Springer Series on Atomic, Optical, and Plasma Physics 94 Slobodan Danko Bosanac Electromagnetic Interactions Springer Series on Atomic, Optical, and Plasma Physics Volume 94 Editor-in-chief Gordon W.F. Drake, Windsor, Canada Series editors James Babb, Cambridge, USA Andre D. Bandrauk, Sherbrooke, Canada Klaus Bartschat, Des Moines, USA Philip George Burke, Belfast, UK Robert N. Compton, Knoxville, USA Tom Gallagher, Charlottesville, USA Charles J. Joachain, Bruxelles, Belgium Peter Lambropoulos, Iraklion, Greece Gerd Leuchs, Erlangen, Germany Pierre Meystre, Tucson, USA The Springer Series on Atomic, Optical, and Plasma Physics covers in a comprehensive manner theory and experiment in the entire field of atoms and molecules and their interaction with electromagnetic radiation. Books in the series provide a rich source of new ideas and techniques with wide applications in fields such as chemistry, materials science, astrophysics, surface science, plasma technology, advanced optics, aeronomy, and engineering. Laser physics is a particular connecting theme that has provided much of the continuing impetus for new developments in the field, such as quantum computation and Bose-Einstein condensation. The purpose of the series is to cover the gap between standard undergraduate textbooks and the research literature with emphasis on the fundamental ideas, methods, techniques, and results in the field. More information about this series at http://www.springer.com/series/411 Slobodan Danko Bosanac Electromagnetic Interactions 123 SlobodanDanko Bosanac Physical Chemistry RuđerBoškovićInstitute Zagreb Croatia ISSN 1615-5653 ISSN 2197-6791 (electronic) SpringerSeries onAtomic, Optical, andPlasma Physics ISBN978-3-662-52876-1 ISBN978-3-662-52878-5 (eBook) DOI 10.1007/978-3-662-52878-5 LibraryofCongressControlNumber:2016944478 ©Springer-VerlagBerlinHeidelberg2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringer-VerlagGmbHBerlinHeidelberg Preface Electromagnetic field carries the bulk of information about the structure of matter, atomsandmolecules,nucleiorevenelementaryparticles.Theoryinthisrespecthas thebasictask:toretrieve thisinformation fromtheavailable datathat areobtained from experiments. The task is, therefore, finding a connection between the density of charges and the intensity of emitted radiation that they produce when subjected to external agitation; in short the task is solving the inverse intensity problem. However, the task as described is not that simple, although the problem of direct inverse intensity problem is immensely difficult one to solve. Charges that one discusses are basically electrons and protons, or their conglomerates, but they are quantum objects for which the concept of charge density must be fundamentally modified from that for classical charges. Electrons and protons are individually, as classical objects, point-likecharges butasquantumobjectstheymustbetreated as delocalized particles and therefore treated as charge densities. The problem, how- ever, is more difficult than that, because being treated as delocalized particles they aredescribed essentiallybyprobabilitydensitiesandthereforebyassumingthatas charge densities must be taken with great caution. There are circumstances when probabilitydensitycouldbetreatedaschargedensitybuttherearewhenthisisnot a correct assumption. The choice when and how to distinguish between these two concepts, probability density versus charge density, depends on the problem to analyze, which is also sometimes not a simple task. There are essentially four problems to analyze in electromagnetic interactions. One is dynamics of charges under the impact of electromagnetic force, another is radiationthatisproducedbymovingcharges,thethirdisstructureofconglomerates ofcharges,andthefourthistheproblemoffieldinteraction,essentially unification of electromagnetic force and the force that results from radiation. All of these problems have been thoroughly studied, perhaps with exception of the last one, however, extreme states under which charges are placed, extreme states of elec- tromagnetic field that interact with charges and fine details of this interaction have room for further investigations. Placing the field and charges under extreme con- ditions requires theoretical tool that adequately could describe these situations. v vi Preface Relativistic classical and quantum theory are the foremost tools, description of electromagnetic field of finite extent in all dimensions, and also having accurate description in nonrelativistic theoretical tools. Electromagnetic field itself is also essential to be understood because this has directimpactonhowfromexperimentsoneinterpretsstructureofmatter.Thebasic principles of electrodynamics are well established, but with the development of quantum principles and applied on the scale of atoms and smaller another of its feature emerged, which is universally accepted: electromagnetic interaction is mediated by photons, manifestation of particle-like interaction on charges. Origins of the idea for the particle nature of electromagnetic field go back to explaining black body radiation, photoelectric effect, and finally the Compton effect. Success ofthemodelisundisputablebuttherearelimitationsonhowfaritcouldbeapplied, for example in the case of very strong electromagnetic fields. There is an obvious question and this is what is the true nature of the photon model, because despite successfulinexplainingmanyfeaturesofmattertherearesomelimitationsofit,for example in interpretation of the Coulomb law as exchange of photons among charges. The answer to this question is not yet clear, and it should be found with in-depth understanding of solutions of the basic equations of dynamics: Dirac and Maxwell equations, coupled with the relativistic classical equation for particles. Those four mentioned problems are investigated in this book, by giving quali- tative description to get their essence before applying exact tools and these are equations for electromagnetic field and both classical and quantum dynamics, and relativistic and nonrelativistic, for charges. Separate discussion is on equations for relativistic dynamics, Maxwell and Dirac equations, as the essential tools for investigatingthechargesunderextremeconfinementsandtheirinteractionwiththe electromagnetic field. The basic tool for describing photon interaction with matter, quantumelectrodynamics,isnotused;however,itismentionedinthecontextwhen the particle-like exchange of electromagnetic interaction is encountered. Zagreb, Croatia Slobodan Danko Bosanac Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Properties of Elementary Charges . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Charge Density in Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Self Energy of Hydrogen Atom . . . . . . . . . . . . . . . . . . . 4 1.2.2 Charge Density in Molecules . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Charge Density in Hydrogen-Like Atom . . . . . . . . . . . . . 8 1.2.4 Electric Dipole of Molecules . . . . . . . . . . . . . . . . . . . . . 9 1.2.5 Van der Waals Potential. . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Structure of Molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.1 Adiabatic Approximation. . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.2 Hydrogen Atom in Harmonic Oscillator. . . . . . . . . . . . . . 28 2 Relativistic Wave Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1 Unifying Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Homogeneous Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Inhomogeneous Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.1 General Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.2 Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3.3 Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.3.4 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1 Basic Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Vector and Scalar Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.3.1 General Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.3.2 Plane Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3.3 Short Pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.3.4 Finite Width Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3.5 Beam Focusing (Paraxial Approximation). . . . . . . . . . . . . 91 vii viii Contents 4 Charge in Electromagnetic Wave. . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1 Basic Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.1 Classical Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.2 Quantum Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.2 Very Short Electromagnetic Pulse . . . . . . . . . . . . . . . . . . . . . . . 119 4.2.1 Impact on Hydrogen Atom. . . . . . . . . . . . . . . . . . . . . . . 119 4.2.2 Impact On Atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.3 Field Reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5 Confinement of Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.1 General Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.1.1 Uniform Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.1.2 Decay of Two Particle System . . . . . . . . . . . . . . . . . . . . 141 5.2 Confinement by Magnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . 147 5.2.1 Time Dependent Magnetic Field. . . . . . . . . . . . . . . . . . . 150 5.3 Confinement with Electromagnetic Wave . . . . . . . . . . . . . . . . . . 155 5.3.1 Classical Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.3.2 Charge in Standing Wave. . . . . . . . . . . . . . . . . . . . . . . . 157 5.3.3 Generalized Standing Wave . . . . . . . . . . . . . . . . . . . . . . 159 5.3.4 Gaussian Polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.3.5 Quantum Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.4 Extreme Confinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.4.1 One Particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.4.2 Two Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.4.3 Charge Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6 Atom in Electromagnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.1 General Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.2 Atom in Electromagnetic Wave. . . . . . . . . . . . . . . . . . . . . . . . . 219 6.2.1 Basic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 6.2.2 First Order Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.2.3 Second Order Interaction . . . . . . . . . . . . . . . . . . . . . . . . 236 7 Radiation by Charge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 7.1 Radiation Zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 7.2 Radiation by Created Charge . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.3 Radiation by a Bound Charge. . . . . . . . . . . . . . . . . . . . . . . . . . 254 7.3.1 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 7.3.2 Radiation by Rotating Molecule . . . . . . . . . . . . . . . . . . . 260 7.3.3 Radiation by Vibrating Molecule. . . . . . . . . . . . . . . . . . . 269 7.3.4 Spectral Line Shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Appendix A: Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Appendix B: Nonrelativistic Green Functions. . . . . . . . . . . . . . . . . . . . 291 Contents ix Appendix C: Useful Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Appendix D: System of N Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331