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Journal of Applied Physics

Analytic Wigner distribution function for tunneling and trajectory models

Kevin L. Jensen, Donald A. Shiffler, Joel L. Lebowitz, Marc M. Cahay, John J. Petillo

Published Mar 19, 2019• 9 citations
Publisher

Abstract

The Wigner function is assembled from analytic wave functions for a one-dimensional closed system (well with infinite barriers). A sudden change in the boundary potentials allows for the investigation of time-dependent effects in an analytically solvable model. A trajectory model is developed to account for tunneling when the barrier is finite. The behavior of the density (the zeroth moment of the Wigner function) after an abrupt change in potential shows net accumulation and depletion over time for a weighting of energy levels characteristic of the supply function in field emission. However, for a closed system, the methods have application to investigations of tunneling and transmission associated with field and photoemission at short time scales.

DOI
10.1063/1.5086434

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The Wigner function is assembled from analytic wave functions for a one-dimensional closed system (well with infinite barriers). A sudden change in the boundary potentials allows for the investigation of time-dependent effects in an analytically solvable model. A trajectory model is developed to account for tunneling when the barrier is finite. The behavior of the density (the zeroth moment of the Wigner function) after an abrupt change in potential shows net accumulation and depletion over time for a weighting of energy levels characteristic of the supply function in field emission. However, for a closed system, the methods have application to investigations of tunneling and transmission associated with field and photoemission at short time scales.
The analytic Wigner function for a single well with infinite walls is extended to the configuration where half of the well is raised and the weighting of the wave functions is in accordance with a thermal Fermi-Dirac distribution. This requires a method for determining the energy eigenstates particularly near the height of the higher half-well. The methods provide a means for constructing analytic and accurate trajectories that can be used to model tunneling times. The fidelity of the trajectory model is assessed with respect to absorption of quanta related to changes within the allowable eigenstates.
The analytic nature of the transmission coefficient for a δ-function barrier makes it a useful tool to examine a variety of technologically important applications, such as photoemission from semiconductors with an alkali coating, the examination of tunneling times for wave packets incident on a barrier, and for parameterizing tunneling through the narrow barrier of a normal-superconducting point contact. The analytic model of a δ-function barrier inside a confining well is extended to the finite height and width rectangular barrier (a delta-function sequence). Methods to exactly evaluate the eigenstates are given and their dependencies are examined. The time evolution of a superposition of the lowest eigenstates is considered for barriers having comparable Gamow tunneling factors so as to quantify the impact of barrier height and shape on time evolution in a simple and exact system and, therefore, serve as a proxy for tunneling time. Last, density profiles and associated quantum potentials are examined for coupled wells to show changes induced by weaker and wider barriers.
A numerical solution to the time evolution equation of the Wigner distribution function (WDF) with an accuracy necessary to simulate the passage of a wave packet past a barrier is developed, where quantum effects require high accuracy and fine discretization. A wave packet incident on a barrier, a portion of which tunnels through, demonstrates behavior that can define various characteristic transmission and reflection delay (TARD) times useful in the simulation of electron emission. A model for the TARD times is proposed that relies only on the asymptotic maxima of the position $\ensuremath{\rho}(x,t)$ and wave number $\ensuremath{\rho}(k,t)$ densities given by the WDF and applied to a ballistic trajectory model for the (faster) transmitted and (slower) reflected parts. The dependence of the TARD times on barrier width, symmetry, and abruptness is analyzed. For symmetrical barriers with characteristics similar to field emission barrier heights and widths, TARD times are on the order of a fraction of a femtosecond. The TARD times for when tunneling predominates are contrasted to tunneling times in the literature. Use of the TARD times in simulations of field emission in nanogap devices or to model ultrashort pulses generated under rapidly changing conditions for electron sources are proposed.
In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.
A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.