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