Table Of ContentSpringer 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
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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
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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
