Scattering Cross Section Formula Derived From Macroscopic Model of Detectors

TL;DR

This paper derives the scattering cross section formula for quantum particles using macroscopic detector models and compares it with Bohmian mechanics, extending the results to various scenarios including non-spherical surfaces and relativistic cases.

Contribution

It provides two rigorous derivations of the scattering cross section formula from macroscopic detector models and compares these with Bohmian mechanics predictions.

Findings

Derivation using imaginary potential model in the limit R→∞, λ→0, Rλ→∞.

Derivation using repeated nearly-projective measurements with R→∞, T→∞, T/R→0.

Comparison showing detector effects are negligible in the far-field regime.

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 $\Psi_0(\boldsymbol{x})$, 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) $\sigma(\boldsymbol{x},t)= m^3 \hbar^{-3} R t^{-4} |\widehat{\Psi}_0(m\boldsymbol{x}/\hbar t)|^2$ with $\widehat{\Psi}_0$ the Fourier transform of $\Psi_0$. 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 $\lambda>0$ in the detector volume (i.e., outside the sphere of radius $R$) in the limit $R\to\infty,\lambda\to 0, R\lambda\to \infty$. 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 $\sigma$, 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.