* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download on the canonical formulation of electrodynamics and wave mechanics
Matter wave wikipedia , lookup
Perturbation theory (quantum mechanics) wikipedia , lookup
Coupled cluster wikipedia , lookup
Two-body Dirac equations wikipedia , lookup
Wave–particle duality wikipedia , lookup
Dirac equation wikipedia , lookup
Quantum chromodynamics wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum field theory wikipedia , lookup
Hidden variable theory wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Perturbation theory wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Renormalization wikipedia , lookup
Noether's theorem wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Path integral formulation wikipedia , lookup
Higgs mechanism wikipedia , lookup
Renormalization group wikipedia , lookup
BRST quantization wikipedia , lookup
Scale invariance wikipedia , lookup
Gauge fixing wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Yang–Mills theory wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Dirac bracket wikipedia , lookup
Canonical quantization wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
History of quantum field theory wikipedia , lookup
ON THE CANONICAL FORMULATION OF ELECTRODYNAMICS AND WAVE MECHANICS By DAVID JOHN MASIELLO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004 For Katie. ACKNOWLEDGMENTS Since August of 1999, I have had the privilege of conducting my Ph.D. research in the group of Prof. Yngve Öhrn and Dr. Erik Deumens at the University of Florida’s Quantum Theory Project. During my time in their group I learned a great deal on the theory of dynamics, in particular, the Hamiltonian approach to dynamics and its applications in electrodynamics and atomic and molecular collisions. I also learned a new appreciation for scientific computing, of which I was previously ignorant. Most importantly, Prof. Öhrn and Dr. Deumens taught me how to think through a physical problem, sort out its underlying dynamical equations, and solve them in a mathematically well-defined manner. I especially want to thank Dr. Erik Deumens, with whom I worked most closely during my Ph.D. research. Erik had a vision when I began my graduate studies and has promoted my work since then to successfully realize it. Along the way, he challenged my creative, mathematical, and physical intuitions and imparted on me a love for theoretical physics. Erik has always taken time to listen to and carefully answer my questions and has always respected my ideas. I thank him for being such an excellent mentor to me. My understanding of physics has also been broadened by many others. Firstly, I would like to thank Dr. Remigio Cabrera-Trujillo, who was a post doctoral associate in the Öhrn-Deumens group, for his guidance especially during my first few years. He has been a great source for advice on many topics from the details of quantum scattering theory to simple computer problems like clearing printer jams. I have joked on many occasions that he was my personal postdoc because he was always so willing to help when I had questions. I would also like to thank the past and present members of my research group, in particular, Dr. Anatol Blass, Dr. Maurı́cio iii Coutinho Neto, Mr. Ben Killian, and Mr. Virg Fermo. In addition, I would like to thank my officemates with whom I have spent almost five years. I thank Ms. Ariana Beste, Mr. Igor Schweigert, and Mr. Tom Henderson for their friendship and camaraderie. I have especially benefited from many conversations with Tom Henderson on aspects of quantum mechanics, quantum field theory, and classical electrodynamics. Several other faculty and staff at the Quantum Theory Project, and the Departments of Chemistry, Physics, and Mathematics at the University of Florida have also encouraged and promoted my Ph.D. research. At the Quantum Theory Project, I thank Prof. Jeff Krause for taking sincere interest in my research and always finding time to listen to me and provide guidance. I have taught with Jeff on a few occasions and have known him to be a great teacher as well as mentor. I thank Prof. Henk Monkhorst for his kindness and good humor. I will especially miss all of the LATEX battles that we have fought over the past several years. In addition, I would like to thank Dr. Ajith Perera for his friendship and patience. I thank the staff, especially Ms. Judy Parker and Ms. Coralu Clements, for keeping all of the administrative aspects of my graduate studies running smoothly. I would also like to thank the custodians Sandra and Rhonda who have been so friendly to me and who keep the Quantum Theory Project impeccably clean. In the Department of Chemistry, I would like to thank the late Prof. Carl Stoufer, who was my undergraduate advisor during my first year, for his friendship, wisdom, and advice. Throughout my entire undergraduate career we would meet a few times per year to catch up over coffee and donuts. It was due to Carl’s support that I was given the opportunity to study at the Quantum Theory Project. In the Department of Physics, I would like to thank Prof. Richard Woodard, from whom I learned quantum field theory. Richard is very passionate about physics and is perhaps the best teacher that I have known. From him I gained a deeper understanding of perturbation theory and its applications in iv quantum electrodynamics. In the Department of Mathematics, I would like to thank Prof. Scott McCullough, who was effectively my undergraduate advisor. While I was an undergraduate student of Scott’s, he imparted to me a deep appreciation for mathematics and a particular interest in analysis. Scott was an excellent teacher and mentor, and under his guidance, my undergraduate research was awarded by the College of Liberal Arts and Sciences. Outside of the University of Florida, many others have contributed to my scientific career. At the University of Central Florida’s Center for Research and Education in Optics and Lasers, I would like to thank Prof. Leonid Glebov, Prof. Kathleen Richardson, and Prof. Boris Zel’dovich for first introducing me to the world of quantum physics. In particular, Prof. Glebov and Prof. Richardson greatly stimulated and encouraged my interests. With their recommendation, I received a fellowship to study at the University of Bordeaux’s Department of Physics and Centre de Physique Moléculaire Optique et Herzienne. While in Bordeaux, France, I had the pleasure of working in the research group of Prof. Laurent Sarger. I wish to thank Prof. Sarger as well as his colleagues for their hospitality during my time in France and for introducing me to the field of atomic and molecular physics, which is the setting for this dissertation. Lastly, I would like to thank my family. My mother and father have always provided unconditional love, support, and guidance to me. They have encouraged my inquisitiveness of Nature and have promoted my education from kindergarten to Ph.D. I thank my inlaws for their love and support and for providing a home away from home while in graduate school. In conclusion, I would like to thank my wonderful wife Katie for her encouragement, companionship, and unending love. v TABLE OF CONTENTS page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 1.2 1.3 1.4 1.5 1.6 2 Physical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Historical and Mathematical Background . . . . . . . . . . . . . . 2 1.2.1 Gauge Symmetry of Electrodynamics . . . . . . . . . . . . 3 1.2.2 Gauge Symmetry of Electrodynamics and Wave Mechanics 5 Approaches to the Solution of the Maxwell-Schrödinger Equations 7 Canonical Formulation of the Maxwell-Schrödinger Equations . . . 11 Format of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 14 Notation and Units . . . . . . . . . . . . . . . . . . . . . . . . . . 15 THE DYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 2.2 3 1 Lagrangian Formalism . . . . . . . . . . . . . . . . 2.1.1 Hamilton’s Principle . . . . . . . . . . . . . . 2.1.2 Example: The Harmonic Oscillator in (q k , q̇ k ) 2.1.3 Geometry of TQ . . . . . . . . . . . . . . . . Hamiltonian Formalism . . . . . . . . . . . . . . . . 2.2.1 Example: The Harmonic Oscillator in (q a , pa ) 2.2.2 Symplectic Structure and Poisson Brackets . 2.2.3 Geometry of T∗ Q . . . . . . . . . . . . . . . ELECTRODYNAMICS AND QUANTUM MECHANICS 3.1 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 19 21 22 24 25 26 27 . . . . . . . . 29 Quantum Mechanics in the Presence of an Electromagnetic Field 3.1.1 Time-Dependent Perturbation Theory . . . . . . . . . . . 3.1.2 Fermi Golden Rule . . . . . . . . . . . . . . . . . . . . . 3.1.3 Absorption of Electromagnetic Radiation by an Atom . . 3.1.4 Quantum Electrodynamics in Brief . . . . . . . . . . . . . Classical Electrodynamics Specified by the Sources ρ and J . . . 3.2.1 Electromagnetic Radiation from an Oscillating Source . . 3.2.2 Electromagnetic Radiation from a Gaussian Wavepacket . vi . . . . . . . . . . . . . . . . 29 30 33 34 36 40 41 47 4 CANONICAL STRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1 4.2 4.3 4.4 4.5 4.6 4.7 5 . . . . . . 56 56 57 59 61 66 . 66 . . . . . . 69 70 71 78 79 81 NUMERICAL IMPLEMENTATION . . . . . . . . . . . . . . . . . . . . 84 5.1 5.2 5.3 5.4 5.5 6 Lagrangian Electrodynamics . . . . . . . . . . . . . . . . . . . . 4.1.1 Choosing a Gauge . . . . . . . . . . . . . . . . . . . . . . 4.1.2 The Lorenz and Coulomb Gauges . . . . . . . . . . . . . Hamiltonian Electrodynamics . . . . . . . . . . . . . . . . . . . 4.2.1 Hamiltonian Formulation of the Lorenz Gauge . . . . . . 4.2.2 Poisson Bracket for Electrodynamics . . . . . . . . . . . . Hamiltonian Electrodynamics and Wave Mechanics in Complex Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hamiltonian Electrodynamics and Wave Mechanics in Real Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Coulomb Reference by Canonical Transformation . . . . . . 4.5.1 Symplectic Transformation to the Coulomb Reference . . 4.5.2 The Coulomb Reference by Change of Variable . . . . . . Electron Spin in the Pauli Theory . . . . . . . . . . . . . . . . . Proton Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . Maxwell-Schrödinger Theory in a Complex Basis . . . . Maxwell-Schrödinger Theory in a Real Basis . . . . . . 5.2.1 Overview of Computer Program . . . . . . . . . 5.2.2 Stationary States: s- and p-Waves . . . . . . . . 5.2.3 Nonstationary State: Mixture of s- and p-Waves 5.2.4 Free Electrodynamics . . . . . . . . . . . . . . . 5.2.5 Analysis of Solutions in Numerical Basis . . . . Symplectic Transformation to the Coulomb Reference . 5.3.1 Numerical Implementation . . . . . . . . . . . . 5.3.2 Stationary States: s- and p-Waves . . . . . . . . 5.3.3 Nonstationary State: Mixture of s- and p-Waves 5.3.4 Free Electrodynamics . . . . . . . . . . . . . . . 5.3.5 Analysis of Solutions in Coulomb Basis . . . . . Asymptotic Radiation . . . . . . . . . . . . . . . . . . Proton Dynamics in a Real Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 88 90 93 93 93 95 99 101 102 102 103 103 103 108 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 APPENDIX A GAUGE TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . . 113 A.1 A.2 Gauge Symmetry of Electrodynamics . . . . . . . . . . . . . . . . 113 Gauge Symmetry of Quantum Mechanics . . . . . . . . . . . . . . 115 vii B GREEN’S FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 B.1 B.2 B.3 C The Dirac δ-Function . . . . . . . . . . . . . . . . . . . . . . . . . 117 The ∇2 Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 The ∂ 2 Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 THE TRANSVERSE PROJECTION OF A(x, t) . . . . . . . . . . . . . 122 C.1 C.2 C.3 Tensor Calculus . . T 0kk (x0 , t) Integrals C.2.1 Inside Step . C.2.2 Outside Step Building AT (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 127 129 131 133 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 viii Figure LIST OF FIGURES page 2–1 The configuration manifold Q = S2 is depicted together with the tangent plane Tqk Q at the point q k ∈ Q. . . . . . . . . . . . . . . . 22 3–1 The coefficient 1 of the unscattered plane wave exp(ik · x) is analogous to the 1 part of the S-matrix, while the scattering amplitude fk (Ω) which modulates the scattered spherical wave exp(ikr)/r is analogous to the iT part. . . . . . . . . . . . . . . . . . . . . . . . 38 3–2 In the radiation zone, the observation point x is located far from the source J. In this case the distance |x − x0 | ≈ r − n̂ · x0 . . . . . . . . 44 3–3 The differential power dP/dΩ or radiation pattern corresponding to an oscillating electric dipole verifies that no radiation is emitted in the direction of the dipole moment. . . . . . . . . . . . . . . . . . . 46 3–4 The norms of J and A are plotted with different velocities along the x-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3–5 The trajectory or world line r(t) of the charge is plotted. . . . . . . . 49 3–6 The bremsstrahlung radiation from a charged gaussian wavepacket moves out on the smeared light cone with maximum at x = ct. . . . 50 3–7 The radiation pattern given by (3.63) shows the characteristic dipole pattern at lowest order. . . . . . . . . . . . . . . . . . . . . . . . . 53 4–1 A limited but relevant portion of the gauge story in the Lagrangian formalism is organized in this picture. . . . . . . . . . . . . . . . . 59 4–2 The Hamiltonian formulation of the gauge story is organized in this picture with respect to the previous Lagrangian formulation. . . . . 65 4–3 Commutivity diagram representing the change of coordinates (q, p) to (p̃, q̃) at both the Lagrangian and equation of motion levels. . . . . 79 5–1 Schematic overview of ENRD computer program. . . . . . . . . . . . 92 5–2 Phase space contour for the coefficients of the vector potential A and its momentum Π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5–3 Phase space contour for the coefficients of the real-valued Schrödinger field Q and its momentum P. . . . . . . . . . . . . . . . . . . . . . 94 ix 5–4 Phase space contour for the coefficients of the scalar potential Φ and its momentum Θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5–5 Real part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. . . . . . . . . . . . . . . 97 5–6 Imaginary part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. . . . . . . . . . . . 97 5–7 Probability for the electron to be in a particular basis eigenstate. . . . 98 5–8 Phase space of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. . . . . . . . . . . . . . . 98 5–9 Real part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. . . . . . . . . . . . . . . 104 5–10 Imaginary part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. . . . . . . . . . . . 104 5–11 Probability for the electron to be in a particular basis eigenstate. . . . 105 5–12 Phase space of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. . . . . . . . . . . . . . . 105 5–13 Schematic picture of the local and asymptotic basis proposed for the description of electromagnetic radiation and electron ionization. . . 107 B–1 The trajectory or world line r(t) of a massive particle moves from past to future within the light cone. . . . . . . . . . . . . . . . . . . . . 120 C–1 Since à = h̃v, the transverse vector potential Ã⊥ = [v − k(k · v)/k 2 ]h̃ and the longitudinal vector potential Ãk = [k(k · v)/k 2 ]h̃, where h̃ is a scalar function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 x Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ON THE CANONICAL FORMULATION OF ELECTRODYNAMICS AND WAVE MECHANICS By David John Masiello May 2004 Chair: Nils Yngve Öhrn Major Department: Chemistry The interaction of electromagnetic radiation with atoms or molecules is often understood when the timescale for the electromagnetic decay of an excited state is separated by orders of magnitude from the timescale of the excited state’s dynamics. In these cases, the two dynamics may be treated separately and a perturbative Fermi golden rule analysis is appropriate. However, there do exist situations where the dynamics of the electromagnetic field and the atomic or molecular system occurs on the same timescale, e.g., photon-exciton dynamics in conjugated polymers and atom-photon dynamics in cold atom collisions. Nonperturbative methods for the solution of the coupled nonlinear MaxwellSchrödinger differential equations are developed in this dissertation which allow for the atomic or molecular and electromagnetic dynamics to occur on the same timescale. These equations have been derived within the Hamiltonian or canonical formalism. The canonical approach to dynamics, which begins with the Maxwell and Schrödinger Lagrangians together with a Lorenz gauge fixing term, yields a set of first order Hamilton equations which form a well-posed initial value problem. That is, their solution is uniquely determined and known in principle once xi the initial values for each of the associated dynamical variables are specified. The equations are also closed since the Schrödinger wavefunction is chosen to be the source for the electromagnetic field and the electromagnetic field reacts back upon the wavefunction. In practice, the Maxwell-Schrödinger Lagrangian is represented in a basis of gaussian functions with different widths and centers. Application of the calculus of variations leads to a set of Euler-Lagrange equations that, for that choice of basis, form and represent the coupled first order Maxwell-Schrödinger equations. In the limit of a complete basis these equations are exact and for any finite choice of basis they provide an approximate system of dynamical equations that can be integrated in time and made systematically more accurate by enriching the basis. These equations are numerically implemented for a basis of arbitrary finite rank. The dynamics of the basis-represented Maxwell-Schrödinger system is investigated for the spinless hydrogen atom interacting with the electromagnetic field. xii CHAPTER 1 INTRODUCTION Chemistry encompasses a broad range of Nature that varies over orders of magnitude in energy from the ultracold nK Bose-Einstein condensation temperatures [1, 2] to the keV collision energies that produce the Earth’s aurorae [3–5]. At the most fundamental level, the study of chemistry is the study of electrons and nuclei. The interaction of electrons and nuclei throughout this energy regime is mediated by the photon which is the quantum of the electromagnetic field. The equations which govern the dynamics of electrons, nuclei, and photons are therefore the same equations which govern all of chemistry [6]. They are the Schrödinger equation [7, 8] iΨ̇ = HΨ (1.1) and Maxwell’s equations [9] ∇ · E = 4πρ ∇×B = 4π Ė J+ c c ∇·B = 0 ∇×E+ Ḃ = 0. (1.2) c As they stand these equations are uncoupled. The solutions of the Schrödinger equation (1.1) do not a priori influence the solutions of the Maxwell equations (1.2) and vice versa. The development of analytic and numerical methods for the solution of the coupled Maxwell-Schrödinger equations is the main purpose of this dissertation. Before delving into the details of these methods a physical motivation as well as a historical and mathematical background is provided. 1.1 Physical Motivation Many situations of physical interest are described by the system of MaxwellSchrödinger equations. Often these situations involve electromagnetic processes that occur on drastically different timescales from that of the matter. An example of such 1 2 a situation is the stimulated absorption or emission of electromagnetic radiation by a molecule. The description of this process by (1.1) and (1.2) accounts for a theoretical understanding of all of spectroscopy, which has provided an immense body of chemical knowledge. However, there do exist situations where the dynamics of the electromagnetic field and the matter occur on the same timescale. For example, in solid state physics certain electronic wavepackets exposed to strong magnetic fields in semiconductor quantum wells are predicted to demonstrate rapid decoherence [10]. The dynamics of the incident field, the electronic wavepacket, and the phonons that it emits is coupled and occurs on the same femtosecond timescale. In atomic physics, the long timescale for the dynamics of cold and ultracold collisions of atoms in electromagnetic traps has been observed to exceed lifetimes of excited states, which are on the order of 10 −8 s. This means that spontaneous emission can occur during the course of collision and may significantly alter the atomic collision dynamics [11, 12]. Cold atom phenomena are also being merged with cavity quantum electrodynamics to realize single atom lasers [13–15]. The function of these novel devices is based on strong coupling of the atom to a single mode of the resonant cavity. Lastly, in polymer chemistry, ultrafast light emission has been detected in certain ladder polymer films following ultrafast laser excitation [16]. A fundamental understanding of the waveguiding process that occurs in these polymers is unknown. It is precisely these situations, where the electromagnetic and matter dynamics occur on the same timescale and are strongly coupled, that are the motivation for this dissertation. 1.2 Historical and Mathematical Background The history of the Maxwell-Schrödinger equations dates back to the early twentieth century when the founding fathers of quantum mechanics worked out the theoretical details of the interaction of electrodynamics with quantum mechanics [17]. It was realized early on that the electromagnetic coupling to matter was through 3 the potentials Φ and A, and not the fields E and B themselves [6, 18, 19]. The potentials and fields are related by E = −∇Φ − Ȧ/c B=∇×A (1.3) which can be confirmed by inspecting the homogeneous Maxwell equations in (1.2). Unlike in classical theory where the potentials were introduced as a convenient mathematical tool, the quantum theory requires the potentials and not the fields. That is, the potentials are fundamental dynamical variables of the quantum theory but the fields are not. A concrete demonstration of this fact was presented in 1959 by Aharonov and Bohm [20]. 1.2.1 Gauge Symmetry of Electrodynamics It was well known from the classical theory of electrodynamics [9] that working with the potentials leads to a potential form of Maxwell’s equations that is more flexible than that in terms of the fields alone (1.2). In potential form, Maxwell’s equations become ∇2 A − h Φ̇ i 4π Ä − ∇ ∇ · A + = − J 2 c c c ∇ · Ȧ ∇2 Φ + = −4πρ. c (1.4a) (1.4b) The homogeneous Maxwell equations are identically satisfied. These potential equations enjoy a symmetry that is not present in the field equations (1.2). This symmetry is called the gauge symmetry and can be generated by the transformation A → A0 = A + ∇F Φ → Φ0 = Φ − Ḟ /c, (1.5) where F is a well-behaved but otherwise arbitrary function called the gauge generator. Applying this gauge transformation to the potentials in (1.4) leads to exactly the same set of potential equations. In other words, these equations are invariant under arbitrary gauge transformation or are gauge invariant. They possess the 4 full gauge symmetry. Notice also that the electric and magnetic fields are gauge invariant. In fact, it turns out that all physical observables are gauge invariant. That electrodynamics possesses gauge symmetry places it in a league of theories known as gauge theories [21]. These theories include general relativity [22, 23] and Yang-Mills theory [24–26]. Gauge theories all suffer from an indeterminateness due to their gauge symmetry. In an effort to deal with this indeterminateness, it is common to first eliminate the symmetry (usually up to the residual symmetry; see Chapter 4) by gauge fixing and then work within that particular gauge. That is, the flexibility implied by the gauge transformation (1.5) allows for the potentials to satisfy certain constraints. These constraints imply a particular choice of gauge and gauge generator. Gauge fixing is the act of constraining the potentials to satisfy a certain constraint throughout space-time. For example, in electrodynamics the potential equations (1.4) ∇2 A − h Ä Φ̇ i 4π − ∇ ∇ · A + =− J 2 c c c ∇ · Ȧ = −4πρ ∇2 Φ + c (1.4) form an ill-posed initial value problem. However, they can be converted to a welldefined initial value problem by adding an equation of constraint to them. For example, adding the constraint Φ̇/c + ∇ · A = 0 leads to the well-defined Lorenz gauge equations ∇2 A − Ä 4π =− J 2 c c ∇2 Φ − Φ̈ = −4πρ c2 (1.6) while adding ∇ · A = 0 leads to the well-defined Coulomb gauge equations ∇2 A − Ä 4π = − JT 2 c c ∇2 Φ = −4πρ, (1.7) where JT is the transverse projection of the current J (see Appendix A). There are many other choices of constraint, each leading to a different gauge. It is always 5 possible to find a gauge function that will transform an arbitrary set of potentials to satisfy a particular gauge constraint. The subject of the gauge symmetry of electrodynamics, which is a subtle but fundamental aspect of this dissertation, is discussed in detail in Chapter 4. In particular, it will be argued that fixing a particular gauge, which in turn eliminates the gauge from the theory, is not necessarily optimal. Rather, it is stressed that the gauge freedom is a fundamental variable of the theory and has its own dynamics. 1.2.2 Gauge Symmetry of Electrodynamics and Wave Mechanics Since the gauge symmetry of electrodynamics was well known, it was noticed by the founding fathers that if quantum mechanics is to be coupled to electrodynamics, then the Schrödinger equation (1.1) needs to be gauge invariant as well. The most simple way of achieving this is to require the Hamiltonian appearing in (1.1) to be of the form H= [P − qA/c]2 + V + qΦ, 2m (1.8) where P is the quantum mechanical momentum, V is the potential energy, and m is the mass of the charge q. This is in analogy with the Hamiltonian for a classical charge in the presence of the electromagnetic field [27, 28]. The coupling scheme embodied in (1.8) is known as minimal coupling, since it is the simplest possible gauge invariant coupling imaginable. The gauge symmetry inherent in the combined system of Schrödinger’s equation and Maxwell’s equations in potential form can be generated by the transformation A → A0 = A + ∇F Φ → Φ0 = Φ − Ḟ /c Ψ → Ψ0 = exp(iqF/c)Ψ. (1.9) The transformation on the wavefunction is called a local gauge transformation and differs from the global gauge transformation exp(iθ), where θ is a constant. These 6 global gauge transformations are irrelevant in quantum mechanics where the wavefunction is indeterminate up to a global phase. Application of the gauge transformation (1.9) to the Schrödinger equation with Hamiltonian (1.8) and to Maxwell’s equations in potential form leads to exactly the same equations after the transformation. Therefore, like the potential equations (1.4) by themselves, the system of Maxwell-Schrödinger equations iΨ̇ = ∇2 A − [P − qA/c]2 Ψ + V Ψ + qΦΨ 2m h Φ̇ i 4π Ä − ∇ ∇ · A + = − J 2 c c c ∇ · Ȧ = −4πρ ∇2 Φ + c (1.10) (1.11a) (1.11b) is invariant under the gauge transformation (1.9). There are several other symmetries that are enjoyed by this system of equations. For example, they are invariant under spatial rotations, nonrelativistic (Galilei) boosts, and time reversal. As a result, the Maxwell-Schrödinger equations enjoy charge, momentum, angular momentum, and energy conservation. That each continuous symmetry gives rise to an associated conservation law was proven by Emmy Noether in 1918 (see Goldstein [27], José and Saletan [28], and Abraham and Marsden [29], and the references therein). This issue is discussed in Chapter 2 in greater detail. It is worthwhile mentioning that the Maxwell-Schrödinger equations are obtainable as the nonrelativistic limit of the Maxwell-Dirac equations iΨ̇D = βmc2 ΨD + cα · [P − qA/c]ΨD + qΦΨD h 4π Ä Φ̇ i ∇ A− 2 −∇ ∇·A+ = − J c c c ∇ · Ȧ = −4πρ ∇2 Φ + c 2 (1.12) (1.13a) (1.13b) 7 which are the equations of quantum electrodynamics (QED) [19, 24, 30]. Here the wavefunction ΨD is a 4-component spinor where the first two components represent the electron and the second two components represent the positron, each with spin1/2. The matrices β and α are related to the Pauli spin matrices [7, 8] and c is the velocity of light. This system of equations possesses each of the symmetries of the Maxwell-Schrödinger equations and in addition is invariant under relativistic boosts. 1.3 Approaches to the Solution of the Maxwell-Schrödinger Equations Solving the Maxwell-Schrödinger equations as a coupled and closed system em- bodies the theory of radiation reaction [9, 26, 31], which is a main theme of this dissertation. However, it should first be pointed out that (1.1) and (1.2) are commonly treated separately. In these cases, the effects of one system on the other are handled in one of the following two ways: • The arrangement of charge and current is specified and acts as a source for the electromagnetic field according to (1.2). • The dynamics of the electromagnetic field is specified and modifies the dynamics of the matter according to (1.1). It is not surprising that either of these approaches is valid in many physical situations. Most of the theory of electrodynamics, in which the external sources are prescribed, fits into the first case, while all of classical and quantum mechanics in the presence of specified external fields fits into the second. As a further example of the first case, the dipole power radiated by oscillating dipoles generated by charge transfer processes in the interaction region of p − H collisions can be computed in a straightforward manner [32, 33]. It is assumed that the dynamics of the oscillating dipole is known and is used to compute the dipole radiation, but this radiation does not influence the p − H collision. As a result energy, momentum, and angular momentum are not conserved between the proton, hydrogen atom, and electromagnetic field system. As a further example of the second case, the effects of stimulated absorption or emission of electromagnetic 8 radiation by a molecular target can be added to the molecular quantum mechanics as a first order perturbative correction. The electrodynamics is specified and perturbs the molecule but the molecule does not itself influence the electrodynamics. This approach, which is known as Fermi’s golden rule (see Chapter 3 and Merzbacher [7], Craig and Thirunamachandran [34], and Schatz and Ratner [35]) is straightforward and barring certain restrictions can be applied to many physical systems. The system of Maxwell-Schrödinger equations or its relativistic analog can be closed and is coupled when the Schrödinger wavefunction Ψ, which is the solution of (1.1), is chosen to be the source for the scalar potential Φ and vector potential A in (1.4). In particular, the sources of charge ρ and current J, which produce the electromagnetic potentials according to (1.4), involve the solutions Ψ of the Schrödinger equation according to ρ = qΨ∗ Ψ J = q Ψ∗ [−i∇ − qA/c]Ψ + Ψ[i∇ − qA/c]Ψ∗ /2m. (1.14) On the other hand, the wavefunction Ψ is influenced by the potentials that appear in the Hamiltonian H in (1.8). The interpretation of the Schrödinger wavefunction as the source for the electromagnetic field was Schrödinger’s electromagnetic hypothesis, which dates back to 1926. The discovery of the quantum mechanical continuity equation and its similarity to the classical continuity equation of electrodynamics only reinforced the hypothesis. However, it implied the electron to be smeared out throughout the atom and not located at a discrete point, which is in contradiction to the accepted Born probabilistic or Copenhagen interpretation. Schrödinger’s wave mechanics had some success, especially with the interaction of the electromagnetic field with bound states, but failed to properly describe scattering states due to the probabilistic nature of measurement of the wavefunction. In addition, certain properties of electromagnetic radiation were found to be inconsistent with experiment. 9 Schrödinger’s electromagnetic hypothesis was extended by Fermi in 1927 and later by Crisp and Jaynes in 1969 [36] to incorporate the unquantized electromagnetic self-fields into the theory. That is, the classical electromagnetic fields produced by the atom were allowed to act back upon the atom. The solutions of this extended semiclassical theory captured certain aspects of spontaneous emission as well as frequency shifts like the Lamb shift. However, it was quickly noticed that some deviations from QED existed [37]. For example, Fermi’s and Jaynes’s theories predicted a time-dependent form for spontaneous decay that is not exponential. There are many properties that are correctly predicted by this semiclassical theory and are also in agreement with QED. In the cases where the semiclassical theory disagrees with QED [37], it has always been experimentally verified that QED is correct. Nevertheless, the semiclassical theory does not suffer from the mathematical and logical difficulties that are present in QED. To this end, the semiclassical theory, when it is correct, provides a useful alternative to the quantum field theory. It is generally simpler and its solutions provide a more detailed dynamical description of the interaction of an atom with the electromagnetic field. Since 1969 many others have followed along the semiclassical path of Crisp and Jaynes. Nesbet [38] computed the gauge invariant energy production rate from a many particle system. Cook [39] used a density operator approach to account for spontaneous emission without leaving the atomic Hilbert space. Barut and Van Huele [40] and Barut and Dowling [41, 42] formulated a self-field quantum electrodynamics for Schrödinger, Pauli, Klein-Gordon, and Dirac matter theories. They were able to eliminate all electromagnetic variables in favor of Green’s function integrals over the sources and were able to recover the correct exponential spontaneous decay from an excited state. Some pertinent critiques of this work are expressed by Bialynicki-Birula in [43] and by Crisp in [44]. Bosanac [45–47] and Došlić and 10 Bosanac [48] argued that the instantaneous effects of the self interaction are unphysical. As a result, they formulated a theory of radiation reaction based on the retarded effects of the self-fields. Milonni, Ackerhalt, and Galbraith [49] predicted chaotic dynamics in a collection of two-level atoms interacting with a single mode of the classical electromagnetic field. Crisp himself has contributed some of the finest work in semiclassical theory. He computed the radiation reaction associated with a rotating charge distribution [50], the atomic radiative level shifts resulting from the solution of the semiclassical nonlinear integro-differential equations [51], the interaction of an atomic system with a single mode of the quantized electromagnetic field [52, 53], and the extension of the semiclassical theory to include relativistic effects [54]. Besides semiclassical theory, a vast amount of research has been conducted in the quantum theory of electrodynamics and matter. QED [19, 24, 30, 55] (see Chapter 3), which is the fully relativistic and quantum mechanical theory of electrons and photons, has been found to agree with all associated experiments. The coupled equations of QED can be solved nonperturbatively [56, 57], but are most often solved by resorting to perturbative methods. As was previously mentioned, there are some drawbacks to these methods that are not present in the semiclassical theory. In addition to pure QED in terms of electrons and photons, there has also been an increasing interest in molecular quantum electrodynamics [34]. Power and Thirunamachandran [58, 59], Salam and Thirunamachandran [60], and Salam [61] have used perturbative methods within the minimal-coupling and multipolar formalisms to study the quantized electromagnetic field surrounding a molecule. In particular, they have clarified the relationship between the two formalisms and in addition have calculated the Poynting vector and spontaneous emission rates for magnetic dipole and electric quadrupole transitions in optically active molecules. 11 In both the semiclassical and quantum mechanical context the self-energy of the electron has been studied [62–65]. The self-energy arises naturally in the minimal coupling scheme as the qΦ term in the Hamiltonian (1.8). More specifically, the electron’s self-energy in the nonrelativistic theory is defined as U= R d3 xqΦ(x, t)Ψ∗ (x, t)Ψ(x, t) = V R d3 x V R d 3 x0 V ρ(x, t)ρ(x0 , t) . |x − x0 | (1.15) As a result of the qΦ term, the Schrödinger equation (1.10) is nonlinear in Ψ. It resembles the nonlinear Schrödinger equation [66] iu̇ = −a(d2 u/dx2 ) + b|u|2 u (1.16) which arises in the modeling of Bose-Einstein condensates with the Gross-Pitaevskii equation and in the modeling of superconductivity with the Ginzburg-Landau equation. In the relativistic theory, the electron is forced to have no structure due to relativistic invariance. As a result, the corresponding self-energy is infinite. On the other hand, the electron may have structure in the nonrelativistic theory. Consequently, the self-energy is finite. The self-energy of the electron will be discussed in Chapter 4 in more detail. 1.4 Canonical Formulation of the Maxwell-Schrödinger Equations The work presented in this dissertation [67] continues the semiclassical story originally formulated by Fermi, Crisp, and Jaynes. Unlike other semiclassical and quantum mechanical theories of electrodynamics and matter where the gauge is fixed at the beginning, it will be emphasized that the gauge is a fundamental degree of freedom in the theory and should not be eliminated. As a result, the equations of motion are naturally well-balanced and form a well-defined initial value problem when the gauge freedom is retained. This philosophy was pursued early on by Dirac, Fock, and Podolsky [68] (see Schwinger [19]) in the context of the Hamiltonian 12 formulation of QED. However, their approach was quickly forgotten in favor of the more practical Lagrangian based perturbation theory that now dominates the QED community. More recently, Kobe [69] studied the Hamiltonian approach in semiclassical theory. Unfortunately, he did not recognize the dynamical equation associated with the gauge and refers to it as a meaningless equation. It is believed that the Hamiltonian formulation of dynamics offers a natural and powerful theoretical approach to the interaction of electrodynamics and wave mechanics that has not yet been fully explored. To this end, the Hamiltonian or canonical formulation of the Maxwell-Schrödinger dynamics is constructed in this dissertation. (Canonical means according to the canons, i.e. standard or conventional.) The associated work involves nonperturbative analytic and numerical methods for the solution of the coupled and closed nonlinear system of Maxwell-Schrödinger equations. The flexibility inherent in these methods captures the nonlinear and nonadiabatic effects of the coupled system and has the potential to describe situations where the atomic and electromagnetic dynamics occur on the same timescale. The canonical formulation is set up by applying the time-dependent variational principle to the Schrödinger Lagrangian LSch = iΨ∗ Ψ̇ − [i∇ − qA/c]Ψ∗ · [−i∇ − qA/c]Ψ − V Ψ∗ Ψ − qΦΨ∗ Ψ, 2m (1.17) and Maxwell Lagrangian together with a Lorenz gauge fixing term, i.e., [Φ̇/c + ∇ · A]2 8π [−Ȧ/c − ∇Φ]2 − [∇ × A]2 [Φ̇/c + ∇ · A]2 = − . 8π 8π LLMax = LMax − (1.18) This yields a set of coupled nonlinear first order differential equations of the form ω η̇ = ∂H/∂η (1.19) 13 where ω is a symplectic form, η is a column vector of the dynamical variables, and H is the Maxwell-Schrödinger Hamiltonian (see Chapter 4). These matrix equations form a well-defined initial value problem. That is, the solution to these equations is uniquely determined and known in principle once the initial values for each of the dynamical variables η are specified. These equations are also closed since the Schrödinger wavefunction acts as the source, which is nonlinear (see J in 1.14), for the electromagnetic potentials and these potentials act back upon the wavefunction. By representing each of the dynamical variables in a basis of gaussian functions GK , P i.e., η(x, t) = K GK (x)ηK (t), where the time-dependent superposition coefficients ηK (t) carry the dynamics, the time-dependent variational principle generates a hierarchy of approximations to the coupled Maxwell-Schrödinger equations. In the limit of a complete basis these equations recover the exact Maxwell-Schrödinger theory, while in any finite basis they form a basis representation that can systematically be made more accurate with a more robust basis. The associated basis equations have been implemented in a Fortran 90 computer program [70] that is flexible enough to handle arbitrarily many gaussian basis functions, each with adjustable widths and centers. In addition, a novel numerical convergence accelerator has been developed based on removing the large Coulombic fields surrounding a charge (that can be computed analytically from Gauss’s law, i.e., ∇ · E = −∇2 Φ = 4πρ, once the initial conditions are provided) by applying a certain canonical transformation to the dynamical equations. The canonical transformation separates the dynamical radiation from the Coulombic portion of the field. This in turn allows the basis to describe only the dynamics of the radiation fields and not the large Coulombic effects. The canonical transformed equations, which are of the form ω̃ η̃˙ = ∂ H̃/∂ η̃, have been added to the existing computer program and the convergence of the solution of the Maxwell-Schrödinger equations is studied. 14 The canonical approach to dynamics enjoys a deep mathematical foundation and permits a general application of the theory to many physical problems. In particular, the dynamics of the hydrogen atom interacting with its electromagnetic field has been investigated for both stationary and superpositions of stationary states. Stationary state solutions of the combined hydrogen atom and electromagnetic field system as well as nonstationary states that produce electromagnetic radiation have been constructed. This radiation carries away energy, momentum, and angular momentum from the hydrogen atom such that the total energy, momentum, angular momentum, and charge of the combined system are conserved. A series of plots are presented to highlight this atom-field dynamics. 1.5 Format of Dissertation A tour of the Lagrangian and Hamiltonian dynamics is presented in Chapter 2. Hamilton’s principle is applied to the derivation of the Euler-Lagrange equations of motion. Emphasis is placed on the Hamiltonian formulation of dynamics, which is presented from the modern point of view which makes connection with symplectic geometry. To this end, both configuration space and phase space geometries are discussed. In Chapter 3, the Schrödinger and Maxwell dynamics will be presented from the point of view of perturbation theory. In the Schrödinger theory, the electromagnetic field is treated as a perturbation on the stationary states of an atomic or molecular system. In the long time limit, the Fermi golden rule accounts for stimulated transitions between these states. As an example, the absorption cross section is calculated for an atom in the presence of an external field. QED is discussed to emphasize the success of perturbation theory. In the Maxwell theory, the electromagnetic fields arising from specified sources of charge and current are presented. The first order (electric dipole) multipolar contributions to the electromagnetic field are calculated. Lastly, the bremsstrahlung from a gaussian charge distribution is analyzed. 15 Chapter 4 contains the main body of the dissertation, which is on the Hamiltonian or canonical approach to the Maxwell-Schrödinger dynamics. Nonperturbative analytic methods are constructed for the solution of the associated coupled and nonlinear equations. The gauge symmetry is discussed in detail and exploited to cast the Maxwell-Schrödinger equations into a well-defined initial value problem. The theory of canonical or symplectic transformations is used to construct a special transformation to remove the Coulombic contribution to the dynamical variables. The well-defined Maxwell-Schrödinger theory from Chapter 4 is numerically implemented in Chapter 5. The associated equations of motion are expanded into a basis of gaussian functions, which renders the partial differential equations as ordinary differential equations. These equations are coded in Fortran 90. In addition, the (canonical transformed) equations associated with the Coulomb reference are incorporated into the existing code. The dynamics of the spinless hydrogen atom interacting with the electromagnetic field are presented in a series of plots. A summary and conclusion of the dissertation are presented in Chapter 6. 1.6 Notation and Units A brief statement should be made about notation. All work will be done in the (1+3)-dimensional background of special relativity with diagonal metric tensor gαβ = g αβ with elements g00 = g 00 = 1 and g11 = g22 = g33 = −1. All 3-vectors will be written in bold faced Roman while all 4-vectors will be written in italics. As usual, Greek indices run over 0, 1, 2, 3 or ct, x, y, z and Roman indices run over 1, 2, 3 or x, y, z. The summation convention is employed over repeated indices. For example, the 4-potential Aµ = (A0 , Ak ) = (Φ, A) and Aµ = gµν Aν = (Φ, −A). The D’Alembertian operator = ∇2 − ∂ 2 /∂(ct)2 = −∂ 2 is used at times in favor of ∂ 2 . Fourier transforms will be denoted with tildes, e.g., F̃ is the Fourier transform of F. The representation independent Dirac notation |hi will be used in the discussion of time-dependent perturbation theory, but for the most part functions h(x) = hx|hi 16 or h̃(k) = hk|hi will be used. (It will be assumed that all of the functions of physics are in C ∞ and in L2 ∩ L1 over either the real or complex field.) Since it is the radiation effects present on the atomic scale that are of interest, it is beneficial to work in natural (gaussian atomic) units where ~ = −|e| = me = 1. In these units the speed of light c ≈ 137 atomic units of velocity. CHAPTER 2 THE DYNAMICS A dynamical system may be well-defined once its Lagrangian and associated dynamical variables as well as their initial values are specified. This information together with the calculus of variations [71] generates the equations which govern the dynamics. Chapter 2 will detail the aspects associated with generating equations of motion for dynamical systems. Many different variational methods exist by which to generate dynamical equations, each having subtle differences [72]. However, all methods rely on the machinery inherent in variational calculus. Given a starting and ending point for the dynamics, the calculus of variations determines the path connecting them. The dynamics is determined by extremizing (either minimizing or maximizing) a certain function of these initial and final points. In this chapter, the Lagrangian and Hamiltonian formalisms [27–29] are presented for discrete and continuous systems. The Lagrangian approach leads to second order equations of motion in time, while the Hamiltonian or canonical approach leads to first order equations of motion in time. The resulting dynamics are equivalent in either case. However, the Hamiltonian approach enjoys a rich mathematical foundation connecting differential geometry and dynamics [28, 29]. Much of the remainder of this dissertation will be devoted to the canonical formulation of Maxwell and Schrödinger theories. The time-dependent variational principle [73], which has its origin in nuclear physics [74], is the variational approach to the determination of the Schrödinger equation. The Hamiltonian dynamics associated with the Schrödinger equation evolves in a generalized phase space endowed with a Poisson bracket. With the 17 18 time-dependent variational principle, many-body dynamics may be consistently described in terms of a few efficiently chosen dynamical variables (see Deumens et. al. [75]). Additionally, the variational technology provides a means by which to construct approximations to the resulting equations of motion in a systematic and well-balanced way. As will be seen in Chapters 4 and 5, these approximations will be of utmost importance in the numerical solutions of the coupled nonlinear MaxwellSchrödinger equations. 2.1 Lagrangian Formalism Before delving into a detailed account of Lagrangian dynamics it is instructive to say a few words about the Lagrangian itself. The Lagrangian is a scalar function of the vectors q k and q̇ k (k = 1, . . . , N ) with dimensions of energy. However, it is not the energy nor is it physically observable. The Lagrangian is a fundamental ingredient in the determination of a dynamical system. That is, the dynamics of a system may be known in principle once the system’s Lagrangian is known and the dynamical variables are given at some time. The Lagrangian may have a number of symmetries. In 1918, Emmy Noether (see Goldstein [27] and the references therein) proved that to each continuous symmetry there is an associated conservation law. For example, since all observations indicate that Nature is invariant under time and space translations as well as spatial rotations, so should be the Lagrangian. If the Lagrangian possesses time translation invariance, then the energy of the system is conserved. If the Lagrangian is invariant to space translations (rotations), then the linear (angular) momentum of the system is conserved. One last symmetry of significance in this dissertation is the gauge symmetry. Since Nature is invariant to the choice of gauge, the Lagrangian should maintain this symmetry as well. If the gauge symmetry is preserved, then the system enjoys conservation of charge. Depending on the particular system at 19 hand, other symmetries may be of importance and should also be respected by the Lagrangian. 2.1.1 Hamilton’s Principle Given a Lagrangian L(q k , q̇ k , t) dependent upon the N position vectors q k , the N velocity vectors q̇ k , and also the time t, the action I is defined by the path integral I(q k , q̇ k , t) ≡ R t2 t1 L(q k , q̇ k , t)dt k = 1, . . . , N. (2.1) That the variation of this integral between the fixed times t1 and t2 leads to a stationary point is a statement of Hamilton’s Principle [27, 28]. Moreover, this stationary point is the correct path for the motion. In mathematical symbols, the motion is a solution of δI = δ R t2 t1 Ldt = 0, (2.2) where δI is the variation of the action I. Only those paths are varied for which δq k (t1 ) = 0 = δq k (t2 ). A particular form of the variational path parametrized by the infinitesimal parameter α is given by q k (t, α) = q k (t, 0) + αη k (t), (2.3) where q k (t) = q k (t, 0) is the correct path of the motion and the vectors η k (t) are well-behaved and vanish at the boundaries t1 and t2 . By continuously deforming q k (t, α) until it is extremized, the correct path can be found. This parametrization of the path in turn parametrizes the action itself. Equation (2.2) may now be rewritten more precisely as δI(α) = ∂I(α) dα = 0 ∂α α=0 (2.4) 20 which represents infinitesimal variations from the correct path. The calculus of variations yields ∂L ∂ q̇ k o ∂I(α) R t2 n ∂L ∂q k = t1 dt + k ∂α ∂q k ∂α ∂ q̇ ∂α n ∂L k t2 R d ∂L o ∂q k ∂L ∂q t2 − , = k + t1 dt ∂ q̇ ∂α t1 ∂q k dt ∂ q̇ k ∂α (2.5) where a partial integration was performed in the second line. Since δq k (t1 ) = 0 = δq k (t2 ), the surface term vanishes. The stationary point of the variation is therefore determined by R t2 n ∂L d ∂L o ∂q k = 0. − dt t1 ∂q k dt ∂ q̇ k ∂α α=0 (2.6) But since the vectors ∂q k /∂α are arbitrary (choose in particular ∂q k /∂α > 0 and continuous on [t1 , t2 ]), the integral is zero only when ∂L d ∂L − =0 k ∂q dt ∂ q̇ k (2.7) by the fundamental lemma of the calculus of variations. Equation (2.7) defines the system of N second order Euler-Lagrange differential equations in terms of the local coordinates (q k , q̇ k ). Since these equations are valid on every coordinate chart, the Euler-Lagrange equations are coordinate independent. It is demonstrated in [28] that (2.7) can be written in a coordinate free or purely geometric form. If these equations admit a solution, then the action has a stationary value. It is this stationary value which determines the motion. The second order form of the EulerLagrange equations can be seen be expanding the total time derivative to give ∂L n ∂ 2 L l ∂2L l ∂2L o − q̈ + q̇ + = 0. ∂q k ∂ q̇ l ∂ q̇ k ∂q l ∂ q̇ k ∂t∂ q̇ k (2.8) It will always be assumed unless otherwise noted that the Hessian condition is satisfied. That is det{∂ 2 L/∂ q̇ l ∂ q̇ k } 6= 0. Lastly, notice that the Lagrangian is arbitrary up to the addition of a total time derivative. That is, if L → L0 = L + (d/dt)K for K a well-behaved function of the 21 dynamical variables, then the action δI → δ R t2 t1 {L + (d/dt)K}dt = δK(t2 ) − δK(t1 ) + R t2 t1 δL dt = R t2 t1 δL dt = δI (2.9) since δK(t2 ) = 0 = δK(t1 ). Thus the same Euler-Lagrange equations (2.7) are generated for L0 as for L. In other words, there are many Lagrangians that lead to the same equations of motion. There is no unique Lagrangian for a particular dynamical system. All Lagrangians differing by only a time derivative will lead to the same dynamics. More generally, in the dynamics of continuous systems two equivalent Lagrangians may differ by a purely surface term in time and space. 2.1.2 Example: The Harmonic Oscillator in (q k , q̇ k ) It is now useful to present a brief illustrative example. In two freedoms, the dynamics of a scalar mass subjected to the force of a harmonic potential with frequency ωk is determined by the Lagrangian (no summation) 1 1 L(q k , q̇ k ) = mq̇ k q̇ k − mωk2 q k q k 2 2 k = 1, 2 (2.10) which is a function of the real-valued vectors q k and q̇ k . Application of the calculus of variations to the associated action functional leads to (2.7) with ∂L/∂ q̇ k = mq̇ k and ∂L/∂q k = −mωk2 q k . The second order Euler-Lagrange equations of motion are 1 k 1 mq̈ + mωk2 q k = 0 2 2 k = 1, 2 (2.11) with initial value solution q k (t) = q k (t0 ) cos(ωk t) + q̇ k (t0 ) sin(ωk t)/ωk . It is said that q k is an integral curve of the dynamical equation (2.11). Once the initial values q k (t0 ) and q̇ k (t0 ) are provided, the dynamics of the harmonic oscillator is known. This dynamics occurs in a space whose coordinates are not just the q k , but both the q k and q̇ k . Some geometric aspects of this space will now be presented. g replacements 22 Tq k Q qk Q Figure 2–1: The configuration manifold Q = S2 is depicted together with the tangent plane Tqk Q at the point q k ∈ Q. 2.1.3 Geometry of TQ In the Lagrangian formalism, the dynamics unfolds in a velocity phase space whose points are of the form (q k , q̇ k ). The position vectors q k lie in a differential manifold called the configuration manifold Q, while the velocity vectors lie in the manifold of vectors tangent to Q. The space formed by attaching the space spanned by all vectors tangent to the point q k ∈ Q is called the tangent fiber above q k or the tangent plane at q k and is denoted by Tqk Q. The union of the configuration manifold Q and the collection of all fibers Tqk Q for each point q k ∈ Q (together with local charts on Tqk Q) is called the velocity phase space, tangent bundle, or tangent manifold of Q and is denoted by TQ. It is that manifold that carries the Lagrangian dynamics, not the configuration manifold Q. A picture is presented in Figure 2–1 corresponding to the case where Q is the two-dimensional surface S2 of the unit ball in R3 . The tangent plane at the point q k reaches out of S2 and into R3 . This larger manifold is where the associated Lagrangian dynamics occurs. 23 The integral curves of a dynamical system are vector fields and are called the dynamics or the dynamical vector fields. The velocity phase space dynamics is a vector field on TQ denoted by ∆L ≡ q̇ k (∂/∂q k ) + q̈ k (∂/∂ q̇ k ), where q̇ k and q̈ k are the components of ∆L and ∂/∂q k and ∂/∂ q̇ k form a local basis for ∆L . The time dependence of a dynamical variable F (q k , q̇ k ), which is an implicitly time-dependent function on TQ, is determined by its variation along the dynamics. That is Ḟ (q k , q̇ k ) ≡ ∆L (F ) = ∂F k ∂F k q̇ + k q̈ . ∂q k ∂ q̇ (2.12) The accelerations q̈ k can be substituted directly from the dynamical equations. Thus, the time dependence of a dynamical variable is determined by the equations of motion themselves without even the knowledge of their solution. Beyond functions and vector fields on TQ, there is another important geometrical quantity called the one-form that is worth defining. One-forms on TQ are linear functionals that map vector fields to functions. That is, if the one-form α = A1a dq a + A2a dq̇ a is applied to the vector field X = X1b (∂/∂q b ) + X2b (∂/∂ q̇ b ), then their inner product results in hα|Xi = A1a X1b dq a (∂/∂q b ) + A1a X2b dq a (∂/∂ q̇ b ) + A2a X1b dq̇ a (∂/∂q b ) + A2a X2b dq̇ a (∂/∂ q̇ b ) = A1a X1b δ ab + A2a X2b δ ab = A1a X1a + A2a X2a , (2.13) where dq a (∂/∂q b ) = dq̇ a (∂/∂ q̇ b ) = δ ab and dq a (∂/∂ q̇ b ) = dq̇ a (∂/∂q b ) = 0, and where Aa and X b are the local components, which are functions, of the one-form α and vector field X. It is common to write hα|Xi ≡ α(X). It should be pointed out that the differential of a function is a one-form. That is dF = ∂F k ∂F k dq + k dq̇ ∂q k ∂ q̇ (2.14) 24 is a one-form and may be applied to the dynamical vector field ∆L to give dF (∆L ) ≡ hdF |∆L i = ∂F k ∂F k q̇ + k q̈ = Ḟ . ∂q k ∂ q̇ (2.15) The one-forms are also called covariant vectors or covectors and are dual to the vector fields which are sometimes called contravariant vectors. 2.2 Hamiltonian Formalism The Lagrangian formalism set up N second order dynamical equations which required 2N initial values to fix the dynamics. Alternatively, and equivalently, the dynamics may be described in terms of 2N first order equations of motion with 2N initial values. This so called Hamiltonian dynamics evolves in a different tangent manifold or phase space with generalized coordinates q a and pa , which are governed by the dynamical equations q̇ a = ∂H ∂pa and − ṗa = ∂H , ∂q a (2.16) where the function H is called the Hamiltonian (see (2.18) below). It is itself a dynamical variable and for many physical systems it is the energy. Since (2.16) are of first order, the associated trajectories are separated on the new phase space. The change of variables from (q a , q̇ a ) to (q a , pa ) is accomplished by a Legendre transformation [27, 28]. The momentum conjugate to the vector q a is defined in terms of the Lagrangian L by pa ≡ ∂L(q a , q̇ a ) . ∂ q̇ a (2.17) Notice that this conjugate momentum is not a vector as is the velocity q̇ a and does not lie in the tangent manifold TQ. Rather the momentum pa is dual to the position vector q a . It is a one-form and lies in the cotangent manifold T∗ Q. This difference will soon be elaborated on. With the momentum pa and the Lagrangian, the Hamiltonian 25 function is constructed according to H(q a , pa ) = pa q̇ a (q a , pa ) − L(q a , pa ). (2.18) Here it is assumed that the relation (2.17) can be inverted to solve for the velocity q̇ a . Hamilton’s canonical equations of motion (2.16), which are first order differential equations in time, can now be obtained from an argument similar to that presented in Section 2.1.1 on Hamilton’s Principle. That is, if the Lagrangian in the action integral (2.1) is replaced by L = pa q̇ a − H from (2.18), i.e., I(q a , pa ) ≡ R t2 t1 [pa q̇ a (q a , pa ) − H(q a , pa )]dt, (2.19) then Hamilton’s equations follow in a straightforward manner. 2.2.1 Example: The Harmonic Oscillator in (q a , pa ) It is now useful to compare the Lagrangian and Hamiltonian dynamics for a simple dynamical system. Recall the Lagrangian for the two freedom harmonic oscillator in (2.10). That is 1 1 L(q a , q̇ a ) = mq̇ a q̇ a − mωa2 q a q a 2 2 a = 1, 2. (2.10) The momentum conjugate to q a is pa ≡ ∂L/∂ q̇ a = mq̇ a and with it the Hamiltonian becomes H(q a , pa ) = pa q̇ a − o p p np p 1 mωa2 q a q a a a a a − mωa2 q a q a = + . 2m 2 2m 2 (2.20) With this Hamiltonian the equations of motion are: ∂H = pa /m ∂pa ∂H −ṗa = a = mωa2 q a ∂q q̇ a = (2.21) and have the initial value solutions q a (t) = q a (t0 ) cos(ωa t) + q̇ a (t0 ) sin(ωa t)/ωa and pa (t) = pa (t0 ) cos(ωa t) − mωa q a (t0 ) sin(ωa t). These are the integral curves of the first 26 order differential equations (2.21) and may be compared to those in the Lagrangian formulation. 2.2.2 Symplectic Structure and Poisson Brackets One of the many powerful aspects of the Hamiltonian or canonical approach to dynamics is the flexibility and ability to treat positions and momenta similarly. This similarity among the coordinates is made explicit by the following notation: ξa = qa a = 1, . . . , N ξ a = pa−N (2.22) a = N + 1, . . . , 2N. Similarly, the forces become ∂H/∂pa ≡ ∂H/∂ξ a+N and ∂H/∂q a ≡ ∂H/∂ξ a so that the equations of motion are: ∂H ξ˙a = a+N ∂ξ ∂H −ξ˙a = a−N ∂ξ a = 1, . . . , N (2.23) a = N + 1, . . . , 2N. These Hamilton equations may be written more compactly as ∂H ωab ξ˙b = a , ∂ξ (2.24) where ωab are the matrix elements of the symplectic form ω. The symplectic form is an antisymmetric 2N × 2N -dimensional matrix of the form ω= 0N − 1 N , 1N 0N (2.25) where 0N and 1N are the N × N -dimensional zero and identity matrices respectively. The matrix (2.25) is also referred to as the canonical symplectic form because it satisfies the properties: ω 2 = −1 and ω T = −ω, (2.26) or equivalently ω ab ωbc = δ ac and ωab = −ωba . The matrix element ω ab with both indices up is the inverse of ωab . 27 In (2.12), the time derivative or variation of a (implicitly time-dependent) dynamical variable F on TQ was demonstrated. In a similar fashion, Ḟ can be viewed in the momentum phase space T∗ Q, which will be discussed shortly. It is Ḟ (q b , ṗb ) = ∂F ˙b ∂F ba ∂H ξ = bω , ∂ξ b ∂ξ ∂ξ a (2.27) where the equation of motion (2.24) was inverted and substituted for ξ˙b . The right hand side of this equation is called the Poisson bracket of F with H. In general, it may be written for any two functions in T∗ Q as {F, G} ≡ ∂F ∂G ∂F ∂G ∂F ba ∂G ω = a − . b a ∂ξ ∂ξ ∂q ∂pa ∂pa ∂q a (2.28) In particular, an alternative form of Hamilton’s equations is derived when the Poisson bracket is applied to the coordinate ξ. That is ξ˙a = {ξ a , H}. (2.29) Since the Poisson bracket is bilinear, antisymmetric, and satisfies the Jacobi identity {f, gh} = g{f, h} + {f, g}h, the set of functions on T∗ Q forms a Lie algebra under Poisson bracket {·, ·}. In fact, the Hamiltonian dynamics can naturally be studied from this point of view [29, 73]. 2.2.3 Geometry of T∗ Q As was previously mentioned, the dynamics associated with Hamilton’s equations of motion (2.24) do not unfold in the same velocity phase space TQ that was defined in Section 2.1.3. These equations of motion define a vector field ξ˙ on a different phase space whose components are the functions ω ba (∂H/∂ξ a ). The integral curves of this vector field are the dynamics. Recall that the points of TQ are made up of q k and q̇ k . The velocities q̇ k are the local components of the vector field q̇ k (∂/∂q k ). However, the momenta are the local components of the one-form pa dq a ≡ (∂L/∂ q̇ a )dq a , which are not the components 28 of a vector field. Since one-forms are dual to vector fields, pa dq a lies in the dual space of Tqa Q. This space is the cotangent space at q a and is denoted by T∗qa Q. In analogy with TQ, the cotangent bundle or cotangent manifold T∗ Q is made up of Q together with its cotangent spaces T∗qa Q. Consequently, the carrier manifold for the Hamiltonian dynamics is not TQ, but rather it is the phase space T∗ Q. The dynamical vector field on T∗ Q is given by ∆H ≡ ξ˙b ∂ ∂ ∂H ∂ ∂H ∂ a ∂ = q̇ + ṗ = − , a ∂ξ b ∂q a ∂pa ∂pa ∂q a ∂q a ∂pa (2.30) where Hamilton’s equations of motion (2.16) were substituted for the q̇ a and ṗa . There is one last geometric quantity that needs to be defined. The symplectic form ω is a two-form on T∗ Q. Two-forms are bilinear, antisymmetric forms that map pairs of vector fields to functions. That is, if X = X a (∂/∂ξ a ) and Y = Y b (∂/∂ξ b ) are vector fields on T∗ Q, then ω(X, Y ) = X a Y b ω(∂/∂ξ a , ∂/∂ξ b ) = X a ωab Y b = X a Ya − Y a Xa . (2.31) The matrix elements ωab = −ωba are identical to those presented earlier. Since ω is nonsingular and the differential dω = 0, i.e., ω is closed, the two-form ω is called a symplectic form. In general, phase space is naturally endowed with a symplectic form or structure. For this reason T∗ Q is also a symplectic manifold [29]. Lastly, it should be mentioned that ω(X, Y ) is a measure of the area between the vectors X and Y. In fact, there is a powerful theorem attributed to Liouville [27–29] that states that the phase space volume must be invariant under canonical transformations in phase space. Canonical transformations are those transformations that maintain the symplectic structure of Hamilton’s dynamical equations ∂H ωab ξ˙b = a . ∂ξ More will be said on canonical transformations in Chapter 4. (2.24) CHAPTER 3 ELECTRODYNAMICS AND QUANTUM MECHANICS The coupling of electrodynamics to charged matter is a complicated problem. This complexity is compounded by the fact that the fields produced by charges in motion react back upon the charges, thus causing a modification of their trajectory. As mentioned in the introduction, the corresponding physics is often analyzed in one of two ways. Either: • The electromagnetic field is taken as an influence on the dynamics of the charges. • The sources of charge and current are used to calculate the dynamics of the electromagnetic field. Chapter 3 will discuss both of these cases in detail. The first portion of this chapter will set up the time-dependent perturbation theory which will be used to make calculations in quantum mechanics under the influence of an electromagnetic field. The second portion of this chapter will explore the electrodynamics resulting from a given ρ and J. In particular, the multipole expansion will be introduced and used to calculate the power radiated from an oscillating electric dipole. Additionally, the electromagnetic fields corresponding to a gaussian wavepacket will be presented. In the narrow width limit of the gaussian, the resulting physics reduces to the expected textbook results for a point source. 3.1 Quantum Mechanics in the Presence of an Electromagnetic Field The dynamics of charges in an external electromagnetic field may be studied at varying levels of sophistication from a purely classical description of both charge and field to a fully quantum treatment. Various semiclassical or mixed quantummatter/classical-field descriptions are available as well as fully quantum and relativistically invariant treatments such as quantum electrodynamics. 29 30 Time-dependent perturbation theory [7] is a systematic method by which to calculate (among other things) properties of the dynamics of charges in an external electromagnetic field. In this section, the time-dependent perturbation theory is introduced for a general perturbation in the context of quantum mechanics. Emphasis is then placed on the classical electromagnetic field as a particular time-dependent perturbation V. Within this framework the perturbation is seen as causing transitions between two stationary states |Ψk i and |Ψm i of an atomic system, and is symbolized to lowest order by the matrix element Vkm = hΨk |V |Ψm i. Experimental observables such as the rate of transition or absorption cross section may be calculated from Vkm . Additionally, time-dependent perturbation theory gives a prescription for calculating successively higher order corrections to Vkm , which may in turn provide better and better agreement with experiment. This section concludes with a discussion of quantum electrodynamics, in which both matter and fields are quantized and the description is relativistically invariant. Here again the time-dependent perturbation theory (often in the form of Feynman diagrams) is the essential machinery used in calculations. 3.1.1 Time-Dependent Perturbation Theory An important class of solutions to the Schrödinger equation (1.1) are those which are eigenfunctions of the Hamiltonian operator H. These solutions |Ψm i satisfy the time-independent Schrödinger equation H|Ψm i = Em |Ψm i (3.1) and are called stationary states. A general solution |Ψ(t)i of the Schrödinger equation (1.1) may be constructed from these stationary states according to |Ψ(t)i = e−iH(t−t0 ) |Ψ(t0 )i = P me −iEm (t−t0 ) |Ψm ihΨm |Ψ(t0 )i, (3.2) 31 where |Ψ(t0 )i is an initial state vector and where the sum over m may imply integration if the energy spectrum is continuous. Equation (3.2) is only applicable when the Hamiltonian is time-independent. For if H ≡ H(t), then the energy of the system is not conserved and H admits no strictly stationary states. However, it may be possible to split a time-dependent Hamiltonian into the sum of two terms: H = H0 + V (t), (3.3) where H0 is time-independent and describes the unperturbed system while V (t) accounts for the time-dependent perturbation. To fix ideas, consider for example the electronic transition induced by a passing electromagnetic disturbance that is localized in both space and time. In other words, the system is initially unperturbed for some long time and is in an eigenstate of H0 . While in the interaction region the system is perturbed by V after which it settles down into another unperturbed eigenstate of H0 for a sufficiently long time. Time-dependent perturbation theory seeks to connect the stationary states of the unperturbed system, i.e., those states satisfying H0 |Ψm i = Em |Ψm i, (3.4) with the time-dependent perturbation V (t). These calculations are most clearly demonstrated in the interaction picture. In the interaction picture the perturbation is singled out by applying the unitary operator U0 = exp(iH0 t) to |Ψ(t)i. That is |ΨI (t)i = eiH0 t |Ψ(t)i (3.5) and the time-dependent Schrödinger equation (1.1) becomes i(d/dt)|ΨI (t)i = VI (t)|ΨI (t)i, (3.6) 32 where VI (t) = U0 V (t)U0† . In other words, the interaction picture separates the physics that depends upon the perturbation from the physics that depends upon only the unperturbed system. The state vector at time t is obtained from that at time t0 via |ΨI (t)i = UI (t, t0 )|ΨI (t0 )i, (3.7) where UI is the time evolution operator which satisfies UI (t, t0 ) = 1 − i Rt t0 dt0 VI (t0 )UI (t0 , t0 ). (3.8) The time evolution operator connects the (orthonormal) stationary states |Ψk i and |Ψm i according to where P Rt dt0 hΨk |VI (t0 )UI (t0 , t0 )|Ψm i P Rt (3.9) = δkm − i n t0 dt0 hΨk |VI (t0 )|Ψn ihΨn |UI (t0 , t0 )|Ψm i P Rt 0 = δkm − i n t0 dt0 eiωkn t hΨk |V (t0 )|Ψn ihΨn |UI (t0 , t0 )|Ψm i, hΨk |UI (t, t0 )|Ψm i = δkm − i n t0 |Ψn ihΨn | = 1 and ωkn = Ek − En were used. The time-dependent perturbation theory is now set up by iterating on (3.8). If the perturbation V is small then the time evolution operator becomes a power series in V. That is UI (t, t0 ) = 1 − i Rt dt0 VI (t0 ) + (−i)2 t0 Rt dt0 VI (t0 ) t0 R t0 t0 dt00 VI (t00 ) + · · · . (3.10) And so at first order the transition amplitude between two distinct states of energy Ek and Em (with k 6= m) is hΨk |UI (t, t0 )|Ψm i = −i Rt t0 0 dt0 eiωkm t hΨk |V (t0 )|Ψm i. (3.11) 33 Assuming that the perturbation is sufficiently small, the probability of finding the system in the state |Ψk i is given by 2 2 Rt 0 Pk←m (t) = hΨk |UI (t, t0 )|Ψm i = − i t0 dt0 eiωkm t hΨk |V (t0 )|Ψm i . (3.12) If the perturbation is localized in time then t0 and t may be naively extended to infinity to yield the transition probability 2 2 R ∞ Pk←m (+∞) = hΨk |UI (+∞, −∞)|Ψm i = − i −∞ dt eiωkm t hΨk |V (t)|Ψm i (3.13) which involves a Fourier integral of the matrix element Vkm = hΨk |V (t)|Ψm i. 3.1.2 Fermi Golden Rule The formalism set up thus far is also applicable for time-independent perturbations V 6= V (t). In this case the transition probability can be obtained from (3.12) as Pk←m (t) = 2|hΨk |V |Ψm i|2 1 − cos(ωkm t) (Ek − Em )2 (3.14) which is proportional to t2 if Ek ≈ Em . Now consider the situation in which there is a near continuum of final states available having energies in the interval (Em − ∆E/2, Em + ∆E/2). If the density of the near continuum states is denoted by ρF (E), then the transition probability to all of these states is given by P k∈F Pk←m (t) = R Em +∆E/2 Em −∆E/2 2|hΨk |V |Ψm i|2 1 − cos(ωkm t) ρF (Ek )dEk , (Ek − Em )2 (3.15) where the sum runs over all states |Ψk i belonging to the near continuum of final states. The quotient [1−cos(ωkm t)]/(Ek −Em )2 is sharply peaked at Ek = Em which confirms that the dominant transitions are those that conserve the unperturbed energy. Since both |hΨk |V |Ψm i|2 and ρF (Ek ) are approximately constant around Em and t is such that ∆E 2π/t (i.e., long time behavior), the transition probability 34 becomes P k∈F Pk←m (t) ≈ 2|hΨk |V |Ψm i|2 ρF (Ek ) 2 R ∞ 1 − cos(ωkm t) dωkm 2 −∞ ωkm (3.16) = 2πt|hΨk |V |Ψm i| ρF (Ek ) which increases linearly with time. The total transition probability per unit time or transition rate Γ is given by Γ = (d/dt) P k∈F Pk←m (t) = 2π|hΨk |V |Ψm i|2 ρF (Ek ) (3.17) and is constant. Fermi’s golden rule of time-dependent perturbation theory [7, 34, 35] embodies the tendency for the perturbed system to make energy conserving transitions for which the probability increase as t2 or to make nearly energy conserving transitions which oscillate in time. Either way the transition rate Γ is constant. Fermi’s golden rule may be extended to include perturbations that vary harmonically in frequency ω. An electromagnetic disturbance of a charge would be an example. In this case the golden rule generalizes to Γ = 2π|hΨk |V |Ψm i|2 ρF (Em + ω). 3.1.3 (3.18) Absorption of Electromagnetic Radiation by an Atom Recall the electromagnetic field coupling to quantum mechanics is given by the minimal coupling prescription i(∂/∂t) → i(∂/∂t) − qΦ (3.19) −i∇ → −i∇ − qA/c, where Aα = (Φ, A) are the dynamical variables of the electromagnetic field. Applying this transformation to the Schrödinger equation iΨ̇ = P2 Ψ/2m + V0 Ψ results in the Schrödinger equation coupled to the electromagnetic field iΨ̇ = [P − qA/c]2 Ψ + V0 Ψ + qΦΨ 2m (3.20) 35 with Hamiltonian H= [P − qA/c]2 + V0 + qΦ. 2m (3.21) In the Coulomb gauge (see Appendix A for details) this Hamiltonian becomes H= q q2 P2 + V0 − A·P+ A2 . 2m mc 2mc2 (3.22) The external free electromagnetic field evolves according to ∇2 A − Ä/c2 = 0 with ∇·A = 0 and Φ = 0 since it is assumed that the charges do not themselves contribute to the field. By ignoring the quadratic term in A, the Hamiltonian H separates into an unperturbed portion H0 = P2 + V0 2m (3.23) V =− q A · P. mc (3.24) plus the perturbation It should be pointed out here that substantial confusion has existed in the literature over the A · P appearing in the perturbation V. This confusion was due the widespread use of E · r and its higher order approximations [59, 60] instead of A · P. The relationship between these two approaches have been thoroughly investigated in [76–79]. The cross section for stimulated absorption (or emission) of radiation by an atom may be calculated via Fermi’s golden rule. If the external field varies harmonically in frequency as a plane wave, then the perturbation becomes V (x, t) = − q A0 ei(k·x−ωt) + A∗0 e−i(k·x−ωt) ˆ · P mc (3.25) where ˆ is the field’s polarization. The rate of energy absorption by the atom is 2 q2ω 2 ik·x Γω = 2 2 |A0 | hΨk |e ˆ · P|Ψm i ρF (Em + ω). mc (3.26) 36 If the density of the near continuum states is narrow then ρF (Em + ω) = δ(Ek − Em + ω) = δ(ωkm + ω) (3.27) and the absorption cross section σabs (ω) = Γ/I0 becomes σabs (ω) = 2 q 2 |A0 |2 hΨk |eik·x ˆ · P|Ψm i δ(ωkm + ω)/m2 c2 ω|A0 |2 /2πc 2 4π 2 q 2 2 ik·x |A0 | hΨk |e ˆ · P|Ψm i δ(ωkm + ω), = 2 ωm c (3.28) where I0 = ω|A0 |2 /2πc is the incident flux of photons of frequency ω. Similarly the emission cross section is σem (ω) = 2 4π 2 q 2 2 −ik·x ˆ |A | hΨ |e · P|Ψ i 0 k m δ(ωkm − ω). ωm2 c (3.29) Notice that the time-dependent perturbation theory gives properties of the solution but not the solution. That is, the cross section is easily accessible but the wavefunction and 4-potential are not. The cross section is a property of the solution and can be calculated from knowledge of the solution. Of course, the wavefunction and 4-potential constitute the actual solution. 3.1.4 Quantum Electrodynamics in Brief The quantum theory of electrodynamics [19, 24, 30], also known as QED, is the interacting quantum field theory of electron and photon fields. The relativistically invariant QED is one of the most successful physical theories to date, in that there is no evidence for any discrepancy between experiment and prediction. However, QED is beset by many mathematical and logical difficulties. These difficulties are in some cases avoided by physical arguments or simply concealed from view as in the renormalization of mass and charge. Putting aside its inconsistencies, QED is a prime example of the success of time-dependent perturbation theory. A combination of the free Dirac theory and the free Maxwell theory provide the unperturbed states on which the interaction 37 Lint = −Jµ Aµ /c operates. The free QED Lagrangian density Lfree QED = ψ̄[iγ µ ∂µ − mc]ψ − 1 Fµν F µν 16π (3.30) gives the equations of motion for the free electron [iγ µ ∂µ − mc]ψ = 0 (3.31) and the equations of motion for the free electromagnetic field ∂µ F µν = ∂ 2 Aν − ∂ ν (∂ · A) = 0ν , (3.32) where the Dirac γ-matrices are related to the Pauli spin matrices (4.66), ψ̄ = ψ † γ 0 is the Dirac adjoint of the four component spinor ψ, and Fµν = ∂µ Aν − ∂ν Aµ is the electromagnetic field tensor. This noninteracting theory sets up the free unperturbed in-state |p1 · · · pn iin and out-state |k1 · · · km iout , which will be connected R by UI (+∞, −∞) = T{exp[i d4 xLint ]}, where T is the time-ordering operator. The resulting matrix elements will yield some properties of the dynamics. Working in the interaction picture, the machinery of time-dependent perturbation theory is used to construct the scattering matrix or S-matrix out hk1 · · · km |p1 · · · pn iin = in hΩ|a(k1 ) · · · a(km )UI (+∞, −∞)a† (p1 ) · · · a† (pn )|Ωiin (3.33) where S = 1 + iT and |Ωiin is the in-vacuum. The respective fermion creation and annihilation operators a† (k) and a(k), create and annihilate single fermions of momentum k according to a† (k)|Ωi = |ki and a(k)|Ωi = 0 where the spin has been neglected for simplicity. The situation in which the particles do not interact at all (the 1 part) as well as the interesting interactions (the iT part) are both included in the S-matrix. The interacting components are commonly collected and are referred 38 to as the T-matrix. Together 1 + iT is used to define the invariant amplitude M as out hk1 · · · km |UI (+∞, −∞)|p1 · · · pn iin = (2π)4 δ(p1 + · · · + pn − k1 − · · · − km ) · iM(p1 , · · · , pn → k1 , · · · , km ). (3.34) This invariant amplitude is analogous to the scattered wavefunction of quantum PSfrag replacements mechanics, i.e., h eikr i Ψk (x) ∼ N eik·x + fk (Ω) , r (3.35) where the unscattered field exp(ik · x) and the spherically scattered field exp(ikr)/r are indicated schematically in Figure 3–1. In fact all of quantum mechanics is Scattered field Incident field Figure 3–1: The coefficient 1 of the unscattered plane wave exp(ik · x) is analogous to the 1 part of the S-matrix, while the scattering amplitude fk (Ω) which modulates the scattered spherical wave exp(ikr)/r is analogous to the iT part. just the nonrelativistic limit of QED. Unfortunately, while QED is suitable for the scattering of single particle states to single particle states, it requires great effort to deal with bound states. The probability of finding |k1 · · · km iout in |p1 · · · pn iin is given by 2 3 d3 km out in d k1 · · · P (+∞) ∼ hk1 · · · km |p1 · · · pn i (2π)3 (2π)3 2 d3 k d3 km 1 · · · = in hΩ|a(k1 ) · · · a(km )UI (+∞, −∞)a† (p1 ) · · · a† (pn )|Ωiin (2π)3 (2π)3 (3.36) 39 which is analogous to (3.13). A similar connection can be made in the cross section. If n = 2 in the in-state, then the differential scattering cross section dσ becomes 2 d 3 k d3 km 1 · · · (2π)4 δ(p1 + p2 − k1 − · · · − km ) dσ ∼ iM(p1 , p2 → k1 , · · · , km ) (2π)3 (2π)3 (3.37) which is analogous to (3.28). As in quantum mechanics, time-dependent perturbation theory in QED gives a prescription by which to calculate properties of the solution which rely on scattering amplitudes, e.g., cross sections, decay rates, and probabilities. It is considerably more difficult to compute the actual solution, which in this case would be the states on which the field operators d3 k P n e−ik·x eik·x o ∗ † Aµ (x, t) = V µ (k, λ)c(k, λ) √ + µ (k, λ)c (k, λ) √ (2π)3 λ 2k 2k −ik·x ik·x o R d3 k P n e e ψi (x, t) = V + vi (k, s)b† (k, s) √ s ui (k, s)a(k, s) √ 3 (2π) 2ω 2ω R (3.38) act. In (3.38), µ and ∗µ are helicity eigenstates of Aµ , and {c, c† } are photon creation and annihilation operators. Similarly ui and vi are eigenspinors of ψi , and {a, a† } and {b, b† } are electron and positron [80] creation and annihilation operators. Lastly it should be pointed out that a beautiful representation of the timedependent perturbation theory was introduced by Feynman [55]. These so called Feynman diagrams provide a pictorial version of the invariant amplitude iM = C · out hk1 · · · km |UI (+∞, −∞)|p1 · · · pn iin n R 4 o out = C · hk1 · · · km |T ei d xLint |p1 · · · pn iin h i R 4 out in = hk1 · · · km | 1 + i d xT Lint + · · · |p1 · · · pn i (3.39) connected . 40 For example, the invariant amplitude for Bhabha scattering, i.e., e+ e− → e+ e− , is iMBhabha = h out p2 i R hk1 k2 |p1 p2 iin + out hk1 k2 |i d4 xT Lint |p1 p2 iin + O(L2int ) p2 k2 k2 + O(L4int ), + = p1 k1 connected p1 k1 (3.40) where each of the above diagrams corresponds to a term (or portion thereof) in the perturbative expansion of iMBhabha . These tree order diagrams are the lowest order nonvanishing diagrams that contribute to and are the largest part of the Bhabha scattering invariant amplitude. Higher order perturbative corrections to the amplitude also have pictorial representations and may be systematically constructed using Feynman’s prescription. In this manner the time-dependent perturbation theory may be diagrammatically written to any order, translated into mathematical expressions, and computed. While this is by no means an easy task, the invariant amplitude may in principle be calculated to any order. Notice again that this machinery produces the amplitude iM, which is a property of the solution but not the actual solution. 3.2 Classical Electrodynamics Specified by the Sources ρ and J If the sources of charge and current are known, then the dynamics of the resulting electromagnetic field can be calculated from Maxwell’s equations at each point in space-time. These fields may behave quite differently depending on the motion of their source. For example, a static source gives rise to a purely electrostatic field, while a uniformly moving source creates both an electric field and a magnetic field. More importantly, if the source is accelerated then electromagnetic radiation is produced. Electromagnetic radiation is a unique kind of electromagnetic field in that 41 it carries away energy, momentum, and angular momentum from its source. The radiation field is not bound to the charge as are the static fields. In this section the electromagnetic fields produced by an oscillating electric dipole are calculated to lowest order via the multipole expansion. The corresponding power and radiation pattern are also presented. Then, the dynamics of the bremsstrahlung produced by a wavepacket source is analyzed. It is shown that the wavepacket’s fields reduce in the narrow width limit to the usual point source results. The consistent coupling of electrodynamics and quantum mechanics is needed because the sources of charge and current produce electromagnetic fields and these fields act back upon the sources. The understanding of this process requires the inclusion of recoil effects on the charges due to the electromagnetic field. These effects, known as radiation reaction effects, are a main aspect of this dissertation and will be discussed in detail in Chapter 4. 3.2.1 Electromagnetic Radiation from an Oscillating Source In this section the Lorenz gauge (see Appendix A for details) is used to investigate the electromagnetic radiation produced by a localized system of charge and current [9] which vary sinusoidally in time according to (the real part of): ρ(x, t) = ρ(x)e−iωt J(x, t) = J(x)e −iωt (3.41) . It is assumed that the electomagnetic potentials and fields also have the same timedependence. The general solutions to the wave equations of (A.5) are given by |x − x0 | i ρ(x0 , t0 ) 0 h δ t − t+ Φ(x, t) = |x − x0 | c 1 R 3 0R ∞ 0 J(x0 , t0 ) 0 h |x − x0 | i A(x, t) = d x dt δ t − t + , −∞ c V |x − x0 | c R R 3 0 ∞ d x dt0 V −∞ (3.42) where G(+) (x, t; x0 , t0 ) = δ(t0 − [t + |x − x0 |/c])/4π|x − x0 | is the retarded Green’s function for the wave operator ∂ 2 = − = ∂ 2 /∂(ct)2 − ∇2 (see Appendix B). It is 42 assumed that there are no boundary surfaces present. With the oscillating sources from (3.41), it will be seen that all of the dynamics of the electromagnetic field for which ω 6= 0 can be described in terms of the A alone. The component of the electromagnetic field for which ω = 0 is just the static electric monopole field Φmonopole (x, t) = q . |x| (3.43) The vector potential for all other frequencies is 0 A(x) = 1 R 3 0 eik|x−x | d x J(x0 ), c V |x − x0 | (3.44) where the wavevector k = ω/c and it is understood that A(x, t) = A(x)e−iωt . For a given charge density J, (3.44) could in principle be computed. With the resulting vector potential the electromagnetic field may be calculated from Ampére’s law. That is B = ∇×A (3.45a) E = (3.45b) i ∇×B k in a region outside the source. Instead of evaluating (3.44) exactly, general properties of its solution may be determined whenever the dimensions of J are much smaller than a wavelength. That is, if the dimensions of the charge density are of order d and the wavelength λ = 2π/k, then d λ. From these distances, the following three spatial regions may be constructed: The near or static zone: drλ The intermediate zone: d r∼λ The far or radiation zone: dλr In each region the electromagnetic field behaves quite differently. For example in the near zone, the fields behave as if they were static fields which show strong 43 dependence on their source. On the other hand in the far zone, the fields display properties of radiation fields which are transverse and fall off as r −1 . The static near zone fields may be obtained from (3.44) by noting that kr 1 since r λ. In this case exp(ik|x − x0 |) ∼ 1 and the vector potential becomes 1 R 3 0 J(x0 ) dx c V |x − x0 | o nP 1R 3 0 4π r 0l ∗ 0 0 Y (Ω)Y (Ω ) = d x J(x ) lm lm lm c V 2l + 1 r l+1 ikp +··· , =− r Anear (x) = (3.46) where |x − x0 |−1 has been expanded into the spherical harmonics Ylm and an integration by parts was performed with all surface terms vanishing. The equation of continuity iωρ + ∇ · J = 0 was also used in the computation as well as the definition R of the dipole moment p = V d3 xxρ(x). From (3.45), the resulting magnetic and electric fields are: ik n̂ × p + · · · r2 3n̂(n̂ · p) − p = +··· , r3 Bnear = (3.47a) Enear (3.47b) where n̂ is the unit vector in the direction of the observation point x. Notice that Enear is independent of the frequency ω and is thus purely static. As expected Bnear is zero in the static limit ω → 0. A multipole expansion of the near zone vector potential can now be made and successively better results may be obtained by going to higher orders in (l, m). 44 At the other extreme, the far zone fields for which kr 1 may be obtained from (3.44) by noticing that p (x − x0 ) · (x − x0 ) s 2x · x0 |x0 |2 + = |x| 1 − |x|2 |x|2 x · x0 ≈ |x| 1 − |x|2 |x − x0 | = (3.48) PSfrag replacements = r − n̂ · x0 since |x0 | |x| = r|n̂| = r. A picture of the corresponding situation is shown in Figure 3–2, where the x0 -integration runs over the domain of the source J. With the x r = |x| n̂ d O Figure 3–2: In the radiation zone, the observation point x is located far from the source J. In this case the distance |x − x0 | ≈ r − n̂ · x0 . approximation (3.48), the far zone vector potential becomes eikr 1 R 3 0 0 d x J(x0 )e−ikn̂·x V r c ∞ ikr X e (−ik)m 1 R 3 0 = d x J(x0 )(n̂ · x0 )m r m=0 m! c V Afar (x) = = −ikp (3.49) eikr +··· , r where |x − x0 |−1 ≈ r −1 if only the leading term in kr is kept. It can now be seen that the vector potential is an outgoing spherical wave with mth-angular coefficient R 3 0 d x J(x0 )(−ik n̂ · x0 )m /cm!. From (3.45), the corresponding fields are: V Bfar Efar ikr h 1 i (n̂ × p) 1 − = k +··· r ikr h1 eikr ik i (n̂ × p) × n̂ + 3 − 2 eikr [3n̂(n̂ · p) − p] + · · · . = k2 r r r 2e (3.50a) (3.50b) 45 The magnetic field is transverse to the radius vector x = r n̂ while the electric field has components longitudinal and transverse to x. Both fields fall off like r −1 at leading order. The r −1 -fields are the true radiation fields which carry energy, momentum, and angular momentum to infinity. This can be seen from the timeaveraged differential power radiated per unit solid angle i dP 1 h c = Re r 2 n̂ · E × B∗ dΩ 2 4π 4 ck = |n̂ × p|2 8π ck 4 2 2 = |p| sin θ 8π (3.51) which in this case is a measure of the energy radiated per unit time per unit solid angle by an oscillating electric dipole p. Integrating this expression over Ω = (θ, φ) gives the total power radiated, i.e., ck 4 2 dP = |p| . P = dΩ dΩ 3 R (3.52) The corresponding radiation pattern is shown in Figure 3–3. In general, the power radiated by an l-pole goes like k 2(l+1) . Notice that it is the r −1 -fields whose power makes it to infinity in three dimensions. This is because E × B ∼ r −2 which exactly cancels the r 2 in the measure factor d3 x = r 2 drdΩ. In two dimensions, it is the r −1/2 -fields whose power makes it to infinity since d2 x = rdrdθ. As before, a more accurate description of the radiation field is obtained by including higher order terms in the sum (3.49). The lowest order (nonvanishing) multipole contributes the most to the field. In the intermediate zone, neither of the previous approximations are valid. In fact all terms in the previous series expansions would have to be kept. The understanding of the behavior of the fields in this zone requires the more sophisticated machinery of vector multipole fields. The interested reader is referred to [9] for a detailed discussion of multipole fields of arbitrary order (l, m). 46 x p θ Figure 3–3: The differential power dP/dΩ or radiation pattern corresponding to an oscillating electric dipole verifies that no radiation is emitted in the direction of the dipole moment. Rather the dipole radiation is a maximum in the direction transverse to p. Outside of the physics literature there is also a large amount of engineering literature in the field of computational electrodynamics. In this area, Maxwell’s field equations are often solved numerically by finite element methods (see Jiao and Jin [81] and references therein). Many applications of this work lie in electromagnetic scattering, waveguiding, and antenna design. The inverse source problem [82] is also another area of interest in engineering. Here, the goal is to determine the sources of charge and current with only the knowledge of the electromagnetic fields outside of the source’s region of support. This problem has benefited from the work of Goedecke [83], Devaney and Wolf [84], Marengo and Ziolkowski [85], and Hoenders and Ferwerda [86], who have demonstrated the decomposition of the electromagnetic field into nonradiating and purely radiating components. 47 3.2.2 Electromagnetic Radiation from a Gaussian Wavepacket Consider the gaussian wavepacket with initial position r moving with constant velocity v Ψ(x, t) = h 2`2 i3/4 π e−` 2 [x−(r+vt)]2 eimv·x , (3.53) √ where b = 1/ 2` is the wavepacket width. The corresponding probability current is given by J(x, t) = vρ(x, t) = q [Ψ∗ (−i∇Ψ) + Ψ(i∇Ψ∗ )]. 2m (3.54) In Fourier space this current becomes J̃(k, t) = qv exp(−ik · [r + vt] − k 2 /8`2 ) and (+) the vector potential is obtained by integrating against the Green’s function Dk for the wave operator (see Appendix B). The vector potential becomes d3 k ik·x R ∞ 0 (+) e dt Dk (t, t0 )4πcJ̃(k, t0 ) V (2π)3 −∞ R d3 k ik·(x−r)−k2 /8`2 R ∞ 0 Θ(t − t0 ) sin ck(t − t0 ) −ik·vt0 e dt e = 4πcqv V (3.55) −∞ (2π)3 ck R d3 k eik·[x−(r+vt)]−k2 /8`2 = 4πcqv V (2π)3 c2 k 2 − (k · v)2 A(x, t) = R which is difficult to perform analytically due to the complicated angular dependence of the integrand. For nonrelativistic velocities, A can be approximated by 2 2 d3 k eik·[x−(r+vt)]−k /8` V (2π)3 c2 k 2 [1 − (v/c)2 cos2 θ] qv R d3 k 2 2 ≈ 2 V 2 eik·[x−(r+vt)]−k /8` 2π c √k qv erf 2`|x − (r + vt)| = , c |x − (r + vt)| A(x, t) = 4πcqv R (3.56) where Gradshteyn and Ryzhik [87] was used. The norms of this vector potential and its associated current density J are plotted along the x-axis in Figure 3–4 for two different velocities. The charge q is taken to be negative. Notice that A follows the charge distribution and that A will generate an electromagnetic field. For v/c 1 this result is equivalent to a Galilei boost of the fields from the rest frame of the 48 1 0.5 0 -5 0 5 10 15 20 x Figure 3–4: The norms of J and A are plotted with different velocities along the x-axis. source. Only the electrostatic field remains by going to the rest frame. And so, there is little difference between uniform motion and no motion. As stated previously, the more interesting field dynamics is created whenever the source is accelerated. To this end, consider the vector potential arising from a moving charge whose current has the simple time dependence J̃(k, t) = qv(t)e−ik·(r+vt)−k = qve 2 /8`2 −ik·(r+vt)−k 2 /8`2 (3.57) Θ(t − t0 )Θ(t1 − t), where v is constant. This time dependence corresponds to a situation in which a source is suddenly accelerated from a standstill to a uniform movement with velocity v and is then instantaneously decelerated again to a standstill (see Figure 3–5). In each of the three temporal regions of the current, the vector potential has a different 49 ct r(t) future PSfrag replacements elsewhere t2 t1 x past Figure 3–5: The trajectory or world line r(t) of the charge is plotted. Electromagnetic radiation is produced at t1 and t2 and moves out on the light cone. behavior. Obviously for t < t0 , A(x, t) = 0. For t0 ≤ t < t1 , √ qv n erf 2`|x − (r + vt)| A(x, t) ≈ c |x − (r + vt)| √ 1 erf 2`[c(t − t0 ) − |x − (r + vt0 )|] + 2 |x − (r + vt0 )| √ 1 erf 2`[c(t − t0 ) + |x − (r + vt0 )|] o − 2 |x − (r + vt0 )| and for t ≥ t1 , √ 2`[c(t − t0 ) − |x − (r + vt0 )|] |x − (r + vt0 )| √ erf 2`[c(t − t0 ) + |x − (r + vt0 )|] − |x − (r + vt0 )| √ erf 2`[c(t − t1 ) + |x − (r + vt1 )|] + |x − (r + vt1 )| √ erf 2`[c(t − t1 ) − |x − (r + vt1 )|] o − . |x − (r + vt1 )| qv n erf A(x, t) ≈ 2c (3.58) (3.59) Again nonrelativistic velocities are assumed. A space-time plot of the norm of this piecewise vector potential is shown in Figure 3–6. Note that the charge was at rest until the time t0 , where it was instantaneously accelerated to a velocity of magnitude acements 50 2 0 -2 -10 -5 x 0 5 10 0 5 10 t Figure 3–6: The bremsstrahlung radiation from a charged gaussian wavepacket moves out on the smeared light cone with maximum at x = ct. v. Then the charge moved uniformly with v until the time t1 , when it was instantaneously decelerated to rest again. Since electromagnetic radiation is produced whenever the velocity changes in time, electromagnetic ripples are produced at t0 and t1 . The ripples move out as radiation at the velocity c of light. Figure 3–6 shows the light cone, which is smeared out due to the nonpointlike structure of the charge. The vector potential presented so far has both longitudinal and transverse components. For the time being, the tranversality of the A is not important. It turns out that the only fields which contribute to the Poynting vector or to the power are the transverse fields. And so it does no harm to keep the full vector potential. For the interested reader, the transverse vector potential AT associated with (3.58) and (3.59) is calculated in Appendix C by analogy to the quadrupole moment tensor. 51 The electric and magnetic fields corresponding to (3.59) are Ȧ c r 2 2 2 2 qv` 2 h e−2` [c(t−t0 )−R(t0 )] − e−2` [c(t−t0 )+R(t0 )] =− c π R(t0 ) E≈− + qv` =− c r e−2` 2 [c(t−t 2 − e−2` R(t1 ) 1 )+R(t1 )] 2 [c(t−t 1 )−R(t1 )] 2 2 − g0 (t) + g1− (t) π| {z } i (3.60) h− (t) neglecting the purely longitudinal −∇Φ, and B=∇×A q` = −v × c r h 2 2 2 2 e−2` [c(t−t1 )+R(t1 )] + e−2` [c(t−t1 )−R(t1 )] 2 û(t1 ) π R(t1 ) e−2` 2 [c(t−t 2 + e−2` − û(t0 ) R(t0 ) r i h q` 2 =− v × û(t1 )g1+ (t) − û(t0 )g0+ (t) , c π 0 )−R(t0 )] 2 [c(t−t 0 )+R(t0 )] 2 i (3.61) where R(t) ≡ |R(t)| = |x − (r + vt)| and where the unit vectors û(t0 ) = R(t0 )/R(t0 ) and û(t1 ) = R(t1 )/R(t1 ). With (3.60) and (3.61), the Poynting vector is c E×B 4π io n h q 2 `2 = 2 h− (t) v × v × û(t1 )g1+ (t) − û(t0 )g0+ (t) . 2π c S= (3.62) The differential power radiated into the solid angle dΩ at time t becomes dP (x, t) = R(t)2 n̂ · S(t) dΩ nh i q 2 `2 v · û(t1 ) g1+ (t) − v · û(t0 ) g0+ (t) n̂ · v = 2 h− (t)R(t)2 2π c h io − v 2 n̂ · û(t1 ) g1+ (t) − n̂ · û(t0 ) g0+ (t) n h i q 2 `2 = 2 h− (t)R(t)2 g0+ (t) v × n̂ · v × û(t0 ) 2π c h io + − g1 (t) v × n̂ · v × û(t1 ) (3.63) 52 where the unit vector n̂ = R(t)/R(t) is normal to the surface of the ball that emanates from the radiation source. The vectors v, n̂, û(t0 ), and û(t1 ) are all constant in time. By choosing the z-axis along the velocity v, the angles between v and the unit vectors û(t0 ) and û(t1 ) are δ0 and δ1 respectively. With a little geometry, it can be verified that sin θ sin θ sin δ0 ≈ p and sin δ1 ≈ p (3.64) 1 + (v/ct)(t1 − t0 ) cos θ 1 − (v/ct)(t1 − t0 ) cos θ by suppressing terms of quadratic order and higher in c−1 , where θ is the angle between v and n̂. In terms of the angles δ0 , δ1 , and θ, the differential power becomes h i q 2 `2 v 2 − dP 2 + + = h (t)R(t) g (t) sin δ sin θ − g (t) sin δ sin θ 0 1 0 1 dΩ 2π 2 c (3.65) which is independent of the polar angle φ. The corresponding radiation pattern is shown in Figure 3–7 and shows that power is radiated in all directions except along the direction of motion. Notice that the “dipole-like” pattern is modified by contributions arising from the expansion of the square roots in the angles δ0 and δ1 . That is h i sin2 θ v 2 2 2 p = sin θ 1 ∓ (t1 − t0 ) cos θ + O (v/c) cos θ . (3.66) 2ct 1 ± (v/ct)(t1 − t0 ) cos θ These contributions are more significant at higher velocities. The quadrupole pattern in Figure 3–7 is obviously overemphasized. By integration over the unit sphere, the total power is found to be n o R q 2 `2 v 2 − 2 π + 2 + 2 P (x, t) = h (t)R(t) 0 dθ g0 (t) sin δ0 sin θ − g1 (t) sin δ1 sin θ πc (3.67) which is equivalent to dE/dt where E is the total field energy. Both of the integrals in (3.67) can be done analytically. Since both h− and g + are proportional to 1/R, the power does not decay with the radius x. 53 PSfrag replacements v x Figure 3–7: The radiation pattern given by (3.63) shows the characteristic dipole pattern at lowest order. Keeping O(v/c) terms reveals the quadrupole pattern. Higher order multipole patterns are generated by O(v 2 /c2 ) and higher terms. For an electron whose charge distribution has a width corresponding to the Bohr radius a0 and has a velocity of ve = 1 a.u. between the times t0 = 0 a.u. and t1 = 1 a.u., the instantaneous power is P ≈ 2 × 10−3 a.u. ≈ 3 × 10−4 J/s at the maximum of the peak from t0 . The power from the t1 peak is the same. In order to put the previous results into perspective it is useful to make a comparison with the Larmor result. The Larmor power P (t) = 2q 2 v̇(t)2 3c3 (3.68) is the instantaneous power radiated by an accelerated point charge that is observed in a reference frame where the velocity of the charge is significantly less than that of light. The angular behavior of the emitted radiation may be determined by examining the differential power q2 dP (t) = v̇(t)2 sin2 θ dΩ 4πc3 (3.69) which is the dipole radiation pattern. If the result of (3.65) is correct, then it should reduce to the Larmor formula in the limit of the wavepacket width b going to zero 54 (point charge). Making use of the identity ` 2 2 δ(x) = lim √ e−` x , `→+∞ π (3.70) √ where ` = 1/ 2b, the differential power in (3.65) becomes h i2 q2 dP 2 = sin θ v{δ(t − t ) − δ(t − t )} . 0 1 dΩ 4πc3 | {z } (3.71) a2 Again v/c 1 was assumed. The term in square brackets has the dimensions of acceleration. And so, (3.71) reduces to the Larmor result (3.69) for the stepwise velocity v(t) = vΘ(t − t0 )Θ(t1 − t). These results are presented in [88]. CHAPTER 4 CANONICAL STRUCTURE The governing equation of quantum mechanics is the Schrödinger equation [7, 8]. In the minimal coupling prescription it is iΨ̇ = [−i∇ − qA/c]2 Ψ + V Ψ + qΦΨ. 2m (1.1) The dynamics of the scalar potential Φ and vector potential A are not described by this linear equation. Specification of these potentials as well as the initial values for the wavefunction Ψ casts the Schrödinger equation into a well-defined boundary value problem that is also a well-defined initial value problem. The governing equations of electrodynamics are Maxwell’s equations [9]: ∇ · E = 4πρ ∇×B = Ė 4π J+ c c ∇·B = 0 ∇×E+ Ḃ = 0. (1.2) c The dynamics of the charge density ρ and current density J are not described by these linear equations. Specification of the external sources as well as the initial values for the electric and magnetic fields E and B satisfying ∇ · E = 4πρ and ∇ · B = 0 casts the Maxwell equations into a well-defined boundary value problem that is also a well-defined initial value problem. Each of these theories are significant in and of themselves. Given a particular arrangement of sources throughout space-time and the initial values for E and B, the Maxwell equations govern the dynamics of the resulting electromagnetic field. Likewise, given a particular external field throughout space-time and the initial value for Ψ, the Schrödinger equation governs the dynamics of the sources. However, notice that the Maxwell equations do not say anything about the dynamics of the sources and the Schrödinger equation does not say anything about the electrodynamics. 55 56 It is possible to couple the linear Maxwell and Schrödinger equations. The resulting nonlinear Maxwell-Schrödinger theory accounts for the dynamics of the charges and the electromagnetic field as well as their mutual interaction. For example, given an initial source and its corresponding Coulomb field, a wavefunction and electromagnetic field are generated. The electromagnetic field has its own dynamics and acts back upon the wavefunction. This in turn causes different fields to be generated. It will be demonstrated that these coupled nonlinear Maxwell-Schrödinger equations can be cast into a well-defined initial value problem and solved in an efficient numerical manner. 4.1 Lagrangian Electrodynamics Consider the Maxwell Lagrangian density LMax = [−Ȧ/c − ∇Φ]2 − [∇ × A]2 J·A − ρΦ + 8π c (4.1) with external sources ρ and J. Variation of this Lagrangian leads to the governing equations of electrodynamics, i.e., ∇2 A − h Φ̇ i 4π Ä − ∇ ∇ · A + =− J 2 c c c ∇ · Ȧ = −4πρ ∇2 Φ + c (1.4) These Maxwell equations (in terms of the potentials) do not form a well-defined initial value problem. But, by choosing a particular gauge they can be turned into one. In other words, these equations are ill-posed as they stand. However, they do enjoy both Lorentz and gauge invariance as does the Lagrangian (4.1). 4.1.1 Choosing a Gauge Working in a particular gauge can be organized into the following hierarchy: 1. At the solution level, a gauge generator F can be chosen so that a gauge transformation of the solutions, i.e., Φ → Φ0 = Φ − Ḟ /c and A → A0 = A + ∇F, maps them to new solutions that satisfy the gauge condition. 57 2. At the equation level, the set consisting of (1.4) together with a gauge constraint has only solutions that satisfy the gauge condition. 3. At the Lagrangian level, a gauge fixing term can be added to (4.1) so that the resulting Euler-Lagrange equations automatically include the gauge constraint. 4.1.2 The Lorenz and Coulomb Gauges The first two tiers can be elaborated on as follows. With a gauge function F satisfying ∇2 F − F̈ /c2 = −[Φ̇/c + ∇ · A] a solution Aα = (Φ, A) of the potential equations (1.4) can be mapped to the Lorenz gauge solution AαLorenz according to the gauge transformation: Φ → ΦLorenz = Φ − Ḟ /c A → ALorenz = A + ∇F. (4.2) Alternatively, adding the gauge constraint Φ̇/c + ∇ · A = 0 to (1.4) leads to the Lorenz gauge equations of motion: ∇2 A − 4π Ä =− J 2 c c ∇2 Φ − Φ̈ = −4πρ. c2 (4.3) With ρ and J specified throughout space-time, the Lorenz gauge equations of motion are well-defined once the initial values for A, Ȧ, Φ, and Φ̇ are known. There is some symmetry left in the solutions to these equations. Namely, the residual gauge freedom left in the homogeneous equation ∇2 F − F̈ /c2 = 0 allows for gauge transformations on the solutions such that the new solutions do not leave the Lorenz gauge. However, these gauge transformed solutions do correspond to different initial conditions. Note that the Lorenz gauge enjoys relativistic or Lorentz invariance. It will be shown, that the Lorenz gauge is the most appropriate gauge for dynamics. With another gauge function G satisfying ∇2 G = −∇·A a solution Aα = (Φ, A) of the potential equations (1.4) can be mapped to the Coulomb gauge solution AαCoulomb according to: Φ → ΦCoulomb = Φ − Ġ/c A → ACoulomb = A + ∇G. (4.4) 58 Alternatively, adding the gauge constraint ∇ · A = 0 to (1.4) leads to the Coulomb gauge equations of motion: ∇2 A − 4π ∇Φ̇ Ä = − J + c2 c c ∇2 Φ = −4πρ. (4.5) Again with ρ and J specified throughout space-time, the Coulomb gauge equations of motion are well-defined once the initial values for A, Ȧ, Φ, and Φ̇ are known. As before, there remains a symmetry or residual gauge freedom from the homogeneous equation ∇2 G = 0. Note that in the Coulomb gauge Gauss’s law reduces to ∇2 Φ = −4πρ. Inverting this equation specifies Φ in terms of ρ. That is Φ = (1/∇2 )[−4πρ]. The scalar potential can now be totally removed from the theory by substitution of this Green’s function integral. This may be done at the expense of Lorentz invariance. In practice, where the equations are to be expanded in a basis of sgaussians, either transverse basis functions would have to be used or the transverse fields would have to be generated from a standard basis. The former case would require a major revision of most existing integral codes, which are in direct space, while the latter would require the instantaneous transverse projection PTab = δ ab − ∂ a ∂ b /∇2 (see Appendix B) This operation, which is over all space, is difficult to describe in terms of a local set of basis functions. Lastly, for the third tier, consider the Lagrangian density (4.1) together with a gauge fixing term for the Lorenz gauge, i.e., [Φ̇/c + ∇ · A]2 8π [−Ȧ/c − ∇Φ]2 − [∇ × A]2 J · A [Φ̇/c + ∇ · A]2 = − ρΦ + − . 8π c 8π LLMax = LMax − (1.18) The resulting Euler-Lagrange equations obtained from LLMax are identical to the Lorenz gauge wave equations in (4.3) which are equivalent to the general potential equations (1.4) together with the constraint Φ̇/c + ∇ · A = 0. LMax g replacements LC Max 59 LLMax d ∂LMax dt ∂ ξ˙ d ∂LMax d ∂LMax Max Max Max =0 =0 =0 − ∂L∂ξ − ∂L∂ξ − ∂L∂ξ dt ∂ ξ˙ dt ∂ ξ˙ gauge invariant ∇ · A + Φ̇/c = 0 ∇·A=0 add add well-posed IVP constraint ill-posed IVP constraint well-posed IVP unique solution many solutions gauge gauge transformation transformation unique solution Maxwell’s equations well-posed IVP Figure 4–1: A limited but relevant portion of the gauge story in the Lagrangian formalism is organized in this picture. The middle column (i.e., the column below LMax ) enjoys full gauge freedom. The far left (Coulomb gauge) and far right (Lorenz gauge) columns have limited gauge freedom. That is, there are a limited class of gauge transformations that can be made on the solutions such that they remain in the same gauge. This symmetry is due to the residual gauge freedom. Note that these solutions correspond to different initial conditions within the gauge. Also note that the Euler-Lagrange equations together with a particular gauge constraint are equivalent to the Euler-Lagrange equations derived from that particular gauge fixed Lagrangian. There are many other known gauges, the choice of which is arbitrary. All choices of gauge lead to the same physically observable electromagnetic fields E and B. Together with the definitions E = −Ȧ/c − ∇Φ and B = ∇ × A, the Lorenz and Coulomb gauge equations of motion as well as the general potential equations (1.4) imply Maxwell’s equations (1.2). A diagram of this gauge story in the Lagrange formulation is presented in Figure 4–1. 4.2 Hamiltonian Electrodynamics In the Hamiltonian prescription, the momentum conjugate to A with respect to the Maxwell Lagrangian (4.1) is Π≡ ∂LMax 1 [Ȧ/c + ∇Φ]. = 4πc ∂ Ȧ (4.6) 60 The momentum conjugate to Φ is identically zero, i.e., Θ≡ ∂LMax = 0. ∂ Φ̇ (4.7) A Hamiltonian density can still be defined as the time-time component of the αβ Maxwell stress-energy tensor TMax = {∂LMax /∂(∂α ξ)}∂ β ξ − g αβ LMax . It is 00 HMax ≡ TMax = Π · Ȧ + ΘΦ̇ − LMax = [−4πcΠ]2 + [∇ × A]2 J·A − c∇Φ · Π + ρΦ − 8π c (4.8) and the resulting equations of motion are: Ȧ ≡ ∂HMax = 4πc2 Π − c∇Φ ∂Π −Π̇ ≡ ∂HMax ∇[∇ · A] − ∇2 A J = − + c∇Θ ∂A 4π c (4.9) Φ̇ ≡ ∂HMax =0 ∂Θ −Θ̇ ≡ ∂HMax = ρ + c∇ · Π. ∂Φ Since the momentum Θ defined in (4.7) is identically zero, so is its time derivative Θ̇ and gradient ∇Θ. Notice that these Hamilton equations form a well-posed initial value problem. The machinery inherent in the Hamiltonian formalism automatically adds a momentum and automatically adds the additional equation of constraint Φ̇ = 0. It turns out that this extra equation fixes a particular gauge where Φ̇ = 0. This gauge can always be fixed by a gauge transformation whose generator satisfies F̈ /c = Φ̇. The residual gauge freedom left in the homogeneous equation F̈ = 0 does allow for a gauge transformation on the solutions to (4.9). These new gauge transformed solutions do not leave the Φ̇ = 0 gauge, but do correspond to a different initial value problem within this gauge. In other words, they are solutions to (4.9) with different initial values. Pay careful attention to the fact that these Hamilton equations of motion form a well-posed initial value problem even though a gauge fixed Lagrangian was not knowingly used. The Hamiltonian formalism automatically added the extra equation Φ̇ = 0. 61 4.2.1 Hamiltonian Formulation of the Lorenz Gauge Rather than fixing the Coulomb gauge at the equation level it may be beneficial to work in a more general theory where a gauge is chosen at the Lagrangian level and retains all of the 4-potential, is Lorentz invariant, and does not require any instantaneous or nonlocal operations. To this end, consider the Lorenz gauge Lagrangian density from (1.18), i.e., LLMax = [−Ȧ/c − ∇Φ]2 − [∇ × A]2 J · A [Φ̇/c + ∇ · A]2 − ρΦ + − . 8π c 8π (1.18) It will be shown that the equations of motion derived from LLMax are well-defined because of the addition of the last term in this expression. It turns out that this term is known in the literature [24, 68] and is a gauge fixing term for the Lorenz gauge. From (1.18), the momentum conjugate to A is Π≡ ∂LLMax 1 = [Ȧ/c + ∇Φ] 4πc ∂ Ȧ (4.10) and the momentum conjugate to Φ is Θ≡ 1 ∂LLMax [Φ̇/c + ∇ · A]. =− 4πc ∂ Φ̇ (4.11) With these momenta and coordinates, electrodynamics is given a symplectic structure. The Hamiltonian density is L HMax = [−4πcΠ]2 + [∇ × A]2 − [4πcΘ]2 J·A − c∇Φ · Π − cΘ∇ · A + ρΦ − (4.12) 8π c and the resulting equations of motion are: Ȧ = 4πc2 Π − c∇Φ −Π̇ = ∇[∇ · A] − ∇2 A J − + c∇Θ 4π c (4.13) Φ̇ = −4πc2 Θ − c∇ · A −Θ̇ = ρ + c∇ · Π. 62 These equations, which are a generalization of (4.3), together with the initial values for A, Π, Φ, and Θ form a well-posed initial value problem. The residual gauge freedom resulting from the homogeneous equation F = 0 does allow for a gauge transformation on the solutions to (4.13). These new gauge transformed solutions do not leave the Lorenz gauge, but do correspond to a different initial value problem within the Lorenz gauge. In other words, they are solutions to (4.13) with different initial values. Notice that a relationship exists between the momentum Θ and the gauge function F leading to the Lorenz gauge. That is, from Θ = −[Φ̇/c + ∇ · A]/4πc and F̈ /c2 − ∇2 F = Φ̇/c + ∇ · A notice that Θ ≡ F/4πc. So the D’Alembertian of the gauge function F acts a generalized coordinate in this phase space. It is the momentum conjugate to the scalar potential Φ. In matrix form, the dynamical equations in (4.13) are 0 0 −1 0 Ȧ ∇ × [∇ × A]/4π − J/c + c∇Θ 0 0 0 −1 Φ̇ ρ + c∇ · Π = 0 Π̇ 4πc2 Π − c∇Φ 1 0 0 0 1 0 0 Θ̇ −4πc2 Θ − c∇ · A , (4.14) where 1 is the 3 × 3 identity matrix. Notice that (4.14) is of the Hamiltonian form ω η̇ = ∂H/∂η. (4.15) More specifically ωab η̇ b = ∂H/∂η a , where η b is a column matrix of the generalized positions and momenta, i.e., k A Φ η b (x, t) = , k Π Θ (4.16) 63 where k = 1, 2, 3. The antisymmetric matrix ωab is the (canonical) symplectic form associated with the phase space of electrodynamics in the Lorenz gauge. By substitution, these first order Hamiltonian equations of motion can be shown to be equivalent to the second order Lorenz gauge equations Φ = −4πρ and A = −4πJ/c. Together with the definition of the electric and magnetic fields, (4.13) imply ∇ · E = 4πρ + 4π Θ̇ ∇×B= 4π Ė J + − 4πc∇Θ c c (4.17) ∇·B =0 ∇×E+ Ḃ = 0. c These equations are not equivalent to Maxwell’s equations unless Θ(x, t) remains constant in space-time throughout the dynamics. In order to analyze this question, the dynamics of the sources must be considered. It should be noticed that the inhomogeneous equations in (4.17) imply Θ ≡ ∇2 Θ − Θ̈ 1 = 2 [ρ̇ + ∇ · J]. 2 c c (4.18) If the matter theory is such that the equation of continuity ρ̇ = −∇ · J is satisfied, then Θ = 0. So if Θ(t = 0) = Θ̇(t = 0) = 0, then Θ(t) = 0 at all times t. In other words, if the sources of charge and current satisfy the equation of continuity, then the dynamical theory arising from the Lagrangian (1.18) is the Maxwell theory of electrodynamics. Note that while (4.9) and (4.13) do not enjoy the full gauge symmetry as do the general potential equations (1.4), this does not mean that the observables resulting from (4.9) or (4.13) are not gauge invariant. Any observable that is calculated will be invariant to the choice of gauge generator. Moreover, once the solutions to these well-defined equations are constructed, these solutions belong to the many solutions of (1.4). This family of solutions is the most general solutions of the potential form of Maxwell’s equations. In fact, gauge transformations can even be made from one 64 particular gauge to another [89]. A diagram depicting the relevant gauge story in the Hamiltonian formulation is presented in Figure 4–2. Notice that there is no Hamiltonian theory that enjoys the full gauge symmetry of (1.4). The Hamiltonian HMax in the far right column is obtained by a Legendre transformation of the gauge invariant Lagrangian LMax in (4.1). However, the Hamiltonian dynamics stemming from the gauge invariant LMax is not gauge invariant, but rather occurs in the gauge where Φ̇ = 0. PSfrag replacements LC Max LLMax LMax d ∂LMax d ∂LMax Max Max Max − ∂L∂ξ =0 − ∂L∂ξ =0 − ∂L∂ξ =0 dt ∂ ξ˙ dt ∂ ξ˙ gauge invariant ∇ · A + Φ̇/c = 0 ∇·A=0 add add ill-posed IVP constraint well-posed IVP well-posed IVP constraint d ∂LMax dt ∂ ξ˙ unique solution gauge transformation many solutions gauge transformation unique solution unique solution gauge transformation Maxwell’s equations well-posed IVP ∂HC ω η̇ = ∂ηMax well-posed IVP C HMax = ∂LC Max ˙ ξ ∂ ξ˙ − LC Max ∂HL ω η̇ = ∂ηMax well-posed IVP L HMax = ∂LL Max ˙ ξ ∂ ξ˙ − LLMax ω η̇ = ∂H∂ηMax well-posed IVP HMax = ∂LMax ˙ ξ ∂ ξ˙ − LMax Figure 4–2: The Hamiltonian formulation of the gauge story is organized in this picture with respect to the previous Lagrangian formulation. Figure 4–1 is depicted in the box with dotted borders. It can now be seen how the Coulomb and Lorenz gauges connect in both formalisms. 65 66 4.2.2 Poisson Bracket for Electrodynamics The phase space that carries the associated dynamics is naturally endowed with a Poisson bracket {·, ·} (recall Chapter 2). This may be seen by considering the variation of ξ along the dynamics ∆H ≡ (∂/∂η)η̇. That is −1 ∆H (ξ) ≡ (d/dt)ξ = (∂ξ/∂η b )η̇ b = (∂ξ/∂η b )ωab (∂H/∂η a ) ≡ {ξ, H}, (4.19) where η are the generalized coordinates. In general, the Poisson bracket of the dynamical variable F with the dynamical variable G is T 0 ∂F/∂A ∂F/∂Φ 0 {F, G} = ∂F/∂Π 1 0 ∂F/∂Θ 0 −1 0 0 0 0 1 0 0 −1 −1 0 0 ∂G/∂A ∂G/∂Φ . ∂G/∂Π ∂G/∂Θ (4.20) Since the symplectic form ω is canonical its inverse is trivial, i.e., ω −1 = ω T = −ω. Also notice that ω 2 = −1, ω T ω = 1, and det ω = 1. 4.3 Hamiltonian Electrodynamics and Wave Mechanics in Complex Phase Space Consider the matter theory associated with the Schrödinger Lagrangian (~ = 1) LSch = iΨ∗ Ψ̇ − [i∇ − qA/c]Ψ∗ · [−i∇ − qA/c]Ψ − V Ψ∗ Ψ − qΦΨ∗ Ψ 2m (1.17) where Ψ is the wavefunction for a single electron, V = q q̄/|x| is the static Coulomb potential energy of a proton, and (Φ, A) are the electron’s scalar and vector potentials. Notice that this Lagrangian is already written in phase space. The momentum conjugate to the wavefunction Ψ is iΨ∗ . Together with the previous Maxwell Lagrangian, the coupled nonlinear dynmical theory arising from the Lagrangians LMax n [−4πcΠ]2 + [∇ × A]2 o 1 = [Π · Ȧ − Π̇ · A] − − c∇Φ · Π 2 8π (4.21) 67 n [i∇ − qA/c]Ψ∗ · [−i∇ − qA/c]Ψ o i + V Ψ∗ Ψ + qΦΨ∗ Ψ LSch = [Ψ∗ Ψ̇ − Ψ̇∗ Ψ] − 2 2m (4.22) 1 Lgauge = [ΘΦ̇ − Θ̇Φ] − {−2πc2 Θ2 − cΘ∇ · A} 2 (4.23) yields the following equations of motion: Ȧ = 4πc2 Π − c∇Φ Φ̇ = −4πc2 Θ − c∇ · A 2 iΨ̇ = [−i∇ − qA/c] Ψ + V Ψ + qΦΨ 2m −Π̇ = ∇[∇ · A] − ∇2 A J − + c∇Θ 4π c −Θ̇ = ρ + c∇ · Π −iΨ̇∗ = (4.24) 2 ∗ [i∇ − qA/c] Ψ + V Ψ∗ + qΦΨ∗ . 2m Surface terms of the form (d/dt){pq/2} have been added in the above Lagrangians in order to symmetrize them, i.e., L = pq̇ − H − (d/dt){pq/2} becomes L = [pq̇ − R R ṗq]/2 − H. This can always be done since the action I = Ldt = [L + (d/dt)g]dt is invariant to the addition of a pure surface term to the Lagrangian. Note that the Schrödinger wavefunctions Ψ and Ψ∗ are complex-valued while the remaining electromagnetic variables are all real-valued. These dynamical equations may be put into matrix form as 2 0 Ψ̇ [−i∇ − qA/c] Ψ/2m + V Ψ + qΦΨ i 0 0 0 0 0 −i 0 0 0 Ψ̇∗ [i∇ − qA/c]2 Ψ∗ /2m + V Ψ∗ + qΦΨ∗ 0 0 0 0 0 −1 0 Ȧ ∇ × [∇ × A]/4π − J/c + c∇Θ = , 0 0 0 0 0 −1 Φ̇ ρ + c∇ · Π 0 0 1 0 0 Π̇ 2 0 4πc Π − c∇Φ 0 0 0 1 0 0 Θ̇ −4πc2 Θ − c∇ · A (4.25) where the symplectic form is canonical. The electromagnetic sector of it is identical to (4.14). These dynamical equations define the coupled Maxwell-Schrödinger 68 theory. This theory is well-defined and closed. In other words, the dynamics of the charges, currents, and fields are all specified as well as their mutual interaction. Given initial values for Ψ, Ψ∗ , A, Π, Φ, and Θ determines their coupled dynamics throughout space-time. With the dynamics of the charges defined, the problem in (4.17) can now be addressed. The Schrödinger equation in (4.24) implies the continuity equation (d/dt)qΨ∗ Ψ = −∇ · q Ψ∗ [−i∇ − qA/c]Ψ + Ψ[i∇ − qA/c]Ψ∗ /2m (4.26) which may be written more compactly as ρ̇ = −∇ · J. From the definition of the momentum Θ in (4.11) and the wave equations Φ = −4πρ and A = −4πJ/c, notice that Θ = 1 1 −1 [(d/dt)Φ/c + ∇ · A] = [(d/dt)4πρ/c + ∇ · 4πJ/c] = 2 [ρ̇ + ∇ · J] = 0 4πc 4πc c (4.27) by appealing to (4.26). So if Θ(t = 0) = Θ̇(t = 0) = 0, then the electrodynamics stays in the Lorenz gauge for all time since the only solution of Θ = 0 with Θ(t = 0) = Θ̇(t = 0) = 0 is Θ(t) = 0. It is worth mentioning that if Θ̇(t = 0) = 0 for all time, then the electronelectron self interaction makes no contribution to the Schrödinger energy. This is true since the self interaction term qΦΨ∗ Ψ in the above Schrödinger Lagrangian cancels exactly with −c∇Φ·Π in the Maxwell Lagrangian. The cancellation requires a partial integration of −c∇Φ · Π to cΦ∇ · Π followed by a substitution of 0 = ρ + c∇ · Π from Θ̇(t = 0) = 0 in (4.24). However, there is still a contribution from the self-energy arising in the Maxwell energy of the Coulombic field. 69 4.4 Hamiltonian Electrodynamics and Wave Mechanics in Real Phase Space The dynamical equations (5.16) are mixed, real and complex. For consistency these equations are put into real form with the Lagrangian densities: n [−4πcΠ]2 + [∇ × A]2 o 1 − c∇Φ · Π LMax = [Π · Ȧ − Π̇ · A] − 2 8π LSch {[∇Q + qAP/c]2 + [−∇P + qAQ/c]2 }/4m 1 = [P Q̇ − Ṗ Q] − 2 +V [Q2 + P 2 ]/2 + qΦ[Q2 + P 2 ]/2 1 Lgauge = [ΘΦ̇ − Θ̇Φ] − {−2πc2 Θ2 − cΘ∇ · A} 2 (4.28) (4.29) (4.30) The functions P and Q are related to the real and imaginary parts of Ψ and Ψ∗ √ √ according to Ψ = [Q + iP ]/ 2 and Ψ∗ = [Q − iP ]/ 2. The equations of motion that are associated with these Lagrangians are: Ȧ = 4πc2 Π − c∇Φ −Π̇ = ∇[∇ · A] − ∇2 A J − + c∇Θ 4π c (4.31a) Φ̇ = −4πc2 Θ − c∇ · A Q̇ = −Θ̇ = ρ + c∇ · Π −∇2 P + q∇ · (AQ)/c + qA · ∇Q/c + q 2 A2 P/c2 + V P + qΦP 2m (4.31b) −Ṗ = − ∇2 Q + q∇ · (AP )/c + q∇P · A/c − q 2 A2 Q/c2 + V Q + qΦQ. 2m 70 These 0 0 0 1 0 0 dynamical equations may be put into matrix form as 0 0 −1 0 0 Ȧ Φ̇ 0 0 0 −1 0 0 0 0 0 −1 Q̇ Π̇ 0 0 0 0 0 1 0 0 0 0 Θ̇ Ṗ 0 1 0 0 0 ∇ × [∇ × A]/4π − J/c + c∇Θ ρ + c∇ · Π −[∇2 Q + q∇ · (AP )/c + q∇P · A/c − q 2 A2 Q/c2 ]/2m + V Q + qΦQ = 4πc2 Π − c∇Φ −4πc2 Θ − c∇ · A [−∇2 P + q∇ · (AQ)/c + qA · ∇Q/c + q 2 A2 P/c2 ]/2m + V P + qΦP , (4.32) where the symplectic form is again canonical. Note that the equation of continuity ρ̇ = −∇ · J still holds with the real charge and current densities ρ = q[Q2 + P 2 ]/2 4.5 J= q {Q∇P − P ∇Q − qQAQ/c − qP AP/c}. (4.33) 2m The Coulomb Reference by Canonical Transformation As was mentioned previously the numerical implementation of the theory can be made to converge more quickly if the basis is chosen judiciously. Recall that the electromagnetic field generated by any charge contains a Coulombic contribution. This monopole term accounts for a large portion of the local electromagnetic field surrounding the charge. It would be advantageous to not describe this large 71 contribution in terms of the basis but rather to calculate it analytically. The remaining smaller portion of the radiative or dynamical electromagnetic field can then be described in terms of the basis. To this end, notice that the scalar potential Φ = ΦC + (Φ − ΦC ) ≡ ΦC + ΦD may be split into a Coulombic portion satisfying ∇2 ΦC = −4πρ that can be calculated analytically and a remainder ΦD regardless of the choice of gauge. The Coulombic potential is not itself a dynamical variable but depends on the dynamical variables R Q and P. That is ΦC (x, t) = V d3 x0 |x − x0 |−1 q[Q(x0 , t)Q(x0 , t) + P (x0 , t)P (x0 , t)]/2. The dynamical portion ΦD is a generalized coordinate and is represented in the basis. Similarly, the momentum conjugate to A may be split into a Coulombic and dynamical piece according to Π≡ 1 ∇ΦC ∂LMax + [Ȧ/c + ∇ΦD ]. = ΠC + ΠD = 4πc 4πc ∂ Ȧ (4.34) Like ΦD , the dynamical portion ΠD is a generalized coordinate and is represented in the basis. 4.5.1 Symplectic Transformation to the Coulomb Reference The transformation to these new coordinates, i.e., ΦD and ΠD , is obtained by the canonical or symplectic transformation Ã(A) A Φ̃(Φ, Q, P ) Φ Q̃(Q) Q → T: Π̃(Π, Q, P ) Π Θ Θ̃(Θ) P̃ (P ) P A Φ − Φ (Q, P ) C Q = Π − ΠC (Q, P ) Θ P , (4.35) where Φ̃ ≡ ΦD and Π̃ ≡ ΠD . The variables Q, P, A, and Θ are unchanged by T. Since both ΦC and ΠC are complicated functions of Q and P, the inversion of T may 72 be quite involved. However, it will be shown that the inverse of T does exist. In fact both the T and T−1 are differentiable mappings on symplectic manifolds. Therefore the canonical transformation is a symplectic diffeomorphism or symplectomorphism [29]. The theory of restricted (i.e., explicitly time-independent) canonical transformations [27, 28] gives the general prescription for the transformation of the old Hamilton equations (4.32) to the new Hamilton equations in terms of T (and TT ) only. In symbols, that is η̇ = ω −1 ∂H ∂η ∂ H̃ → η̃˙ = ω̃ −1 , ∂ η̃ (4.36) where the new Hamiltonian H̃ is equivalent to the old Hamiltonian H expressed in terms of the new variables η̃. (For simplicity H̃ will be written as H from this point forward.) To this end, consider the time derivative of the new column matrix ∂ η̃i η̃˙ i = η̇j ≡ Tij η̇j ∂ηj or η̃˙ = Tη̇. (4.37) Substituting η̇ from (4.36) results in −1 ∂H η̃˙ i = Tij ωjk ∂ηk ∂H or η̃˙ = Tω −1 . ∂η (4.38) Lastly the column matrix ∂H/∂η can be written as ∂H ∂H ∂ η̃l ∂H = ≡ TklT ∂ηk ∂ η̃l ∂ηk ∂ η̃l or ∂H ∂H = TT ∂η ∂ η̃ (4.39) so that the new equations of motion (4.38) become −1 T ∂H Tkl η̃˙ i = Tij ωjk ∂ η̃l ∂H ∂H or η̃˙ = Tω −1 TT ≡ ω̃ −1 . ∂ η̃ ∂ η̃ (4.40) 73 This canonical transformation on the equations of motion leaves only the computation of ω̃ −1 ≡ Tω −1 TT since the Hamiltonian automatically becomes H= [−4πc{Π̃ + ΠC (Q̃, P̃ )}]2 + [∇ × Ã]2 − [4πcΘ̃]2 Q̃2 + P̃ 2 + q[Φ̃ + ΦC (Q̃, P̃ )] 8π 2 − c∇[Φ̃ + ΦC (Q̃, P̃ )] · [Π̃ + ΠC (Q̃, P̃ )] − cΘ̃∇ · à + Q̃2 + P̃ 2 [∇Q̃ + q ÃP̃ /c]2 + [−∇P̃ + q ÃQ̃/c]2 +V 4m 2 (4.41) in terms of the new coordinates. However, the inversion of ω is not simple in practice. It turns out that the equations of motion (4.36) are most practically written as ∂H = ω̃ η̃˙ = (T−1 )T ωT−1 η̃˙ ∂ η̃ ∂H ∂H not η̃˙ = ω̃ −1 = Tω −1 TT , ∂ η̃ ∂ η̃ (4.42) where the inverse transformation T−1 is the transformation of the inverse mapping, i.e., Tij−1 ≡ ∂ηi /∂ η̃j . It will be shown that det T 6= 0 so the mapping is well-defined. These equations of motion are of the desired form because they involve ω and not ω −1 . That ω −1 is undesirable is seen by going to the basis. In the basis, the canonical symplectic form becomes 0 −1 1 0 → 0 ∂ 2 hP |Qi ∂pK ∂qJ 2 i − ∂∂qKhQ|P ∂pJ 0 (4.43) which is not easily inverted. As a result it is simpler to compute (T−1 )T ωT−1 than Tω −1 TT even though T−1 is needed in the former case. It will be shown that the explicit evaluation of T−1 is not necessary. To continue with the transformed equations of motion in (4.42), which only require ω, the mapping (T−1 )T : ∂/∂η → ∂/∂ η̃ must first be set up. The transposed inverse transformation (T−1 )T is defined on the vector fields themselves according 74 to ∂/∂ à ∂/∂ Φ̃ ∂/∂ Q̃ ∂/∂ Π̃ ∂/∂ Θ̃ ∂/∂ P̃ 0 0 0 0 0 ∂A/∂ à 0 ∂Φ/∂ Φ̃ 0 0 0 0 0 ∂Φ/∂ Q̃ ∂Q/∂ Q̃ ∂Π/∂ Q̃ 0 0 = 0 0 0 ∂Π/∂ Π̃ 0 0 0 0 0 0 ∂Θ/∂ Θ̃ 0 0 ∂Φ/∂ P̃ 0 ∂Π/∂ P̃ 0 ∂P/∂ P̃ ∂/∂A ∂/∂Φ ∂/∂Q . ∂/∂Π ∂/∂Θ ∂/∂P (4.44) Notice that det(T−1 )T = det T−1 = (det T)−1 ≡ ∂(A, Φ, Q, Π, Θ, P ) ∂(Ã, Φ̃, Q̃, Π̃, Θ̃, P̃ ) (4.45) = (∂ Ã/∂A)(∂ Φ̃/∂Φ)(∂ Q̃/∂Q)(∂ Π̃/∂Π)(∂ Θ̃/∂Θ)(∂ P̃ /∂P ) = 1 so that the transformation is canonical and symplectic or area preserving. In other words, the new infinitesimal volume element dη̃ is related to the old infinitesimal volume element dη by dη̃ = det T dη = dη (4.46) since the determinant of the Jacobian is unity. Thus, the volume element of phase space is the same before and after the transformation. It is a canonical invariant. 75 With (T−1 )T the similarity transformation of the canonical symplectic form in (4.32) is −1 T ω̃ ≡ (T ) ωT −1 = ΩM Ω> , Ω∨ ΩG (4.47) where 0 −(∂A/∂ Ã)(∂Π/∂ P̃ ) (∂A/∂ Ã)(−1)(∂Π/∂ Π̃) Ω> = 0 (∂Φ/∂ Φ̃)(−1)(∂Θ/∂ Θ̃) 0 0 −(∂Φ/∂ Q̃)(∂Θ/∂ Θ̃) (∂Q/∂ Q̃)(−1)(∂P /∂ P̃ ) (4.48) 0 0 (∂Π/∂ Π̃)(1)(∂A/∂ Ã) ∨ Ω = 0 (∂Θ/∂ Θ̃)(1)(∂Φ/∂ Φ̃) (∂Θ/∂ Θ̃)(∂Φ/∂ Q̃) (∂Π/∂ P̃ )(∂A/∂ Ã) 0 (∂P /∂ P̃ )(1)(∂Q/∂ Q̃) (4.49) ΩM = ΩG = 0 0 −(∂A/∂ Ã)(∂Π/∂ Q̃) 0 0 0 (∂Π/∂ Q̃)(∂A/∂ Ã) 0 0 0 0 0 0 0 (∂Θ/∂ Θ̃)(∂Φ/∂ P̃ ) . 0 −(∂Φ/∂ P̃ )(∂Θ/∂ Θ̃) 0 (4.50) (4.51) The factors of 1 and −1 are explicitly written in Ω> and Ω∨ to bring out their similarity to the canonical symplectic form in (4.32). After computing the derivates 76 in ω̃ it can be shown that ω̃ equals 0 0 −∂ΠC (Q̃, P̃ )/∂ Q̃ −1 0 −∂ΠC (Q̃, P̃ )/∂ P̃ 0 0 0 0 −1 0 0 0 −∂ΦC (Q̃, P̃ )/∂ Q̃ −1 ∂ΠC (Q̃, P̃ )/∂ Q̃ 0 1 0 0 0 0 0 0 1 ∂Φ ( Q̃, P̃ )/∂ Q̃ 0 0 ∂Φ ( Q̃, P̃ )/∂ P̃ C C ∂ΠC (Q̃, P̃ )/∂ P̃ 0 1 0 −∂ΦC (Q̃, P̃ )/∂ P̃ 0 (4.52) with ΦC (x, t) = and ΠC (x, t) = q R Q(x0 , t)2 + P (x0 , t)2 3 0 dx 2 V |x − x0 | q R Q(x0 , t)2 + P (x0 , t)2 3 0 ∇ V d x. 8πc |x − x0 | (4.53) (4.54) And so the new symplectic form contains extra elements that are not present in the canonical ω. These extra elements add additional time-dependent couplings to the theory. As before, the associated phase space is naturally endowed with the Poisson bracket {F, G} ˜ = (∂F/∂ η̃)T ω̃ −1 (∂G/∂ η̃). (4.55) 77 The transformed equations of motion with symplectic form (4.52) may be written in full as: ∂ΠC ˙ ˙ − ∂ΠC P̃˙ = ∂H = ∇ × [∇ × Ã] + c∇Θ̃ Q̃ − Π̃ 4π ∂ Q̃ ∂ P̃ ∂ à o q n q q − Q̃∇P̃ − P̃ ∇Q̃ − Q̃ÃQ̃ − P̃ ÃP̃ 2mc c c 2 2 Q̃ + P̃ ∂H ˙ = + c∇ · [Π̃ + ΠC ] =q − Θ̃ 2 ∂ Φ̃ ∂ΠC ˙ ∂ΦC ˙ −∇2 Q̃ − q∇ · (ÃP̃ )/c − q∇P̃ · Ã/c + q 2 Ã2 Q̃/c2 ∂H · à − = Θ̃ − P̃˙ = 2m ∂ Q̃ ∂ Q̃ ∂ Q̃ − + V Q̃ + q[Φ̃ + ΦC ]Q̃ ∂ΠC ∂ Q̃ n Q̃2 + P̃ 2 o ∂Φ C + q + c∇ · [Π̃ + ΠC ] 2 ∂ Q̃ + {4πc2 [Π̃ + ΠC ] − c∇[Φ̃ + ΦC ]} · ˙ = ∂H = 4πc2 [Π̃ + Π ] − c∇[Φ̃ + Φ ] à C C ∂ Π̃ ˙ + ∂ΦC Q̃˙ + ∂ΦC P̃˙ = ∂H = −4πc2 Θ̃ − c∇ · à Φ̃ ∂ Q̃ ∂ P̃ ∂ Θ̃ −∇2 P̃ + q∇ · (ÃQ̃)/c + q à · ∇Q̃/c + q 2 Ã2 P̃ /c2 ∂H ∂ΠC ˙ ∂ΦC ˙ · à + Q̃˙ − = Θ̃ = 2m ∂ P̃ ∂ P̃ ∂ P̃ + V P̃ + q[Φ̃ + ΦC ]P̃ ∂ΠC ∂ P̃ n Q̃2 + P̃ 2 o ∂Φ C , + q + c∇ · [Π̃ + ΠC ] 2 ∂ P̃ + {4πc2 [Π̃ + ΠC ] − c∇[Φ̃ + ΦC ]} · (4.56) where ΠC ≡ ΠC (Q̃, P̃ ) and ΦC ≡ ΦC (Q̃, P̃ ). The forces appearing on the right hand side of these equations have become more complicated, especially those in the Schrödinger equations. There are new nonlinear terms. However, it is possible to substitute these equations among themselves in order to simplify them. Notice that parts of the ∂H/∂ Φ̃ and ∂H/∂ Π̃ equations appear in the forces of the Schrödinger 78 equations. Substitution of ∂H/∂ Φ̃ and ∂H/∂ Π̃ into the Schrödinger equations results in the following simplified equations: − ∂ΠC ˙ ˙ − ∂ΠC P̃˙ = ∇ × [∇ × Ã] + c∇Θ̃ Q̃ − Π̃ 4π ∂ Q̃ ∂ P̃ o q q q n Q̃∇P̃ − P̃ ∇Q̃ − Q̃ÃQ̃ − P̃ ÃP̃ − 2mc c c 2 2 ˙ = q Q̃ + P̃ + c∇ · [Π̃ + Π ] − Θ̃ C 2 ∇2 Q̃ + q∇ · (ÃP̃ )/c + q∇P̃ · Ã/c − q 2 Ã2 Q̃/c2 −P̃˙ = − + V Q̃ 2m + q[Φ̃ + ΦC ]Q̃ ˙ = 4πc2 [Π̃ + Π ] − c∇[Φ̃ + Φ ] à C C ˙ + ∂ΦC Q̃˙ + ∂ΦC P̃˙ = −4πc2 Θ̃ − c∇ · à Φ̃ ∂ Q̃ ∂ P̃ −∇2 P̃ + q∇ · (ÃQ̃)/c + q à · ∇Q̃/c + q 2 Ã2 P̃ /c2 Q̃˙ = + V P̃ 2m + q[Φ̃ + ΦC ]P̃ . (4.57) The generalized forces appearing on the right hand side are now very similar to the forces in (4.31). In fact, the equations of motion (4.57) can be further simplified as: −[ Π̇ + Π̇C (Q, P )] = ∂H/∂A Ȧ = ∂H/∂Π − Θ̇ = ∂H/∂Φ Φ̇ + Φ̇C (Q, P ) = ∂H/∂Θ −Ṗ = ∂H/∂Q Q̇ = ∂H/∂P , (4.58) where the tildes were omitted to show the resemblance between (4.58) and (4.31). 4.5.2 The Coulomb Reference by Change of Variable It can be shown that the new equations of motion ω̃ η̃˙ = ∂H/∂ η̃, which were obtained by a symplectic transformation in phase space, may also be obtained by a change of variable in the Lagrangians (4.28)-(4.30). The new Lagrangian density is: R δ [pq̇ − H(p, q)]dt = 0 ω η̇ = ∂H/∂η Change of Variables L̃(p̃, q̃) R δ [p̃q̃˙ − H(p̃, q̃)]dt = 0 L(p, q) ω̃ η̃˙ = ∂H/∂ η̃ Canonical Transformation Figure 4–3: Commutivity diagram representing the change of coordinates (q, p) to (p̃, q̃) at both the Lagrangian and equation of motion levels. L̃Max {[−4πc(Π̃ + ΠC )]2 + [∇ × Ã]2 }/8π 1 ˙ ˙ ˙ = [(Π̃ + ΠC ) · à − ( Π̃ + Π̃C ) · Ã] − 2 −c∇[Φ̃ + ΦC ] · [Π̃ + ΠC ] (4.59) L̃Sch {[∇Q̃ + q ÃP̃ /c]2 + [−∇P̃ + q ÃQ̃/c]2 }/4m 1 ˙ ˙ = [P̃ Q̃ − P̃ Q̃] − 2 +[V + q(Φ̃ + ΦC )]{Q̃2 + P̃ 2 }/2 1 ˙ + Φ̇ ) − Θ̃( ˙ Φ̃ + Φ )] − {−2πc2 Θ̃2 − cΘ̃∇ · Ã}. L̃gauge = [Θ̃( Φ̃ C C 2 (4.60) (4.61) That the transformation to the Coulomb reference holds at both Lagrangian and equation of motion level demonstrates the commutivity of the diagram in Figure 4–3. 4.6 Electron Spin in the Pauli Theory The electron field used so far in the nonrelativistic Schrödinger theory is a field of spin zero, i.e., a scalar field. It is a simple generalization of the theory to add in the electron’s spin. The electron field would then be a two component spinor field, g replacements 79 80 i.e., a spin-1/2 field, and would be of the form ΨP (x, t) = Ψ↑ (x, t) . Ψ↓ (x, t) (4.62) The first component Ψ↑ is spin up and the second component Ψ↓ is spin down. The dynamics of ΨP is governed by the Pauli equation [30] iΨ̇P = [−i∇ − qA/c]2 ΨP q + V ΨP + qΦΨP − σ · [∇ × A]ΨP 2m 2mc (4.63) which is the nonrelativistic limit of the Dirac equation iΨ̇D = βmc2 ΨD + cα · [−i∇ − qA/c]ΨD + qΦΨD (4.64) in terms of the four component spinor ΨD , where the β and α matrices are β= and σx = 0 1 1 0 I 0 0 −I σy = α= 0 −i i 0 0 σ σ 0 σz = (4.65) 1 0 . 0 −1 (4.66) Notice that taking the nonrelativistic limit of the Dirac equation involves the elimination of the two component positron field from ΨD . Also note that the current density associated with the Pauli theory [90] is different from that in the Schrödinger theory (see (1.14)). It is JP = q Ψ†P [−i∇ − qA/c]ΨP + ΨP [i∇ − qA/c]Ψ†P + ∇ × [Ψ†P σΨP ] /2m, (4.67) where Ψ†P = (Ψ∗↑ Ψ∗↓ ) is the adjoint of ΨP . This can be derived by taking the nonrelativistic limit of the Dirac current density. The last term in (4.67) is only present in the Pauli current. This term does not effect the continuity equation ρ̇ = −∇ · J since ∇ · ∇ × [Ψ†P σΨP ] = 0. 81 4.7 Proton Dynamics In the theory set up so far, the matter dynamics was entirely described by the electronic wavefunction Ψ. The proton had no dynamics whatsoever. Only the electrostatic scalar potential Φq̄ = q̄/|x| of the structureless proton of charge q̄ entered so as to bind the electron in the hydrogen atom. A first step in the direction of atomic and molecular collisions requires the dynamics of the proton as well (and eventually a few other particles). Suppose the proton is described by its own wavefunction Ω and Lagrangian density Lq̄Sch = iΩ∗ Ω̇ − [i∇ − q̄A/c]Ω∗ · [−i∇ − q̄A/c]Ω − q̄ΦΩ∗ Ω, 2mq̄ (4.68) where (Φ, A) are the scalar and vector potentials arising from the charge and current densities ρ = qΨ∗ Ψ + q̄Ω∗ Ω J = q Ψ∗ [−i∇ − qA/c]Ψ + Ψ[i∇ − qA/c]Ψ∗ /2mq + q̄ Ω∗ [−i∇ − q̄A/c]Ω + Ω[i∇ − q̄A/c]Ω∗ /2mq̄ . (4.69) (4.70) These densities are just the sum of the individual electronic and proton densities. The proton density is not a delta function. Thus, the proton wavefunction is not a delta function either. Rather it is described by a wavepacket and has some structure. With (4.68) the total Lagrangian is n [−4πcΠ]2 + [∇ × A]2 o 1 − c∇Φ · Π = [Π · Ȧ − Π̇ · A] − 2 8π (4.71) o n [i∇ − qA/c]Ψ∗ · [−i∇ − qA/c]Ψ i ∗ ∗ ∗ = [Ψ Ψ̇ − Ψ̇ Ψ] − + qΦΨ Ψ 2 2mq (4.72) LMax LqSch 82 n [i∇ − q̄A/c]Ω∗ · [−i∇ − q̄A/c]Ω o i Lq̄Sch = [Ω∗ Ω̇ − Ω̇∗ Ω] − + q̄ΦΩ∗ Ω 2 2mq̄ 1 Lgauge = [ΘΦ̇ − Θ̇Φ] − {−2πc2 Θ2 − cΘ∇ · A}. 2 (4.73) (4.74) Notice that the electron Lagrangian (4.72) does not explicitly contain the static proton potential energy V = q q̄/|x| as did the previous Schrödinger Lagrangian (1.17). The two matter fields are coupled entirely through electrodynamics. That is, the electron-proton interaction is mediated by the electrodynamics. The Coulombic potential is included implicitly in qΦΨ∗ Ψ and q̄ΦΩ∗ Ω in the above matter Hamiltonians. In other words, the scalar potential Φ contains (in any gauge) a Coulomb piece of the form ρ(x0 , t) 3 0 dx V |x − x0 | R qΨ∗ (x0 , t)Ψ(x0 , t) 3 0 R q̄Ω∗ (x0 , t)Ω(x0 , t) 3 0 d x + V d x. = V |x − x0 | |x − x0 | ΦC (x, t) = Φ + (ΦC − Φ) = R (4.75) With this potential, the qΦΨ∗ Ψ term in the electron Hamiltonian contains the electron-proton attraction as well as electron-electron self interaction. Similarly the q̄ΦΩ∗ Ω in the proton Hamiltonian contains the electron-proton attraction and proton-proton self interaction. The self-energies that are computed from the aforementioned self interactions are finite because Ψ and Ω are square integrable functions. That is Eint = R ρ(x, t)ΦC (x, t)d3 x = V R d3 x V R d 3 x0 V ρ(x, t)ρ(x0 , t) <∞ |x − x0 | (4.76) for both the cross terms (electron-proton attraction) and the direct terms (electronelectron and proton-proton repulsion). Note that in the relativistic quantum theory the direct terms are infinite and there are infinitely many Coulomb states of the 83 bare problem to sum over [40]. These infinities do not arise in the semiclassical theory presented in this dissertation. While the self interactions do appear in the above matter Hamiltonians, the resulting self-energies are finite and moreover do not even contribute to the electron or proton portions of the energy. This is due to −c∇Φ · Π in the above Maxwell Hamiltonian. After a partial integration this term becomes cΦ∇ · Π. Substitution of −Θ̇ = ρ + c∇ · Π = 0 from (4.31) turns cΦ∇ · Π into −ρΦ, which cancels +ρΦ in the electron and proton energies. However, the self interactions do remain in the Coulomb energy E2 /8π of the electromagnetic field. Note that the self interactions do appear in the Hamiltonians and therefore do make a contribution to the overall dynamics. It should be mentioned that this theory of electron-proton dynamics can be applied to electron-positron dynamics as well. While there is a 2000-fold difference in mass between the proton and the positron, the two theories are otherwise identical. In either case, the theory may be rich enough to capture bound states of hydrogen or positronium. CHAPTER 5 NUMERICAL IMPLEMENTATION The formal theory of Maxwell-Schrödinger dynamics was constructed in the previous chapter. In particular, the coupled and nonlinear Maxwell-Schrödinger equations [P − qA/c]2 Ψ + V Ψ + qΦΨ 2m h Ä Φ̇ i 4π 2 ∇ A− 2 −∇ ∇·A+ = − J c c c ∇ · Ȧ ∇2 Φ + = −4πρ c iΨ̇ = (5.1) (5.2a) (5.2b) were recognized to be ill-posed unless an extra equation of constraint is added to them. Using the Hamiltonain approach to dynamics, this extra equation was automatically generated by adding a Lorenz gauge fixing term at the Lagrangian level. It was emphasized in Chapter 4 that the resulting Hamiltonian system of differential equations, which are of first order in time, form a well-defined initial value problem. That is, the Maxwell-Schrödinger dynamics are known in principle once the initial values are specified for each of the dynamical variables. The details of converting the formal mathematics of Chapter 4 to a form that can be practically implemented in a computer are presented in this chapter. The Hamiltonian system of partial differential equations will be reduced to a Hamiltonian system of ordinary differential equations in time by introducing a spatial basis for each of the dynamical variables. The resulting basis equations are coded in a Fortran 90 computer program. With this program, various pictures are made to depict the dynamics of the hydrogen atom interacting with the electromagnetic field. 84 85 5.1 Maxwell-Schrödinger Theory in a Complex Basis Each of the Maxwell-Schrödinger dynamical variables, which are themselves fields, may be expanded into a complete basis of functions GK according to Ψ(x, t) = Ak (x, t) = Φ(x, t) = P P P K GK (x)ψK (t) Ψ∗ (x, t) = K GK (x)akK (t) Πk (x, t) = K GK (x)φK (t) Θ(x, t) = P P P ∗ K GK (x)ψK (t) K GK (x)πkK (t) (5.3) K GK (x)θK (t), where the index K runs over the basis and the index k runs over 1, 2, 3 or x, y, z. Any complete set of functions such as the oscillator eigenstates will suffice. In the following work the set of gaussian functions of the form GK (x) = G∗K (x) = NK exp(−`K [x − rK ]2 ) (5.4) are used. These functions are centered on rK , normalized to unity by NK , and are real-valued. Additionally, they span L2 so that any square integrable function may be represented in this basis. In principle the sums in (5.3) are to infinity. However, a complete basis cannot be realized in practice. But for all practical purposes the numerical results can be shown to converge to within an arbitrary accuracy in a finite basis. In fact with a smart choice of basis, the numerical results may converge with just a few terms. Here the basis coefficients, which are complex- and real-valued as well as time-dependent, carry the dynamics. The basis representation of the previous Lagrangians is LMax = P KM LSch = Lgauge = 1 [(∂/∂amM )ȧmM − (∂/∂πmM )π̇mM ]SMax − HMax 2 P i ∗ ∗ K [(∂/∂ψK )ψ̇K − (∂/∂ψK )ψ̇K ]SSch − HSch 2 P KM 1 [(∂/∂φK )φ̇K − (∂/∂θK )θ̇K ]Sgauge − Hgauge 2 (5.5) (5.6) (5.7) 86 with integrals SMax = R V Π · Ad3 x SSch = R V Ψ∗ Ψd3 x Sgauge = R V ΘΦd3 x. (5.8) The calculus of variations leads to the following dynamical equations: ∂ 2 SMax ∂H ȧnN = ∂πmM ∂anN ∂πmM − (5.9) ∂H ∂ 2 Sgauge φ̇L = ∂θI ∂φL ∂θI or NIL φ̇L = ∇θI H (5.10) ∂H ∂ 2 iSSch ψ̇L = ∗ ∂ψI ∂ψL ∂ψI∗ or iCIL ψ̇L = ∇ψI∗ H (5.11) ∂ 2 SMax ∂H π̇mM = ∂anN ∂πmM ∂anN or T − MnN ,mM π̇mM = ∇anN H (5.12) ∂H ∂ 2 Sgauge θ̇K = ∂φJ ∂θK ∂φJ or − NJT K θ̇K = ∇φJ H (5.13) ∂ 2 iSSch ∗ ∂H ψ̇ = ∗ K ∂ψJ ∂ψK ∂ψJ or ∗ − iCJ∗ K ψ̇K = ∇ ψJ H (5.14) − − or MmM,nN ȧnN = ∇πmM H which are of the Hamiltonian form ω η̇ = ∂H/∂η. The summation convention is used throughout. These equations may be written more compactly as M ȧ = ∇π H − M T π̇ = ∇a H N φ̇ = ∇θ H − N T θ̇ = ∇φ H iC ψ̇ = ∇ψ∗ H − iC ∗ ψ̇ ∗ = ∇ψ H (5.15) 87 and can be cast into matrix form as 0 0 0 0 0 iC 0 −iC ∗ 0 0 0 0 0 0 0 −M T 0 0 0 0 0 0 0 −N T 0 0 M 0 0 0 0 0 0 N 0 0 ψ̇ ψ̇ ∗ ȧ = φ̇ π̇ θ̇ ∗ ∂H/∂ψ ∂H/∂ψ ∂H/∂a , ∂H/∂φ ∂H/∂π ∂H/∂θ (5.16) where the matrices M, N, and C and defined in (5.9)-(5.14). This symplectic form almost has the canonical structure of (4.25). In a basis of rank N, the contractions involving a and π run to 3N while the contractions involving the remaining dynamical variables run to N. This is because a and π are spatial vectors that have (x, y, z)-components whereas the remaining dynamical variables are scalars. With the choice of representation in (5.3) and the choice of basis in (5.4) all approximations are specified. The equations of motion in (5.16) are the basis representation of the coupled Maxwell-Scrödinger equations. They are automatically obtained by applying the time-dependent variational principle to the Lagrangians (5.5)-(5.7). In the limit of a complete basis these equations are exact. The complex phase space that carries the associated dynamics is endowed with the Poisson bracket ∗ ∂F/∂ψ ∂F/∂ψ ∂F/∂a {F, G} = ∂F/∂φ ∂F/∂π ∂F/∂θ T iC 0 0 0 0 0 0 −iC ∗ 0 0 0 0 0 0 0 0 −M T 0 0 0 0 0 0 −N T 0 0 M 0 0 0 0 0 0 N 0 0 −1 ∗ ∂G/∂ψ ∂G/∂ψ ∂G/∂a . ∂G/∂φ ∂G/∂π ∂G/∂θ (5.17) 88 Even though the symplectic form is not canonical, its inversion is simple. The matrix R elements in ω involve gaussian overlap integrals like hGI |GK i = V hGI |xihx|GK id3 x. 5.2 Maxwell-Schrödinger Theory in a Real Basis As was done previously, each dynamical variable may be expanded into a complete basis of functions GK as P Q(x, t) = P Ak (x, t) = P Φ(x, t) = K GK (x)qK (t) P (x, t) = K GK (x)akK (t) Πk (x, t) = K GK (x)φK (t) Θ(x, t) = P P P K GK (x)pK (t) K GK (x)πkK (t) (5.18) K GK (x)θK (t), where the index K runs over the basis and the index k runs over 1, 2, 3 or x, y, z. Unlike in (5.3), the coefficients in (5.18) that carry the dynamics are all real-valued. In this basis, the real Lagrangian densities become LMax = P KM 1 [(∂/∂amM )ȧmM − (∂/∂πmM )π̇mM ]SMax − HMax 2 LSch = Lgauge = with integrals SMax = R P 1 K [(∂/∂qK )q̇K − (∂/∂pK )ṗK ]SSch − HSch 2 P KM Π · Ad3 x V 1 [(∂/∂φK )φ̇K − (∂/∂θK )θ̇K ]Sgauge − Hgauge 2 SSch = R P Qd3 x V Sgauge = R V ΘΦd3 x. (5.19) (5.20) (5.21) (5.22) Applying the calculus of variations to the above Lagrangians leads to the equations of motion: ∂H ∂ 2 SMax ȧnN = ∂πmM ∂anN ∂πmM ∂H ∂ 2 Sgauge φ̇L = ∂θI ∂φL ∂θI or MmM,nN ȧnN = ∇πmM H (5.23) or NIL φ̇L = ∇θI H (5.24) 89 ∂ 2 SSch ∂H q̇L = ∂pI ∂qL ∂pI − ∂ 2 SMax ∂H π̇mM = ∂anN ∂πmM ∂anN − or CIL q̇L = ∇pI H (5.25) T − MnN ,mM π̇mM = ∇anN H or (5.26) ∂H ∂ 2 Sgauge θ̇K = ∂φJ ∂θK ∂φJ or − NJT K θ̇K = ∇φJ H (5.27) ∂H ∂ 2 SSch ṗK = ∂qJ ∂pK ∂qJ or − CJT K ṗK = ∇qJ H (5.28) − which are of the Hamiltonian form ω η̇ = ∂H/∂η. These equations may be written more compactly as M ȧ = ∇π H − M T π̇ = ∇a H N φ̇ = ∇θ H − N T θ̇ = ∇φ H C q̇ = ∇p H − C T ṗ = ∇q H and can be cast into matrix form as T 0 0 0 0 0 −M 0 0 0 0 −N T 0 0 0 −C T 0 0 0 M 0 0 0 0 0 0 N 0 0 0 0 0 0 C 0 0 0 ȧ φ̇ q̇ = π̇ θ̇ ṗ (5.29) ∂H/∂a ∂H/∂φ ∂H/∂q , ∂H/∂π ∂H/∂θ ∂H/∂p (5.30) where the matrices M, N, and C and defined in (5.23)-(5.28). Again they are the basis representation of the coupled Maxwell-Schrödinger equations of motion. 90 The real phase space that carries the associated dynamics is endowed with the Poisson bracket ∂F/∂a ∂F/∂φ ∂F/∂q {F, G} = ∂F/∂π ∂F/∂θ ∂F/∂p T 0 0 0 −M 0 0 0 0 0 M T 0 0 0 −N T 0 0 0 0 −C T 0 0 0 0 0 0 N 0 0 0 0 0 0 C 0 0 0 −1 ∂G/∂a ∂G/∂φ ∂G/∂q . ∂G/∂π ∂G/∂θ ∂G/∂p (5.31) Even though the symplectic form is not canonical, its inversion is simple once again. The matrix elements in ω involve gaussian overlap integrals like hGI |GK i = R hGI |xihx|GK id3 x. V 5.2.1 Overview of Computer Program The equations of motion (5.30) are coded in Fortran 90. The computer program is called Electron Nuclear Radiation Dynamics or ENRD. Each matrix element in the symplectic form and in the forces is performed analytically. The program is flexible enough to handle a rank N basis of s-gaussians, each with an adjustable width and an arbitrary center. A numerical solution to (5.30) is determined once the initial value data is specified for η b . The forces ∂H/∂η a are constructed from this data. The symplectic form ωab is then inverted with the LAPACK [91] subroutine DGESVX, which is the expert driver for the AX = B solver DGESV. This −1 establishes a first order system of differential equations of the form η̇ b = ωab ∂H/∂η a which may be solved, for example, with an Euler stepping method. That is −1 ∂H/∂η a ](t). η b (t + ∆t) = η b (t) + (∆t)η̇ b (t) = η b (t) + (∆t)[ωab (5.32) In practice, the Euler method is not accurate enough so the more sophisticated RK4 method [92] is implemented in the code. The equations of motion are advanced 91 at a fixed stepsize of 10−3 au. For typical basis function widths and centers, the estimated condition number reported by DGESVX is about 30. Lastly, it should be pointed out that the equations of motion (5.30) are numerically implemented in terms of the electric field E ≡ −4πcΠ rather than the momentum Π. It was found that working in terms of this new (scaled) coordinate provides a more balanced set of dynamical equations. Nevertheless, the electromagnetic radiation is still quite small compared to the dynamics of the matter. An overview of the ENRD program is presented in Figure 5–1. MAXENERGY.f90 RHS EM.f90 RIRJ.f90 • Maxwell energy. • Compute hηM |η(t)i. • RHS Maxwell. • Create library for vector arguments. ∂H ∂H , ∂Π ∂A enrd.f90 • Input deck. GAENERGY.f90 RHS GA.f90 DERIVS.f90 • Build vector arguments. • Gauge energy. • RHS Gauge. • Build forces ∂H ∂η on RHS. • Create library for four index matrix elements. • Build symplectic form ω. • Call integrator. • Compute hηM |η(t)i. ∂H ∂H ∂Θ , ∂Φ SCHENERGY.f90 RHS SH.f90 • Schrödinger energy. • Compute hηM |η(t)i. • RHS Schrödinger. ∂H ∂H ∂P , ∂Q • Call LAPACK routine DGESVX. • Invert ω η̇ = ∂H/∂η to get η̇ = ω −1 ∂H/∂η. • Write η(t + ∆t) to PS.dat. RK4.f90 METRIC.f90 • Symplectic form ω. • Runge-Kutta 4 ODE solver. η(t + ∆t) = η(t) + (∆t)η̇(t) 92 Figure 5–1: Schematic overview of ENRD computer program. 93 5.2.2 Stationary States: s- and p-Waves The ENRD program was first tested with a stationary state of the hydrogen atom. In a basis of six gaussians an s-wave was constructed as well as the corresponding basis representation of the Coulombic scalar potential and the Coulombic electric field. In fact, any spherically symmetric distribution of charge along with the corresponding Coulomb fields would suffice. This delicate balance of charges and fields proved to be a stationary state of the combined system. No electromagR netic radiation was produced. The total charge q = V ρ(x)d3 x remained constant. A px -wave and its associated Coulombic fields were also created in the same basis. This again is a stationary state. 5.2.3 Nonstationary State: Mixture of s- and p-Waves After identifying some stationary states, a nonstationary state that is a mixture of s- and px -waves was constructed in the same rank six basis. Both the Coulombic scalar potential and the Coulombic electric field that are associated with this charge distribution were created as well. Electromagnetic radiation was produced as the electron oscillated between stationary states. Energy, momentum, and angular momentum were exchanged between the electron and the electromagnetic field. It was shown that the total energy and total Hamiltonian are conserved to four deciR mal places. The total charge q = V ρ(x)d3 x remained constant. The phase space contours for the electromagnetic field, matter field, and gauge field are presented in Figures 5–2, 5–3, and 5–4 respectively. 5.2.4 Free Electrodynamics Lastly a free electromagnetic field was constructed. In this case no charge was created. Energy, momentum, and angular momentum were exchanged only between the electromagnetic and gauge degrees of freedom. The total energy remained constant. frag replacements 94 20 10 0 πkK (t) -10 -20 PSfrag replacements -0.5 0.5 0 akK (t) Figure 5–2: Phase space contour for the coefficients of the vector potential A and its momentum Π. 5 pK (t) 0 -5 -5 0 qK (t) 5 Figure 5–3: Phase space contour for the coefficients of the real-valued Schrödinger field Q and its momentum P. 20 g replacements 95 0.0005 θK (t) 0 -0.0005 -30 -20 -10 φK (t) 0 10 Figure 5–4: Phase space contour for the coefficients of the scalar potential Φ and its momentum Θ. 5.2.5 Analysis of Solutions in Numerical Basis The solutions η(t) of the equations of motion (5.30) are further analyzed by expansion into the basis eigenstates ηM . The Schrödinger eigenstates are found by diagonalizing the time-independent Schrödinger equation HC = SC, (5.33) where H is the basis representation of the Hamiltonian H = −∇2 /2m + V, C is the matrix of basis expansion coefficients, S is the basis overlap matrix, and is the matrix of energy eigenvalues. Similarly, the Maxwell eigenstates are found by diagonalizing the free wave equation ∇2 φ − φ̈/c2 = 0, where φ can be the scalar potential Φ or any component of the vector potential A. Fourier inversion of the free wave equation results in −c2 ∇2 φ̃ = ω 2 φ̃, where ω is the frequency. In a basis this equation turns into the matrix equation HC = ω 2 SC, (5.34) 96 where H is the basis representation of the Hamiltonian-like quantity H = −c2 ∇2 , C is the matrix of basis expansion coefficients, S is the basis overlap matrix, and ω 2 is the matrix of frequencies squared. Recall that energy is related to frequency by E = ~ω, so that in atomic units energy is equivalent to frequency. Both of these basis equations (5.33) and (5.34) are recognized as belonging to the generalized eigenvalue problem Aη = λBη, which can be inverted with the LAPACK routine DSYGV. The ENRD program employs DSYGV to solve both (5.33) and (5.34) for their corresponding eigenvalues λM and eigenvectors ηM . With the eigenvectors ηM , the evolving state vector η(t) can be expanded according to |η(t)i = P M |ηM ihηM |η(t)i, (5.35) where CM (t) ≡ hηM |η(t)i are the basis expansion coefficients. The real and imaginary parts of the Schrödinger coefficients for a superposition of s- and px -waves are plotted versus time in Figures 5–5 and 5–6 respectively. The squares of these coefficients are plotted versus time in Figure 5–7. Notice in Figures 5–5 and 5–6 that there are three frequencies involved in the dynamics, which correspond to excitations of the s-, px -, and dy2 −z 2 -waves. Figure 5–7 suggests that the electron decays from px to s in under 10 au of time. However, due to the finite size of the basis, the electron is excited back to the px -state as the electromagnetic fields reflect off of the artificial basis boundaries. Lastly, the phase space contour of the Schrödinger coefficients are presented in Figure 5–8. rag replacements 97 s px py pz 1 d x2 2 dy2 −z RehΨM |Ψ(t)i 0 0 10 t 20 30 Figure 5–5: Real part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. s px py pz 1 d x2 2 dy2 −z ImhΨM |Ψ(t)i 0 −1 0 10 t 20 30 Figure 5–6: Imaginary part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. PSfrag replacements −1 −10 −20 −30 ImhΨM |Ψ(t)i pK (t) 1 0.001 −0.001 −1 RehΨN |Ψ(t)i qK (t) P 98 2 M |hΨM |Ψ(t)i| s px 0.8 0.6 |hΨM |Ψ(t)i|2 0.4 0.2 0 0 10 t 20 30 Figure 5–7: Probability for the electron to be in a particular basis eigenstate. d x2 2 dy2 −z pK (t) 0.5 ImhΨM |Ψ(t)i 0 −0.5 −1 −1 −0.5 0 RehΨM |Ψ(t)i 0.5 1 Figure 5–8: Phase space of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. 30 1 −10 −20 −30 0.001 −0.001 qK (t) s px py pz 99 5.3 Symplectic Transformation to the Coulomb Reference Recall the basis representation of the Maxwell-Schrödinger equations of motion in (5.30). They are T 0 0 0 0 0 −M 0 0 0 0 −N T 0 0 0 0 0 0 −C T M 0 0 0 0 0 0 N 0 0 0 0 0 0 C 0 0 0 where MKM ≡ ∂ 2 hΠ|Ai ∂πK ∂aM NKM ≡ ∂ 2 hΘ|Φi ∂θK ∂φM ȧ φ̇ q̇ = π̇ θ̇ ṗ ∂H/∂a ∂H/∂φ ∂H/∂q , ∂H/∂π ∂H/∂θ ∂H/∂p CKM ≡ ∂ 2 hP |Qi ∂pK ∂qM (5.30) (5.36) and where the integrals h·|·i involve only gaussian functions. In analogy to the transformation (T−1 )T in (4.44) that was defined on the vector fields ∂/∂η in function space, a basis representation of (T−1 )T can be made. This basis representation is defined in terms of the coefficients according to 0 0 0 0 0 ∂/∂ã ∂a/∂ã ∂/∂ φ̃ 0 ∂φ/∂ φ̃ 0 0 0 0 0 ∂φ/∂ q̃ ∂q/∂ q̃ ∂π/∂ q̃ 0 0 ∂/∂ q̃ = ∂/∂ π̃ 0 0 0 ∂π/∂ π̃ 0 0 ∂/∂ θ̃ 0 0 0 0 ∂θ/∂ θ̃ 0 0 ∂φ/∂ p̃ 0 ∂π/∂ p̃ 0 ∂p/∂ p̃ ∂/∂ p̃ ∂/∂a ∂/∂φ ∂/∂q ∂/∂π ∂/∂θ ∂/∂p (5.37) 100 so that the symplectic form in (5.30) transforms as T 0 0 0 0 0 −M 0 0 0 0 −N T 0 T 0 0 −C 0 0 0 M 0 0 0 0 0 0 N 0 0 0 0 0 0 C 0 0 0 0 −MÃQ̃ −MÃΠ̃ 0 −MÃP̃ 0 0 0 0 0 −NΦ̃Θ̃ 0 0 0 0 −NQ̃Θ̃ −CQ̃P̃ MQ̃à → MΠ̃à 0 0 0 0 0 0 0 0 NΘ̃P̃ NΘ̃Φ̃ NΘ̃Q̃ MP̃ à 0 CP̃ Q̃ 0 −NP̃ Θ̃ 0 where the new matrix elements are: MX̃ Ỹ NX̃ Ỹ CX̃ Ỹ ∂a ∂π ∂ 2 hΠ̃ + ΠC |Ãi ∂a ∂ 2 hΠ̃ + ΠC |Ãi ∂π M = = ≡ ∂π∂a ∂ X̃ ∂ Ỹ ∂ X̃ ∂ Ỹ ∂ X̃∂ Ỹ 2 2 ∂θ ∂ hΘ|Φ̃ + ΦC i ∂φ ∂ hΘ̃|Φ̃ + ΦC i ∂θ ∂φ N = = ≡ ∂θ∂φ ∂ X̃ ∂ Ỹ ∂ X̃ ∂ Ỹ ∂ X̃∂ Ỹ 2 2 ∂p ∂q ∂p ∂ hP |Qi ∂q ∂ hP̃ |Qi ≡ C = = ∂ X̃ ∂ Ỹ ∂ X̃ ∂p∂q ∂ Ỹ ∂ X̃∂ Ỹ (5.38) , (5.39) for X and Y an arbitrary dynamical variable. The remaining elements are determined by transposition. Again the extra terms in ω̃ add new time-dependent couplings to the theory. These new terms can all be performed analytically. The 101 resulting equations 0 0 0 0 MQ̃à 0 MΠ̃à 0 0 NΘ̃Φ̃ MP̃ à 0 5.3.1 of motion are −MÃQ̃ −MÃΠ̃ 0 0 0 −NΦ̃Θ̃ 0 0 −NQ̃Θ̃ 0 0 0 NΘ̃Q̃ 0 0 CP̃ Q̃ 0 −NP̃ Θ̃ −MÃP̃ 0 −CQ̃P̃ 0 NΘ̃P̃ 0 ã˙ ˙ φ̃ q̃˙ = π̃˙ θ̃˙ p̃˙ ∂H/∂ã ∂H/∂ φ̃ ∂H/∂ q̃ . ∂H/∂ π̃ ∂H/∂ θ̃ ∂H/∂ p̃ (5.40) Numerical Implementation Recall that the forces appearing in the canonical transformed equations (4.57) could be simplified by substituting these equations among themselves. As a result 0 −MÃQ̃ −MÃΠ̃ 0 −MÃP̃ 0 0 0 0 0 −NΦ̃Θ̃ 0 0 0 0 0 0 −CQ̃P̃ . (5.41) ω̃ → MΠ̃à 0 0 0 0 0 0 0 0 NΘ̃P̃ NΘ̃Φ̃ NΘ̃Q̃ 0 0 CP̃ Q̃ 0 0 0 Note that (5.41) is not a symplectic form. After making this substitution, the Hamiltonian structure is lost. However, the numerical implementation is greatly facilitated with the simplified equations (4.57) instead of those in (4.56). Since the ENRD program does not rely on a symplectic integrator scheme to advance the dynamics, the symplectic structure is not numerically important anyway. The equations of motion (4.57) has been added to the ENRD code. The Coulomb reference can be conveniently turned on or off (resulting in (4.32)) with an optional flag. As before, the program is flexible enough to handle a rank N basis of s-gaussians, each with an adjustable width and an arbitrary center. A solution 102 to (4.57) may be obtained once the initial value data is specified for η̃ b . The forces ∂H/∂ η̃ a are constructed from this data. The new terms appearing on left hand side of (4.57) are coded analytically. Notice that these terms make up the elements of a matrix that is not a symplectic form. Nevertheless, the resulting matrix equations are integrated with the same RK4 stepping method and the DGESVX subroutine of LAPACK. For typical basis function widths and centers, the condition number reported by DGESVX is on the order of one thousand. 5.3.2 Stationary States: s- and p-Waves The ENRD program with the Coulomb reference was first tested with a stationary state of the hydrogen atom. In a basis of six gaussians an s-wave was constructed. The corresponding basis representation of the Coulombic scalar potential and the Coulombic electric field were not needed. All Coulombic properties are treated analytically once the Coulomb reference is chosen. Again, it was found that any spherically symmetric distribution of charge will suffice to produce an s-wave that is a stationary state of the combined system. No electromagnetic radiation was R produced. The total charge q = V ρ(x)d3 x remained constant. A px -wave was also created in the same basis. This again was a stationary state of the combined system. 5.3.3 Nonstationary State: Mixture of s- and p-Waves After identifying some stationary states, a nonstationary state that is a mixture of s- and px -waves was constructed in the same rank six basis. Both the Coulombic scalar potential and the Coulombic electric field that are associated with this charge distribution were done analytically by the canonical transformation to the Coulomb reference. Electromagnetic radiation was produced as the electron oscillated between stationary states. Energy, momentum, and angular momentum were exchanged between the electron and the electromagnetic field. It was shown that the total energy and total Hamiltonian are conserved to two decimal places. The total charge R q = V ρ(x)d3 x remained constant. 103 5.3.4 Free Electrodynamics Lastly a free electromagnetic field was constructed. In this case no charge was created. Energy, momentum, and angular momentum were exchanged only between the electromagnetic and gauge degrees of freedom. The total energy remained constant. 5.3.5 Analysis of Solutions in Coulomb Basis As done previously, the evolving state vector η̃(t) in the Coulomb basis is expanded in terms of the stationary eigenbasis η̃M ≡ ηM according to |η̃(t)i = P M |η̃M ihη̃M |η̃(t)i. (5.35) As before, the real and imaginary parts of the Schrödinger coefficients for a superposition of s- and px -waves are plotted versus time in Figures 5–9 and 5–10 respectively. The squares of these coefficients are plotted versus time in Figure 5–11. Notice in Figures 5–9 and 5–10 that there are again three frequencies involved in the dynamics, which correspond to excitations of the s-, px -, and dy2 −z 2 -waves. Figure 5–11 suggests that the electron decays from px to s in under 15 au of time. However, due to the same aforementioned basis effects, the electron oscillates between the sand px -states. Lastly, the phase space contour of the Schrödinger coefficients are presented in Figure 5–12. 5.4 Asymptotic Radiation It has been demonstrated that the dynamics of the hydrogen atom’s electron in the presence of the electromagnetic field was quasiperiodic. This unphysical behavior is due to the fact the electromagnetic radiation produced by the electron cannot escape to infinity and carry away energy, momentum, and angular momentum. Rather, the radiation reflects off of the artificial boundaries of the finite spatial basis and reexcites the electron. rag replacements 104 s px py pz 1 d x2 2 dy2 −z RehΨM |Ψ(t)i 0 0 10 t 20 30 Figure 5–9: Real part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. s px py pz 1 d x2 2 dy2 −z ImhΨM |Ψ(t)i 0 −1 0 10 t 20 30 Figure 5–10: Imaginary part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. PSfrag replacements −1 −10 −20 −30 ImhΨM |Ψ(t)i pK (t) 1 0.001 −0.001 −1 RehΨN |Ψ(t)i qK (t) P 105 2 M |hΨM |Ψ(t)i| s px 0.8 0.6 |hΨM |Ψ(t)i|2 0.4 0.2 0 0 10 t 20 30 Figure 5–11: Probability for the electron to be in a particular basis eigenstate. pK (t) 0.5 ImhΨM |Ψ(t)i 0 −0.5 −1 −1 −0.5 0 RehΨM |Ψ(t)i 0.5 1 Figure 5–12: Phase space of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t) is a superposition of s- and px -waves. 30 d x2 2 dy2 −z −10 −20 −30 0.001 −0.001 qK (t) s px py pz 1 106 The asymptotic problem, be it electromagnetic radiation or free (ionized) electrons, has posed a difficult numerical challenge. Free electromagnetic radiation in vacuum does not spread in time, since there is no dispersion, but does travel at the speed of light c ≈ 137 au. However, the velocity of the sources of charge and current, e.g., the electron in the hydrogen atom, is on the order of 1 au. This drastically different velocity scale makes a numerical description of the time-dependent theory in direct space quite demanding. On the other hand, the description of the free nonrelativistic electron is made difficult by a combination of its large velocity v (v < c), the spreading of its wavepacket, and the rapid oscillation of its phase. Even in vacuum, the Schrödinger equation is dispersive so that the electronic wavepacket width grows proportionally with time and its phase grows quadratically with the distance from the center of the wavepacket. Several techniques have been developed to partially treat these problems. In 1947, Wigner and Eisenbud [93] developed the R-matrix method, which provides a technique for matching the solutions on some surface separating the inner bound state region and outer scattering state region. More recently, masking functions, repetitive projection and complex rotation methods, and Siegert pseudostates are common theoretical tools. These techniques are discussed by Yoshida, Watanabe, Reinhold, and Burgdörfer in [94] and by Tolstikhin, Ostrovsky, and Nakamura in [95]. A scaling transformation method that eliminates the rapid phase variation and wavepacket expansion and requires no matching at infinity has been presented by Sidky and Esry in [96]. Lastly, McCurdy and collaborators [97–99] have effectively implemented an exterior complex scaling method [100] in the time-independent formulation of scattering theory. The exterior complex scaling method maps all coordinates beyond a certain radius to a contour that is rotated by some fixed angle into the complex plane. This technique damps all purely outgoing scattered waves to zero exponentially which permits a numerical treatment on a finite domain or grid. g replacements 107 Asymptotic basis Local basis φ(t) t Figure 5–13: Schematic picture of the local and asymptotic basis proposed for the description of electromagnetic radiation and electron ionization. The amplitude from the asymptotic basis is dumped into the free field φ, which acts as a storage tank for energy and probability. A formulation of the asymptotic numerical problem that falls more in line with the canonical treatment presented in this dissertation would begin at the Lagrangian level with a Lagrangian of the form L = LENRD + Lcoupling + Lfree . (5.42) The ENRD Lagrangian LENRD would be the Maxwell-Schrödinger Lagrangian from (1.17) and (1.18). The dynamics of this system would be described by two different types of basis functions. As pictured in Figure 5–13, the atomic or molecular system would have a local basis representation in terms of real gaussian basis functions of the form GK (x) = G∗K (x) = NK exp(−`K [x − rK ]2 ). Further away, a set of complex basis functions of the form (5.4) 108 Gk (x) = P lm Clm Ylm (x̂) eikr exp(−ar 2 ) r (5.43) would be used, where the wavevector magnitude k = ω/c could be chosen to lie in some range kmin ≤ k ≤ kmax and most likely only a few l would be necessary. These complex basis functions will require the calculation of new matrix elements. The free Lagrangian Lfree would be the free particle Lagrangian iψ ∗ ψ̇ − {i∇ψ ∗ · −i∇ψ}/2m or the free field Lagrangian ∂α φ∗ ∂ α φ. The solutions of the free equations of motion derived from these free Lagrangians are known analytically and are of the form exp(i[k · x − ωt]). The coupling Lagrangian should be a Lorentz scalar that is made up of a certain combination or product of dynamical variables of LENRD and of Lfree . If amplitude is put into the coefficients of the asymptotic basis functions Gk , then the amplitude will transfer to the free solutions ψ or φ. This amplitude will provide an initial condition for the free fields, thereby defining ψ or φ throughout spacetime. The free fields will store the energy and probability (and momentum and angular momentum) radiated at infinity, which is needed to maintain the various conservation laws. 5.5 Proton Dynamics in a Real Basis The previous complex Schrödinger Lagrangians may be written in real form by √ taking the electronic wavefunction Ψ = [Q + iP ]/ 2 and the protonic wavefunction √ Ω = [U + iW ]/ 2. In terms of these real dynamical variables the Hamiltonian density becomes H= [−4πcΠ]2 + [∇ × A]2 − [4πcΘ]2 − c∇Φ · Π − cΘ∇ · A 8π Q2 + P 2 [∇Q + qAP/c]2 + [−∇P + qAQ/c]2 + qΦ + 4mq 2 2 2 2 [∇U + q̄AW/c] + [−∇W + q̄AU/c] U + W2 + + q̄Φ 4mq̄ 2 (5.44) 109 As in (5.3), each of the dynamical variables may be expanded into a basis. In particular, the real and imaginary components of Ψ and Ω are expanded as Q(x, t) = P (x, t) = P P K GK (x)qK (t) U (x, t) = K GK (x)pK (t) W (x, t) = P P K GK (x)uK (t) (5.45) K GK (x)wK (t). The basis functions GK are chosen to be simple s-gaussians. The Hamilton equations of motion associated with these real dynamical variables are 0 0 0 −M T 0 0 0 0 0 −N 0 0 0 0 0 0 0 M 0 0 0 T 0 0 0 0 −C T 0 0 0 0 −K T 0 0 0 0 0 N 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 K 0 0 0 ȧ φ̇ q̇ u̇ π̇ θ̇ ṗ ẇ = ∂H/∂a ∂H/∂φ ∂H/∂q ∂H/∂u ∂H/∂π ∂H/∂θ ∂H/∂p ∂H/∂w , (5.30) where MKM ≡ ∂ 2 hΠ|Ai ∂πK ∂aM NKM ≡ ∂ 2 hΘ|Φi ∂θK ∂φM CKM ≡ ∂ 2 hP |Qi ∂pK ∂qM and where the integrals h·|·i involve only gaussian functions. KKM ≡ ∂ 2 hW |U i ∂wK ∂uM (5.46) CHAPTER 6 CONCLUSION Nonperturbative analytical and numerical methods for the solution of the nonlinear Maxwell-Schrödinger equations have been presented including the complete coupling of both systems. The theory begins by applying the calculus of variations to the Maxwell and Schrödinger Lagrangians together with a gauge fixing term for the Lorenz gauge. Within the Hamiltonian or canonical prescription, this yields a set of first order differential equations in time of the form ωab η̇ b = ∂H/∂η a . (6.1) These Maxwell-Schrödinger equations are closed when the Schrödinger wavefunction is chosen as a source for the electromagnetic field and the electromagnetic field acts back upon the wavefunction. Moreover, this system of equations forms a welldefined initial value problem. That is, the entire dynamics is known in principle once the initial values for each of the dynamical variables η are specified. The resulting dynamics enjoys conservation of energy, momentum, angular momentum, and charge between the matter and the electromagnetic field. In practice, the Maxwell-Schrödinger equations are represented in a finite basis of gaussian functions GK (x) and solved numerically. That is, each dynamical variable is expanded in this basis according to η(x, t) = P K GK (x)ηK (t), (6.2) where the time-dependent coefficients ηK (t) carry the dynamics. As a result, a hierarchy of approximate equations of motion are generated that basis-represent the exact Maxwell-Schrödinger equations and can be made systematically more and more 110 111 accurate by enriching the basis. In the limit of a complete basis, these approximate equations would be exact since the gaussian functions span L2 . The basis representation of the Maxwell-Schödinger equations of motion has been numerically implemented in a Fortran 90 computer program. This program allows for an arbitrary rank basis of s-gaussians of varying widths and centers. It has been used to explore the dynamics of the hydrogen atom interacting with the electromagnetic field. In particular, stationary states of the combined atomfield system have been constructed as well as nonstationary states that radiate. This radiation carries away energy, momentum, and angular momentum from the hydrogen atom. A series of plots are presented to document the radiative decay of hydrogen’s electron from a superposition of s and px states to the s ground state. In order to improve numerical convergence, a canonical transformation was performed on the Maxwell-Schrödinger equations to isolate the Coulombic or electrostatic contribution to the scalar potential ΦC and electric field EC . This portion of the fields can be performed analytically once the source ρ is specified by solving the Poisson equation ∇2 ΦC = −4πρ and then calculating EC = −∇ΦC . By removing the burden of describing both the Coulombic and radiative contributions to the electrodynamics, the efforts of the basis are focused entirely on the description of the radiation. The canonical transformed equations of motion have been represented in a gaussian basis as done previously and have been added to the existing Fortran 90 computer program. With an optional flag the Coulomb reference can be used. Otherwise the raw numerical basis is used by default. As before, a series of plots are presented to document the dynamics of the hydrogen atom interacting with the electromagnetic field. The results in both cases are analyzed. The work presented in this dissertation is particularly applicable to physical situations where the dynamics of the sources of charge and current occurs on the same timescale as the dynamics of the electromagnetic field. In these situations, adiabatic 112 and perturbative approaches may be insufficient to describe the strongly coupled matter-field dynamics. Possible applications of the Maxwell-Schrödinger theory lie in photon-electron-phonon dynamics in semiconductor quantum wells [10], spontaneous emission in cold atom collisions [11, 12], atom-photon interaction in single atom laser cavities [14, 15], and photon-exciton dynamics in fluorescent polymers [16]. APPENDIX A GAUGE TRANSFORMATIONS A.1 Gauge Symmetry of Electrodynamics The basic equations of electrodynamics [9] are: 4π Ė J+ c c ∇ · E = 4πρ ∇×B= ∇·B=0 Ḃ = 0, ∇×E+ c (A.1) where ρ and J are the charge and current densities. These Maxwell equations may be rewritten in terms of the scalar and vector potentials Φ and A as ∇ · Ȧ = −4πρ c h Ä Φ̇ i 4π 2 ∇ A− 2 −∇ ∇·A+ = − J c c c ∇2 Φ + (A.2a) (A.2b) by defining the electric field E = −Ȧ/c − ∇Φ and the magnetic field B = ∇ × A. Maxwell’s equations written in either field or potential forms are invariant under the gauge transformation Φ → Φ0 = Φ − Ġ c (A.3) A → A0 = A + ∇G, where G is an arbitrary and well-behaved function called the gauge function or gauge generator. It is said that Maxwell’s equations enjoy the full gauge freedom. However, a particular gauge may be chosen with an appropriate choice of the gauge function. For example, a gauge transformation can always be made on the potentials so that the Lorenz gauge condition is satisfied. That is 0 = ∇ · A0 + Φ̇ G̈ Φ̇0 = ∇ · A + ∇2 G + − c c c 113 (A.4) 114 implies that the Lorenz gauge function satisfies the equation G = −[∇ · A + Φ̇/c]. This choice of gauge function leads to the manifestly Lorentz invariant equations of motion: Φ̈ = −4πρ c2 Ä 4π ∇2 A − 2 = − J. c c ∇2 Φ − (A.5a) (A.5b) Another choice is the Coulomb (radiation) gauge, which leads to the equations of motion: ∇2 Φ = −4πρ ∇2 A T − ÄT 4π = − JT 2 c c (A.6a) (A.6b) with the gauge function satisfying ∇2 G = −∇ · A. The Coulomb gauge vector potential AT is the transverse projection of the Lorenz gauge vector potential A, as is the current JT . That is AaT (x, t) R d3 k ik·x h ab k a k b i b ∂a∂b i b δ − 2 à (k, t), e = δ − 2 A (x, t) = V ∇ (2π)3 k h ab (A.7) where PTab = δ ab − ∂ a ∂ b /∇2 is the transverse projection operator (see Appendix R C) and 1/∇2 is shorthand for the Green’s function V d3 x0 [−4π|x − x0 |]−1 of the Laplacian operator (see Appendix B), where (∇2 /∇2 )g = g for g a well-behaved function. More precisely R 3 0 −1 1 2 ∇ g(x) = d x g(x0 )∇02 = g(x). V 2 ∇ 4π|x − x0 | (A.8) The dynamical radiation fields associated with AT are almost separated from the instantaneous or static fields associated with Φ, as seen in (A.6). A closer examination of the transverse projection operator PTab will show that even AT contains instantaneous components. That these instantaneous effects exactly cancel between AT 115 and Φ to produce the causal E and B is one of the often misunderstood properties of the Coulomb gauge [101]. Notice that the Lorenz gauge equations of motion (A.5) enjoy a limited gauge freedom known as the residual gauge symmetry. In other words, within the Lorenz gauge there is still a limited family of gauge transformations that can be made on the potentials that leave them in the Lorenz gauge. Such gauge generators satisfy the homogeneous equation G = 0. Similarly, there is residual gauge freedom left in the Coulomb gauge. That is, there is a limited class of gauge transformations that can be made on (A.6) that leave them unchanged as well. Such gauge generators satisfy the homogeneous equation ∇2 G = 0. Many other choices of gauge function are possible, each leading to a different gauge. A particular gauge is often chosen in accordance with a given physical situation so as to simplify the associated mathematics. See Cohen-Tannoudji et. al. [30] for a discussion of other gauges pertinent in the context of atomic and molecular physics. A.2 Gauge Symmetry of Quantum Mechanics In addition to the electrodynamic gauge symmetries of (A.3), quantum mechanics exhibits the additional symmetry Ψ → Ψ0 = ΨeiqG/c . (A.9) The coupled system (3.20) and (A.2) are invariant under gauge transformations (A.3) and (A.9) with the same gauge function G. While there are other gauge invariant coupling schemes, the minimal coupling prescription embodied in (3.20) is the simplest. APPENDIX B GREEN’S FUNCTIONS The Green’s function or fundamental solution G associated with the inhomogeneous (partial) differential equation Lu(k) = h(k) satisfies the equation LG(k, k0 ) = δ (n) (k − k0 ) (B.1) where L is a differential operator, u and h are C ∞ and L2 ∩ L1 functions, and δ is the Dirac δ-function defined below. The inhomogeneous term h is often called a source for u. In this language, the Green’s function G(k, k0 ) = G(k0 , k) is a solution of the differential equation corresponding to a point-like source, i.e., the δ-function. Loosely speaking the Green’s function is the inverse of the operator L. The particular solution u of the differential equation Lu = h may be obtained in principle by integration against G over all of space. That is R u(k) = d(n) k 0 h(k0 )G(k0 , k) (B.2) which along with the solution of Lu = 0 constitutes the full solution. In other words, the differential equation Lu = h has been transformed into an integral equation in which the Green’s function is the kernel. Substitution of this solution into the differential equation leads to R h(k) = L d(n) k 0 h(k0 )G(k0 , k) R = d(n) k 0 h(k0 )δ (n) (k0 − k). (B.3) A brief discussion of the δ-function is presented in the next section. With this knowledge, the Green’s functions for the Laplacian ∇2 and the wave operator ∂ 2 = −2 = ∂ 2 /∂(ct)2 − ∇2 are derived. 116 117 B.1 The Dirac δ-Function The Dirac δ-function may be defined in n-dimensions by the volume integral R d(n) k ik·x 1e = δ (n) (x), n (2π) (B.4) where k·x is the Euclidean scalar product. This integral is just the Fourier inverse of the unit 1. Notice that the δ-function is not a function but rather it is a distribution which is zero everywhere except at x = 0. It may be used as an integration kernel to “pluck out” the value of a function at a particular point. For example R g(x) = d(n) y g(y)δ (n) (y − x), (B.5) where δ (n) (y − x) = δ (n) (x − y). A particular choice for the function g such as g(y) = 1 shows that 0, if x lies outside Ω R (n) (n) d y δ (y − x) = , Ω 1, if x lies inside Ω (B.6) where Ω is a closed region of integration. Furthermore that if g is well-behaved in Ω, then R d(n) y g(y) Ω ∂ (n) ∂g(y) δ (y − x) = − ∂y ∂y y=x (B.7) after an integration by parts. If Ω is all of space, then the integral in (B.6) is always 1 and the function g in (B.7) need only vanish at infinity. Notice that the dimensions of the δ-function must cancel the dimensions of the differential d(n) y to give a dimensionless result. Hence δ (n) has dimensions of (length)−n . There are a number of additional properties of the δ-function which will not be elaborated on here. The interested reader is referred to [7] for a detailed discussion. 118 The ∇2 Operator B.2 Consider the Poisson equation from electrodynamics ∇2 Φ = −4πρ, (B.8) where Φ is the scalar potential and ρ is the charge density. The corresponding Green’s function satisfies the equation ∇2 G(x, x0 ) = δ(x − x0 ), (B.9) where the dimensionality of the δ-function has been omitted. By going to the Fourier space (B.9) is diagonalized and becomes (ik)2 G̃ = exp(−ik · x0 ). By another Fourier inversion 0 G(x, x0 ) = − R −1 d3 k eik·(x−x ) = . (2π)3 k2 4π|x − x0 | V (B.10) With (B.10) the Poisson equation (B.8) is recast as the integral equation Φ(x, t) = = R R V d3 x0 [−4πρ(x0 , t)]G(x0 , x) d 3 x0 V ρ(x0 , t) . |x − x0 | (B.11) Together with the homogeneous solutions of the Laplace equation ∇2 Φ = 0, the total solution is obtained. B.3 The ∂ 2 Operator Consider the wave equation from electrodynamics −∂ 2 Φ = Φ = ∇2 Φ − Φ̈ = −4πρ, c2 (B.12) where Φ is the scalar potential and ρ is the charge density as before. The corresponding Green’s function satisfies h 1 ∂2 i ∇ − 2 2 G(x, t; x0 , t0 ) = δ(x − x0 )δ(t − t0 ). c ∂t 2 (B.13) 119 The solution of (B.13) will require the Fourier inversion in both x and t to obtain ˜ = − exp(−i[k · x0 − ωt0 ]). Division by [k2 − ω 2 /c2 ] followed by another [k2 − ω 2 /c2 ]G̃ Fourier inversion in both k and ω will give the desired result. Rather than performing this task in one step, it is beneficial to split the effort in half. First consider the Fourier inverse of (B.12) in x. That is ¨ + (ck)2 Φ̃ = 4πc2 ρ̃, Φ̃ (B.14) with Green’s function satisfying h i (d2 /dt2 ) + (ck)2 Dk (t, t0 ) = δ(t − t0 ). (B.15) Fourier inversion in t results in [ω 2 − c2 k 2 ]D̃k = − exp(iωt0 ) from which the retarded Green’s function becomes (+) Dk (t, t0 ) 0 R ∞ dω e−iω(t−t0 ) 0 sin ck(t − t ) = − −∞ = Θ(t − t ) , 2π ω 2 − c2 k 2 ck (B.16) where the integration was performed in lower half complex plane. Similarly the (−) advanced solution is Dk (t, t0 ) = Θ(t0 − t) sin ck(t − t0 )/ck by integrating in the upper half plane. The solution to (B.13) is more clear, now that this first step has been accom˜ = − exp(−i[k · x0 − ωt0 ]) must plished. It has already been shown that [k2 − ω 2 /c2 ]G̃ be Fourier inverted twice in order to obtain G. The resulting Green’s function is 0 d3 k ik·(x−x0 ) R ∞ dω e−iω(t−t ) G(x, t; x , t ) = c V e −∞ 2π ω 2 − c2 k 2 (2π)3 0 R cΘ(t − t ) ∞ (B.17) = 2 dk sin ck(t − t0 ) sin(k|x − x0 |) 0 0 2π |x − x | i cΘ(t − t0 ) h 0 0 0 0 δ |x − x | − c(t − t ) − δ |x − x | + c(t − t ) , = 4π|x − x0 | 0 0 2 R where the retarded Green’s function D (+) in (B.16) was used in the first equality and a change of variables was used in the last equality. The first term in (B.17) is 120 referred to as the retarded solution G(+) (x, t; x0 , t0 ) = cΘ(t − t0 ) 0 0 δ |x − x | − c(t − t ) 4π|x − x0 | (B.18) while the second term is just zero since both |x − x0 | and c(t − t0 ) are positive. An analogous computation with the advanced Green’s function D (−) leads to the advanced solution PSfrag replacements G(−) (x, t; x0 , t0 ) = cΘ(t0 − t) 0 0 δ |x − x | + c(t − t ) . 4π|x − x0 | (B.19) The retarded solution exhibits the causal properties of field propagation (see Figure B–1). That is, a disturbance that is observed at the point (x, t) in space-time ct r(t) future elsewhere x past Figure B–1: The trajectory or world line r(t) of a massive particle moves from past to future within the light cone. A massless particle such as a photon propagates on the light cone. originated from a point that is a distance |x − x0 | = c(t − t0 ) away and at a time t0 = t − |x − x0 |/c earlier. The opposite is true for the advanced solution. With (B.18) or (B.19) the wave equation (B.12) is recast as the integral equation R 3 0 ∞ d x dt0 [−4πρ(x0 , t)]G(±) (x0 , t0 ; x, t) V −∞ R 3 0R ∞ 0 ρ(x0 , t0 ) 0 h |x − x0 | i = V d x −∞ dt δ t − t∓ . |x − x0 | c Φ(x, t) = R (B.20) 121 Together with the homogeneous solutions of the wave equation Φ = 0, the total solution is obtained. APPENDIX C THE TRANSVERSE PROJECTION OF A(x, t) It can be seen in both (3.56) and (3.59) that the full vector potential generated by a charge q moving with velocity v also points in the direction of v. When working PSfrag replacements in the Coulomb gauge it is not the full vector potential that is needed but the transverse projection thereof. The transverse vector potential AT does not flow in the direction perpendicular to v, but rather the direction perpendicular to the wavevector k as seen in Figure C–1. The true meaning of transverse and longitudinal k vk ∼ k(k · v)/k 2 v v⊥ −k(k · v)/k 2 r(t) Figure C–1: Since à = h̃v, the transverse vector potential Ã⊥ = [v − k(k · v)/k 2 ]h̃ and the longitudinal vector potential Ãk = [k(k·v)/k 2 ]h̃, where h̃ is a scalar function. is easily visualized by going to the Fourier space. There ik · Ã⊥ = 0 and ik × Ãk = 0, where Ã⊥ is the Fourier inverse of AT and Ãk is the Fourier inverse of the longitudinal AL . The transverse and longitudinal projections satisfy AT + AL = A. Each component AiT of the AT can be obtained by contraction of the transverse 122 123 projection tensor T ij with the velocity vj . That is AiT (x, t) = T ij (x, t)vj (C.1) where T ij (x, t) is related to the transverse projection operator PTij = δ ij − ∂ i ∂ j /∇2 according to h n −h(x, t) o ∂i∂j i PTij h(x, t)vj = δ ij − 2 h(x, t)vj = [∂ i ∂ j − ∇2 δ ij ] vj = T ij (x, t)vj . 2 | {z } ∇ ∇ {z } | Aj (x,t) g(x,t) (C.2) Note that PTik = PTij PTjk and P† = P. In Fourier space T ij is of the form T̃ ij (k, t) = [k i k j − k 2 δ ij ]g̃(k, t) = [3k i k j − k 2 δ ij ]g̃(k, t) + [−2k i k j ]g̃(k, t) | {z } | {z } Q̃ij (k,t) (C.3) M̃ ij (k,t) where g̃ distinguishes between the Fourier inverses of A(0) , A(1) , or A(2) . These vector potentials are the pieces which make up the full potential A = Θ(t − t1 )Θ(t2 − t)[A(0) + A(1) ] + Θ(t − t2 )[A(1) − A(2) ] (C.4) generated by the current J̃ = qvΘ(t − t1 )Θ(t2 − t) exp(−ik · [r + vt] − k 2 /8`2 ). The first instantaneous or Coulomb-like piece is √ erf 2`|x − (r + vt)| qv v A(0) (x, t) = Φ(0) (x, t) = c c |x − (r + vt)| while the remaining three pairs of radiating terms are of the form √ qv n erf 2`[c(t − tk ) − |x − (r + vtk )|] v (k) (k) A (x, t) = Φ (x, t) = c 2c |x − (r + vtk )| √ erf 2`[c(t − tk ) + |x − (r + vtk )|] o − |x − (r + vtk )| (C.5) (C.6) 124 for k = 1, 2. The first term in (C.3) resembles the traceless quadrupole moment tensor and is labeled Q̃ij , while the second term M̃ ij is the remainder. It will be seen that the M̃ ij tensor can be completely determined from Q̃ij . Solving for each component of T ij first and then contracting with the velocity eliminates the Fourier inversion of Ã⊥ (k, t) = [k(k · v) − k 2 v]g̃(k, t) (C.7) in favor of the angularly inferior T ij (x, t) = R V d3 k i j [k k − k 2 δ ij ]g̃(k, t)eik·x , (2π)3 (C.8) followed by a simple multiplication of vj . In matrix notation the transverse vector potential is just AT = Tv = [Q + M]v. (C.9) Looking only at Q̃ij (although the same is true for both M̃ ij and T̃ ij ), it can be shown that while Q̃ij 6= 0 for all i and j, its Fourier inverse Qij is diagonal in a certain frame. For example (for A(0) ) (0) (0) −4πq −ik·(r+vt)−k2 /8`2 e 6= 0, ck 4 (C.10) −4πq ik·[x−(r+vt)]−k2 /8`2 d3 k 3kx ky e =0 3 (2π) ck 4 (C.11) Q̃12 (k, t) = 3kx ky g̃C (k, t) = 3kx ky while (0) Q12 (x, t) = R V in the boosted frame of origin r + vt where x − (r + vt) is rotated about the origin to lie along the êz -axis. As expected, if x − (r + vt) is placed in general along the constant vector n̂0 = êx sin θ0 cos φ0 + êy sin θ0 sin φ0 + êz cos θ0 , then the angular (0) part of Q12 becomes R dΩ sin θ cos φ sin θ sin φ eik|x−(r+vt)|[cos θ cos θ0 +sin θ sin θ0 cos(φ−φ0 )] 6= 0. (C.12) 125 C.1 Tensor Calculus It will be seen that the transverse projection tensor T ij in (C.8) is diagonal in the boosted frame of origin r + vt where the general vector x − (r + vt) → x is rotated about the origin to the new vector x0 = r(0, 0, 1) which lies along the êz -axis. This spatial rotation is performed via the rotation matrix cos θ cos φ cos θ sin φ − sin θ Λkl (Ω) = cos φ 0 − sin φ , sin θ cos φ sin θ sin φ cos θ (C.13) where x0k = Λkl xl or in matrix notation x0 = Λx. With Λkl , the transverse projection tensor in (C.8) may be rotated to the diagonal frame by T kl (x) → T 0kl (x0 ) = = ∂x0k ∂x0l ij −1 0 T (Λ x ) ∂xi ∂xj (C.14) Λki Λl j T ij (x) where Λki Λl j T ij = Λki T ij [Λ−1 ]j l or T0 = ΛTΛ−1 . The explicit time-dependence which is unaffected by the rotation has been dropped for brevity. Applying this similarity transformation to T ij results in T 0kl (x0 ) = Λki T ij (x)[Λ−1 ]j l = R V R d3 k 0 ik0 ·x0 0kl 0 d3 k ik·x k ij −1 l e Λ T̃ (k)[Λ ] = e T̃ (k ), i j V (2π)3 (2π)3 (C.15) where d3 k 0 = det Λ d3 k = d3 k and k0 · x0 = k · x. In this frame the three diagonal elements of T 0kl are known. They are 011 0 011 0 012 0 013 0 0 0 T (x ) T̃ (k ) T̃ (k ) T̃ (k ) R d3 k 0 ik0 ·x0 = 0 T̃ 021 (k0 ) T̃ 022 (k0 ) T̃ 023 (k0 ) . 022 0 e T (x ) 0 V (2π)3 0 0 T 033 (x0 ) T̃ 031 (k0 ) T̃ 032 (k0 ) T̃ 033 (k0 ) (C.16) Each of the elements T 0ii (x0 ) will be computed in the diagonal frame. However the physics is not correct until x0 is rotated back to the general position 126 x = r(sin θ cos φ, sin θ sin φ, cos θ). The reverse rotation is obtained by inversion of the transformation (C.14). The resulting transverse projection tensor becomes 2 2 2 cos θ sin φ cos φ − sin θ cos θ cos φ cos θ cos φ 011 0 2 T kl (x) = cos2 θ sin2 φ − sin θ cos θ sin φ cos θ sin φ cos φ T (x ) 2 − sin θ cos θ cos φ − sin θ cos θ sin φ sin θ 2 − sin φ cos φ 0 sin φ 022 0 + cos2 φ 0 − sin φ cos φ T (x ) 0 0 0 2 2 2 sin θ sin φ cos φ sin θ cos θ cos φ sin θ cos φ 033 0 T (x ) 2 2 2 + sin θ sin φ cos φ sin θ sin φ sin θ cos θ sin φ sin θ cos θ cos φ sin θ cos θ sin φ cos2 θ (C.17) which may be written in matrix form as T = Λ−1 T0 Λ. The transverse vector potential AkT (x) is now obtained by the simple contraction T kl (x)vl . Note that the T 0kk (x0 ) appearing in (C.17) are scalar functions that only depend upon the norm of x0 . A proper treatment of the tensor calculus reveals that ∇0 · A0T (x0 , t) = ∇ · AT (x, t). Componentwise that is ∂k0 T 0kl (x0 , t)vl0 = ∂i T ij (x, t)vj . (C.18) It must still be verified that AT is divergenceless in either frame. It is sufficient to show that ∂i T ij vj = 0. This result is most easily shown by working in the (r, θ, φ) basis with unit vectors êx = êr sin θ cos φ + êθ cos θ cos φ − êφ sin φ êy = êr sin θ sin φ + êθ cos θ sin φ + êφ cos φ êz = êr cos θ − êθ sin θ. (C.19) 127 In these spherical coordinates the the velocity becomes cos θ sin θ sin φ sin θ cos φ v x = v1 cos θ cos φ + v2 cos θ sin φ + v3 − sin θ v= v y 0 cos φ − sin φ vz and the transverse vector potential is just 33 33 33 T cos θ T sin θ sin φ T sin θ cos φ 11 11 11 AT = v 1 T cos θ cos φ + v2 T cos θ sin φ + v3 −T sin θ . 22 22 0 T cos φ −T sin φ (C.20) (C.21) It is not difficult to verify that ∇ · AT = 0 using the spherical divergence. To summarize the work so far, it was stated that a frame exists where the transverse projection tensor T ij is diagonal. A spatial rotation was performed to go to that diagonal frame. The tensor was then rotated back to the arbitrary frame. There the matrix elements of the general T ij involve the diagonal elements T 0ii as seen in (C.21). In the following section the three T 0ii terms corresponding to the three terms in the vector potential A = Θ(t−t1 )Θ(t2 −t)[A(0) +A(1) ]+Θ(t−t2 )[A(1) −A(2) ] will be computed. C.2 T 0kk (x0 , t) Integrals In (C.3) and (C.9) it was shown that the transverse vector potential may be obtained by contraction of the transverse projection tensor T = Q + M with the velocity. That is d3 k [Q̃ij (k, t) + M̃ ij (k, t)]eik·x vj V 3 (2π) R d3 k R d3 k i j 2 ij ik·x = V [3k k − k δ ]g̃(k, t)e v + [−2k i k j ]g̃(k, t)eik·x vj , j V (2π)3 (2π)3 AiT (x, t) = R (C.22) 128 where it was noticed that Q̃ij is analogous to the traceless quadrupole moment tensor from electrodynamics [9]. In the diagonal frame Q̃0ij looks like 011 0 0 0 Q̃ (x , t) 0ij 0 Q̃ (x , t) = 0 Q̃022 (x0 , t) 0 033 0 0 0 Q̃ (x , t) (C.23) with Q̃011 + Q̃022 + Q̃033 = 0. The elements of the M tensor can all be found from Q. Recall from (3.58)-(3.59) or (C.4)-(C.6) that the piecewise vector potential A = Θ(t − t1 )Θ(t2 − t)[A(0) + A(1) ] + Θ(t − t2 )[A(1) − A(2) ] (C.24) is made up of a Coulomb-like piece √ v (0) qv erf 2`|x − (r + vt)| A (x, t) = Φ (x, t) = c c |x − (r + vt)| (0) (C.25) and three pairs of radiating terms n erf √2`[c(t − t ) − |x − (r + vt )|] qv v k k A(k) (x, t) = Φ(k) (x, t) = c 2c |x − (r + vtk )| √ erf 2`[c(t − tk ) + |x − (r + vtk )|] o − |x − (r + vtk )| (C.26) for k = 1, 2. For each of these pieces there is a corresponding transverse projection tensor. For example, when t1 ≤ t ≤ t2 there is a T(0) = Q(0) + M(0) associated with A(0) and a T(1) = Q(1) + M(1) associated with A(1) . Each of these tensors involve a Fourier inversion. The resulting integrals are computed below. 129 C.2.1 Inside Step When t1 ≤ t ≤ t2 , the vector potential Ain = A(0) + A(1) . The traceless part of the transverse projection tensor is d3 k (0) (1) [3k i k j − k 2 δ ij ]{g̃C (k, t) + g̃R (k, t)}eik·x V (2π)3 −4πq −ik·(r+vt)−k2 /8`2 ck 4 e R d3 k [3k i k j − k 2 δ ij ] = V (2π)3 + 4πq e−ik·(r+vt1 )−k2 /8`2 cos ck(t − t1 ) ck 4 Qij (x, t) = R eik·x (C.27) and M ij (x, t) = (0) R V d3 k (0) (1) [−2k i k j ]{g̃C (k, t) + g̃R (k, t)}eik·x 3 (2π) (C.28) (k) where −k 2 g̃C v is the Fourier inverse of A(0) and −k 2 g̃R v is the Fourier inverse of A(k) . Higher order terms in v/c are omitted from (C.27). (0) (0) Taking g̃ in (C.22) as g̃C , the Qij tensor becomes (0) Qij d3 k −4πq ik·[x−(r+vt)]−k2 /8`2 2 = V [3k k − k δ ] e . i j ij (2π)3 ck 4 R (C.29) (0) Using cylindrical symmetry it is found that all off-diagonal elements of Qij are zero (0) (0) and Q11 = Q22 when the vector x − (r + vt) is rotated to the frame where it lies (0) (0) (0) (0) along the z-axis. But since Qij is traceless, Q33 = −2Q11 . Thus, solving for Q33 determines Q(0) entirely. That is (0) Q33 d3 k −4πq ik·[x−(r+vt)]−k2 /8`2 = V [3kz2 − k 2 ] e 3 (2π) ck 4 z 2 Φ(0) 3q R ∞ −z 2 /16`2 dz cos(|x − (r + vt)|z)e W =− + −1,−1/2 πc 0 8`2 c (0) 2q` Φ 1 3 1 5 , ; , ; −2`2 |x − (r + vt)|2 + = −√ 2 F2 2 2 2 2 c 2πc √ 2 [x−(r+vt)]2 −2` 3q erf 2`|x − (r + vt)| 3q e Φ(0) √ =− 2 , + + 8` c |x − (r + vt)|3 c 8π`c |x − (r + vt)|2 R (C.30) 130 where the primes which denote this diagonal frame have been temporarily omitted (0) and where Gradshteyn and Ryzhik [87] was used. Notice that Q33 not only de(0) (0) (0) (0) (0) (0) termines Q11 and Q22 , it also determines M11 , M22 , and M33 . Taking M̃ij as (0) M̃ij = [−2ki kj ] (0) (0) −4πq −ik·(r+vt)−k2 /8`2 e ck 4 (0) (C.31) (0) (0) it is found that M11 = M22 = [cQ33 + 2Φ(0) ]/3c and M33 = −2[cQ33 − Φ(0) ]/3c (0) with Mij = 0 for i 6= j. With this, the vector potential transverse to A(0) becomes (0) AT = [Q(0) + M(0) ]v. (1) (C.32) (1) Taking g̃ in (C.22) as g̃R , the Qij tensor becomes (1) Qij = R V 4πq d3 k 2 2 [3ki kj − k 2 δij ] 4 eik·[x−(r+vt1 )]−k /8` cos ck(t − t1 ). 3 (2π) ck (C.33) (1) Using spherical symmetry it is again found that all off-diagonal elements of Qij are (1) (1) zero and Q11 = Q22 when the vector x − (r + vt1 ) is chosen to lie along the z-axis. (1) (1) (1) (1) Again since Qij is traceless, Q33 = −2Q11 . Thus, solving for Q33 determines Q(1) entirely. That is (1) d3 k 4πq 2 2 [3kz2 − k 2 ] 4 eik·[x−(r+vt1 )]−k /8` cos ck(t − t1 ) V 3 (2π) ck 12q` −2`2 c2 (t−t1 )2 R 1 2 2 2 du u2e−2` [x−(r+vt1 )] u e =√ 0 2πc Q33 = R Φ(1) · cosh 4`2 c(t − t1 )|x − (r + vt1 )|u + c n −2`2 [c(t−t1 )+|x−(r+vt1 )|]2 3q [c(t − t1 ) − |x − (r + vt1 )|]e =√ |x − (r + vt1 )|3 8π`c 2 2 [c(t − t1 ) + |x − (r + vt1 )|]e−2` [c(t−t1 )−|x−(r+vt1 )|] o − |x − (r + vt1 )|3 √ 3q 1 + 4`2 c2 (t − t1 )2 n erf 2`[c(t − t ) + |x − (r + vt )|] + 2 1 1 8` c |x − (r + vt1 )|3 √ o Φ(1) , − erf 2`[c(t − t1 ) − |x − (r + vt1 )|] + c (C.34) 131 (1) (1) (1) where Gradshteyn and Ryzhik [87] was used. Similarly Q33 determines M11 , M22 , (1) (1) and M33 . Taking M̃ij as (1) M̃ij = [−2ki kj ] (1) 4πq −ik·(r+vt1 )−k2 /8`2 e cos ck(t − t1 ) ck 4 (1) (1) (1) (C.35) (1) it is found that M11 = M22 = [cQ33 + 2Φ(1) ]/3c and M33 = −2[cQ33 − Φ(1) ]/3c (1) with Mij = 0 for i 6= j. With this, the vector potential transverse to A(1) becomes (1) AT = [Q(1) + M(1) ]v (0) (C.36) (1) so that within the step ATin = AT + AT . C.2.2 Outside Step When t > t2 , the vector potential Aout = A(1) − A(2) . The traceless part of the transverse projection tensor is d3 k (1) (2) [3k i k j − k 2 δ ij ]{g̃R (k, t) − g̃R (k, t)}eik·x V 3 (2π) 4πq −ik·(r+vt1 )−k2 /8`2 cos ck(t − t1 ) e R d3 k ck 4 i j 2 ij = V [3k k − k δ ] (2π)3 − 4πq e−ik·(r+vt2 )−k2 /8`2 cos ck(t − t2 ) ck 4 Qij (x, t) = R eik·x (C.37) and M ij (x, t) = (k) R V d3 k (1) (2) [−2k i k j ]{g̃R (k, t) − g̃R (k, t)}eik·x , 3 (2π) where −k 2 g̃R v is the Fourier inverse of A(k) . (C.38) 132 (1) (1) Taking g̃ in (C.22) as g̃R , the zz-component of the tensor Qij is 2 (1) Q33 2 3q n [c(t − t1 ) − |x − (r + vt1 )|]e−2` [c(t−t1 )+|x−(r+vt1 )|] =√ |x − (r + vt1 )|3 8π`c 2 2 [c(t − t1 ) + |x − (r + vt1 )|]e−2` [c(t−t1 )−|x−(r+vt1 )|] o − |x − (r + vt1 )|3 √ 3q 1 + 4`2 c2 (t − t1 )2 n + 2 2`[c(t − t ) + |x − (r + vt )|] erf 1 1 8` c |x − (r + vt1 )|3 √ o Φ(1) − erf 2`[c(t − t1 ) − |x − (r + vt1 )|] + c (C.39) (1) (1) (1) (1) (1) and as before M11 = M22 = [cQ33 + 2Φ(1) ]/3c and M33 = −2[cQ33 − Φ(1) ]/3c with (1) Mij = 0 for i 6= j. (2) (2) Lastly, taking g̃ in (C.22) as g̃R , the zz-component of Qij is found to be 2 2 3q n [c(t − t2 ) − |x − (r + vt2 )|]e−2` [c(t−t2 )+|x−(r+vt2 )|] |x − (r + vt2 )|3 8π`c 2 2 [c(t − t2 ) + |x − (r + vt2 )|]e−2` [c(t−t2 )−|x−(r+vt2 )|] o − |x − (r + vt2 )|3 √ 3q 1 + 4`2 c2 (t − t2 )2 n erf 2`[c(t − t2 ) + |x − (r + vt2 )|] + 2 3 8` c |x − (r + vt2 )| √ o Φ(2) . − erf 2`[c(t − t2 ) − |x − (r + vt2 )|] + c (C.40) (2) Q33 = √ (2) Taking M̃ij as (2) M̃ij = [−2ki kj ] (2) 4πq −ik·(r+vt1 )−k2 /8`2 cos ck(t − t1 ) e ck 4 (2) (2) (2) (C.41) (2) it is found that M11 = M22 = [cQ33 + 2Φ(2) ]/3c and M33 = −2[cQ33 − Φ(2) ]/3c (2) with Mij = 0 for i 6= j. With this, the vector potential transverse to A(2) becomes (2) AT = [Q(2) + M(2) ]v (1) (2) so that outside the step ATout = AT − AT . (C.42) 133 C.3 Building AT (x, t) With the result of equations (C.30), (C.34), (C.39), and (C.40), the transverse vector potential can be determined by contraction of Qij + Mij with the velocity vj . As a result AT becomes AT = [Q + M]v = Θ(t − t1 )Θ(t2 − (C.43) (0) t)[AT These results are presented in [88]. + (1) AT ] + Θ(t − (1) t2 )[AT − (2) AT ]. REFERENCES [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science, 269(5221):198–201, 1995. [2] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle. Bose-Einstein condensation in a gas of sodium atoms. Physical Review Letters, 75(22):3969–3973, 1995. [3] A. Keiling, J. R. Wygant, C. A. Cattell, F. S. Mozer, and C. T. Russell. The global morphology of wave Poynting flux: Powering the aurora. Science, 299:383–386, 2003. [4] G. T. Marklund, N. Ivchenko, T. Karlsson, A. Fazakerley, M. Dunlop, P.-A. Lindqvist, S. Buchert, C. Owen, M. Taylor, A. Vaivalds, P. Carter, M. André, and A. Balogh. Temporal evolution of the electric field accelerating electrons away from the auroral ionosphere. Nature, 414:724–727, 2001. [5] Patrick T. Newell. Rhythms of the auroral dance. Nature, 414:700–701, 2001. [6] P. A. M. Dirac. The Principles of Quantum Mechanics. Oxford University Press, Oxford, 4th edition, 1958. [7] Eugen Merzbacher. Quantum Mechanics. J. Wiley & Sons, New York, 3rd edition, 1998. [8] Albert Messiah. Quantum Mechanics. J. Wiley & Sons, Amsterdam, 1st edition, 1961. [9] John David Jackson. Classical Electrodynamics. J. Wiley & Sons, New York, 3rd edition, 1999. [10] K. El Sayed, J. A. Kenrow, and C. J. Stanton. Femtosecond relaxation kinetics of highly excited electronic wavepackets in semiconductors. Physical Review B, 57(19):12369–12377, 1998. [11] C. Orzel, S. D. Bergeson, S. Kulin, and S. L. Rolston. Time-resolved studies of ultracold ionizing collisions. Physical Review Letters, 80(23):5093–5096, 1998. [12] S. D. Gensemer and P. L. Gould. Ultracold collisions observed in real time. Physical Review Letters, 80(5):936–939, 1998. 134 135 [13] Yi Mu and C. M. Savage. One-atom lasers. Physical Review A, 46(9):5944– 5954, 1992. [14] J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble. Experimential realization of a one-atom laser in the regime of strong coupling. Nature, 425:268–271, 2003. [15] Howard Carmichael and Luis A. Orozco. Single atom lases orderly light. Nature, 425:246–247, 2003. [16] M. Nisoli, S. Stagira, M. Zavelani-Rossi, and S. DeSilvesti. Ultrafast lightemission processes in poly(para-phenylene)-type ladder polymer films. Physical Review B, 59(17):11328–11332, 1999. [17] W. Heitler. The Quantum Theory of Radiation, volume 3rd. Oxford University Press, Oxford, 1954. [18] Enrico Fermi. Quantum theory of radiation. Reviews of Modern Physics, 4:87–132, 1932. [19] Julian Schwinger. Selected Papers on Quantum Electrodynamics. Dover Publications, New York, 1958. [20] Y. Aharonov and D. Bohm. Significance of electromangetic potentials in quantum theory. Physical Review, 115(3):485–491, 1959. [21] Mike Guidry. Gauge Field Theories. J. Wiley & Sons, New York, 1980. [22] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Gravitation. W. H. Freeman & Company, San Francisco, 1973. [23] Hans C. Ohanian and Remo Ruffini. Gravitation and Spacetime. W.W. Norton & Company, New York, 2nd edition, 1976. [24] Michael E. Peskin and Daniel V. Schroeder. An Introduction to Quantum Field Theory. Perseus Books, Cambridge, 1995. [25] David Griffiths. Introduction to Elementary Particles. J. Wiley & Sons, New York, 1987. [26] A. O. Barut. Electrodynamics and Classical Theory of Fields and Particles. Dover, New York, 1964. [27] Herbert Goldstein. Classical Mechanics. Addison Wesley, Reading, 2nd edition, 1980. [28] Jorge V. José and Eugene J. Saletan. Classical Dynamics: A Contemporary Approach. Cambridge University Press, Cambridge, 1998. 136 [29] Ralph Abraham and Jerrold E. Marsden. Foundations of Mechanics. Benjamin/Cummings, Reading, 2nd edition, 1978. [30] Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg. Photons and Atoms. J. Wiley & Sons, New York, 1989. [31] Hendrik A. Lorentz. Theory of Electrons. Dover Publications, New York, 2nd edition, 1952. [32] Herbert F. M. DaCosta, David A. Micha, and Keith Runge. Spectra of light emitted during slow ion-atom collisions. Physical Review A, 56(5):3334–3337, 1997. [33] Herbert F. M. DaCosta, David A. Micha, and Keith Runge. Time- and frequency-domain properties of light emitted in slow ion-atom collisions. Journal of Chemical Physics, 107(21):9018–9027, 1997. [34] D. P. Craig and T. Thirunamachandran. Molecular Quantum Electrodynamics. Dover Publications, New York, 1984. [35] George C. Schatz and Mark A. Ratner. Quantum Mechanics in Chemistry. Prentice Hall, Englewood Cliffs, 1993. [36] M. D. Crisp and E. T. Jaynes. Radiation effects in semiclassical theory. Physical Review, 179(5):1253–1261, 1969. [37] R. K. Nesbet. Where semiclassical radiation theory fails. Physical Review Letters, 27(9):553–556, 1971. [38] R. K. Nesbet. Spontaneous emission in semiclassical radiation theory. Physical Review A, 4(1):259–264, 1971. [39] Richard J. Cook. Spontaneous emission without field quantization. Physical Review A, 27(4):2265–2267, 1983. [40] A. O. Barut and J. F. Van Huele. Quantum electrodynamics based on selfenergy: Lamb shift and spontaneous emission without field quantization. Physical Review A, 32(6):3187–3195, 1985. [41] A. O. Barut and Jonathan P. Dowling. Self-field quantum electrodynamics: The two-level atom. Physical Review A, 41(5):2284–2294, 1990. [42] A. O. Barut and J. P. Dowling. Interpretation of self-field quantum electrodynamics. Physical Review A, 43(7):4060, 1991. [43] Iwo Bialynicki-Birula. Comment on “Quantum electrodynamics based on selfenergy: Lamb shift and spontaneous emission without field quantization”. Physical Review A, 34(4):3500–3501, 1986. 137 [44] Michael D. Crisp. Comment on “Self-field quantum electrodynamics: The two-level atom”. Physical Review A, 43(7):4058–4059, 1991. [45] S. Danko Bosanac. Classical radiation by a free spinless particle when radiation reaction force is included. Journal of Physics A, 27:1723–1741, 1994. [46] S. Danko Bosanac. Nonrelativistic theory of the radiation reaction interaction. Physical Review A, 50(4):2899–2907, 1994. [47] S. Danko Bosanac. General classical solution for dynamics of charges with radiation reaction. Journal of Physics A, 34:473–490, 2001. [48] N. Došlić and S. Danko Bosanac. Harmonic oscillator with the radiation reaction interaction. Physical Review A, 51(5):3485–3494, 1995. [49] P. W. Milonni, J. R. Ackerhalt, and H. W. Galbraith. Chaos in the semiclassical N -atom Jaynes-Cummings model: Failure of the rotating-wave approximation. Physical Review Letters, 50(13):966–969, 1983. [50] Michael D. Crisp. Magnetic effects in radiation reaction theory. Physical Review A, 39(12):6224–6231, 1989. [51] Michael D. Crisp. Self-fields in semiclassical radiation theory. Physical Review A, 42(7):3703–3717, 1990. [52] Michael D. Crisp. Interaction of a charged harmonic oscillator with a single quantized electromagnetic field mode. Physical Review A, 44(1):563–573, 1991. [53] Michael D. Crisp. Application of the displaced oscillator basis in quantum optics. Physical Review A, 46(7):4138–4149, 1992. [54] Michael D. Crisp. Relativistic neoclassical radiation theory. Physical Review A, 54(1):87–92, 1996. [55] R. P. Feynman. Space-time approach to quantum electrodynamics. Physical Review, 76(6):769–789, 1949. [56] Jay R. Ackerhalt, Peter L. Knight, and Joseph H. Eberly. Radiation reaction and radiative frequency shifts. Physical Review Letters, 30(10):456–460, 1973. [57] John V. Shebalin. Numerical solution of the coupled Dirac and Maxwell equations. Physics Letters A, 226:1–6, 1997. [58] E. A. Power and T. Thirunamachandran. Quantum electrodynamics with nonrelativistic sources. I. Transformation to the multipolar formalism for second-quantized electron and Maxwell interacting fields. Physical Review A, 28(5):2649–2662, 1983. 138 [59] E. A. Power and T. Thirunamachandran. Quantum electrodynamics with nonrelativistic sources. II. Maxwell fields in the vicinity of a molecule. Physical Review A, 28(5):2663–2670, 1983. [60] A. Salam and T. Thirunamachandran. Maxwell fields and Poynting vector in the proximity of a chiral molecule. Physical Review A, 50(6):4755–4766, 1994. [61] A. Salam. Maxwell field operators, the energy density, and the Poynting vector calculated using the minimal-coupling framework of molecular quantum electrodynamics in the Heisenberg picture. Physical Review A, 56(4):2579– 2591, 1997. [62] E. J. Moniz and D. H. Sharp. Absence of runaways and divergent self-mass in nonrelativistic quantum electrodynamics. Physical Review D, 10(4):1133– 1136, 1974. [63] S. M. Blinder. Structure and self-energy of the electron. International Journal of Quantum Chemistry, 90:144–147, 2002. [64] Elliott H. Lieb and Michael Loss. Self-energy of electrons in non-perturbative QED. arXiv:math-ph/9908020 v2, September 1999. [65] Marcel Griesemer, Elliott H. Lieb, and Michael Loss. Ground states in nonrelativistic quantum electrodynamics. arXiv:math-ph/0007014 v2, March 2001. [66] Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse. Springer Verlag, 1999. [67] D. Masiello, E. Deumens, and Y. Öhrn. On the canonical formulation of electrodynamics and wave mechanics I: Theory. in preparation. [68] P. A. M. Dirac, V. A. Fock, and Boris Podolsky. On quantum electrodynamics. In Julian Schwinger, editor, Selected Papers on Quantum Electrodynamics, pages 29–40. Dover Publications, New York, 1958. [69] Donald H. Kobe. Gauge-invariant classical Hamiltonian formulation of the electrodynamics of nonrelativistic particles. American Journal of Physics, 49(6):581–588, 1981. [70] D. Masiello, E. Deumens, and Y. Öhrn. On the canonical formulation of electrodynamics and wave mechanics II: Numerical implementation. in preparation. [71] I. M. Gelfand and S. V. Fomin. Calculus of Variations. Dover, Mineola, 1963. [72] J. Broeckhove, L. Lathouwers, E. Kesteloot, and P. Van Leuven. On the equivalence of time-dependent variational principles. Chemical Physics Letters, 149(5,6):547–550, 1988. 139 [73] Peter Kramer and Marcos Saraceno. Geometry of the Time-Dependent Variational Principle in Quantum Mechanics. Number 140 in Lecture Notes in Physics. Springer-Verlag, Berlin, 1981. [74] Peter Ring and Peter Schuck. The Nuclear Many-Body Problem. SpringerVerlag, New York, 1980. [75] E. Deumens, A. Diz, R. Longo, and Y. Öhrn. Time-dependent theoretical treatments of the dynamics of electrons and nuclei in molecular systems. Reviews of Modern Physics, 66(3):917–984, 1994. [76] E. A. Power and T. Thirunamachandran. On the nature of the Hamiltonian for the interaction of radiation with atoms and molecules: (e/mc)p·A, −µ·E, and all that. American Journal of Physics, 46(4):370–378, 1978. [77] Donald H. Kobe and Arthur L. Smirl. Gauge invariant formulation of the interaction of electromagnetic radiation and matter. American Journal of Physics, 46(6):624–633, 1978. [78] Donald H. Kobe. Gauge-invariant resolution of the controversy over length versus velocity forms of the interaction with electric dipole radiation. Physical Review A, 19(1):205–214, 1979. [79] Willis E. Lamb Jr., Rainer R. Schlicher, and Marlan O. Scully. Matterfield interaction in atomic physics and quantum optics. Physical Review A, 36(6):2763–2772, 1987. [80] R. P. Feynman. The theory of positrons. Physical Review, 76(6):749–759, 1949. [81] Dan Jiao and Jian-Ming Jin. Three-dimensional orthogonal vector basis functions for time-domain finite element solution of vector wave equations. IEEE Transactions on Antennas and Propagation, 51(1):59–66, 2003. [82] Edwin A. Marengo and Anthony J. Devaney. The inverse source of electromagnetics: Linear inversion formulation and minimum energy solution. IEEE Transactions on Antennas and Propagation, 47(2):410–412, 1999. [83] G. H. Goedecke. Classically radiationless motions and possible implications for quantum theory. Physical Review, 135(1B):B281–B288, 1964. [84] A. J. Devaney and E. Wolf. Radiating and nonradiating classical current distributions and the fields they generate. Physical Review D, 8(4):1044–1047, 1973. [85] Edwin A. Marengo and Richard W. Ziolkowski. On the radiating and nonradiating components of scalar, electromagnetic, and weak gravitational sources. Physical Review Letters, 83(17):3345–3349, 1999. 140 [86] B. J. Hoenders and H. A. Ferwerda. Identification of the radiative and nonradiative parts of a wave function. Physical Review Letters, 87(6):060401, 2001. [87] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, London, 6th edition, 2000. [88] D. Masiello, E. Deumens, and Y. Öhrn. Bremsstrahlung from a gaussian wavepacket. in preparation. [89] J. D. Jackson. From Lorenz to Coulomb and other explicit gauge transformations. American Journal of Physics, 70(9):917–928, 2002. [90] Marek Nowakowski. The quantum mechanical current of the Pauli equation. American Journal of Physics, 67(10):916–919, 1999. [91] E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, S. Osrtouchov, and D. Sorensen. LAPACK Users Guide. Society for Industrial and Applied Mathematics, Philadelphia, 1992. [92] William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes. Cambridge University Press, Cambridge, 1986. [93] E. P. Wigner and L. Eisenbud. Higher angular momenta and long range interaction in resonance reactions. Physical Review, 72(1):29–41, 1947. [94] S. Yoshida, S. Watanabe, C.O. Reinhold, and J. Burgdörfer. Reflection-free propagation of wave packets. Physical Review A, 60(2):1113–1123, 1999. [95] Oleg I. Tolstikhin, Valentin Ostrovsky, and Hiroki Nakamura. Siegert pseudostate formulation of scattering theory: One-channel case. Physical Review A, 58(3):2077–2096, 1998. [96] E. Y. Sidky and B. D. Esry. Boundary-free propagation with the timedependent Schrodinger equation. Physical Review Letters, 85(24):5086–5089, 2000. [97] M. Baertschy, T.N. Rescigno, W.A. Isaacs, X. Li, and C.W. McCurdy. Electron-impact ionization of atomic hydrogen. Physical Review A, 63:022712, 2001. [98] C.W. McCurdy, D.A. Horner, T.N. Rescigno, and F. Martı́n. Theoretical treatment of double photoionization of helium using a b-spline implementation of exterior complex scaling. Physical Review A, 69:032707, 2004. [99] T. N. Rescigno, M. Baertschy, W. A. Isaacs, and C. W. McCurdy. Collisional breakup in a quantum system of three charged particles. Science, 286:2474– 2479, 1999. 141 [100] Barry Simon. The definition of molecular resonance curves by the method of exterior complex scaling. Physics Letters A, 71:211–214, 1979. [101] O. L. Brill and B. Goodman. Causility in the Coulomb gauge. American Journal of Physics, 35(9):832–837, 1967. BIOGRAPHICAL SKETCH David John Masiello was born on October 8th, 1977, in Providence, Rhode Island and was the only child of John Alfred Masiello and Norma Jean Masiello. Although his immediate family was small, David was part of a large Italian-American family that gathered religiously every Sunday for dinner. David’s many cousins were more like brothers and sisters. Together they spent endless days catching butterflies, building tree forts, and practicing their artwork under the supervision of their grandfather. In search of warm sunshine and blue skies, David’s parents decided to leave New England. They moved to sunny Florida just in time for David to begin high school. While in high school David became interested in chemistry and biology, and entered the University of Florida in 1995 with the intentions of pursuing a career in medicine. These intentions quickly changed as David found that his deeper questions could not be answered by these disciplines. In May of 1999, David received a B.S. degree in mathematics from the University of Florida. During his undergraduate career, David became interested in the applications of mathematics in the physical sciences. This interest led him to carry out research in optical physics over three university campuses worldwide. Always striving for a deeper more fundamental understanding of Nature, David decided to stay at the University of Florida to earn a Ph.D. under the advisement of Prof. Yngve Öhrn and Dr. Erik Deumens at the Quantum Theory Project. During his third year of graduate school David married his college sweetheart, Kathryn Allida Masiello. 142