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