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Non-Cartesian Distributed Approximating Functional Troy Gerald Konshak Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2005 Non-Cartesian distributed approximating functional Troy Gerald Konshak Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons, and the Physical Chemistry Commons Recommended Citation Konshak, Troy Gerald, "Non-Cartesian distributed approximating functional " (2005). Retrospective Theses and Dissertations. 1747. https://lib.dr.iastate.edu/rtd/1747 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. NOTE TO USERS This reproduction is the best copy available. ® UMI Non-Cartesian distributed approximating functional by Troy Gerald Konshak A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Physical Chemistry Program of Study Committee: David K. Hoffman, Major Professor Mark S. Gordon James W. Evans William S. Jenks Xueyu Song Iowa State University Ames, Iowa •2005 Copyright © Troy Gerald Konshak, 2005. All rights reserved. UMI Number: 3200434 Copyright 2005 by Konshak, Troy Gerald All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI UMI Microform 3200434 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 ii Graduate College Iowa State University This is to certify that the doctoral dissertation of Troy Gerald Konshak has met the dissertation requirements of Iowa State University Signature was redacted for privacy. Committee Member Signature was redacted for privacy. Commi; Mej Signature was redacted for privacy. Committee Member Signature was redacted for privacy. Member Signature was redacted for privacy. Majop Professor Signature was redacted for privacy. For the Major Program This work is dedicated to my wife and son who have proven that it is possible to have both a family and a Ph.D. iv Contents List of Acronyms ix Mathematical Notation x List of Common Symbols xiii Abstract xvi 1 Introduction 1 2 Non-Cartesian Distributed Approximating Functionals as Constrained Minimum States of the Heisenberg Total Uncertainty Product 8 2.1 Heisenberg Total Uncertainty Product 10 2.2 Minimum Uncertainty States of the Heisenberg Total Uncertainty Product 16 2.3 Constrained Minimum Uncertainty States of the Heisenberg Total Uncertainty Product 26 2.4 Non-Cartesian Distributed Approximating Functional 38 Appendix 2A: Phase and Normalization of Gaussian State in Position Domain 44 Appendix 2B: Possibility of Non-Negative Valued Initial Change Functions 47 Appendix 2C: Points of Inflection, Rapid Transition, and Maximum Curvature in the NCDAF Momentum Window 49 Appendix 2D: Square Variance in Position and Momentum of the NCDAF 53 Figures 58 3 The Properties of the Non-Cartesian Distributed Approximating Functional Poly­ nomials and Minimum Uncertainty Wavelets 70 3.1 Identification and Characterization of the NCDAF Polynomials 71 3.1.1 First Few [Z]zl1D/2~1+0(Z2) 78 3.1.2 Recursion and Derivative Relationships of [Z]1 (Z2) 78 V 3.1.3 Vàluea of at ^ = 0 83 3.1.4 Orthogonality of [Z]1lI?^2 1+l\z2) 84 3.1.5 Completeness of [Z}1L^>/2~~1+1\Z2) 85 3.2 Characterization of the NC//-Wavelets 86 3.2.1 Generator Expressions for [Z]lLif^2~1+l\z2)e~z2 86 3.2.2 Derivative Relations of [Z]1 L\^^2^1+l\z2)e~z 88 3.2.3 Lowering Relations of [Z]lL^^2 1+l\z2)e~z 92 3.3 The Ladder Operators of the NC/z-Wavelets 98 Appendix 3A: Associated Laguerre Polynomials 109 Appendix 3B: Proof of Recursion Relationship 3.39 112 Appendix 3C: Proof of Recursion Relationship 3.40 113 Appendix 3D: Ladder Operators of the D-Dimensional Isotropic Harmonic Oscillator 116 4 The Numerical Properties of the Non-Cartesian Distributed Approximating Func­ tional 122 4.1 Expressions for NCDAF Function and Derivative Approximations 124 4.2 NCDAF Approximation of Linear Operators 130 4.3 Variational Derivation of NCDAF Approximation 133 4.4 Discrete NCDAF Approximation 140 4.5 General Numerical Properties of NCDAF Approximation 143 4.5.1 Measurement of Error 144 4.5.2 NCDAF Approximation of Polynomials 144 4.5.3 General Characteristics of the NCDAF Residual of Functions 148 4.5.4 NCDAF Approximation of Derivatives 150 4.5.5 Uniform Convergence 152 4.5.6 Repeated Application of the NCDAF Approximation 154 vi 4.5.7 NCDAF Approximation of Discontinuities 155 4.6 Discretization Effects 157 4.6.1 Approximation of Functions Between Grid Points 157 4.6.2 Discrete NCDAF in the Momentum Domain 159 4.6.3 A Further Limitation of the Discrete NCDAF Approximation 160 Appendix 4A: NCDAF Approximation of the Arbitrary Order Laplacian and Gradient of Arbitrary Order Laplacian 163 Appendix 4B: Two Dimensional Test Polynomials 166 Appendix 4C: Table of Optimal DAF Parameters 168 Figures 169 5 Comparison of Non-Cartesian and Cartesian Product Distributed Approximating Functionals 203 5.1 Cartesian Product DAF 204 5.2 Interrelationship Between CP and NCDAFs 205 5.3 Equivalence of CPDAF and NCDAF Approximations 208 5.3.1 Momentum Plateau Viewpoint 209 5.3.2 Polynomial Term Diagram Viewpoint 210 5.4 Computational Efficiency 211 5.5 Approximation of Polynomials 214 5.6 Coordinate Axis Bias 217 5.7 Induced Artificial Rectangular Structure 219 Appendix 5A: Derivation of the Basic NCDAF and CPDAF Interrelationships 221 Appendix 5B: Two Dimensional Test Polynomials 224 Figures 226 vii 6 Constructing Missing Data with Non-Cartesian Distributed Approximating Func­ tionals 241 6.1 Basic Principles 242 6.2 Equations 244 6.2.1 Derivation 244 6.2.2 Practical Considerations 245 6.3 Example Calculations 248 6.3.1 One-Dimensional Example 248 6.3.2 Two Dimensional Example 249 6.3.3 Image Example 250 Appendix 6A: Edge Treatment and Extrapolation 252 6A.1 Basis Function Tail 253 6A.2 Periodic Tail 256 Figures 259 Table 266 7 The Non-Cartesian Distributed Approximating Functional Representation of the Quantum Mechanical Free Evolution Propagator 267 7.1 Free Evolution Propagator 269 7.2 NCDAF Free Evolution Propagator 275 7.3 Discretization and Numerical Concerns 280 7.4 Numerical Demonstrations 282 7.4.1 Gaussian Wavepacket with Zero Group Momentum 282 7.4.2 Gaussian Wavepacket with Nonzero Group Momentum 283 Appendix 7A: Real and Imaginary Parts of the Free Propagator 285 Appendix 7B: NCDAF Diffusion Propagator 287 viii Appendix 7C: NCDAF Representation of Metaplectic Operators 290 Figures 293 8 Conclusion 305 References 308 Acknowledgments 322 ix List of Acronyms AIK Approximating Identity Kernel CMUS Constrained Minimum Uncertainty State CPDAF Cartesian Product Distributed Approximating Functional DAF Distributed Approximating Functional FEP Free Evolution Propagator FFT Fast Fourier Transform HT UP Heisenberg Total Uncertainty Product HUP Heisenberg Uncertainty Product MTUS Minimum Total Uncertainty State MUS Minimum Uncertainty State NCDAF Non-Cartesian Distributed Approximating Functional NC/t-Wavelet Non-Cartesian Minimum Uncertainty Wavelet RUP Robertson Uncertainty Product TDSE Time Dependent Schrôdinger Equation X Mathematical Notation It will be assumed the reader is comfortable with the notation of undergraduate calculus [1], including elementary vector calculus [2] [3], and Hilbert space methods used in quantum mechanics [4]. Discussed here is the shorthand employed in this thesis for handling quantities in arbitrary number of dimensions and coordinate systems1. The symbol , will be used to indicate a vector quantity of D dimensions. A couple examples include using p*' to indicate the momentum and using T\ to indicate the location of the ith sampled point. Components of a vector will be indicated with the same letter without the arrow and with a subscript. For example the components of p will be written as Pi iPa, • • • pjj. An important example to note is, the components of r i will be written as • • • r^jj where the reader needs to keep in mind that it is not a second rank tensor. It will be often be convenient to abbreviate the components of a vector that appear in the argument of a function by the vector symbol. For example, r /(^) = /( i,r2,--- ,rD) . Two important vector operators that will be encountered often in this thesis are the gradient and the Laplacian • It is important to note that this is in a coordinate system independent manner. For example, the l Note, attention will be restricted to Euclidean spaces. Laplacian could be express in D dimensional rectangular coordinates as or in hyperspherical coordinates [5] as Vt = —— — rD~x— — r,- —V r rD~1 dr dr r2 \ drj dri ) ' i>3 where r is the magnitude of 7". Integrals over all components of a (real) vector will be symbolized by / dPr f{rx,r2,---rD) , JRD where ZD symbolizes all of the real D dimensional space. In rectangular coordinates for three dimensions this means p poo POO POO / d~t /(r1,r2r3) = / dx dy dz g(x,yz) , «/R3 J — oo J—oo J — oo and in spherical coordinates it is p P2TT PIT POO 1 2 / d rf(rl,r2ri)= sin<pd<p dff r dr h(r,6,<j>) , Je3 JO JO JO where g and A are the function / expressed in rectangular and spherical coordinates respectively.
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