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Hartman effect and bohmian mechanics

Said Lantigua

Published Oct 14, 2024• 0 citations
OpenAlex

Abstract

In this work, we present a study of the intriguing Hartman Effect, through two approaches. The first one is a relativistic approach of quantum theory using the Dirac equation, and the second one is a geometrodynamic approach using Bohmian mechanics. Therefore, we first developed a mathematical model from the Dirac equation in an alternative representation, which allowed us to deduce a group (or tunneling) time as the sum of the residence times inside the potential barrier for particles with spin up and spin down, and the self-interaction time between the incident and reflected wave functions of the spin-up particles. In other words, we obtained a model that allows us to observe the influence of spin on the tunneling time, in the absence of an external magnetic field, which offers a less satisfactory explanation of the Hartman Effect in terms of group time. Subsequently, through a geometrodynamic approach of Bohmian mechanics, we constructed a general solution for the problem of scattering of incident particles on a constant potential barrier under the hypothesis that such particles undergo quantum tunneling along geodesic trajectories in an Alcubierre-type spacetime. Consequently, from the solution, we determined the mathematical expressions for the quantum potential, momentum, position, and tunneling time in terms of the spacetime geometry. That is, we obtained a mathematical model that allows us to explain the Hartman Effect as a consequence of the spacetime distortion generated by the quantum potential inside the barrier.

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In this work, we present a study of the intriguing Hartman Effect, through two approaches. The first one is a relativistic approach of quantum theory using the Dirac equation, and the second one is a geometrodynamic approach using Bohmian mechanics. Therefore, we first developed a mathematical model from the Dirac equation in an alternative representation, which allowed us to deduce a group (or tunneling) time as the sum of the residence times inside the potential barrier for particles with spin up and spin down, and the self-interaction time between the incident and reflected wave functions of the spin-up particles. In other words, we obtained a model that allows us to observe the influence of spin on the tunneling time, in the absence of an external magnetic field, which offers a less satisfactory explanation of the Hartman Effect in terms of group time. Subsequently, through a geometrodynamic approach of Bohmian mechanics, we constructed a general solution for the problem of scattering of incident particles on a constant potential barrier under the hypothesis that such particles undergo quantum tunneling along geodesic trajectories in an Alcubierre-type spacetime. Consequently, from the solution, we determined the mathematical expressions for the quantum potential, momentum, position, and tunneling time in terms of the spacetime geometry. That is, we obtained a mathematical model that allows us to explain the Hartman Effect as a consequence of the spacetime distortion generated by the quantum potential inside the barrier.
For the relativistic tunneling effect described using Dirac's equation, Winful et al. [Phys. Rev. A 70, 052112 (2004)] presented the deduction of a general result that allows for the determination of the phase time (group delay) as the sum of the particle dwell time inside a potential barrier and of the self-interference delay associated with the incident and reflected wave functions interaction. In this paper, a mathematical model is derived through a construction analogous to the proposal mentioned above, but based on an alternative representation for Dirac's equation. This representation is similar to the one introduced by Ajaib [Found. Phys. 45, 1586 (2015)]. Thus, from the application of this model in the study of the tunneling effect in the absence of an external magnetic field, the influence of spin on the tunneling times is described. More specifically, the tunneling time is obtained as the sum of the dwell times inside the potential barrier for particles with spin up and spin down and the self-interaction time associated with the incident and reflected wave functions for particles with spin up.
We study the relativistic quantum mechanical problem of a Dirac particle tunneling through two successive electrostatic barriers. Our aim is to study the emergence of the so-called generalized Hartman effect, an effect observed in the context of nonrelativistic tunneling as well as in its electromagnetic counterparts and which is often associated with the possibility of superluminal velocities in the tunneling process. We discuss the behavior of both the phase (or group) tunneling time and the dwell time, and show that in the limit of opaque barriers the relativistic theory also allows the emergence of the generalized Hartman effect. We compare our results with the nonrelativistic ones and discuss their interpretation.
This research aims to study the behavior of quantum tunneling within the framework of polymer quantization, where spacetime has a discrete structure with a fundamental length μ0\mu_0μ0​. From the Hamiltonian difference equation, we obtained analytical wave functions in each region of the potential barrier and numerically calculated the transmission coefficient and Wigner time delay τw\tau_wτw​ for comparison with standard quantum mechanics. The calculation results show that when the particle energy is lower than the potential barrier E<W0E < W_0E<W0​, the tunneling time τw(n)\tau_w(n)τw​(n) increases rapidly within the narrow barrier and saturates when the width is sufficiently large, confirming that the Hartman effect phenomenon persists even when the spacetime structure is discontinuous. The value of μ0\mu_0μ0​ affects the saturation time but does not change the qualitative nature of this behavior. In the case of E>W0E > W_0E>W0​, the time increases continuously with width, which is consistent with standard quantum mechanics. This result suggests that the semi-classical properties of potential barrier penetration, particularly the Hartman effect, are stable against modifications to spacetime structure and provide important insights into understanding the behavior of quantum systems within a framework inspired by Loop Quantum Gravity (LQG).
The phenomenon of quantum tunneling challenges classical intuitions about particle behavior and offers profound insights into the nature of quantum entities. This article examines tunneling through the lens of an effect predicted by Thomas E. Hartman in 1962: the tunneling time-delay becomes independent of the barrier width as the latter increases, implying superluminal effective velocities that appear to conflict with relativistic constraints. Exploiting the inherent symmetries of transmitted and reflected time-delays, our analysis shows that a tunneling particle, like its reflected counterpart, completely avoids the barrier region. However, since it can be detected on the transmitted side of the barrier, the only logical conclusion is that the classical notion of spatiality is violated during the tunneling process. In other words, quantum tunneling and the associated Hartman effect strongly support the idea that quantum non-locality should be interpreted as quantum non-spatiality. The latter, according to the conceptuality interpretation of quantum mechanics, expresses the conceptual-like nature of quantum entities, which during measurement transition from an abstract state to a maximally concrete one, in full compatibility with the Hartman effect.
We show that scattering a quantum particle on a one-dimensional potential barrier as well as scattering the electromagnetic wave on a quasi-one-dimensional layered structure (both represent scattering problems with one 'source' and two 'sinks') violate the superposition principle; the role of nonlinear elements is played here by the potential barrier and the layered structure, splitting the incident (probability and electromagnetic) wave into two parts (transmitted and reflected). This explains why all attempts to solve the tunneling time problem within the framework of the standard (linear) models of these processes, both in quantum mechanics and in classical electrodynamics, have been unsuccessful. We revise the traditional formulation of the superposition principle, present a new (nonlinear) wave model, by the example of the quantum-mechanical scattering process, and show that concepts of the tunneling time developed on its basis are free from the Hartman paradox.
I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.