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The Journal of Chemical Physics

Reflection and transmission of waves by a complex potential—a semiclassical Jeffreys–Wentzel–Kramers–Brillouin treatment

Ágnes Vibók, Gabriel G. Balint‐Kurti

Published May 15, 1992• 144 citations

Abstract

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.

DOI: 10.1063/1.462414

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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.

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.

Wave-packet scattering from a stationary potential is significantly modified when the wave packet is subject to an external time-dependent force during the interaction. In the semiclassical limit, wave-packet motion is simply described by Newtonian equations, and the external force can, for example, cancel the potential force, making a potential barrier transparent. Here we consider wave-packet scattering from reflectionless potentials, where in general the potential becomes reflective when probed by an accelerated wave packet. In the particular case of the recently introduced class of complex Kramers-Kronig potentials we show that a broad class of time-dependent forces can be applied without inducing any scattering, while there is a breakdown of the reflectionless property when there is a broadband distribution of initial particle momentum, involving both positive and negative components.

The reflection of wave packets from a quantum well bounded by rectangular potential barriers, and which has a tunneling transmission resonance, is discussed. Numeric solutions to Schrödinger’s time-dependent equation show that the shape of the probability function ψψ* for a reflected wave packet can be quite different from that of the incident packet. Either a symmetric double hump or an asymmetric shape are possible for the reflected ψψ*, depending on the sharpness of the resonance. These shapes are explained analytically in terms of the reflection coefficient.

Numerical simulations are frequently required for quantum scattering problems and often face difficulties with finite grids and unwanted, unphysical reflections. For decades, improved complex absorbing potentials (CAPs) have been sought. Today, the rise of ultracold physics makes a solution essential as CAP errors increase at lower energies. We present a method that provides a physical, semiclassical picture of how to improve CAPs based on the behavior of classical trajectories. The method does not rely on the mathematical formalism often required by existing methods and reduces the error associated with CAP-based calculation of the low-energy scattering wavefunctions by up to several orders of magnitude relative to the standard Woods-Saxon approach, as demonstrated via the distorted-wave Born approximation. This indicates the method may be applied to the numerical simulation of collisions in the ultracold regime.

A more general treatment than has been available of the Wentzel-Brillouin-Kramers method of solving Schr\"odinger's equation for one degree of freedom is given. Wentzel's original energy-level condition is shown to be an asymptotic expression, good for large masses and large values of the quantum number. The connection between this method and that of Young is discussed. Finally the formulas are written in a form convenient for application to the calculation of energy levels from actual potential functions.