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A Wigner Distribution Analysis of One Dimensional Scattering

Brent R. Lacy

Published Mar 1, 2009• 0 citations

Abstract

We applied the Wigner Distribution Function, a distribution function of time and frequency based on an initial function of either of those variables, to a series of time based correlation functions. These time based correlation functions were the result of a 1-dimensional free particle wave packet, the reactant wave function, which had propagated through a quantum potential well and then had components of the reactant wave function that exited the opposite side of the well auto-correlated in time with a stationary 1-dimensional free particle wave packet, the product wave function. This process was undertaking in order to generate a 3-dimensional depiction, in time and frequency, of the reactant wave functions interaction with the quantum potential well. Fortran 77 code was utilized to generated the time propagation of the reactant wave function by means of the Split Operator Method, which was given the following initial set of conditions; x sub 0 = -20 (Bohr radii), k sub 0 = 3 (atomic units), and = 1 (Bohr radius). A series of potential wells with variable depths were implemented into the code. The code then computed the correlation in time of the exiting reactant wave function with a stationary wave function before applying the Wigner Distribution Function. When Wigner Distribution Function was applied to the time correlation function many recognizable features on the potential well were observed from the 3-dimensional plot generated including transmission resonance energy levels. The classical time of arrival was also captured by the Wigner Distribution Function. As a useful tool the Wigner Distribution Function provides more insight into the quantum interactions of chemical reactions in terms of time and frequency than traditional spectrographic analysis.

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We applied the Wigner Distribution Function, a distribution function of time and frequency based on an initial function of either of those variables, to a series of time based correlation functions. These time based correlation functions were the result of a 1-dimensional free particle wave packet, the reactant wave function, which had propagated through a quantum potential well and then had components of the reactant wave function that exited the opposite side of the well auto-correlated in time with a stationary 1-dimensional free particle wave packet, the product wave function. This process was undertaking in order to generate a 3-dimensional depiction, in time and frequency, of the reactant wave functions interaction with the quantum potential well. Fortran 77 code was utilized to generated the time propagation of the reactant wave function by means of the Split Operator Method, which was given the following initial set of conditions; x sub 0 = -20 (Bohr radii), k sub 0 = 3 (atomic units), and = 1 (Bohr radius). A series of potential wells with variable depths were implemented into the code. The code then computed the correlation in time of the exiting reactant wave function with a stationary wave function before applying the Wigner Distribution Function. When Wigner Distribution Function was applied to the time correlation function many recognizable features on the potential well were observed from the 3-dimensional plot generated including transmission resonance energy levels. The classical time of arrival was also captured by the Wigner Distribution Function. As a useful tool the Wigner Distribution Function provides more insight into the quantum interactions of chemical reactions in terms of time and frequency than traditional spectrographic analysis.

We calculate the Wigner quasi-probability distribution for position and momentum, P_W^(n)(x,p), for the energy eigenstates of the standard infinite well potential, using both x- and p-space stationary-state solutions, as well as visualizing the results. We then evaluate the time-dependent Wigner distribution, P_W(x,p;t), for Gaussian wave packet solutions of this system, illustrating both the short-term semi-classical time dependence, as well as longer-term revival and fractional revival behavior and the structure during the collapsed state. This tool provides an excellent way of demonstrating the patterns of highly correlated Schrodinger-cat-like `mini-packets' which appear at fractional multiples of the exact revival time.

In the second part of this paper on micro-canonical ensemble a new numerical approach for consideration of quantum dynamics and calculations of the average values of quantum operators and time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weyl's symbol of the considered operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For the latter function the integral Wigner–Liouville's type equation has been derived. The initial condition for this equation has been obtained in the form of the Fourier transform of the Wiener path integral representation of the matrix elements of the propagators at initial time. A numerical procedure for solving this equation combining both molecular dynamics and Monte Carlo methods has been developed. Numerical results have been obtained for series of average values of the quantum operators as well as for the time correlation function characterizing the energy level structure, the momentum flow of tunnelling particles at barrier crossing and the absorption spectra of electrons in a potential well. The developed quantum dynamics method was tested by comparison of numerical results with analytical estimations. Tunnelling transitions and the effect of the quasi stationary state have been considered as the reasons for the peculiarities in behaviour of the time correlation functions and position and momentum dispersions. The possibility of applying the developed approach to the theory of classical wave propagation in random media has also been considered. For classical waves some results have been obtained for Gaussian beam propagation in 2D and 3D waveguides.

The quantum dynamics of an ensemble of interacting electrons in an array of random scatterers is treated using a numerical approach for the calculation of average values of quantum operators and time correlation functions in the Wigner representation. The Fourier transform of the product of matrix elements of the dynamic propagators obeys an integral Wigner-Liouville-type equation. Initial conditions for this equation are given by the Fourier transform of the Wiener path-integral representation of the matrix elements of the propagators at the chosen initial times. This approach combines both molecular dynamics and Monte Carlo methods and computes numerical traces and spectra of the relevant dynamical quantities such as momentum-momentum correlation functions and spatial dispersions. Considering, as an application, a system with fixed scatterers, the results clearly demonstrate that the many-particle interaction between the electrons leads to an enhancement of the conductivity and spatial dispersion compared to the noninteracting case.

We calculate and visualize the Wigner quasi-probability distribution for the position and momentum, PW(n)(x,p), for the energy eigenstates of the infinite square well. We evaluate the time-dependent Wigner distribution, PW(x,p;t), for Gaussian wave packet solutions of this system, and illustrate the short-term semi-classical time dependence and the longer-term revival and fractional revival behavior. Our results indicate how the Wigner distribution can be used to examine the highly correlated dynamical position-momentum structure of quantum states. In particular, this tool provides an excellent way of demonstrating the patterns of highly correlated Schrödinger-cat-like “mini-packets” which appear at fractional multiples of the exact revival time.

In two previous studies, the time-dependent scattering of a wave packet from a Gaussian barrier was investigated computationally in the complex z-plane. One of these involved the 'direct' propagation of the wave packet in the complex space, and the other used numerical analytic continuation techniques to generate the dynamics in the complex plane from the wave function computed on the real-axis. In the current study, the dynamics of exact quantum trajectories are analyzed for the same barrier scattering problem. Thousands of quantum trajectories were launched from positions near the center of the initial wave packet. These trajectories were computed by integrating equations-of-motion involving the quantum momentum function, which was obtained from the time-dependent wave function and its derivative. In order to analyze the dynamics, many trajectories were plotted on space-time diagrams. Particular emphasis was placed upon trajectories undergoing reflection in the barrier region. Some groups of strongly correlated trajectories form long-lived highly organized patterns, including helical wrappings around a series of stagnation filaments. These curves alternate with quasi-nodes where the amplitude of the wave function reaches low values. In addition, other trajectories for short times follow hyperbolic paths as they propagate near vorticity tubes surrounding these quasi-nodes.

A correlation function formulation for the state-selected total reaction probability, Nα(E), is suggested. A wave packet, correlating with a specific set of internal reactant quantum numbers, α, is propagated forward in time until bifurcation is complete at which time the nonreactive portion of the amplitude is discarded. The autocorrelation function of the remaining amplitude is then computed and Fourier transformed to obtain a reactivity spectrum. Dividing by the corresponding spectrum of the original, unfiltered, wave packet normalizes the reactivity spectrum, yielding the total reaction probability from the internal state, α. The procedure requires negligible storage and just one time-energy Fourier transform for each initial reactant state, independent of the number of open channels of products. The method is illustrated numerically for the one-dimensional Eckart barrier, using both quantum-mechanical and semiclassical propagation methods. Summing over internal states of reactants gives the cumulative reaction probability, N(E). The relation to the trace formula [W. H. Miller, S. D. Schwartz, J. W. Tromp, J. Chem. Phys. 79, 4889 (1983)], N(E)=12(2πℏ)2 tr(F̄δ(H−E)F̄δ(H−E)), is established, and a new variant of the trace formula is presented.