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.

## REFERENCES

*Z*= 0, the line for −

*a*<

*x*<

*a*has to be replaced by

*A*

_{0}+

*A*

_{1}

*x*; for

*Z*= −Δ, ψ(

*x*) =

*D*

_{−}+

*E*

_{−}

*x*for

*x*< −

*a*and ψ(

*x*) =

*D*

_{+}+

*E*

_{+}

*x*for

*x*>

*a*.

*Z*’s that we excluded in this definition: for $Z\u2208(-\u221e,0]\{-\Delta}$ there exist no nonzero functions with

*C*

_{±}= 0 satisfying the eigenvalue Equation (36) for all

*x*≠ ±

*a*such that ψ and ψ′ are continuous. We have not included the proof in this paper. For

*Z*= −Δ, the coefficients

*C*

_{±}are not defined, so the condition (43) makes no sense.

^{z}= ζ has infinitely many solutions

*z*, all of which have real part ln |ζ|; the imaginary parts differ by integer multiples of 2π; by ln ζ we denote that

*z*which has −π < Im

*z*≤ π.

*V*is bounded, by an application of the Kato-Rellich theorem, Ref. 22, Theorem X.15, the Hamiltonian $H=-(\u210f2/2m)\u2202x2+V$ is self-adjoint on the domain of $-\u2202x2$. It can be easily checked that for any

*t*the derivative ∂

_{x}

*f*(

*x*,

*t*) is absolutely continuous in

*x*, and thus the function $f(\xb7,t)$ belongs to the domain of

*H*. These properties can be used to justify all the manipulations made here. We also stress that if we had not chosen the constants

*A*

_{±}and

*B*

_{±}in Eq. (41) so that the function is continuously differentiable, then the addition of the Gaussian cut-off would have resulted in functions which are in $L2(R)$ but which do not belong to the domain of

*H*. Thus our estimates are not valid for such initial states. For more sophisticated mathematical methods to study such problems, see, for instance, Refs. 15 and 16.

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