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Journal of Yunnan Normal University

The application of the Finite Difference Time Domain method in the problem of potential barrier

Qing Zhou

Published Jan 1, 2007• 0 citations

Abstract

The application of the Finite Difference Time Domain(FDTD) method in quantum-mechanical problems becomes more and more interesting.The postulate,the boundary cut-off condition and the initial trial wave function are described.Based on the FDTD method,three different potential-barrier problems including rectangular potential-barrier,deltoid potential-barrier and syntonic potential-barrier are investigated.The energy variation and the probability variation are calculated and modeled,the important phenomenon of quantum tunnel effect reappeared.It is convenient to apply the FDTD method in quantum-mechanical problems and the results are acceptable while using the boundary cut-off based on the expectation energy of the system.And it is potentially further extended to other quantum-mechanical problems.

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The application of the Finite Difference Time Domain(FDTD) method in quantum-mechanical problems becomes more and more interesting.The postulate,the boundary cut-off condition and the initial trial wave function are described.Based on the FDTD method,three different potential-barrier problems including rectangular potential-barrier,deltoid potential-barrier and syntonic potential-barrier are investigated.The energy variation and the probability variation are calculated and modeled,the important phenomenon of quantum tunnel effect reappeared.It is convenient to apply the FDTD method in quantum-mechanical problems and the results are acceptable while using the boundary cut-off based on the expectation energy of the system.And it is potentially further extended to other quantum-mechanical problems.

This paper introduced the FDTD method,the basic principles of the boundary and initial treatment selection of the trial wave function.On this basis,the matrix calculation to simulate the barrier change the energy of the system were given,the probability of changes in the quantum tunneling effect reproduce the important physical phenomena.Further demonstrates the use of FDTD method to solve the problem of quantum mechanics is to facilitate workable and systematic approach to cut the average energy of the border can be more satisfactory results.

Quantum tunneling is a phenomenon where particles have a probability of penetrating a potential barrier despite having total energy llower than the barrier height. This study analyzes the tunneling effect by solving the one-dimensional (1D) Schrodinger equation using the finite difference method to obtain the wave function evolution for various potential barrier configurations. The solution is implemented in a Graphical User Interface (GUI) MATLAB to facilitate analysis and visualization, allowing users to interactively adjust potential parameters, energy, and other conditions. Simualation results demonstratehow transmission probability depends on energy, height and width of potential barrier. This GUI provides an intiuitive tool for exploring quantun tunneling, making it valuable for both education and research in quantum physics.

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.