w, w ⟩ \neq ⟨ σ w, w ⟩$ , furnished by step 1.1, and put $P : = T_{w, w}$ , an orthogonal projection by [F2]. The map $E^{'} (B) : = 1_{1 \in B} P + 1_{0 \in B} (I - P)$ is a regular PVM on the Borel sets of $R$ with $\int λ d E^{'} (λ) = 1 \cdot P + 0 \cdot (I - P) = P$ : the values are orthogonal projections with $E^{'} (\emptyset) = 0$ and $E^{'} (R) = I$ , multiplicativity holds because $P (I - P) = 0$ , and countable additivity holds on the two-point space ${0, 1}$ . By uniqueness in the spectral theorem [F6], $E^{'}$ is the spectral PVM of $P$ , so $E_{P} ({1}) = P$ and $μ_{τ}^{P} ({1}) = tr (τ P)$ for every density operator $τ$ by [F3]. Step 1.2 identifies these numbers as $⟨ ρ w, w ⟩$ and $⟨ σ w, w ⟩$ , which differ by the choice of $w$ ; this proves (b).

3.1step 1.1step 1.2step 2.1F2F4algebra∎

[The uniform bound] Let $w$ be a unit vector and let $T$ be self-adjoint trace class. By [F2] and [F4], $∥ T_{w, w} ∥ \leq ∥ T_{w, w} ∥_{1} = 1$ and $∣ tr (T T_{w, w}) ∣ \leq ∥ T T_{w, w} ∥_{1} \leq ∥ T ∥_{1}$ ; applying this to $T = ρ - σ$ and using step 1.2 gives $∣ ⟨ ρ w, w ⟩ - ⟨ σ w, w ⟩ ∣ = ∣ tr ((ρ - σ) T_{w, w}) ∣ \leq ∥ ρ - σ ∥_{1}$ , which is (c); the existence statement of (a) and the separating probability identity of (b) are steps 1.1 and 2.1.

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Assigned sharp observables and the outcome law of the framework

Physical scope

The observable and outcome-law vocabulary of the non-relativistic quantum framework: the reading of the observable and probability postulate P2 as assigned sharp self-adjoint quantities with a declared unit, their spectral projection-valued measures, the outcome law of a preparation, the spectral-support reading of outcome sets, the conditional readings of mean and variance, and the affinity of the law under statistical mixing. This item fixes terminology and readings only: it adopts no assumption beyond P1 and P2, asserts no detector or apparatus realization, makes no joint-measurement claim, records no empirical data, and excludes the finite unsharp-effect extension of P2(iv), which belongs to the measurement-and-instruments page.

Definition

(a) Assigned sharp observables. Adopt the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2) for a declared system whose state space is the model's The state space of the framework: preparations, pure rays and density operators. An assigned sharp observable of the system is a pair $(A, u_{A})$ consisting of a densely defined self-adjoint operator $A : D (A) \subset H \to H$ on the assigned Hilbert space $H$ (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators, Hilbert space) and a declared unit $u_{A}$ in which the real spectral coordinate of $A$ is read, in the sense of Dimensions, SI unit conventions and quantity types of the framework. The domain $D (A)$ is part of the assignment: symmetry on a smaller domain, or a formal differential expression, does not by itself supply an assigned observable (Symmetric, self-adjoint and essentially self-adjoint operators). Whether a concrete laboratory quantity of a given apparatus is described by a particular pair $(A, u_{A})$ is a tested identification with uncertainty under the preparation and apparatus postulate P7 and is not produced by this definition.

(b) Spectral measure and outcome law. Assume 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. The self-adjoint operator $A$ has a unique spectral projection-valued measure $E_{A}$ on the Borel $σ$ -algebra of $R$ , with $D (A) = {x : \int λ^{2} d E_{A, x} < \infty}$ and $A x = \int λ d E_{A} (λ) x$ , where $E_{A, x} (B) = ⟨ E_{A} (B) x, x ⟩$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, The Borel sigma-algebra of a topological space). For a preparation $ρ$ , that is a state $ρ \in D (H)$ of The state space: density operators, pure and mixed states, and purity, the outcome law of the pair $(A, u_{A})$ in the preparation $ρ$ is the assignment $μ_{ρ}^{A} (B) : = tr (ρ E_{A} (B)) (B \subseteq R Borel),$ the reading of clause (ii) of P2; for the pure preparation $[ψ]$ of a unit vector $ψ$ it is $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ , which depends only on the ray by State vectors, unit vectors, rays, normalization and phases of a Hilbert space. The law is a probability measure on the Borel sets by Density operators and scalar spectral probability clause (b), and it is affine in the preparation: for a statistical mixture $\sum_{n} w_{n} ρ_{n}$ one has $μ_{\sum_{n} w_{n} ρ_{n}}^{A} = \sum_{n} w_{n} μ_{ρ_{n}}^{A}$ by Affinity and trace-norm continuity of the scalar spectral law clause (b).

(c) Outcome sets and exact values. A declared finite-resolution outcome of the quantity is a Borel set $B$ ; its probability in the preparation $ρ$ is $μ_{ρ}^{A} (B)$ , and a "value in $B$ " is read only through this number. The reading of an exact value $λ$ is the singleton $B = {λ}$ , with probability $μ_{ρ}^{A} ({λ})$ ; by Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a) and (b) this number is zero for every preparation unless $λ$ is an eigenvalue of $A$ , and when $λ$ is an eigenvalue it is the total weight of the eigenspace. In particular the model provides no normalizable eigenvector at a non-eigenvalue of $A$ . If $A$ has no eigenvalues, every preparation has an atomless outcome law: each singleton has probability zero, while $R$ remains the outcome space and the law is concentrated on $σ (A)$ as in clause (d). Zero singleton probability does not exclude real point values as realizations of this law; it supplies no normalized exact-value eigenstate. A finite-resolution record is read through its declared Borel bin, in the sense of clause (iii) of P2.

(d) Spectral support. The outcomes of the observable lie in its spectrum: with $σ (A)$ the spectrum of Resolvent and spectrum of an unbounded operator, one has $μ_{ρ}^{A} (B) = μ_{ρ}^{A} (B \cap σ (A))$ and $μ_{ρ}^{A} (σ (A)) = 1$ for every preparation by Support of the spectral measure, the spectrum and the values of the scalar law clause (b). This is the precise reading of the statement that the result of a measurement lies in the spectrum of the observable.

(e) Moments. The mean and the variance of the outcome law in a preparation are the numbers $\int λ d μ_{ρ}^{A} (λ)$ and $\int (λ - m)^{2} d μ_{ρ}^{A} (λ)$ with $m$ the mean, read in the declared unit $u_{A}$ and its square respectively; they exist only under the integrability hypotheses of Mean, second moment and variance of the scalar spectral law, which also supplies the formulas in terms of $A$ and the state. For a pure preparation of a unit vector $ψ \in D (A)$ the mean and the second moment are $⟨ A ψ, ψ ⟩$ and $∥ A ψ ∥^{2}$ ; membership of $ψ$ in $D (A)$ or in $D (A^{2})$ is a genuine hypothesis and is not asserted for an arbitrary preparation.

(f) Unsharp finite outcomes are separate. Clause (iv) of P2 permits a finite family of bounded effects $F_{a}$ with $0 \leq F_{a} \leq I$ and $\sum_{a} F_{a} = I$ and outcome probabilities $tr (ρ F_{a})$ as an operational description of a finite-resolution apparatus. That extension is tied to an apparatus instrument and is developed, together with state updates, on the measurement-and-instruments page; this definition fixes only the sharp outcome law of clause (b) and makes no claim that every laboratory procedure realizes it.

(g) Status. Clauses (a)-(f) record the readings of 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 P1 and P2 and the choice hypotheses of the spectral theorem, assert no dynamics, no joint measurement of several observables, no detector realization and no empirical result, and make no claim that any physical system is exactly described by an assigned pair $(A, u_{A})$ . The additive additivity, support, atom and moment statements cited are mathematical consequences proved in the cited items; their physical reading is exactly the reading of P2 fixed here.

Remarks

Mixtures versus superpositions. The affinity of clause (b) concerns statistical mixtures with classical weights, not superpositions: a coherent superposition is a normalized vector combination of unit representatives, and its outcome law need not be the weighted average of their laws. The interference terms can vanish for a particular observable, so equality of these laws is possible. The distinction is carried by the state-space items of the preceding page.
No discrete basis is assumed. Nothing in this definition asserts that $A$ has eigenvalues or a complete system of eigenvectors. The outcome law exists for every assigned self-adjoint $A$ through its spectral PVM; discrete sums are the special case of a purely atomic law.

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The Born outcome law: probability measure, mixtures, atoms and spectral support

Physical scope

Conditional consequences of the adopted observable and probability postulate P2 for the sharp observables of the framework: the outcome law of an assigned self-adjoint observable and preparation is a probability measure on the Borel sets; it is affine under statistical mixtures; it assigns zero to an exact value that is not an eigenvalue; it is supported in the spectrum of the observable; and it carries no claimed density. The item is conditional on P1 and P2 and on the choice hypotheses of the spectral theorem; it asserts nothing about concrete apparatuses, joint measurements, moments, state updates or empirical data.

Statement

Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1) and the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2), and assume 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 spectral suppliers. Let $H$ be the state space assigned to a declared system, let $(A, u_{A})$ be an assigned sharp observable of the framework with spectral projection-valued measure $E_{A}$ and spectrum $σ (A)$ , and let $μ_{ρ}^{A}$ be the outcome law of Assigned sharp observables and the outcome law of the framework. Then for every preparation $ρ$ (The state space of the framework: preparations, pure rays and density operators) and every Borel set $B \subseteq R$ :

(a) Probability measure. $0 \leq μ_{ρ}^{A} (B) \leq 1$ and $μ_{ρ}^{A} (R) = 1$ ; for pairwise disjoint Borel sets $B_{n}$ with union $B$ one has $μ_{ρ}^{A} (B) = \sum_{n} μ_{ρ}^{A} (B_{n})$ ; and for the pure preparation of a unit vector $ψ$ the law is $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ . The declared outcome law of the model is therefore a probability measure on the Borel $σ$ -algebra of $R$ , not a formal weight attached to eigenvectors.

(b) Mixtures. If $ρ = \sum_{n} w_{n} ρ_{n}$ is a statistical mixture of preparations with weights $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , then $μ_{ρ}^{A} (B) = \sum_{n} w_{n} μ_{ρ_{n}}^{A} (B)$ for every Borel $B$ , the series converging absolutely: the outcome law of a mixture is the weighted average of the outcome laws, with classical weights.

(c) Exact values. For $λ \in R$ the probability of the exact value $λ$ is $μ_{ρ}^{A} ({λ}) = tr (ρ E_{A} ({λ}))$ ; it is zero for every preparation whenever $λ$ is not an eigenvalue of $A$ , and when $λ$ is an eigenvalue it equals the trace weight of the eigenspace $\ker (A - λ)$ , with $μ_{T_{u, u}}^{A} ({λ}) = 1$ for every unit eigenvector $u$ .

(d) Spectral support. $μ_{ρ}^{A} (B) = μ_{ρ}^{A} (B \cap σ (A))$ and $μ_{ρ}^{A} (σ (A)) = 1$ : outcomes outside the spectrum of the observable have probability zero.

(e) No density and no discreteness are asserted. The postulate asserts that $μ_{ρ}^{A}$ is a probability measure on the Borel sets and does not assert that it is a density with respect to Lebesgue measure or a finite or countable sum of point masses; the operational object named there is the probability of a declared Borel set, such as a finite-resolution bin, not the value of a density at a point (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clause (iii)). Which structure a particular pair $(A, ρ)$ exhibits is a property of that pair and is not fixed by the adopted law.

(f) What is not claimed. No mean, variance or higher moment is asserted by this item; their existence is conditional and is treated separately. No claim is made about the joint statistics of two or more observables, about the state after the outcome, about the physical mechanism producing an outcome, or about any concrete apparatus. The statement is conditional on P1 and P2 and on the cited mathematics.

Facts & Assumptions

Given: The adopted postulates P1 and P2 (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 Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice); the assigned state space $H$ with preparation class $D (H)$ of The state space of the framework: preparations, pure rays and density operators; an assigned sharp observable $(A, u_{A})$ with spectral PVM $E_{A}$ and spectrum $σ (A)$ ; preparations $ρ, ρ_{n}$ ; Borel sets $B, B_{n} \subseteq R$ ; and the readings of Assigned sharp observables and the outcome law of the framework.

[F1]

P1. Pure preparations are the unit rays of the assigned Hilbert space, general preparations are the density operators $ρ$ with $tr ρ = 1$ , and a statistical mixture $\sum_{n} w_{n} ρ_{n}$ with $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ is a preparation (State and composition postulate P1: Hilbert spaces, preparations and composite systems clauses (i)-(iii), The state space: density operators, pure and mixed states, and purity).

[F2]

P2. An ideal sharp real-valued quantity is a self-adjoint operator with a declared unit; its spectral PVM assigns to a Borel set $B$ the probability $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ , equal to $∥ E_{A} (B) ψ ∥^{2}$ for a unit pure preparation; the law is a measure and not necessarily a density or a discrete sum; an exact continuum value is an idealization, while a real detector's finite-resolution outcome is represented by a declared Borel set (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clauses (i)-(iii)).

[F3]

Scalar law. For every density operator $ρ$ with expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ and every PVM $E$ , the assignment $B \mapsto tr (ρ E (B))$ is a probability measure with $tr (ρ E (B)) = \sum_{n} w_{n} ∥ E (B) u_{n} ∥^{2}$ ; for a unit vector $u$ it equals $∥ E (B) u ∥^{2}$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F4]

Atoms and eigenvalues. $E_{A} ({λ})$ is the projection onto $\ker (A - λ)$ ; it is nonzero exactly when $λ$ is an eigenvalue of $A$ ; consequently $tr (ρ E_{A} ({λ})) = 0$ for every preparation when $λ$ is not an eigenvalue, and $μ_{T_{u, u}}^{A} ({λ}) = 1$ for every unit eigenvector $u$ at $λ$ (Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a) and (b)).

[F5]

Spectral support. $E_{A} (R ∖ σ (A)) = 0$ and $tr (ρ E_{A} (B)) = tr (ρ E_{A} (B \cap σ (A)))$ with $μ_{ρ}^{A} (σ (A)) = 1$ for every preparation (Support of the spectral measure, the spectrum and the values of the scalar law clauses (a) and (b)).

[F6]

Affinity. For a trace-norm convergent mixture $ρ = \sum_{n} w_{n} ρ_{n}$ of density operators one has $tr (ρ E_{A} (B)) = \sum_{n} w_{n} tr (ρ_{n} E_{A} (B))$ for every Borel set $B$ , the series converging absolutely (Affinity and trace-norm continuity of the scalar spectral law clauses (b) and (c)).

[F7]

Nature of the law (P2 clause (iii)). The postulate asserts that the outcome law is a measure, not necessarily a density with respect to Lebesgue measure and not necessarily a finite or countable sum of point masses; an exact value of a continuum quantity is an idealization, and a real detector reports the probability of a declared Borel set such as a bin (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clause (iii)).

[F8]

Spectral theorem. The spectral PVM $E_{A}$ exists, is unique among regular PVMs on the Borel sets of $R$ representing $A$ , and is the object whose scalar measures enter [F3] (Spectral theorem for unbounded self-adjoint operators (PVM form)).

Proof

1.1givenF2F3F8algebra

[Clause (a)] By [F2] the outcome law is defined using the spectral PVM of [F8], and by $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ ; by [F3] it is a probability measure on the Borel sets with values in $[0, 1]$ , normalization $μ_{ρ}^{A} (R) = tr ρ = 1$ and countable additivity, and for a unit vector $ψ$ it equals $∥ E_{A} (B) ψ ∥^{2}$ . This is clause (a).

1.2givenF1F6algebra

[Clause (b)] Let $ρ = \sum_{n} w_{n} ρ_{n}$ be a statistical mixture; by [F1] it is a preparation, and [F6] gives $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B)) = \sum_{n} w_{n} tr (ρ_{n} E_{A} (B)) = \sum_{n} w_{n} μ_{ρ_{n}}^{A} (B)$ for every Borel $B$ with absolute convergence.

1.3givenF4algebra

[Clause (c)] By [F4], $E_{A} ({λ})$ is the projection onto $\ker (A - λ)$ and vanishes exactly when $λ$ is not an eigenvalue; hence $μ_{ρ}^{A} ({λ}) = tr (ρ E_{A} ({λ})) = 0$ for every preparation in that case, while for a unit eigenvector $u$ at $λ$ the same reference gives $μ_{T_{u, u}}^{A} ({λ}) = 1$ , and the value is the trace weight of the eigenspace.

1.4givenF5algebra

[Clause (d)] By [F5], $E_{A} (R ∖ σ (A)) = 0$ and hence $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B)) = tr (ρ E_{A} (B \cap σ (A))) = μ_{ρ}^{A} (B \cap σ (A))$ , with $μ_{ρ}^{A} (σ (A)) = tr ρ = 1$ ; this is the spectral-support reading of [F2].

1.5givenF2F7

[Clause (e)] By [F2] the law is asserted to be a probability measure, and by [F7] the postulate neither asserts that it is a density with respect to Lebesgue measure nor that it is a finite or countable sum of point masses, the operational object being the probability of a declared Borel set. Therefore clause (e) is a reading of P2 clause (iii) alone, and no further structure of $μ_{ρ}^{A}$ is claimed.

2.1step 1.1step 1.2step 1.3step 1.4step 1.5∎

[Clause (f) and conclusion] Steps 1.1-1.4 prove clauses (a)-(d) from P1, P2 and the cited mathematics, and step 1.5 records the limits of clause (e). Nothing in the argument uses moments, joint statistics, state updates, apparatus models or data, so none of these is claimed.

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Mean, second moment and variance of an assigned observable in a preparation

Physical scope

Conditional statements, within the framework's observable and probability postulate, about the moments of an assigned sharp observable in a preparation. For a pure preparation represented by a unit vector ψ, the second moment is finite exactly when ψ lies in D(A), the mean is finite exactly when ψ lies in D(|A|^{1/2}), and for ψ in D(A) the mean is the quadratic form and the variance is the squared norm of the centred vector. For a density-operator expansion ρ=ΣₙwₙT_{uₙ,uₙ}, the mean is finite exactly when Σₙwₙ∫|λ|dE_{A,uₙ}<∞, and the second moment is finite exactly when every positive-weight uₙ lies in D(A) and Σ_{n:wₙ>0}wₙ||Auₙ||²<∞. A preparation need not have finite moments automatically. The item asserts no apparatus error, no disturbance and no empirical result.

Statement

Adopt the postulates P1 and P2 (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) 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 $(A, u_{A})$ be an assigned sharp observable with outcome law $μ^{A}$ in the reading of Assigned sharp observables and the outcome law of the framework, and let $ρ$ be a preparation. Then:

(a) Pure preparations. Let $ρ = T_{ψ, ψ}$ be the pure preparation of a unit vector $ψ \in D (A)$ . Then the mean of the outcome law exists and equals $⟨ A ψ, ψ ⟩$ , and the second moment exists and equals $∥ A ψ ∥^{2}$ : $m (ψ) = \int_{R} λ d μ_{ψ}^{A} (λ) = ⟨ A ψ, ψ ⟩, \int_{R} λ^{2} d μ_{ψ}^{A} (λ) = ∥ A ψ ∥^{2},$ both read in the powers of the declared unit $u_{A}$ of the observable. If $ψ \in D (A^{2})$ , the variance is $\int_{R} (λ - m (ψ))^{2} d μ_{ψ}^{A} (λ) = ⟨ A^{2} ψ, ψ ⟩ - ⟨ A ψ, ψ ⟩^{2} .$

(b) Variance and its hypotheses. For every unit $ψ \in D (A)$ the variance exists and equals ${Var}_{ψ} (A) = ∥ A ψ ∥^{2} - ⟨ A ψ, ψ ⟩^{2} = ∥ (A - ⟨ A ψ, ψ ⟩ I) ψ ∥^{2} \geq 0,$ without any hypothesis beyond $ψ \in D (A)$ ; no appeal to a discrete eigenbasis is made.

(c) Non-existence is possible. The identities of (a) and (b) are conditional on membership of $ψ$ in the stated domains. If $ψ \notin D (∣ A ∣^{1 / 2})$ then $\int_{R} ∣ λ ∣ d μ_{ψ}^{A} (λ) = \infty$ , so the mean of the outcome law does not exist as a finite real number; the mean is finite exactly when $ψ \in D (∣ A ∣^{1 / 2})$ , and then $ψ$ need not lie in $D (A)$ . If $ψ \notin D (A)$ then the second moment $\int λ^{2} d μ_{ψ}^{A}$ is infinite. In particular every preparation of the model is a unit ray or a density operator, and no preparation is equipped with finite moments automatically.

(d) Mixtures. Let $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ be a spectral expansion of a preparation, $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , $(u_{n})$ orthonormal. In every weighted moment sum, zero-weight terms are defined to be zero before evaluating a component integral; equivalently, only positive-weight indices are summed. This applies also to component means that are undefined at zero-weight indices. Then the mean of $μ_{ρ}^{A}$ is finite if and only if $\sum_{n} w_{n} \int ∣ λ ∣ d E_{A, u_{n}} < \infty$ , and then $\int λ d μ_{ρ}^{A} = \sum_{n} w_{n} \int λ d E_{A, u_{n}}$ . For the second moment, put $M_{n} : = \int λ^{2} d E_{A, u_{n}}$ , so $M_{n} = ∥ A u_{n} ∥^{2}$ for $u_{n} \in D (A)$ and $M_{n} = + \infty$ otherwise; a zero weight contributes zero. Then $\int λ^{2} d μ_{ρ}^{A} = \sum_{n} w_{n} M_{n} \in [0, \infty]$ , and it is finite exactly when every $u_{n}$ of positive weight lies in $D (A)$ and $\sum_{n : w_{n} > 0} w_{n} ∥ A u_{n} ∥^{2} < \infty$ . Thus componentwise domain membership alone does not guarantee a finite mixed-state second moment. Whenever the mixture has a finite absolute first moment, the identity of clause (b) of The Born outcome law: probability measure, mixtures, atoms and spectral support gives its mean as the absolutely convergent weighted average of the component means.

(e) Status. Clauses (a)-(d) are conditional consequences of P1 and P2 together with the mathematical moment results cited below; they assert no apparatus resolution, no measurement error, no disturbance and no empirical value. Whether a concrete preparation has a finite mean or variance for a concrete assigned observable is a question about the pair $(ρ, A)$ and the tested identification of the observable, not a consequence of the framework's postulates alone.

Facts & Assumptions

Given: The adopted postulates P1 and P2 (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 Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice); an assigned sharp observable $(A, u_{A})$ with spectral law $μ^{A}$ and declared unit $u_{A}$ in the reading of Assigned sharp observables and the outcome law of the framework; a unit vector $ψ$ ; a preparation $ρ$ with spectral expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ (The state space: density operators, pure and mixed states, and purity, Density operators and scalar spectral probability clause (a)).

[F1]

The outcome law. For an assigned observable the preparation-dependent law is $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ , equal to $∥ E_{A} (B) ψ ∥^{2}$ for a pure preparation of a unit vector $ψ$ , and $B \mapsto μ_{ρ}^{A} (B)$ is a probability measure on the Borel sets of $R$ (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 clause (a)).

[F2]

Moments of the scalar law. For a unit vector $ψ$ : the second moment satisfies $\int λ^{2} d μ_{ψ}^{A} = ∥ A ψ ∥^{2}$ if $ψ \in D (A)$ and equals $+ \infty$ otherwise; the mean satisfies $\int ∣ λ ∣ d μ_{ψ}^{A} < \infty$ exactly when $ψ \in D (∣ A ∣^{1 / 2})$ , and is then finite and real with $\int λ d μ_{ψ}^{A} = ⟨ A ψ, ψ ⟩$ for $ψ \in D (A)$ ; for $ψ \in D (A)$ the variance is $∥ A ψ ∥^{2} - ⟨ A ψ, ψ ⟩^{2} = ∥ (A - ⟨ A ψ, ψ ⟩ I) ψ ∥^{2}$ , and for $ψ \in D (A^{2})$ it is $⟨ A^{2} ψ, ψ ⟩ - ⟨ A ψ, ψ ⟩^{2}$ (Mean, second moment and variance of the scalar spectral law clauses (a)-(c)).

[F3]

Mixtures. For preparations $ρ_{n}$ with weights $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , the mixture $ρ = \sum_{n} w_{n} ρ_{n}$ is a preparation and $μ_{ρ}^{A} (B) = \sum_{n} w_{n} μ_{ρ_{n}}^{A} (B)$ for every Borel $B$ (Affinity and trace-norm continuity of the scalar spectral law clause (b), clauses (b) and (d)).

[F4]

State moments of a mixture. All sums use clause (d)'s zero-weight convention. For a density operator with expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ , the mean of $μ_{ρ}^{A}$ is finite exactly when $\sum_{n} w_{n} \int ∣ λ ∣ d E_{A, u_{n}} < \infty$ and is then $\sum_{n} w_{n} \int λ d E_{A, u_{n}}$ , while $\int λ^{2} d μ_{ρ}^{A} = \sum_{n} w_{n} M_{n} \in [0, \infty]$ , where $M_{n} = \int λ^{2} d E_{A, u_{n}}$ equals $∥ A u_{n} ∥^{2}$ on $D (A)$ and $+ \infty$ otherwise (Mean, second moment and variance of the scalar spectral law clause (d)).

[F5]

Preparations and domains. A preparation is a unit ray of the assigned Hilbert space or a density operator on it; being a preparation imposes no membership in $D (A)$ or $D (A^{2})$ beyond what is stated for the assigned observable $A$ (The state space of the framework: preparations, pure rays and density operators clauses (a)-(c), State and composition postulate P1: Hilbert spaces, preparations and composite systems clauses (ii)-(iii)).

Proof

1.1givenF1F2algebra

[Clause (a): mean and second moment of a pure preparation] Let $ρ = T_{ψ, ψ}$ with $∥ ψ ∥ = 1$ . By [F1] the law of the preparation is $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ ; by [F2] its second moment is $∥ A ψ ∥^{2}$ if $ψ \in D (A)$ (and $+ \infty$ otherwise), and its mean is $⟨ A ψ, ψ ⟩$ when $ψ \in D (A)$ ; when $ψ \in D (A^{2})$ , [F2] gives the displayed variance value. This proves (a).

1.2givenF2F5algebra

[Clause (c): domains and non-existence] By [F2], $\int ∣ λ ∣ d μ_{ψ}^{A}$ is finite exactly for $ψ \in D (∣ A ∣^{1 / 2})$ , and then the mean is finite; if $ψ \notin D (∣ A ∣^{1 / 2})$ the integral of $∣ λ ∣$ diverges and no finite mean exists, while $\int λ^{2} d μ_{ψ}^{A} = \infty$ whenever $ψ \notin D (A)$ . Membership of $ψ$ in these domains is not implied by being a preparation: [F5] says only that preparations are unit rays or density operators.

2.1givenstep 1.1F2algebra

[Clause (b): variance under the single hypothesis $ψ \in D (A)$ ] Let $ψ \in D (A)$ be a unit vector. By [F2] the variance equals $∥ A ψ ∥^{2} - ⟨ A ψ, ψ ⟩^{2}$ and also $∥ (A - m) ψ ∥^{2}$ with $m = ⟨ A ψ, ψ ⟩$ , a real number because $A$ is self-adjoint; this squared norm is nonnegative, so the variance exists, is finite and is nonnegative, and no eigenbasis or additional domain hypothesis is used.

3.1givenstep 1.1step 2.1F2F3F4algebra

[Clause (d): mixtures] Let $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ be the spectral expansion, omitting zero-weight terms before evaluating any component moment as stipulated in (d). By [F4] the mean of $μ_{ρ}^{A}$ is finite exactly when $\sum_{n} w_{n} \int ∣ λ ∣ d E_{A, u_{n}} < \infty$ and then has the displayed value. The same series identity gives $\int λ^{2} d μ_{ρ}^{A} = \sum_{n} w_{n} M_{n}$ , with zero-weight terms omitted and $M_{n} = \int λ^{2} d E_{A, u_{n}} = ∥ A u_{n} ∥^{2}$ on $D (A)$ and $+ \infty$ off it by [F2]. A nonnegative series is finite exactly when all its positive-weight terms are finite and their weighted sum converges, so membership of each positive-weight $u_{n}$ in $D (A)$ is necessary but not sufficient. The mixture identity for the law itself is [F3]; taking $g = ∣ λ ∣$ and $λ^{2}$ in [F4] and combining with [F2] yields the stated criteria. Under the finite absolute first-moment condition, the component absolute first moments have finite weighted sum, so summing positive and negative parts separately gives the absolutely convergent weighted-mean identity.

4.1step 1.1step 1.2step 2.1step 3.1∎

[Clause (e) and conclusion] Steps 1.1, 1.2, 2.1 and 3.1 prove (a)-(d) as conditional statements from P1, P2 and the cited mathematics; the reminders recorded in (e) are visible in the hypotheses of those steps, and no apparatus, error, disturbance or empirical claim enters the argument.

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Position and momentum outcome laws and the Born density of the massive particle model

Physical scope

Conditional consequences of the adopted observable and probability postulate for the spinless massive-particle model of the kinematics postulate: the position operators are self-adjoint with maximal domains and pure-preparation coordinate laws given by the marginals of |ψ|²; their unique joint position PVM has density |ψ|² for pure preparations and mixture density Σₙwₙ|uₙ|² for a density-operator preparation ρ=ΣₙwₙT_{uₙ,uₙ}; the momentum operators are self-adjoint with Fourier multipliers; |F₂ψ(ξ)|² is the frequency density, and the momentum-coordinate density is its marginal scaled by (2πℏ)⁻¹; exact values of each position and momentum coordinate have probability zero for every preparation although the spectra are all of R. The item is spinless, asserts no joint position-momentum statistics, no dynamics and no empirical result.

Statement

Adopt the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2) and the kinematics and massive-particle model postulate Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators (P5) for a spinless particle of mass $m > 0$ and action constant $ℏ > 0$ , with state space $H = L^{2} (R^{3}; C)$ and position and momentum operators $Q_{j}, P_{j}$ as in Position and momentum operators on Euclidean L2, 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 $F_{2}$ be the unitary Plancherel transform with the convention $\hat{f} (ξ) = \int f (x) e^{- 2 π i x \cdot ξ} d x$ on $L^{1}$ (Fourier transform on complex L1 classes, Plancherel theorem), and for a unit vector $ψ \in H$ let $μ_{ψ}^{Q_{j}}$ and $μ_{ψ}^{P_{j}}$ be the outcome laws of the assigned observables $Q_{j}$ and $P_{j}$ in the reading of Assigned sharp observables and the outcome law of the framework. Then, using the numerical SI coordinates of Dimensions, SI unit conventions and quantity types of the framework: $x_{j}$ and the marginal coordinate $t$ are read in metres, $ξ_{j}$ in $m^{- 1}$ , $p$ in $k g m s^{- 1}$ , $m$ in kilograms and $ℏ$ in $J s$ . The assigned pairs are $(Q_{j}, m)$ and $(P_{j}, k g m s^{- 1})$ ; the factor $2 π ℏ ξ_{j}$ has momentum units and $x \cdot ξ$ is dimensionless. Normalized position wave functions have units $m^{- 3 / 2}$ , their Fourier transforms $m^{3 / 2}$ , and the marginal densities below have the inverse units of their respective outcome coordinates.

(a) Position observable. $Q_{j}$ is self-adjoint with domain $D (Q_{j}) = {ψ \in H : x_{j} ψ \in H}$ , its spectral PVM is given by $E_{Q_{j}} (B) ψ = 1_{x_{j}^{- 1} (B)} ψ$ , and $μ_{ψ}^{Q_{j}} (B) = \int_{x_{j}^{- 1} (B)} ∣ ψ (x) ∣^{2} d x (B \subseteq R Borel) .$ The law is absolutely continuous with respect to one-dimensional Lebesgue measure. Its density is the $L^{1} (R)$ equivalence class $f_{j}$ given for almost every $t$ by $f_{j} (t) : = \int_{R^{2}} ∣ ψ (x_{⊥}, t) ∣^{2} d x_{⊥};$ this class is independent of the chosen representative of $ψ$ and is the $j$ -th marginal density of $∣ ψ ∣^{2}$ . The law is atomless: $μ_{ψ}^{Q_{j}} ({λ}) = 0$ for every $λ \in R$ . Moreover $σ (Q_{j}) = R$ .

(b) Joint position law. There is a unique regular projection-valued measure $G$ on the Borel $σ$ -algebra of $R^{3}$ with $G (B_{1} \times B_{2} \times B_{3}) = E_{Q_{1}} (B_{1}) E_{Q_{2}} (B_{2}) E_{Q_{3}} (B_{3})$ for all Borel $B_{1}, B_{2}, B_{3} \subseteq R$ , and its scalar law in the preparation $ψ$ is the measure of the density $∣ ψ ∣^{2}$ : $μ_{ψ}^{G} (B) = ∥ G (B) ψ ∥^{2} = \int_{B} ∣ ψ (x) ∣^{2} d x (B \subseteq R^{3} Borel) .$ For a density-operator preparation $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ , its joint scalar law is $μ_{ρ}^{G} (B) : = tr (ρ G (B)) = \sum_{n} w_{n} \int_{B} ∣ u_{n} (x) ∣^{2} d x = \int_{B} f_{ρ} (x) d x, f_{ρ} (x) : = \sum_{n} w_{n} ∣ u_{n} (x) ∣^{2},$ where $f_{ρ} \in L^{1} (R^{3})$ is a nonnegative density class of total integral $1$ , by [F8] and Tonelli. Thus the model's joint position law in a spatial region $B$ has density $∣ ψ ∣^{2}$ for a pure preparation and $f_{ρ}$ for a density-operator preparation.

(c) Momentum observable. $P_{j}$ is self-adjoint with domain $D (P_{j}) = {ψ \in H : ξ_{j} F_{2} ψ \in L^{2}}$ and Fourier representation $F_{2} (P_{j} ψ) (ξ) = 2 π ℏ ξ_{j} F_{2} ψ (ξ)$ ; its spectral PVM is $E_{P_{j}} (B) = F_{2}^{- 1} M_{1_{m^{- 1} (B)}} F_{2}$ with $m (ξ) = 2 π ℏ ξ_{j}$ , and $μ_{ψ}^{P_{j}} (B) = \int_{m^{- 1} (B)} ∣ F_{2} ψ (ξ) ∣^{2} d ξ, m^{- 1} (B) = {ξ \in R^{3} : 2 π ℏ ξ_{j} \in B} .$ The law is absolutely continuous with respect to Lebesgue measure in the momentum variable $p$ . Its density is the $L^{1} (R)$ equivalence class $g_{j}$ given for almost every $p$ by $g_{j} (p) = \frac{1}{2 π ℏ} \int_{R^{2}} ∣ F_{2} ψ (ξ_{⊥}, \frac{p}{2 π ℏ}) ∣^{2} d ξ_{⊥}$ where the class is independent of the chosen representative of $F_{2} ψ$ . The law is atomless; moreover $σ (P_{j}) = R$ .

(d) Exact values. For every $λ \in R$ one has $μ_{ψ}^{Q_{j}} ({λ}) = 0$ and $μ_{ψ}^{P_{j}} ({λ}) = 0$ for every preparation, although both observables have spectrum $R$ . This says that each coordinate law is atomless; it does not remove $R$ from the law's Borel outcome space. Finite-resolution readings are the Borel bin probabilities of (a)-(c).

(e) Scope. The statement is conditional on P2, on the massive spinless model of P5 and on the cited mathematics. The componentwise extension to the spin- $s$ model is not proved here; no joint statistics of a position and a momentum observable, no dynamics and no apparatus or empirical claim are made. The identification of the operators $Q_{j}, P_{j}$ with concrete laboratory position and momentum measurements is a tested identification under P7 and is not asserted by this item.

Facts & Assumptions

Given: The adopted postulates P2 and P5 for a spinless particle of mass $m > 0$ with state space $H = L^{2} (R^{3}; C)$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators); the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice); the operators $Q_{j}, P_{j}$ of Position and momentum operators on Euclidean L2; the Plancherel transform $F_{2}$ of Fourier transform on complex L1 classes and Plancherel theorem; a unit vector $ψ \in H$ ; a density-operator preparation $ρ$ and its finite or countable spectral expansion when it is used; Borel sets $B, B_{1}, B_{2}, B_{3}$ ; and the outcome-law reading of Assigned sharp observables and the outcome law of the framework.

[F1]

Outcome law of an assigned observable. For an assigned observable $A$ and a unit vector $ψ$ the law is $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ , where $E_{A}$ is the spectral PVM of $A$ (Assigned sharp observables and the outcome law of the framework clauses (b) and (c), clause (ii)).

[F2]

The massive-particle model. P5 assigns the state space $H = L^{2} (R^{3}; C)$ to a spinless particle of mass $m > 0$ and defines $(Q_{j} ψ) (x) = x_{j} ψ (x)$ on the classes with $x_{j} ψ \in L^{2}$ and $(P_{k} ψ) = - i ℏ D_{k} ψ$ on the classes whose componentwise weak derivative lies in $L^{2}$ (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators clause (iii)).

[F3]

Self-adjointness and Fourier representation. $Q_{j}$ is self-adjoint with $D (Q_{j}) = {ψ : x_{j} ψ \in L^{2}}$ ; $P_{j}$ is self-adjoint with $D (P_{j}) = {ψ : ξ_{j} F_{2} ψ \in L^{2}}$ and $F_{2} (P_{j} ψ) (ξ) = 2 π ℏ ξ_{j} F_{2} ψ (ξ)$ almost everywhere (Self-adjointness, Fourier representation and translation identities of position and momentum clauses (a) and (b)).

[F4]

Multiplication operators. Let $(X, Σ, μ)$ be a $σ$ -finite measure space and $m : X \to R$ measurable and finite $μ$ -a.e.; then $M_{m} f = m f$ on $D (M_{m}) = {f : \int ∣ m ∣^{2} ∣ f ∣^{2} d μ < \infty}$ is self-adjoint with regular spectral PVM $E (B) = M_{1_{m^{- 1} (B)}}$ , $\int λ d E = M_{m}$ and $σ (M_{m})$ equal to the essential range of $m$ (Spectral measure of a real L2 multiplication operator clauses (a)-(d)).

[F5]

Unitary transport of a spectral PVM. If $U$ is unitary and $F$ is a regular PVM, then $B \mapsto U^{- 1} F (B) U$ is a regular PVM with $U^{- 1} (\int λ d F (λ)) U = \int λ d (U^{- 1} F U) (λ)$ on the transported domain; the spectral PVM of a self-adjoint operator is the unique regular PVM representing it, so a transported regular PVM representing the transported operator is its spectral PVM (Spectral theorem for unbounded self-adjoint operators (PVM form)).

[F6]

Tonelli, null sets and indefinite integrals. For $σ$ -finite measure spaces, integrals of product-measurable nonnegative functions may be iterated (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product); for nonnegative measurable $f$ the set function $A \mapsto \int_{A} f d μ$ is a measure (The indefinite integral of a nonnegative measurable function is a measure); countable subsets of $R^{n}$ are Lebesgue-null (Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ ); the substitution $p = 2 π ℏ s$ has positive Jacobian $2 π ℏ$ by A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions.

[F7]

Measure uniqueness on a pi-system. Finite measures on the $σ$ -algebra generated by a $π$ -system that agree on the $π$ -system and have equal total mass are equal (Finite measures agreeing on a generating pi-system and on the whole space are equal); the Euclidean Borel $σ$ -algebra of $R^{3}$ is generated by the rectangles $B_{1} \times B_{2} \times B_{3}$ (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n},).

[F8]

Density-operator law. Every density-operator preparation has a finite or countable expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ with $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ and $(u_{n})$ orthonormal; for any PVM $E$ its law is $μ_{ρ} (B) = \sum_{n} w_{n} ∥ E (B) u_{n} ∥^{2}$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F9]

Regularity on $R^{3}$ . Every finite Borel measure on the second-countable locally compact Hausdorff space $R^{3}$ is regular (Locally finite Borel measures on second-countable LCH spaces are regular); a regular PVM is a PVM whose finite scalar measures $B \mapsto ⟨ E (B) χ, χ ⟩$ are regular (Projection valued measure, Regular Borel measure on an LCH space).

[F10]

Borel representatives and product measures. Under Countable Choice, Lebesgue measure is the completion of its Borel restriction, and completion-measurable real functions have Borel representatives equal almost everywhere ( $L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets, A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra). Apply this separately to the real and imaginary parts and replace any infinite values on their Borel null sets by zero to obtain finite complex Borel representatives. Product Lebesgue measure agrees with Euclidean Lebesgue measure on Borel sets (On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}), and Borel functions on the split coordinates are product-measurable by [F7]. If a representative is changed on a Lebesgue-null set $N$ , enclose $N$ in a Borel null set $Z$ using the completion description; Tonelli [F6] applied to $1_{Z}$ makes almost every section $Z_{t}$ null. Completeness then makes $N_{t}$ measurable and null for those $t$ , so the slice integrals are unchanged almost everywhere.

Proof

1.1givenF1F2F3F4F6F7F10algebra

[The position observable] Choose finite Borel representatives of all wave functions used below by [F10]; their integrals agree with those of the original Lebesgue classes. On the $σ$ -finite space $(R^{3}, Borel, Lebesgue)$ the function $x \mapsto x_{j}$ is real, measurable and finite everywhere, so [F4] applies to $M_{x_{j}}$ ; by [F2] and [F3] the operator $Q_{j}$ is exactly $M_{x_{j}}$ with its displayed domain, hence is self-adjoint with spectral PVM $E_{Q_{j}} (B) = M_{1_{x_{j}^{- 1} (B)}}$ and $σ (Q_{j})$ , the essential range of $x_{j}$ , equal to $R$ (every open interval in the $j$ -th coordinate has a slab preimage of positive Lebesgue measure). For a unit vector $ψ$ , [F1] gives $μ_{ψ}^{Q_{j}} (B) = ∥ E_{Q_{j}} (B) ψ ∥^{2} = \int_{x_{j}^{- 1} (B)} ∣ ψ (x) ∣^{2} d x$ . Tonelli [F6] gives a nonnegative measurable slice integral $f_{j} (t) = \int_{R^{2}} ∣ ψ (x_{⊥}, t) ∣^{2} d x_{⊥}$ for almost every $t$ and $μ_{ψ}^{Q_{j}} (B) = \int_{B} f_{j} (t) d t$ for every Borel $B$ . If the representative of $ψ$ changes on a Lebesgue-null subset of $R^{3}$ , the Borel-null envelope and section argument of [F10] shows that the slice function changes only on a null subset of $R$ , so it defines a representative-independent $L^{1} (R)$ class with integral $1$ . For the atomless claim, Tonelli gives $μ_{ψ}^{Q_{j}} ({λ}) = \int_{{λ}} f_{j} (t) d t = 0$ because ${λ}$ is countable hence Lebesgue-null [F6]. This proves clause (a).

1.2givenF1F3F4F5F6F10algebra

[The momentum observable] By [F3] the operator $P_{j}$ is self-adjoint with the stated domain and Fourier representation. Let $M : = M_{2 π ℏ ξ_{j}}$ be the multiplication operator of [F4] on $L^{2} (R^{3}; C)$ with multiplier $m (ξ) = 2 π ℏ ξ_{j}$ , whose spectral PVM is $E^{M} (B) = M_{1_{m^{- 1} (B)}}$ ; then $P_{j} = F_{2}^{- 1} M F_{2}$ on $D (P_{j}) = F_{2}^{- 1} D (M)$ by [F3], so by the transport rule [F5] the operator family $E_{P_{j}} (B) : = F_{2}^{- 1} E^{M} (B) F_{2}$ is a regular PVM representing $P_{j}$ ; since the spectral PVM is unique among regular PVMs representing a self-adjoint operator, $E_{P_{j}}$ is the spectral PVM of $P_{j}$ and $σ (P_{j}) = σ (M) = R$ is the essential range of $2 π ℏ ξ_{j}$ . For a unit $ψ$ , [F1] and unitarity of $F_{2}$ give $μ_{ψ}^{P_{j}} (B) = ∥ E_{P_{j}} (B) ψ ∥^{2} = ∥ E^{M} (B) F_{2} ψ ∥^{2} = \int_{m^{- 1} (B)} ∣ F_{2} ψ (ξ) ∣^{2} d ξ$ . Substituting $p = 2 π ℏ ξ_{j}$ and applying Tonelli [F6] gives a nonnegative measurable slice density $g_{j} (p) = \frac{1}{2 π ℏ} \int_{R^{2}} ∣ F_{2} ψ (ξ_{⊥}, p / (2 π ℏ)) ∣^{2} d ξ_{⊥}$ for almost every $p$ , with $μ_{ψ}^{P_{j}} (B) = \int_{B} g_{j} (p) d p$ for every Borel $B$ . The section argument of [F10], followed by the positive scaling $p = 2 π ℏ s$ , shows that changes of representative of $F_{2} ψ$ change this slice function only on a null set of $p$ , so it defines a representative-independent $L^{1} (R)$ class. For $c = λ / (2 π ℏ)$ , Tonelli [F6] and the nullity of the singleton ${c} \subset R$ give $μ_{ψ}^{P_{j}} ({λ}) = \int_{{c}} (\int_{R^{2}} ∣ F_{2} ψ (ξ_{⊥}, s) ∣^{2} d ξ_{⊥}) d s = 0$ . Thus the momentum coordinate law is atomless. This proves clause (c).

2.1step 1.1step 1.2F8algebra

[Exact values for density-operator preparations] Let $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ be the finite or countable expansion of a density-operator preparation from [F8]. For every $j$ and $λ$ , the law formula of [F8] gives $μ_{ρ}^{Q_{j}} ({λ}) = \sum_{n} w_{n} μ_{u_{n}}^{Q_{j}} ({λ}) = 0, μ_{ρ}^{P_{j}} ({λ}) = \sum_{n} w_{n} μ_{u_{n}}^{P_{j}} ({λ}) = 0,$ because each unit vector $u_{n}$ has zero singleton probabilities by steps 1.1 and 1.2. Every preparation is either a pure unit ray or a density operator in the state-space assignment The state space of the framework: preparations, pure rays and density operators, so this proves clause (d) for every preparation.

2.2step 1.1F1F4F6F7F8F9F10algebra

[The joint position law] Define $G (B) : = M_{1_{B}}$ for Borel $B \subseteq R^{3}$ . Each $G (B)$ is an orthogonal projection, $G (\emptyset) = 0$ , $G (R^{3}) = I$ , and $G (B) G (C) = G (B \cap C)$ . If pairwise disjoint $B_{n}$ have union $B$ , then for every $χ \in H$ the squared norm of $G (B) χ - \sum_{n \leq N} G (B_{n}) χ$ is $\int_{B ∖ \cup_{n \leq N} B_{n}} ∣ χ ∣^{2} d x$ , which tends to zero by countable additivity of the measure $C \mapsto \int_{C} ∣ χ ∣^{2} d x$ [F6]; hence $G$ is a PVM. For each $χ \in H$ , its scalar measure is $⟨ G (B) χ, χ ⟩ = \int_{B} ∣ χ (x) ∣^{2} d x$ , a finite Borel measure of total mass $∥ χ ∥^{2}$ by [F6], and it is regular by [F9] because $R^{3}$ is second-countable, locally compact and Hausdorff. By the regularity definition in [F9], $G$ is a regular PVM. Its rectangle values satisfy $G (B_{1} \times B_{2} \times B_{3}) = M_{1_{B_{1}} (x_{1}) 1_{B_{2}} (x_{2}) 1_{B_{3}} (x_{3})} = E_{Q_{1}} (B_{1}) E_{Q_{2}} (B_{2}) E_{Q_{3}} (B_{3})$ by step 1.1. For a unit $ψ$ , the scalar law is $μ_{ψ}^{G} (B) = ∥ G (B) ψ ∥^{2} = \int_{B} ∣ ψ (x) ∣^{2} d x$ . For a density operator $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ , [F8] gives $μ_{ρ}^{G} (B) = tr (ρ G (B)) = \sum_{n} w_{n} \int_{B} ∣ u_{n} (x) ∣^{2} d x$ ; Tonelli [F6] identifies this with $\int_{B} f_{ρ} (x) d x$ for $f_{ρ} : = \sum_{n} w_{n} ∣ u_{n} ∣^{2}$ , whose integral is $\sum_{n} w_{n} = 1$ . If $H^{'}$ is another regular PVM with the same values on all rectangles, then for every unit $φ$ the finite measures $B \mapsto ⟨ H^{'} (B) φ, φ ⟩$ and $B \mapsto ⟨ G (B) φ, φ ⟩$ agree on rectangles, which form a $π$ -system generating the Borel $σ$ -algebra of $R^{3}$ by [F7], and have the common total mass $∥ φ ∥^{2}$ ; [F7] makes them equal on all Borel sets, and polarization recovers equality of the operator values, so $H^{'} = G$ . Thus $G$ is the unique regular PVM with the displayed rectangle values and the joint position density is $∣ ψ ∣^{2}$ for a pure preparation and $f_{ρ}$ for a density-operator preparation.

2.3step 1.1step 1.2F1algebra

[Exact values for pure preparations] For a pure preparation represented by a unit vector $ψ$ , steps 1.1 and 1.2 give $μ_{ψ}^{Q_{j}} ({λ}) = μ_{ψ}^{P_{j}} ({λ}) = 0$ for every $λ \in R$ . The spectra remain $σ (Q_{j}) = σ (P_{j}) = R$ by those same steps; no finite-resolution reading is affected, since the bin probabilities remain those of (a)-(c).

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

[Conclusion] Steps 1.1 and 1.2 prove (a) and (c), step 2.2 proves (b), and steps 2.3 and 2.1 prove (d). Each is conditional on P2, P5 and the cited mathematical hypotheses; the limits in (e) are exactly those used in the argument.

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Distinct preparations are separated by the statistics of a bounded sharp observable

Physical scope

A conditional separation statement for the framework's preparations: two distinct preparations of a declared system assign different probabilities to some outcome of some bounded sharp observable of the framework, namely to the outcome 1 of a rank-one orthogonal projection; the difference is bounded by the trace-norm distance of the two states. The item asserts no finite-measurement procedure that identifies a preparation, no apparatus realization and no empirical result; it concerns the model's state space under the state and observable postulates.

Statement

Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1) and the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2), 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 $H$ be the state space assigned to a declared system and let $ρ \neq σ$ be two preparations of the system in the sense of The state space of the framework: preparations, pure rays and density operators clauses (a)-(c). Then there is a bounded sharp observable of the framework, namely an orthogonal projection $P$ of rank one, such that the outcome laws of Assigned sharp observables and the outcome law of the framework differ at the outcome set ${1}$ : $μ_{ρ}^{P} ({1}) = tr (ρ P) \neq tr (σ P) = μ_{σ}^{P} ({1}) .$ Moreover for every rank-one projection $P$ one has $∣ μ_{ρ}^{P} ({1}) - μ_{σ}^{P} ({1}) ∣ \leq ∥ ρ - σ ∥_{1}$ ; consequently, if $μ_{ρ}^{P} = μ_{σ}^{P}$ for every bounded sharp observable $P$ (or merely for every rank-one projection), then $ρ = σ$ .

The statement is conditional on P1 and P2 and asserts no procedure that determines a preparation from finitely many measurements, no apparatus realization, no error rate and no empirical result.

Facts & Assumptions

[F1]

Mathematical separation. For distinct density operators $ρ, σ$ there is a unit vector $w \in H$ with $⟨ ρ w, w ⟩ \neq ⟨ σ w, w ⟩$ ; for the rank-one projection $P = T_{w, w}$ the spectral law of $P$ satisfies $μ_{ρ}^{P} ({1}) = tr (ρ P) = ⟨ ρ w, w ⟩$ and likewise for $σ$ , and $∣ tr ((ρ - σ) T_{w, w}) ∣ \leq ∥ ρ - σ ∥_{1}$ (Distinct density operators are separated by the law of a bounded rank-one projection clauses (a)-(c)).

[F2]

The framework's outcome law and its reading. An assigned sharp observable has spectral PVM $E_{A}$ and outcome law $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ , equal to $∥ E_{A} (B) ψ ∥^{2}$ for a pure preparation; the law is a probability measure on the Borel sets of $R$ , and the outcome-set reading of the model is that of Assigned sharp observables and the outcome law of the framework (The Born outcome law: probability measure, mixtures, atoms and spectral support clauses (a)-(c), clause (ii)).

[F3]

Projections are assigned bounded observables. An orthogonal projection $P$ is a bounded self-adjoint operator with $0 \leq P \leq I$ on $H$ (The Hilbert orthogonal projection onto a closed subspace, Hilbert projections are linear, self-adjoint and contractive), hence qualifies as a sharp observable of the framework when assigned with the dimensionless unit of its spectral coordinate.

Proof

1.1givenF1F2F3algebra

[Existence of a separating bounded observable] Let $w$ and $P = T_{w, w}$ be as in [F1]: such a unit vector exists because $ρ \neq σ$ and $ρ - σ$ is a nonzero self-adjoint trace-class operator. By [F3] the operator $P$ is a bounded self-adjoint projection, hence an assigned sharp observable of the framework once its unit (the dimensionless spectral coordinate, which here takes the values $0$ and $1$ ) is declared; its spectral law at ${1}$ is $tr (ρ P)$ and $tr (σ P)$ by [F2], and [F1] gives $tr (ρ P) = ⟨ ρ w, w ⟩ \neq ⟨ σ w, w ⟩ = tr (σ P)$ .

2.1step 1.1givenF1F2algebra∎

[Uniform bound and separation] The bound $∣ μ_{ρ}^{P} ({1}) - μ_{σ}^{P} ({1}) ∣ = ∣ tr ((ρ - σ) P) ∣ \leq ∥ ρ - σ ∥_{1}$ is [F1] applied to the rank-one projection $P$ . If all rank-one projections have equal laws for $ρ$ and $σ$ , then $⟨ ρ w, w ⟩ = ⟨ σ w, w ⟩$ for every unit $w$ by the same identification, contradicting the choice of $w$ in [F1] unless $ρ = σ$ ; hence statistical identity for all such observables forces the preparations to coincide. This proves the statement, and the assertion involves no measurement procedure beyond the model's outcome law.

RemarkRemark: AI-adaptedProof: Not applicableOpen item page →

What the observable and Born-probability page adopts, proves and does not claim

Physical scope

A scope audit of the observable and Born-probability page: it records that the page reads the state and composition postulate P1 and the observable and probability postulate P2 in observable vocabulary, proves conditional consequences about the probability law, its atoms, its support and its moments, and about position and momentum under the massive-particle postulate P5, and asserts no empirical, apparatus, joint-measurement or determination of a preparation from finite empirical data. It proves nothing and reports no observation.

Scope of the page

Adopted, not restated. The physical content of this page adopts the state and composition postulate P1 (State and composition postulate P1: Hilbert spaces, preparations and composite systems), assigning unit rays and density operators as preparations, and the observable and probability postulate P2 (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes), read in the vocabulary of Assigned sharp observables and the outcome law of the framework: assigning to a declared system a self-adjoint operator with a declared unit, its spectral projection-valued measure, and the outcome law $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ . The general outcome-law, moment and state-separation statements are conditional on P1 and P2; the choice hypotheses of the spectral theorem are carried explicitly, and the position and momentum statements additionally adopt the massive-particle model postulate P5 (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), assigning the spinless state space $L^{2} (R^{3}; C)$ and its position and momentum operators, as read through the magnetic-and-detection operator definitions. No additional physical postulate is introduced on this page.

What is proved. Atoms of the spectral measure and eigenvalues of a self-adjoint operator identifies the atoms of the spectral measure with the eigenspace projections, so that an exact value has zero probability unless it is an eigenvalue. Support of the spectral measure, the spectrum and the values of the scalar law identifies the support of the spectral measure with the spectrum, giving the precise reading of the statement that an outcome lies in the spectrum. Affinity and trace-norm continuity of the scalar spectral law proves affinity under statistical mixtures and uniform trace-norm continuity. Mean, second moment and variance of the scalar spectral law assembles the mean, second moment and variance formulas with their exact domain criteria, and Distinct density operators are separated by the law of a bounded rank-one projection proves that distinct states are separated by the law of a rank-one projection. The physics theorems read these results as conditional statements about assigned observables: The Born outcome law: probability measure, mixtures, atoms and spectral support (the law is a probability measure, affine in mixtures, atomic only at eigenvalues, supported in the spectrum and not assumed to have a density), Mean, second moment and variance of an assigned observable in a preparation (moments exist exactly under the stated domain hypotheses), Position and momentum outcome laws and the Born density of the massive particle model (the joint position density $∣ ψ ∣^{2}$ and the momentum-coordinate density obtained from the frequency density $∣ F_{2} ψ ∣^{2}$ by taking its marginal and the Jacobian factor $(2 π ℏ)^{- 1}$ ) and Distinct preparations are separated by the statistics of a bounded sharp observable (distinct preparations differ in some bounded observable's statistics; all such exact laws uniquely determine the preparation within the model).

No complete eigenbasis. Following the correction recorded in the source audit, the page never asserts that a general self-adjoint operator has a complete discrete system of eigenvectors. The spectral projection-valued measure replaces the eigenbasis; discrete sums are the special case of an atomic law, and continuous parts are handled through measures. The expectation formula $⟨ \hat{O} ⟩ = ⟨ ψ, \hat{O} ψ ⟩$ is proved here only for $ψ$ in the domain of the operator and through the spectral integral, not through an assumed eigenfunction expansion.

What is not claimed. No item of this page asserts an apparatus realization of any observable, a joint measurement of two observables, a state update after an outcome, a density for an arbitrary outcome law, the existence of a mean or variance for an arbitrary preparation, a measured value, determination of a preparation from finitely many measurements, or a comparison with data. The finite unsharp-effect extension of P2(iv) and all instrument content belong to the measurement-and-instruments page, the componentwise spin extension of the position and momentum laws belongs to the kinematics page, and the identification of concrete laboratory procedures with assigned observables is a tested identification under P7, not a consequence of the postulates. Agreement of the conditional predictions with data would test the combined model and apparatus assumptions; it is not established or claimed here.

5 · Examples, experiments and counterexamples

None yet.

Qm Observables And Born Probabilities

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Atoms of the spectral measure and eigenvalues of a self-adjoint operator

Statement

Facts & Assumptions

Proof

Affinity and trace-norm continuity of the scalar spectral law

Statement

Facts & Assumptions

Proof

Mean, second moment and variance of the scalar spectral law

Statement

Facts & Assumptions

Proof

Distinct density operators are separated by the law of a bounded rank-one projection

Statement

Facts & Assumptions

Proof

Assigned sharp observables and the outcome law of the framework

Definition

Remarks

The Born outcome law: probability measure, mixtures, atoms and spectral support

Statement

Facts & Assumptions

Proof

Mean, second moment and variance of an assigned observable in a preparation

Statement

Facts & Assumptions

Proof

Position and momentum outcome laws and the Born density of the massive particle model

Statement

Facts & Assumptions

Proof

Distinct preparations are separated by the statistics of a bounded sharp observable

Statement

Facts & Assumptions

Proof

What the observable and Born-probability page adopts, proves and does not claim

Scope of the page

5 · Examples, experiments and counterexamples

Sources