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Molecular Physics

Quantum-mechanical tunnelling and the unitarity of the scattering matrix

Zelek S. Herman, Erkki Brändas

Published May 1, 1975• 9 citations
Publisher

Abstract

The one-dimensional tunnel effect is discussed within the framework of Weyl's theory of differential equations. A two-channel scattering experiment is formulated in terms of Riccati's differential equation, and the fundamental quantities are evaluated by means of Wronskians, logarithmic derivatives and the Weyl-Titchmarsh m-coefficient. No approximations are invoked. It is also demonstrated that the scattering matrix S is characterized by SS* = 1 and is unitary if and only if flux is conserved.

DOI
10.1080/00268977500101351

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The one-dimensional tunnel effect is discussed within the framework of Weyl's theory of differential equations. A two-channel scattering experiment is formulated in terms of Riccati's differential equation, and the fundamental quantities are evaluated by means of Wronskians, logarithmic derivatives and the Weyl-Titchmarsh m-coefficient. No approximations are invoked. It is also demonstrated that the scattering matrix S is characterized by SS* = 1 and is unitary if and only if flux is conserved.
We establish the general form of the evolution equation and study scattering and tunneling in real quantum mechanics. We prove an analogue of Stone's theorem for a real Kahler space and derive the corresponding general evolution equation, whose evolution operator is both orthogonal and symplectic. We formulate scattering theory in terms of real wave operators and the associated S-matrix. For the class of potentials considered, we show that the differential scattering cross section is equivalent to that obtained in standard complex quantum mechanics. We also show that the tunneling probability in real quantum mechanics is identical to that in complex quantum mechanics.
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.
A two-dimensional scattering potential represents the quantum extension of a diffractive lattice: a Dirac delta function with a modulated permeability along the y -axis. This model does not have an explicit classical analogue and quantum effects such as tunneling and diffraction play an important role. An analytical solution for the one-harmonic case is found. For the general case of an arbitrary number of harmonics a simple criterion is derived for the range of parameters where quantum chaos is permitted (but does not necessarily occur). The statistical properties of the S -matrix for the given model have been investigated. The deviations from the usual predictions for irregular scattering in the random matrix theory (RMT) framework have been found and are discussed.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.
Starting from the time-independent Schrodinger equation for elastic scattering a theory of wave-particle duality is developed which takes into account effects due to the quantum potential by means of a generalised ray equation. The duality between waves and particles is expressed in terms of two basic equations. Wave effects are described by a nonlinear amplitude equation which is coupled to a generalized ray equation that describes particle effects. The coupling between these two aspects of duality occurs by means of a function called the quantum expansion coefficient which describes the way in which a bundle of ray paths expand as the propagation distance increases. For systems in which the phase of the one-body wavefunction is a separable function of its coordinates, the authors derive simplified forms for the ray equations. As an example the authors show how the theory may be used to calculate tunnelling ray paths in planar electron channelling. The reflected rays in this case originate from beyond the classical turning points, thereby demonstrating the existence of a Goos-Hanchen effect in electron channeling.
A recurrence method for analytical and numerical evaluation of tunneling, transmission, and reflection amplitudes is developed. As the first step, a rule for composition of two arbitrary scatterers separated by a region of constant potential is obtained. Transmission and reflection amplitudes for this double-barrier potential are expressed in terms of transmission and reflection amplitudes for its subparts. As the length of the constant-potential region can be arbitrary and the subparts of a potential may, in turn, be arbitrary segmented potentials, one obtains formulas recurerence formulas which express the scattering amplitudes for the arbitrary segmented potential via the scattering amplitudes for the subparts into which the complete potential can be divided. The efficiency of the method is demonstrated by solving analytically the problem of scattering from locally periodic potentials. Since an arbitrary potential can be approximated by a set of infinitely narrow rectangular barriers, the recurrence formulas can be applied to any potential, giving, in the limit of zero-width segments, differential equations for transmission, and reflection amplitudes.