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Physical Review A

Reflection above potential steps

Robin Côté, Hanno Friedrich, Johannes Trost

Published Sep 1, 1997• 87 citations
Publisher

Abstract

Classically forbidden reflection by a potential step is analyzed by matching WKB waves to solutions of a Schr\"odinger equation involving the tail of the potential on the up side of the step. Analytic expressions for the leading deviation of the reflection probability from unity and the phase of the reflection amplitude at the top of the step are derived for potentials decaying as an inverse power of the coordinate $V(x)\ensuremath{\propto}\ensuremath{-}{1/x}^{\ensuremath{\alpha}}, \ensuremath{\alpha}>2$.

DOI
10.1103/physreva.56.1781

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Physical Review Letters•Jun 24, 1996•69 citations
Classically forbidden reflection by a potential step is analyzed by matching WKB waves to solutions of a Schr"odinger equation involving the tail of the potential on the up side of the step. Analytic expressions for the leading deviation of the reflection probability from unity and the phase of the reflection amplitude at the top of the step are derived for potentials decaying as an inverse power of the coordinate V(x)\ensuremath∝\ensuremath−1/x\ensuremathα,\ensuremathα>2V(x)\ensuremath{\propto}\ensuremath{-}{1/x}^{\ensuremath{\alpha}}, \ensuremath{\alpha}>2V(x)\ensuremath∝\ensuremath−1/x\ensuremathα,\ensuremathα>2.
The reflection and transmission of wave functions at a potential step is a well-known issue in a textbook of quantum mechanics. We studied the reflection and transmission characteristics analytically when the potential step is moving at a constant velocity vvv in the same direction as an incident wave function by means of solving the time-dependent Schrödinger equation. As for an infinite potential step, it is known that group velocity is the same as the moving velocity of the potential step. We found two interesting results when the potential step has a finite height of V0V_0V0​. The transmission occurs when the kinetic energy of incident wave function is larger than the effective potential hight of (m2v+V0)2(\sqrt{\frac{m}{2}}v + \sqrt{V_0})^2(2m​​v+V0​​)2. The other result is that the reflectivity depends on xxx, which derives from the interference between the incident and the reflected wave functions.
We investigate tunneling through a barrier and reflection over a step potential in the limit of long waves, i.e., near the base of the barrier, or for barriers with a small action integral over the forbidden region, and at the top of the ramp where the usual semiclassical approximation breaks down systematically. General approximations of the probability of transmission and reflection are given and compared to competing ones showing that the present approach based on the comparison method can give far better results.
In this paper, we calculate the transmission and reflection amplitudes of wave functions for different potentials such as the delta function, the rectangular barrier, the Eckart potential, and the Hulthen potential. We describe the relationship between these amplitudes and compute the reflection resonances between each potential. We describe the transmission and reflection probabilities using the WKB formula and compare the results with ones obtained from matching the boundary conditions. Furthermore, we use a two by two transfer matrix to calculate a rigorous bound on the transmission and reflection probabilities.
This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a downward potential step. In contrast, classical particles get reflected only at upward steps. The conditions for this effect are that the wave length is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. This phenomenon is suggested by non-normalizable solutions to the time-independent Schrödinger equation, and we present evidence, numerical and mathematical, that it is also indeed predicted by the time-dependent Schrödinger equation. Furthermore, this paradoxical reflection effect suggests, and we confirm mathematically, that a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau. PACS: 03.65.-w, 03.65.Nk, 01.30.Rr. Key words: Schrödinger equation; potential step; confining potential; reflection and transmission coefficients. 1
We discuss a phenomenon of elementary quantum mechanics that is counterintuitive, non-classical, and apparently not widely known: the reflection of a particle at a downward potential step. In contrast, classically, particles are reflected only at upward steps. The conditions for this effect are that the wavelength is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. The phenomenon is suggested by non-normalizable solutions to the time-independent Schrödinger equation. We present numerical and mathematical evidence that it is also predicted by the time-dependent Schrödinger equation. The paradoxical reflection effect suggests and we confirm mathematically that a particle can be trapped for a long time (though not indefinitely) in a region surrounded by downward potential steps, that is, on a plateau.
We study the WKB method away from the short-wave limit. Incorporating the correct phase loss due to reflection at a classical turning point, in place of the usual choice $\ensuremath{\pi}/2$, can greatly improve the accuracy of the WKB wave function in the classically allowed region. For a repulsive 1/x2{1/x}^{2}1/x2 potential this leads to far more accurate WKB wave functions that the usual Langer modification of the potential.