QUANTUM MECHANICS with Basic Field Theory BIPIN R.DESAI
Quantum Mechanics with Basic Field Theory Students and instructors alike will find this organized and detailed approach to quantum mechanics ideal for a two-semester graduate course on the subject. This textbook covers,step-by-step,important topics in quantum mechanics,from tra- ditional subjects like bound states,perturbation theory and scattering,to more current topics such as coherent states,quantum Hall effect,spontaneous symmetry breaking,super conductivity,and basic quantum electrodynamics with radiative corrections.The large number of diverse topics are covered in concise,highly focused chapters,and are explained in simple but mathematically rigorous ways.Derivations of results and formulas are carried out from beginning to end,without leaving students to complete them. With over 200 exercises to aid understanding of the subject,this textbook provides a thorough grounding for students planning to enter research in physics.Several exercises are solved in the text,and password-protected solutions for remaining exercises are available to instructors at www.cambridge.org/9780521877602. Bipin R.Desai is a Professor of Physics at the University of California,Riverside,where he does research in elementary particle theory.He obtained his Ph.D.in Physics from the University of California,Berkeley.He was a visiting Fellow at Clare Hall,Cambridge University,UK,and has held research positions at CERN,Geneva,Switzerland,and CEN Saclay,France.He is a Fellow of the American Physical Society
Quantum Mechanics with Basic Field Theory Students and instructors alike will find this organized and detailed approach to quantum mechanics ideal for a two-semester graduate course on the subject. This textbook covers, step-by-step, important topics in quantum mechanics, from traditional subjects like bound states, perturbation theory and scattering, to more current topics such as coherent states, quantum Hall effect, spontaneous symmetry breaking, superconductivity, and basic quantum electrodynamics with radiative corrections. The large number of diverse topics are covered in concise, highly focused chapters, and are explained in simple but mathematically rigorous ways. Derivations of results and formulas are carried out from beginning to end, without leaving students to complete them. With over 200 exercises to aid understanding of the subject, this textbook provides a thorough grounding for students planning to enter research in physics. Several exercises are solved in the text, and password-protected solutions for remaining exercises are available to instructors at www.cambridge.org/9780521877602. Bipin R. Desai is a Professor of Physics at the University of California, Riverside, where he does research in elementary particle theory. He obtained his Ph.D. in Physics from the University of California, Berkeley. He was a visiting Fellow at Clare Hall, Cambridge University, UK, and has held research positions at CERN, Geneva, Switzerland, and CEN Saclay, France. He is a Fellow of the American Physical Society
Contents Preface page xvii Physical constants 1 Basic formalism 1.1 State vectors 1.2 Operators and physical observables 13 Eigenstates 14 Hermitian conjugation and Hermitian operators Hermitian operators:their eigenstates and eigenvalues 6 1.6 Superposition principle 1.7 Completeness relation 18 Unitary operators 1.9 Unitary operators as transformation operators 10 1.10 Matrix formalism 1.11 Eigenstates and dia agonalization of matrices 1.12 Density operator 1618 1.13 Measurement 20 1.14 Problems 21 2 Fundamental commutator and time evolution of state vectors and operators 24 2.1 Conti s variables:and P operators Canonical commutator [X,P] 246 2.3 P as a derivative operator:another way 29 2.4 X and P as Hermitian operators 5 Uncertainty principle 32 2.6 Some interesting applications of uncertainty relations 27 Space displacement operator 36 28 Time evolution operat Appendix to Chapter 2 4 2.10 Problems 3 Dynamical equations 55 3.1 Schrodinger picture 3.2 Heisenberg picture 57
Contents Preface page xvii Physical constants xx 1 Basic formalism 1 1.1 State vectors 1 1.2 Operators and physical observables 3 1.3 Eigenstates 4 1.4 Hermitian conjugation and Hermitian operators 5 1.5 Hermitian operators: their eigenstates and eigenvalues 6 1.6 Superposition principle 7 1.7 Completeness relation 8 1.8 Unitary operators 9 1.9 Unitary operators as transformation operators 10 1.10 Matrix formalism 12 1.11 Eigenstates and diagonalization of matrices 16 1.12 Density operator 18 1.13 Measurement 20 1.14 Problems 21 2 Fundamental commutator and time evolution of state vectors and operators 24 2.1 Continuous variables: X and P operators 24 2.2 Canonical commutator [X , P] 26 2.3 P as a derivative operator: another way 29 2.4 X and P as Hermitian operators 30 2.5 Uncertainty principle 32 2.6 Some interesting applications of uncertainty relations 35 2.7 Space displacement operator 36 2.8 Time evolution operator 41 2.9 Appendix to Chapter 2 44 2.10 Problems 52 3 Dynamical equations 55 3.1 Schrödinger picture 55 3.2 Heisenberg picture 57
Contents Interaction picture 59 3.4 Superposition of time-dependent states and energy-time 63 35 Probability conservation 3.7 Ehrenfest's theorem 68 3.8 Problems 70 Free particles 73 4.1 Free particle in one dimension 73 4.2 Normalization 75 4.3 Momentum eigenfunctions and Fourier transforms Minimum uncertainty wave packet 13 45 Group velocity of a superposition of plane waves 3 4.6 Three dimensions-Cartesian coordinates 84 Three dimensions-spherical coordinates The radial wave equation 9 4.9 Properties of Yim(,) 92 4.10 Angular momentum 9 4.11 Determining L2from th ngular variables Commutator Li,L and L2 98 4.13 Ladder operators 100 4.14 Problems 102 5 Particles with spin V 103 5.1 Spin⅓system 103 5.2 Pauli matrices 104 The spin⅓eigenstates Matrix representation of ox and oy 8 5.5 Eigenstates of o and o. 108 5.6 eigenstates of spin in an arbitrary direction 109 5.7 e important relations for 110 Arbitrary 2 x 2 matrices in terms of Pauli matrices 5.9 Projection operator for spin systems 112 5.10 Density matrix for spin states and the ensemble average 114 Compete wavefunctio 116 Pauli exclusion principle and Fermi energy 116 5.13 Problems 118 6 Gauge invariance,angular momentum,and spin 6 Gauge invariance 18 6.2 Quantum mechanics 121 Canonical and kinematic momenta 123 64 Probability conservation 124
viii Contents 3.3 Interaction picture 59 3.4 Superposition of time-dependent states and energy–time uncertainty relation 63 3.5 Time dependence of the density operator 66 3.6 Probability conservation 67 3.7 Ehrenfest’s theorem 68 3.8 Problems 70 4 Free particles 73 4.1 Free particle in one dimension 73 4.2 Normalization 75 4.3 Momentum eigenfunctions and Fourier transforms 78 4.4 Minimum uncertainty wave packet 79 4.5 Group velocity of a superposition of plane waves 83 4.6 Three dimensions – Cartesian coordinates 84 4.7 Three dimensions – spherical coordinates 87 4.8 The radial wave equation 91 4.9 Properties of Ylm(θ, φ) 92 4.10 Angular momentum 94 4.11 Determining L2 from the angular variables 97 4.12 Commutator Li, Lj and L2, Lj 98 4.13 Ladder operators 100 4.14 Problems 102 5 Particles with spin ½ 103 5.1 Spin ½ system 103 5.2 Pauli matrices 104 5.3 The spin ½ eigenstates 105 5.4 Matrix representation of σx and σy 106 5.5 Eigenstates of σx and σy 108 5.6 Eigenstates of spin in an arbitrary direction 109 5.7 Some important relations for σi 110 5.8 Arbitrary 2 × 2 matrices in terms of Pauli matrices 111 5.9 Projection operator for spin ½ systems 112 5.10 Density matrix for spin ½ states and the ensemble average 114 5.11 Complete wavefunction 116 5.12 Pauli exclusion principle and Fermi energy 116 5.13 Problems 118 6 Gauge invariance, angular momentum, and spin 120 6.1 Gauge invariance 120 6.2 Quantum mechanics 121 6.3 Canonical and kinematic momenta 123 6.4 Probability conservation 124
Contents 6.5 Interaction with the orbital angular momentum 125 Interaction with spin:intrinsic magnetic moment 126 Spin-orbit interaction 128 6.8 Aharonov-Bohm effect 129 6.9 Problems 7 Stern-Gerlach experiments 133 7.1 Experimental set-up and electron's magnetic moment 133 7.2 Discussion of the results 134 2 Problems 136 8 Some exactly solvable bound-state problems 137 8.1 Simple one-dimensional systems 137 8.2 Delta-function potential 145 Properties of a symmetric potential 147 8.4 The ammonia molecule 148 85 Periodic potentials 151 Problems in three dimensions 1 Simple systems . Hydrogen-like atom 16 89 Problems 70 9 Harmonic oscillator 174 9.1 Harmonic oscillator in one dimension 174 9.2 Problems 184 10 Coherent states 187 10.1 Eigenstates of the lowering operator 187 10.2 Coherent states and semiclassical description 192 10.3 Interaction of a harmonic oscillator with an electric field 194 104 Appendix to Chapter 10 10.5 Problems 200 11 Two-dimensional isotropic harmonic oscillator 203 11.1 The two-dimensional Har iltonian 11.2 Problems 207 2 Landau levels and quantum Hall effect 208 Landau levels in symmetric gauge 12.2 Wavefunctions for the LLL 212 12.3 Landau levels in Landau gauge 214 12.4 Quantum Hall effect 216 12. avefur ction for filled LLLs in a Fermi system 12.6 Problems
ix Contents 6.5 Interaction with the orbital angular momentum 125 6.6 Interaction with spin: intrinsic magnetic moment 126 6.7 Spin–orbit interaction 128 6.8 Aharonov–Bohm effect 129 6.9 Problems 131 7 Stern–Gerlach experiments 133 7.1 Experimental set-up and electron’s magnetic moment 133 7.2 Discussion of the results 134 7.3 Problems 136 8 Some exactly solvable bound-state problems 137 8.1 Simple one-dimensional systems 137 8.2 Delta-function potential 145 8.3 Properties of a symmetric potential 147 8.4 The ammonia molecule 148 8.5 Periodic potentials 151 8.6 Problems in three dimensions 156 8.7 Simple systems 160 8.8 Hydrogen-like atom 164 8.9 Problems 170 9 Harmonic oscillator 174 9.1 Harmonic oscillator in one dimension 174 9.2 Problems 184 10 Coherent states 187 10.1 Eigenstates of the lowering operator 187 10.2 Coherent states and semiclassical description 192 10.3 Interaction of a harmonic oscillator with an electric field 194 10.4 Appendix to Chapter 10 199 10.5 Problems 200 11 Two-dimensional isotropic harmonic oscillator 203 11.1 The two-dimensional Hamiltonian 203 11.2 Problems 207 12 Landau levels and quantum Hall effect 208 12.1 Landau levels in symmetric gauge 208 12.2 Wavefunctions for the LLL 212 12.3 Landau levels in Landau gauge 214 12.4 Quantum Hall effect 216 12.5 Wavefunction for filled LLLs in a Fermi system 220 12.6 Problems 221
Contents 13 Two-level problems 223 13.1 Time-independent problems 223 132 Time-dependent problems 234 13.3 Problems 246 14 Spin systems in the presence of magnetic fields 141 Constant magnetic field 251 14.2 Spin precession 14.3 Time-dependent magnetic field:spin magnetic resonance 14.4 Problems 258 15 Oscillation and regeneration in neutrinos and neutral K-mesons as two-level systems 15.1 Neutrinos 260 15.2 The solar neutrino puzzle 260 Neutrino oscillations 5 Decay and regeneration 15.5 Oscillation and regeneration of stable and unstable systems 269 15.6 Neutral &-mesons 273 15.7 Problems 276 16 Time-independent perturbation for bound states 16.1 Basic formalism 277 16.2 Harm nic oscilla r:perturbative vs.exact results l63 Second-order Stark effect 16.4 Degenerate states 287 165 Linear Stark effect 289 16.6 Problems 290 17 Time-dependent perturbation 293 17.1 Basic formalism 293 17.2 Harmonic perturbation and Fermi's golden rule 17 Transitions into a group of states and scattering cross-section 286 17.4 Resonance and decav 30 17.5 Appendix to Chapter 17 310 17.6 Problems 315 18 Interaction of charged particles and radiation in perturbation theory 318 18.1 Electron in an electromagnetic field:the absorption cross-section 318 182 18.3 Coulomb excitations of an atom 184 lonization 328 18.5 Thomson.Ravleigh.and raman scattering in second-order perturbation 331 18.6 Problems 339
x Contents 13 Two-level problems 223 13.1 Time-independent problems 223 13.2 Time-dependent problems 234 13.3 Problems 246 14 Spin ½ systems in the presence of magnetic fields 251 14.1 Constant magnetic field 251 14.2 Spin precession 254 14.3 Time-dependent magnetic field: spin magnetic resonance 255 14.4 Problems 258 15 Oscillation and regeneration in neutrinos and neutral K-mesons as two-level systems 260 15.1 Neutrinos 260 15.2 The solar neutrino puzzle 260 15.3 Neutrino oscillations 263 15.4 Decay and regeneration 265 15.5 Oscillation and regeneration of stable and unstable systems 269 15.6 Neutral K-mesons 273 15.7 Problems 276 16 Time-independent perturbation for bound states 277 16.1 Basic formalism 277 16.2 Harmonic oscillator: perturbative vs. exact results 281 16.3 Second-order Stark effect 284 16.4 Degenerate states 287 16.5 Linear Stark effect 289 16.6 Problems 290 17 Time-dependent perturbation 293 17.1 Basic formalism 293 17.2 Harmonic perturbation and Fermi’s golden rule 296 17.3 Transitions into a group of states and scattering cross-section 299 17.4 Resonance and decay 303 17.5 Appendix to Chapter 17 310 17.6 Problems 315 18 Interaction of charged particles and radiation in perturbation theory 318 18.1 Electron in an electromagnetic field: the absorption cross-section 318 18.2 Photoelectric effect 323 18.3 Coulomb excitations of an atom 325 18.4 Ionization 328 18.5 Thomson, Rayleigh, and Raman scattering in second-order perturbation 331 18.6 Problems 339
Contents 19 Scattering in one dimension 342 19.1 Reflection and transmission coefficients 19.2 Infinite barrier 19.3 Finite barrier with infinite range 345 19.4 Rigid wall preceded by a potential well 348 19.5 19.6 Square-well potential and sonances Tunneling 19.7 Problems 356 20 Scattering in three dimensions-a formal theory 20. Formal solutions in terms of Green's function 20.2 Lippmann-Schwinger equation 20.3 Born approximation 204 205 g from a Yukawa potential Rutherford scattering 20.6 Charge distribution 6 207 Probability conservation and the optical theorem 36 20.8 209 Absorption 370 Relation between the T-matrix and the scattering amplitude 20.10 The S-matrix 374 20.11 Unitarity of the S-matrix and the relation between S and T 378 20.12 Properties of the T-matrix and the optical theorem(again) 380 20.13 Appendix to Chapter 20 383 20.14 Problems 384 21 Partial wave amplitudes and phase shifts 3 21. Scattering amplitude in terms of phase shifts 386 21.2 y.Ki.and T 39 213 Integral relations for x.K.and T 393 214 Wronskian 21.5 Calculation of phase shifts:some examples 21.6 Problems 405 22 Analytic structure of the S-matrix 407 S-matrix poles 22.2 Jost function formalism 413 22.3 Levinson's theorem 420 ))4 Explicit calculation of the Jost function for/=0 421 22.5 Integral representation of Fo(k 22.6 Problems 23 Poles of the Green's function and composite systems 427 23.1 Relation between the time-evolution operator and the Green's function 427 23.2 Stable and unstable states 429
xi Contents 19 Scattering in one dimension 342 19.1 Reflection and transmission coefficients 342 19.2 Infinite barrier 344 19.3 Finite barrier with infinite range 345 19.4 Rigid wall preceded by a potential well 348 19.5 Square-well potential and resonances 351 19.6 Tunneling 354 19.7 Problems 356 20 Scattering in three dimensions–aformal theory 358 20.1 Formal solutions in terms of Green’s function 358 20.2 Lippmann–Schwinger equation 360 20.3 Born approximation 363 20.4 Scattering from a Yukawa potential 364 20.5 Rutherford scattering 365 20.6 Charge distribution 366 20.7 Probability conservation and the optical theorem 367 20.8 Absorption 370 20.9 Relation between the T-matrix and the scattering amplitude 372 20.10 The S-matrix 374 20.11 Unitarity of the S-matrix and the relation between S and T 378 20.12 Properties of the T-matrix and the optical theorem (again) 382 20.13 Appendix to Chapter 20 383 20.14 Problems 384 21 Partial wave amplitudes and phase shifts 386 21.1 Scattering amplitude in terms of phase shifts 386 21.2 χl, Kl, and Tl 392 21.3 Integral relations for χl, Kl, and Tl 393 21.4 Wronskian 395 21.5 Calculation of phase shifts: some examples 400 21.6 Problems 405 22 Analytic structure of the S-matrix 407 22.1 S-matrix poles 407 22.2 Jost function formalism 413 22.3 Levinson’s theorem 420 22.4 Explicit calculation of the Jost function for l = 0 421 22.5 Integral representation of F0(k) 424 22.6 Problems 426 23 Poles of the Green’s function and composite systems 427 23.1 Relation between the time-evolution operator and the Green’s function 427 23.2 Stable and unstable states 429
Contents 23.3 Scattering amplitude and resonance 430 23.4 Complex poles 431 235 Two types of resonances 431 The reaction matrix Composite systems 238 Appendix to Chapter 23 447 24 Approximation methods for bound states and scattering 24.1 WKB approximation 58 24.2 Variational method 458 24.3 Eikonal approximation 461 24.4 Problems 466 25 Lagrangian method and Feynman path integrals 25.1 Euler-Lagrange equations 469 N oscillators and the continuum limit Feynman path integrals 25.4 Problems 478 26 Rotations and angular momentum 479 26.1 Rotation of coordinate axes 479 26.2 Scalar functions and orbital angular momentum 483 26.3 State vectors 485 26.4 Transformation of matrix elements a and representations of the rotation operator 487 26.5 Generators of infinitesimal rotations:their eigenstates and eigenvalues 26.6 Repres tions of J2 and J for j=andj=1 26.7 Spherical harmonics 26.8 Problems 501 27 Symmetry in quantum mechanics and symmetry groups 27.1 Rotational symmetry 27.2 Parity transformation 505 27.3 Time reversal 507 27.4 Symmetry groups 27.5 D/(R)forj=and j=1:examples of SO(3)and SU(2)groups 514 27.6 Problems 516 28 Addition of angular momenta 28.1 Combining eigenstates:simple examples 28.2 Clebsch-Gordan coefficients and their recursion relations 522 28.3 Combining spin %and orbital angular momentum 524 28.4 Appendix to Chapter 28 527 28.5 Problems 528
xii Contents 23.3 Scattering amplitude and resonance 430 23.4 Complex poles 431 23.5 Two types of resonances 431 23.6 The reaction matrix 432 23.7 Composite systems 442 23.8 Appendix to Chapter 23 447 24 Approximation methods for bound states and scattering 450 24.1 WKB approximation 450 24.2 Variational method 458 24.3 Eikonal approximation 461 24.4 Problems 466 25 Lagrangian method and Feynman path integrals 469 25.1 Euler–Lagrange equations 469 25.2 N oscillators and the continuum limit 471 25.3 Feynman path integrals 473 25.4 Problems 478 26 Rotations and angular momentum 479 26.1 Rotation of coordinate axes 479 26.2 Scalar functions and orbital angular momentum 483 26.3 State vectors 485 26.4 Transformation of matrix elements and representations of the rotation operator 487 26.5 Generators of infinitesimal rotations: their eigenstates and eigenvalues 489 26.6 Representations of J 2 and Ji for j = 1 2 and j = 1 494 26.7 Spherical harmonics 495 26.8 Problems 501 27 Symmetry in quantum mechanics and symmetry groups 502 27.1 Rotational symmetry 502 27.2 Parity transformation 505 27.3 Time reversal 507 27.4 Symmetry groups 511 27.5 Dj (R) for j = 1 2 and j = 1: examples of SO(3) and SU(2) groups 514 27.6 Problems 516 28 Addition of angular momenta 518 28.1 Combining eigenstates: simple examples 518 28.2 Clebsch–Gordan coefficients and their recursion relations 522 28.3 Combining spin ½ and orbital angular momentum l 524 28.4 Appendix to Chapter 28 527 28.5 Problems 528
Xiii Contents 29 Irreducible tensors and Wigner-Eckart theorem 529 29.1 Irreducible spherical tensors and their properties 529 29.2 The irreducible tensors:Yim(,and D/(x) 533 29.3 Wigner-Eckart theorem 536 29.4 Applications of the Wigner-Eckart theorem 538 29.5 Appendix to Chapter 29:SO(3),SU(2)groups and Young's tableau 541 29.6 Problems 548 30 Entangled states 549 30.1 Definition of an entangled state 549 30.2 The singlet state 551 30.3 Differentiating the two approaches 552 30.4 Bell's inequality 553 30.5 Problems 555 31 Special theory of relativity:Klein-Gordon and Maxwell's equations 556 31.1 Lorentz transformation 556 31.2 Contravariant and covariant vectors 557 31.3 An example of a covariant vector 560 31.4 Generalization to arbitrary tensors 561 31.5 Relativistically invariant equations 563 31.6 Appendix to Chapter 31 569 31.7 Problems 572 32 Klein-Gordon and Maxwell's equations 575 32.1 Covariant equations in quantum mechanics 575 32.2 Klein-Gordon equations:free particles 576 32.3 Normalization of matrix elements 578 32.4 Maxwell's equations 579 32.5 Propagators 581 32.6 Virtual particles 586 32.7 Static approximation 586 32.8 Interaction potential in nonrelativistic processes 587 32.9 Scattering interpreted as an exchange of virtual particles 589 32.10 Appendix to Chapter 32 593 33 The Dirac equation 597 33.1 Basic formalism 597 33.2 Standard representation and spinor solutions 600 33.3 Large and small components of u(p) 601 33.4 Probability conservation 605 33.5 Spin for the Dirac particle 607 34 Dirac equation in the presence of spherically symmetric potentials 611 34.1 Spin-orbit coupling 611
xiii Contents 29 Irreducible tensors and Wigner–Eckart theorem 529 29.1 Irreducible spherical tensors and their properties 529 29.2 The irreducible tensors: Ylm(θ, φ) and Dj (χ ) 533 29.3 Wigner–Eckart theorem 536 29.4 Applications of the Wigner–Eckart theorem 538 29.5 Appendix to Chapter 29: SO(3), SU(2) groups and Young’s tableau 541 29.6 Problems 548 30 Entangled states 549 30.1 Definition of an entangled state 549 30.2 The singlet state 551 30.3 Differentiating the two approaches 552 30.4 Bell’s inequality 553 30.5 Problems 555 31 Special theory of relativity: Klein–Gordon and Maxwell’s equations 556 31.1 Lorentz transformation 556 31.2 Contravariant and covariant vectors 557 31.3 An example of a covariant vector 560 31.4 Generalization to arbitrary tensors 561 31.5 Relativistically invariant equations 563 31.6 Appendix to Chapter 31 569 31.7 Problems 572 32 Klein–Gordon and Maxwell’s equations 575 32.1 Covariant equations in quantum mechanics 575 32.2 Klein–Gordon equations: free particles 576 32.3 Normalization of matrix elements 578 32.4 Maxwell’s equations 579 32.5 Propagators 581 32.6 Virtual particles 586 32.7 Static approximation 586 32.8 Interaction potential in nonrelativistic processes 587 32.9 Scattering interpreted as an exchange of virtual particles 589 32.10 Appendix to Chapter 32 593 33 The Dirac equation 597 33.1 Basic formalism 597 33.2 Standard representation and spinor solutions 600 33.3 Large and small components of u(p) 601 33.4 Probability conservation 605 33.5 Spin ½ for the Dirac particle 607 34 Dirac equation in the presence of spherically symmetric potentials 611 34.1 Spin–orbit coupling 611
XIV Contents 34.2 K-operator for the spherically symmetric potentials 613 34.3 Hydrogen atom 616 34.4 Radial Dirac equation 618 34.5 Hydrogen atom states 623 34.6 Hydrogen atom wavefunction 624 34.7 Appendix to Chapter 34 626 35 Dirac equation in a relativistically invariant form 631 35.1 Covariant Dirac equation 631 35.2 Properties of the y-matrices 632 35.3 Charge-current conservation in a covariant form 633 35.4 Spinor solutions:ur(p)and v(p) 635 35.5 Normalization and completeness condition for ur(p)and vr(p) 636 35.6 Gordon decomposition 640 35.7 Lorentz transformation of the Dirac equation 642 35.8 Appendix to Chapter 35 644 36 Interaction of a Dirac particle with an electromagnetic field 647 36.1 Charged particle Hamiltonian 647 36.2 Deriving the equation another way 650 36.3 Gordon decomposition and electromagnetic current 651 36.4 Dirac equation with EM field and comparison with the Klein-Gordon equation 653 36.5 Propagators:the Dirac propagator 655 36.6 Scattering 657 36.7 Appendix to Chapter 36 661 37 Multiparticle systems and second quantization 663 37.1 Wavefunctions for identical particles 663 37.2 Occupation number space and ladder operators 664 37.3 Creation and destruction operators 666 37.4 Writing single-particle relations in multiparticle language:the operators,N,H,and P 670 37.5 Matrix elements of a potential 671 37.6 Free fields and continuous variables 672 37.7 Klein-Gordon/scalar field 674 37.8 Complex scalar field 678 37.9 Dirac field 680 37.10 Maxwell field 683 37.11 Lorentz covariance for Maxwell field 687 37.12 Propagators and time-ordered products 688 37.13 Canonical quantization 690 37.14 Casimir effect 693 37.15 Problems 697
xiv Contents 34.2 K-operator for the spherically symmetric potentials 613 34.3 Hydrogen atom 616 34.4 Radial Dirac equation 618 34.5 Hydrogen atom states 623 34.6 Hydrogen atom wavefunction 624 34.7 Appendix to Chapter 34 626 35 Dirac equation in a relativistically invariant form 631 35.1 Covariant Dirac equation 631 35.2 Properties of the γ -matrices 632 35.3 Charge–current conservation in a covariant form 633 35.4 Spinor solutions: ur(p) and vr(p) 635 35.5 Normalization and completeness condition for ur(p) and vr(p) 636 35.6 Gordon decomposition 640 35.7 Lorentz transformation of the Dirac equation 642 35.8 Appendix to Chapter 35 644 36 Interaction of a Dirac particle with an electromagnetic field 647 36.1 Charged particle Hamiltonian 647 36.2 Deriving the equation another way 650 36.3 Gordon decomposition and electromagnetic current 651 36.4 Dirac equation with EM field and comparison with the Klein–Gordon equation 653 36.5 Propagators: the Dirac propagator 655 36.6 Scattering 657 36.7 Appendix to Chapter 36 661 37 Multiparticle systems and second quantization 663 37.1 Wavefunctions for identical particles 663 37.2 Occupation number space and ladder operators 664 37.3 Creation and destruction operators 666 37.4 Writing single-particle relations in multiparticle language: the operators, N, H, and P 670 37.5 Matrix elements of a potential 671 37.6 Free fields and continuous variables 672 37.7 Klein–Gordon/scalar field 674 37.8 Complex scalar field 678 37.9 Dirac field 680 37.10 Maxwell field 683 37.11 Lorentz covariance for Maxwell field 687 37.12 Propagators and time-ordered products 688 37.13 Canonical quantization 690 37.14 Casimir effect 693 37.15 Problems 697