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The paper derives the standard scattering cross-section formula as the actual detection distribution from two explicit macroscopic detector models, showing it is not merely a heuristic.

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2026-08-03 12:45 UTC pith:AHIIZEQS

load-bearing objection Kaimal and Tumulka do deliver a derivation of the scattering cross-section formula from two explicit macroscopic detector models; the core math is clean, but the absorption-rate rule is assumed, not derived, and the empirical link is only as strong as that model. the 3 major comments →

arxiv 2601.01625 v3 pith:AHIIZEQS submitted 2026-01-04 quant-ph

Scattering Cross Section Formula Derived From Macroscopic Model of Detectors

Rashi Kaimal , Roderich Tumulka This is my paper
classification quant-ph MSC 81U0581P1581Q05
keywords scattering cross sectiondetection timeimaginary potentialquantum Zeno effectfar-field regimePOVMDirac equationstar-shaped surfaces
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper targets the standard quantum-mechanical scattering cross section σ(x,t) = m³R/(ℏ³t⁴) |Ψ̂₀(mx/ℏt)|², which is commonly used to describe when and where a distant detector clicks. It claims that this formula is not just a plausible flux argument, but the leading-order joint distribution of detection time and place produced by two concrete models of a detector: one in which detectors act as a negative imaginary potential outside a large sphere, and one in which they are repeated soft position measurements (a 'Zeno-dynamics' model). Both derivations share a mechanism: in the appropriate limits, the detector's influence on the incoming wave is a small perturbation on the scale of the sphere radius R, so the detection statistics coincide with the free-particle current across the sphere. The paper also extends the result to arbitrary star-shaped surfaces, several noninteracting particles, time-dependent surfaces, and the Dirac equation.

Core claim

The paper's central claim is that the distribution of the detection time T_D and place X_D converges, after rescaling by R, to δ(ρ−1) (m³/ℏ³τ⁴) |Ψ̂₀(mu/ℏτ)|² d²u dρ dτ (equation 5), i.e., the scattering cross section (1). In the imaginary-potential model, the detector region outside |x|=R carries V=−iλ; the limit R→∞, λ→0, λR→∞ makes the transmitted wave decay on a scale short compared to R while the reflected wave is O(λ), so the absorption-rate formula (2λ/ℏ)|Ψ|² yields the delta function in the radial variable. In the repeated-measurement model, soft steps of width σₙ are applied at times nT; the limit R→∞, T→∞, T/R→0 makes each step move outward faster than it spreads, leaving the interi

What carries the argument

The load-bearing construction is a soft absorbing boundary: either a negative imaginary potential step V=−iλ 1_{|x|>R} or a repeated nearly-projective measurement with 'no-detection' operator P_σ = (1/4)(1−erf((|x|−R)/2σ))². The key underlying identity is that the Fourier modes of Ψ₀ separate spatially at late times, so the wave near radius R is locally a plane wave with wave vector k=mx/ℏt; the reflection and transmission coefficients at the step are B=O(λ) and C=1+O(λ), and the transmitted wave decays as exp(−λ(t−t_cross)/ℏ). The joint distribution then follows from the absorption-rate formula P(X_D∈d³x, T_D∈dt)=(2λ/ℏ)1_{|x|>R}|Ψ|²d³x dt. In the Zeno model, the analogous identity is the fr

Load-bearing premise

The load-bearing premise is that a detector's click probability is literally proportional to the local wave-function density outside the sphere—the absorption-rate rule (2λ/ℏ)|Ψ|²—and that the fuzzy position measurement in the Zeno model has exactly the stated POVM; real detectors may not obey either idealization.

What would settle it

A direct test: prepare a free quantum particle with known initial wave function Ψ₀, surround it with single-particle detectors on a sphere of radius R, and record T_D and X_D. Rescale by R and compare the empirical joint distribution with (5). A systematic disagreement that grows with R beyond the paper's predicted o(R) corrections would falsify the derivation for those detectors.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The standard scattering formula (1) is justified as a genuine prediction about detector clicks, not just a plausible guess, for both macroscopic detector models.
  • The effect of detectors on the particle's wave function is controllable: reflected or backward components are O(λ) or exponentially small, and the deviations of detection time and place from undisturbed arrival time are o(R) in the limit.
  • For arbitrary star-shaped detector surfaces, the formula generalizes to σ(x,t)= m³ n(x)·x / (ℏ³t⁴) |Ψ̂₀(mx/ℏt)|².
  • For N noninteracting (possibly entangled) particles, the joint detection distribution is given by a tensor product of single-particle POVMs, yielding the product formula (10).
  • In the Dirac case, the leading detection distribution is covariant and implies a no-signaling theorem in the far-field regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If real detectors' click statistics depart from the absorption-rate rule (2λ/ℏ)|Ψ|², the derivation's conclusion may still hold but would require a different model; a direct calibration of detector response against incident wave amplitude would test this premise.
  • The limit λR→∞ suggests a trade-off for finite-size experiments: too weak a detector leaves particles undetected, while too strong a detector causes larger reflection; an extension could optimize λ for a given R.
  • The Zeno-dynamics model suggests a general design principle: any detection scheme whose soft step moves outward faster than it spreads will reproduce the free flux distribution, which could be tested with engineered measurement pulses in ultracold or photonic settings.
  • The Dirac analysis quantifies how the helical motion of trajectories in superpositions of positive and negative energies averages out of detection statistics, offering a reason why Zitterbewegung has not appeared in scattering experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper claims to derive the standard far-field scattering cross-section formula (1) for the joint distribution of detection time and place from two explicit macroscopic detector models: (i) a negative imaginary potential in the detector region (Section 3.1), in the limit R→∞, λ→0, λR→∞; and (ii) repeated nearly-projective measurements of 1_{|x|>R} (Section 3.2), in the limit R,T→∞, T/R→0. In both models the paper shows that the leading-order joint distribution converges to (5), i.e., to the scattering cross-section density. The paper also compares the detection time with the Bohmian arrival time in the absence/presence of detectors, showing that deviations are small compared to R but not absolutely, and extends the results to non-spherical surfaces, N non-interacting particles, time-dependent surfaces, and the Dirac equation.

Significance. If the two detector models are accepted as adequate macroscopic descriptions, the paper addresses a long-standing gap in scattering theory: it derives the standard formula (1) from concrete detection mechanisms rather than simply identifying probability flux with detection rate. The derivation has several strengths: the passage from the absorption model to the delta-function limit (63) is explicit and clean; the repeated-measurement model provides a genuinely different route using a soft-step POVM; and the Bohmian comparison quantifies the disturbance caused by detectors. The extension to the Dirac equation and the no-signaling observation in Section 6.3 are interesting consequences. The paper is transparent that a fully microscopic detector derivation is not attempted; this honesty is a strength, but it also means the empirical claim is conditional on the model assumptions.

major comments (3)
  1. [Section 4, Eqs. (55)–(56)] The central derivation of (5) from the imaginary-potential model hinges on the asymptotic forms (55) and (56) for the reflected and transmitted waves. These are asserted with '≈' and the parenthetical remark that they 'could be obtained through a stationary phase calculation,' but no derivation or error estimate is provided. This is a load-bearing gap: without a rigorous or at least detailed asymptotic justification, the main result is a formal derivation. Please either provide a stationary-phase derivation in an appendix with explicit conditions on Ψ0 and the order of the error terms, or clearly state that the result is formal and that rigorous justification is left open.
  2. [Section 3.1, Eq. (21) and Introduction] The Introduction states the goal as deriving that (1) is 'the actual probability density of when and where detectors click.' However, Eq. (21) simply assumes that the detection probability equals the absorption rate (2λ/ℏ)1_{|x|>R}|Ψ(x,t)|^2 d^3x dt. This is the observable content of the imaginary-potential model, not a consequence of a microscopic detector description; the Introduction explicitly defers such a derivation to refs. [19,43]. Thus the claim is conditional on the validity of this absorption-rate model. This limitation should be stated more prominently — for example in the abstract or conclusions — and the phrase 'actual probability density' should be qualified as 'actual within the macroscopic models considered.' The paper already acknowledges this, but the framing could easily be misread.
  3. [Section 5.2, Eq. (96) and surrounding text] For the repeated-measurement model, the paper first treats a heuristic 'first approximation' (Eq. (81)) and then uses the soft-step calculation to argue that boundary effects are negligible. The control of errors relies on the inequality w_n(T) ≪ R/n (Eq. (96)) and the step moving outward faster than it spreads. However, the argument remains qualitative: no explicit bound is given on the difference between the actual detection distribution and the leading-order formula (5) in terms of the parameters T, R, and σ1. Since this is the core of the Zeno-model derivation, a more quantitative error estimate would substantially strengthen the claim.
minor comments (5)
  1. [Figure 1 caption] There is a typo: 'the curved boundary appears g flat' should be 'appears flat.'
  2. [Section 5.2, after Eq. (92)] The bound on |erf(e^{iθ}x) − erf(x cos θ)| contains a factor exp(−(cos^2 θ − sin^2 θ)x^2) but is missing a temperature-independent constant; the inequality as written is not dimensionally transparent. A precise statement would improve readability.
  3. [Section 6.2, Eq. (117)] The assertion that the POVM of joint outcomes is the tensor product of single-particle POVMs assumes that the detectors act independently on the N particles. This assumption should be stated explicitly.
  4. [Introduction, Remark 2] The term 'Zeno dynamics' is used for the limit T→∞, T/R→0, whereas the standard quantum Zeno effect corresponds to T→0. The paper notes the difference, but readers may still find the terminology confusing; a brief remark clarifying the analogy would help.
  5. [General] The paper contains numerous informal '≈' statements in key asymptotic formulas (e.g., Eq. (4), Eq. (55), Eq. (94)). A consistent convention for what '≈' means (asymptotic equality, uniform error bounds, etc.) would make the mathematical status of the claims clearer.

Circularity Check

0 steps flagged

No significant circularity: the target formula (1) is derived from explicitly postulated detector models, not assumed; the main caveat is a deferred microscopic derivation of the detector models themselves.

full rationale

The paper does not assume the scattering cross section formula (1) in its derivations. In the imaginary-potential model, detection probability is defined by Eq. (21) as the absorption rate (2λ/ℏ)|Ψ(x,t)|^2 in the detector region; this is a postulated macroscopic detector model, not a restatement of (1). The subsequent calculation, Eqs. (60)–(63), uses only the free far-field asymptotic form (4), the stationary-phase reflection/transmission coefficients (52)–(56), and the elementary identity (64) for a delta-sequence. No parameter is fitted to (1); λ merely controls the softness of the detector and is taken to zero with λR→∞. In the Zeno model, detection is defined by the explicit POVM (33)–(34) and the collapse rules (39)–(41), and the limiting distribution (5) is derived from free Schrödinger evolution plus these measurement postulates. The self-citations (e.g., [19,43,24,25,38]) are used as context, for model definitions, or for auxiliary results; they are not used to assume the target formula. The genuine limitation is that the link from a microscopic quantum-mechanical detector description to Eq. (21) and to the Zeno POVM is not proved here — the paper states that such a derivation "will not be attempted here" and points to outlines in [19,43]. That is a missing justification of the empirical relevance of the detector models, but it is not circularity: the central derivation from each model to (1) is mathematically explicit and self-contained. The score of 2 reflects only the conditional nature and the presence of non-load-bearing self-citations.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The paper introduces no new particles or forces; its detector models are effective evolutions (imaginary potential, POVM soft-step) treated as axioms. The main free parameters are the model parameters λ, T, and σ1, all chosen in the limit regime rather than fitted to data.

free parameters (3)
  • imaginary potential strength λ = λ→0, λR→∞
    Model parameter controlling detector softness; no data fitting, but chosen by hand to define the limit.
  • stroboscopic measurement interval T = T→∞, T/R→0
    Model parameter for repeated-measurement detector; chosen by hand.
  • soft-step width scale σ1 = O(T/R) ≪ σ1 ≪ O(T^2/R), equivalently Rσ1/T→∞ and Rσ1/T^2→0
    Chosen in Eq. (38) so that the softened step moves outward faster than its width grows; not fitted to data.
axioms (6)
  • domain assumption Free Schrödinger evolution with imaginary potential −iλ 1_{|x|>R} is a valid macroscopic detector model, with detection probability rate (2λ/ℏ)|Ψ|^2 outside the sphere.
    Sections 2.3 and Eq. (21); the paper motivates but does not derive this from a microscopic detector model.
  • domain assumption Repeated nearly-projective measurements of the POVM (33)-(34) model detectors, with the Bohmian outcome rule (43)-(44).
    Section 3.2 and Eqs. (33)-(34); replaces a sharp projection by a soft error-function step.
  • standard math Large-time asymptotic form (4) of the free Schrödinger propagator: locally a plane wave with Fourier-mode separation, with error O(1/R).
    Cited to [4, Prop. 3.17], [33, Thm. IX.31], [15, Eq. (9.20)]; used throughout to evaluate detection distributions.
  • domain assumption Initial wave function Ψ0 is fixed and effectively supported near the origin while R→∞.
    Section 3 setup; required for the spherical detector to be in the far-field regime.
  • domain assumption For non-spherical surfaces, the outward normal satisfies n(x)·x>0, so the region is star-shaped and the normal current is outward.
    Section 6.1 Eq. (108); needed for formula (9) to have non-negative density.
  • domain assumption In the Dirac case, the imaginary potential step produces a reflected amplitude O(λ), and positive- and negative-energy contributions can be treated independently after dropping cross terms.
    Appendix C and Remark 14; needed for the covariant formulas (12) and (145).

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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 $\Psi_0(\boldsymbol{x})$, 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.

Figures

Figures reproduced from arXiv: 2601.01625 by Rashi Kaimal, Roderich Tumulka.

Figure 1
Figure 1. Figure 1: Figure (a) illustrates a ball of radius R, denoted by Ω, which is enclosed by detectors positioned along its boundary ∂Ω. The detectors detect particles that reach this boundary. Figure (b) shows a zoomed-in view near the boundary. In the limit R → ∞, 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 t… view at source ↗
Figure 2
Figure 2. Figure 2: Shape of the error function, erf(x), as defined in Eq. (35). 3.2 Second Model: Zeno Dynamics Secondly, we study the same setup but we model the detectors in a different way. We use re￾peated nearly-projective measurements of the observable 1Ωc at non-random times T , 2T , 3T , . . . , nT , . . .. The reason we use a nearly-projective measurement rather than an exactly pro￾jective one is to make smaller (th… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the nearly-projective measurement used in the Zeno dynamics at time [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of Ψ at times (a) t = nT + (no detection), (b) t = (n + 1)T −, (c) t = (n + 1)T + (detection). As in [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗

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Reference graph

Works this paper leans on

46 extracted references · 23 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun (eds.):Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables. Washington, D.C.: National Bureau of Standards (1964)

    1964

  2. [2]

    Aharonov and D

    Y. Aharonov and D. Bohm: Time in the Quantum Theory and the Uncertainty Relation for Time and Energy.Physical Review122: 1649 (1961)

    1961

  3. [3]

    Allcock: The time of arrival in quantum mechanics

    G.R. Allcock: The time of arrival in quantum mechanics. II. The individual measurement. Annals of Physics53: 286–310 (1969) 35

    1969

  4. [4]

    Amrein, J.M

    W.O. Amrein, J.M. Jauch, and K.B. Sinha:Scattering Theory in Quantum Mechanics. London: Benjamin (1977)

    1977

  5. [5]

    Beck and D

    C. Beck and D. Lazarovici: Particle Detection in the Scattering Regime. Preprint (2025)

    2025

  6. [6]

    Bethe: A Continuum Theory of the Compound Nucleus.Physical Review57: 1125– 1144 (1940)

    H.A. Bethe: A Continuum Theory of the Compound Nucleus.Physical Review57: 1125– 1144 (1940)

    1940

  7. [7]

    Cavendish and S

    W. Cavendish and S. Das: Absorbing detectors meet scattering theory.Physical Review A112: 062217 (2025)http://arxiv.org/abs/2509.07518

    arXiv 2025

  8. [8]

    Coplan, J.H

    M.A. Coplan, J.H. Moore, and J.P. Doering: (e,2e) spectroscopy.Reviews of Modern Physics66: 985–1014 (1994)

    1994

  9. [9]

    Daumer, D

    M. Daumer, D. Dürr, S. Goldstein, and N. Zanghì: On the quantum probability flux through surfaces.Journal of Statistical Physics88: 967–977 (1997). Reprinted in [12]. http://arxiv.org/abs/quant-ph/9512016

    Pith/arXiv arXiv 1997

  10. [10]

    D. Dürr, S. Goldstein, T. Moser, and N. Zanghì: A microscopic derivation of the quantum mechanical formal scattering cross section.Communications in Mathematical Physics266: 665–697 (2006)http://arxiv.org/abs/quant-ph/0509010

    Pith/arXiv arXiv 2006

  11. [11]

    D. Dürr, S. Goldstein, and N. Zanghì: Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory.Journal of Statistical Physics116: 959–1055 (2004) Reprinted in [12].http://arxiv.org/abs/quant-ph/0308038

    Pith/arXiv arXiv 2004

  12. [12]

    D. Dürr, S. Goldstein, and N. Zanghì:Quantum Physics Without Quantum Philosophy. Heidelberg: Springer (2013)

    2013

  13. [13]

    Dürr and P

    D. Dürr and P. Pickl: Flux-Across-Surfaces Theorem for a Dirac Particle.Journal of Mathematical Physics44: 423–465 (2003)http://arxiv.org/abs/math-ph/0207010

    Pith/arXiv arXiv 2003

  14. [14]

    Dürr and S

    D. Dürr and S. Teufel: On the exit statistics theorem of many particle quantum scattering. In P. Blanchard and G. Dell’Antonio (ed.s):Multiscale Methods in Quantum Mechanics, Basel: Birkhäuser (2004)http://arxiv.org/abs/quant-ph/0405007

    Pith/arXiv arXiv 2004

  15. [15]

    Dürr and S

    D. Dürr and S. Teufel:Bohmian Mechanics.Heidelberg: Springer (2009)

    2009

  16. [16]

    L. Frolov: Detecting screens modeled by extensions of Schrödinger operators that generate C0 contraction semigroups.Reviews in Mathematical Physicsonline ready (2025)http: //arxiv.org/abs/2506.00231

    Pith/arXiv arXiv 2025

  17. [17]

    Goldstein, R

    S. Goldstein, R. Tumulka, and N. Zanghì: On the Spin Dependence of Detection Times and the Nonmeasurability of Arrival Times.Scientific Reports14: 3775 (2024)http: //arxiv.org/abs/2309.11835

    Pith/arXiv arXiv 2024

  18. [18]

    Goldstein, R

    S. Goldstein, R. Tumulka, and N. Zanghì: Arrival Times Versus Detection Times.Foun- dations of Physics54: 63 (2024)http://arxiv.org/abs/2405.04607

    Pith/arXiv arXiv 2024

  19. [19]

    Halliwell: Arrival Times in Quantum Theory from an Irreversible Detector Model.Progress of Theoretical Physics102(4): 707–717 (1999)http://arxiv.org/abs/ quant-ph/9805057

    J.J. Halliwell: Arrival Times in Quantum Theory from an Irreversible Detector Model.Progress of Theoretical Physics102(4): 707–717 (1999)http://arxiv.org/abs/ quant-ph/9805057

    Pith/arXiv arXiv 1999

  20. [20]

    Holland:The Quantum Theory of Motion.Cambridge University Press (1993) 36

    P.R. Holland:The Quantum Theory of Motion.Cambridge University Press (1993) 36

    1993

  21. [21]

    Kiukas, A

    J. Kiukas, A. Ruschhaupt, P.O. Schmidt, and R.F. Werner: Exact Energy–Time Uncer- tainty Relation for Arrival Time by Absorption.Journal of Physics A: Mathematical and Theoretical45: 185301 (2012)http://arxiv.org/abs/1109.5087

    Pith/arXiv arXiv 2012

  22. [22]

    Komech: On Dynamical Justification of Quantum Scattering Cross Section.Journal of Mathematical Analysis and Applications432(1): 583–602 (2015)http://arxiv.org/ abs/1206.3677

    A. Komech: On Dynamical Justification of Quantum Scattering Cross Section.Journal of Mathematical Analysis and Applications432(1): 583–602 (2015)http://arxiv.org/ abs/1206.3677

    Pith/arXiv arXiv 2015

  23. [23]

    Lienert, S

    M. Lienert, S. Petrat, and R. Tumulka:Multi-time Wave Functions: An Introduction. Heidelberg: Springer (2020)

    2020

  24. [24]

    Lienert and R

    M. Lienert and R. Tumulka: Born’s Rule for Arbitrary Cauchy Surfaces.Letters in Math- ematical Physics110: 753–804 (2020)http://arxiv.org/abs/1706.07074

    Pith/arXiv arXiv 2020

  25. [25]

    Lill and R

    S. Lill and R. Tumulka: Another Proof of Born’s Rule on Arbitrary Cauchy Surfaces. Annales Henri Poincaré23: 1489–1524 (2022)http://arxiv.org/abs/2104.13861

    Pith/arXiv arXiv 2022

  26. [26]

    Maccone and K

    L. Maccone and K. Sacha: Quantum Measurements of Time.Physical Review Letters124: 110402 (2020)http://arxiv.org/abs/1810.12869

    Pith/arXiv arXiv 2020

  27. [27]

    Misra and E.C.G

    B. Misra and E.C.G. Sudarshan: The Zeno’s paradox in quantum theory.Journal of Mathematical Physics18: 756–763 (1977)

    1977

  28. [28]

    Muga and R

    J.G. Muga and R. Leavens: Arrival Time in Quantum Mechanics.Physics Reports338: 353 (2000)

    2000

  29. [29]

    Narayanan, A

    K. Narayanan, A. Perryman, and S. Tahvildar-Zadeh: Asymptotic momentum of Dirac particles in one space dimension. Preprint (2025)http://arxiv.org/abs/2512.21423

    Pith/arXiv arXiv 2025

  30. [30]

    Newton:Scattering Theory of Waves and Particles, 2nd edition

    R.G. Newton:Scattering Theory of Waves and Particles, 2nd edition. New York: Springer (1982)

    1982

  31. [31]

    Pauli: Die allgemeinen Prinzipien der Wellenmechanik

    W. Pauli: Die allgemeinen Prinzipien der Wellenmechanik. Pages 83–272 in: H. Geier and K. Scheel (editors),Handbuch der Physik, 2nd ed., vol. XXIV/1: Quantentheorie, Berlin: Springer (1933)

    1933

  32. [32]

    Peters, S

    C. Peters, S. Tahvildar-Zadeh, and R. Tumulka: No-Signaling Theorem Covering also Absorbing Boundary Conditions and Imaginary Potentials. In preparation (2026)

    2026

  33. [33]

    Reed and B

    M. Reed and B. Simon:Methods of Modern Mathematical Physics, Vol. II: Fourier Anal- ysis, Self-Adjointness.San Diego: Academic Press (1975)

    1975

  34. [34]

    Römer, D

    S. Römer, D. Dürr, and T. Moser: Asymptotic Behavior of Bohmian Trajectories in Scattering Situations.Journal of Physics A: Mathematical and General38: 8421–8443 (2005)http://arxiv.org/abs/math-ph/0505074

    Pith/arXiv arXiv 2005

  35. [35]

    E. Schrödinger: Über die kräftefreie Bewegung in der relativistischen Quanten- mechanik.Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch- mathematische Klasse24: 418–428 (1930)

    1930

  36. [36]

    Tahvildar-Zadeh and S

    S. Tahvildar-Zadeh and S. Zhou: Detection time of Dirac particles in one space dimension. Pages 187–201 in S. Goldstein, R. Tumulka, and N. Zanghì (ed.s),Physics and the Nature of Reality: Essays in Memory of Detlef Dürr, Heidelberg: Springer (2024)http://arxiv. org/abs/2112.07366 37

    Pith/arXiv arXiv 2024

  37. [37]

    Thaller:The Dirac Equation

    B. Thaller:The Dirac Equation. Heidelberg: Springer (1992)

    1992

  38. [38]

    Hidden Variable Interpretation of Spontaneous Localization Theory

    R. Tumulka: Comment on “Hidden Variable Interpretation of Spontaneous Localization Theory.”Journal of Physics A: Mathematical and Theoretical44: 478001 (2011)http: //arxiv.org/abs/1108.1520

    Pith/arXiv arXiv 2011

  39. [39]

    Tumulka: Distribution of the Time at Which an Ideal Detector Clicks.Annals of Physics442: 168910 (2022)http://arxiv.org/abs/1601.03715

    R. Tumulka: Distribution of the Time at Which an Ideal Detector Clicks.Annals of Physics442: 168910 (2022)http://arxiv.org/abs/1601.03715

    Pith/arXiv arXiv 2022

  40. [40]

    Tumulka: Detection Time Distribution for Several Quantum Particles.Physical Review A106: 042220 (2022)http://arxiv.org/abs/1601.03871

    R. Tumulka: Detection Time Distribution for Several Quantum Particles.Physical Review A106: 042220 (2022)http://arxiv.org/abs/1601.03871

    Pith/arXiv arXiv 2022

  41. [41]

    Tumulka: Detection Time Distribution for the Dirac Equation

    R. Tumulka: Detection Time Distribution for the Dirac Equation. Preprint (2016)http: //arxiv.org/abs/1601.04571

    Pith/arXiv arXiv 2016

  42. [42]

    Tumulka:Foundations of Quantum Mechanics

    R. Tumulka:Foundations of Quantum Mechanics. Heidelberg: Springer (2022)

    2022

  43. [43]

    Tumulka: On a Derivation of the Absorbing Boundary Rule.Physics Letters A494: 129286 (2024)http://arxiv.org/abs/2310.01343

    R. Tumulka: On a Derivation of the Absorbing Boundary Rule.Physics Letters A494: 129286 (2024)http://arxiv.org/abs/2310.01343

    Pith/arXiv arXiv 2024

  44. [44]

    Ullrich, R

    J. Ullrich, R. Moshammer, A. Dorn, R. Dörner, L.P.H. Schmidt, and H. Schmidt- Böcking: Recoil-ion and electron momentum spectroscopy: reaction-microscopes.Reports on Progress in Physics66(9): 1463–1545 (2003)

    2003

  45. [45]

    N. Vona, G. Hinrichs, and D. Dürr: What does one measure when one measures the arrival time of a quantum particle?Physical Review Letters111: 220404 (2013)http: //arxiv.org/abs/1307.4366

    Pith/arXiv arXiv 2013

  46. [46]

    Werner: Arrival time observables in quantum mechanics.Annales de l’Institute Henri Poincaré, section A47: 429–449 (1987) 38

    R. Werner: Arrival time observables in quantum mechanics.Annales de l’Institute Henri Poincaré, section A47: 429–449 (1987) 38

    1987