Scattering Cross Section Formula Derived From Macroscopic Model of Detectors

TL;DR

The paper derives the non-relativistic scattering cross section density $\sigma(\boldsymbol{x},t)$ for detectors arranged on a large sphere, starting from two macroscopic detector models: an outside imaginary potential and a sequence of nearly-projective measurements (Zeno dynamics). It shows that, in the appropriate limits, both models reproduce the leading cross-section density $\sigma(\boldsymbol{x},t)= \frac{m^3 R}{\hbar^3 t^4} |\widehat{\Psi}_0(\frac{m\boldsymbol{x}}{\hbar t})|^2$, while providing controlled subleading corrections to arrival times and positions and analyzing the detector-induced disturbance. The work extends the framework to arbitrary detector surfaces, many-particle systems, time-dependent boundaries, and the Dirac equation, and it establishes a covariant Dirac cross-section formula along with a no-signaling property in the detection regime. By linking microscopic wave dynamics to macroscopic detection statistics through explicit macroscopic models, the paper clarifies the detectors’ back-action and solidifies the theoretical foundation of the standard scattering cross section. The results offer practical justification for the use of the cross-section density in far-field scattering and provide generalizable tools for more complex geometries and relativistic settings.

Abstract

We are concerned with the justification of the statement, commonly (explicitly or implicitly) used in quantum scattering theory, that for a free non-relativistic quantum particle with initial wave function $Ψ_0(\boldsymbol{x})$, surrounded by detectors along a sphere of large radius $R$, the probability distribution of the detection time and place has asymptotic density (i.e., scattering cross section) $σ(\boldsymbol{x},t)= m^3 \hbar^{-3} R t^{-4} |\widehatΨ_0(m\boldsymbol{x}/\hbar t)|^2$ with $\widehatΨ_0$ the Fourier transform of $Ψ_0$. We give two derivations of this formula, based on different macroscopic models of the detection process. The first one consists of a negative imaginary potential of strength $λ>0$ in the detector volume (i.e., outside the sphere of radius $R$) in the limit $R\to\infty,λ\to 0, Rλ\to \infty$. The second one consists of repeated nearly-projective measurements of (approximately) the observable $1_{|\boldsymbol{x}|>R}$ at times $\mathscr{T},2\mathscr{T},3\mathscr{T},\ldots$ in the limit $R\to\infty,\mathscr{T}\to\infty,\mathscr{T}/R\to 0$; this setup is similar to that of the quantum Zeno effect, except that there one considers $\mathscr{T}\to 0$ instead of $\mathscr{T}\to\infty$. We also provide a comparison to Bohmian mechanics: while in the absence of detectors, the arrival times and places of the Bohmian trajectories on the sphere of radius $R$ have asymptotic distribution density given by the same formula as $σ$, their deviation from the detection times and places is not necessarily small, although it is small compared to $R$, so the effect of the presence of detectors on the particle can be neglected in the far-field regime. We also cover the generalization to surfaces with non-spherical shape, to the case of $N$ non-interacting particles, to time-dependent surfaces, and to the Dirac equation.

Figures (4)

• Figure 1: Figure (a) illustrates a ball of radius $R$, denoted by $\Omega$, which is enclosed by detectors positioned along its boundary $\partial\Omega$. The detectors detect particles that reach this boundary. Figure (b) shows a zoomed-in view near the boundary. In the limit $R \to \infty$, the curved boundary appears g flat, effectively resembling a straight wall. For the error estimates in the model with imaginary potentials, it plays a role that an incident plane wave $\Psi_{\text{inc}}$ will, due to the potential step at the boundary $\partial\Omega$, be partly reflected and partly transmitted.
• Figure 2: A plot illustrating the shape of an error function, $\operatorname{erf}(x)$, as defined in Eq. \ref{['erfdef']}.
• Figure 3: Illustration of the nearly-projective measurement used in the Zeno dynamics at time $n\mathscr{T}$. Shown are $\pm|\Psi|$ (dashed) and $\mathrm{Re} \, \Psi$ (black) as a function of $|\boldsymbol{x}|$ in a neighborhood of $\boldsymbol{x}_0$ with $|\boldsymbol{x}_0|=R$ in several cases with $\Psi_{WOD}=\exp(i\boldsymbol{k}\cdot \boldsymbol{x})$: (a) for $\Psi= 1_\Omega \Psi_{WOD}$ i.e., after a projective measurement (no detection); (b) $\Psi=\hat{P}_\sigma \Psi_{WOD}$ as in \ref{['Pdef']}, i.e., after a nearly-projective measurement (no detection); (c) $\Psi=(\hat{I}-\hat{P}_\sigma)\Psi_{WOD}$ as in \ref{['IPdef']}, i.e., after a nearly-projective measurement (detection).
• Figure 4: Illustration of $\Psi$ at times (a) $t=n\mathscr{T}+$ (no detection), (b) $t=(n+1)\mathscr{T}-$, (c) $t=(n+1)\mathscr{T}+$ (detection). As in Figure \ref{['softstep']}, $\pm |\Psi|$ (dashed) is shown along with $\mathrm{Re}\,\Psi$ (black) in a neighborhood of $\boldsymbol{x}_0$ with $|\boldsymbol{x}_0|=R$. Between the two measurements, i.e., between (a) and (b), the step moves outward by $R/n$, and its width grows to \ref{['wnRn']}.

Theorems & Definitions (15)

• Remark 1
• Remark 1
• Remark 2
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• Remark 5
• Remark 6
• Remark 7
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• Remark 9