Bohmian mechanics: A legitimate hydrodynamic picture for quantum mechanics, and beyond

TL;DR

This work argues that Bohmian mechanics should be regarded as a legitimate, operational representation of quantum mechanics rather than a metaphysical hidden-variable theory. By recasting Schrödinger’s equation in a hydrodynamic form, it derives a continuity equation for ρ and a velocity field v = j/ρ = ∇S/m, with a quantum potential Q, showing that Bohmian trajectories emerge naturally from standard QM. It further explores weak measurements and weak values to distinguish observable versus detectable quantities, illustrating how phase coherence and the velocity field encode quantum dynamics, including interference and entanglement. Beyond quantum systems, the paper demonstrates Schrödinger-like dynamics in paraxial optics, enabling trajectory-based analyses of light beams and phenomena such as Airy-beam propagation, thereby broadening the applicability and utility of Bohmian-inspired hydrodynamic methods.

Abstract

Since its inception, Bohmian mechanics has been surrounded by a halo of controversy. Originally proposed to bypass the limitations imposed by von Neumann's theorem on the impossibility of hidden-variable models in quantum mechanics, it faced strong opposition from the outset. Over time, however, its use in tackling specific problems across various branches of physics has led to a gradual shift in attitude, turning the early resistance into a more moderate acceptance. A plausible explanation for this change may be that, since the late 1990s and early 2000s, Bohmian mechanics has been taking on a more operational and practical role. The original hidden-variable idea has gradually faded from its framework, giving way to a more pragmatic approach that treats it as a suitable analytical and computational tool. This discussion explores how and why such a shift in perspective has occurred and, therefore, answers questions such as whether Bohmian mechanics should be considered once and for all a legitimate quantum representation (i.e., worth being taught in elementary quantum mechanics courses) or, by extension, whether these ideas can be transferred to and benefit other fields. Here, the Schrödinger equation and several specific numerical examples are re-examined in the light of a less restrictive view than the standard one usually adopted in quantum mechanics.

Figures (6)

• Figure 1: Probability density (a) and velocity field (b) for a linear superposition of two Gaussian wave packets simulating the transverse diffraction undergone by two Gaussian beams and subsequent appearance of interference traits. The Bohmian trajectories associated with the velocity field displayed in panel (a) are superimposed in both graphs to show evidence of the local dynamics induced and, more specifically, how they try to avoid those regions with high values of the velocity and stay in those other with lower values. This explains why their statistics results in a series of populated regions surrounded by other regions with negligible density, that is, a series of interference channels that concentrate the average flow of particles. In panel (a), the color code associates blue for negligible values of the density and red to higher values; in panel (b); the color code associates blue to negative values of the velocity, red to positive values of the density, and green to values around zero.
• Figure 2: Transverse cuts of the velocity field at various times for a two-Gaussian wave-packet superposition (a) and a single Gaussian wave packet (b). The color code for each transverse cut is indicated in the legend provided in panel (b), which is the same in both cases. Although both scenarios exhibit a clockwise rotation of the slope characterizing the velocity field at a given given time, the presence of an intermediate shear at $x=0$ in the two wave-packet superposition is a clear indication that the superposition principle does not hold in the case of the velocity field. In this latter case, the system works as a wholeness regardless of whether, mathematically, we describe it in terms of a linear superposition.
• Figure 3: Propagation of a bipartite continuous entangled state (upper row) and a factorizable two-Gaussian superposition state (lower row). The contour plots of the probability density at $t=0$ and $t=10$ are shown in the first and second columns, respectively. The velocity field felt by the $X$ and $Y$ systems at $t=10$ are shown in the third and fourth columns, respectively. In each plot, the dots represent sets of initial conditions. Specifically, the white dots indicate various initial positions for system $X$ keeping fixed the position for system $Y$ at either $A$ or $B$, while dots denote initial positions for system $Y$ keeping fixed, at $A$ or $B$, the position for system $X$. In each plot the horizontal axis represents the position coordinates for system $X$, while the vertical axis specifies the position coordinates for system $Y$. The color code associated with the contour plots follows the same specifications as in Fig. \ref{['fig1']}.
• Figure 4: Bohmian trajectories associated with a Bell-type bipartite continuous variable state (a)-(b), and with a factorizable two-Gaussian superposition state (c)-(d). In the left column, the time-evolution of the trajectories in $x$ and $y$ is shown together with the projections onto the $X$- and $Y$-planes, which represent the corresponding subspaces for the $X$ and $Y$ subsystems. In the right column, $x$-coordinate of the trajectories for sysbsystem $X$, which represent the projection of this subsystem's trajectories in its corresponding reduced subspace.
• Figure 5: Propagation of an ideal Airby beam (a) and a finite-energy Airy beam (b). The contour plots represent how the transverse intensity distribution varies along the longitudinal coordinate, both given in physical units (i.e., following data from the experiment sanz:JOSAA:2022). Superimposed to the contour plots, solid lines represent Bohmian trajectories illustrating the beam propagation. Dashed lines, on the other hand, indicate the position of the nodal point as a function of $z$ in the ideal case (a), which serve to better understand the kinks observed in the trajectories associated with the finite-energy Airy beam (b). These trajectories have also been computed using Eq. (\ref{['eq45']}), although it is clear that, in regions where the intensity vanishes, there cannot be trajectories. Finally, the black solid line represents the trajectory that starts with initial condition at the position where the initial beam reaches its main intensity maximum. Concerning the plots below and on top each contour plot, they represent, respectively, the initial intensity distribution and the final one at $z=30$ cm.