Elastic scattering of a wave can be quantified by a shift in the phase with respect to the phase of the incoming wave. A qualitative measure of the time during which the effect occurs is given by the Wigner time delay. The tunneling time is known to saturate with increasing tunneling barrier width (Hartman effect). Here, we analyze the elastic quantum mechanical scattering in a deformed one-dimensional Riemannian manifold, particularly with respect to the Wigner time delay and conclude on the Hartman effect. It is shown that scattering due to local curvature variations results in imperfect conduction and leads to a Wigner time delay that at low energies, is in variance with the classical time delay that is inferred from the arc length. At moderate and high energies, however, classical and quantum time delays coincide.
Elastic scattering of a wave can be quantified by a shift in the phase with respect to the phase of the incoming wave. A qualitative measure of the time during which the effect occurs is given by the Wigner time delay. The tunneling time is known to saturate with increasing tunneling barrier width (Hartman effect). Here, we analyze the elastic quantum mechanical scattering in a deformed one-dimensional Riemannian manifold, particularly with respect to the Wigner time delay and conclude on the Hartman effect. It is shown that scattering due to local curvature variations results in imperfect conduction and leads to a Wigner time delay that at low energies, is in variance with the classical time delay that is inferred from the arc length. At moderate and high energies, however, classical and quantum time delays coincide.
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).
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 develop a new quantum-mechanical approach to scattering a particle on a one-dimensional (1D) system of two identical rectangular potential barriers, which implies modelling the dynamics of its subprocesses -- transmission and reflection -- at all stages of scattering. On its basis we define, for each subprocess, the dwell time as well as the local (exact) and asymptotic (extrapolated) group times. Our concept of the asymptotic transmission group time confirms the validity of the Wigner phase time in the opaque limit, as well as the existence of the usual and generalized Hartman effects predicted on its basis. On the energy scale, this concept is valid everywhere in the high energy region as well as in the low energy region, excepting resonance points and their neighborhoods. On the contrary, the Buttiker dwell time is valid, as the transmission time, just only at the resonance points. Our concept of the transmission dwell time predicts monotonous growth of the tunneling time when the distance between the opaque barriers increases. By our approach only this time scale yields the true time spent, on average, by transmitted particles in the region occupied by the system. We explain why the asymptotic and local transmission group times cannot play this role and why the concept of {\it transmission group} velocity lies beyond the scope of special relativity. And else, all the transmission times admit only indirect measurements. Hence the unambiguous interpretation of all tunneling-time experiments is impossible when the transmission dynamics at all stages of scattering is unknown.
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.
We develop a new quantum-mechanical approach to scattering a particle on a one-dimensional (1D) system of two identical rectangular potential barriers, which implies modelling the dynamics of its subprocesses -- transmission and reflection -- at all stages of scattering. On its basis we define, for each subprocess, the dwell time as well as the local (exact) and asymptotic (extrapolated) group times. Our concept of the asymptotic transmission group time confirms the validity of the Wigner phase time in the opaque limit, as well as the existence of the usual and generalized Hartman effects predicted on its basis. On the energy scale, this concept is valid everywhere in the high energy region as well as in the low energy region, excepting resonance points and their neighborhoods. On the contrary, the Buttiker dwell time is valid, as the transmission time, just only at the resonance points. Our concept of the transmission dwell time predicts monotonous growth of the tunneling time when the distance between the opaque barriers increases. By our approach only this time scale yields the true time spent, on average, by transmitted particles in the region occupied by the system. We explain why the asymptotic and local transmission group times cannot play this role and why the concept of {\it transmission group} velocity lies beyond the scope of special relativity. And else, all the transmission times admit only indirect measurements. Hence the unambiguous interpretation of all tunneling-time experiments is impossible when the transmission dynamics at all stages of scattering is unknown.
We study the nature of tunneling phase time for various quantum mechanical structures such as networks and rings having potential barriers in their arms. We find the generic presence of the Hartman effect, with superluminal velocities as a consequence, in these systems. In quantum networks, it is possible to control the "super arrival" time in one of the arms by changing the parameters on another arm which is spatially separated from it. This is yet another quantum nonlocal effect. Negative time delays (time advancement) and "ultra Hartman effect" with negative saturation times have been observed in some parameter regimes. In the presence and absence of Aharonov-Bohm (AB) flux, quantum rings show the Hartman effect. We obtain the analytical expression for the saturated phase time. In the opaque barrier regime, this is independent of even the AB flux thereby generalizing the Hartman effect. We also briefly discuss the concept of "space collapse or space destroyer" by introducing a free space in between two barriers covering the ring. Further, we show in presence of absorption that the reflection phase time exhibits the Hartman effect in contrast to the transmission phase time.