# Detlef Dürr, arrival-time distributions, and spin in Bohmian mechanics:
Personal recollections and state-of-the-art

Siddhant Das<EMAIL_ADDRESS>Mathematisches Institut,
Ludwig-Maximilians-Universität München, Theresienstr. 39, D-80333 München,
Germany

(April. 30, 2022)

###### Abstract

I recount here my association with Prof. Detlef Dürr leading to our memorable
research collaboration on arrival-time distributions in quantum mechanics. He
influenced my life, both personally and professionally, as few others have or
ever will. Detlef is my role model for what a brilliant, discerning scientist,
academic, and mentor can and should be.

The “arrival-time problem” in quantum mechanics is examined selectively, with
an emphasis on the arrival-time distributions of Bohmian particles. In what
follows, the “exotic” Bohmian arrival-time distributions of spin-polarized
electrons accelerating down a cylindrical waveguide [S. Das and D. Dürr, Sci.
Rep. 9: 2242 (2019)], and some variations thereof are discussed. I shall not
go into the mathematical treatment more than is necessary to spell out the key
results. The intention is to document the circumstances and motivations
underlying the ideas.

> _“It is our choices, Harry, that show what we truly are, far more than our
> abilities.”_ — Albus Dumbledore (J. K. Rowling Rowling (2015))

It was my good fortune to know Prof. Dr. Detlef Dürr as a mentor, research
collaborator, and close confidant from October 2016 until January 2021. We had
regular interactions following his enriching and uniquely crafted course
“Introduction to Bohmian Mechanics,” offered during the Summer Semester of
2016. This course attracted a large number of enthusiastic participants of all
ages. As it happened, Detlef was officially teaching Bohmian mechanics for the
very first time at LMU Munich.111And also the very last time, owing to his
retirement in November of that year. His lectures were generously seasoned
with witty comments, historical anecdotes, and philosophical insight, while a
palpable sense of seriousness was ever-present. By the end of the semester,
his passion for Bohmian mechanics rubbed off on some of us. What more could
one ask for?

Bohmian mechanics (de Broglie-Bohm or pilot-wave theory, BM in the sequel) is
an understanding of quantum physics featuring _both_ waves and particles,
first formulated by Louis de Broglie in the 1920s. It was rediscovered,
refined, and elucidated by David Bohm in the 50s, popularised and defended by
John S. Bell in the 80s, and augmented by modern-day theorists. In BM,
particles move in a highly non-Newtonian way, obeying the _guiding equation_.
This equation expresses their velocities in terms of the wave function, whose
primary role in the theory is to guide particles along well-defined
trajectories. Its statistical significance is a derived consequence, and it
evolves by Schrödinger’s equation at _all_ times.

This deterministic, matter-in-motion worldview of BM is at such odds with
those suggested by orthodox/standard/textbook quantum mechanics (QM) that
newcomers are typically and understandably perplexed. Detlef thus took great
care in the BM course to set the stage. He began by reappraising material
(handpicked from the “red book” Dürr and Teufel (2009)) on dynamical systems,
analytical mechanics, Boltzmann’s statistical mechanics, and Brownian motion,
emphasizing the conceptual precision of the “classical” theories over their
known limitations in the quantum domain. The lectures on “chance in physics”
(Dürr and Teufel, 2009, Ch. 4) were genuinely excellent and insightful.

BM proper was only discussed in the second half of the course. Except for the
double-slit experiment, Detlef did not focus much attention on concrete
examples/applications of the theory.222Nor did he address (or speculate on)
the well-documented historical shortcomings responsible for physicists’
disengagement from BM; see, e.g., Becker (2018); Bacciagaluppi and Valentini
(2009); Freire Junior (2015). He preferred to not lose sight of the forest
(derivation of Born’s rule, quantum nonlocality, measurement formalism) for
the trees. If asked “[i]s it in fact useful to compute Bohmian trajectories in
various quantum mechanical situations we deem of interest?”, he candidly
asserted, “[i]n general, it is not!” (Dürr and Teufel, 2009, p. 155). Yet,
Bohmian trajectories for benchmark QM problems, e.g., diffraction Dubertrand
_et al._ (2018), two-slit interference (with Tastevin and Laloë (2018) and
without Philippidis  _et al._ (1979) monitoring), barrier tunneling Norsen
(2013); Dewdney and Hiley (1982), EPR correlations (Bohm and Hiley, 1993, Sec.
10.6), etc., “remain among the most striking illustrations so far of the
insight provided by this theory into quantum phenomena” Holland and
Philippidis (2003). In particular, a principled treatment of the classical
limit Allori  _et al._ (2002), and a dynamical underpinning of the Pauli
repulsion (or degeneracy pressure) (Holland, 1995, p. 284), which these
trajectories facilitate, impressed me early on.

It soon became evident, despite Detlef’s tongue-in-cheek admonishment (“every
lecture in quantum physics is an act of brainwashing”), that BM grounds the
formalism and empirical predictions of standard QM coherently and
compellingly; see Bohm and Hiley (1993); Dürr and Teufel (2009); Holland
(1995); Goldstein (2021); Bricmont (2016); Cushing  _et al._ (2013); Oriols
and Mompart (2012). Fortunately, in so doing, “it neither needs nor is
embarrassed by an observer” (Bell, 2004, Ch. 15). Understanding how BM
explains and computes the results of standard quantum experiments without
deploying ill-defined measurement axioms Bell (1990), or a laundry list of
quantum observables, was indeed a revelation.

One couldn’t shake the feeling that “the significance of the [quantum
observables] has been exaggerated, in the sense that elements entering as
useful _mathematical techniques_ have been raised to the level of fundamental
concepts in the _physical theory_ ” (Bohm, 1953, p. 278).333D. Bohm’s close
friend Richard P. Feynman couldn’t have agreed more vehemently: “If anyone
tells me that ‘to every observable there corresponds a Hermitian operator for
which the eigenvalues correspond to observed values,’ _I will defeat him! I
will cut his feet off!_ ” Hestenes (2003). Besides, “[a]ny physicist who is
ready to quantize everything under his pen” (Dürr and Teufel, 2009, p. vi)
must take the _canonical quantization_ procedure (for engineering quantum
observables) with a grain of salt, as it is known (Groenewold-Van Hove theorem
Carosso (2022); Ali and Engliš (2005)) that a procedure associating (i) every
classical phase-space function $f(x,p)$ to a quantum operator $F$, in
particular, (ii) $x$ and $p$ to the usual position and momentum operators of
QM, and simultaneously (iii) classical Poisson brackets to quantum
commutators:
$\\{f,g\\}\mapsto(1\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}i\hbar)\kern 1.00006pt[F,G]$, as Dirac envisioned,
_cannot_ exist.444Ironically, Groenewold discovered his no-go result while
attempting to demonstrate the impossibility of certain “hidden variable”
accounts of quantum mechanics Carosso (2022), BM being the prime example of a
“hidden variable” theory (see also Bell (1966)). Indeed, in real-world
applications, when the canonical quantization procedure leads to an impasse
(see below), practitioners are forced to guess the relevant quantum
observables from ad hoc symmetry principles, experience, or good taste.

That being said, in all fairness, while operators as quantum observables play
_no_ role in the formulation of BM, their effectiveness as empirical
shorthands is best appreciated through a Bohmian lens. The _emergence_ of the
“momentum operator” in a first principles Bohmian analysis of asymptotic
momentum measurements (Dürr and Teufel, 2009, Sec. 15.1.2.) is a classic
example (see, however, Dürr  _et al._ (2004) for a general argument applicable
also to generalized quantum observables as positive-operator-valued measures,
or POVMs). More to the point, the concept of a particle trajectory is
universally invoked in actual experiments when _reconstructing_ particle
momentum from directly observable quantities, such as arrival (or flight)
times. While the final momentum distribution agrees with the QM (also BM)
prediction,555Although “[i]n quantum mechanics there is no such concept as the
path of a particle” (Landau and Lifshitz, 1977, p. 2). the adequacy/limits of
this time-honoured _time-of-flight momentum spectroscopy_ technique can only
be analyzed within the context of a trajectory-containing quantum theory such
as BM.

Against this backdrop, and observing that the BM literature preemptively
discusses experiments inspired by standard QM, I became interested in “new”
experimental predictions of BM early on666As was, notably, Jean-Pierre Vigier
Cufaro-Petroni and Vigier (1992); Dewdney  _et al._ (1985, 1984), a prominent
collaborator of L. de Broglie and D. Bohm.—predictions that could help “draw
inferences from experiment regarding the particle track[s]” (Holland, 1995,
Sec. 5.5) within established limits Dürr  _et al._ (1992). But since there is
a compelling argument Dürr  _et al._ (1992) that BM is “experimentally
equivalent” to regular QM, at least “insofar as the latter is unambiguous”
(Bell, 2004, Ch. 14), the prospects for definitive experiments were grim.
Moreover, given that QM has been tested exhaustively, the existence of
_feasible_ experiments (i.e., not of Wigner’s-friend-type) for which its
recipes would prove to be wanting and, concomitantly, the conceptual resources
of BM could be brought to bear, was unprecedented. It must be mentioned that
David Bohm speculated early on that the theory may “lead to completely
different kinds of predictions” at distances “of the order of $10^{-13}\kern
1.00006pt$cm or less” Bohm (1952), though no concrete experimental ideas were
outlined (see also (Bohm, 1962, Sec. 13)). In addition, cosmological
fingerprints of BM-type field theories suggested in Valentini (2010) remain
somewhat controversial.

Figure 1: Detlef and his disciples at the Mensa Arcisstraße (circa July 2019).
From left, Arun Ravishankar, Detlef Dürr, Nikolai Leopold, Siddhant Das, and
Manuela Feistl.

A silver lining appeared in the final week of the BM course when Nikolai
Leopold (see Fig. 1) briefly introduced the “arrival-time problem” in QM,
which I was completely ignorant of. I feel certain that my worldline would
have been very different had I missed Nikolai’s talk. He was a Ph.D. student
in Detlef’s “Workgroup Mathematical Foundations of Physics” at the time,
working on “Effective Evolution Equations from Quantum Mechanics” Leopold
(2017). Nikolai was exploring as a side project the parameter space of an
arrival-time experiment involving photons. This experiment, requiring a
coherent superposition of two Gaussian wave packets, was previously considered
by Nicola Vona in his Ph.D. thesis Vona (2014).777Recently, Rafsanjani _et
al._ Ayatollah Rafsanjani  _et al._ (2023) have suggested a very practical
realization of this type of experiment using a double-slit interferometer. At
the same time, an experimental implementation of this work was being discussed
with the group of Prof. Harald Weinfurter, whose expertise in quantum optics
meant the calculations had to be specifically tailored for photons.

Arrival-time or time-of-flight (ToF) experiments are scattering experiments
concerned with “ _when_ the detector clicks” rather than “ _where_ the
detector clicks.” The latter is a mainstay of standard QM (and quantum field
theory) with specialized theoretical tools at its service, e.g., the
$S$-matrix formalism. The former, on the other hand, has been the subject of
much controversy since nearly the beginning of QM.888In (Pauli, 1979, Nov. 17,
1925), Wolfgang Pauli wrote in a letter to Niels Bohr: “Perhaps the following
points in the right direction of possible further progress. In the new theory
[matrix mechanics], not all physically observable quantities really occur,
namely, the time instants of the transition processes are still missing, which
are certainly observable in principle (e.g., the instants of ejections of
electrons in the photoelectric effect).” In a typical theoretical analysis of
a ToF experiment, one considers a particle with a well-localized wave function
$\psi_{0}(\mathbf{r})$ at time zero,999For example, the moment of releasing a
particle from a trap after state preparation. propagating either freely or in
specified external potentials. Critical to any such discussion is the
probability $\smash{\Pi(\tau)\,d\tau}$ that the particle’s time-of-arrival on
a _specified surface_ $\mathcal{D}$ (representing the surface of a detector,
e.g., a scintillation screen) is between times $\smash{\tau}$ and
$\smash{\tau+d\tau}$.101010Not to be confused with the probability of locating
the particle in a volume element $d^{3}r$ surrounding $\mathbf{r}$ at a
_specified time_ $t$, viz., $|\psi_{t}(\mathbf{r})|^{2}\kern 1.00006ptd^{3}r$
(Born’s rule!), where $\psi_{t}$ is the particle’s wave function at time $t$.
Today, the computation of the probability density $\Pi(\tau)$, which is
empirically well-accessible, is one of the last areas where physicists
disagree about what QM should predict Muga and Leavens (2000); Muga  _et al._
(1998a); Das and Dürr . This, in a nutshell, is the “arrival-time problem”.

ToF experiments had long been on Detlef’s radar, considering how
straightforward it was to describe them using the inventory of BM. (In
particular, the particle trajectories defined by the guiding equation.) That
BM is naturally suited for handling arrival- and tunneling-time problems has
long been recognized Daumer  _et al._ (1997); Leavens (1998a, b), (Holland,
1995, Sec. 5.5). But Detlef’s interest in this subject went back to his work
on quantum scattering formalism (Dürr and Teufel, 2009, Ch. 16), Dürr  _et
al._ (2000); Vona and Dürr (2015); Dürr and Teufel (2004); Dürr  _et al._
(2010), especially the flux-across-surfaces theorems Daumer  _et al._ (1997,
1996); Dürr  _et al._ (2005) and the time-dependent analysis of Gamow decay
Dürr  _et al._ (2011a); *GamowEr; Grummt (2014); Grummt and Vona (2021). A
grounding in scattering theory enabled him to appreciate the significance of
the probability current (or quantum flux) density in ToF experiments early on.
Before I met Detlef, relevant research on arrival times was conducted in
collaboration with his former students Helmut Satzger Satzger (2002), Nicola
Vona Vona (2014), and Günter Hinrichs Vona  _et al._ (2013). So, in
retrospect, it is somewhat surprising that he did not bring up arrival-time
experiments earlier in the course when pressed about scenarios where BM would
be particularly useful.

The issue of arrival times got under my skin. For an outstanding problem in QM
and an area of considerable theoretical activity (see below), the conspicuous
lack of experimental contact or guidance seemed especially striking. That
situation seems to have changed very little even today (see, however, Das  _et
al._ (2022)).111111In contrast, the “tunneling-time problem,” which asks “how
long does it take for a particle to tunnel?” has received much experimental
attention in recent years Yu  _et al._ (2022); Kheifets (2020); Ramos  _et
al._ (2020); Spierings and Steinberg (2021); Field (2022). One possible
explanation is that the available ToF predictions have not previously been
worked out in sufficient detail to be compared with actual experiments.

ToF measurements, on the other hand, are central to experimental methods
determining energies and momenta in both single- and few-particle scattering
experiments featuring wave packet coherence Wolf and Helm (2000), including,
e.g., the temperature of trapped ions Stopp  _et al._ (2021), band structure
Tusche  _et al._ (2016); Medjanik  _et al._ (2017), Wigner’s function
Kurtsiefer  _et al._ (1997), chemical reaction dynamics (as in the Rydberg
tagging ToF technique Yang and Zhang (2013)), etc. However, Newtonian or
geometric optics Arndt  _et al._ (1996) _Ansätze_ frequently suffice for
interpreting the ToF measurements in these instances.121212This is hardly
surprising from a Bohmian standpoint as the appearance of Newtonian behaviour
under scattering conditions is a common characteristic of BM Römer  _et al._
(2005). As a result, in many cases, the arrival times of Bohmian particles may
be approximated via Newtonian assumptions. The “unreasonable effectiveness” of
semiclassical heuristics in laboratory ToF experiments has, however, also
played a key role in _masking_ the underlying concerns posed by the “arrival-
time problem”.

I had a strong urge to improve matters on this front. As it happened, given
Detlef’s approaching retirement, his limited time shared between existing
Ph.D. students, and some health-care needs of his family, he had somewhat
reluctantly decided not to supervise me for a master’s thesis. In the
meantime, I continued corresponding with Nikolai about his work and tried
suggesting ways to relax various idealizations in the theoretical treatments.
I soon realized that the association with Nikolai was a bit like “killing two
birds with one stone.” First, I saw Detlef more often, long after the BM
course had ended, and second, I could focus on the arrival-time problem that
had so enticed me. As time went on, Detlef’s attention and interest in the
problem seemed to be reinvigorated. I soon became an integral part of the
discussions and even started developing the ideas in ways I felt might be more
promising—something both Detlef and Nikolai welcomed very graciously.

During Detlef’s retirement celebrations in November, Nikolai introduced me to
Prof. Roderich Tumulka (or Rodi, as he is fondly called) over lunch, who had
already independently worked on the quantum “arrival-time problem” Tumulka
(2022a, b, c). This meeting turned out to be a critical turning point. After
briefly listening to what I was pursuing, Rodi suggested that if I was going
to make the effort to analyze realistic ToF experiments, it should be
refocused on massive particles rather than photons (Bohmian theories for
photons were somewhat contentious131313For this reason I believe the impact of
the weakly measured average photon trajectories Kocsis  _et al._ (2011) was
slightly diminished, even though “trajectories measured in this fashion
reproduce those predicted in the Bohm-de Broglie interpretation of quantum
mechanics.” A similar experiment with Argon atoms was ongoing a few years ago
Morley  _et al._ (2016).). This proved to be invaluable advice, for which I
remain grateful to him. That evening, I stopped working on photon arrival
times and started hunting for feasible experiments with massive particles.

Before continuing my personal story further, let me make a few observations
about certain theoretical efforts surrounding the “arrival-time problem”.
Various ToF distributions suggested in the literature can be divided into two
broad categories. We limit our discussion here to the category of ideal (or
intrinsic) arrival-time distributions that are _apparatus-independent_
theoretical predictions given by some functional of the initial wave function
$\psi_{0}$ and the geometrical surface of the detector $\mathcal{D}$
(typically a single point on a line in one-dimensional discussions). The
second category of non-ideal (or measurement-inspired) ToF distributions
typically features simplified detector models characterized by new
phenomenological parameters. Various suggestions have been put forward, e.g.,
simple absorbing boundary conditions Werner (1987); Tumulka (2022a), 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,
2001), a variety of quantum clocks (Muga  _et al._ , 2008, Ch. 8), and even a
timeless formulation of quantum measurement Maccone and Sacha (2020); Gambini
and Pullin (2022).

In the category of ideal ToF distributions—insofar as such a description may
be warranted—those based on arrival-time _observables_ are widely studied in
the literature. The observables in question are typically motivated via
classical arrival-time expressions, e.g., the well-known Aharonov-Bohm
operator Aharonov and Bohm (1961); Paul (1962)

$T_{\text{AB}}=\frac{m}{2}\kern
1.00006pt\Big{[}(L-X)P^{-1}+P^{-1}(L-X)\Big{]},$ (1)

obtained by a symmetric quantization of the classical ToF expression
$\smash{\tau_{\text{cl}}(x,p)=m\kern
1.00006pt(L-x)\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}p}$ of a freely moving particle that at
$\smash{t=0}$ had position $x$ and momentum $p$, and arrived after time
$\tau_{\text{cl}}$ at a distant point $L$ on a line.141414Though it is rarely
recognized that the correct classical ToF formula valid for an arbitrary
initial point $\smash{(x,p)}$ in phase space is
$\smash{\tau_{\text{cl}}(x,p)=m\kern
1.00006pt(L-x)\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}p}$ for $\smash{\text{sgn}\kern
1.00006ptp=\text{sgn}\kern 1.00006pt(L-x)}$ and $\infty$ otherwise; the
quantization of which seems nothing short of impossible. In (1), $X$ and $P$
are the standard position and momentum operators of QM, respectively.

A preliminary observation that justified interest in this kind of operator was
that $T_{\text{AB}}$ is (formally) canonically conjugate to the free-motion
Hamiltonian, raising expectations for the long-sought “time-energy uncertainty
relation”. However, $T_{\text{AB}}$ is not a self-adjoint operator (Pauli,
1933, p. 60), hence fails to provide a quantum observable à la Dirac and von
Neumann. As it happens, this is not a showstopper because the (doubly
degenerate) eigenstates of this operator, denoted $\ket{\tau,\pm}$, can be
used to define a generalized observable or POVM Egusquiza and Muga (1999);
Muga  _et al._ (1998b), and in turn the arrival-time distribution

$\Pi_{\text{AB}}(\tau)=|\\!\innerproduct{\tau,+}{\psi_{0}}\\!|^{2}+|\\!\innerproduct{\tau,-}{\psi_{0}}\\!|^{2},\qquad\innerproduct{\tau,\pm}{\psi_{0}}=\int_{0}^{\infty}\\!\\!\\!dp\,\sqrt{\frac{p}{2\kern
1.00006pt\pi\hbar\,m}}\,{\tilde{\psi\mkern
3.0mu}\mkern-3.0mu}{}_{\tau}(\pm\kern 1.00006ptp)\,e^{\pm\kern
0.70004ptipL/\hbar},$ (2)

where $\smash{{\tilde{\psi\mkern
3.0mu}\mkern-3.0mu}{}_{t}(p)=\innerproduct{p}{\psi_{t}}}$ is the momentum-
space representation of the freely-evolved wave function $\psi_{t}(x)$.
Several authors arrived at this ToF distribution (see Das and Nöth (2021) for
a critical review), notably J. Kijowski Kijowski (1974), whose _axiomatic_
derivation also applies to a three-dimensional setting for an infinite, planar
$\mathcal{D}$.

The above method of generating arrival-time operators from classical
expressions seldom works for scenarios other than free, one-dimensional
motion, owing to the fact that analogues of $\tau_{\text{cl}}(x,p)$ are rarely
available for motion in external potentials. This has resulted in a rather
long, circuitous, and near endless proliferation of (inequivalent) arrival-
time observables Hegerfeldt and Muga (2010); Hegerfeldt  _et al._ (2010); León
_et al._ (2000); Baute  _et al._ (2001, 2000); Galapon (2004); Galapon and P.
Magadan (2018); Gotō  _et al._ (1981a, b); Dias and Parisio (2017); Brunetti
and Fredenhagen (2002); Sombillo and Sombillo (2023); Przanowski  _et al._
(2013), not to mention the many technical and interpretative issues
accompanying them151515For example, the suggestion of assigning probabilities
to “negative arrival times” ($T_{\text{AB}}$ has negative eigenvalues), which
causes the predicted mean ToF,
$\matrixelement{\psi_{0}}{T_{\text{AB}}}{\psi_{0}}$, to vanish for _any_ real-
valued $\psi_{0}$ regardless of its form, or of the distance from the detector
Das and Nöth (2021). discussed at length in Das and Nöth (2021); Mielnik and
Torres-Vega (2005). To this end, a gauge-invariant generalization of
$T_{\text{AB}}$ applicable in the presence of external magnetic fields is yet
to be found, see (Das and Nöth, 2021, Sec. 3). In light of these issues the
value of a theory such as BM, whose framework offers a starting point
completely different from relying on guessing a self-adjoint operator or POVM,
becomes evident.

We turn now to the quantum flux distribution

$\Pi_{\text{QF}}(\tau)=\int_{\mathcal{D}}\mathbf{j}(\mathbf{r},\tau)\cdot
d\mathbf{s},$ (3)

where $\mathbf{j}(\mathbf{r},t)$ is the quantum flux (or probability current)
density associated with Schrödinger’s equation, and $\smash{d\mathbf{s}}$ the
surface measure at $\smash{\mathbf{r}\in\mathcal{D}}$. As indicated earlier,
this distribution would be a natural guess for an arrival-time distribution
from scattering considerations (see also (Das and Nöth, 2021, p. 6)), and
prima facie has a very wide domain of applicability including three-
dimensional settings, generic detection surfaces $\mathcal{D}$ and, most
importantly, ToF experiments featuring external electromagnetic fields.
However, it ceases to be a meaningful ToF distribution for some wave
functions, particularly those for which (3) is negative.161616For this reason,
the quantum flux distribution does not follow from a POVM Vona  _et al._
(2013). This problem is remedied by the Bohmian arrival-time distribution, to
which we turn next.

Numerous authors have discussed the arrival-time distribution of Bohmian
particles Daumer  _et al._ (1997); Grübl and Rheinberger (2002), (Holland,
1995, Sec. 5.5), most notably C. Richard Leavens Leavens (1998a, b), (Muga
_et al._ , 2008, Ch. 5). To specify this ToF distribution, consider a particle
with position $\mathbf{R}$, obeying the de Broglie-Bohm (guiding) equation

$\dot{\mathbf{R}}(t)=\mathbf{v}_{\text{dBB}}^{\psi_{t}}\big{(}\mathbf{R}(t),t\big{)},$
(4)

where $\dot{\mathbf{R}}$ denotes the particle’s velocity and $\psi_{t}$ its
wave function, a solution of Schrödinger’s equation:

$i\hbar\,\frac{\partial\psi_{t}}{\partial t}=H\psi_{t}.$ (5)

The precise form of the velocity field
$\mathbf{v}_{\text{dBB}}^{\psi_{t}}(\mathbf{r},t)$ and Hamiltonian $H$ depends
on particle spin, external potentials, and on whether the motion is
relativistic or non-relativistic Bohm and Hiley (1993); Dürr and Teufel
(2009); Holland (1995); Goldstein (2021); Bricmont (2016); Cushing  _et al._
(2013); Oriols and Mompart (2012). As soon as these are specified, particle
trajectories may be calculated by solving the dynamical equations subject to
suitable initial conditions $\mathbf{R}_{0}$ and $\psi_{0}$.

With these theoretical resources and only these, the _first_ arrival time (or
hitting time) of a Bohmian trajectory $\mathbf{R}(t)$, starting at some
$\smash{\mathbf{R}_{0}\in{\text{supp}}(\psi_{0})}$171717The support of
$\psi_{0}$, supp($\psi_{0}$), is the region for which $\psi_{0}$ is nonzero.
at time zero, and arriving at $\mathcal{D}$ at time $\tau$, can be defined as
follows:

$\tau(\mathbf{R}_{0})=\text{inf}\kern 1.00006pt\big{\\{}\,t\geq 0\kern
1.00006pt:\kern 1.00006pt\mathbf{R}(t)\in\mathcal{D}\,\text{ and
}\,\mathbf{R}(0)=\mathbf{R}_{0}\in\text{supp}(\psi_{0})\kern
1.00006pt\big{\\}}$ (6)

—a quantity that has no counterpart in standard QM.181818Or, for that matter
in either spontaneous collapse (wave-function-reduction) or many-world
theories. In standard QM, the detection event is tied to the observation-
induced collapse of the wave function, whose time of occurrence (relevant to
the measured ToF) is ambiguous. N.B: The infimum in (6) ensures that the ToF
of Bohmian particles not intercepting $\mathcal{D}$, e.g., those back-
scattered at an intervening potential barrier giving rise to non-detection
events, is infinity, since $\text{inf}\kern 1.00006pt\emptyset:=\infty$.

Now, given some statistical hypothesis about the initial conditions
$\mathbf{R}_{0}$ and $\psi_{0}$, the Bohmian arrival-time distribution
$\Pi_{\text{BM}}(\tau)$ can be determined using (6). In a sequence of
identically prepared ToF experiments with a fixed initial wave function
$\psi_{0}$, the $\mathbf{R}_{0}$s are typically
$|\psi_{0}|^{2}$-distributed,191919In such an ensemble, the Bohmian particle
positions remain $|\psi_{t}|^{2}$-distributed at _any_ later time $t$ by
virtue of the guiding equation. This property, dubbed _equivariance_ ,
expresses the compatibility between the evolution of the wave function given
by Eq. (5) and the evolution of the particle’s position given by Eq. (4).
therefore, the probability distribution of (6) can be written as

$\Pi_{\text{BM}}(\tau)=\int_{\text{supp}(\psi_{0})}\\!\\!d^{3}\kern-1.00006ptR_{0}~{}\,\delta\big{(}\tau-\tau(\mathbf{R}_{0})\big{)}\kern
1.00006pt|\psi_{0}(\mathbf{R}_{0})|^{2},$ (7)

where $\delta(\cdot)$ is Dirac’s delta function. For completeness, a _non-
detection probability_

$P_{\text{BM}}(\infty)=\int_{\tau^{-1}(\infty)}\\!\\!\\!d^{3}\kern-1.00006ptR_{0}~{}|\psi_{0}(\mathbf{R}_{0})|^{2}$
(8)

is defined, which is the $|\psi_{0}|^{2}$-measure of precisely those initial
conditions for which the Bohmian particle does _not_ intercept
$\smash{\mathcal{D}}$ at any finite time. It follows that, together with
definition (8), the integral of (7) for all $\smash{\tau\geq 0}$ evaluates to
one, as it must for any well-behaved probability distribution. (Equations
(7-8) can be easily adapted to instances where $\psi_{0}$ is chosen at random
from a statistical mixture encoded in a density matrix.)

The distribution $\Pi_{\text{BM}}$ is well-defined for generic external
electromagnetic (even time-dependent) potentials as well as generic (finite,
non-finite, even multipiece) detection surfaces $\mathcal{D}$. It also
naturally generalizes to multi-particle arrival-time problems unlike
$\Pi_{\text{QF}}$, see Kazemi and Hosseinzadeh (2023); Dürr  _et al._ (2010);
Dürr and Teufel (2004), as well as to a joint probability distribution of
impact positions _and_ arrival times Das  _et al._ (2022). These attributes
explain why the Bohmian trajectories are widely used for addressing arrival-
and tunneling-time problems in the literature; see Das  _et al._ (2022);
Kazemi and Hosseinzadeh (2023); Xie  _et al._ (2022); Leavens (1996, 1993);
Douguet and Bartschat (2018); Oriols  _et al._ (1996); Mousavi and Golshani
(2008a); Mousavi (2015); Ruggenthaler  _et al._ (2005); Pan and Home (2012);
Home and Majumdar (1999); Majumdar and Home (2002); Mousavi and Golshani
(2008b); Demir (2022); Field (2022); Nogami  _et al._ (2000) for many
applications over the years.

A word of caution: the Bohmian first-arrival times discussed here, which do
not take into account the physical interaction with the detectors
(characteristic of other ideal arrival-time distributions discussed above),
concern first and foremost the arrival times of Bohmian _trajectories_ ,
rather than the times registered by macroscopic measurement devices. Whether
the former are good approximations to the latter demands further study. In
most familiar settings the agreement is highly persuasive; see, e.g., Dürr
_et al._ (2000); Das  _et al._ (2022). However, It is conceivable that the
differences between the two may be significant in some cases. In such
circumstances, a deeper Bohmian analysis that goes well beyond the pragmatic
assumptions underpinning $\Pi_{\text{BM}}$ would be required.

When I began working with Detlef and Nikolai in 2017, I was unfamiliar with
much of the literature on ToF distributions, but $\Pi_{\text{BM}}$ “smelled
right” and was an easy starting point for studying ToF experiments utilizing
BM. Yet, from a practical standpoint, it is rarely directly
computable202020However, in one-dimensional settings at least, an explicit
formula (Kreidl  _et al._ , 2003, Eq. (12)) for $\Pi_{\text{BM}}$ has been
directly computed. except, of course, when it reduces _exactly_ to the quantum
flux distribution $\Pi_{\text{QF}}(\tau)$ under the current positivity
condition (CPC) (Das and Nöth, 2021, p. 6):

$\forall\kern 1.00006ptt>0\quad\text{and}\quad\forall\kern
1.00006pt\mathbf{r}\,\in\,\mathcal{D},\quad\mathbf{j}(\mathbf{r},t)\cdot
d\mathbf{s}\kern 1.00006pt\geq\kern 1.00006pt0.$ (9)

Note that whenever (9) holds, $\Pi_{\text{QF}}(\tau)$ is guaranteed to be a
meaningful probability distribution. If the CPC does _not_ hold, a closed-form
expression for $\Pi_{\text{BM}}$ involving only the wave function
$\smash{\psi_{t}}$ and/or its partial derivatives cannot, in general, be
given; one must therefore evaluate it numerically. However, it turns out that
(9) is practically always met in the far-field or scattering regime and is
central to the flux-across-surfaces theorems mentioned earlier Daumer  _et
al._ (1997, 1996); Dürr  _et al._ (2005).

Since $\Pi_{\text{QF}}$ _is_ the Bohmian arrival-time distribution in the
absence of backflow, Detlef sometimes preferred to express $\Pi_{\text{BM}}$
itself as a flux integral over $\mathcal{D}$, à la Eq. (3), introducing a so-
called _truncated flux_ density $\tilde{\mathbf{j}}$ (Daumer  _et al._ , 1997,
Eq. (10)), implicitly defined in terms of the first arrivals of Bohmian
trajectories at $\mathcal{D}$. Despite being mathematically identical to (7),
the truncated-flux formulation of $\Pi_{\text{BM}}$ is a bit too formal212121I
do not mean to imply here that Detlef was a formalist. For the most part he
“stayed well clear of purely academic questions like, for instance, the domain
of self-adjointness of observable operators”, firmly believing that “whoever
sees in those a key to understanding quantum mechanics is certainly wrong”
(Dürr and Lazarovici, 2020, p. 217). for most physics students, and is not
especially suited for numerical calculations; I therefore typically avoid
using it.

This “cornucopia of existing ToF predictions,” of which we have introduced but
a few, was seen by Detlef as a clear sign that the “chapter on time
measurements in quantum mechanics” is still substantially unwritten. “Quantum
mechanics, arguably the most thoroughly tested theory of nature, surely
deserves better,” he once declared emphatically. Evidently, in order to draft
this unwritten chapter, we must _do_ ToF experiments using present-day
technology and not just theorize about them. Detlef, of course, was well aware
that discerning competing ToF predictions in actual experiments presents
several technical and practical challenges.

Foremost among them was one stemming from the early observation that “several
theoretical treatments converge to essentially the same arrival-time
distribution at asymptotic distances and times from the particle source and
intervening scatterers” (Muga and Leavens, 2000, p. 358). So, in practical
experiments probing the far-field regime, the Bohmian arrival-time predictions
for massive particles would be extremely difficult to distinguish from
competing predictions, particularly those based on semiclassical (or, better,
mixed quantum-classical) considerations, such as Das and Struyve (2021)

$\Pi_{\text{SC}}(\tau)=\left(\frac{m}{\tau}\right)^{3}\\!\int_{\mathcal{D}}\,\left|{\tilde{\psi\mkern
3.0mu}\mkern-3.0mu}{}_{0}\\!\left(\frac{m\kern
1.00006pt\mathbf{r}^{\prime}}{\tau}\right)\right|^{2}\frac{\mathbf{r}^{\prime}\cdot
d\mathbf{s}}{\tau},$ (10)

implicit in the ToF momentum measurements mentioned above.222222Before I
became involved, Detlef and Nicola had been actively seeking suitable
experimental scenarios for distinguishing $\Pi_{\text{SC}}$ from the quantum
flux (equivalently, Bohmian) ToF distribution (Vona, 2014, Ch. 4).

Considering this, one is generally advised to relocate “the detectors closer
to the region of coherent wave packet production, or closer to the interaction
region” (Muga and Leavens, 2000, p. 419), i.e., to devise _near-field_
arrival-time measurements, where semiclassical approximations are invalid and
discernible differences between different theoretical proposals are manifest.
When actual experimental parameters are considered, it rapidly becomes clear
that the particle should arrive nearly instantly at the detector; therefore,
resolving ToF distributions at short distances demands impeccable time-
resolution capabilities.

_Back to the story._ Drawing on this background, Detlef proposed that it might
be fruitful to investigate (the considerably more challenging) few-particle
arrival-time problem, where the quantum flux distribution was inapplicable
Dürr and Teufel (2004) but a natural many-body generalization of
$\Pi_{\text{BM}}(\tau)$ would still be usable.232323I should note in passing
that there is presently much interest in experimentally probing the interplay
of Coulomb repulsion and the degeneracy pressure in few-electron experiments
Keramati  _et al._ (2021); Meier  _et al._ (2023)—a problem Detlef was
particularly interested in.

I subsequently began examining a two-particle arrival-time problem. Most ideas
had to be fleshed out from scratch because I could find no earlier work on
few-body arrival-time problems based on BM. At times I managed to overwork
myself, attempting to coax the theory into revealing something unexpected, to
the point that Detlef often intervened with remarks like “Siddhant, don’t
forget to live” …I must add that Detlef enthusiastically supported my
applications for _DAAD-Abschlussstipendium_ , the _LMU-Scholarship_ , as well
as the _Bayern State Scholarship_ , all of which offered valuable financial
support during the period 2016-2017. It cannot be overstated how wildly
efficient Detlef was at drafting grant applications, writing recommendation
letters, referee reports, rebuttals, etc., and concurrently taking great care
of his many pet animals (no pun intended).242424Prof. Tim Maudlin has shared
an amusing anecdote about how Detlef once politely excused himself in the
middle of someone’s talk, reasoning that it was time to walk his dog.

A few weeks went by with no tangible progress. Enter a second, watershed
moment (the first being the already stated counsel from Rodi): an opportunity
to give a talk at the group “Oberseminar.” Detlef encouraged me to present a
brief update on my current work at this weekly gathering, which Detlef and his
associates took very seriously at the time. I was quite terrified by the
invitation since I had really nothing to report; no results, nothing! Detlef
convinced me that doing an overview presentation and providing some
perspective on the literature would suffice. But, at that time I was too
impatient to closely engage with the vast “arrival-time problem” literature,
and I was also afraid of committing the scandal of delivering a dull seminar.

Therefore, one afternoon I decided to undertake something that seemed far more
interesting, and likely manageable in about a week’s time. The seminar was to
be held on July 12, 2017. I focused my attention on outlining a single-
particle ToF experiment that could be presented to the group members in under
an hour. The precise experimental setting I considered is as follows: A
spin-1/2 particle is confined to move along a long waveguide, modelled as a
semi-infinite cylinder. It is first trapped between the end face of the
waveguide and an impenetrable potential barrier placed at a distance $d$, as
shown in Fig. 2. The particle is prepared in the ground state $\psi_{0}$ of
this cylindrical box, then the barrier at $d$ is suddenly switched off at
$\smash{t=0}$, enabling the particle to move freely within the waveguide. The
arrival surface $\mathcal{D}$ is the plane situated a distance
$\smash{L~{}(\gg d)}$ from the end face of the waveguide, where arrival times
are recorded. This experiment undergoes several repetitions under identical
conditions, yielding the arrival-time distribution $\Pi(\tau)$.

In what follows, I describe the main motivations that underpin this particular
experimental setup. First, as discussed in Lamb (1969), trapping the particle
in a box and then abruptly releasing it was intended to produce a well-defined
and dependably preparable wave function,252525If the barrier at $d$ is
released sufficiently rapidly, the particle’s wave function immediately after
is $\psi_{0}$. as well as a well-defined starting time for the ToF experiment.
Since only the front wall is removed rather than both walls, the wave packet
expands unidirectionally rather than bidirectionally. In the latter instance,
two detector plates would be required, otherwise, detection events in one of
the directions would go unreported, contributing to an appreciable non-
detection probability.

Figure 2: _Left panel:_ A semi-infinite cylindrical waveguide mounted on the
$xy$-plane of a right-handed coordinate system, the axis of the cylinder
defining the $z$-axis. A spin-1/2 particle is prepared in a ground state of a
cylindrical box of width $d$. This state is spin-polarized along the unit
vector $\hat{\mathbf{s}}$. At time zero, the particle is released into the
waveguide leading to detection events on a detection plate (at $\smash{z=L}$)
downstream, where its arrival times are recorded. _Right panel:_ Snapshots of
the function $W(z,t)$: $\mathrm{Re}[W]$ (red), $\mathrm{Im}[W]$ (blue), $|W|$
(black) for select values of $t$.

The waveguide was incorporated in order to allow the particle to travel a long
distance, thereby facilitating _far-field_ detections, while concomitantly
preventing scattering-regime characteristics from setting in. The latter
manifest whenever the potentials are of short range. This, I hoped, would
provide measurable differences between $\Pi_{\text{BM/QF}}$ and, say,
$\Pi_{\text{SC}}$, given that these distributions normally agree when the
Bohmian trajectories become asymptotically Newtonian.

Finally, why specify spin-1/2 particles? Introducing spin into the discussion
may appear to some readers an unnecessary complication, as spin is easily
ignored in most situations. The main reason for considering particles with
nonzero spin here was that the Bohmian guiding equation, Eq. (4), for a
spin-1/2 particle, is more-or-less uniquely pinned down by relativistic
covariance considerations Holland (1992); Holland and Philippidis (2003);
Holland (2003), whereas there was still some concern about the uniqueness of
the guiding equation for spin-0 particles Deotto and Ghirardi
(1998).262626These issues arise because the goal of BM is to underpin the
statistical predictions of QM with deterministic dynamics of point particles,
and there are a priori far too many options for formulating such theories.

The Bohmian velocity field (cf. Eq. (4)) for spin-1/2 particles of mass $m$
moving at non-relativistic speeds, adapted here to the special case of space-
spin factorized or spin-polarized wave functions (those of the form
$\Psi_{t}=\psi_{t}(\mathbf{r})\chi$ with “spatial part” $\psi_{t}$ and “spin
part” $\chi$—a two-spinor), is given by (Bohm and Hiley, 1993, Ch. 10)

$\mathbf{v}_{\text{dBB}}^{\psi_{t}}(\mathbf{r},t)=\frac{\hbar}{m}\kern
1.00006pt\imaginary\kern-1.00006pt\left[\frac{\boldsymbol{\nabla}\psi_{t}}{\psi_{t}}\right]+\frac{\hbar}{m}\kern
1.00006pt\real\kern-1.00006pt\left[\frac{\boldsymbol{\nabla}\psi_{t}}{\psi_{t}}\right]\times\hat{\mathbf{s}},$
(11)

where $\hat{\mathbf{s}}:=\chi^{\dagger}\bm{\sigma}\chi$ is a unit vector
defined with the help of Pauli’s spin-vector matrix
$\bm{\sigma}=\sigma_{x}\kern 1.00006pt\hat{\mathbf{x}}+\sigma_{y}\kern
1.00006pt\hat{\mathbf{y}}+\sigma_{z}\kern 1.00006pt\hat{\mathbf{z}}$. This
velocity formula differs from that for a spin-0 particle by an additional
term, the so-called “spin term”, which is unfortunately often omitted in most
presentations of spin in BM, perhaps following an early proposal of John S.
Bell (Bell, 1971, pp. 171-181).272727This term, it turns out, is entirely
responsible for electron mobility in the ground state of a hydrogen atom
Colijn and Vrscay (2002, 2003a, 2003b), and it may even lead to particle
trajectories crossing the symmetry axis in a double-slit experiment Holland
and Philippidis (2003). Finally, the so-called “ring currents” arising out of
this term are central to the study of aromaticity, see Sundholm  _et al._
(2021).

Let me spell out a few steps involved in a Bohmian treatment of the proposed
experiment featuring a spin-dependent guiding equation. For concreteness, I
chose a harmonic potential $V_{\perp}(x,y)=m\kern 1.00006pt\omega^{2}\kern
1.00006pt(x^{2}+y^{2})\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}2$ to model the waveguide, which acts on the wave
function at all times. The barriers at $\smash{z=0}$ and $\smash{z=d}$ were
modelled by the (time-dependent) potential
$\smash{V_{\parallel}(z)=v(z)+\theta(-\,t)\kern 1.00006ptv(d-z)}$, where
$\smash{v(z)=0}$ for $\smash{z>0}$ and infinite otherwise, and $\theta$ is
Heaviside’s unit step function that eliminates the barrier $v(d-z)$ for
$\smash{t>0}$.

In units for which $\smash{\hbar=m=d=1}$,282828In other words, when masses,
lengths, and times are expressed in units of $m$, $d$, and
$\smash{md^{2}\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}\hbar}$, respectively. the parameter space of this
problem is comprised of the trapping frequency $\omega$, the flight-path-
length $L$, and two “spin-polarization angles”, $\alpha$, $\beta$, specifying
the orientation of $\hat{\mathbf{s}}$ (see Fig. 2).

The ground state wave functions of the trapped spin-1/2 particle assume a
space-spin factorized form $\Psi_{0}(\mathbf{r})=\psi_{0}(\mathbf{r})\chi$,
with spatial and spin parts given by

$\psi_{0}(\mathbf{r})=\sqrt{\frac{2\kern
1.00006pt\omega}{\pi}}\,\theta(z)\theta(1-z)\sin(\pi
z)\,e^{-\frac{\omega}{2}(x^{2}+y^{2})},\quad\text{ respectively
}\quad\chi=\matrixquantity(\cos\frac{\alpha}{2}\\\\[5.0pt]
e^{i\beta}\sin\frac{\alpha}{2}).$ (12)

These wave functions vanish in the regions $z\leq 0$ and $z\geq d$
($\smash{=1}$ in our units), differing only by the values of the spin-part
parameters $\smash{\alpha\in[0,\pi]}$, $\smash{\beta\in[0,2\kern
1.00006pt\pi)}$.292929This particular parameterization of $\chi$ is done in
order that
$\hat{\mathbf{s}}:=\chi^{\dagger}\bm{\sigma}\chi=\sin\alpha\big{(}\cos\beta\kern
1.00006pt\hat{\mathbf{x}}+\sin\beta\kern
1.00006pt\hat{\mathbf{z}}\big{)}+\cos\alpha\kern 1.00006pt\hat{\mathbf{z}}$.
Lacking time, I chose the ground states corresponding to $\smash{\alpha=0}$,
$\smash{\alpha=\pi}$ and
$\smash{\alpha=\pi\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}2}$, with $\smash{\beta=0}$, for the seminar. With
these, we obtain what may be referred to as the “spin-parallel”, “spin-
antiparallel” and “spin-perpendicular” wave functions, respectively, which
determines the orientation of the unit (spin) vector $\hat{\mathbf{s}}$
relative to the waveguide axis associated with these wave functions.

Once the barrier is turned off, a prepared ground state wave function spreads
dispersively into the waveguide, obeying Eq. (5) with Hamiltonian
$\smash{H=(1\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}2)\kern 1.00006pt\laplacian+V_{\perp}(x,y)+v(z)}$.
It continues to remain space-spin factorized with “spin part” $\chi$, the same
as that set at time zero, and “spatial part” also identical to $\psi_{0}$, Eq.
(12), except that $\theta(z)\theta(1-z)\sin(\pi z)$ is replaced by
$\theta(z)W(z,t)\kern 1.00006pte^{-\kern 0.70004pti\omega t}$. I could obtain
this function $W(z,t)$ in closed-form (Das and Dürr, 2019, Eq. (11)) by
solving Schrödinger’s equation, which led to a considerable simplification in
the numerical treatment of the arrival-time experiment. A few snapshots of
$W(z,t)$ are depicted in Fig. 2 (right panel). It shows the unfolding of a
truly remarkable wave phenomenon called _diffraction in time_ Moshinsky
(1952), manifesting in response to a sudden change in the boundary conditions
of the wave function at a given surface (in this case, the plane
$\smash{z=1}$). Observe that the prepared wave function breaks into an
infinite collection of tiny ripples or wavelets close to this surface,
nucleating into _self-similar_ wave packets, each propagating down the
waveguide in succession.

This wave function determines the particle velocity in the guidance equation
(11), in turn, the Bohmian trajectories $\smash{\mathbf{R}(t)=X(t)\kern
1.00006pt\kern 1.00006pt\hat{\mathbf{x}}+Y(t)\kern 1.00006pt\kern
1.00006pt\hat{\mathbf{y}}+Z(t)\kern 1.00006pt\kern
1.00006pt\hat{\mathbf{z}}}$, which can be used to express the first arrival-
time, (6), of an individual trajectory, _viz._ ,
$\smash{\tau(\mathbf{R}_{0})=\text{inf}\kern 1.00006pt\big{\\{}\,t\geq 0\kern
1.00006pt:\kern 1.00006ptZ(t)=L\,\text{ and }\,\smash{0<Z_{0}<1}\kern
1.00006pt\big{\\}}}$, as a function of the initial position $\mathbf{R}_{0}$
and the flight-distance $L$. And, assuming that the initial positions are
randomly distributed according to the $|\psi_{0}|^{2}$-distribution,303030Note
that
$\smash{\Psi_{0}^{\dagger}\Psi_{0}^{\phantom{\dagger}}\overset{\eqref{initspinor}}{=}|\psi_{0}|^{2}}$
for any $\alpha$, $\beta$. the desired Bohmian ToF distribution is obtained
via Eq. (7). Since the parameters in the initial wave function are, in view of
(12), $\alpha$ and $\beta$, we denote $\Pi_{\text{BM}}(\tau)$ by
$\Pi^{\alpha|\beta}_{\text{BM}}(\tau)$.

Histograms of the BM distributions for spin-parallel
$\smash{(\alpha=\beta=0)}$ and spin-perpendicular $(\alpha=\pi/2,\kern
1.00006pt\beta=0)$ to the waveguide-axis are shown in Fig. 3 for the
parameters $\smash{\omega=10^{3}}$ and $\smash{L=100}$. To obtain these
graphs, I sampled $\approx 10^{5}$ random initial positions from the
$|\psi_{0}|^{2}$-distribution, solved the Bohmian guiding equations
numerically313131The assistance of Grzesio Gradziuk and Leopold Kellers with
the numerical simulations using Mathematica was essential and greatly
appreciated, and provided me with an opportunity to learn Mathematica
programming for numerical calculations. for each point in this ensemble,
continued until the trajectory hit $\smash{z=L}$, recorded the arrival time
and plotted the histogram for $\Pi_{\text{BM}}(\tau)$.

Figure 3: _Left panel_ : Bohmian arrival-time distributions for particles
spin-polarized parallel $\Pi^{0|0}(\tau)$ and perpendicular
$\Pi^{\frac{\pi}{2}|0}(\tau)$ to the waveguide axis. The solid, dashed curves
are the quantum flux (QF) resp. semiclassical (SC) ToF predictions. Arrival
times on the horizontal axis are reported in the units of $md^{2}/\hbar$. Each
histogram has been generated with $10^{5}$ Bohmian trajectories. We see
agreement between $\Pi^{0|0}(\tau)$ and $\Pi_{\text{QF}}(\tau)$. The
distribution $\Pi^{\frac{\pi}{2}|0}(\tau)$ predicts no first arrivals after
time $\tau_{\text{max}}$. Note the disagreement of all distributions with
$\Pi_{\text{SC}}(\tau)$. _Left panel Inset:_ Mean first-arrival-time
$\expectationvalue{\tau}$ vs. spin orientation angle $\alpha$ for
$\smash{L=10}$ and $\smash{\beta=0}$. _Right panel_ : Typical Bohmian
trajectories for $\smash{\alpha=\pi/2}$, $\smash{\beta=0}$ (solid) and
$\smash{\beta=3\pi/4}$ (dashed). Both trajectories are plotted for
$\smash{t\in[0,5]}$ with (common) initial condition $\mathbf{R}_{0}=0.05\kern
0.80005pt(\hat{\mathbf{x}}+\sqrt{3}\kern 0.80005pt\hat{\mathbf{y}})+0.5\kern
0.80005pt\hat{\mathbf{z}}$. Here, $\smash{z=1}$ denotes the hard-wall
potential barrier, which was switched off at $\smash{t=0}$.

For the spin-parallel (also antiparallel) wave function,
$\Pi^{0|0}_{\text{BM}}(\tau)$ (blue histogram in Fig. 3) agrees with
$\Pi_{\text{QF}}$ (i.e., Eq. (3) evaluated with the Pauli-current Wilkes
(2020)) owing to the CPC (9) being satisfied. (This was an important
consistency check that the numerical calculations were on track.) It could be
shown that this ToF distribution is independent of the trapping frequency
$\omega$, which sets the effective diameter of the waveguide
$\smash{\approx\sqrt{\hbar\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}m\omega}}$, and has an asymptotic $\tau^{-4}$ fall-
off.323232A direct consequence of the back-wall at $\smash{z=0}$, without
which it would have an asymptotic $\tau^{-2}$ fall-off.

On the other hand, for the spin-perpendicular wave function, the results of
the simulations were striking! In this case, no Bohmian particle struck
$\smash{z=L}$ beyond a characteristic maximum arrival-time $\tau_{\text{max}}$
regardless of the initial position $\mathbf{R}_{0}$. Consequently, the density
$\Pi^{\frac{\pi}{2}|0}_{\text{BM}}(\tau)$ vanished for
$\smash{\tau>\tau_{\text{max}}}$, strongly disagreeing with the QF
distribution. This was because, to my complete surprise, _the CPC failed to
apply for this wave function_. (Typical Bohmian trajectories for this wave
function are depicted in Fig. 3, right panel.) Furthermore, this feature
remained intact for all values of $L$ and $\omega$, except when
$\smash{\omega=0}$, in which case there is no radial confinement.

In fact, in the limit $\smash{\omega\to 0}$,
$\smash{\Pi_{\text{BM}}^{\alpha|\beta}=\Pi_{\text{QF}}}$ for _all_
$\alpha,\kern 1.00006pt\beta$ hence, in particular,
$\smash{\tau_{\text{max}}\to\infty}$, and the spin-polarization-dependence is
lost. This demonstrates the importance of the waveguide confinement to
observing backflow (i.e., the violation of CPC), and in turn the spin-
dependence of $\Pi_{\text{BM}}$.333333While the waveguide does play a crucial
role in provoking and even influencing the amount of backflow in this setup,
it would manifest neither for a spin-0 particle moving within the waveguide
nor for a spin-1/2 particle moving freely. (For a discussion of the limit
$\smash{\omega\to 0}$, see (Das  _et al._ , 2019, Sec. V. C.).)

Finally, a mention should be made of the smaller secondary lobes of the ToF
distributions (visible for $\tau\lesssim 20$ in Fig. 3). They are created by a
particle propagating within the support of the smaller wavelets created at the
moment the barrier at $d$ is removed, as explained previously and seen in Fig.
2 (right panel). Because of the nodes separating these wave packets, the
Bohmian particle remains inside the support of the wave packet for which its
random initial position was realized at $\smash{t=0}$. The wave packets
advance in time carrying the particle with them, hence the arrival times are
recorded in bunches. Such secondary lobes caused by diffraction in time have
been detected experimentally in the ToF distributions of atoms Szriftgiser
_et al._ (1996) and neutrons Hils  _et al._ (1998).

So, serendipitously, I had stumbled upon something completely unexpected. The
results would not have been expected based on the wave function alone, since
the Hamiltonian describing the particle’s propagation through the waveguide is
_spin-independent_.343434Although in quantum chemistry one often encounters
spin-dependent answers to questions for which the Hamiltonian could well be
spin-independent (Walters, 1995, p. 38). The spin-dependent arrival-time
distributions found in this setup are a genuine consequence of the particle’s
dynamics determined by Eqs. (4) and (11).353535We note in passing that spin-
dependent arrival-time distributions would also be predicted by some of the
absorbing Boundary Conditions (ABCs) Tumulka (2016), but there it would not be
particularly unexpected since a spin-dependent boundary condition is
explicitly imposed on the wave function at $\smash{z=L}$.

Needless to say, participants at the seminar were unpreparedly surprised. Of
course, Detlef did not buy into the perplexing results right away, but he was
maximally pleased by the disagreement with the semiclassical prediction
(dashed curve in Fig. 3) which drops asymptotically as
$\tau^{-2}$.363636Actually, (10) is only applicable for free propagation, so
it had to be generalized in order to account for the presence of the waveguide
as well as the back wall; see (Das and Dürr, 2019, Eq. (18)). Soon after this
oberseminar both Detlef and I worked very intensely on this arrival-time
setup, slowly learning a great deal about the role of spin in the process. We
had regular weekly meetings (every Wednesday). He was very attentive to
details and always had a listening ear.

Subsequently, a short article detailing these findings was published Das and
Dürr (2019), most of which transformed into my master’s thesis that I defended
on October 25, 2017. (Detlef personally invited Prof. Weinfurter, who
graciously probed me until 19:00 that evening, despite the fact that the
defence began at 16:15.) In terms of conveying research, Detlef’s philosophy
was “one idea per paper.” He was also a very talented writer373737With a
healthy appetite for Latin idioms, e.g., “cum grano salis”, “experimentum
crucis”, etc., capable of masterfully intertwining physics with formalism, as
anybody who has read his many books and research articles can attest.

Such “unexpected and very well-articulated arrival-time distributions,” Detlef
insisted, “almost demand experimental inspection.” Therefore, in the meantime,
we also contacted a number of experimental groups to realize this arrival-time
experiment.

As I was wrapping up my master’s program, Detlef invited me to continue
working on arrival times as his doctoral student and arranged funds to support
me. This is when I earned the distinction of being his last Ph.D. student.
Following a short break with my family in India, I returned to Munich in
mid-2018 and resumed our collaboration. Over the following two years, a very
detailed understanding of the waveguide setup and variants thereof was
acquired, thanks especially to the contributions of Markus Nöth—a gifted
mathematical physicist, close friend, and collaborator.

The predicted spin-dependent arrival-time distributions would readily manifest
for various kinds of confining potentials in waveguides, and in waveguides of
different cross-sections, e.g., elliptical ones. In addition, it was found
that a parallel plate waveguide suppresses the arrival-time distributions for
a significantly greater range of $\alpha$s about
$\pi\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}2$ than a circular one, making it the most
promising arrangement for experimentally probing the spin-dependent
distributions.383838I worked out the parallel-plate setup to answer Ward
Struyve’s concern that the suppression in arrival times in the cylindrical
waveguide setup manifests extremely close to $\smash{\alpha=\pi/2}$, as
evidenced by the sharp drop in the $\expectationvalue{\tau}$ vs. $\alpha$
curve shown in Fig. 3 (left panel inset). These results also do not depend on
the presence or precise form of the back wall. They hold for particles of spin
$\smash{\geq 1/2}$ and, somewhat surprisingly, for composite particles such as
individual hydrogen atoms prepared in triplet states as well. Similar results
arise in a relativistic treatment based on Dirac’s equation and the Bohm-Dirac
guiding equation, whose validity extends to relativistic Bohmian velocities
Das (2021). Interestingly, nodes in the initial wave function $\psi_{0}$
induce _gaps_ (or “no arrival windows”) in the distribution for
$\Pi^{\frac{\pi}{2}|0}_{\text{BM}}(\tau)$. A striking illustration of this
occurs if one prepares the particle in excited states
$\smash{\propto\sin(n\pi)}$ instead of the ground state (12). It will take
some time to prepare all of these extensions for publication, and work is
currently underway.

In the middle of this euphoric collaboration, all hell was unleashed by the
Covid-19 pandemic. Detlef’s untimely passing on January 3, 2021, was a seismic
event for me. I recall with some sadness a moment in December 2019 when he
said to me as we were waiting for a train to Frankfurt that, in his opinion “I
had put my finger on something very profound,” and that he “would not be
around when they see it.” In retrospect, these were truly crushing words, and
I had no clue that the unthinkable would come to pass so soon. I had developed
such a close bond with him, and so many aspects of my life had been shared
with him, that it was of course very difficult to move on. To whom would I
turn in pursuing research now? It seemed for some time afterwards to be a
pointless endeavour.

Our work had come to a point where he had begun one of his presentations by
saying “I shall report on a recent result by Siddhant Das for arrival time
distributions in QM, which may provide the elements for a paradigm change to a
quantum world which is understandable” Dürr (2018). I eventually came to
realize that for me to give up at this stage would seem a betrayal of both the
confidence he had in me, and in the work we had done together. I suppose that
the primary reason I continue it is that, though I loved Detlef immensely, I
love the pursuit of physics research with the same passion that he had for it,
even more. I am very thankful for the support of my family and friends, many
of them former students and colleagues of his (I would mention especially
Anirudh Chandrasekaran, James M. Wilkes, Serj Aristarhov, Markus Nöth, Paula
Reichert, Ward Struyve, Dirk Deckert, Dustin Lazarovici, Prof. Hemalatha
Thiagarajan, Prof. Subhendra D. Mahanti, Prof. Jean Bricmont, Prof. A. Shadi
Tahvildar-Zadeh, Prof. Sheldon Goldstein, and Prof. Tim and Vishnya Maudlin),
who have helped me hold my personal and academic life together and slowly
resume work.

Clearly, much remains to be done to turn these preliminary findings into more
realistic experimental proposals, and there may well be more surprises to come
in the future (some that Detlef may already have anticipated). Only time will
tell, of course. Reflecting on this time in my life, I consider myself
extremely privileged to have been a small part of Detlef’s scientific legacy.
His warmth, contagious enthusiasm, collegial grace, and, above all, his
willingness to welcome nascent scientists to the field have improved our
community in ways that will not be forgotten.

I thank James M. Wilkes and Dustin Lazarovici for carefully reviewing this
article and for their invaluable editorial input. Thank you also to Stephen N.
Lyle for pointing out a few typos.

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