ADVANCED QUANTUM MECHANICS Yuli V.Nazarov and Jeroen Danon CAMBRIDGE CAMBRIDGE more information-www.cambridge.org/9780521761505
more information – www.cambridge.org/9780521761505
Advanced Quantum Mechanics An accessible introduction to advanced quantum theory,this graduate-level textbook focuses on its practical applications rather than on mathematical technicalities.It treats real-life examples,from topics ranging from quantum transport to nanotechnology.to equip student with a toolbox of the cal techniques Beginning with second quantization,the authors illustrate its use with different con- densed matter physics examples.They then explain how to quantize classical fields,with a focus on the electromagnetic field,taking students from Maxwell's equations to photons, coherent states,and absorption and emission of photons.Following this is a unique master- level presentation on dissip tive quantum mechanics,before the textbook concludes with a short introduction to relativistic quantum mechanics,covering the Dirac equation and a relativistic second quantization formalism. The textbook includes 70 end-of-chapter problems.Solutions to some problems are given at the end of the chapter,and full solutions to all problems are available for instructors at www.cambridge.org/9780521761505. ssor in the Quantum Nanoscience Department,the Kavli Institute of Nanoscience,Delft University of Technology.He has worked in quantum transport since the emergence of the field in the late 1980s. JEROEN DANON is a Researcher at the Dahlem Center for Complex Quantum Systems,Free University of Berlin.He works in the fields of quantum transport and mesoscopic physics
Advanced Quantum Mechanics An accessible introduction to advanced quantum theory, this graduate-level textbook focuses on its practical applications rather than on mathematical technicalities. It treats real-life examples, from topics ranging from quantum transport to nanotechnology, to equip students with a toolbox of theoretical techniques. Beginning with second quantization, the authors illustrate its use with different condensed matter physics examples. They then explain how to quantize classical fields, with a focus on the electromagnetic field, taking students from Maxwell’s equations to photons, coherent states, and absorption and emission of photons. Following this is a unique masterlevel presentation on dissipative quantum mechanics, before the textbook concludes with a short introduction to relativistic quantum mechanics, covering the Dirac equation and a relativistic second quantization formalism. The textbook includes 70 end-of-chapter problems. Solutions to some problems are given at the end of the chapter, and full solutions to all problems are available for instructors at www.cambridge.org/9780521761505. YULI V. NAZAROV is a Professor in the Quantum Nanoscience Department, the Kavli Institute of Nanoscience, Delft University of Technology. He has worked in quantum transport since the emergence of the field in the late 1980s. JEROEN DANON is a Researcher at the Dahlem Center for Complex Quantum Systems, Free University of Berlin. He works in the fields of quantum transport and mesoscopic physics
Contents page PARTI SECOND QUANTIZATION 1 Elementary quantum mechanics 1.1 Classical mechanics 3 2 Schrodinge Dirac formulation Schrodinger and Heisenberg pictures 1.5 Perturbation theory 1.6 Time-dependent perturbation theory 1 1.6.1 Fermi's golden rule 8 1.7 and momentum 1.7.2 045 Iwo spins 1.8 Two-level system:The qubit 6 1.9 Harmonic oscillator 2 1.10 The density matrix 31 Exercises 38 Solutions 41 2 ldentical partides 43 2.1 Schrodinger equation for identical particles 4 2.2 The symme etry postulate 47 Quantum fields 2.3 Solutions of the N-particle Schrodinger equation 80 2.3.1 Symmetric wave function:Bosons 5 232 Antisymmetric wave function:Fermions 233 Fock space 56 Exercises Solutions 96 3 Second quantization 6 3.1 Second quantization for bosons 3.1.1 Commutation relations 3.1.2 The cture of Fock space
Contents Figure Credits page x Preface xi PART I SECOND QUANTIZATION 1 1 Elementary quantum mechanics 3 1.1 Classical mechanics 3 1.2 Schrödinger equation 4 1.3 Dirac formulation 7 1.4 Schrödinger and Heisenberg pictures 11 1.5 Perturbation theory 13 1.6 Time-dependent perturbation theory 14 1.6.1 Fermi’s golden rule 18 1.7 Spin and angular momentum 20 1.7.1 Spin in a magnetic field 24 1.7.2 Two spins 25 1.8 Two-level system: The qubit 26 1.9 Harmonic oscillator 29 1.10 The density matrix 31 1.11 Entanglement 33 Exercises 38 Solutions 41 2 Identical particles 43 2.1 Schrödinger equation for identical particles 43 2.2 The symmetry postulate 47 2.2.1 Quantum fields 48 2.3 Solutions of the N-particle Schrödinger equation 50 2.3.1 Symmetric wave function: Bosons 52 2.3.2 Antisymmetric wave function: Fermions 54 2.3.3 Fock space 56 Exercises 59 Solutions 61 3 Second quantization 63 3.1 Second quantization for bosons 63 3.1.1 Commutation relations 64 3.1.2 The structure of Fock space 65 v
Contents 3.2 Field operators for bosons 3.2.1 3.2.2 of field opetator Hamilto nian int 3.2.3 Why second quantization? 170224 34 Second quantization for fermions 3.4.1 Creation and annihilation operators for fermions 3.4.2 Field operators 3.5 Summary of second quantization Exercises 15870888 Solutions PART II EXAMPLES 4 Magnetism 90 4.1 Non-interacting Fermi gas 4.2 Magnetic ground state 4.2.1 Trial wave function 4.3 Ene Kinetic energy 32 Potential energy 4.3.3 Energy balance and phases 4.4 Broken symmetry 4.5 Excitations in ferromagnetic metals Single-part e excitations Electron-hole pairs Magnons 99999999991010161910 45.4 Magnon spectrum Exercises Solutions 5 Superconductivity 5.1 Attractive interaction and Cooper pairs 5.1.1 Trial wave function 5.12 Nambu boxes 5.2 Energy minimization 5.3 Particles and quasiparticles 5.4 Broken symmetry Exercises Solutions 6 Superfluidity 6.1 Non-interacting Bose gas
vi Contents 3.2 Field operators for bosons 66 3.2.1 Operators in terms of field operators 67 3.2.2 Hamiltonian in terms of field operators 70 3.2.3 Field operators in the Heisenberg picture 72 3.3 Why second quantization? 72 3.4 Second quantization for fermions 74 3.4.1 Creation and annihilation operators for fermions 75 3.4.2 Field operators 78 3.5 Summary of second quantization 79 Exercises 82 Solutions 83 PART II EXAMPLES 87 4 Magnetism 90 4.1 Non-interacting Fermi gas 90 4.2 Magnetic ground state 92 4.2.1 Trial wave function 92 4.3 Energy 93 4.3.1 Kinetic energy 93 4.3.2 Potential energy 94 4.3.3 Energy balance and phases 97 4.4 Broken symmetry 98 4.5 Excitations in ferromagnetic metals 99 4.5.1 Single-particle excitations 99 4.5.2 Electron–hole pairs 102 4.5.3 Magnons 103 4.5.4 Magnon spectrum 105 Exercises 109 Solutions 110 5 Superconductivity 113 5.1 Attractive interaction and Cooper pairs 114 5.1.1 Trial wave function 116 5.1.2 Nambu boxes 118 5.2 Energy 119 5.2.1 Energy minimization 120 5.3 Particles and quasiparticles 123 5.4 Broken symmetry 125 Exercises 128 Solutions 132 6 Superfluidity 135 6.1 Non-interacting Bose gas 135
Contents 6.2 Field theory for interacting Bose gas 6.2.1 Hamiltonian and Heisenberg equation 63 The condensate 139 6.3.1 Broken symmetry 139 Excitations as oscillation 641 Particles and qua asiparticles 6.5 Topological excitations 6.5.1 Vortices 146 652 Vortices as quantum states 149 653 Vortex lines 151 Exercises Solutions PART III FIELDS AND RADIATION 159 7 Classical fields 71 Chain of coupled oscillators 7.2 Continuous elastic string 6 7.2.1 Hamiltonian and equation of motion 64 7.2.2 Solution of the equation of motion 1 723 The elastic string as a set of oscillators 166 73 c field Useful relations 60 7.3.3 Vector and scalar potentials 170 734 Gauges 171 7.3.5 Electromagnetic field as a set of oscillators 172 7.3.6 The LC-oseillator Exercises Solutions 8 Quantization of fields 183 81 81.1 or and oscillator 82 The elastic string:phonons 88 8.3 Fluctuations of magnetization:magnons 8.4 Quantization of the electromagnetic field 18 8.4.1 Photons 8.4.2 Field operators 8.4.3 Zero-point energy,uncertainty relations. and vacuum fluctuations 94 844 The simple oscillator 1 Exercises Solutions
vii Contents 6.2 Field theory for interacting Bose gas 136 6.2.1 Hamiltonian and Heisenberg equation 138 6.3 The condensate 139 6.3.1 Broken symmetry 139 6.4 Excitations as oscillations 140 6.4.1 Particles and quasiparticles 141 6.5 Topological excitations 142 6.5.1 Vortices 146 6.5.2 Vortices as quantum states 149 6.5.3 Vortex lines 151 Exercises 154 Solutions 157 PART III FIELDS AND RADIATION 159 7 Classical fields 162 7.1 Chain of coupled oscillators 162 7.2 Continuous elastic string 163 7.2.1 Hamiltonian and equation of motion 164 7.2.2 Solution of the equation of motion 165 7.2.3 The elastic string as a set of oscillators 166 7.3 Classical electromagnetic field 167 7.3.1 Maxwell equations 168 7.3.2 Useful relations 170 7.3.3 Vector and scalar potentials 170 7.3.4 Gauges 171 7.3.5 Electromagnetic field as a set of oscillators 172 7.3.6 The LC-oscillator 174 Exercises 177 Solutions 181 8 Quantization of fields 183 8.1 Quantization of the mechanical oscillator 183 8.1.1 Oscillator and oscillators 185 8.2 The elastic string: phonons 187 8.3 Fluctuations of magnetization: magnons 189 8.4 Quantization of the electromagnetic field 191 8.4.1 Photons 191 8.4.2 Field operators 192 8.4.3 Zero-point energy, uncertainty relations, and vacuum fluctuations 194 8.4.4 The simple oscillator 198 Exercises 201 Solutions 203
Contents 9 Radiation and matter 205 9 1 Transition rates 9.2 921 Master equations 9.2.2 Equilibrium and black-body radiation 0301124 9.3 Interaction of matter and radiation 9.4 Spontaneous emission by atoms 9.4.1 Dipole approxim tion 9.42 Transition rates 93 Selection rules 9.5 Blue glow:Cherenkov radiation 9.5.1 Emission rate and spectrum of Cherenkov radiation 9.6 Bremsstrahlung 97 Processes in lasers 9.7.1 Master equation for lasers 9.7.2 Photon number distribution Exercises Solutions 10 Coherentstates 10.1 Superpositions 10.2 Excitation of an oscillator 44404 10.3 Properties of the coherent state 10.4 Back to the laser 10.4.1 Optical coherence time 10.4.2 Maxwell-Bloch equations 10.5 Coherent states of matter 10.5.1 Cooper pair box 49555 Exercises Solutions 265 PART IV DISSIPATIVE QUANTUM MECHANICS 267 11 Dissipative quantum mechanics 11.1 Clas ical damped oscillato 11.1.Dynam cal susceptibility 11.1.2 Damped electric oscillator 11.2 Quantum description 11.2.1 Difficulties with the quantum description 11.2.2 Solution:Many degrees of freedom 112.3 Boson bath 11.2.4 Quantum equations of motion 11.2.5 Diagonalization
viii Contents 9 Radiation and matter 205 9.1 Transition rates 206 9.2 Emission and absorption: General considerations 207 9.2.1 Master equations 210 9.2.2 Equilibrium and black-body radiation 211 9.3 Interaction of matter and radiation 214 9.4 Spontaneous emission by atoms 218 9.4.1 Dipole approximation 218 9.4.2 Transition rates 219 9.4.3 Selection rules 222 9.5 Blue glow: Cherenkov radiation 223 9.5.1 Emission rate and spectrum of Cherenkov radiation 225 9.6 Bremsstrahlung 227 9.7 Processes in lasers 229 9.7.1 Master equation for lasers 231 9.7.2 Photon number distribution 232 Exercises 235 Solutions 238 10 Coherent states 240 10.1 Superpositions 240 10.2 Excitation of an oscillator 241 10.3 Properties of the coherent state 244 10.4 Back to the laser 249 10.4.1 Optical coherence time 250 10.4.2 Maxwell–Bloch equations 252 10.5 Coherent states of matter 256 10.5.1 Cooper pair box 257 Exercises 262 Solutions 265 PART IV DISSIPATIVE QUANTUM MECHANICS 267 11 Dissipative quantum mechanics 269 11.1 Classical damped oscillator 269 11.1.1 Dynamical susceptibility 270 11.1.2 Damped electric oscillator 272 11.2 Quantum description 273 11.2.1 Difficulties with the quantum description 273 11.2.2 Solution: Many degrees of freedom 274 11.2.3 Boson bath 274 11.2.4 Quantum equations of motion 275 11.2.5 Diagonalization 277
Contents 11.3 Time-dependent fluctuations 279 11.3.1 Fluctuation-dissipation theorem 280 11.3.2 Kubo formula 11.4 Heisenberg uncertainty relation 282 11.4.1 Density matrix of a damped oscillator 12 Transitions and dissipatio 12.1 Complicating the damped oscillator:Towards a qubit 12.1.1 Delocalization criterion 12 2 Spin-boson model 12 3 Shifted oscillators 12 4 Shake e-up and P(E 2.5 Orthogonality catastrophe 12.6 Workout of P(E) 12.7 Transition rates and delocalization 38 12.8 Classification of environments 12.8.1 Subohmic 12.820hmic 12.8.3 Superohmic 12.9 Vacuum as an environment Exercises 310 Solutions 312 PARTV RELATIVISTIC QUANTUM MECHANICS 315 13 Relativisticquantum mechanics 13.1 Principles of the theory of relativity 13.1.1 Lorentz transformation 13.1.2 Minkowski spacetime 13.1.3 The Minkowski metric 323 13.1 Four-vectors 13.2 Dirac equation 13.2.1 Solutions of the Dirac equation 13.2.2 Second quantization 13.2.3 Interaction with the electromagnetic field 3.3.1 13.3.2 Perturbation theory and divergences 13.4 Renormalization Exercises 348 Solutions 351 Index 352
ix Contents 11.3 Time-dependent fluctuations 279 11.3.1 Fluctuation–dissipation theorem 280 11.3.2 Kubo formula 281 11.4 Heisenberg uncertainty relation 282 11.4.1 Density matrix of a damped oscillator 283 Exercises 286 Solutions 288 12 Transitions and dissipation 290 12.1 Complicating the damped oscillator: Towards a qubit 290 12.1.1 Delocalization criterion 292 12.2 Spin–boson model 292 12.3 Shifted oscillators 294 12.4 Shake-up and P(E) 296 12.5 Orthogonality catastrophe 297 12.6 Workout of P(E) 298 12.7 Transition rates and delocalization 301 12.8 Classification of environments 302 12.8.1 Subohmic 304 12.8.2 Ohmic 305 12.8.3 Superohmic 306 12.9 Vacuum as an environment 307 Exercises 310 Solutions 312 PART V RELATIVISTIC QUANTUM MECHANICS 315 13 Relativistic quantum mechanics 317 13.1 Principles of the theory of relativity 317 13.1.1 Lorentz transformation 318 13.1.2 Minkowski spacetime 321 13.1.3 The Minkowski metric 323 13.1.4 Four-vectors 324 13.2 Dirac equation 326 13.2.1 Solutions of the Dirac equation 330 13.2.2 Second quantization 333 13.2.3 Interaction with the electromagnetic field 336 13.3 Quantum electrodynamics 337 13.3.1 Hamiltonian 338 13.3.2 Perturbation theory and divergences 339 13.4 Renormalization 343 Exercises 348 Solutions 351 Index 352
Figure Credits Photo of Erwin Schrodinger:Robertson,obtained via Flickr The Commons from the Smithsonian Institution,www.si.edu page 5 Photo of Werner Heisenberg:Bundesarchiv.Bild183-R57262/CC-BY-SA Photo of Enrico Fermi:courtesy National Archives,photo no.434-OR-7(24) Photo of Vladimir Fock:AIP Emilio Segre Visual Archives.gift of Tatiana Yudovin Photo of Pascual Jordan:SLUB Dresden/Deutsche Fotothek.Grossmann Photo of Yoichiro Nambu:Betsy Devine 100 Photo of John Bardeen:AIP Emilio Segre Visual Archives 117 Photo of Leon Cooper:AIP Emilio Segre Visual Archives,gift of Leon Cooper 117 Photo of John Rob rSchrieffer:AIP Emilio Segre Visual Archives 117 Photo of Richard Feynman:Christopher Sykes courey AIP Emilio Visual Archives Photo of Paul Dirac:Science Service 193 Photo of Max Planck:obtained via Flickr The Commons from the Smithsonian Institution.www si.edu 213 Photo of Roy Glauber:Markus Possel Photo of Philip Anderson:Kenneth C.Zirkel
Figure Credits Photo of Erwin Schrödinger: Robertson, obtained via Flickr The Commons from the Smithsonian Institution, www.si.edu page 5 Photo of Werner Heisenberg: Bundesarchiv, Bild183-R57262 / CC-BY-SA 12 Photo of Enrico Fermi: courtesy National Archives, photo no. 434-OR-7(24) 55 Photo of Vladimir Fock: AIP Emilio Segrè Visual Archives, gift of Tatiana Yudovina 68 Photo of Pascual Jordan: SLUB Dresden/Deutsche Fotothek, Grossmann 73 Photo of Yoichiro Nambu: Betsy Devine 100 Photo of John Bardeen: AIP Emilio Segrè Visual Archives 117 Photo of Leon Cooper: AIP Emilio Segrè Visual Archives, gift of Leon Cooper 117 Photo of John Robert Schrieffer: AIP Emilio Segrè Visual Archives 117 Photo of Richard Feynman: Christopher Sykes, courtesy AIP Emilio Segrè Visual Archives 144 Photo of Paul Dirac: Science Service 193 Photo of Max Planck: obtained via Flickr The Commons from the Smithsonian Institution, www.si.edu 213 Photo of Roy Glauber: Markus Pössel 250 Photo of Philip Anderson: Kenneth C. Zirkel 299 x
Preface Courses on advanced quantum mechanics have a long tradition.The tradition is in fact so long that the word "advanced"in this context does not usually mean"new"or"up-to- date.The basic quantum mechanic were developed in the wenties of the ast century,initially to explain experiments in atomi fast and great advance in the thirties and forties,when a quantum theory for large numbers of identical particles was developed.This advance ultimately led to the modern concepts of elementary particles and quantum fields that concern the underlying structure of our Universe.At a less fundamental and more practical level,it has also laid the basis for our present understanding of solid state andc ndensed ma ter physics s and,at a later stage,for artificially made quantum systems.The basics of this leap forward of quantum theory are what is usually covered by a course on advanced quantum mechanics. Most courses and textbooks are designed for a fundamentally oriented education:build- ing on basic quantum theory.they provide an introduction for students who wish to h advanc ed quantum theor y of elementary particles and quantum fields.In order to do this in a"right"way.there is usually a strong emphasis on technicalities relate to relativity and on the underlying mathematics of the theory.Less frequently,a course serves as a brief introduction to advanced topics in advanced solid state or condensed matter. Such presentation style does not necessarily reflect the taste and interests of the mod- ern studen The last 20 years brought enormous progress in applying qu tum mechanics in a very different context.Nanometer-sized quantum devices of different kinds are being manufactured in research centers around the world.aiming at processing quantum infor mation or making elements of nano-electronic circuits.This development resulted in a fascination of the present generation of students with topics like quantum computing and technology. Ma ny ould like to put this fa inatio on olid and base their understan ling of t ese topics on scientific fundamentals grounds. These are usu ally people with a practical attitude,who are not immediately interested in brain-teasing concepts of modern string theory or cosmology.They need fundamental knowledge to work with and to apply to"real-life"quantum mechanical problems arising in an unusual context.This book is mainly aimed at this category of students. The present book is based on the contents of the course Advanced Quantum Mechanics a part of the master program of the Applied Physics curriculum of the Delft University of Technology.The DUT is a university for practically inclined people,jokingly called "bike-repairmen"by the students of more traditional universities located in nearby cities. While probably meant to be belitling.the joke does capture the essence of the research in Delft.Indeed,the structure of the Unive center of the physics
Preface Courses on advanced quantum mechanics have a long tradition. The tradition is in fact so long that the word “advanced” in this context does not usually mean “new” or “up-todate.” The basic concepts of quantum mechanics were developed in the twenties of the last century, initially to explain experiments in atomic physics. This was then followed by a fast and great advance in the thirties and forties, when a quantum theory for large numbers of identical particles was developed. This advance ultimately led to the modern concepts of elementary particles and quantum fields that concern the underlying structure of our Universe. At a less fundamental and more practical level, it has also laid the basis for our present understanding of solid state and condensed matter physics and, at a later stage, for artificially made quantum systems. The basics of this leap forward of quantum theory are what is usually covered by a course on advanced quantum mechanics. Most courses and textbooks are designed for a fundamentally oriented education: building on basic quantum theory, they provide an introduction for students who wish to learn the advanced quantum theory of elementary particles and quantum fields. In order to do this in a “right” way, there is usually a strong emphasis on technicalities related to relativity and on the underlying mathematics of the theory. Less frequently, a course serves as a brief introduction to advanced topics in advanced solid state or condensed matter. Such presentation style does not necessarily reflect the taste and interests of the modern student. The last 20 years brought enormous progress in applying quantum mechanics in a very different context. Nanometer-sized quantum devices of different kinds are being manufactured in research centers around the world, aiming at processing quantum information or making elements of nano-electronic circuits. This development resulted in a fascination of the present generation of students with topics like quantum computing and nanotechnology. Many students would like to put this fascination on more solid grounds, and base their understanding of these topics on scientific fundamentals. These are usually people with a practical attitude, who are not immediately interested in brain-teasing concepts of modern string theory or cosmology. They need fundamental knowledge to work with and to apply to “real-life” quantum mechanical problems arising in an unusual context. This book is mainly aimed at this category of students. The present book is based on the contents of the course Advanced Quantum Mechanics, a part of the master program of the Applied Physics curriculum of the Delft University of Technology. The DUT is a university for practically inclined people, jokingly called “bike-repairmen” by the students of more traditional universities located in nearby cities. While probably meant to be belittling, the joke does capture the essence of the research in Delft. Indeed, the structure of the Universe is not in the center of the physics curriculum xi
Preface in Delft,where both research and education rather concentrate on down-to-earth topics. ntum dots.super ronics,and ma others.The theoretical part of the curriculum is designed to support this research in the most efficient way:after a solid treatment of the basics,the emphasis is quickly shifted to apply the theory to understand the essential properties of quantum devices.This book is written with the same philosophy.It presents the fundamentals of advanced quantum theory at an operational level:we have tried to keep the technical and mathematical basis as simple as pos sible,and as soon as we have enough theoretical tools at hand we move on and give examples how to use them. The book starts with an introductory chapter on basic quantum mechanics.Since this book is intended for a course on advanced quantum mechanics.we assume that the reader is already familiar with all concepts discussed in this chapter.The reason we included it was to make the book more"self-contained."s well as tom ake sure that we all understand the basics in the same way when we discuss advanced topics.The following two chapters introduce new material:we extend the basic quantum theory to describe many (identical) particles,instead of just one or two,and we show how this description fits conveniently in the framework of second quantization. then have all the to effects in many-particle the cd part(Chapter)epro vide some examples and show how we can understand magnetism,superconductivity,and superfluidity by straightforward use of the theoretical toolbox presented in the previous chapters. After focusing exclusively particle quantum theory in the first parts of the book.we then m ve on to in into our theoretical fra work.In Chapters and 8 we explain in very general terms how almost any classical field can be"quantized and how this procedure naturally leads to a very particle-like treatment of the excitations of the fields.We give many examples,but keep an emphasis on the electromagnetic field because of its fundamental importance.In Chapter 9 we then provide the last"missing epuzzle":we explain how to describe the interaction between particles and efield.With this knowledge at hand,we construct simple models t describe several phenomena from the field of quantum optics:we discuss the radiative decay of excited atomic states,as well as Cherenkov radiation and Bremsstrahlung,and we give a simplified picture of how a laser works.This third part is concluded with a short introduction on coherent states:a very general concept,but in particular very important in he field of quantu optic In the fourth part of the book follows anique master-level introduction to dissipativ quantum mechanics.This field developed relatively recently (in the last three decades) and is usually not discussed in textbooks on quantum mechanics.In practice,however,the concept of dissipation is as important in quantum mechanics as it is in classical mechanics The idea of a e.g.a harn onic oscillator which is brought into a station ary excited eigenstate and will stay there forever,is in reality too idealized interaction with a(possibly very complicated)environment can dissipate energy from the system and can ultimately bring it to its ground state.Although the problem seems inconceivably hard
xii Preface in Delft, where both research and education rather concentrate on down-to-earth topics. The DUT is one of the world-leading centers doing research on quantum devices such as semiconductor quantum dots, superconducting qubits, molecular electronics, and many others. The theoretical part of the curriculum is designed to support this research in the most efficient way: after a solid treatment of the basics, the emphasis is quickly shifted to apply the theory to understand the essential properties of quantum devices. This book is written with the same philosophy. It presents the fundamentals of advanced quantum theory at an operational level: we have tried to keep the technical and mathematical basis as simple as possible, and as soon as we have enough theoretical tools at hand we move on and give examples how to use them. The book starts with an introductory chapter on basic quantum mechanics. Since this book is intended for a course on advanced quantum mechanics, we assume that the reader is already familiar with all concepts discussed in this chapter. The reason we included it was to make the book more “self-contained,” as well as to make sure that we all understand the basics in the same way when we discuss advanced topics. The following two chapters introduce new material: we extend the basic quantum theory to describe many (identical) particles, instead of just one or two, and we show how this description fits conveniently in the framework of second quantization. We then have all the tools at our disposal to construct simple models for quantum effects in many-particle systems. In the second part of the book (Chapters 4–6), we provide some examples and show how we can understand magnetism, superconductivity, and superfluidity by straightforward use of the theoretical toolbox presented in the previous chapters. After focusing exclusively on many-particle quantum theory in the first parts of the book, we then move on to include fields into our theoretical framework. In Chapters 7 and 8, we explain in very general terms how almost any classical field can be “quantized” and how this procedure naturally leads to a very particle-like treatment of the excitations of the fields. We give many examples, but keep an emphasis on the electromagnetic field because of its fundamental importance. In Chapter 9 we then provide the last “missing piece of the puzzle”: we explain how to describe the interaction between particles and the electromagnetic field. With this knowledge at hand, we construct simple models to describe several phenomena from the field of quantum optics: we discuss the radiative decay of excited atomic states, as well as Cherenkov radiation and Bremsstrahlung, and we give a simplified picture of how a laser works. This third part is concluded with a short introduction on coherent states: a very general concept, but in particular very important in the field of quantum optics. In the fourth part of the book follows a unique master-level introduction to dissipative quantum mechanics. This field developed relatively recently (in the last three decades), and is usually not discussed in textbooks on quantum mechanics. In practice, however, the concept of dissipation is as important in quantum mechanics as it is in classical mechanics. The idea of a quantum system, e.g. a harmonic oscillator, which is brought into a stationary excited eigenstate and will stay there forever, is in reality too idealized: interactions with a (possibly very complicated) environment can dissipate energy from the system and can ultimately bring it to its ground state. Although the problem seems inconceivably hard
