^{1,a)}, Sheldon Goldstein

^{2,b)}, Jani Lukkarinen

^{3,c)}and Roderich Tumulka

^{4,d)}

### Abstract

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.

The authors thank the Institut des Hautes Études Scientifiques at Bures-sur-Yvette, France, where the idea for this article was conceived, for hospitality. For discussions on the topic the authors thank, in particular, Federico Bonetto (Georgia Tech), Ovidiu Costin (Ohio State University), and Herbert Spohn (TU München). The work of S. Goldstein was supported in part by NSF Grant (No. DMS-0504504). The work of J. Lukkarinen was supported by the Academy of Finland and by the Deutsche Forschungsgemeinschaft project Sp 181/19-2. The work of R. Tumulka was supported by the European Commission through its 6th Framework Programme “Structuring the European Research Area” and contract Nr. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action.

I. INTRODUCTION

II. STATIONARY ANALYSIS OF THE RECTANGULAR STEP

III. SOFT STEP

IV. WAVE PACKETS

A. Numerical simulation

B. Is it for real?

V. PARAMETER DEPENDENCE

VI. THE CLASSICAL LIMIT

VII. A PLATEAU AS A TRAP

VIII. EIGENFUNCTIONS WITH COMPLEX ENERGY

IX. WAVE PACKETS ON THE PLATEAU

X. CONCLUSIONS

### Key Topics

- Eigenvalues
- 24.0
- Reflection coefficient
- 12.0
- Wave functions
- 12.0
- Bound states
- 5.0
- Quantum mechanics
- 5.0

## Figures

A potential *V* (*x*) containing a downward step.

A potential *V* (*x*) containing a downward step.

A potential containing a soft step.

A potential containing a soft step.

Numerical simulation of the time-dependent Schrödinger equation for the hard step potential of Eq. (1). The picture shows ten snapshots of |ψ|^{2} (black lines) at different times before, during, and after passing the potential step (order: left column top to bottom, then right column top to bottom). The straight lines in the figures depict the potential in arbitrary units. It can be seen that there is a transmitted wave packet and a reflected wave packet. The initial wave function is a Gaussian wave packet centered at *x* = 0.4 with σ = 0.01 and *k* _{0} = 500π. The simulation assumes infinite potential walls at *x* = 0 and *x* = 1. The step height is Δ = 15*E*, and the *x*-interval is resolved with a linear mesh of *N* = 10^{4} points. The snapshots are taken at times 6, 7, 8,…,15 in appropriate time units.

Numerical simulation of the time-dependent Schrödinger equation for the hard step potential of Eq. (1). The picture shows ten snapshots of |ψ|^{2} (black lines) at different times before, during, and after passing the potential step (order: left column top to bottom, then right column top to bottom). The straight lines in the figures depict the potential in arbitrary units. It can be seen that there is a transmitted wave packet and a reflected wave packet. The initial wave function is a Gaussian wave packet centered at *x* = 0.4 with σ = 0.01 and *k* _{0} = 500π. The simulation assumes infinite potential walls at *x* = 0 and *x* = 1. The step height is Δ = 15*E*, and the *x*-interval is resolved with a linear mesh of *N* = 10^{4} points. The snapshots are taken at times 6, 7, 8,…,15 in appropriate time units.

An example of how numerical error may lead to wrong predictions. The simulation shown in Fig. 3 was repeated with a soft step potential as in Eq. (9) with *L* = 0.005 for different values of the step height Δ. The plot shows the values for the reflection probability . These values cannot be correct; for the parameters used in this simulation (see the following), *R* cannot become close to 1 and must stay between 0 and 10^{−17} for every Δ > 0. The simulation used a standard algorithm for simulating the Schrödinger equation,^{6} a grid of *N* = 10^{4} sites, and as the initial wave function a Gaussian packet with parameters *k* _{1} = 400π, *x* _{0} = 0.4, and σ = 0.005. The bound of 10^{−17} follows from Eq. (18) and the fact that the reflection coefficient (10) is bounded by , which here is exp(−4π^{2}) < 10^{−17}.

An example of how numerical error may lead to wrong predictions. The simulation shown in Fig. 3 was repeated with a soft step potential as in Eq. (9) with *L* = 0.005 for different values of the step height Δ. The plot shows the values for the reflection probability . These values cannot be correct; for the parameters used in this simulation (see the following), *R* cannot become close to 1 and must stay between 0 and 10^{−17} for every Δ > 0. The simulation used a standard algorithm for simulating the Schrödinger equation,^{6} a grid of *N* = 10^{4} sites, and as the initial wave function a Gaussian packet with parameters *k* _{1} = 400π, *x* _{0} = 0.4, and σ = 0.005. The bound of 10^{−17} follows from Eq. (18) and the fact that the reflection coefficient (10) is bounded by , which here is exp(−4π^{2}) < 10^{−17}.

The region (shaded) in the plane of the parameters *u* and *v*, defined in Eq. (24), in which the reflection probability (25) exceeds 99%. The horizontally shaded subset is the region in which Eq. (26) holds.

The region (shaded) in the plane of the parameters *u* and *v*, defined in Eq. (24), in which the reflection probability (25) exceeds 99%. The horizontally shaded subset is the region in which Eq. (26) holds.

Potential plateau.

Potential plateau.

Potential well.

Potential well.

Plot of |ψ_{ n }(*x*)|^{2} for an eigenfunction ψ_{ n } with complex eigenvalue according to Eq. (36) with *V* (*x*) the plateau potential as in Fig. 6. The parameters are *n* = 4 and Δ = 64 *W*, corresponding to Δ = 32π^{2} = 315.8 in units with *a* = 1, *m* = 1, and .

Plot of |ψ_{ n }(*x*)|^{2} for an eigenfunction ψ_{ n } with complex eigenvalue according to Eq. (36) with *V* (*x*) the plateau potential as in Fig. 6. The parameters are *n* = 4 and Δ = 64 *W*, corresponding to Δ = 32π^{2} = 315.8 in units with *a* = 1, *m* = 1, and .

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