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Qm Framework And Quantities — Examples

1 · Prerequisites

2 · Summary

The companion page to qm-framework-and-quantities records worked checks of the framework's conventions and refutations of specific statements about them. The dimension bookkeeping of Dimension bookkeeping for the quantities and laws of the framework checks the coordination, wave-function, operator, commutator, Hamiltonian, Schroedinger-equation, current and boost-phase quantities against the unit conventions of the framework page, and the type checks of Type checks: rays, density operators, mixtures, local quantities and outcome laws verify on finite-dimensional data that a pure preparation is a unit ray and a rank-one density operator, that a mixture of orthogonal pure preparations is a density operator that is not pure, that a local observable acts on product states through its tensor factor, and that a finite-spectrum outcome law is a probability measure rather than a number.

Three counterexamples then refute precise statements. The massive free-particle family has no strong zero-mass limit on Schwartz states refutes the specified strong action limit on every Schwartz state together with a finite free-frequency limit at fixed nonzero wave number: one explicit normalized compactly supported smooth state has diverging kinetic-operator norm, and every fixed normalized operator-domain state with nonzero momentum second moment has divergent kinetic expectation. Its model root-mean-square speed depends on mass and is not held fixed as mass tends to zero. This does not decide other notions of convergence or rule out independently specified massless models. A smooth time-dependent translation chart need not be an inertial frame exhibits a smooth time-dependent translation chart of the arena that is not a Galilean transformation and assigns nonzero coordinate acceleration to a curve of zero coordinate acceleration in the standard chart. The mean position of a normalized state need not exist displays a normalized state in a one-dimensional P1/P2 instance whose first absolute and second position moments both diverge, so its mean position exists only under the moment condition recorded in the framework's quantity conventions. A closing scope audit (What the example and counterexample checks of this companion page establish) records what each check establishes and that none of them reports or predicts a measured outcome.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Dimension bookkeeping for the quantities and laws of the framework

Physical scope

A worked application of the dimension and unit conventions of the framework to the quantities and displayed laws of this page: coordinates, wave function, position and momentum operators, the canonical commutator, the kinetic and potential terms of the Hamiltonian, the Schroedinger equation, the probability current and the candidate dimensionless exponent that could be used as a boost phase in a separately specified model. The item checks conventions only, asserts no frame-transformation law, and makes no empirical claim.

Example

Apply the dimension conventions of Dimensions, SI unit conventions and quantity types of the framework to the quantities of the framework: the time and space coordinates of an inertial frame of Inertial frames of the framework and the class of frame changes, a position wave function of the one-particle model, the position and momentum operators and their canonical commutator of Canonical commutation relations of position and momentum on the Schwartz core, the kinetic and potential terms of the Hamiltonian and the Schroedinger equation of Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators and Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, the probability density and current of the continuity calculation, and the candidate phase exponent $m v \cdot x / ℏ$ if used in a separately specified boost model.

Facts & Assumptions

Given: The dimension symbols $ℓ, τ, μ, ϑ$ with the product, quotient and derivative rules of Dimensions, SI unit conventions and quantity types of the framework; the readings $[t] = τ$ , $[x_{i}] = ℓ$ , $[ψ] = ℓ^{- 3 / 2}$ for a position wave function in three spatial dimensions, $[ℏ] = μ ℓ^{2} τ^{- 1}$ , $[m] = μ$ , $[E] = μ ℓ^{2} τ^{- 2}$ , $[v_{i}] = ℓ τ^{- 1}$ and $[R_{i j}] = 1$ .

[L1]

Position and momentum acts: $Q_{j} f = x_{j} f$ and $P_{k} f = - i ℏ D_{k} f$ for one component, with the Schwartz core and the canonical commutator $(Q_{j} P_{k} - P_{k} Q_{j}) f = i ℏ δ_{j k} f$ (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Canonical commutation relations of position and momentum on the Schwartz core).

[L2]

Hamiltonian and evolution: for $m, ℏ > 0$ and a real bounded measurable potential $V$ in three spatial dimensions, $H_{o p} = - \frac{ℏ^{2}}{2 m} Δ + V (Q)$ on $D (H_{o p}) = W^{2, 2} (R^{3}; C)$ componentwise. Its unitary evolution is $ψ (t) = e^{- i t H_{o p} / ℏ} ψ (0)$ for every initial vector; the norm derivative and the equation $i ℏ \partial_{t} ψ = H_{o p} ψ$ are asserted exactly when $ψ (0) \in D (H_{o p})$ , in which case this domain is invariant under the evolution (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution).

[L3]

Probability current of a smooth solution: $j = \frac{ℏ}{m} Im (\overline{ψ} \nabla ψ)$ (Smooth local continuity equation and its integrated forms).

[L4]

Frame parameters: $[b] = τ$ , $[a_{i}] = ℓ$ and $[v_{i}] = ℓ τ^{- 1}$ ; conditionally, if a separately specified model uses $m v \cdot x / ℏ$ as a phase exponent, these units make that exponent dimensionless (Inertial frames of the framework and the class of frame changes, Dimensions, SI unit conventions and quantity types of the framework).

Verification

technique · direct; evaluate the dimension rules on each displayed quantity and combine

1.1givenalgebra

The coordinates have $[t] = τ$ and $[x_{i}] = ℓ$ , so a three-dimensional position wave function has $[ψ] = ℓ^{- 3 / 2}$ , its squared modulus has $[∣ ψ ∣^{2}] = ℓ^{- 3}$ , and the product $∣ ψ ∣^{2} d^{3} x$ is dimensionless, as required of a probability density.

1.2givenL1algebra

For the operators of [L1], $[Q_{j} ψ] = [x_{j}] [ψ] = ℓ \cdot ℓ^{- 3 / 2} = ℓ^{- 1 / 2} = [ψ] \cdot ℓ$ gives $[Q_{j}] = ℓ$ , and $[P_{k} ψ] = [ℏ] \cdot [\partial_{k} ψ] = μ ℓ^{2} τ^{- 1} \cdot ℓ^{- 5 / 2} = μ ℓ^{- 1 / 2} τ^{- 1} = [ψ] \cdot μ ℓ τ^{- 1}$ gives $[P_{k}] = μ ℓ τ^{- 1}$ ; the commutator identity is consistent, since $[Q_{j} P_{k}] = ℓ \cdot μ ℓ τ^{- 1} = μ ℓ^{2} τ^{- 1} = [ℏ] = [i ℏ δ_{j k}]$ .

1.3givenL2algebra

For the Hamiltonian of [L2], the kinetic term acts as $[(- ℏ^{2} / 2 m) Δ ψ] = μ^{2} ℓ^{4} τ^{- 2} μ^{- 1} ℓ^{- 2} ℓ^{- 3 / 2} = μ ℓ^{2} τ^{- 2} ℓ^{- 3 / 2} = [E] [ψ]$ , and the potential term satisfies $[V ψ] = [V] [ψ]$ with $[V] = μ ℓ^{2} τ^{- 2} = [E]$ ; both terms therefore carry the dimension of energy times wave function, as required for their sum to define an operator.

1.4givenL2algebra

The Schroedinger equation is dimensionally consistent: $[i ℏ \partial_{t} ψ] = μ ℓ^{2} τ^{- 1} \cdot τ^{- 1} \cdot ℓ^{- 3 / 2} = μ ℓ^{2} τ^{- 2} ℓ^{- 3 / 2} = [H_{o p} ψ]$ , and the phase $H_{o p} t / ℏ$ of the evolution operator is dimensionless because $[E] [t] [ℏ]^{- 1} = 1$ .

1.5givenL3algebra

The current of [L3] has $[j] = [ℏ] [m]^{- 1} [∣ ψ ∣^{2}] [\nabla ψ] [ψ]^{- 1} = ℓ^{2} τ^{- 1} \cdot ℓ^{- 3} \cdot ℓ^{- 1} = ℓ^{- 2} τ^{- 1}$ , which is $ℓ^{1 - d} τ^{- 1}$ for $d = 3$ : a probability flux per unit area and time, not a charge current.

2.1step 1.1step 1.2step 1.3step 1.4step 1.5L4algebra∎

The expression $m v \cdot x / ℏ$ in [L4] is dimensionless, $[m v \cdot x] / [ℏ] = μ ℓ τ^{- 1} \cdot ℓ \cdot (μ ℓ^{2} τ^{- 1})^{- 1} = 1$ . Thus it can serve as a dimensionless argument if a separately specified model assigns it as a phase; this dimensional calculation alone proves no frame-transformation or covariance law. Combining with steps 1.1-1.5, every quantity and displayed law checked here is dimensionally consistent under the stated conventions.

Remarks

The check is bookkeeping inside the conventions of Dimensions, SI unit conventions and quantity types of the framework: it verifies dimensional consistency and nothing else. It contains no measured value, tests no physical equation against data and makes no claim about the applicability of the model.
The internal index of the spin factor is dimensionless and does not enter these computations; the spin label and the factor $C^{2 s + 1}$ of Identical-particle and spin assignments P6: permutation sectors and finite-spin representations carry no length, duration or mass dimension, and the spin operators have the dimension $μ ℓ^{2} τ^{- 1}$ of action, as fixed in the conventions.

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Type checks: rays, density operators, mixtures, local quantities and outcome laws

Physical scope

Worked type checks on the smallest nontrivial instances of the framework's state and observable assumptions: a pure spin-one-half preparation as a unit ray and as a rank-one density operator, a statistical mixture of two orthogonal pure preparations, the action of a local observable on a product state of two spin factors, and the outcome law of a finite-spectrum observable as a probability measure. The item verifies the mathematical types of the adopted objects and makes no empirical claim.

Example

Check the mathematical types of the objects introduced by the postulates of this page on explicit finite-dimensional data: the two-dimensional complex Hilbert space $H = C^{2}$ with its standard inner product and the orthonormal basis $(e_{+}, e_{-})$ , the spin operators $S_{i} = ℏ J_{i}$ of Finite spin matrices for $s = \frac{1}{2}$ , and the composite $H \hat{\otimes} H$ with the product vectors and local operators of Bounded tensor maps: extension, norm estimate, adjoint and commutation.

Facts & Assumptions

Given: The state postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems, the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, the spin assignment of Identical-particle and spin assignments P6: permutation sectors and finite-spin representations, the Hilbert space $C^{2}$ with orthonormal basis $(e_{+}, e_{-})$ and the operators $S_{z} = ℏ J_{z}$ with $J_{z} e_{\pm} = \pm \frac{1}{2} e_{\pm}$ of Finite spin matrices, and the number conventions of Dimensions, SI unit conventions and quantity types of the framework.

[L1]

Density operators and their scalar laws: a density operator is a self-adjoint positive trace-class operator with trace one; finite orthonormal expansions $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ with $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ are density operators, and for a projection-valued measure $E$ the assignment $B \mapsto tr (ρ E (B))$ is a probability measure reducing to $∥ E (B) ψ ∥^{2}$ for the pure operator $ρ = T_{ψ, ψ}$ with $∥ ψ ∥ = 1$ (Density operators and scalar spectral probability clauses (a) and (b)).

[L2]

Rank-one calculus: $T_{u, v} x = ⟨ x, v ⟩ u$ is trace class with $tr T_{u, v} = ⟨ u, v ⟩$ , so $T_{ψ, ψ}$ is a rank-one positive operator with trace $∥ ψ ∥^{2}$ (Density operators and scalar spectral probability, Rank-one operator calculus).

[L3]

Bounded tensor maps act on product vectors by $S \otimes T (ψ_{A} \otimes ψ_{B}) = (S ψ_{A}) \otimes (T ψ_{B})$ , and $(S \otimes T)^{*} = S^{*} \otimes T^{*}$ ; hence tensoring the self-adjoint $S_{z}$ with $I$ gives a bounded self-adjoint operator (Bounded tensor maps: extension, norm estimate, adjoint and commutation clauses (a), (c)).

[L4]

The spectral family of $S_{z}$ is $E_{S_{z}} ({ℏ / 2}) = P_{+}$ and $E_{S_{z}} ({- ℏ / 2}) = P_{-}$ , the orthogonal projections onto the lines $C e_{+}$ and $C e_{-}$ , and $σ (S_{z}) = {ℏ / 2, - ℏ / 2}$ (Finite spin matrices clauses (a) and (e), Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clause (ii)).

Verification

technique · direct; compute the objects on the displayed data

1.1givenL1L2algebra

Let $ψ : = \frac{1}{\sqrt{2}} (e_{+} + e_{-})$ , so $∥ ψ ∥ = 1$ by orthonormality. The pure preparation is the unit ray $[ψ] = {z ψ : z \in C ∖ {0}}$ : the vector $e^{i θ} ψ$ has the same norm and defines the same ray, and the associated density operator $T_{ψ, ψ}$ satisfies $tr T_{ψ, ψ} = ∥ ψ ∥^{2} = 1$ by [L2]; hence it is a density operator in the sense of [L1].

1.2givenL1algebra

Let $0 < w < 1$ and let $ψ_{1}, ψ_{2}$ be orthonormal vectors of $H$ , and put $ρ : = w T_{ψ_{1}, ψ_{1}} + (1 - w) T_{ψ_{2}, ψ_{2}}$ . By [L1] $ρ$ is a density operator with $tr ρ = w + (1 - w) = 1$ ; orthogonality gives $ρ ψ_{1} = w ψ_{1}$ and $ρ ψ_{2} = (1 - w) ψ_{2}$ . Both eigenvalues are positive, so the range of $ρ$ contains the two linearly independent vectors $ψ_{1}, ψ_{2}$ ; hence $ρ$ has rank at least two and is not a rank-one operator $T_{φ, φ}$ for any $φ$ . The mixture of two orthogonal pure preparations is therefore a preparation of a different type from either component.

1.3givenL3L4algebra

On the composite $H \hat{\otimes} H$ consider the product state $Ψ : = e_{+} \otimes e_{+}$ and the local observable $S_{z} \otimes I$ . By [L3] and the self-adjointness of $S_{z}$ in [L4], this tensor operator is self-adjoint, and $(S_{z} \otimes I) Ψ = (S_{z} e_{+}) \otimes e_{+} = \frac{ℏ}{2} Ψ$ . On the product basis of $H \hat{\otimes} H$ , the spectral projection for $ℏ / 2$ is $P_{+} \otimes I$ , so the local outcome probability for ${ℏ / 2}$ is $∥ (P_{+} \otimes I) Ψ ∥^{2} = 1$ .

1.4givenL1L4algebra

For the state $ψ = \frac{1}{\sqrt{2}} (e_{+} + e_{-})$ the outcome law of $S_{z}$ is $B \mapsto μ_{ψ}^{S_{z}} (B) = tr (T_{ψ, ψ} E_{S_{z}} (B)) = ∥ E_{S_{z}} (B) ψ ∥^{2}$ . By [L4] the projections are $P_{\pm}$ onto the lines $C e_{\pm}$ , so $μ_{ψ}^{S_{z}} ({ℏ / 2}) = ∥ P_{+} ψ ∥^{2} = \frac{1}{2}$ and $μ_{ψ}^{S_{z}} ({- ℏ / 2}) = ∥ P_{-} ψ ∥^{2} = \frac{1}{2}$ . Thus the law is $\frac{1}{2} δ_{ℏ / 2} + \frac{1}{2} δ_{- ℏ / 2}$ on the two-point set $σ (S_{z})$ and is not concentrated at a single outcome; its finite mean is $\frac{1}{2} (ℏ / 2) + \frac{1}{2} (- ℏ / 2) = 0$ .

2.1step 1.1step 1.2step 1.3step 1.4algebra∎

Combining steps 1.1-1.4: rays, density operators, mixtures, local tensor actions and outcome laws of the adopted postulates all have the mathematical types claimed for them, and the displayed computations are the ones the definitions and lemmas give.

Remarks

The checks are internal consistency checks of the adopted types: they verify that the objects named by the postulates are of the claimed mathematical kind and that the displayed computations are the ones the definitions give. They assert nothing about any laboratory preparation and record no measurement.
The number conventions of Dimensions, SI unit conventions and quantity types of the framework enter only through the reading of the outcomes $\pm ℏ / 2$ as angular-momentum values and through the dimensionless character of the weights $w$ , $1 - w$ and of the ray parameter $e^{i θ}$ .

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The massive free-particle family has no strong zero-mass limit on Schwartz states

Physical scope

A refutation, in the spin-zero massive free-particle model, of the specific limit asserted below: convergence K_m f to K_0 f in L^2 for every Schwartz vector to a self-adjoint K_0, together with a finite free-frequency limit at each fixed nonzero wave number. One explicit Schwartz vector has divergent kinetic-operator norm; for every fixed normalized operator-domain state with nonzero momentum second moment the kinetic expectation diverges as m tends to zero. For each m>0, this class has finite nonzero model root-mean-square speed v_rms,m=m^{-1}(sum_j ||P_j psi||^2)^{1/2}; that speed is not held fixed as m varies. The formal free frequency at each fixed nonzero wave number also diverges. The item excludes that specified limit, not other mathematical notions of convergence or independently specified massless quantum models, and asserts no empirical claim. The radiation extension is a separate adopted model.

Spin-sector specialization. Throughout this counterexample, specialize P5 to the spin label $s = 0$ , so the one-particle Hilbert space is the scalar space $L^{2} (R^{3}; C)$ . The argument concerns this explicit sector of the massive model.

Statement refuted

In the one-particle model of Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators with potential $V = 0$ , the free kinetic operators $K_{m} : = - \frac{ℏ^{2}}{2 m} Δ, D (K_{m}) = W^{2, 2} (R^{3}; C), m > 0,$ have a massless strong-action limit: there exist a self-adjoint operator $K_{0} \geq 0$ on $L^{2} (R^{3}; C)$ with $S (R^{3}; C) \subseteq D (K_{0})$ such that $K_{m} f \to K_{0} f$ in $L^{2}$ for every $f$ in the Schwartz class $S (R^{3}; C)$ as $m ↓ 0$ , and the formal free-plane-wave frequencies of this massive family have a finite pointwise limit at every fixed nonzero wave vector. Precisely, for $k \in R^{3} ∖ {0}$ with $[k] = ℓ^{- 1}$ , define $ω_{m} (k) : = ℏ ∣ k ∣^{2} / (2 m)$ by the plane-wave solution $e^{i (k \cdot x - ω_{m} (k) t)}$ of the free equation; the additional claim is that $\lim_{m ↓ 0} ω_{m} (k)$ is finite, where $[ω_{m}] = τ^{- 1}$ under Dimensions, SI unit conventions and quantity types of the framework.

Facts & Assumptions

Given: $ℏ > 0$ ; the spin-zero one-particle sector of P5, with $V = 0$ ; for $m > 0$ the operator $K_{m} = - \frac{ℏ^{2}}{2 m} Δ$ with domain $W^{2, 2} (R^{3}; C)$ (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators clause (iv), Integer-order Sobolev spaces and their norms); the free momentum operators $P_{j}$ of Position and momentum operators on Euclidean L2; and the explicit normalized state $ψ_{L_{0}}$ of [L3].

[L1]

The free-plus-bounded-potential theorem with $V = 0$ and $c = 2 π^{2} ℏ^{2} / m$ gives: $- \frac{ℏ^{2}}{2 m} Δ$ is self-adjoint on $W^{2, 2} (R^{3}; C)$ , and for every Schwartz function $g$ one has $- \frac{ℏ^{2}}{2 m} Δ g = - \frac{ℏ^{2}}{2 m} (Δ g)$ evaluating the Laplacian classically (Real multiplication operators and free plus bounded-potential Schrödinger operators clauses (b) and (c), Schwartz space and its seminorms).

[L2]

Momentum operators: $P_{j} = - i ℏ D_{j}$ is self-adjoint with domain $D (P_{j}) = {u : D_{j} u \in L^{2}}$ , and $- \frac{ℏ^{2}}{2 m} Δ = \frac{1}{2 m} \sum_{j} P_{j}^{2}$ as operators on the classes $S \subseteq W^{2, 2}$ ; for $u \in D (P_{j})$ one has $⟨ P_{j} u, u ⟩ = ⟨ u, P_{j} u ⟩$ and $∥ P_{j} u ∥^{2} = ⟨ P_{j} u, P_{j} u ⟩$ (Self-adjointness, Fourier representation and translation identities of position and momentum clause (b), Position and momentum operators on Euclidean L2).

[L3]

An explicit length-scaled test state is obtained by putting $η_{L_{0}} (x) = \exp ⁣ (- 1 / (1 - ∣ x ∣^{2} / L_{0}^{2}))$ for $∣ x ∣ < L_{0}$ and $η_{L_{0}} (x) = 0$ for $∣ x ∣ \geq L_{0}$ , where $L_{0} > 0$ has dimension $ℓ$ . Every interior derivative is a finite sum of a polynomial in $(1 - ∣ x ∣^{2} / L_{0}^{2})^{- 1}$ and the bounded coordinates times this exponential; the exponential beats every polynomial at the boundary, so all derivatives extend by zero and $η_{L_{0}} \in C_{c}^{\infty}$ . For $g (s) = e^{- 1 / (1 - s)}$ , $g^{'} (s) = - e^{- 1 / (1 - s)} (1 - s)^{- 2} \neq 0$ for $0 \leq s < 1$ , so $\nabla η_{L_{0}} (x) = 2 x L_{0}^{- 2} g^{'} (∣ x ∣^{2} / L_{0}^{2})$ is nonzero for $0 < ∣ x ∣ < L_{0}$ . With $J : = \int_{∣ y ∣ < 1} \exp ⁣ (- 2 / (1 - ∣ y ∣^{2})) d^{3} y \in (0, \infty)$ , set $N : = J^{- 1 / 2}$ and $ψ_{L_{0}} (x) : = N L_{0}^{- 3 / 2} η_{L_{0}} (x)$ . Change of variables $x = L_{0} y$ gives $∥ ψ_{L_{0}} ∥_{2} = 1$ and $[ψ_{L_{0}}] = ℓ^{- 3 / 2}$ ; direct differentiation at $x = 0$ gives $Δ ψ_{L_{0}} (0) = - 6 N e^{- 1} L_{0}^{- 7 / 2} \neq 0$ .

[L4]

Frequencies of the free model: for $k \in R^{3}$ , $ω \in R$ , and $[k] = ℓ^{- 1}$ , the smooth formal plane-wave mode $ψ_{k, ω} (x, t) = e^{i (k \cdot x - ω t)}$ satisfies $i ℏ \partial_{t} ψ_{k, ω} = - \frac{ℏ^{2}}{2 m} Δ ψ_{k, ω}$ at every point exactly when $ℏ ω = \frac{ℏ^{2} ∣ k ∣^{2}}{2 m}$ , that is $ω = \frac{ℏ ∣ k ∣^{2}}{2 m}$ , because $\partial_{t} ψ_{k, ω} = - i ω ψ_{k, ω}$ and $Δ ψ_{k, ω} = - ∣ k ∣^{2} ψ_{k, ω}$ and the exponential does not vanish (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators clause (iv)); the frequency has dimension $τ^{- 1}$ in the conventions of Dimensions, SI unit conventions and quantity types of the framework. For $k \neq 0$ this plane wave is not in $L^{2} (R^{3})$ , so it is a formal mode rather than a normalized state.

[L5]

For $ψ \in W^{2, 2} (R^{3}; C)$ , with $F_{2}$ the unitary Plancherel transform, the free kinetic operator has Fourier multiplier $2 π^{2} ℏ^{2} ∣ ξ ∣^{2} / m$ and $P_{j}$ has multiplier $2 π ℏ ξ_{j}$ ; hence $⟨ K_{m} ψ, ψ ⟩ = \frac{2 π^{2} ℏ^{2}}{m} \int ∣ ξ ∣^{2} ∣ F_{2} ψ (ξ) ∣^{2} d ξ = \frac{1}{2 m} \sum_{j} ∥ P_{j} ψ ∥_{2}^{2}$ (Real multiplication operators and free plus bounded-potential Schrödinger operators clauses (b)-(c), Self-adjointness, Fourier representation and translation identities of position and momentum, Plancherel theorem).

Counterexample

technique · direct; compute the operator values and frequencies on explicit smooth data under the stated assumptions

1.1givenL1L3algebra

[No strong limit on a smooth class] Take the normalized $ψ_{L_{0}} \in C_{c}^{\infty} (R^{3}; C)$ of [L3], for which $Δ ψ_{L_{0}} \neq 0$ . By [L1], $K_{m} ψ_{L_{0}} = - \frac{ℏ^{2}}{2 m} Δ ψ_{L_{0}}$ , so $∥ K_{m} ψ_{L_{0}} ∥_{2} = \frac{ℏ^{2}}{2 m} ∥ Δ ψ_{L_{0}} ∥_{2} \to + \infty$ as $m ↓ 0$ . Therefore this family is not Cauchy in $L^{2}$ on this Schwartz vector and cannot converge there to $K_{0} ψ_{L_{0}}$ for any operator $K_{0}$ ; in particular it cannot converge on every Schwartz state.

1.2givenL2L5algebra

[Every fixed state with nonzero momentum second moment has divergent kinetic expectation] Let $ψ \in W^{2, 2} (R^{3}; C)$ be fixed and normalized, and suppose $p_{2} (ψ) : = \sum_{j} ∥ P_{j} ψ ∥_{2}^{2} > 0$ . Each first weak derivative of $ψ$ lies in $L^{2}$ by its $W^{2, 2}$ membership, so [L2] places $ψ$ in every $D (P_{j})$ and makes each $∥ P_{j} ψ ∥_{2}^{2}$ finite. For each mass $m > 0$ , its model root-mean-square speed is $v_{r m s, m} (ψ) : = m^{- 1} p_{2} (ψ)^{1 / 2}$ , which is finite and nonzero; this speed is not held fixed when $m$ varies. By [L5], its kinetic expectation is $⟨ K_{m} ψ, ψ ⟩ = p_{2} (ψ) / (2 m) \to + \infty$ as $m ↓ 0$ . Thus the claimed divergence holds for exactly the stated fixed-state class, with no claim for vectors outside this operator domain.

1.3givenL4algebra

[Frequency divergence] By [L4], $[ℏ ∣ k ∣^{2} / m] = τ^{- 1}$ , and for each fixed wave number $k \neq 0$ the formal free frequency $ω_{m} (k) = \frac{ℏ ∣ k ∣^{2}}{2 m}$ satisfies $ω_{m} (k) \to + \infty$ as $m ↓ 0$ . Thus no finite frequency law arises at a fixed nonzero wave number in this limit, and in particular it does not produce a law linear in $∣ k ∣$ .

2.1step 1.1step 1.2step 1.3∎

Step 1.1 refutes the asserted strong convergence for every Schwartz state, step 1.2 establishes divergence for the stated nonzero-velocity class, and step 1.3 refutes a finite limiting frequency at each fixed nonzero wave number. Hence the mass-zero limiting inference fails for the displayed massive family; this does not rule out a separately specified massless model.

Remarks

What is refuted is the specific limiting assertion in the Statement: strong action convergence on every Schwartz vector to a self-adjoint $K_{0}$ together with a finite free-frequency limit at each fixed nonzero wave number. The calculation does not decide other operator topologies or separately defined families of massless models. The radiation extension is adopted with its own assumptions and dispersion and detection models.
The computations are carried out on the smooth classes where the operator action is the classical one; no spectral convergence theorem and no limiting evolution is asserted, because steps 1.1 and 2.1 show that the relevant limits do not exist on those classes.

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A smooth time-dependent translation chart need not be an inertial frame

Physical scope

A refutation, on explicit data of the model, of the claim that every chart of the Newtonian arena obtained by a smooth time-dependent spatial translation is an inertial frame of the framework. It exhibits a smooth chart that is a bijection of the coordinate space but not a Galilean transformation, and shows that a parametrized curve of zero coordinate acceleration in the standard chart has nonzero coordinate acceleration in that chart. The item refutes the stated frame-class claim; it adopts no force law and makes no empirical claim.

Statement refuted

Fix the spacetime arena of The Newtonian spacetime arena of the framework together with an inertial frame $F$ of Inertial frames of the framework and the class of frame changes with coordinates $(t, x)$ . Every smooth chart of the arena of the form $G_{w} : R \times R^{3} \to R \times R^{3}, G_{w} (t, x) = (t, x + w (t)),$ with $w : R \to R^{3}$ of class $C^{\infty}$ and $[w_{i}] = ℓ$ for every component, is an inertial frame of the model; that is, the map $G_{w}$ belongs to the Galilean transformation class $G$ of The Galilean transformations of the coordinate space for every such $w$ .

Facts & Assumptions

Given: The Galilean transformations $g_{(b, a, v, R)} (t, x) = (t + b, R x + v t + a)$ and their parameters (The Galilean transformations of the coordinate space); the inertial frames of the arena, whose changes are exactly the Galilean transformations (Inertial frames of the framework and the class of frame changes, Transformation between adapted charts and its kinematic consequences clause 1); a nonzero vector $c \in R^{3}$ with $[c_{i}] = ℓ τ^{- 2}$ ; a velocity vector $u \in R^{3}$ with $[u_{i}] = ℓ τ^{- 1}$ ; a positive time scale $T_{0}$ with $[T_{0}] = τ$ ; and the shear map $G (t, x) : = (t, x + t^{2} c)$ .

[L1]

Galilean transformations and their parameter map: a map $R \times R^{3} \to R \times R^{3}$ is Galilean exactly when it equals $g_{(b, a, v, R)}$ for some $b \in R$ , $a, v \in R^{3}$ , $R \in S O (3)$ ; their spatial terms have dimension $ℓ$ under the frame-coordinate conventions (The Galilean transformations of the coordinate space, Dimensions, SI unit conventions and quantity types of the framework, The orthogonal group $O (3)$ and the rotation group $S O (3)$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

[L3]

Coordinate velocities and accelerations of parametrized curves are the componentwise derivatives of the coordinate curves, and the second coordinate of the standard chart of the arena is identified with $x$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Transformation between adapted charts and its kinematic consequences).

Counterexample

technique · direct; compare the shear chart with the Galilean class on explicit data and compute one coordinate acceleration

1.1givenL1L3algebra

[The chart is a bijection] The map $G (t, x) = (t, x + t^{2} c)$ is a bijection of $R \times R^{3}$ with inverse $(t, y) \mapsto (t, y - t^{2} c)$ , since $G (t, y - t^{2} c) = (t, y)$ and $G = (G^{- 1})^{- 1}$ is smooth as well; hence $G$ is a legitimate chart of the coordinate space in the smooth structure of the arena.

1.2givenL1algebra

[The shear is not Galilean] Suppose $G = g_{(b, a, v, R)}$ . Comparing time coordinates gives $t = t + b$ for every $t$ , so $b = 0$ . Then $x + t^{2} c = R x + v t + a$ for all $(t, x) \in R \times R^{3}$ . At $t = 0$ this gives $x = R x + a$ for all $x$ ; taking $x = 0$ gives $a = 0$ , and $(R - I_{3}) x = 0$ for all $x$ gives $R = I_{3}$ by comparing the action on $e_{1}, e_{2}, e_{3}$ . The terms $t^{2} c$ and $v t$ both have dimension $ℓ$ . Evaluating at $t = T_{0}$ yields $v = T_{0} c$ ; at $t = 2 T_{0}$ the same identity would then give $4 T_{0}^{2} c = 2 T_{0} v = 2 T_{0}^{2} c$ , contradicting $T_{0} > 0$ and $c \neq 0$ . Thus no parameter quadruple represents $G$ , so by the set definition in [L1] it is not Galilean.

1.3givenL3algebra

[A zero-acceleration curve becomes accelerating] Let $γ : R \to R \times R^{3}$ be $γ (s) : = (s, u s)$ , where $s$ is a time coordinate. In the standard chart its coordinate curve is $x (s) = u s$ with $x^{''} = 0$ ; in the chart $G$ it is $x^{'} (s) = u s + s^{2} c$ , whose second derivative is $2 c \neq 0$ . The terms $u s$ and $s^{2} c$ both have dimension $ℓ$ , and their first and second derivatives have dimensions $ℓ τ^{- 1}$ and $ℓ τ^{- 2}$ . Hence zero coordinate acceleration is not preserved by this chart, although it is preserved by every Galilean change of frame by clause 3 of Transformation between adapted charts and its kinematic consequences.

2.1givenL3step 1.1step 1.2step 1.3∎

Step 1.1 makes $G$ a smooth chart. Every change between the inertial frames in the Given paragraph belongs to the Galilean class, whereas step 1.2 proves that $G$ does not; hence this chart is not an inertial frame. Independently, step 1.3 shows that it fails preservation of zero coordinate acceleration, which clause 3 of Transformation between adapted charts and its kinematic consequences supplies as a necessary property of inertial-frame changes, without asserting sufficiency. Thus the refuted statement fails: not every smooth time-dependent spatial translation is an inertial frame of the model, and the inertial frames form a proper subclass of the smooth charts of the arena.

Remarks

The shear chart is a legitimate coordinate description of the arena, and the coordinate acceleration computed in step 2.1 is a coordinate effect, not a force: no force law is adopted here, and no statement is made about the motion of any physical body. What fails is the claim that the chart is an inertial frame of the model.
The example separates two notions that the framework keeps apart: an admissible coordinate chart is not automatically an inertial frame, and an inertial frame is defined by membership in the Galilean class of the model, not by smoothness of its transition map.

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The mean position of a normalized state need not exist

Physical scope

A refutation, on an explicit normalized state in the one-dimensional specialization of the P1 state assignment and P2 position-probability rule, of the claim that every normalized preparation has a finite position mean. The displayed state has unit norm but neither a finite second moment nor a finite first absolute moment. The item defines the mean as the Lebesgue expectation and states its absolute-integrability condition; it asserts no measured value and no empirical claim.

Statement refuted

Specialize P1 to a one-dimensional system with state space $H = L^{2} (R; C)$ and specialize P2 to the position observable $Q ψ = x ψ$ , with outcome coordinate measured in metres. Its spectral projection for a Borel set $B \subseteq R$ is multiplication by $1_{B}$ , so a pure preparation $[ψ]$ has position law $μ_{ψ}^{Q} (B) = \int_{B} ∣ ψ (x) ∣^{2} d x$ . Define its mean position to be the Lebesgue expectation $E_{μ_{ψ}^{Q}} [X] = \int_{R} x d μ_{ψ}^{Q} (x)$ when it is a finite real number; this is so exactly when $x \in L^{1} (μ_{ψ}^{Q})$ , equivalently $\int_{R} ∣ x ∣ ∣ ψ (x) ∣^{2} d x < \infty$ . The refuted claim is that every normalized $ψ \in L^{2} (R; C)$ has such a finite mean.

Facts & Assumptions

Given: Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). The one-dimensional instance of P1 with $H = L^{2} (R; C)$ and a normalized vector $ψ$ ; the P2 position observable $Q$ as multiplication by $x$ , with the spectral PVM and law of [L6]; and the quantity conventions of Dimensions, SI unit conventions and quantity types of the framework.

[L1]

Let $L_{0} > 0$ be a length and define $ψ_{L_{0}} (x) = \frac{1}{2 \sqrt{L_{0}}}$ for $∣ x ∣ \leq L_{0}$ and $ψ_{L_{0}} (x) = \frac{\sqrt{L_{0}}}{2 ∣ x ∣}$ for $∣ x ∣ \geq L_{0}$ . The formulas agree at $∣ x ∣ = L_{0}$ , and $[ψ_{L_{0}}] = ℓ^{- 1 / 2}$ . It is measurable and represents an almost-everywhere class; its $L^{2}$ membership is established in step 1.1 using [L5] and monotone convergence. Its squared tail is $L_{0} / (4 x^{2})$ , whose improper Riemann integral converges by the rational $p$ -test (Improper integrals over unbounded intervals, The improper $p$ -test for rational exponents, The space $L^{p} (μ)$ as the quotient by null functions).

[L2]

For $N > L_{0}$ the fundamental theorem and power rule give $\int_{L_{0}}^{N} x^{- 2} d x = L_{0}^{- 1} - N^{- 1}$ , $\int_{L_{0}}^{N} x^{- 1} d x = \ln (N / L_{0})$ , and $\int_{L_{0}}^{N} 1 d x = N - L_{0}$ ; the improper integrals are the limits as $N \to \infty$ (Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ , For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, Improper integrals over unbounded intervals).

[L3]

Monotone convergence for the truncated absolute moments: $\int_{- N}^{N} ∣ ψ ∣^{2} \to \int_{R} ∣ ψ ∣^{2}$ and $\int_{- N}^{N} ∣ x ∣ ∣ ψ ∣^{2} \to \int_{R} ∣ x ∣ ∣ ψ ∣^{2} \in [0, + \infty]$ along the increasing sequence of sets $[- N, N]$ , and similarly for the second moments (Monotone convergence for the integral).

[L4]

Second moments and Cauchy-Schwarz: $ψ \in D (Q)$ means $\int_{R} x^{2} ∣ ψ (x) ∣^{2} d x < \infty$ , and for a probability density with finite second moment the first absolute moment is finite by the Cauchy-Schwarz inequality applied to the probability measure $∣ ψ (x) ∣^{2} d x$ (Position and momentum operators on Euclidean L2, Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

[L5]

For each $N > L_{0}$ , the three global integrands $∣ ψ_{L_{0}} ∣^{2}$ , $∣ x ∣ ∣ ψ_{L_{0}} ∣^{2}$ and $x^{2} ∣ ψ_{L_{0}} ∣^{2}$ are real and continuous on $[- N, N]$ , including at $x = \pm L_{0}$ , because their defining formulas agree there. The tail functions $x^{- 2}$ , $x^{- 1}$ and $1$ of [L2] are real and continuous on $[L_{0}, N]$ , which excludes zero. Each of these restrictions is therefore bounded and Riemann integrable by A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. Under the stated Countable Choice hypothesis, A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral identifies its finite-interval Riemann and Lebesgue integrals. Thus [L2] evaluates the truncated Lebesgue integrals used by [L3]; the tail notation $\int_{L_{0}}^{\infty}$ below denotes the monotone limit of these nonnegative Lebesgue integrals, including $+ \infty$ .

[L6]

Choice is carried from P2 and is an explicit hypothesis of Spectral measure of a real L2 multiplication operator clauses (a)-(c). On the positive, $σ$ -finite Lebesgue measure space $R$ , the measurable finite multiplier $m (x) = x$ satisfies that lemma's hypotheses. Under the stated Choice assumption (The Axiom of Choice), it identifies the self-adjoint position operator $Q$ on $D (Q) = {f : x f \in L^{2}}$ and its spectral PVM $E_{Q} (B) = M_{1_{B}}$ , with pure-state law $μ_{ψ}^{Q} (B) = ∥ E_{Q} (B) ψ ∥^{2} = \int_{B} ∣ ψ ∣^{2} d x$ for every unit $ψ$ . The explicit moment calculations require no additional arbitrary choice.

Counterexample

technique · direct; compute the norm, the second moment and the first absolute moment of the displayed state

1.1givenL1L2L3L5algebra

[Norm one] The squared modulus is $1 / (4 L_{0})$ on $[- L_{0}, L_{0}]$ and $L_{0} / (4 x^{2})$ for $∣ x ∣ \geq L_{0}$ . Its integral is $2 L_{0} / (4 L_{0}) + 2 \int_{L_{0}}^{\infty} L_{0} / (4 x^{2}) d x = \frac{1}{2} + \frac{L_{0}}{2} \lim_{N \to \infty} (L_{0}^{- 1} - N^{- 1}) = 1$ by [L2] and [L3], so $ψ_{L_{0}}$ is a normalized state.

1.2givenL1L2L3L5algebra

[Second moment infinite] The second moment is $\int_{R} x^{2} ∣ ψ_{L_{0}} ∣^{2} d x = \frac{L_{0}^{2}}{6} + 2 \int_{L_{0}}^{\infty} x^{2} \cdot \frac{L_{0}}{4 x^{2}} d x = \frac{L_{0}^{2}}{6} + \frac{L_{0}}{2} \lim_{N \to \infty} (N - L_{0}) = + \infty$ by [L2] and [L3]; equivalently $x ψ_{L_{0}} \notin L^{2}$ and $ψ_{L_{0}} \notin D (Q)$ , although it is a normalized state.

1.3givenL1L2L3L5algebra

[First absolute moment infinite] The first absolute moment is $\int_{R} ∣ x ∣ ∣ ψ_{L_{0}} ∣^{2} d x = \frac{L_{0}}{4} + 2 \int_{L_{0}}^{\infty} x \cdot \frac{L_{0}}{4 x^{2}} d x = \frac{L_{0}}{4} + \frac{L_{0}}{2} \lim_{N \to \infty} \ln (N / L_{0}) = + \infty$ by [L2] and [L3]. Both one-sided integrals diverge, so $x$ is not integrable for the probability measure $μ_{ψ_{L_{0}}}^{Q}$ and its Lebesgue expectation is not a finite real mean.

2.1step 1.1step 1.2step 1.3L4L6∎

By steps 1.1-1.3 the P1 preparation $[ψ_{L_{0}}]$ is normalized and its P2 position law has infinite first absolute moment, so the refuted universal claim fails. By the definition in the Statement its mean exists exactly when $\int_{R} ∣ x ∣ d μ_{ψ_{L_{0}}}^{Q} (x) = \int_{R} ∣ x ∣ ∣ ψ_{L_{0}} (x) ∣^{2} d x < \infty$ ; and by [L4], the second-moment condition $ψ_{L_{0}} \in D (Q)$ implies this integrability by Cauchy-Schwarz.

Remarks

The example concerns the existence of a moment, not the definition of the preparation: the displayed $ψ$ is a legitimate normalized state, and statements about its position distribution are computed from $∣ ψ ∣^{2} d x$ . What fails is only the claim that the first moment is automatically finite.
Moments are not part of the probability postulate of the page; each mean, variance or higher moment of an outcome law is asserted only where its integrability has been established, exactly as the quantity conventions of the framework require.

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What the example and counterexample checks of this companion page establish

Physical scope

A scope audit of the worked checks of this companion page: what the dimension and type checks establish, which specific statements the counterexamples refute, and the empirical neutrality of all of them. The remark records status only, proves nothing new and asserts no observation.

Scope of the worked checks

This companion page contains worked checks of the framework's conventions and refutations of specific statements about them. The status of each is as follows.

Dimension bookkeeping. Dimension bookkeeping for the quantities and laws of the framework verifies that the quantities and displayed laws of the framework are dimensionally consistent under the conventions of Dimensions, SI unit conventions and quantity types of the framework: coordinates, wave function, position and momentum acts, the canonical commutator, the kinetic and potential terms, the Schroedinger equation, the probability current and the boost phase. The check is bookkeeping inside the conventions: it contains no measured value and tests no physical equation against data.

Type checks. Type checks: rays, density operators, mixtures, local quantities and outcome laws verifies the mathematical types of the adopted objects on explicit finite-dimensional data: a pure preparation as a unit ray and as a rank-one density operator, a mixture of two orthogonal pure preparations as a density operator that is not pure, the action of a local observable on a product state, and the outcome law of a finite-spectrum observable as a probability measure on its possible values rather than a single deterministic outcome. These are internal consistency checks of the postulates; they establish no laboratory fact.

Counterexample: the specified mass-zero limit. The massive free-particle family has no strong zero-mass limit on Schwartz states refutes the specific assertion of a self-adjoint strong action limit on every Schwartz state together with finite limiting frequency at each fixed nonzero wave number: one explicit normalized compactly supported smooth state has diverging kinetic-operator norm, and each fixed normalized operator-domain state with nonzero momentum second moment has divergent kinetic expectation. Its model root-mean-square speed depends on mass and is not held fixed during this limit. The argument does not decide other notions of convergence or exclude independently specified massless models; the radiation extension is a separately adopted model.

Counterexample: the frame class. A smooth time-dependent translation chart need not be an inertial frame refutes the claim that every smooth time-dependent spatial translation chart of the arena is an inertial frame: the quadratic shear is a smooth bijection of the coordinate space that is not a Galilean transformation, and a curve of zero coordinate acceleration in the standard chart acquires nonzero coordinate acceleration in it. It refutes that specific frame-class claim; it adopts no force law and makes no statement about physical bodies.

Counterexample: existence of moments. The mean position of a normalized state need not exist refutes, in a one-dimensional instance of P1 and P2, the claim that every normalized preparation has a finite position mean: an explicit normalized state with infinite second and first absolute moments is displayed. It records that the mean is the Lebesgue expectation and requires absolute integrability; it does not question the definition of the preparation or the probability law.

Empirical neutrality. None of the five items reports, compares or predicts a measured outcome. No experiment is described, no uncertainty is quoted and no statistical inference is drawn; the checks are mathematical consequences of the framework's stated assumptions or refutations of stated claims under explicit hypotheses, and no empirical conclusion is drawn from them.

Remarks

A refuted statement in the counterexamples is a specific claim about the framework's objects, not a claim about nature; a reader should not read a refutation of "every smooth translation chart is inertial" as evidence for or against any physical model.
The checks assume the choice hypotheses recorded in their items; where a consequence uses Countable Choice or the Axiom of Choice, the assumption is declared in the item that states the consequence.

Qm Framework And Quantities — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Dimension bookkeeping for the quantities and laws of the framework

Example

Facts & Assumptions

Verification

Remarks

Type checks: rays, density operators, mixtures, local quantities and outcome laws

Example

Facts & Assumptions

Verification

Remarks

The massive free-particle family has no strong zero-mass limit on Schwartz states

Statement refuted

Facts & Assumptions

Counterexample

Remarks

A smooth time-dependent translation chart need not be an inertial frame

Statement refuted

Facts & Assumptions

Counterexample

Remarks

The mean position of a normalized state need not exist

Statement refuted

Facts & Assumptions

Counterexample

Remarks

What the example and counterexample checks of this companion page establish

Scope of the worked checks

Remarks

Sources