Wave propagation algorithm is a common method to study electromagnetic field propagation. Quantum wave propagation algorithm is an improved algorithm based on wave propagation algorithm to calculate the evolution of schrodinger equation in quantum mechanics. This paper mainly introduces the general idea of quantum wave propagation algorithm, and presents the general analysis process by taking one-dimensional infinite deep potential well, one-dimensional square potential barrier and time-dependent Schrodinger equation as examples. By comparing with the strict analytical solution of Schrodinger equation, the evolution image of probability wave function of micro particles with time is given, which verifies the correctness and effectiveness of the algorithm. The programming of quantum wave algorithm is not complicated, and has the characteristics of simple structure and strong practicability.
Wave propagation algorithm is a common method to study electromagnetic field propagation. Quantum wave propagation algorithm is an improved algorithm based on wave propagation algorithm to calculate the evolution of schrodinger equation in quantum mechanics. This paper mainly introduces the general idea of quantum wave propagation algorithm, and presents the general analysis process by taking one-dimensional infinite deep potential well, one-dimensional square potential barrier and time-dependent Schrodinger equation as examples. By comparing with the strict analytical solution of Schrodinger equation, the evolution image of probability wave function of micro particles with time is given, which verifies the correctness and effectiveness of the algorithm. The programming of quantum wave algorithm is not complicated, and has the characteristics of simple structure and strong practicability.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.
The purpose of this article is to deeply study the evolution process and measurement behavior of wave function in quantum mechanics by computer simulation. Based on Schrodinger equation and quantum measurement postulate, a set of numerical simulation framework is constructed, and typical models such as one-dimensional infinite potential well, simple harmonic oscillator, double-slit interference and two-level system are simulated by Crank-Nicolson method. Probability sampling mechanism is introduced in the simulation process to reproduce the wave function collapse caused by measurement, and the key results in the evolution process are shown. It includes the change of maximum probability density at different time steps (0.78 → 0.63 → 0.78), the exchange of kinetic energy and potential energy under the condition of energy conservation, the distribution characteristics of measurement positions in the double-slit interference experiment and the statistical frequency of the measurement results of the two-level system (0.503:0.497). The research shows that the designed simulation method can effectively reproduce the dynamic evolution law of quantum system and the statistical characteristics of measurement, and provide a visual auxiliary tool for understanding quantum phenomena.
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.
Quantum tunneling is an important model in elementary quantum mechanics. However, since solution of its wave function involves transcendental equation, many elementary quantum mechanics textbooks often focus on the explanation of transmission coefficient, and rarely mention the evolution of wave function. Although this process is mentioned in some textbooks, it is often solved by graphic method, which is not conducive to beginners' understanding of the process. In this paper, a numerical calculation method for beginners is proposed to solve this problem. According to Dirac's vectorization idea, the state vectors and operators in the Schrodinger equation are discretized in the form of vectors and matrices, respectively, and the time-dependent evolution of wave function tunneling under several potential barriers is solved by Julia numerically.
In this paper, the dynamic evolution process of Gaussian wave packet which describes a microscopic particle in one-dimensional and two-dimensional potential barriers are numerically studied by virtue of the split operator method. Moreover, the expected values of coordinates, momentum and Hamiltonian operators of the particle are calculated, and the Heisenberg uncertainty relationship is verified throughout the quantum tunneling process. Specifically, the time evolution operator is split into two parts which independently include the contributions of kinetic energy and potential energy. The wave function between coordinate and momentum representations is converted arbitrarily by utilizing the fast Fourier transform, and the numerical solution of the time-dependent Schrödinger equation is achieved. Accordingly, the results indicate that if the potential barrier is time independent, the time evolution of average total energy of the system is conserved on a large scale, but not strictly conserved on a small scale. When microscopic particle tunnels through a potential barrier, the total energy fluctuates rapidly. While leaving the potential barrier, it restores to the total energy value before quantum tunneling. Furthermore, in a one-dimensional double well potential, Gaussian wave packet oscillates back and forth within the double potential well, frequently crossing the middle potential barrier. Then the evolution of average energy oscillates rapidly and periodically with small amplitude. For the two-dimensional potential barrier, the reflection direction of the wave packet may not be the same as the incident direction, and after a period of time, particle mainly appears on both sides of the potential barrier. As a consequence, this paper presents a numerical foundation for the visualization research of quantum mechanics teaching and the in-depth exploration of quantum tunneling dynamics, deepening the understanding of wave-particle duality for the microscopic particle and Heisenberg's uncertainty principle.
A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.