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The plane wave and the point-localized delta are not preparations of the model

Physical scope

A refutation under P1, P2 and P5, with Countable Choice and the Axiom of Choice for the cited spectral suppliers, of the informal claim that a free plane wave, or the point-localized distribution symbol delta(x minus X), is a preparation of the massive-particle model with the idealized position density one: the plane wave is not an element of the state space and the exact-position symbol is not an element of the state space either, while the model's own exact-position statement is that the probability of an exact value is zero and finite bins carry the probability. The item refutes that specific informal claim; it asserts no empirical result and no claim about approximation schemes in which such objects are formally useful.

Statement refuted

Adopt the state and composition postulate P1, the observable and probability postulate P2, and the massive-particle model postulate P5 of the framework (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, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), with Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice) for the cited spectral suppliers. Then: every bounded smooth function $ψ : R^{3} \to C$ that solves the free time-independent Schrödinger equation $- \frac{ℏ^{2}}{2 m} Δ ψ = E ψ$ pointwise determines a pure preparation of the model with position density $∣ ψ ∣^{2}$ ; in particular the plane wave $ψ_{k} (x) = e^{i k \cdot x}$ with $k \neq 0$ is a preparation with constant position density $∣ ψ_{k} ∣^{2} = 1$ , and the point-localized object $δ^{3} (x - X)$ is the preparation of a particle exactly at $X$ .

Point-localized symbol. Here $δ^{3} (x - X)$ denotes the evaluation functional $δ_{X} : C_{c} (R^{3}; C) \to C$ , $δ_{X} (η) : = η (X)$ , where $C_{c}$ is the space of continuous complex functions with compact support. An $L^{2}$ representation would be a class $f$ with $\int f (x) η (x) d x = η (X)$ for every such $η$ ; these integrals are defined by Cauchy–Schwarz because $η \in L^{2}$ .

Facts & Assumptions

Given: The adopted postulates P1, P2 and P5 for a spinless particle with state space $H = L^{2} (R^{3}; C)$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems clauses (i)-(iii), Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clauses (i)-(iii), Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators clauses (iii)-(iv)); the position operators $Q_{j}$ with their spectral facts (Position and momentum operators on Euclidean L2, Position and momentum outcome laws and the Born density of the massive particle model); a wave vector $k \in R^{3}$ , $k \neq 0$ ; a point $X \in R^{3}$ ; the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice).

[F1]

Preparations are rays of the state space. A pure preparation of the model is a unit ray $[ψ] = {z ψ : z \in C ∖ {0}}$ of a vector $ψ \in H$ with $∥ ψ ∥ = 1$ ; the zero vector is excluded, and a general preparation is a density operator on $H$ . The model assigns no preparation to an object that is not an element of $H$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems clause (ii), The state space of the framework: preparations, pure rays and density operators clauses (a)-(c)).

[F2]

The plane wave has constant modulus and infinite norm. For every real $θ$ one has $∣ e^{i θ} ∣^{2} = \cos^{2} θ + \sin^{2} θ = 1$ , hence $∣ e^{i k \cdot x} ∣^{2} = 1$ for every $x$ ; consequently for the truncations $1_{[- N, N]^{3}}$ one has $\int_{[- N, N]^{3}} ∣ ψ_{k} ∣^{2} d x = (2 N)^{3}$ , and monotone convergence gives $\int_{R^{3}} ∣ ψ_{k} ∣^{2} d x = + \infty$ , so $ψ_{k}$ is not an element of $L^{2} (R^{3}; C)$ (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ , Parity and the Pythagorean identity for sine and cosine, Monotone convergence for the integral,).

[F3]

The position observable has no eigenvalues and no atoms. The spectrum of $Q_{j}$ is all of $R$ , its outcome law in every preparation is atomless, and by the atom/eigenvalue correspondence it has no eigenvectors: there is no nonzero $x \in H$ with $Q_{j} x = λ x$ for any $λ$ (Position and momentum outcome laws and the Born density of the massive particle model clauses (a) and (d), Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a) and (b)).

[F4]

Outcome-law reading. The model reads an exact position value $λ$ through the singleton ${λ}$ and a finite-resolution outcome through a Borel bin $B$ , with probabilities given by the spectral law of Assigned sharp observables and the outcome law of the framework; a vector outside $H$ carries no such law.

[F5]

Cauchy–Schwarz. In the $L^{2}$ inner product, $∣ \int f \overline{h} ∣ \leq ∥ f ∥_{2} ∥ h ∥_{2}$ for $f, h \in L^{2}$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs); the integral pairing convention is that of the position/momentum operator definition.

Counterexample

technique · direct; show that the objects named in the refuted statement are not elements of the state space and that the model's exact-position statement is atomless

1.1givenF2algebra

[The plane wave is not in the state space] By [F2], $∣ ψ_{k} (x) ∣^{2} = 1$ for every $x$ and $\int_{R^{3}} ∣ ψ_{k} ∣^{2} d x = + \infty$ ; hence $ψ_{k} \notin H = L^{2} (R^{3}; C)$ . For every nonzero scalar $z$ the same computation gives $\int ∣ z ψ_{k} ∣^{2} = ∣ z ∣^{2} \cdot \infty = \infty$ , so no nonzero scalar multiple of $ψ_{k}$ lies in $H$ either, and the formal normalization " $ψ_{k} / \sqrt{V}$ " depends on a box volume $V$ and is also not an element of $H$ .

1.2givenF3F4F5constructalgebra

[The exact-position symbol is not a state] If an $L^{2}$ class $f$ represented $δ_{X}$ , use the real continuous compactly supported functions $η_{n} (x) : = \prod_{j = 1}^{3} \max {0, 1 - n ∣ x_{j} - X_{j} ∣}$ , for integers $n \geq 1$ . They satisfy $η_{n} (X) = 1$ , $0 \leq η_{n} \leq 1$ , and are supported in the cube $\prod_{j} [X_{j} - 1 / n, X_{j} + 1 / n]$ , so $∥ η_{n} ∥_{2} \leq (2 / n)^{3 / 2}$ . The representation and [F5] would give $1 = ∣ \int f η_{n} ∣ \leq ∥ f ∥_{2} (2 / n)^{3 / 2} \to 0$ , a contradiction. Thus $δ_{X}$ has no $L^{2}$ representation. Independently, [F3] rules out nonzero position eigenvectors. The model's exact-coordinate law is $μ_{ρ}^{Q_{j}} ({X_{j}}) = 0$ for every preparation $ρ$ , with $X_{j} \in R$ the $j$ -th coordinate of $X \in R^{3}$ .

2.1step 1.1step 1.2F1F4algebra

[Hence no preparation is assigned to either object] By [F1] pure preparations are unit rays of $H$ and general preparations are density operators on $H$ ; since neither $ψ_{k}$ nor $δ^{3} (x - X)$ determines an element of $H$ , neither determines a preparation, and no position density can be attached to them within the model. The identities $∣ ψ_{k} ∣^{2} = 1$ and " $δ$ is localized" are formal statements outside the state space, not probabilities of the model.

3.1step 1.1step 1.2step 2.1F1F3F4algebra∎

[The pointwise solution property does not supply a state] Granting that the free plane wave satisfies $- \frac{ℏ^{2}}{2 m} Δ ψ_{k} = E ψ_{k}$ pointwise with $E = ℏ^{2} ∣ k ∣^{2} / (2 m)$ , and that the time-dependent expression $e^{i (k \cdot x - ω t)}$ with $ω = E / ℏ$ solves the free Schrödinger equation classically, those are statements about a smooth function on $R^{3}$ and do not place it in $H$ ; the model's dynamics and its Born law are defined on the state space, and by steps 1.1-2.1 the plane wave is not in it. The finite-resolution content of the model is the bin probability of a normalizable packet, not the idealized plane wave. Specifically, under the stated hypotheses of Position-bin probabilities of a Gaussian packet clauses (i)-(ii), for its Gaussian first-coordinate packet, every bin $(a, b]$ with $a < b$ has positive probability, whereas every exact first-coordinate value has probability zero. Positivity in every positive-length bin is not claimed for arbitrary normalizable packets. This refutes the displayed claim.

Remarks

The counterexample concerns the state assignment of this model, not the usefulness of plane waves or delta symbols as distributional tools: they are standard idealizations in formal manipulations, where any probability conclusion requires a separate mathematical statement about normalizable approximants. Nothing here asserts that such approximation schemes fail, only that the raw objects are not preparations and carry no Born law.
Solving the free equation and being bounded do not by themselves imply membership in $H$ ; the plane wave above is one explicit bounded solution that is not square-integrable. Every proposed preparation must separately satisfy the model's state-space and normalization requirements.

The plane wave and the point-localized delta are not preparations of the model

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels

Sources