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Journal of Physics A Mathematical and Theoretical

On the quantum mechanical scattering from a potential step

Victor F. Los, Andrei V. Los

Published Jan 14, 2010• 12 citations
Publisher

Abstract

The problem of finding the exact spacetime particle's propagator in the presence of a potential step (interface between different materials) is revisited. In contrast to the conventional Feynman path-integral approach, integration over all energy values of the particle's spectral density matrix (discontinuity of the energy-dependent Green's function across the real energy axis) is suggested for obtaining the exact spacetime propagator. The energy-dependent Green's functions are found in the framework of the multiple scattering theory (MST). The problem of finding the step-localized energy-dependent potentials responsible for the particle's reflection from and transmission through a potential step, which are needed for MST application, is solved. The obtained exact result for the particle's propagator is expressed in terms of integrals of elementary functions and has a significantly simpler form than that reported earlier. The obtained expressions allow easy evaluation of all limiting cases, including the case of the infinitely large potential step, as well as simple numerical visualization. The square of the absolute value of the propagator, which represents the relative transition probability density between two spacetime points, is plotted and discussed in detail for the cases of particle reflection and transmission.

DOI
10.1088/1751-8113/43/5/055304

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The problem of finding the exact spacetime particle's propagator in the presence of a potential step (interface between different materials) is revisited. In contrast to the conventional Feynman path-integral approach, integration over all energy values of the particle's spectral density matrix (discontinuity of the energy-dependent Green's function across the real energy axis) is suggested for obtaining the exact spacetime propagator. The energy-dependent Green's functions are found in the framework of the multiple scattering theory (MST). The problem of finding the step-localized energy-dependent potentials responsible for the particle's reflection from and transmission through a potential step, which are needed for MST application, is solved. The obtained exact result for the particle's propagator is expressed in terms of integrals of elementary functions and has a significantly simpler form than that reported earlier. The obtained expressions allow easy evaluation of all limiting cases, including the case of the infinitely large potential step, as well as simple numerical visualization. The square of the absolute value of the propagator, which represents the relative transition probability density between two spacetime points, is plotted and discussed in detail for the cases of particle reflection and transmission.
A propagator for the one dimensional time-dependent Schrödinger equation with an asymmetric rectangular potential is obtained using the multiple-scattering theory approach. It allows for the consideration of the reflection and transmission processes as the particle scattering at the potential jump (in contrast to the conventional wave-like picture) and for accounting for the nonclassical counterintuitive contribution of the backward-moving component of the wave packet attributed to the particle. This propagator completely resolves the corresponding time-dependent Schrödinger equation (defines the wave function ψ(x,t)ψ(x,t)ψ(x,t)) and allows for considering the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of time. These results are related to fundamental issues such as measuring time in quantum mechanics (tunneling time, time of arrival, dwell time). For imaginary time, which represents an inverse temperature (t→−iℏ/kBT)(t\rightarrow-i{\hbar}/k_{B}T)(t→−iℏ/kB​T), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to a rectangular potential. If one shifts to imaginary time (t→−it)(t\rightarrow-it)(t→−it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals from elementary functions and thus allow for a numerical visualization of the probability density ∣ψ(x,t)∣2|ψ(x,t)|^2∣ψ(x,t)∣2, the density matrix and the solution of the diffusion-like equation. The results obtained may also be useful for spintronics applications due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.
A propagator for the one-dimensional time-dependent Schr¨odinger equation with an asymmetric rectangular potential is obtained, by using the multiple-scattering theory. It allows the consideration of the reflection and transmission processes as the scattering of a particle at the potential jump (in contrast to the conventional wave-like picture) and the account for the non-classical counterintuitive contribution of the backward-moving component of the wave packet attributed to a particle. This propagator completely resolves the corresponding time-dependent Schr¨odinger equation (defines the wave function w(x, t)) and allows the consideration of the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of the time. These results are related to fundamental issues such as measuring the time in quantum mechanics (tunneling time, time of arrival, dwell time). For the imaginary time, which represents an inverse temperature (t → −ih/kBT), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to the action of a rectangular potential. This density matrix provides information about particles’ density in the different spatial areas relative to the potential location and on the quantum coherence of different particle spatial states. If one passes to the imaginary time (t → −it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals of elementary functions and thus allow a numerical visualization of the probability density |w(x, t)|2, the density matrix, and the solution of the diffusion-like equation. The results obtained may also be applied to spintronics due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.
An exact time-dependent solution for the wave function (r, t) of a particle moving in the presence of an asymmetric rectangular well/barrier potential varying in one dimension is obtained by applying a novel for this problem approach using multiple scattering theory (MST) for the calculation of the space-time propagator.This approach, based on the found effective potentials localized at the potential jumps and responsible for transmission through and reflection from the considered rectangular potential, enables considering these processes from standpoint of a particle (rather than a wave).The solution describes these quantum phenomena as time dependent and is related to both the fundamental issues (such as time measurement) of quantum mechanics and the kinetic theory of nanostructures due to the fact that the considered potential can model the spin-dependent potential profile of the magnetic multilayers used in spintronics devices.It is presented in terms of integrals of elementary functions and is a sum of the forward-and backward-moving components of the wave packet.The relative contribution of these components and their interference as well as of the potential asymmetry to both the probability density (x, t) 2 and the particle dwell time is considered and numerically visualized for the narrow and broad energy (momentum) distributions of the initial Gaussian wave packet.As shown, in the case of a broad initial wave packet, the quantum-mechanical counterintuitive
A simple exact solution for the spacetime particle propagator in the presence of a one-dimensional rectangular potential is obtained. This propagator solves the Cauchy problem for the corresponding time-dependent Schrodinger equation in terms of the forward-moving (positive momentum) and backward-moving (negative momentum) components of the wavefunction. The exact formulae for the time-dependent wavefunction in different spatial regions (before, inside and after the barrier) are found. These results demonstrate that generally the forward- and backward-moving components of the wavefunction as well as their interference contribute to the probability density of finding a particle in different spacetime regions. The time of arrival to and the dwell time in the above-mentioned spatial regions are also obtained. The relative contribution of the positive- and negative-momentum components of the wavefunction as well as those of the below- and above-barrier energy regions are discussed for the case of an initial Gaussian wave packet. Numerical visualization of the probability density to find a particle in the different spacetime regions is performed and discussed in the tunneling regime.
By calculating the reflection and transmission coefficients for a square potential barrier in terms of a series of scatterings at step potential discontinuities, a multiple scattering approach to potential problems is presented. This approach is applied to WKB potential problems by interpreting the WKB connection formulas in terms of reflection and transmission coefficients at each turning point. An algorithm for the systematic construction of the energy Green’s function is given. The significance of complex coordinate turning points is demonstrated. The equivalence of this approach and that of the Gaussian approximation of the path integral is mentioned briefly.
The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of \ensuremath{\delta}-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the \ensuremath{\delta}-function perturbation infinite repulsive produces a totally reflecting boundary, hence giving a path integral solution in half spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step potential and the finite potential well in the half space, respectively.