# Accession Number:

## ADA496104

# Title:

## A Wigner Distribution Analysis of One Dimensional Scattering

# Descriptive Note:

## Master's thesis

# Corporate Author:

## AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH SCHOOL OF ENGINEERING AND MANAGEMENT

# Personal Author(s):

# Report Date:

## 2009-03-01

# Pagination or Media Count:

## 87.0

# 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.

# Descriptors:

- *DISTRIBUTION FUNCTIONS
- *QUANTUM WELLS
- *ELECTRON SCATTERING
- *QUANTUM THEORY
- THESES
- CORRELATION
- CHEMICAL REACTIONS
- QUANTUM EFFICIENCY
- SPLITTING
- WAVE FUNCTIONS
- OPERATORS(PERSONNEL)
- ENERGY LEVELS
- REACTANTS(CHEMISTRY)
- FUNCTIONS(MATHEMATICS)
- ONE DIMENSIONAL
- TIME DEPENDENCE
- SPECTROGRAPHS
- PROPAGATION
- METHODOLOGY

# Subject Categories:

- Atomic and Molecular Physics and Spectroscopy
- Quantum Theory and Relativity