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7 - The Continuous Spectrum and Scattering States

Published online by Cambridge University Press:  11 May 2023

Uri Peskin
Uri Peskin
Affiliation:
Technion - Israel Institute of Technology, Haifa
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Summary

The Schrödinger equation is reformulated as a universal continuity equation, which connects between changes in the particles probability density distribution to probability current densities (fluxes). The formulation of particle conservation in terms of stationary fluxes enables one to associate stationary wave functions also to open quantum systems characterized by stationary particle currents. These functions are (improper) solutions of the stationary Schrödinger equation, obtained under scattering boundary conditions. These boundary conditions can be fulfilled for any positive asymptotic kinetic energy, hence, the energy spectrum of the scattering states is continuous. We demonstrate flux calculations in scattering through a one-dimensional potential energy well/barrier, focusing on transmission and reflection probabilities. Nonclassical phenomena such as transmission at energies below a potential energy barrier (quantum tunneling), or reflections at energies above a potential energy well are analyzed. The phenomenon of full transmission through a double barrier structure (resonant tunneling) is introduced in the context of nanoscale transport.

Keywords

quantum scatteringprobability current densityprobability fluxcontinuity equationscattering wave functionstransmission probabilityreflection probabilityresonant tunneling
Type
Chapter
Information
Quantum Mechanics in Nanoscience and Engineering , pp. 61 - 73
Publisher: Cambridge University Press
Print publication year: 2023

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References

Taylor, J. R., “Scattering Theory: The Quantum Theory of Nonrelativistic Collisions” (Dover, 2006).Google Scholar
Hernández, G., “Fabry-Perot Interferometers” (Cambridge University Press, 1988).Google Scholar
Moiseyev, N., “Non-Hermitian Quantum Mechanics” (Cambridge University Press, 2011).CrossRefGoogle Scholar

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