The Supersymmetric Dirac Equation: The Application to Hydrogenic Atoms

216 Pages·2011·1.917 MB·English

by  Allen Hirshfeld

THE SUPERSYMMETRIC
DIRAC EQUATION
The Application to Hydrogenic Atoms
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
THE SUPERSYMMETRIC
DIRAC EQUATION
The Application to Hydrogenic Atoms
Allen Hirshfeld
Technical University of Dortmund, Germany
Imperial College Press
ICP
Published by
Imperial College Press
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Paul Dirac photograph by A. Börtzells Tryckeri, courtesy of AIP Emilio Segre Visual Archives,
E. Scott Bar and Weber Collections.
THE SUPERSYMMETRIC DIRAC EQUATION
The Application to Hydrogenic Atoms
Copyright © 2012 by Imperial College Press
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN-13 978-1-84816-797-1
ISBN-10 1-84816-797-0
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Printed in Singapore.
Catherine - The Supersymmetric Dirac.pmd 1 9/30/2011, 3:24 PM
August27,2011 9:21 9inx6in TheSupersymmetricDiracEquation b1217-fm
Iwishtodedicatethisbooktothememoryofmyfather,Dr.MartinA.Hirshfeld,
whotaughtmemanythings,amongthemthemeaningofproof.
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August27,2011 9:21 9inx6in TheSupersymmetricDiracEquation b1217-fm
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August27,2011 9:21 9inx6in TheSupersymmetricDiracEquation b1217-fm
Preface
Onemorning,whenIwasanundergraduate,overacupofcoffeeinthecafeteria
of theWeizman Institute, where he was working on his PhD in physics, Charles
Robinson opened my eyes to the wonders of the Dirac equation, and sowed the
seedforthisbook.
ImetPaulDiracintheearly1960s,ontheoccasionofhisvisittothePhysics
DepartmentoftheTel-AvivUniversity.Wetalkedaboutmyphysicsprojectatthat
time (I was working on my Master’s thesis), and I attended his talk, where he
explained the difficulties in the foundations of quantum field theory.Afterwards
heautographedhisbookforme.
I am very grateful to Jens Peder Dahl, Professor Emeritus of the Chemistry
Department of the Technical University of Denmark, for the fruitful and
encouragingcorrespondenceonhiswork.
I wish to thank Dirk Fischer for his assistance with the technical aspects of
thepreparationofthemanuscript,andespeciallyforhishelpinthepreparationof
thefigures.
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August27,2011 9:21 9inx6in TheSupersymmetricDiracEquation b1217-fm
Contents
Preface vii
ListofFigures xiii
1. Introduction 1
2. TheClassicalKeplerProblem 5
2.1 CentralForces . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 TheLaplaceVector . . . . . . . . . . . . . . . . . . . . . . . 8
3. SymmetryoftheClassicalProblem 15
3.1 LieGroupsandLieAlgebras . . . . . . . . . . . . . . . . . 15
3.2 SomeSpecialLieAlgebras . . . . . . . . . . . . . . . . . . 18
3.3 PoissonBrackets . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 TheInverseSquareLaw . . . . . . . . . . . . . . . . . . . . 25
4. FromSolarSystemstoAtoms 31
4.1 RutherfordScattering . . . . . . . . . . . . . . . . . . . . . 31
4.2 ConservationoftheLaplaceVector . . . . . . . . . . . . . . 32
4.3 TheDifferentialCrossSection . . . . . . . . . . . . . . . . . 33
5. TheBohrModel 35
5.1 SpectroscopicSeries . . . . . . . . . . . . . . . . . . . . . . 35
5.2 ThePostulatesoftheModel . . . . . . . . . . . . . . . . . . 35
5.3 ThePredictionsoftheModel . . . . . . . . . . . . . . . . . 37
5.4 CorrectionforFiniteNuclearMass . . . . . . . . . . . . . . 42
6. InterpretationoftheQuantumRules 47
6.1 TheSommerfeld–WilsonQuantizationConditions . . . . . . 47
6.2 deBroglie’sWaveInterpretation . . . . . . . . . . . . . . . . 49
ix