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Physique
Franck Laloë and Geneviève Tastevin
The outcomes of measurements in the de Broglie–Bohm theory II
Volume 26 (2025), p. 479-482
Online since: 4 July 2025
https://doi.org/10.5802/crphys.255
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Comptes Rendus. Physique
2025, Vol. 26, p. 479-482
https://doi.org/10.5802/crphys.255
Research article / Article de recherche
The outcomes of measurements in the de
Broglie–Bohm theory II
Les résultats de mesure dans la théorie de de Broglie et
Bohm II
Franck Laloë a and Geneviève Tastevin a
a Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France,
24 rue Lhomond 75005 Paris, France
E-mails: laloe@lkb.ens.fr, tastevin@lkb.ens.fr
Abstract. Part I of this article discussed the quantum measurement process within the de Broglie–Bohm
theory. In the experiment considered, the outcome of the measurements was primarily determined by the
initial Bohmian positions within the measurement apparatus. This is nevertheless not always the case. Part II
is a short addendum with a more general discussion: cases where the measurement results reveal the values
of the positions attached to the measured system, the measurement apparatus, or both, are considered.
Résumé. La partie I de cet article contenait une discussion du processus de mesure dans le cadre de la théorie
de de Broglie et Bohm. Dans l’expérience étudiée, les résultats de mesure étaient principalement déterminés
par les positions bohmiennes des particules au sein de l’appareil de mesure. Ce n’est cependant pas toujours
le cas. La partie II est un court addendum qui propose une discussion plus générale, et envisage les cas dans
lesquels les résultats de mesure révèlent les positions du système quantique mesuré, de l’appareil de mesure,
ou des deux.
Keywords. Quantum measurement, de Broglie theory, Bohmian theory, Additional quantum variables.
Mots-clés. Mesure quantique, Théorie de de Broglie, Théorie bohmienne, Variables supplémentaires.
Manuscript received 7 June 2025, accepted 25 June 2025.
In [1], we discussed the quantum measurement process within the de Broglie–Bohm (dBB)
theory in order to emphasize that, in some cases, the outcome of the measurements was primar-
ily determined by the initial Bohmian positions within the measurement apparatus M; contrary
to what intuition suggests, the positions attached to the measured system S then play no role in
the determination of the outcome. This is nevertheless not always the case: in other conditions,
the opposite is true, and what determines the results of measurement are the positions of S, not
those of M; as in classical physics, performing a measurement then reveals a pre-existing prop-
erty (within the dBB theory) of S. See also for instance the discussion in [2]. The present text is
an addendum to [1] containing a more general discussion of the measurement problem, and dis-
tinguishing between cases where the measurement results are determined by the Bohmian po-
sitions of S, or M, or both. This can be seen as a discussion of the mechanism of contextuality
within the dBB theory.
1. A first, very simple, example is illustrated in Figure 1: a beam splitter separates the quantum
wave of a single particle into two components, which are then directed towards two detectors D1
and D2. The particles may be for instance metastable Helium atoms detected with microchannel
ISSN (electronic): 1878-1535 https://comptes-rendus.academie-sciences.fr/physique/
480 Franck Laloë and Geneviève Tastevin
plates [3], while the atomic beam splitter can be obtained optically with near-resonant lasers [4].
When a metastable atom reaches the entrance of a microchannel, an electron is ejected from the
solid surface which, under the effect of an applied electric field, triggers an electronic cascade
inside the channel. This amplification effect eventually leads to a macroscopic electric current,
and to a click of the detector. If the atoms are sent one by one into the system, in each realization
of the experiment only one of the detectors clicks, while the other remains silent.
In this simple experiment, which detector clicks is determined by the trajectory of the particle
in the optical device. This is because the Bohmian trajectory is continuous and cannot jump
from one path to the other; when it reaches one of the detectors, the single particle trajectory
determines which component of the many particle state vector is active and contains the cascade
process, and which remains an “empty wave”. Mathematically, the state vector describing the
whole system of the atom and the two detectors contains no component where the atom is in
one arm while the variables of the detector in the other arm are subject to the cascade effect; the
Bohmian positions cannot explore regions of the configuration space where the wave function
vanishes. As a consequence, in this case, and as one could naively expect, the outcome of the
measurement is indeed determined by the trajectory of the measured atom, that is by its position
within its wave packet just after the crossing of the beam splitter.
D
1
D
2
Figure 1. A very simple experiment where the Schrödinger wave of a single particle is split
into two components, which are then sent on two detectors D1 and D2.
A similar situation occurs when a beam of particles is sent onto a cloud chamber in order to
detect the trajectories of particles. When the wave packet of a particle enters the cloud chamber
and initiates a track by a cascade of condensation effects, it is the Bohmian position of the
particle that determines the location of the start of the track, and therefore its continuation in
the chamber, as discussed for instance by Darwin and Mott [5]. Again, in this case the result
of measurement gives an information on the position of the particle before measurement; the
Bohmian positions of the particles in the cloud play no essential role.
2. Maybe the example of quantum measurement most frequently discussed in textbooks re-
mains the Stern–Gerlach experiment – even if in practice this category experiment is rarely per-
formed in laboratories. This case is special since no cascade or amplification effect is involved,
so that no single particle detection really occurs; what is done is to repeat the experiment a very
large number of times, until visible spots of atoms are created on the detection plate. This allows
the observation of the quantization of the spin of the Ag atoms, without any individual observa-
tion.
Reference [6] provides an interesting discussion of the Stern–Gerlach experiment within dBB
theory. It emphasizes the fact that the trajectory of each atom deviates upwards if its initial
position is initially in the upper part of the wave packet, downwards if it is initially in the lower
Franck Laloë and Geneviève Tastevin 481
part1. The surprising result is that this situation does not change when the direction of the
gradient of the magnetic field is inverted, so that the relation between values of the spin and
trajectory deviations is swapped. As a consequence, in an experimental situation a given initial
position of the atom determines a value +ħ/2 of the spin, in another situation an opposite value
−ħ/2. In other words, the initial atomic position determines the result of measurement (the value
of one component of its spin), but in a way that depends on the design of the apparatus; the result
is contextual.
3. Reference [1] discusses another experiment, an “optical version of the Stern–Gerlach experi-
ment”, where two light beams with opposite circular polarizations create spin dependent recoil
effects on the atoms. In this case, the two spin states of the atom occupy initially the same region
of space; they start to separate only under the effect of the interaction with the measurement
system. Two competing effects then take place: on one hand, the overlap between the two wave
functions of the atom associated with the two spin directions decreases as a function of time, but
on the other hand the overlap between the wave functions of the measurement apparatus associ-
ated with the two results also starts to decrease. Because the measurement apparatus contains a
large number N of particles which, each, contribute to the decrease of the overlap, the N -particle
wave functions become much more rapidly orthogonal in the configuration space of the appara-
tus than those of the atom. At the end, the choice of the result of measurement depends only on
the initial positions of the particles inside the measurement apparatus, while the Bohmian posi-
tion of the atom itself is practically irrelevant. It is the measurement apparatus that determines
the result, and then drives the evolution of the position of the atom accordingly. The conclu-
sion is therefore the exact opposite of that of the preceding paragraph, which is not contradictory
since the experiment is different.
One may notice that the Gordon term was not included in the calculations in [1], which
simplifies the equations. The discussion relies mostly on the motion of wave packets (not affected
by perturbations of the Bohmian trajectories) and on the trajectories of the Bohmian positions
of the measurement apparatus (which are not subject to a Gordon term). The conclusions of [1]
should therefore remain almost unchanged when this term is included.
4. One can also have intermediate situations, for instance if the system S first interacts with
an intermediate quantum system S’ containing N ′ particles, which eventually interacts with a
macroscopic measurement apparatus containing a very large number of particles. Depending on
the dynamics of the interaction between S and S’ and on the value of N ′, the information carried
by S’ to the measurement apparatus may depend primarily on the position of the Bohmian
variables of S, or of those of S’; the dynamics of interaction between S’ and the measurement
apparatus also plays a role. Therefore, in such an indirect measurement using an ancilla quantum
system S’, depending on the conditions the result of measurement may be determined by the
initial positions of either S, S’, or M, or be a compromise between different Bohmian variables.
In conclusion, what matters is the spatial extent (before the measurement) of the parts of the
total wave function that, under the effect of the interaction with the measurement apparatus M,
will become entangled with orthogonal wave functions of S that are situated in different regions
(of its configuration space). In other words, what matters is the spatial extent of the wave func-
tions associated with the eigenvectors of the measured observable of S. If these eigenfunctions
are already spatially separated, the result of measurement is determined by the Bohmian posi-
tion of S; if they overlap spatially, the Bohmian position of M may play an important role, or even
determine the measurement result as in [1].
1Relativistic considerations show that the Bohmian velocity includes a rotational spin-dependent term (Gordon term)
[7–9] that is not taken into account in this discussion. The Gordon term slightly modifies the trajectories [7], but does not
change the main conclusions concerning the strong contextuality of the measurement results.
482 Franck Laloë and Geneviève Tastevin
In experiments where the position of a particle is directly measured, the result of measure-
ment (the place at which the particle materializes, for instance in a cloud chamber) is indeed de-
termined by the Bohmian trajectory of the position of the particle. This is what intuition predicts;
it remains true when the correlations between the positions of several particles are measured, for
example as in [10]. By contrast, in experiment where another observable is measured, such as
an internal variable of the particle (spin), the situation may be the opposite. In this case, the re-
sult of measurement may be a consequence of the initial state of the measurement apparatus (as
defined by the initial position of its Bohmian positions).
Acknowledgments
The authors thank Siddhant Das for interesting comments and discussions.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
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//arxiv.org/abs/2404.09934.
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tion of thermal and quantum depletion in a Bose gas”, Phys. Rev. Lett. 117 (2016), article no. 235303 (5 pages).
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[5] N. F. Mott, “The wave mechanics of α-ray tracks”, Proc. Roy. Soc London A 126 (1929), pp. 79–84.
[6] J. Bricmont, Making sense of quantum mechanics, Springer, 2016, pp. x+331.
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