A method is presented for solving the one-dimensional Schroedinger equation. The method gives the wave function in the form of an infinite series. The first term of the series is the WBK approximation. Each of the higher terms in turn represents the reflections generated by the preceding term. The expansion is obtained by approximating the potential by a "staircase" potential, which is constant over short regions and changes its value discontinuously between these regions. The method is applied to periodic potentials. By taking the WBK approximation and the first reflection term into account, the zones of the periodic potential can be found in terms of a coefficient which gives the probability that a particle will be reflected in passing through one cell of the periodic potential.
A method is presented for solving the one-dimensional Schroedinger equation. The method gives the wave function in the form of an infinite series. The first term of the series is the WBK approximation. Each of the higher terms in turn represents the reflections generated by the preceding term. The expansion is obtained by approximating the potential by a "staircase" potential, which is constant over short regions and changes its value discontinuously between these regions. The method is applied to periodic potentials. By taking the WBK approximation and the first reflection term into account, the zones of the periodic potential can be found in terms of a coefficient which gives the probability that a particle will be reflected in passing through one cell of the periodic potential.
A method, based on the use of Milne's function w, for the numerical determination of quantum-mechanical reflection coefficients in one-dimensional problems to any preassigned degree of accuracy is outlined. The case in which a "barrier" is both preceded and followed by a field-free space is considered in detail, sections 2-4, and a numerical example is worked out, section 5. The procedure in the case in which a "barrier" is preceded by a field-free space and is followed by a potential for which Schroedinger's equation can be solved analytically is outlined, sections 6-7, and the special case in which a "barrier" is followed by a uniform field is considered in some detail, section 8. The case in which a "barrier" is both preceded and followed by potentials for which Schroedinger's equation can be solved analytically is mentioned, section 8. Formulas are given for the evaluation in terms of $\ensuremath{\psi}$ and its first derivative, of the respective densities of the dextral and of the sinistral current flowing past any point x in regions where the total energy is greater than the potential energy, section 6. A procedure is given for finding solutions representing unidirectional beams at infinity, section 6.
This article presents a simple, yet general and exact analytic-numerical method of solving one-dimensional quantum mechanics problems. It is based on the concept of dividing an arbitrary potential into convenient segments and analyzing the multiple reflection of a plane matter wave between neighboring potential walls. Amplitude reflection and transmission coefficients of each segment are the basis for the analysis. A simple iterative technique for calculating them by a square barrier approximation is given. This method is applicable to various potential barriers, wells, and periodic structures including continuous variations of potential energy and particle effective mass.
A recurrence method for analytical and numerical evaluation of tunneling, transmission, and reflection amplitudes is developed. As the first step, a rule for composition of two arbitrary scatterers separated by a region of constant potential is obtained. Transmission and reflection amplitudes for this double-barrier potential are expressed in terms of transmission and reflection amplitudes for its subparts. As the length of the constant-potential region can be arbitrary and the subparts of a potential may, in turn, be arbitrary segmented potentials, one obtains formulas recurerence formulas which express the scattering amplitudes for the arbitrary segmented potential via the scattering amplitudes for the subparts into which the complete potential can be divided. The efficiency of the method is demonstrated by solving analytically the problem of scattering from locally periodic potentials. Since an arbitrary potential can be approximated by a set of infinitely narrow rectangular barriers, the recurrence formulas can be applied to any potential, giving, in the limit of zero-width segments, differential equations for transmission, and reflection amplitudes.
The Schrödinger equation is a fundamental equation in physics for describing quantum mechanical behavior. It is a partial differential equation that shows how a wave function of a physical system evolves over time. In addition, it is associated with the kinetic energy and the potential energy, both of which contribute towards the total energy. This paper explains mathematical aspect of the Schrödinger equation. The concept of the WKB approximation has also been reviewed, which is an important method in finding the approximate solutions for the wave function. In addition, the paper also shows how to derive the transmission and reflection probabilities in relation to a one-dimensional quantum problem, called a “tunneling problem”, which occurs when a classical particle passes through a region where total energy is less than its potential energy. The WKB approximation can be used as a basis for formally writing down the exact solutions. Furthermore, it gives a high accuracy for the propagation of waves with high frequency or short wavelength.
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.
This chapter solves the eigen-problems for a few simple one-dimensional systems to acquire experience in solving the Schrodinger eigen-equation. One-dimensional problems are physically interesting not only because they are simple, but also because a number of more complicated higher-dimensional problems can be reduced to the solutions of equations in one dimension. For a symmetric potential, an eigenfunction of the Schrödinger equation can be always chosen to have a definite parity. The probability density of the plane wave solution is a constant, independent of the spatial coordinate of the particle. The eigensolution of a particle in an infinite square well consists of an infinite number of discrete energy levels. In classical mechanics, a particle cannot penetrate a potential barrier whose height is higher than the energy of the particle.