The Supersymmetric Dirac Equation: The Application to Hydrogenic Atoms PDF
Preview The Supersymmetric Dirac Equation: The Application to Hydrogenic Atoms
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 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE 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, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-1-84816-797-1 ISBN-10 1-84816-797-0 Typeset by Stallion Press Email: [email protected] 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. v August27,2011 9:21 9inx6in TheSupersymmetricDiracEquation b1217-fm TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk 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. vii August27,2011 9:21 9inx6in TheSupersymmetricDiracEquation b1217-fm TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk 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
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