Paradoxical reflection in quantum mechanics
A potential V (x) containing a downward 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 = 15E, and the x-interval is resolved with a linear mesh of N = 104 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 = 104 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(−4π2) < 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.
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 = 32π2 = 315.8 in units with a = 1, m = 1, and .
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