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arXiv:1408.6603 · Revista Mexicana de Física

Tunneling of Polymer Particles

A. Martín-Ruiz, E. Chan-López, Adrián Carbajal-Domı́nguez, Bernal, J.

Published Aug 28, 2014• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1408.6603

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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.
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).
Numerical simulations of quantum tunneling with time-dependent barriers show that there is a resonance, with a marked increase in the transmission coefficient. For a raised cosine potential, and for low energies with square barriers, the resonance occurs when a modulation quantum can take a tunneling particle to the top of the barrier. For energies near the top of a square barrier the resonance may be understood by hypothesizing that a tunneling particle may travel from end to end of the barrier until it is ultimately either transmitted or reflected.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.