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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.

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

Statement

Facts & Assumptions

Proof

Depends on

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