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Formal nonrelativistic limit of the repulsive Coulomb scattering angle

Physical scope

Formal c-to-infinity limit of exact model formulas at fixed rest mass, Coulomb coupling, Newtonian kinetic energy, and impact parameter. It is not a measured comparison or empirical validation.

Example

Fix $m, κ, K, b > 0$ and let $c \to \infty$ with $W_{c} = m c^{2} + K$ and $J_{c} = b \sqrt{A_{c}}$ , where $A_{c} = (W_{c}^{2} - m^{2} c^{4}) / c^{2}$ . For sufficiently large $c$ , $J_{c} > κ / c$ because $J_{c}^{2} - κ^{2} / c^{2} \to 2 m K b^{2} > 0$ .

The relativistic scattering angle tends to the nonrelativistic repulsive Coulomb angle:

$θ_{c} ⟶ 2 \arctan (\frac{κ}{2 K b}) .$

This agrees with Repulsive Coulomb scattering: the unique outer turning point, global non-colliding motion, the asymptotic impact distance and $Δ = π / 2 - \arctan (κ / (2 E b))$ at kinetic energy $K$ and impact parameter $b$ .
The limit compares ideal mathematical model formulas while holding the indicated parameters fixed; it supplies no experimental evidence for either model.

Facts & Assumptions

Given: Fixed $m, κ, K, b > 0$ , the adopted fixed-source SI Coulomb action model with $q Q > 0$ , the flat massive-particle momentum postulate, ${A C}_{ω}$ for the canonical tangent-bundle smooth structure, the relativistic branch formula, and the exact nonrelativistic repulsive Coulomb scattering formula.

[A1]

Assume exactly countable choice ${A C}_{ω}$ for the canonical tangent-bundle smooth structure, as required by the action-variation and local-IVP suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

In the flat specialization adopt the massive-particle momentum postulate used by the physical scattering theorem (Adopted massive-particle momentum, observer measurement and isolated-collision conservation).

[F1]

For $W > m c^{2}$ and $J > κ / c$ , the mathematical branch proposition and conditional physical theorem give $θ = π - 2 (J / \sqrt{C}) \arctan (\sqrt{C A} / d)$ (Exact repulsive relativistic Coulomb scattering branch, Exact repulsive relativistic Coulomb scattering in the fixed-source test-body model).

[F2]

The nonrelativistic repulsive Coulomb angle for energy $K > 0$ and impact parameter $b > 0$ is $2 \arctan (κ / (2 K b))$ (Repulsive Coulomb scattering: the unique outer turning point, global non-colliding motion, the asymptotic impact distance and $Δ = π / 2 - \arctan (κ / (2 E b))$ ).

[F3]

For $y > 0$ , $\arctan y + \arctan (1 / y) = π / 2$ (Reciprocal positive arguments of the principal inverse tangent); the principal inverse tangent is continuous (The principal inverse tangent $\arctan : R \to (- π / 2, π / 2)$ ).

Verification

technique · take limits of the three dimensionless factors in the exact relativistic formula and use the positive-argument tangent identity

1.1givenA1A2F1F3algebra

With $W_{c} = m c^{2} + K$ , $A_{c} = 2 m K + \frac{K^{2}}{c^{2}}, d_{c} = \frac{W_{c} κ}{c^{2}} = m κ + \frac{K κ}{c^{2}}, C_{c} = J_{c}^{2} - \frac{κ^{2}}{c^{2}} = 2 m K b^{2} + \frac{K^{2} b^{2} - κ^{2}}{c^{2}} .$ Thus $J_{c} / \sqrt{C_{c}} \to 1$ and $\sqrt{C_{c} A_{c}} / d_{c} \to 2 K b / κ$ . By [F3] and continuity, [F1] gives $θ_{c} \to π - 2 \arctan (2 K b / κ)$ . Apply [F3] with $y = 2 K b / κ > 0$ to rewrite this limit as $2 \arctan (κ / (2 K b))$ . This limit retains [A1] and [A2] only when it is read through the physical model theorem.

2.1step 1.1F2given∎

The final expression is exactly [F2]. The agreement is a formal limit with fixed model parameters, not a measured result or validation of the model assumptions.

Formal nonrelativistic limit of the repulsive Coulomb scattering angle

Example

Facts & Assumptions

Verification

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