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arXiv:2601.01625 · Journal of Physics A Mathematical and Theoretical

Scattering Cross Section Formula Derived From Macroscopic Model of Detectors

Rashi Kaimal, Roderich Tumulka

quant-phSubmitted Jan 4, 2026• Announced Jan 6, 2026• 0 citations
v1 · Jan 4, 2026v2 · Feb 12, 2026v3 · Jun 8, 2026
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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.

Journal reference
Journal of Physics A: Mathematical and Theoretical 59: 295303 (2026)
DOI
10.1088/1751-8121/ae880a
Comments
36 pages LaTeX, 4 figures

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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 Ψ0(x)Ψ_0(\boldsymbol{x})Ψ0​(x), surrounded by detectors along a sphere of large radius RRR, the probability distribution of the detection time and place has asymptotic density (i.e., scattering cross section) σ(x,t)=m3ℏ−3Rt−4∣Ψ^0(mx/ℏt)∣2σ(\boldsymbol{x},t)= m^3 \hbar^{-3} R t^{-4} |\widehatΨ_0(m\boldsymbol{x}/\hbar t)|^2σ(x,t)=m3ℏ−3Rt−4∣Ψ0​(mx/ℏt)∣2 with Ψ^0\widehatΨ_0Ψ0​ the Fourier transform of Ψ0Ψ_0Ψ0​. 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>0λ>0λ>0 in the detector volume (i.e., outside the sphere of radius RRR) in the limit R→∞,λ→0,Rλ→∞R\to\infty,λ\to 0, Rλ\to \inftyR→∞,λ→0,Rλ→∞. The second one consists of repeated nearly-projective measurements of (approximately) the observable 1∣x∣>R1_{|\boldsymbol{x}|>R}1∣x∣>R​ at times T,2T,3T,…\mathscr{T},2\mathscr{T},3\mathscr{T},\ldotsT,2T,3T,… in the limit R→∞,T→∞,T/R→0R\to\infty,\mathscr{T}\to\infty,\mathscr{T}/R\to 0R→∞,T→∞,T/R→0; this setup is similar to that of the quantum Zeno effect, except that there one considers T→0\mathscr{T}\to 0T→0 instead of T→∞\mathscr{T}\to\inftyT→∞. 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 RRR have asymptotic distribution density given by the same formula as σσσ, their deviation from the detection times and places is not necessarily small, although it is small compared to RRR, 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 NNN non-interacting particles, to time-dependent surfaces, and to the Dirac equation.
We derive and explain the key ideas behind a time-dependent formulation of quantum scattering theory, applicable generally to systems with a finite-range scattering potential. The scattering is initiated and probed by plane wave packets, which are localized just outside the range of the potential. The asymptotic limits of conventional scattering theory (initiation in the remote past; detection in the remote future) are not taken. Instead, the differential cross section (DCS) is obtained by projecting the scattered wave packet onto the probe plane wave packets. The projection also yields a time-dependent version of the DCS. Cuts through the wave packet, just as it exits the scattering potential, yield time-dependent and time-independent angular distributions that give a close-up picture of the scattering which complements the DCS. We have previously applied the theory to interpret experimental cross sections of chemical reactions [e.g., S. C. Althorpe, F. Fern\'andez-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, and E. Wrede, Nature (London) 416, 67 (2002)]. This paper gives the derivation of the theory, and explains its relation to conventional scattering theory. For clarity, the derivation is restricted to spherical-particle scattering, though it may readily be extended to general multichannel systems. We illustrate the theory using a simple application to hard-sphere scattering.
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Abstract How to compute the probability distribution of a detection time, i.e., of the time which a detector registers as the arrival time of a quantum particle, is a long-debated problem. In this regard, Bohmian mechanics provides in a straightforward way the distribution of the time at which the particle actually does arrive at a given surface in 3-space in the absence of detectors. However, as we discuss here, since the presence of detectors can change the evolution of the wave function and thus the particle trajectories, it cannot be taken for granted that the arrival time of the Bohmian trajectories in the absence of detectors agrees with the one in the presence of detectors, and even less with the detection time. In particular, we explain why certain distributions that Das and Dürr (Sci. Rep. 9: 2242, 2019) presented as the distribution of the detection time in a case with spin, based on assuming that all three times mentioned coincide, are actually not what Bohmian mechanics predicts.
Conventional scattering theory is incomplete in that it does not adequately describe the behaviour of the wave function at macroscopic distances from the scattering reaction volume. In scattering experiments particles are incident from sources at macroscopic distance and measured at macroscopic distance from the microscopic reaction volume. Standard experimental procedure is to describe asymptotic particle motion by classical mechanics. However, this transition to classical mechanics is not accounted for in standard quantum scattering theory. Hence, we revisit conventional scattering theory with the introduction of semiclassical propagators to describe the wave function in both incident and outgoing asymptotic scattering channels. This description leads to the manifestation of classical trajectories in the quantum wave function both before and after the collision. It includes the case of extraction with external fields as in modern multi-particle detectors. We derive the asymptotic Kemble imaging theorem (IT) limit to demonstrate how formal quantum scattering theory contains implicit classical mechanics threaded throughout the asymptotic wave functions.
In a recent paper Grot, Rovelli, and Tate (GRT) [Phys. Rev. A 54, 4676 (1996)] derived an expression for the probability distribution $\ensuremath{\pi}(T;X)$ of intrinsic arrival times T(X)T(X)T(X) at position x=Xx=Xx=X for a quantum particle with initial wave function $\ensuremath{\psi}(x,t=0)$ freely evolving in one dimension. This was done by quantizing the classical expression for the time of arrival of a free particle at X,X,X, assuming a particular choice of operator ordering, and then regulating the resulting time of arrival operator. For the special case of a minimum-uncertainty-product wave packet at t=0t=0t=0 with average wave number 〈k〉〈k〉〈k〉 and variance $\ensuremath{\Delta}k$ they showed that their analytical expression for $\ensuremath{\pi}(T;X)$ agreed with the probability current density J(x=X,t=T)J(x=X,t=T)J(x=X,t=T) only to terms of order $\ensuremath{\Delta}k/〈k〉.$ They dismissed the probability current density as a viable candidate for the exact arrival time distribution on the grounds that it can sometimes be negative. This fact is not a problem within Bohmian mechanics where the arrival time distribution for a particle, either free or in the presence of a potential, is rigorously given by ∣J(X,T)∣|J(X,T)|∣J(X,T)∣ (suitably normalized) [W. R. McKinnon and C. R. Leavens, Phys. Rev. A 51, 2748 (1995); C. R. Leavens, Phys. Lett. A 178, 27 (1993); M. Daumer et al., in On Three Levels: The Mathematical Physics of Micro-, Meso-, and Macro-Approaches to Physics, edited by M. Fannes et al. (Plenum, New York, 1994); M. Daumer, in Bohmian Mechanics and Quantum Theory: An Appraisal, edited by J. T. Cushing et al. (Kluwer Academic, Dordrecht, 1996)]. The two theories are compared in this paper and a case presented for which the results could not differ more: According to GRT's theory, every particle in the ensemble reaches a point x=X,x=X,x=X, where $\ensuremath{\psi}(x,t)$ and J(x,t)J(x,t)J(x,t) are both zero for all t,t,t, while no particle ever reaches XXX according to the theory based on Bohmian mechanics. Some possible implications are discussed.
There are several inequivalent proposals in the literature for how to compute the probability distribution of the time that a detector registers for the arrival of a quantum particle. For three of these proposals, based on two kinds of absorbing boundary conditions and imaginary potentials, we compute the predicted distribution for an experimental setup involving a single non-relativistic quantum particle with spin 0 or 1/2 in a wave guide along the zzz axis with the detector waiting downstream. We find that the distribution shows signs of partial reflection of the wave function off of the detector; for a spin-1/2 wave function, it is independent of the initial spin orientation for the parameters tested but does depend, for boundary conditions coupling to the spin, on the width of the wave guide. We also compare our predictions with the competing ones of Das and Dürr [arXiv:1802.07141].