Born’s interpretation of the squared modulus of a wave function (|ψ|2) as the probability density of finding a particle within a specified region of space is a key ingredient of the standard framework of quantum mechanics, thereby implying that the standard interpretation of quantum mechanics is inherently epistemological. On the other hand, the possibility of an alternative interpretation of quantum mechanics by interpreting |ψ|2 as the probability density of a particle being actually present within a specified region was first suggested by de Broglie [1]. Later, Bohm [2-4] developed the details of such an ontological model of quantum mechanics by using the notion of an observer-independent spacetime trajectory of an individual particle T&F Cat # K11224, Chapter 8, Page 114, 15-9-2010 that is determined by its wave function through an equation of motion which is formulated in a way consistent with the Schrödinger time evolution. Bohm’s model, thus, explicitly refuted the counterarguments (such as those put forward by Pauli [5] and von Neumann [6]) that claimed to have ruled out the formulation of such a model. Subsequently, much work has been done on various aspects of the Bohmian model [7-10]. That such a model is not unique has been elaborately discussed [11,12] while different versions of the ontological model of quantum mechanics have been proposed [13-21].
The present article focuses on studying the extent to which the nonuniqueness that is inherent in the standard quantum mechanical calculation of arrival/transit time distribution can be exploited to enable an empirical scrutiny of any causal trajectory model such as the Bohmian scheme. For this purpose, we consider the example of spin-1/2 neutral particles corresponding to a wave packet which passes through a spin rotator(SR) that contains constant magnetic field confined within a region - in such a case, the transit time distribution can be measured in terms of the spin distribution of particles emerging from the SR. In particular, we investigate the way one can compare the Bohmian predictions obtained for this example with that using one of the quantum approaches, say, the probability current density based scheme. Here the Bohmian calculational procedure involves a couple of critical subtleties that lead to some specific directions for further studies.
It is argued that in Bohmian mechanics the effective wave function of a subsystem of the universe does not encode the influences of other par-ticles on the subsystem. This suggests that the ontology of Bohmian mechanics does not consist only in Bohmian particles and their posi-tions. It is nonetheless pointed out that since the wave function in configuration space may represent the state of ergodic motion of non-Bohmian particles in three-dimensional space, the ontology of Bohmian mechanics may still consist only in particles. Bohmian mechanics, which is also called the de Broglie-Bohm theory, is a realistic alternative to standard quantum mechanics initially proposed by de Broglie (1928) and later developed by Bohm (1952). According to the theory, a complete realistic description of a quantum system is provided by the configuration defined by the positions of its particles together with its wave function. It has been a hot topic of debate how to interpret the wave function in the theory. Recently Esfeld et al (2013) further developed the nomological interpretation of the wave function, as suggested by Dürr, Goldstein and Zangh̀ı (1997), Goldstein and Teufel (2001) and Goldstein and Zangh̀ı (2012). They argued that the ontology of Bohmian mechanics consists only in Bohmian particles and their positions, and in particular, the effective wave function of a subsystem of the universe encodes the influences of other particles in the universe on the subsystem. In this note, we will argue that this interesting view may be problematic. The Bohmian law of motion is expressed by two equations, a guiding equation for the configuration of particles in three-dimensional space and the Schrödinger equation, describing the time evolution of the wave function, which enters the guiding equation. The law can be formulated as follows: dQ(t) dt = vΨ(t)(Q(t)). (1)
Bohmian mechanics, developed by Bohm in 1952, provides an alternative interpretation to nonrelativistic quantum mechanics [1-3]. In the hydrodynamic formulation of quantum mechanics, the continuity equation and the quantum Hamilton-Jacobi equation (QHJE) are obtained by substituting the wave function expressed in terms of the real amplitude and the real action function into the time-dependent Schrödinger equation. Bohm’s analytical approach has been used to compute and interpret realvalued quantum trajectories from a precomputed wave function for a diverse range of physical processes [3-8]. In the synthetic approach, the quantum trajectory method has been developed as a computational tool to generate the wave function by evolving ensembles of real-valued quantum trajectories through the integration of the hydrodynamic equations on the fly [9]. Remarkable progress has been made in the use of T&F Cat # K11224, Chapter 18, Page 284, 15-9-2010 real-valued quantum trajectories for solving a wide range of quantum mechanical problems [10].
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function
ψ. Given a Schrodinger wave function
ψ, we use its value
ψ0 at some fixed time (say,
t=0) to define the probability measure
∣ψ0∣2dx on the system configuration space
M (
=Rn). On the resulting probability space
M0, we introduce a stochastic process
ξ(t) corresponding to the Heisenberg position operator
XH(t) such that, in the Heisenberg state
∣ψh⟩ corresponding to
ψ0, the expectation value of
XH(t) equals that of
ξ(t) in
M0. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process
ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the
ψ0-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.
The Schrödinger equation and the Born rule constitute two of the most successful structures in quantum mechanics. The former governs the continuous evolution of the wave function, while the latter connects its squared amplitude to the statistical distribution of experimental outcomes. Yet these two structures operate at different descriptive levels: the Schrödinger state is continuous, spatially extended, and phase-sensitive, whereas individual experiments terminate in localized physical events. This paper develops a theoretical framework for reconstructing the physical relation between these two levels. Rather than proceeding only in the conventional direction from Schrödinger dynamics to Born probabilities, we introduce an inverse explanatory strategy: from Born back to Schrödinger. Born statistics are treated as an empirical constraint from which the physical role of the wave function is reconsidered. We propose that, for the single-particle regime examined here, the wave function may be interpreted functionally as a dynamical propagation structure rather than as the localized particle itself. Its amplitude, phase, conserved current, gauge-covariant propagation momentum, and associated velocity field jointly define a structured family of possible motions. This interpretation leads to three separations: propagation from localization, propagation from physical population, and arrival from detector registration. A physical preparation generates a source measure (\mu_0^{\rm phys}); Schrödinger dynamics generates a propagation map (\Phi_\Psi); and microscopic detector dynamics is represented by a response map (\mathcal R_D). The Born problem can then be formulated as the closure condition
RD[(ΦΨ)∗μ0phys]=?μBorn,D. This relation is not claimed here as a derived law, but as the central dynamical problem that a physical reconstruction of quantum probability must solve without inserting the Born measure at either boundary. The paper synthesizes six preceding studies on wave-packet localization, Born's scattering interpretation, the emergence of probability, the physical meaning of the wave function, source-to-detector transport, and unified propagation dynamics. It places the resulting framework in dialogue with Madelung and Hamilton–Jacobi formulations, Bohmian mechanics, decoherence, Quantum Darwinism, structural and symmetry-based derivations of the Born rule, Everettian approaches, objective-collapse theories, and detector models. The resulting proposal is best understood as a propagation-first theoretical reconstruction program: Schrödinger dynamics organizes possible propagation, while Born statistics constrain how that propagation becomes physically realized. Keywords: Schrödinger equation; Born rule; wave function; quantum foundations; propagation structure; quantum probability; quantum trajectories; source-to-detector transport; detector response; wave–particle duality; Bohmian mechanics; quantum measurement
Bohmian mechanics is endowed with an ontological package that supposedly allows to solve the main interpretational problems of quantum mechanics. We are concerned in this work by the emergence of classicality from the quantum mechanical substrate. We will argue that although being superficially attractive, the de Broglie-Bohm interpretation does not shed new light on the quantum-to-classical transition. This is due to nature of the dynamical law of Bohmian mechanics by which the particles follow the streamlines of the probability flow. As a consequence, Bohmian trajectories can be highly non-classical even when the wavefunction propagates along classical trajectories, as happens in semiclassical systems. In order to account for classical dynamics, Bohmian mechanics needs non-spreading and non-interfering wave packets: this is achieved for practical purposes by having recourse to decoherence and dense measurements. However one then faces the usual fundamental problems associated with the meaning of reduced density matrices. Moreover the specific assets of the de Broglie-Bohm interpretation – in particular the existence of point-like particles pursuing well-defined trajectories – would play no role in accounting for the emergence of classical dynamics.