We investigate the tunneling time of a wave packet propagating through a non-Hermitian potential Vr − iVi in space-fractional quantum mechanics. By applying the stationary phase method, we derive a closed-form expression for the tunneling time for this system. This study presents the first investigation of tunneling time at the interplay of non-Hermitian quantum mechanics and space-fractional quantum mechanics. The variation in tunneling time as the system transitions from a real to a complex potential is analyzed. We demonstrate that the tunneling time exhibits a dependence on the barrier width d in the limit d → ∞, showing the absence of the Hartman effect. A particularly striking feature of our findings is the potential manifestation of the Hartman effect for a specific combination of the absorption component Vi and the Levy index α. This behavior arises from the fact that the presence of the absorption component Vi leads to a monotonic increase in tunneling time with barrier thickness, whereas the Levy index α reduces the tunneling time. The interplay of these contrasting influences facilitates the emergence of the Hartman effect under a specific combination of Vi and the fractional parameter α.
We investigate the tunneling time of a wave packet propagating through a non-Hermitian potential Vr − iVi in space-fractional quantum mechanics. By applying the stationary phase method, we derive a closed-form expression for the tunneling time for this system. This study presents the first investigation of tunneling time at the interplay of non-Hermitian quantum mechanics and space-fractional quantum mechanics. The variation in tunneling time as the system transitions from a real to a complex potential is analyzed. We demonstrate that the tunneling time exhibits a dependence on the barrier width d in the limit d → ∞, showing the absence of the Hartman effect. A particularly striking feature of our findings is the potential manifestation of the Hartman effect for a specific combination of the absorption component Vi and the Levy index α. This behavior arises from the fact that the presence of the absorption component Vi leads to a monotonic increase in tunneling time with barrier thickness, whereas the Levy index α reduces the tunneling time. The interplay of these contrasting influences facilitates the emergence of the Hartman effect under a specific combination of Vi and the fractional parameter α.
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. 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 for comparison with standard quantum mechanics. The calculation results show that when the particle energy is lower than the potential barrier E<W0, the tunneling time 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 affects the saturation time but does not change the qualitative nature of this behavior. In the case of E>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).
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.
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 reexamine the Salecker-Wigner-Peres clock formalism and show that it can be directly applied to the phenomenon of tunneling. Then we apply this formalism to the determination of the tunneling time of a nonrelativistic wave packet, sharply concentrated around a tunneling energy, incident on a symmetric double-barrier potential. In order to deepen the discussion about the generalized Hartman effect, we consider the case in which the clock runs only when the particle can be found inside the region between the barriers and show that, whenever the probability to find the particle in this region is non-negligible, the corresponding time (which in this case turns out to be a dwell time) increases with barrier spacing.
Abstract The Hartman effect refers to the rather paradoxical result that the time spent by a quantum mechanical particle or a photon to tunnel through an opaque potential barrier becomes independent of barrier width for long barriers. Such an effect, which has been observed in different physical settings, raised a lively debate and some controversies, owing to the correct definition and interpretation of tunneling times and the apparent superluminal transmission. A rather open question is whether (and under which conditions) the Hartman effect persists for inelastic scattering, that is, when the potential becomes non‐Hermitian and the scattering matrix is not unitary. Here, tunneling through a heterojunction barrier in the tight‐binding picture is considered, where the barrier consists of a generally non‐Hermitian finite‐sized lattice attached to two semi‐infinite nearest‐neighbor Hermitian lattice leads. A simple and general condition is derived for the persistence of the Hartman effect in non‐Hermitian barriers, showing that it can be found rather generally when non‐Hermiticity arises from nonreciprocal couplings, that is, when the barrier displays the non‐Hermitian skin effect, without any special symmetry in the system.
In quantum mechanics, the singularity of the potential is usually removed at first, the desired values are computed for the regularized potential, and then an inverse limit transition is made in the resulting expressions. In this paper, the regularization method is used in the quantum tunneling of a power-law potential of the form V(x)=V0/∣x∣α . Тhe reverse limit transition is possible for the range of power 0<α<1 . The tunneling and reflection time of the wave packet is computed. It is shown that the McColl-Hartman paradox, well-known for ordinary potential barriers, is also valid for the moderately singular potential. The mathematical structure of the uncommon formation of paradox is presented.