The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections

(a) Framework data. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the preparation and apparatus discipline P7 (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, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), with the short-range scattering model of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units: mass $m>0$, spin zero, state space $H_0=L^2({\mathbb{R}}^3;\mathbb{C})$, a declared short-range potential $V$, the free and interacting assigned energy observables $H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ and $H$, its reading of the flow by Assigned energy observable, unitary flow and the two pictures of autonomous evolution, and the units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) Incident plane-wave reading and declared density scale. As an additional model assumption of the pair, adopt the stationary elastic beam model. Fix a wave number $k>0$ in ${\mathrm{m}}^{-1}$, a direction $d\in S^2$, and an incident number-density scale $n_{\mathrm{i}\mathrm{n}\mathrm{c}}>0$ in ${\mathrm{m}}^{-3}$. This scale is an explicitly declared model normalization; no numerical sample density or measured beam value is supplied. Retain the unit-amplitude, dimensionless incident shape $u_{\mathrm{i}\mathrm{n}\mathrm{c}}(x):=e^{ikd\cdot x},$ with energy reading $E:={\mathrm{\hslash }}^2{k}^2\mathrm{/}(2m)$ in joules. The physical incident beam field is ${\mathrm{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{c}}:=\sqrt{n_{\mathrm{i}\mathrm{n}\mathrm{c}}}{u}_{\mathrm{i}\mathrm{n}\mathrm{c}}$, with unit ${\mathrm{m}}^{-3\mathrm{/}2}$ and constant model number density $\mathrm{\mid }{\mathrm{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{c}}{\mathrm{\mid }}^2=n_{\mathrm{i}\mathrm{n}\mathrm{c}}$. For a dimensionless unit-amplitude shape $\psi$, write ${\mathbf{j}}^{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}}[\psi ]:=(\mathrm{\hslash }\mathrm{/}m)\mathrm{Im}(\bar{\psi }\mathrm{\nabla }\psi )$ for the mathematical current of The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings. Its incident value is $(\mathrm{\hslash }k\mathrm{/}m)d$, with velocity unit $\mathrm{m}{\mathrm{s}}^{-1}$. The model's physical number-current density is the current of $\mathrm{\Phi }=\sqrt{n_{\mathrm{i}\mathrm{n}\mathrm{c}}}\psi$, $\mathbf{j}[\mathrm{\Phi }]:=\frac{\mathrm{\hslash }}{m}\mathrm{Im}(\bar{\mathrm{\Phi }}\mathrm{\nabla }\mathrm{\Phi })=n_{\mathrm{i}\mathrm{n}\mathrm{c}}{\mathbf{j}}^{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}}[\psi ]\mathrm{.}$ Thus the incident current density and incident flux density through a plane normal to $d$ are ${\mathbf{j}}_{\mathrm{i}\mathrm{n}\mathrm{c}}=n_{\mathrm{i}\mathrm{n}\mathrm{c}}\frac{\mathrm{\hslash }k}{m}d,J_{\mathrm{i}\mathrm{n}\mathrm{c}}:=n_{\mathrm{i}\mathrm{n}\mathrm{c}}\frac{\mathrm{\hslash }k}{m}>0,$ both in ${\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$ as in Dimensions, SI unit conventions and quantity types of the framework clause (d)5. The mathematical plane-wave current of the cited lemma remains the unit-shape computation; the density factor supplies the declared physical beam normalization.

(c) Scattered-wave reading and generalized stationary solution. For a declared short-range potential $V$ the model declares a generalized stationary solution at energy $E$ and direction $d$ to be a function $\mathrm{\Psi }:{\mathbb{R}}^3\to \mathbb{C}$ in $H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2({\mathbb{R}}^3)$, meaning that all weak derivatives of order at most two are locally square integrable (Integer-order Sobolev spaces and their norms), with $V\mathrm{\Psi }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$, of class $C^2$ on the complement of the essential closed support of $V$, and satisfying the equation on all of ${\mathbb{R}}^3$ distributionally: ${\int }_{{\mathbb{R}}^3}\mathrm{\Psi }(x)[-\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{\Delta }\eta (x)+(V(x)-E)\eta (x)]dx=0(\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3;\mathbb{C}))\mathrm{.}$ Here the essential closed support is the complement of the union of open sets on which $V=0$ almost everywhere; the test functions are smooth and compactly supported; $\mathrm{\Delta }={\sum }_{i=0}^2{\mathrm{\partial }}_i^2$ uses the spatial coordinate convention of the current lemma. Outside the essential closed support, $V=0$ almost everywhere and $\mathrm{\Psi }$ is $C^2$, so the continuous free-equation residual has zero test pairing and hence vanishes pointwise: $-\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{\Delta }\mathrm{\Psi }=E\mathrm{\Psi }$. The equation containing the declared representative $V(x)$ holds almost everywhere there; it holds pointwise there when $V$ is replaced by its zero representative on this exterior open set. No pointwise equation with arbitrary null-set values of $V$ is asserted. The solution has the outgoing asymptotic reading $\mathrm{\Psi }(x)=e^{ikd\cdot x}+\frac{f_k(\mathrm{\Omega })}{r}{e}^{ikr}+\mathcal{E}(x),x=r\mathrm{\Omega },r\to \mathrm{\infty },$ with a scattering amplitude function $f_k:S^2\to \mathbb{C}$, read in metres relative to the unit incident shape, and a physical beam field $\mathrm{\Phi }:=\sqrt{n_{\mathrm{i}\mathrm{n}\mathrm{c}}}\mathrm{\Psi }$ with unit ${\mathrm{m}}^{-3\mathrm{/}2}$. The same constant factor multiplies its incident, outgoing and remainder terms; $f_k$ remains the amplitude relative to the incident field, in metres. Here $\mathrm{\Omega }:=x\mathrm{/}r$ is the outgoing unit direction. Require $f_k\in C^1(S^2)$ in the radial-extension convention of The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings (c), and require $\mathrm{\mid }\mathcal{E}(r,\mathrm{\Omega })\mathrm{\mid }+\mathrm{\mid }{\mathrm{\partial }}_r\mathcal{E}(r,\mathrm{\Omega })\mathrm{\mid }\le C{r}^{-2}$ uniformly in $\mathrm{\Omega }$ for all sufficiently large $r$, with some finite $C$. For the bounded compact-support class the model selects the outgoing integral solution of Stationary outgoing integral and flux, with its potential parameter replaced by $W_x=2mV\mathrm{/}{\mathrm{\hslash }}^2$. Explicitly, choose $R>0$ enclosing the essential support, put $G_k(z)=e^{ik\mathrm{\mid }z\mathrm{\mid }}\mathrm{/}(4\pi \mathrm{\mid }z\mathrm{\mid })$ for $z\mathrm{\ne }0$, and on $L^2(B_R;\mathbb{C})$ put $K_{k}u(x)={\int }_{B_R}{G}_k(x-y)W_x(y)u(y)dy$. Select $u=(I+K_k{)}^{-1}(u_{\mathrm{i}\mathrm{n}\mathrm{c}}{\mathrm{\mid }}_{B_R})$ and $\mathrm{\Psi }(x)=u_{\mathrm{i}\mathrm{n}\mathrm{c}}(x)-{\int }_{B_R}{G}_k(x-y)W_x(y)u(y)dy$, using the locally continuous representative supplied by $H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$. The cited theorem proves invertibility and the global equation, with a smooth amplitude and these uniform remainder bounds. The selection is independent of enlarging $R$: the restriction of a solution constructed in a larger ball solves the smaller invertible integral equation, and $W_x=0$ outside the smaller ball makes both global integrals identical. Hence the compact-support model determines $\mathrm{\Psi }$ and $f_k$ from $V,k,d$; arbitrary exterior fields and arbitrary interior extensions are excluded. For the general resolvent-tail class these requirements define a conditional reading only, without an existence or uniqueness assertion. The scattered physical field is ${\mathrm{\Phi }}_{\mathrm{s}\mathrm{c}}:=\sqrt{n_{\mathrm{i}\mathrm{n}\mathrm{c}}}(\mathrm{\Psi }-u_{\mathrm{i}\mathrm{n}\mathrm{c}})$. The physical scattered radial current density, obtained by applying the quadratic current formula to this scattered component, is $j_r^{\mathrm{s}\mathrm{c}}(r,\mathrm{\Omega })=n_{\mathrm{i}\mathrm{n}\mathrm{c}}(\frac{\mathrm{\hslash }k}{m}\frac{\mathrm{\mid }{f}_k(\mathrm{\Omega }){\mathrm{\mid }}^2}{r^2}+O(r^{-3}))(r\to \mathrm{\infty }),$ in ${\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$ for the fixed declared density scale. The leading unit-shape current is clause (c) of The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings; the current of the remainder is controlled by the stationary theorem. The definition uses the scattered component, with no assumption that incident/scattered interference in the total field vanishes. A generalized stationary solution is generally not an element of $H_0$; it is not a preparation of the model, and clause (d) of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units does not apply to it. It is used only through the declared flux quantities computed from it, and every such use states the idealization.

(d) Differential and total cross-section readings. Under the declared equal-channel elastic flux assumptions -- that the incident and outgoing readings refer to the same elastic channel at the same energy, that the incident flux density is spatially uniform over the transverse extent of the target and the outgoing current is read far outside the support of $V$ -- the model declares the differential cross section $\frac{d\sigma }{d\mathrm{\Omega }}(\mathrm{\Omega }):={\lim }_{r\to \mathrm{\infty }}\frac{r^2{j}_r^{\mathrm{s}\mathrm{c}}(r,\mathrm{\Omega })}{J_{\mathrm{i}\mathrm{n}\mathrm{c}}}=\mathrm{\mid }{f}_k(\mathrm{\Omega }){\mathrm{\mid }}^2,$ read in ${\mathrm{m}}^2\mathrm{s}{\mathrm{r}}^{-1}$, and the total cross section $\sigma :={\int }_{S^2}\frac{d\sigma }{d\mathrm{\Omega }}(\mathrm{\Omega })d\mathrm{\Omega }={\int }_{S^2}\mathrm{\mid }{f}_k(\mathrm{\Omega }){\mathrm{\mid }}^{2}d\mathrm{\Omega },$ read in ${\mathrm{m}}^2$, whenever the displayed limit and integral exist; the existence and finiteness of both are proved on this page for the bounded compact-support class at $k>0$. No stationary existence or finite cross-section result is asserted here for the general resolvent-tail class. The common positive density factor cancels in the ratio, $r^2{j}_r^{\mathrm{s}\mathrm{c}}\mathrm{/}{J}_{\mathrm{i}\mathrm{n}\mathrm{c}}\to \mathrm{\mid }{f}_k{\mathrm{\mid }}^2$, so the cross sections are independent of the declared $n_{\mathrm{i}\mathrm{n}\mathrm{c}}$. This normalization supplies no relation to the energy outcome law of P2 and no assertion that an $L^2$ preparation realizes either stationary field.

(e) Status and scope. Clauses (b)-(d) are declarations of the pair under P1, P2, P4 and P7 together with the short-range model; they are an idealized stationary reading and not a model of an apparatus. The new density scale is a symbolic normalization parameter of that reading, not a measured sample value; multiplication by $\sqrt{n_{\mathrm{i}\mathrm{n}\mathrm{c}}}$ does not turn the stationary shape into an $L^2$ preparation. In particular no beam source, target density, collimation, detector acceptance, efficiency, background or counting procedure appears, and the symbols $d\sigma \mathrm{/}d\mathrm{\Omega }$ and $\sigma$ denote model quantities computed from the declared solution. An identification of a measured count cross section with these quantities requires a separate preparation and detection model and carries the uncertainty of that identification.

• Flux, not probability. The cross-section readings are ratios of current densities of the declared physical stationary fields, whose currents are $n_{\mathrm{i}\mathrm{n}\mathrm{c}}$ times the unit-shape currents computed in The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings; they are not read as probabilities of detecting a single particle, and no such single-particle statement is asserted.
• Optical theorem and other refinements. No optical theorem, dispersion relation or inelastic extension is declared or used on this pair.