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Journal of Mathematical Physics

Propagator for finite range potentials: The case of reflection

Ilaria Cacciari, Paolo Moretti

Published Apr 1, 2007• 1 citation
Publisher

Abstract

Following a previous study on the transmission propagator for a finite range potential, the problem of reflection is considered. It is found that the Laplace transform of the reflection propagator can be expressed in terms of the usual Fredholm determinant Δ and of a new similar determinant Γ, containing the peculiar characteristics of reflection. As an example, an array of delta potentials is considered. Moreover, a possible application to the calculation of quantum traversal time is shown.

DOI
10.1063/1.2728518

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Following a previous study on the transmission propagator for a finite range potential, the problem of reflection is considered. It is found that the Laplace transform of the reflection propagator can be expressed in terms of the usual Fredholm determinant Δ and of a new similar determinant Γ, containing the peculiar characteristics of reflection. As an example, an array of delta potentials is considered. Moreover, a possible application to the calculation of quantum traversal time is shown.
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.
The quantum propagator and characteristic equation in the presence of a chain of δ-potentials are obtained in the rectangular, cylindrical and spherical coordinate systems. The simplicity and efficiency of the method is illustrated via examples. As an application, the characteristic equation of a quantum harmonic oscillator confined to an infinite box is obtained. The roots of the characteristic equation, determining the energy eigenvalues of the restricted oscillator, are calculated approximately and compared with the existing numerical data. Due to the similarity between Schrödinger and diffusion equations, the problem can be considered as exact solutions of the diffusion equation in the presence of a chain of δ-potentials as boundary.
Regarding the quantum mechanical transmission–reflection problem in one dimension with a potential of the form of the derivative of the Dirac delta function δ'(x) = dδ(x)/dx, Christiansen et al recently found that, depending on how δ'(x) is interpreted, there can be a resonance which leads to partial transmission. This is in contrast to the earlier consensus that such a potential allows no transmission. The δ'(x) can be regarded as the narrow-width limit of a certain function Δ'(x) of a finite range. Christiansen et al assumed a rectangular function for Δ'(x). We examine various other forms and how the resonance depends on the shape of Δ'(x). We also present some general observations related to the 'threshold anomaly'.