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Paradoxical Reflection in Quantum Mechanics

Pedro L. Garrido, Sheldon Goldstein, Jani Lukkarinen, Roderich Tumulka

Published Jan 1, 2008• 17 citations
OpenAlex

Abstract

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

Related papers

Paradoxical reflection in quantum mechanics

Pedro L. Garrido, Sheldon Goldstein, Jani Lukkarinen +1 more

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.

arXiv:0808.0610•American Journal of Physics•Nov 29, 2011•20 citations

Quantum reflection in the linearly downward potential

N Chamnan, Boonlit Krunavakarn

In this work, the motion of a particle in one dimension under the influence of the linearly downward potential well is studied within the context of the non-relativistic quantum mechanics. The attention is paid on the paradoxical phenomenon of the reflection of a particle that is in contrast between classical and quantum physics. Classically, the reflection effect occurs only at a potential barrier. To demonstrate such counter-intuitive phenomenon, the Schrödinger equation is solved to obtain the reflection coefficient in the scattering state by considering an incident particle that is represented by a monochromatic plane wave having an energy E > 0, propagates freely from left to right, pass through the potential well. The continuity conditions at boundaries give the desired result that is expressed in terms of the Airy functions which depends on the incident energy E, the strength jV 0 j and the range L of the well. The value of the reflection coefficient R lies in the interval 0 < R < 1, and its behavior is the decreasing function with respect to the range L.

Journal of Physics Conference Series•Sep 1, 2017•1 citation

Analyses of reflection and transmission at moving potential step

Samura, Toshiharu, Masato Ohmukai

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 $v$ 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 $V_0$. The transmission occurs when the kinetic energy of incident wave function is larger than the effective potential hight of $(\sqrt{\frac{m}{2}}v + \sqrt{V_0})^2$. The other result is that the reflectivity depends on $x$, which derives from the interference between the incident and the reflected wave functions.

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Time-dependent versus time-independent probabilities of transmission and reflection of a quantum particle incident upon potentials with large changes

Mark R. A. Shegelski, Kevin Malmgren

We investigate the transmission and reflection of a quantum particle incident upon a step potential increase, a step potential decrease, a square well, and a square barrier, all well studied in undergraduate quantum mechanics. We are especially interested in the extreme where the change in the potential is arbitrarily large, but with the difference in the energy of the particle and the potential held fixed, if possible. We obtain the probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation. In the time-dependent case, we have the particle initially in a Gaussian wave packet or a similar localized state. We obtain results that fall into three categories: results that are not surprising, results where time-dependent and time-independent agree surprisingly well, and results that are very different. We discuss the unexpected results. Our work may be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.

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Time-dependent versus time-independent probabilities of transmission and reflection of a quantum particle incident upon a step potential and a square potential well

Mark R. A. Shegelski, Kevin Malmgren, Logan Salayka-Ladouceur

We investigate the transmission and reflection of a quantum particle incident upon a step potential decrease and a square well. The probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation are in excellent agreement. We explain why the probabilities agree so well. In doing so, we make use of an exact analytical expression for the square well for time-dependent transmission and reflection, which reveals additional interesting and unexpected results. One such result is that transmission of a wave packet can occur with the probability of transmission depending weakly on the initial spread of the packet. The explanations and the additional results will be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.

Canadian Journal of Physics•Oct 2, 2015•1 citation

Quantum and classical limits in a potential step

CLAUDIO O. DIB, Oscar F. Orellana

Abstract We have often found among many of our students and colleagues the common idea that the mathematical expression for a physical quantity that is essentially of quantum nature must contain a dependence on ℏ. Conversely, a phenomenon described by classical physics should contain no explicit reference to ℏ. However, the problem of a particle encountering a discontinuous potential step, which is one of the simplest examples in quantum mechanics, contradicts this common thought: even when the particle carries enough kinetic energy to go across the step, the resulting expression for the reflection probability is non-zero—a purely quantum phenomenon—and yet it contains no reference to ℏ. We show that the absence of ℏ in this purely quantum expression is due to the idealised limit in which the potential rises sharply at a single position, thus losing any reference to a length dimension in the problem. To address the correct classical limit of the phenomenon we first regularise the discontinuity in the potential by taking a continuous linearly rising function over a distance a . We show that the correct classical limit, which is when no reflection occurs, is obtained for large values of a , not for vanishing a . The latter actually corresponds to a purely quantum case, albeit in a limiting case of a discontinuously abrupt change in the potential at a single point in space.

European Journal of Physics•Apr 20, 2017•2 citations
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.
In this work, the motion of a particle in one dimension under the influence of the linearly downward potential well is studied within the context of the non-relativistic quantum mechanics. The attention is paid on the paradoxical phenomenon of the reflection of a particle that is in contrast between classical and quantum physics. Classically, the reflection effect occurs only at a potential barrier. To demonstrate such counter-intuitive phenomenon, the Schrödinger equation is solved to obtain the reflection coefficient in the scattering state by considering an incident particle that is represented by a monochromatic plane wave having an energy E > 0, propagates freely from left to right, pass through the potential well. The continuity conditions at boundaries give the desired result that is expressed in terms of the Airy functions which depends on the incident energy E, the strength jV 0 j and the range L of the well. The value of the reflection coefficient R lies in the interval 0 < R < 1, and its behavior is the decreasing function with respect to the range L.
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 the transmission and reflection of a quantum particle incident upon a step potential increase, a step potential decrease, a square well, and a square barrier, all well studied in undergraduate quantum mechanics. We are especially interested in the extreme where the change in the potential is arbitrarily large, but with the difference in the energy of the particle and the potential held fixed, if possible. We obtain the probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation. In the time-dependent case, we have the particle initially in a Gaussian wave packet or a similar localized state. We obtain results that fall into three categories: results that are not surprising, results where time-dependent and time-independent agree surprisingly well, and results that are very different. We discuss the unexpected results. Our work may be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.
We investigate the transmission and reflection of a quantum particle incident upon a step potential decrease and a square well. The probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation are in excellent agreement. We explain why the probabilities agree so well. In doing so, we make use of an exact analytical expression for the square well for time-dependent transmission and reflection, which reveals additional interesting and unexpected results. One such result is that transmission of a wave packet can occur with the probability of transmission depending weakly on the initial spread of the packet. The explanations and the additional results will be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.
Abstract We have often found among many of our students and colleagues the common idea that the mathematical expression for a physical quantity that is essentially of quantum nature must contain a dependence on ℏ. Conversely, a phenomenon described by classical physics should contain no explicit reference to ℏ. However, the problem of a particle encountering a discontinuous potential step, which is one of the simplest examples in quantum mechanics, contradicts this common thought: even when the particle carries enough kinetic energy to go across the step, the resulting expression for the reflection probability is non-zero—a purely quantum phenomenon—and yet it contains no reference to ℏ. We show that the absence of ℏ in this purely quantum expression is due to the idealised limit in which the potential rises sharply at a single position, thus losing any reference to a length dimension in the problem. To address the correct classical limit of the phenomenon we first regularise the discontinuity in the potential by taking a continuous linearly rising function over a distance a . We show that the correct classical limit, which is when no reflection occurs, is obtained for large values of a , not for vanishing a . The latter actually corresponds to a purely quantum case, albeit in a limiting case of a discontinuously abrupt change in the potential at a single point in space.