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Journal of Physics A Mathematical and General

Reflection above potential steps

Zafar Ahmed

Published Jan 1, 1999• 5 citations
Publisher

Abstract

Disparate behaviour of above-the-step quantal reflection probability for smooth and composite semi-infinite potentials is shown. For a smooth semi-infinite potential (step) the reflectivity is usually a monotonically decreasing function of energy. But the composite two-piece (non-differentiable, e.g., at x = 0) potential step gives rise to a parameter-dependent pronounced single minimum in the reflectivity. Three analytically solvable and several other models of composite semi-infinite (step) potentials are shown supporting such a behaviour of the quantal reflection.

DOI
10.1088/0305-4470/32/15/005

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Disparate behaviour of above-the-step quantal reflection probability for smooth and composite semi-infinite potentials is shown. For a smooth semi-infinite potential (step) the reflectivity is usually a monotonically decreasing function of energy. But the composite two-piece (non-differentiable, e.g., at x = 0) potential step gives rise to a parameter-dependent pronounced single minimum in the reflectivity. Three analytically solvable and several other models of composite semi-infinite (step) potentials are shown supporting such a behaviour of the quantal reflection.
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.
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.
We present the case of quantal transmission through a smooth, single-piece exponential potential, V(x)=−V0ex/aV(x)=-V_0 e^{x/a}V(x)=−V0​ex/a, in contrast to the piece-wise continuous potentials, as a pedagogic model to demonstrate the analytic extraction of transmission (reflection) co-efficient.
We study quantum reflection in a system of two attractive step potentials with a steplike coupling term. The amplitude R11{R}_{11}R11​ for elastic reflection generally has the near-threshold behavior $|{R}_{11}|\stackrel{\stackrel{\ensuremath{\rightarrow}}{k}0}{\ensuremath{\sim}}1\ensuremath{-}2bk$ familiar from the single-channel case. Reflectivity in the elastic channel is quenched in the presence of a second adiabatic channel open for transmission, while it is enhanced when the second transmission channel is closed; there is a pronounced minimum in the reflectivity when the potential in the upper adiabatic channel is zero or close to zero. When the upper adiabatic channel is repulsive, the elastic reflectivity can have an exact zero, at which quantum reflection leads to complete transfer into the excited state. Any degree of entanglement of translational and internal motion can be achieved at an appropriate energy between the threshold for inelastic reflection and this zero of the elastic reflectivity. For degenerate reflection channels, the unusual case of incomplete reflection at threshold occurs when the potential in the upper adiabatic channel vanishes.
In the discussion of temporary behaviors of quantum tunneling, people usually like to focus their attention on rectangular barrier with steep edges, or to deal with smooth barrier with semi-classical or even numerical calculations. Very few discussions on analytic solutions of tunneling through smooth barrier appear in the literature. In this paper, we provide two such examples, a semi-infinite long barrier V(x)=A2[1+tanh(x/a)] and a finite barrier V(x) = A sech2(x/a). To each barrier, we calculate the associated phase time and dwell time after obtaining the analytic solution. The results show that, different from rectangular barrier, phase time or dwell time does increase with the length parameter a controlling the effective extension of the barrier. More interestingly, for the finite barrier, phase time or dwell time exhibits a peak in k-space. A detailed analysis shows that this interesting behavior can be attributed to the strange tunneling probability Ts(k), i.e., Ts(k) displays a unit step function-like profile Θ(k − k0), especially when a is large, say, a ≫ 1/κ, 1/k. And k0≡mA/ħ is exactly where the peak appears in phase or dwell time k-spectrum. Thus only those particles with k in a very narrow interval around k0 are capable to dwell in the central region of the barrier sufficiently long.