The criterion for the validity of the JWKB approximation is used to derive a first order differential equation for a negative imaginary absorbing potential, −iε(r). The resulting absorbing potential switches on as rapidly as possible with increasing r without causing too much reflection, and it has a second order pole at the end of the absorbing region which eliminates all transmission. It is completely specified by a single physical parameter: the minimum energy Emin at which absorption is required; the width of the absorbing region can be chosen to correspond to the de Broglie wavelength at this minimum energy. With this choice, the exact quantum mechanical reflection probability of the potential is <1% at E=Emin, <0.1% at E=2Emin, and <0.01% for all E⩾3Emin. Because of its lack of parameters and its convenient reflection properties, we anticipate that the new potential (and other potentials of a similar form) will find useful application in both time-dependent and time-independent quantum scattering calculations.
The criterion for the validity of the JWKB approximation is used to derive a first order differential equation for a negative imaginary absorbing potential, −iε(r). The resulting absorbing potential switches on as rapidly as possible with increasing r without causing too much reflection, and it has a second order pole at the end of the absorbing region which eliminates all transmission. It is completely specified by a single physical parameter: the minimum energy Emin at which absorption is required; the width of the absorbing region can be chosen to correspond to the de Broglie wavelength at this minimum energy. With this choice, the exact quantum mechanical reflection probability of the potential is <1% at E=Emin, <0.1% at E=2Emin, and <0.01% for all E⩾3Emin. Because of its lack of parameters and its convenient reflection properties, we anticipate that the new potential (and other potentials of a similar form) will find useful application in both time-dependent and time-independent quantum scattering calculations.
In this paper, the reflection and transmission of plane waves are examined from a complex potential. Such potentials have the property of absorbing wave packets incident on them and are used widely in time-dependent quantum scattering theory. The purpose of the study is to determine the optimal form of potential to be used for absorbing wave packets near the edges of finite grids in coordinate space. The best potentials for such purposes lead to the minimum possible transmission and reflection of the incident wave packet. The Jeffreys–Wentzel–Kramers–Brillouin (JWKB) theory is used to address this problem and a new form for the optimal complex potential is proposed. A scaled dimensionless form of the Schrödinger equation is also derived, so that the parameters of any optimized potential obtained for a particular collision energy and mass combination may be readily converted to apply to a new set of masses and energies.
Scattering by a one-dimensional potential V=V(x) which is constant outside (-a, a) (a>0) is investigated. Exact expressions for the transmission and reflection coefficients are obtained in terms of f 0 (a), f 1 (a), f' O (a) and f' 1 (a), where f 0 and f 1 are the solutions in (-a, a) satisfying the boundary conditions f 0 (-a)=1, f 0 '(-a)=0, f 1 (-a)=0 and f 1 '(-a)=1. These expressions are used to show that the first-order WKB approximation conserves particles. Numerical results are obtained from these expressions for the transmission and reflection probabilities by (i) using the WKB formulae and (ii) solving the time-independent Schrodinger equation numerically in (-a, a). The time T for the centre of a wavepacket to traverse the interval (-a, a) is also obtained in terms of f 0 (a), f 1 (a), f 0 '(a) and f 1 '(a), and some of its properties studied. In particular, it is shown that T takes the classical form in the classical limit.
We show that an attractive potential created by an evanescent wave quantum mechanically reflects an atom with nonzero probability if the atom’s incident de Broglie wavelength is larger than the decay length of the evanescent wave. The amplitude reflection coefficient is calculated by use of an analytical solution of the corresponding Schrödinger equation. We also discuss electromagnetic wave analog of the effect, the partial reflection of light at an exponentially increasing index. We interpret the quantum reflection in terms of a virtual turning point, by means of a complex stationary phase approximation. If the potential remains exponential until it becomes as deep as several hundred recoil energies, the reflection coefficient for a low-energy de Broglie wave is independent of the details of the potential far from the virtual turning point. This means that, if the evanescent wave is strong enough, interactions between the atom and the glass surface at which the evanescent wave is created can be neglected.
Abstract The reflection and transmission coefficients describe the behavior of the matter wave incident on a potential barrier. They can be expressed in terms of the probability with which the matter wave can be reflected or transmitted. The central equation accounting for the behavior of the matter wave is the Schrödinger equation. The Schrödinger equation is the second order partial differential equation. However, in a stationary state, the Schrödinger equation is reduced to the time independent Schrödinger equation. This time independent Schrödinger equation is the second order linear ordinary differential equation. Since the time independent Schrödinger equation is linear, superposition of any of the two solutions to the time independent Schrödinger equation is also a solution. In this paper, we focus on the superposition of various potentials. The reflection and transmission coefficients from the superposition of various potentials are obtained. A comparison between the exact coefficients and those obtained by the 2 × 2 transfer matrix is made. The relationship between the transmission coefficient of the superposed potential and that of each individual potential is found. The results show that the transmission coefficient obtained from the 2 × 2 transfer matrix is of a lower bound on the exact transmission coefficient.
The JWKB approximation is widely used in many problems in quantum mechanics for its simplicity and also because a closed-form solution often provides more insight than a numerical one. Unfortunately, like all approximations, the JWKB method has its limitations: Firstly, it may fail near or at a classical turning point, and secondly, it employs solutions for a potential which deviates from the actual potential W(x). In this work, a new method is proposed to avoid the limitations of the JWKB method while keeping its advantages. First, the singularity around the classical turning point is resolved not only for a linear potential as proposed by Langer, but also for any potential with a linear leading term through the turning point, and second, a solution is constructed in closed form which is an exact solution for the actual potential W(x). The new method consists in finding a substitute potential U(x) in such a way that, together with its (DELTA){U,x} it is equal to the actual W(x). The solution is then based on U(x) but actually is the solution for W(x) because W(x) = U(x) + (DELTA){U,x}. The new method is first applied to potentials with only two turning points. The eigenvalue problem is reduced to the solution of a linear equation and the energy eigenvalues are exactly given. Then, potentials with more than two turning points are studied and the eigenvalue problem is again reduced to the solution of a polynomial equation whose order depends upon the number of turning points. Then the method is applied to the hindered or restricted rotation. The energy levels are found to be associated with the N-th roots of 1 in the complex plane, i.e. on the unit circle. The method described here is of rather general nature and can be used to solve a number of problems in quantum mechanics, where a solution for the Schrodinger equation in closed form is needed. The applications provided in this work are only given as examples of the applicability and the relative simplicity of the method.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1980 .S223. Source: Dissertation Abstracts International, Volume: 41-03, Section: B, page: 1002. Thesis (Ph.D.)--University of Windsor (Canada), 1980.
In quantum mechanics, among the proposals for how to compute the probability distribution of detection time and place along a given spatial surface or in a given volume, two leading candidates involve absorbing boundary conditions (ABCs) and complex absorbing potentials (CAPs) for the Schrödinger equation. Here, we extend earlier results on how the two are related. It was previously shown that ABCs arise as a limiting case of CAPs, but for that a (reflecting) Neumann boundary condition was assumed at the end of the region with the CAP. Here, we obtain ABCs in a different limit using a Dirichlet boundary condition at the end of the CAP region, which seems more practically relevant and realizable.