Quantum Tunneling Times and the Hartman Effect in Polymer Quantization

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).