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.
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.
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 article provides a comprehensive review of the quantum tunneling effect, one of the most important phenomena in quantum mechanics. According to the laws of classical physics, a particle cannot overcome a potential barrier if it has insufficient energy. However, in quantum mechanics, due to the wave nature of the particle, the probability of penetrating the barrier is non-zero. This work analyzes in detail the theoretical foundations of the tunneling effect, its description via the Schrödinger equation, probability amplitudes, boundary conditions for wave functions, and their physical meaning. Furthermore, the application of this effect in semiconductor physics, nuclear physics, scanning tunneling microscopy, and nanotechnology is discussed. Based on the work of Kazakh scientists, the place and significance of the tunneling effect in modern physics are determined. The research results provide a deeper understanding of quantum phenomena and promote the development of their practical applications.
The working principle of a large number of modern scientific equipment has been applied to the quantum tunneling effect. Although the explanations for the relevant principles of this effect are clarified, the calculation process is very complex and difficult to understand. Previous studies combine specific applications without theoretical model references. This study adopts the method of using the time-independent Schrodinger equation to solve the wave function, which is more convenient to obtain the expression of transmission probability. Then, one discusses the relationship between transmission probability and potential barrier width, potential barrier spacing, and incident particle energy. The experimental results show that the transmission probability decreases exponentially from 1 to 0 as the potential barrier width gradually increases. As the potential barrier width increases, the transmission probability exhibits a clear periodic oscillation relationship, the oscillation period remains unchanged, and this paper derives the relationship between the oscillation period and the potential barrier width. As the incident energy of particles gradually increases, the transmission probability exhibits periodic changes, and the minimum value of transmission probability increases with the periodic changes. Based on the analysis, the smaller the potential barrier width, the greater the incident energy, and the more likely quantum tunneling occurs.
Many solutions have been propounded for Schrodinger’s equation. The application of these solutions is restricted to a few aspects of physics. One of the methods used for solving the Schrodinger equation is the Finite Difference Time Domain (FDTD) method [1]. It was used to solve numerically the Schrodinger equation for ground state and excited state eigenvalues, and eigenfunctions for a single particle in one-, two-, and three-dimensional potential wells. In the past, analytical methods which have been used to solve the Schrodinger equation are: supersymmetry [1], Nikiforov Uvarov method (NU) [3], asymptotic iteration method (AIM) [4,5], hyperviria perturbation [6], exact quantization rule (EQR) [7], perturbative formalism [8], wave function ansatz method [9], and path integral [10]. In all these methods highlighted above, none has been incorporated with the Bloch NMR to solve nuclear spin among other problems in physics and chemistry. The solution of the Schrodinger equation using the mathematical modeling of Bloch NMR has wide range of applications in many fields of physics and chemistry. It can be used for checking and improving models developed to study quantum mechanical systems, improve on the effective analysis of biological systems, analyzing spin transport, nuclear spin dynamics, improvement of the numerical methods and so on. In Sec. 2, the solution of the Schrodinger is given. In Sec. 3, the solution was validated by applying it to the harmonic solution. The energy
The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.