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