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arXiv:quant-ph/0309176

A practice problem for Schr{ö}dinger transmission

Zafar Ahmed

Published Sep 24, 2003• 1 citation
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Abstract

We present the case of quantal transmission through a smooth, single-piece exponential potential, $V(x)=-V_0 e^{x/a}$, in contrast to the piece-wise continuous potentials, as a pedagogic model to demonstrate the analytic extraction of transmission (reflection) co-efficient.

DOI
10.48550/arxiv.quant-ph/0309176

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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.
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.
FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, with a transmission coefficient governed by exponential decay of the wavefunction inside the barrier; "impossible" relativistic measurements are shown to be possible but non-ideal, revealing a fundamental tension between QFT and measurement theory. MATH: - Transmission coefficient T≈e−2κL T \approx e^{-2\kappa L} T≈e−2κL, where κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0 - E)}/\hbar κ=2m(V0​−E)​/ℏ (rectangular barrier, E<V0 E < V_0 E<V0​). - For a general barrier V(x) V(x) V(x): T≈exp⁡[−2ℏ∫x1x22m(V(x)−E) dx] T \approx \exp\left[-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x)-E)}\, dx\right] T≈exp[−ℏ2​∫x1​x2​​2m(V(x)−E)​dx] (WKB approximation). - Reflection coefficient R=1−T R = 1 - T R=1−T (unitarity). - In QFT impossibility: signaling between spacelike-separated regions is avoided only if measurement operators commute — the paper (arXiv:2311.13644) shows nonlocal joint measurements in non-relativistic QM produce the same paradox, implying a constraint on measurement algebra: \( [\hat{O}_A, \hat{O}_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x)V(x)V(x) exceeds the value of the total energy EEE, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x)V(x)V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a.{V(x),=0 \mathrm{for} x<0 \mathrm{and}\mathrm{for} x>a,}{={V}_{0}>0 \mathrm{for} 0<x<a.}V(x),=0forx<0andforx>a,=V0​>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0x=0x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=ax=ax=a at about the time at which the incident packet reaches the point x=0x=0x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.