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Quantum Tunneling Times and the Hartman Effect in Polymer Quantization

Siriburana, Thanakorn

Published Dec 2, 2025• 0 citations
Publisher

Abstract

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 $\mu_0$. 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 $\tau_w$ for comparison with standard quantum mechanics. The calculation results show that when the particle energy is lower than the potential barrier $E < W_0$, the tunneling time $\tau_w(n)$ 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 $\mu_0$ affects the saturation time but does not change the qualitative nature of this behavior. In the case of $E > W_0$, 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).

DOI
10.5281/zenodo.17785251

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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).
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.
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.
In this paper we study the tunneling using a background independent (polymer) quantization scheme. We show that at low energies, for the tunneling through a single potential barrier, the polymer transmission coefficient and the polymer tunneling time converge to its quantum-mechanical counterparts in a clear fashion. As the energy approaches the maximum these polymer quantities abruptly decrease to zero. We use the transfer matrix method to study the tunneling through a series of identical potential barriers. We obtain that the transmission coefficients (polymer and quantum-mechanical) behave qualitatively in a similar manner, as expected. Finally we show that the polymer tunneling time exhibits anomalous peaks compared with the standard result. Numerical results are also presented.
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.
The time a quantum particle spends traversing a classically forbidden region is among the most persistently contested observables in quantum theory, its definition splintering across the Wigner phase time, the Büttiker–Landauer clock, and complex-time path-integral constructions. The debate has centered almost exclusively on a single number—the mean delay—and on the Hartman effect, whereby that mean saturates to a width-independent constant for an opaque barrier, suggesting an unbounded and apparently superluminal traversal speed. Yet an attoclock measures not a number but a distribution: the observable imprinted on the photoelectron spectrum is a temporal spread, not a delay. The narrow-band limit of this variance is fixed by the time-energy uncertainty relation alone, Var(τ)→ℏ2/4ΔE2\text{Var}(\tau) \to \hbar^2/4\Delta E^2Var(τ)→ℏ2/4ΔE2, a general property of any finite-bandwidth wave packet. What has not been shown is that this floor survives intact through a barrier deep in the Hartman-opaque regime, where the mean delay has already saturated and where naive intuition expects the pulse to broaden. We prove, for the rectangular barrier, that the transmitted variance remains locked onto this incident-packet value independently of the barrier width ddd—a spectral-pinning theorem. Equivalently, the dimensionless product Var(τ)ΔE2\text{Var}(\tau)\Delta E^2Var(τ)ΔE2 collapses onto a single universal plateau shared by all bandwidths, a data collapse confirmed numerically to better than 0.5% across two decades of opacity and verified against the predicted inverse-square bandwidth scaling Varnarrow/Varbroad=(ΔEbroad/ΔEnarrow)2\text{Var}{\text{narrow}}/\text{Var}{\text{broad}} = (\Delta E_{\text{broad}}/\Delta E_{\text{narrow}})^2Varnarrow/Varbroad=(ΔEbroad​/ΔEnarrow​)2 to within a few percent, the small residual excess (28.56 versus 25) being itself a quantitative signature of barrier-induced reshaping. The genuine new result is the finite-bandwidth correction responsible for this excess: the opaque barrier acts as a coherent spectral filter—a tunneling analogue of the gain-narrowing mechanism familiar from ultrafast optics—that reweights the transmitted band and causes the variance to decrease weakly with opacity [log-log exponent α≈+0.070\alpha \approx +0.070α≈+0.070 in the convention Var∝(κd)−α\text{Var} \propto (\kappa d)^{-\alpha}Var∝(κd)−α], while the mean arrival time is pushed above the Hartman plateau. We derive the leading filter coefficient λ2\lambda_2λ2​ in closed form, verify it against an independent, exactly solvable smooth barrier (Eckart), and connect both the pinned floor and its filtering correction to experiment through the attoclock map σθ≃ωστ\sigma_\theta \simeq \omega \sigma_\tauσθ​≃ωστ​. The result recasts the tunneling-time question from the fragile arithmetic of a saturating mean to the robust, universal statistics of a spectrally pinned—and quantifiably filtered—width.
A quantum particle's traversal time through a forbidden barrier has long been debated, with definitions varying across Wigner, Büttiker–Landauer, and path-integral frameworks. Prior work focused on the mean delay and the Hartman effect—where the average time becomes constant for opaque barriers, implying superluminal speed. This study shifts focus from the mean to the variance of arrival time. For a finite-bandwidth wave packet, we prove that in the narrow-band limit, the transmitted arrival-time variance becomes independent of barrier width and equals that of the incident packet—a spectral-pinning effect. This universal plateau, confirmed numerically within 0.5% accuracy, overturns the expectation that opaque barriers broaden pulses. Instead, the width is set by the incident spectrum via time-energy uncertainty. Beyond this limit, the barrier acts as a spectral filter, slightly narrowing the variance with opacity while shifting the mean above the Hartman plateau. These predictions are experimentally testable via attoclock measurements, reframing tunneling time as a robust statistical property rather than a problematic mean value.