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Quantum Tunneling and Hartman Effect

Ali Övgün

Published Jan 1, 2015• 0 citations
Publisher

Abstract

Thıs is my undergraduate thesis. So far I have worked on tunneling and its classi-cal, semi-classical and quantum description, then I looked some unsolved problems . Ihave learned about Hartman e ect , tunneling time, semi-classical approximations ,pathintegral and a classical description of tunneling
DOI
10.13140/rg.2.1.2979.3449

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The non-stationary solution method for a problem of tunneling of nonrelativistic particles and photons through a barrier on the basis of multiple internal re ections of wave packets in relation of barrier boundaries is submitted.The method is described and proved in solving an one-dimentional problem of tunneling of the particle through a rectangular barrier.For problems of tunneling of the particle through a spherically symmetric barrier and the photon through an one-dimensional barrier the amplitudes of transmitted and re ected wave packets in relation to the barrier, times of the tunneling and the re ection are found using of the method.Hartman's and Fletcher's eect is analysed.
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).