# Questioning the adequacy of certain quantum arrival-time distributions

Siddhant Das<EMAIL_ADDRESS>Mathematisches Institut,
Ludwig-Maximilians-Universität München, Theresienstr. 39, D-80333 München,
Germany  Ward Struyve<EMAIL_ADDRESS>Instituut voor Theoretische
Fysica, KU Leuven, Belgium  Centrum voor Logica en Filosofie van de
Wetenschappen, KU Leuven, Belgium

###### Abstract

It is shown that a class of exponentially decaying time-of-arrival probability
distributions suggested by Włodarz, Marchewka and Schuss, and Jurman and
Nikolić, as well as a semiclassical distribution implicit in time-of-flight
momentum measurements, do not show the expected behavior for a Gaussian wave
train. This casts doubts on the physical adequacy of these arrival-time
proposals. In contrast, the quantum flux distribution (a special case of the
Bohmian arrival-time distribution) displays the expected behavior.

##  I Introduction

In view of the experimentally driven genesis of quantum mechanics and its
notable empirical successes, it is a great surprise that a straightforward
question such as “ _how long_ does it take for a quantum particle to strike
the detector surface in a double-slit experiment?” could be even more
problematic than the question “ _where_ , in such an experiment, does the
particle strike the detector surface?”. While the second question pertaining
to the ubiquitous interference pattern is discussed in every quantum mechanics
textbook and is experimentally well-established, the former concerning the
arrival (or detection) time of the particle amenable to laboratory time-of-
flight (TOF) experiments Kurtsiefer  _et al._ (1997); *wig; *Pfaudetector is a
matter of an ongoing debate.

This seems almost paradoxical given that TOF measurements are the quintessence
of methods determining, e.g., energies and momenta of particles Ullrich  _et
al._ (2003); Kothe  _et al._ (2013); Gliserin  _et al._ (2016); Wolf and Helm
(2000), chemical reaction dynamics (as in the Rydberg tagging TOF technique
Schnieder  _et al._ (1997); Xie  _et al._ (2020)), or the temperature of
single trapped atoms/ions Fuhrmanek  _et al._ (2010); Stopp  _et al._ (2021).
However, it is not quantum mechanics that is invoked to interpret the TOF
measurements in these cases. Instead, one employs various ansatzes and
heuristics based on either Newtonian mechanics or geometric optics, whose
capabilities for describing the data are highly questionable (especially in
single-particle experiments featuring wave packet coherence).

That said, over the past decades, an increasing number of physicists have
endeavored to formulate a first-principles description of arrival times within
quantum mechanics, resulting in a multitude of disparate theoretical proposals
for computing the arrival-time
distribution111$\Pi(\tau)\,d\kern-1.00006pt\tau$ is the probability that a
particle prepared in a state $\psi(x,0)$ at time zero is registered on a
specified detector between time $\tau$ and $\tau+d\kern-1.00006pt\tau$.
$\Pi(\tau)$ of a quantum particle Muga and Leavens (2000); Muga  _et al._
(1998, 2008). However, experiments designed to help choose between competing
viewpoints have been slow in coming.

The TOF distributions suggested in the literature can be divided into two
broad categories. First, ideal (or intrinsic) arrival-time distributions that
are _apparatus-independent_ theoretical predictions, given by some functional
of the initial wave function $\psi(x,0)$ and the geometrical surface of the
detector (typically a single point on a line in one-dimensional discussions).
A notable example is the quantum flux distribution

$\Pi_{\text{QF}}(\tau)=\frac{\hbar}{m}\,\text{Im}\big{[}\psi^{*}(L,\tau)\,\partial_{x}\psi(L,\tau)\big{]},$
(1)

applicable for a particle of mass $m$ arriving at the point $x=L$ on a line.
Here, $\psi(x,t)$ denotes its wave function at time $t$, a solution of
Schrödinger’s equation

$i\hbar\kern 1.00006pt\frac{\partial\psi(x,t)}{\partial
t}=-\,\frac{\hbar^{2}}{2m}\kern 1.00006pt\frac{\partial^{2}\psi(x,t)}{\partial
x^{2}}+V(x,t)\kern 1.00006pt\psi(x,t),$ (2)

with initial condition $\psi(x,0)$. The quantum flux distribution has been
arrived at from various theoretical viewpoints, in particular, as the arrival-
time distribution in Bohmian mechanics (de Broglie-Bohm or pilot-wave theory)
Daumer  _et al._ (1997); *DDGZ96; Leavens (1998a); *Leav98; Grübl and
Rheinberger (2002); *Kreidl in the absence of backflow (Das and Nöth, 2021, p.
6). Another well-known example applicable for freely moving particles,
$\smash{V(x,t)=0}$, is the Aharonov-Bohm (Egusquiza and Muga, 1999, Sec. 3)
and Kijowski Kijowski (1974) arrival-time distribution (Das and Nöth, 2021,
Sec. 2), which is typically indistinguishable from $\Pi_{\text{QF}}(\tau)$ in
the far-field or scattering regime accessible to present-day experiments.

Yet another ideal arrival-time distribution often implicit in TOF _momentum_
measurements is the semi-classical distribution

$\Pi_{\text{SC}}(\tau)=\frac{mL}{\hbar\tau^{2}}\kern-1.00006pt\left|{\tilde{\psi\mkern
3.0mu}\mkern-3.0mu}{}\kern-1.49994pt\left(\frac{mL}{\hbar\tau}\right)\right|^{2}\\!,$
(3)

where

${\tilde{\psi\mkern 3.0mu}\mkern-3.0mu}{}(k)=\frac{1}{\sqrt{2\kern
1.00006pt\pi}}\int_{-\infty}^{\infty}\\!\\!\\!dx~{}\psi(x,0)\,e^{-\,ikx}$ (4)

is the Fourier transform of the wave function prepared at _time zero_
Kurtsiefer  _et al._ (1995); Copley and Udovic (1993); Gliserin  _et al._
(2016); Wolf and Helm (2000). This distribution is typically motivated along
the following lines: For a classical trajectory
$\smash{x(t)=x(0)+pt\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}m}$, with $x(0)\ll L$, the arrival-time of the
particle is approximately given by
$\smash{\tau=mL\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}p}$. The above distribution $\Pi_{\text{SC}}(\tau)$
is then obtained by considering this classical arrival-time formula, assuming
that the width of $\psi(x,0)$ is much smaller than $L$ and that the momentum
$p$ is distributed according to the quantum mechanical momentum distribution
$\hbar^{-1}\kern 1.00006pt|{\tilde{\psi\mkern
3.0mu}\mkern-3.0mu}{}(p\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}\hbar)|^{2}$ (Vona, 2014, p. 21). For a suitably
localized $\psi(x,0)$, the semiclassical distribution (3) is also obtained
from $\Pi_{\text{QF}}(\tau)$ for a large $L$ _and_ large $\tau$ Daumer  _et
al._ (1997, 1996).

The second category is that of non-ideal or measurement-inspired TOF
distributions that involve a model of the detector. Various suggestions have
been put forward, e.g., simple absorbing boundary conditions Werner (1987);
Tumulka (2016), complex potentials Halliwell and Yearsley (2009); Muga  _et
al._ (1999); Allcock (1969), wave function collapse (both detector induced
Echanobe  _et al._ (2008); Jurman and Nikolić (2021) and spontaneous Blanchard
and Jadczyk (1996, 1998)), path integrals with absorbing boundaries Marchewka
and Schuss (1998); *PI2; *PI3, a variety of quantum clocks (Muga  _et al._ ,
2008, Ch. 8), and even a timeless formulation of QM Maccone and Sacha (2020).
An overview of these proposals, including a novel experimental set-up for
distinguishing one from another (and in particular from
$\Pi_{\text{QF}}(\tau)$) will appear in Das and Dürr .

In what follows, we focus on a class of non-ideal TOF distributions Włodarz
(2002); Marchewka and Schuss (1998); *PI2; *PI3; Jurman and Nikolić (2021)
that have the form

$\Pi(\tau)=\lambda(\tau)\kern
1.00006pt\exp(-\\!\int_{0}^{\tau}\\!\\!\\!dt~{}\lambda(t)),$ (5)

where $\lambda(t)$ is the so-called “intensity function” for which various
proposals exist, see Table 1. This distribution is normalized as

$\int_{0}^{\infty}\\!\\!\\!d\kern-1.00006pt\tau~{}\Pi(\tau)\,+\,P(\infty)=1,$
(6)

where

$P(\infty)=\lim\limits_{\tau\to\infty}\frac{\Pi(\tau)}{\lambda(\tau)}$ (7)

is a “non-detection probability”, accounting for the fraction of experimental
runs in which the particle never arrives at $L$. To derive (5) a time interval
$[\kern 1.00006pt0,\tau]$ is considered (Włodarz, 2002, p. 2), discretized
into small time steps $\smash{\Delta
t=\tau\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}N}$, where $\lambda(t_{n})\Delta t$ is the
probability for the particle to be detected between time $t_{n}=n\kern
1.00006pt\Delta t$ and $t_{n+1}=(n+1)\kern 1.00006pt\Delta t$,
$n=0,\,1,\,2,\,\dots\kern 1.00006ptN-1$. Further, assuming _independent_
probabilities at each time step222This generates an inhomogeneous Poisson
point process with rate $\lambda(t)$., the probability for the particle to be
detected between time $t_{N-1}$ and $t_{N}\kern 1.00006pt(=\tau)$ is simply

$\lambda(t_{N-1})\kern 1.00006pt\Delta
t\prod_{n=0}^{N-2}\big{(}1-\lambda(t_{n})\kern 1.00006pt\Delta t\big{)}.$ (8)

By taking the limit $\Delta t\to 0$, equivalently $N\to\infty$, of (8), the
time-of-arrival density (5) is obtained. The intensity function $\lambda$ is
supposed to follow from the physics of the detector.

Table 1: Exponentially decaying arrival-time proposals and their intensity functions (the wave functions $\overline{\psi}$ and $\psi_{c}$ are defined in Sections II and III). Proponents | $\smash{\lambda(t)}$ | Ref.
---|---|---
Włodarz (W) | $\displaystyle\smash{\lambda_{0}\kern 1.00006pt|\psi(L,t)|^{2}}$ | Włodarz (2002)
Marchewka & Schuss (MS) | $\displaystyle\smash{(\lambda^{\prime}\epsilon\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern 1.00006pt/\kern 1.00006pt}}\pi)\kern 1.00006pt|\partial_{x}\kern 1.00006pt\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}(L,t)|^{2}}$ | Marchewka and Schuss (1998); *PI2; *PI3
Jurman & Nikolić (JN) | $\displaystyle\smash{\frac{1}{\delta t}\\!\int_{L}^{L+\Delta L}\kern-14.22636ptdx~{}|\psi_{c}(x,t)|^{2}}$ | Jurman and Nikolić (2021)

We will challenge these proposals by considering a train of Gaussian wave
packets that initially have the same width and are moving with the same
velocity towards the detector. By choosing the parameters so that each
Gaussian wave packet reaches the detector one by one without significant
spreading, it is expected on the basis of quasi-classical reasoning that each
packet will contribute in the same way to the arrival-time distribution. In
particular, it is expected that the arrival-time distribution will display
peaks of the same shape and height at times roughly corresponding to the
classically expected arrival times (i.e., the hitting times of individual
packets). However, we will show that this is not the case for the proposals
given in Table 1. While these distributions display peaks at the expected wave
packet arrival times, the peaks are exponentially damped.

The outline of the paper is as follows: In Sec. II the Gaussian train is
introduced. The analysis of the exponential proposals for this wave function
follows next in Sec. III. The semiclassical distribution is treated in Sec. IV
and we conclude in Sec. V.

##  II Gaussian wave train

We direct our attention to the dynamics on a line, the detector occupying the
interval $(L,L+\Delta L)$. Consider first a single Gaussian wave packet,
initially ($t=0$) centered at $x=0$, to the left of the detector, given by

$\phi(x,0)=\frac{1}{\sqrt{\sigma\kern-1.00006pt\sqrt{\pi}}}\exp(-\,\frac{x^{2}}{2\,\sigma^{2}}+\frac{i}{\epsilon}\kern
1.00006ptv\kern 1.00006ptx).$ (9)

Here,
$\epsilon=\hbar\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}m$,

$\sigma\ll L$ (10)

is the width of the wave packet, and $v>0$ is the phase velocity. Under the
free Schrödinger evolution, with Hamiltonian

$H=-\,\frac{\hbar^{2}}{2\kern 1.00006ptm}\,\frac{d^{2}}{dx^{2}},$ (11)

the time-dependent packet is

$\displaystyle\phi(x,t)$ $\displaystyle=e^{-\,it\kern 0.70004ptH/\hbar}\kern
1.00006pt\phi(x,0)$
$\displaystyle=\frac{1}{\sqrt{\sigma(t)\sqrt{\pi}}}\exp\kern-1.00006pt\left\\{-\,\frac{\sigma}{\sigma(t)}\left[\frac{x^{2}}{2\,\sigma^{2}}-\,\frac{iv}{\epsilon}\\!\left(x-\frac{vt}{2}\right)\right]\kern
1.00006pt\right\\}\\!,$ (12)

where

$\sigma(t)=\sigma\left(1+i\frac{\epsilon t}{\sigma^{2}}\right)\\!.$ (13)

The amplitude of this packet is

$|\phi(x,t)|=\frac{1}{\sqrt{|\sigma(t)|\sqrt{\pi}}}\exp(-\,\frac{1}{2}\kern
1.00006pt\frac{(x-vt)^{2}}{|\sigma(t)|^{2}}).$ (14)

From this we see that the center of the packet arrives at the detector at time

$\tau_{0}=\frac{L}{v}.$ (15)

Hence, the corresponding time-of-arrival distribution $\Pi_{0}(\tau)$ is
expected to be approximately peaked around $\tau_{0}$.

We also assume that the packet suffers negligible distortion during
$0<t<\tau_{0}$. This is guaranteed if

$\frac{\epsilon\tau_{0}}{\sigma^{2}}=\frac{\epsilon L}{\sigma^{2}v}\ll 1,$
(16)

which we shall refer to as the “no spreading condition”. In this case,

$|\phi(x,t)|\approx|\phi(x-vt,0)|$ (17)

for $\smash{0<t<\tau_{0}}$.

Consider now an initial superposition of $N$ Gaussian wave packets with the
same width and velocity, but centered at $\smash{x=-\,kL}$, $\smash{k=0,\kern
1.00006pt1,\kern 1.00006pt\dots,\kern 1.00006ptN-1}$,

$\psi(x,0)=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}\phi(x+kL,0),$ (18)

as shown in Fig. 1.

Figure 1: Train of Gaussian wave packets moving towards the detector.

It evolves into

$\psi(x,t)=e^{-\,itH/\hbar}\,\psi(x,0)=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}\phi(x+kL,t),$
(19)

where $\phi(\cdot,t)$ is given by (II). Assuming

$\smash{N\tau_{0}\ll\sigma^{2}\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}\epsilon},$ (20)

the wave packets will remain non-overlapping until time $N\tau_{0}$, the time
at which the $N$th wave packet of the Gaussian train strikes $x=L$. As a
consequence,

$\displaystyle|\psi(x,t)|^{2}$
$\displaystyle\approx\frac{1}{N}\sum_{k=0}^{N-1}|\phi(x+kL,t)|^{2}$
$\displaystyle\overset{\eqref{mainly}}{\approx}\frac{1}{N}\sum_{k=0}^{N-1}|\phi(x+kL-
vt,0)|^{2}$ (21)

for all $0<t<N\tau_{0}$.

In this event, one expects the arrival-time distribution to have $N$ peaks of
nearly identical shape and height centered at times $\smash{k\tau_{0}}$,
$k=1,\kern 1.00006pt2\kern 1.00006pt\dots,\kern 1.00006ptN$, i.e.,

$\Pi(\tau)\approx\frac{1}{N}\\!\sum_{k=1}^{N}\Pi_{0}(\tau-k\tau_{0}).$ (22)

That is, the Gaussian wave packets in the train arrive at the detector one by
one with time delays of $\tau_{0}$, so that there should be a peak in the
arrival-time density, which has the same shape for any Gaussian wave packet.

The quantum flux TOF distribution (1), i.e.,

$\Pi_{\text{QF}}(\tau)\approx\frac{v}{N\\!\sqrt{\pi}\kern
1.00006pt\sigma}\sum_{k=1}^{N}\exp[-\,\frac{v^{2}}{\sigma^{2}}\kern
1.00006pt\big{(}\tau-k\tau_{0}\big{)}^{2}],$ (23)

which happens to agree with the Bohmian distribution in this case (due to the
absence of backflow) is of the expected form (22). Note that the integral of
(23) over all $\tau>0$ is approximately unity, hence it predicts a zero non-
detection probability as per Eq. (6).

##  III Exponential distributions

The arrival-time distributions proposed in Włodarz (2002); Marchewka and
Schuss (1998); *PI2; *PI3; Jurman and Nikolić (2021) do not have the form (22)
for the Gaussian wave train. Instead, the probability density is exponentially
falling off. In particular, assuming only (10) and (20), we will show that the
intensity functions (Table 1) take, for the Gaussian wave train (18), the form

$\lambda(t)\approx\frac{1}{N}\sum_{k=1}^{N}\lambda_{0}(t-k\tau_{0}),$ (24)

where $\lambda_{0}(t)$ is the intensity function corresponding to $\phi(x,t)$
supported on $|t-\tau_{0}|\leq\Delta\tau$, $2\Delta\tau$ being the duration
over which $\phi$ sweeps over the detector $\approx
3\sigma\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}v$. This implies that the distribution $\Pi(\tau)$
decays exponentially over time owing to the exponential factor in (5). In
fact, for $\smash{k\tau_{0}<\tau<(k+1)\tau_{0}}$, we have

$\exp(-\\!\int_{0}^{\tau}\\!\\!\\!dt~{}\lambda(t))\approx\exp(-\kern
1.00006ptk\\!\int_{\tau_{\scaleto{0}{3.5pt}}-\Delta\tau}^{\tau_{\scaleto{0}{3.5pt}}+\Delta\tau}\\!\\!\\!\\!\\!dt~{}\lambda_{0}(t)).$
(25)

It follows that the expected behavior (22) cannot hold.

###  III.1 The Włodarz proposal

Using (II), we readily obtain the intensity function $\lambda_{\text{W}}$
given in Table 1:

$\lambda_{\text{W}}(t)\approx\frac{\lambda_{0}}{N}\sum_{k=1}^{N}|\phi(kL-
vt,0)|^{2},$ (26)

which implies the property (24).

###  III.2 The Marchewka & Schuss proposal

To calculate $\lambda_{\text{MS}}(t)$, cf. Table 1, we need $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}(x,t)$ which is the solution to Schrödinger’s equation on the half-line
$(-\infty,L]$ with initial condition $\psi(x,0)$ and Dirichlet boundary
condition at $\smash{x=L}$. It is given by

$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}(x,t)=\psi(x,t)-\psi(2\kern 1.00006ptL-x,t)$ (27)

for $x\leq L$. (This state is only approximately normalized to unity, since
(18) has support on $[L,\infty)$.) The (left) derivative at $L$ is
$\partial_{x}\kern 1.00006pt\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}(L,t)=2\kern 1.00006pt\partial_{x}\kern 1.00006pt\psi(L,t)$ and

$|\partial_{x}\kern 1.00006pt\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}(L,t)|^{2}\approx\frac{4}{N}\\!\sum_{k=1}^{N}|\partial_{x}\phi(kL,t)|^{2},$
(28)

where we used the fact that the $\phi(kL,t)$ and hence the
$\partial_{x}\phi(kL,t)$ for different $k$s are approximately non-overlapping
due to (20).

To evaluate the summands we use (II), obtaining

$\displaystyle|\partial_{x}\phi(x,t)|^{2}$
$\displaystyle=\left|\frac{\phi(x,t)}{\sigma(t)}\right|^{2}\left[\left(\frac{x}{\sigma}\right)^{\\!2}+\left(\frac{v\kern
1.00006pt\sigma}{\epsilon}\right)^{\\!2}\right]\\!,$ (29)

which is exact, but in view of (20),

$\displaystyle|\partial_{x}\phi(kL,t)|^{2}\approx\left(\frac{v}{\epsilon}\right)^{\\!2}\\!|\phi(kL,t)|^{2}$
(30)

for $k=1,\kern 1.00006pt2,\kern 1.00006pt\dots\kern 1.00006ptN$ and
$0<t<N\tau_{0}$. Then, using (17) and (28), we arrive at

$\lambda_{\text{MS}}(t)\approx\frac{4\kern
1.00006ptv^{2}\lambda^{\prime}}{N\pi\epsilon}\sum_{k=1}^{N}\big{|}\phi(kL-
vt,0)\big{|}^{2}.$ (31)

In this case, the intensity function approximately agrees with
$\lambda_{\text{W}}$ (up to a proportionality factor). Again the property (24)
obtains.

###  III.3 The Jurman & Nikolić proposal

To calculate $\lambda_{\text{JN}}(t)$, given in Table 1, we need
$\psi_{c}(x,t)$, defined by

$\psi_{c}(x,t)=e^{-\,i\delta tH/\hbar}e^{-\,i(t-\delta t)\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{H}/\hbar}\,\psi(x,0),$ (32)

where $H$ is the free Hamiltonian (11) and $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{H}$
refers to free motion on the half-line $(-\infty,L]$ with Dirichlet boundary
conditions at $L$. The initial wave function is assumed to be supported on the
half-line. Our initial wave function (18) is actually nonzero in the region
$[L,\infty)$ but, as before, we will ignore its tail beyond $x=L$. So, using
the notation introduced in (27), we can also write

$\psi_{c}(x,t)=e^{-\,i\delta tH/\hbar}\,\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}(x,t-\delta t).$ (33)

To evaluate (32) for the Gaussian train, consider for $0\leq k<N$, the
function $\phi_{c}$ defined by

$\displaystyle\phi_{c}(x+kL,t)$ $\displaystyle\coloneqq e^{-\,i\delta
tH/\hbar}e^{-\,i(t-\delta t)\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{H}/\hbar}\,\phi(x+kL,0)$ (34) $\displaystyle=e^{-\,i\delta
tH/\hbar}e^{-\,i(t-k\tau_{\scaleto{0}{3pt}}-\delta t)\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{H}/\hbar}$ $\displaystyle\kern
56.9055pt\times\,e^{-\,ik\tau_{\scaleto{0}{3pt}}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{H}/\hbar}\,\phi(x+kL,0)$ $\displaystyle=e^{-\,i\delta
tH/\hbar}e^{-\,i(t-k\tau_{\scaleto{0}{3pt}}-\delta t)\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{H}/\hbar}\,\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}(x+kL,k\tau_{0}).$ (35)

Since $\phi(x+kL,k\tau_{0})$ is centered at $x=0$ and has a width
$|\sigma(k\tau_{0})|\approx\sigma$ in view of the “no spreading” condition
(20) [cf Eq. (13)], we have

$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}(x+kL,k\tau_{0})\approx\phi(x+kL,k\tau_{0}).$

Using (17) and (20), the amplitude of this wave function satisfies

$\big{|}\phi(x+kL,k\tau_{0})\big{|}\approx|\phi(x,0)|.$ (36)

Its phase is

$\displaystyle\text{Arg}\kern 1.00006pt\big{[}\phi(x+kL,k\tau_{0})\big{]}$
$\displaystyle=\text{Arg}\kern 1.00006pt\big{[}\phi(x,0)\big{]}-\frac{\epsilon
k\tau_{0}}{2\,\sigma^{2}}$ $\displaystyle\kern
1.00006pt+\,\frac{k\tau_{0}}{2}\kern-1.00006pt\left[\frac{v^{2}}{\epsilon}+\,\epsilon\,\frac{(x\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}\sigma)^{2}}{|\sigma(k\tau_{0})|^{2}}\right]\\!.$
(37)

Equation (20), together with the condition $\smash{|x|\lesssim 3\kern
1.00006pt\sigma}$ valid within the bulk of the support of the wave function,
allow us to neglect both the second term and the second term in brackets,
hence

$\text{Arg}\kern
1.00006pt\big{[}\phi(x+kL,k\tau_{0})\big{]}\approx\text{Arg}\kern
1.00006pt\big{[}\phi(x,0)\big{]}+k\kern 1.00006pt\frac{vL}{2\kern
1.00006pt\epsilon}.$

It follows that

$\phi(x+kL,k\tau_{0})\approx\phi(x,0)\kern 1.00006pte^{ikvL/2\epsilon}$

and

$\displaystyle\phi_{c}(x+kL,t)$ $\displaystyle\approx e^{-\,i\delta
tH/\hbar}e^{-\,i(t-k\tau_{\scaleto{0}{3pt}}-\delta t)\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{H}/\hbar}\phi(x,0)\kern 1.00006pte^{ikvL/2\epsilon}$
$\displaystyle\overset{\eqref{defn}}{=}\kern
1.00006pte^{ikvL/2\epsilon}\phi_{c}(x,t-k\tau_{0}).$ (38)

Hence, by linearity

$\psi_{c}(x,t)\approx\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}e^{ikvL/2\epsilon}\phi_{c}(x,t-k\tau_{0}).$
(39)

Ignoring the tails of the Gaussians, we have that for $\delta t\ll\tau_{0}$,
at most only one of the wave packets $\phi_{c}(x,t-k\tau_{0})$ will have its
support in $[L,L+\Delta]$ at a given time. (Remember that $\phi_{c}$ is
obtained by free evolution with Dirichlet boundary conditions up until time
$t-\delta t$, and then free evolution for a time $\delta t$.) Therefore we can
ignore cross terms for the density in the interval $[L,L+\Delta]$, and write

$\displaystyle\lambda_{\text{JN}}(t)$ $\displaystyle\approx\frac{1}{\delta
t}\sum_{k=0}^{N-1}\int_{L}^{L+\Delta
L}\\!\\!\\!\\!\\!\\!\\!dx~{}|\phi_{c}(x,t-k\tau_{0})|^{2},$ (40)

so that property (24) obtains.

##  IV The semiclassical distribution

The semiclassical distribution, cf. Eqs. (3-4), is

$\Pi_{\text{SC}}(\tau)=\frac{L}{\epsilon\tau^{2}}\kern-1.00006pt\left|\frac{1}{\sqrt{2\kern
1.00006pt\pi}}\\!\int_{-\infty}^{\infty}\\!\\!\\!dx~{}\psi(x,0)\,e^{-\,iLx/\epsilon\tau}\right|^{2}\\!.$
(41)

Using the Fourier transform

$\displaystyle\frac{1}{\sqrt{2\kern
1.00006pt\pi}}\\!\int_{-\infty}^{\infty}\\!\\!\\!dx~{}\phi(x+x_{0},0)\,e^{-\,iv_{\scaleto{0}{3pt}}\kern
0.70004ptx/\epsilon}$
$\displaystyle\qquad=\sqrt{\frac{\sigma}{\pi^{1/2}}}\,\exp[\frac{i}{\epsilon}\kern
1.00006ptv_{0}\kern 1.00006ptx_{0}-\kern 1.00006pt\frac{\sigma^{2}}{2\kern
1.00006pt\epsilon^{2}}\left(v-v_{0}\right)^{\\!2}],$ (42)

we find (without approximations) that

$\displaystyle\Pi_{\text{SC}}(\tau)$ $\displaystyle=\frac{\sigma
L}{N\kern-1.00006pt\sqrt{\pi}\kern 1.00006pt\epsilon\kern
1.00006pt\tau^{2}}\,\frac{\sin^{2}(NL^{2}\\!\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}2\kern
1.00006pt\epsilon\tau)}{\sin^{2}(L^{2}\\!\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}2\kern 1.00006pt\epsilon\tau)}$ $\displaystyle\kern
56.9055pt\times\exp[-\,\frac{\sigma^{2}v^{2}}{\epsilon^{2}}\left(1-\tau_{0}/\tau\right)^{2}].$
(43)

The distribution is peaked around $\tau_{0}$, contrary to what is expected of
the Gaussian wave train. This should come as no surprise since
$\Pi_{\text{SC}}$ is fully determined by the momentum distribution of the
initial wave function, which in the present example is centered around
$\smash{p=mv}$. While the semiclassical distribution sometimes follows from
the quantum flux/Bohmian distribution, such is not the case here, as can be
seen in Fig. 2. This explains why the latter _does_ show the expected
behavior, unlike the former.

Figure 2: An illustration of $\Pi_{\text{SC}}(\tau)$ and
$\Pi_{\text{QF}}(\tau)$ (dot-dashed) for parameter values $N=5$, $\sigma=5$,
$L=10\kern 1.00006pt\sigma$, $v=1$, and $\epsilon=0.05$ (in arbitrary units).
Inset: Magnified view of $\Pi_{\text{SC}}(\tau)$.

##  V Discussion and outlook

Figure 3: An illustration of the exponential decay of the arrival-time
distributions for a Gaussian wave train satisfying the no spreading condition
(20), with parameters $N=10$, $\sigma=5$, $L=10\kern 1.00006pt\sigma$,
$\epsilon=0.01$, and $v=1$ (in arbitrary units). The red curve is
$\Pi_{\text{JN}}$ with $\delta t=0.5$ and $\Delta L=0.5\kern 1.00006pt\sigma$,
the green curve is $\Pi_{\text{W}}$ with $\lambda_{0}=2$, while the blue curve
denotes $\Pi_{\text{MS}}$ with $\lambda^{\prime}=0.01$. The black curve is
$\Pi_{\text{QF}}$, which does not display an exponential decay. The
exponential TOF curves were calculated numerically without any approximations
(see text for details).

The exponential distributions for the Gaussian train (18) are plotted in Fig.
3 along with the quantum flux/Bohmian distribution. All arrival-time
distributions were produced using the exact analytic expressions without
invoking approximations (10) and (20)333For the Jurman & Nikolić proposal, we
used  $\lambda_{\text{JN}}(t)\approx(\Delta
L\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}\delta t)\kern
1.00006pt\big{|}\psi_{c}(L,t)\big{|}^{2},$ applicable for a small $\Delta L$.
However, $\psi_{c}(x,t)$ [Eq. (33)] was evaluated exactly in terms of error
functions by integrating (27) against the known free-particle propagator for
time $\delta t$.. While the quantum flux distribution displays the expected
behavior (i.e., featuring identical and well-separated peaks centered at times
$\tau_{0}$, $2\kern 1.00006pt\tau_{0}$, …,$\kern 1.00006pt10\kern
1.00006pt\tau_{0}$), the exponential ones do not, thus substantiating our
analysis. The free parameters $\lambda_{0}$, $\lambda^{\prime}$, $\Delta L$,
and $\delta t$ were chosen for best visibility. However, this undesirable
behavior cannot be evaded by tuning these parameters: making them larger
causes a faster decay, while making them smaller moderates the decay at the
cost of increasing the non-detection probability (to the extent of a vanishing
arrival-time density in the case of an appreciable removal of the decay). In
fact, given any choice of these free parameters, the number $N$ of Gaussians
in the train and their velocity $v$ could be so chosen that the exponential
decay practically washes out the arrival-time peaks corresponding to the
trailing Gaussians.

The exponential proposals were aimed at deriving the TOF distribution by means
of a detector model. While different intensity functions $\lambda$ can be
considered, our results show that the failure is not so much attributable to
the particular choice of $\lambda$ but presumably the assumption of
_independence_ that underlies the Poisson process.

While it is, as a matter of principle, necessary to account for the effect of
the detector in _any_ experiment, the extent to which the physics of the
detector needs to be taken seriously for predicting arrival times is not self-
evident. In practice, scattering experiments such as the double-slit and the
Stern-Gerlach experiment are routinely analyzed with no reference whatsoever
to the detector. But since the detectors employed in these experiments are
typically no more specialized than the ones found in TOF experiments (e.g., a
scintillation screen employed in Kurtsiefer  _et al._ (1997); *wig;
*Pfaudetector), it is not a priori obvious why the physics of the detector is
any more relevant for predicting the statistics of arrival times than it is
for predicting the statistics of impact positions. Hence it should not come as
a surprise that the quantum flux distribution gives the expected result
despite ignoring the detector. It has even been shown that
$\Pi_{\text{QF}}(\tau)$ can arise from a careful consideration of a physical
detector, e.g., a laser curtain inducing fluorescence from an incoming atom
Damborenea  _et al._ (2002, 2003); Hannstein  _et al._ (2005). This suggests
that one should turn to realistic TOF experiments if one wants to take the
detector seriously.

Finally, the semiclassical distribution depicted in Fig. 2 also fails to
display the expected behavior since it is largely supported around $\tau_{0}$.
This distribution is often used in the experimental determination of the
momentum distribution. Using the measured arrival-time distribution
$\Pi_{\text{meas}}(\tau)$, the empirical momentum distribution is taken to be

$\frac{mL}{p^{2}}\kern
1.00006pt\Pi_{\text{meas}}\\!\left(\frac{mL}{p}\right)\kern-1.00006pt,$ (44)

corresponding to the quantum mechanical momentum distribution $\hbar^{-1}\kern
1.00006pt|{\tilde{\psi\mkern
3.0mu}\mkern-3.0mu}{}(p\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}\hbar)|^{2}$, thereby tacitly assuming the validity
of (3). However, our results indicate that such reconstructions are
questionable (see also (Vona, 2014, Ch. 4)).

## Acknowledgements

W.S. is supported by the Research Foundation Flanders (Fonds Wetenschappelijk
Onderzoek, FWO), Grant No. G066918N. It is a pleasure to thank J. M. Wilke for
valuable editorial input.

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