Hartman Effect from a Geometrodynamic Extension of Bohmian Mechanics

TL;DR

The paper introduces a geometrodynamic extension of Bohmian mechanics in which quantum tunneling is modeled as geodesic motion within an Alcubierre-type spacetime. By partitioning the problem into three regions and employing a two-energy superposition, it derives a geometry-enabled quantum potential and region-specific momenta that predict the Hartman effect: tunneling times saturate for wide barriers due to a geometric self-regulation where spacetime distortion adjusts to keep a constant traversal time. This framework reconciles superluminal tunneling with relativistic causality and links quantum tunneling to spacetime geometry, offering a unified interpretation and potential pathways to explore geometric contributions in condensed-matter and quantum-gravity analog systems. The approach provides concrete expressions for $Q$, $S$, and $P$ across regions, and demonstrates how the Alcubierre-like metric mediates tunneling dynamics without violating fundamental limits, while suggesting broader implications for topological and geometrical quantum phenomena.

Abstract

This paper develops a geometrodynamic extension of Bohmian mechanics to describe quantum tunneling through a potential barrier, treating particle trajectories as geodesics in an Alcubierre-type spacetime. The model provides analytical expressions for the quantum potential, particle dynamics, and tunneling time, explicitly linked to the underlying spacetime geometry. For narrow barriers, the tunneling time depends on the barrier width, while for sufficiently wide barriers, it saturates to a constant value-recovering the Hartman effect. This behavior arises from a geometric self-regulation mechanism, where the quantum potential dynamically adjusts the spacetime distortion to maintain a fixed tunneling time, consistent with relativistic causality despite effective superluminal propagation. The results establish a direct connection between quantum tunneling and spacetime geometry, offering a unified framework to interpret the Hartman effect. This approach naturally incorporates relativistic constraints while suggesting that similar geometric mechanisms may underlie other quantum phenomena, such as topological phases in condensed matter systems.

Figures (5)

• Figure 1: Schematic representation of the quantum tunneling phenomenon of a particle with energy $E$, incident from the left on a potential barrier of height $V_{0}$ and width $a$. Specifically, images (a), (b) and (c) are presented, where the regions of interest $I$, $II$ and $III$ are considered. In image (a), the incidence of a Bohmian particle in black and its guide wave in yellow on the potential barrier is presented in region $I$. In image (b), in region $II$ the Bohmian particle in black and its guiding wave in blue, inscribed within the hypothetical Alcubierre ball, a larger circumference with a discontinuous line, that is, the hypothetical representation of the tunnel effect as a distortion of space-time. Finally. In image (c) the transmitted Bohmian particle in black and its guide wave in red. In that sense, the regions $I$, $II$ and $III$ are considered in the construction of the general solution presented in section \ref{['sec:4']}.
• Figure 2: Graphical representation of the trajectories followed by particles incident from the left on the potential barrier.
• Figure 3: Tunneling time ($\Delta t = 3 [a^{3}/8 \mathcal{E}_{\leq}]^{1/2}$) for narrow barriers ($a \leq R$) with different energy values ($\mathcal{E}_{\leq}$).
• Figure 4: Constant tunneling time ($\Delta t = 3/n_{0}$) (Hartman effect) for wide barriers ($a > 2R$) with different scale factors ($n_{0}$).
• Figure 5: Linear relationship between $v_s/c$ and barrier width $a$. The blue area indicates the superluminal regime ($v_s/c>1$), reached when $a > c/n_0 \approx 1.4R$.