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Quantum Scattering Theory and Applications

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Quantum Scattering Theory and Applications Quantum Scattering Theory and Applications A thesis presented by Adam LupuSax to The DepartmentofPhysics in partial fulllment of the requirements for the degree of Do ctor of Philosophy in the sub ject of Physics Harvard University Cambridge Massachusetts September c Adam LupuSax All rights reserved Abstract Scattering theory provides a convenient framework for the solution of a varietyof problems In this thesis we fo cus on the combination of b oundary conditions and scattering potentials and the combination of nonoverlapping scattering p otentials within the context of scattering theory Using a scattering tmatrix approach we derivea useful relationship between the scattering tmatrix of the scattering potential and the Green function of the b oundary and the tmatrix of the combined system eectively renormalizing the scatter ing tmatrix to account for the b oundaries In the case of the combination of scattering potentials the combination of tmatrix op erators is achieved via multiple scattering the ealsoderive metho ds primarily for numerical use for nding the Green function of ory W arbitrarily shap ed b oundaries of various sorts These metho ds can b e applied to b oth op en and closed systems In this thesis we consider single and multiple scatterers in two dimensional strips regions which are innite in one direction and b ounded in the other as well as two dimensional rectangles In D strips both the renormalization of the single scatterer strength and the conductance of disordered manyscatterer systems are studied For the case of the single scatterer we see nontrivial renormalization eects in the narrow wire limit In the many scatterer case wenumerically observe suppression of the conductance beyond that which is explained by weak lo calization In closed systems we fo cus primarily on the eigenstates of disordered many scatterer systems There has b een substantial investigation and calculation of prop erties of the eigenstate intensities of these systems We have for the rst time b een able to inves tigate these questions numerically Since there is little exp erimental work in this regime these numerics provide the rst test of various theoretical mo dels Our observations indicate that the probability of large uctuations of the intensity of the wavefunction are explained qualitatively byvarious eldtheoretic mo dels However quantitatively no existing theory accurately predicts the probability of these uctuations Acknowledgments Doing the work which app ears in this thesis has been a largely delightful way to sp end the last veyears The nancial supp ort for my graduate studies was provided bya National Science Foundation Fellowship Harvard University and the HarvardSmithsonian Institute for Theoretical Atomic and Molecular Physics ITAMP Together all of these sources provided me with the wonderful opp ortunity to study without b eing concerned ab out my nances My advisor Rick Heller is a wonderful source of ideas and insights I b egan working with Rick four years ago and I have learned an immense amount from him in that One time From the very rst time wespokeIhave felt not only challenged but resp ected particularly nice asp ect of having Rick as an advisor is his ready availability More than one tricky part of this thesis has b een sorted out in a marathon conversation in Ricks oce I cannot thank him enough for all of his time and energy In the last veyearsIhave had the great pleasure of working not only with Rick himself but also with his p ostdo cs and other students Maurizio Carioli was a p ostdo c when I b egan working with Rick There is much I cannot imagine having learned so quickly or so well without him particularly ab out numerical metho ds Lev Kaplan a student and then p ostdo c in the group is an invaluable source of clear thinking and uncanny insight He has also demonstrated a nearly innite patience in discussing our work My classmate Neepa Maitra and I b egan working with Rick at nearly the same time and have b een partners in this journey Neepas emotional supp ort and p erceptive comments and questions ab out substantially easier Alex Barnett Bill Bies Greg my work have made my last ve years Fiete Jesse Hersch Bill Hosten and Areez Mo dy all graduate students in Ricks group have given me wonderful feedbackonthis and other work The substantial p ostdo c contingent in the group Michael Haggerty Martin Naraschewski and Doron Cohen have b een equally helpful and provided very useful guidance along the way At the time I b egan graduate scho ol I was pleasantly surprised by the co op erative spirit among my classmates Manyofusspentcountless hours discussing physics and sorting out problem sets Among this crowd I must particularly thank Martin Bazant Brian Busch Sheila Kannappan Carla Levy Carol Livermore Neepa Maitra Ron Rubin and Glenn Wong for making many late nights b earable and oftentimes fun Imust particularly thank Martin Carla and Neepa for remaining great friends and colleagues in the years that followed Ihave had the great fortune to make go o d friends at various stages in my life and I am honored to count these three among them It is hard to imagine howIwould have done all of this without my ancee Kiersten Conner Our up coming marriage has been a singular source of joy during the pro cess of writing this thesis Her unagging supp ort and b oundless sense of humor have kept me centered throughout graduate scho ol My parents Chip Lupu and Jana Sax have both been a great source of supp ort and encouragement throughout my life and the last veyears have b een no exception The rest of my family has also b een very supp ortive particularly my grandmothers Sara Lupu and Pauline Sax and my stepmother Nancy Altman It saddens me that neither of my grandfathers Dave Lupu or N Irving Sax are alive to see this moment in my life but I thank them b oth for teaching me things that have help ed bring me this far Citations to Previously Published Work Portions of chapter and App endix B have app eared in Quantum scattering from arbitrary b oundaries MGE da Luz AS LupuSax and EJ Heller Physical Review B no pages Contents Title Page Abstract Acknowledgments Citations to Previously Published Work Table of Contents List of Figures List of Tables Intro duction and Outline of the Thesis Intro duction Outline of the Thesis Quantum Scattering Theory in dDimensions CrossSections Unitarit y and the Optical Theorem Green Functions Zero Range Interactions Scattering in two dimensions Scattering in the Presence of Other Potentials Multiple Scattering Renormalized tmatrices Scattering From Arbitrarily Shap ed Boundaries Intro duction Boundary Wall Metho d I Boundary Wall Metho d I I Perio dic Boundary Conditions Green Function Interfaces Numerical Considerations and Analysis From Wavefunctions to Green Functions Eigenstates Contents Scattering in Wires I One Scatterer One Scatterer in a Wide Wire The Green function of an empty p erio dic wire Renormalization of the ZRI Scattering Strength From the Green function to Conductance Computing the channeltochannel Green function One Scatterer in a Narrow Wire Scattering in Rectangles I One Scatterer Dirichlet b oundaries Perio dic b oundaries Disordered Systems Disorder Averages Mean Free Path Prop erties of Randomly Placed ZRIs as a Disordered Potential Eigenstate Intensities and the PorterThomas Distribution Weak Lo calization Strong Lo calization Anomalous Wav efunctions in Two Dimensions Conclusions Quenched Disorder in D Wires Transp ort in Disordered Systems Quenched Disorder in D Rectangles Extracting eigenstates from tmatrices Intensity Statistics in Small Disordered Dirichlet Bounded Rectangles Intensity Statistics in Disordered Perio dic Rectangle Algorithms Conclusions Bibliography A Green Functions A Denitions A Scaling L A Integration of Energy Green functions A Green functions of separable systems A Examples A The Gorkov bulk sup erconductor Green Function B Generalization of the Boundary Wall Metho d Contents C Linear Algebra and NullSpace Hunting C Standard Linear Solvers
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