Scattering Cross Section Formula Derived From Macroscopic Model of Detectors

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.