QUANTUM MECHANICS Gennaro Auletta Mauro Fortunato Giorgio Parisi :e www.cambridge.org/9780521869638
Quantum Mechanics The important changes quantum mechanics has undergone in recent years are reflected in hisneaprach for A strong narrative and over 300 worked problems lead the student from experiment. through general principles of the theory.to modern applications.Stepping through results allow students to.with bas antum mechan ics,the book moves on to more advanced theory,followed by applications,perturbation methods and special fields,and ending with new developments in the field.Historical, mathematical,and philosophical boxes guide the student through the theory.Unique to this textbook are chapters on measurement and quantum optics.both at the forefront of current research.Advanced undergraduate and graduate students will benefit from this new perspective on the fundamental physical paradigm and its applications. Online resources including solutions to selected problems and 200 figures,with color versions of some figures,are available at www.cambridge.org/Auletta. Gennaro Auletta is Scientific Director of Science and Philosophy at the Pontifical Grego- rian University,Rome.His main areas of research are quantum mechanics,logic,cognitive sciences,information theory,and applications to biological systems. Mauro Fortunato is a Structurer at Cassa depositieprestiti S.p.A..Rome.He is involved in financial engineering,applying mathematical methods of quantum physics to the pricing of complex financial derivatives and the construction of structured products. Giorgio Parisi is Professor of Quantum Theories at the University of Rome"La Sapienza. He has won several prizes,notably the Boltzmann Medal,the Dirac Medal and Prize,and the Daniel Heineman prize.His main research activity deals with elementary particles. very large scale sim behavior
Quantum Mechanics The important changes quantum mechanics has undergone in recent years are reflected in this new approach for students. A strong narrative and over 300 worked problems lead the student from experiment, through general principles of the theory, to modern applications. Stepping through results allows students to gain a thorough understanding. Starting with basic quantum mechanics, the book moves on to more advanced theory, followed by applications, perturbation methods and special fields, and ending with new developments in the field. Historical, mathematical, and philosophical boxes guide the student through the theory. Unique to this textbook are chapters on measurement and quantum optics, both at the forefront of current research. Advanced undergraduate and graduate students will benefit from this new perspective on the fundamental physical paradigm and its applications. Online resources including solutions to selected problems and 200 figures, with color versions of some figures, are available at www.cambridge.org/Auletta. Gennaro Auletta is Scientific Director of Science and Philosophy at the Pontifical Gregorian University, Rome. His main areas of research are quantum mechanics, logic, cognitive sciences, information theory, and applications to biological systems. Mauro Fortunato is a Structurer at Cassa depositi e prestiti S.p.A., Rome. He is involved in financial engineering, applying mathematical methods of quantum physics to the pricing of complex financial derivatives and the construction of structured products. Giorgio Parisi is Professor of Quantum Theories at the University of Rome “La Sapienza.” He has won several prizes, notably the Boltzmann Medal, the Dirac Medal and Prize, and the Daniel Heineman prize. His main research activity deals with elementary particles, theory of phase transitions and statistical mechanics, disordered systems, computers and very large scale simulations, non-equilibrium statistical physics, optimization, and animal behavior
Contents List of figures page xi List of tables List of definitions,principles,etc. xvi List of boxes List of symbols List of abbreviations Introduction Part I Basic features of quantum mechanics 1 From classical mechanics to quantum mechanics 7 1.1 Review of the foundations of classical mechanics 7 10 An interferometry experiment and its consequences 1.3 State as vecto Quantum probability 2028 1.5 The historical need of a new mechanics Summary Problems Further reading 2 Quantum observables and states 3 2.1 Basic features of quantum observables 2.2 Wave function and basic observables 68 2.3 Uncertainty relation 2.4 Quantum algebra and quantum logic Summary 2% Problems Further reading 3 Quantum dynamics 100 3.1 The Schrodinger equation 101 3.2 Properties of the Schrodinger eq ation 107 3.3 One-dimensional free particle in a box 35 Unitary transformations 117
Contents List of figures page xi List of tables xvii List of definitions, principles, etc. xviii List of boxes xx List of symbols xxi List of abbreviations xxxii Introduction 1 Part I Basic features of quantum mechanics 1 From classical mechanics to quantum mechanics 7 1.1 Review of the foundations of classical mechanics 7 1.2 An interferometry experiment and its consequences 12 1.3 State as vector 20 1.4 Quantum probability 28 1.5 The historical need of a new mechanics 31 Summary 40 Problems 41 Further reading 42 2 Quantum observables and states 43 2.1 Basic features of quantum observables 43 2.2 Wave function and basic observables 68 2.3 Uncertainty relation 82 2.4 Quantum algebra and quantum logic 92 Summary 96 Problems 97 Further reading 99 3 Quantum dynamics 100 3.1 The Schrödinger equation 101 3.2 Properties of the Schrödinger equation 107 3.3 Schrödinger equation and Galilei transformations 111 3.4 One-dimensional free particle in a box 113 3.5 Unitary transformations 117
vi Contents 3.6 Different pictures 125 37 Time derivatives and the Ehrenfest theorem 129 8 Energy-time uncertainty relation 39 Towards a time operator 01 Summary 138 Further reading 4 Examples of quantum dynamics 4.1 Finite potential wells 4.2 Potential barrier 4.3 Tunneling 45 4.4 Harmonic oscillator 154 4.5 Quantum particles in simple field Summarv Problems 170 5 Density matrix 5.1 Basic formalism 52 Expectation values and measurement outcomes Tim evolution and density matrix Statistical properties of quantum mechanics 5.5 Compound systems 5.6 Pure-and mixed-state representation mmary Problems Further reading 190 Part ll More advanced topics 6 Angular momentum and spin 6 Orbital angular momentum 6 Speciale amples Spin 0 64 Composition of angular momenta and total angular momentum 226 6.5 Angular momentum and angle 239 Summar Problems Further reading 244 7 Identical particles 245 7.1 Statistics and quantum mechanics 245 7.2 Wave function and symmetry 247 7.3 Spin and statistics 249
vi Contents 3.6 Different pictures 125 3.7 Time derivatives and the Ehrenfest theorem 129 3.8 Energy–time uncertainty relation 130 3.9 Towards a time operator 135 Summary 138 Problems 139 Further reading 140 4 Examples of quantum dynamics 141 4.1 Finite potential wells 141 4.2 Potential barrier 145 4.3 Tunneling 150 4.4 Harmonic oscillator 154 4.5 Quantum particles in simple fields 165 Summary 169 Problems 170 5 Density matrix 174 5.1 Basic formalism 174 5.2 Expectation values and measurement outcomes 177 5.3 Time evolution and density matrix 179 5.4 Statistical properties of quantum mechanics 180 5.5 Compound systems 181 5.6 Pure- and mixed-state representation 187 Summary 188 Problems 189 Further reading 190 Part II More advanced topics 6 Angular momentum and spin 193 6.1 Orbital angular momentum 193 6.2 Special examples 207 6.3 Spin 217 6.4 Composition of angular momenta and total angular momentum 226 6.5 Angular momentum and angle 239 Summary 241 Problems 242 Further reading 244 7 Identical particles 245 7.1 Statistics and quantum mechanics 245 7.2 Wave function and symmetry 247 7.3 Spin and statistics 249
Contents 7.4 Exchange interaction 7.5 Two recent applications Summary 257 Problem Further reading 派 8 Symmetries and conservation laws 259 81 Quantum transformations and symmetries 8.2 Continuous symmetries 83 Discrete symmetries 8.4 A brief introduction to group theory 267 Summary Problems Further reading 276 9 The measurement problem in quantum mechanics 27 9.1 Statement of the problem 9.2 A brief history of the problem 284 93 Schrodinger cats 9.4 Decoherence 9.5 Reversibility/irreversibility 9.6 Interaction-free measurement 315 97 Delayed-choice experim 98 Quantum Zeno effect 9. Conditional measurements or postselection 02 9.10 Positive operator valued measure 9.11 Quantum non-demolition measurements 9.12 Decision and estimation theory Summary 349 Problems 351 Further reading 353 Part Ill Matter and light 10 Perturbations and approximation methods 10.1 Stationary perturbation theory 357 10.2 Time-dependent perturbation theory 36 10.3 Adiabatic theorem 10.4 The variational methoc 10.5 Classical limit 372 106 Semiclassical limit and WKB approximation 10.7 Scattering theory 10.8 Path integrals Summary 398
vii Contents 7.4 Exchange interaction 254 7.5 Two recent applications 255 Summary 257 Problems 257 Further reading 258 8 Symmetries and conservation laws 259 8.1 Quantum transformations and symmetries 259 8.2 Continuous symmetries 264 8.3 Discrete symmetries 266 8.4 A brief introduction to group theory 267 Summary 275 Problems 275 Further reading 276 9 The measurement problem in quantum mechanics 277 9.1 Statement of the problem 278 9.2 A brief history of the problem 284 9.3 Schrödinger cats 291 9.4 Decoherence 297 9.5 Reversibility/irreversibility 308 9.6 Interaction-free measurement 315 9.7 Delayed-choice experiments 320 9.8 Quantum Zeno effect 322 9.9 Conditional measurements or postselection 325 9.10 Positive operator valued measure 327 9.11 Quantum non-demolition measurements 335 9.12 Decision and estimation theory 341 Summary 349 Problems 351 Further reading 353 Part III Matter and light 10 Perturbations and approximation methods 357 10.1 Stationary perturbation theory 357 10.2 Time-dependent perturbation theory 366 10.3 Adiabatic theorem 369 10.4 The variational method 371 10.5 Classical limit 372 10.6 Semiclassical limit and WKB approximation 378 10.7 Scattering theory 384 10.8 Path integrals 389 Summary 398
viⅷi Contents Problems 399 Further reading 11 Hydrogen and helium atoms 401 11.1 Introduction 401 11.2 Quantum theory of the hydrogen atom 403 113 Atom and magnetic field 413 11.4 Relativistic corrections 11.5 Helium atom 11.6 Many-electron effect Summary Problems 437 Further reading 438 12 Hydrogen molecular ion 439 12.1 The molecular problem 439 12.2 Born-Oppenheimer approximation 440 12.3 Vibrational and rotational degrees of freedom 12.4 The Morse potential 12.5 Chemical bonds and further approximations 44g Summary Problems Further reading 454 13 Quantum optics 13.1 Quantization of the electromagnetic field 57 13.2 Thermodynamic equilibrium of the radiation field 462 133 Phase mber ertainty relation 13.4 Special states of the electromagnetic field 465 13.5 Quasi-probability distributions 13.6 Quantum-optical coherence 481 13.7 Atom-field interaction 13.8 Geometric phase 13.9 The Casimir effect 501 Summary 506 Problems Further reading Part IV Quantum information:state and correlations 14 Quantum theory of open systems 513 14.1 General considerations 514 14.2 The master equation 516
viii Contents Problems 399 Further reading 399 11 Hydrogen and helium atoms 401 11.1 Introduction 401 11.2 Quantum theory of the hydrogen atom 403 11.3 Atom and magnetic field 413 11.4 Relativistic corrections 423 11.5 Helium atom 426 11.6 Many-electron effects 431 Summary 436 Problems 437 Further reading 438 12 Hydrogen molecular ion 439 12.1 The molecular problem 439 12.2 Born–Oppenheimer approximation 440 12.3 Vibrational and rotational degrees of freedom 443 12.4 The Morse potential 447 12.5 Chemical bonds and further approximations 449 Summary 453 Problems 453 Further reading 454 13 Quantum optics 455 13.1 Quantization of the electromagnetic field 457 13.2 Thermodynamic equilibrium of the radiation field 462 13.3 Phase–number uncertainty relation 463 13.4 Special states of the electromagnetic field 465 13.5 Quasi-probability distributions 474 13.6 Quantum-optical coherence 481 13.7 Atom–field interaction 484 13.8 Geometric phase 497 13.9 The Casimir effect 501 Summary 506 Problems 507 Further reading 509 Part IV Quantum information: state and correlations 14 Quantum theory of open systems 513 14.1 General considerations 514 14.2 The master equation 516
Contents 14.3 A formal generalization 14.4 Quantum jumps and quantum trajectories 14.5 Quantum optics and Schrodinger cats Summary Problems 9 Further reading 542 15 State measurement in quantum mechanics 15.1 Protective measurement of the state 54 15.2 Quantum cloning and unitarity violation 15.3 Measurement and reversibility Quantum state reconstruction 15.5 The nature of quantum states Summary Problem Further reading 16 Entanglement:non-separability 567 16.1 EPR 16.2 Bohm's version of the EPR state 16.3 HV theories 164 Bell's contribution 16.5 Experimental test 16.6 Bell inequalities with homodyne detection 167 Bell theorem without inequalities 16.8 What is quantum non-locality? 16.9 Further developments about inequalities 16.10 Conclusion Summary Problems Further reading 627 17 Entanglement:quantum information and computation Information and entropy 17.2 Entanglement and information 17.3 Measurement and information 17.4 Qubits 17.5 Teleportation 创 17.6 Quantum cryptography 646 17.7 Elements of quantum computation 650 17.8 Quantum algorithms and d error correctior Summary
ix Contents 14.3 A formal generalization 523 14.4 Quantum jumps and quantum trajectories 528 14.5 Quantum optics and Schrödinger cats 533 Summary 540 Problems 541 Further reading 542 15 State measurement in quantum mechanics 544 15.1 Protective measurement of the state 544 15.2 Quantum cloning and unitarity violation 548 15.3 Measurement and reversibility 550 15.4 Quantum state reconstruction 554 15.5 The nature of quantum states 564 Summary 565 Problems 565 Further reading 566 16 Entanglement: non-separability 567 16.1 EPR 568 16.2 Bohm’s version of the EPR state 573 16.3 HV theories 577 16.4 Bell’s contribution 582 16.5 Experimental tests 595 16.6 Bell inequalities with homodyne detection 605 16.7 Bell theorem without inequalities 613 16.8 What is quantum non-locality? 619 16.9 Further developments about inequalities 623 16.10 Conclusion 625 Summary 625 Problems 626 Further reading 627 17 Entanglement: quantum information and computation 628 17.1 Information and entropy 628 17.2 Entanglement and information 631 17.3 Measurement and information 639 17.4 Qubits 642 17.5 Teleportation 643 17.6 Quantum cryptography 646 17.7 Elements of quantum computation 650 17.8 Quantum algorithms and error correction 659 Summary 671
Contents Problems 672 Further reading 673 Bibliography 674 Author index 710 Subject index 716
x Contents Problems 672 Further reading 673 Bibliography 674 Author index 710 Subject index 716
Figures 1.1 Graphical representation of the Liouville theorem page 12 1.2 Photoelectric effect 13 1.3 Mach-Zender interferometer 15 1.4 The Michelson-Morley interferometer 16 1.5 Interferometer for detecting gravitational waves 17 1.6 Interference in the Mach-Zender interferometer 17 1.7 Results of the experiment performed by Grangier,Roger,and Aspect 18 1.8 Oscillation of electric and magnetic fields 20 1.9 Polarization of classical light 21 1.10 Decomposition of an arbitrary vector la) 22 1.11 Poincare sphere representation of states 28 1.12 Mach-Zender interferometer with the lower path blocked by the screen S 30 1.13 Black-body radiation intensity corresponding to the formula of Rayleigh- Jeans (1),Planck (2),and Wien (3) 32 1.14 Planck's radiation curves in logarithmic scale for the temperatures of liquid nitrogen,melting ice,boiling water,melting aluminium,and the solar surface 33 1.15 Compton effect 34 1.16 Dulong-Petit's,Einstein's,and Debye's predictions for specific heat 36 1.17 Lyman series for ionized helium 37 1.18 The Stern-Gerlach Experiment 39 1.19 Momentum conservation in the Compton effect 41 2.1 Polarization beam splitter 44 2.2 Change of basis 52 2.3 Filters 62 2.4 Two sequences of two rotations of a book 65 2.5 Probability distributions of position and momentum for a momentum eigenfunction 85 2.6 Probability distributions of position and momentum for a position eigenfunction 86 2.7 Time evolution of a classical degree of freedom in phase space and graphical representation of the uncertainty relation 88 2.8 Inverse proportionality between momentum and position uncertainties 89 2.9 Smooth complementarity between wave and particle 90 2.10 Illustration of the distributive law 93 2.11 Proposed interferometry and resulting non-Boolean algebra and Boolean subalgebras 94
Figures 1.1 Graphical representation of the Liouville theorem page 12 1.2 Photoelectric effect 13 1.3 Mach–Zender interferometer 15 1.4 The Michelson–Morley interferometer 16 1.5 Interferometer for detecting gravitational waves 17 1.6 Interference in the Mach–Zender interferometer 17 1.7 Results of the experiment performed by Grangier, Roger, and Aspect 18 1.8 Oscillation of electric and magnetic fields 20 1.9 Polarization of classical light 21 1.10 Decomposition of an arbitrary vector |a 22 1.11 Poincaré sphere representation of states 28 1.12 Mach–Zender interferometer with the lower path blocked by the screen S 30 1.13 Black-body radiation intensity corresponding to the formula of Rayleigh– Jeans (1), Planck (2), and Wien (3) 32 1.14 Planck’s radiation curves in logarithmic scale for the temperatures of liquid nitrogen, melting ice, boiling water, melting aluminium, and the solar surface 33 1.15 Compton effect 34 1.16 Dulong–Petit’s, Einstein’s, and Debye’s predictions for specific heat 36 1.17 Lyman series for ionized helium 37 1.18 The Stern–Gerlach Experiment 39 1.19 Momentum conservation in the Compton effect 41 2.1 Polarization beam splitter 44 2.2 Change of basis 52 2.3 Filters 62 2.4 Two sequences of two rotations of a book 65 2.5 Probability distributions of position and momentum for a momentum eigenfunction 85 2.6 Probability distributions of position and momentum for a position eigenfunction 86 2.7 Time evolution of a classical degree of freedom in phase space and graphical representation of the uncertainty relation 88 2.8 Inverse proportionality between momentum and position uncertainties 89 2.9 Smooth complementarity between wave and particle 90 2.10 Illustration of the distributive law 93 2.11 Proposed interferometry and resulting non-Boolean algebra and Boolean subalgebras 94
Figures 2.12 hasse diagrams of several boolean and non-Boolean algebras 3.1 Positive potential vanishing at infinity Potential function tending to finite values asx →士 0 Potential well 111 3.4 Relation between two different inertial reference frames R and R'under Galilei transformations 3 3.5 Particle in a box of dimensiona Energy levels of a particle in a one-dimensional box 3.7 First three energy eigenfunctions for a one-dimensional particle confined in a box of dimension a 115 3.8 Beam Splitters as unitary operators 119 3.9 proiector as a residue of the closed contour in a complex plane 124 3.10 A graphical representation of the apparatus proposed by Bohr 4.1 Schematic and tric one-dir otential wells 4.2 Solution of Eq.(4.8) 143 4.3 Wave functions and probability densities for the first three eigenfunctions for the symmetric finite-well potential 1 4 Stepwise continuity Potential barrier 64 4.6 Closed surface used to compute the flux of J 4 Delta potential barrier 149 4.8 Classical tuming points and quantum tunneling 4. Tunneling of o-particles 4.10 Carbon atoms shown by scanning tunneling microscopy 153 411 Potential and energy levels of the harmonic oscillator 4.12 Eigenfunctions for the one-dimensional harmonic oscillator 162 4.13 Potential energy corresponding to a particle in a uniform field 1 4.14 Triangular well 167 4.15 A quantum particle with energy E enc ounters a potential step of heigh Vo<E 171 4.16 Rectangular potential barrier with finite width a 171 5 Representation of pure and mixed states on a sphere 187 61 Angular momentum of a classical particle 6. Levi-Civita tensor Relationship between rectangular and spherical coordinates 200 s-and p-states 206 6 Rigid rotato 6.6 Energy levels and transition frequencies for a rigid rotator 67 Cylindrical coordinates 212 Energy levels of the thre -dimensional ha monic oscillator 6.9 Levels in the spectrum of hydrogen atom 6.10 An electric dipole with charges +e and -e in an electric field gradient 218 6.1m Scheme of spin superposition in single-crystal neutron interferometry 223 6.12 Lande vectorial model for angul 227
xii Figures 2.12 Hasse diagrams of several Boolean and non-Boolean algebras 95 3.1 Positive potential vanishing at infinity 109 3.2 Potential function tending to finite values as x → ±∞ 110 3.3 Potential well 111 3.4 Relation between two different inertial reference frames R and R under Galilei transformations 112 3.5 Particle in a box of dimension a 113 3.6 Energy levels of a particle in a one-dimensional box 115 3.7 First three energy eigenfunctions for a one-dimensional particle confined in a box of dimension a 115 3.8 Beam Splitters as unitary operators 119 3.9 Projector as a residue of the closed contour in a complex plane 124 3.10 A graphical representation of the apparatus proposed by Bohr 133 4.1 Schematic and asymmetric one-dimensional potential wells 142 4.2 Solution of Eq. (4.8) 143 4.3 Wave functions and probability densities for the first three eigenfunctions for the symmetric finite-well potential 144 4.4 Stepwise continuity 145 4.5 Potential barrier 147 4.6 Closed surface used to compute the flux of J 148 4.7 Delta potential barrier 149 4.8 Classical turning points and quantum tunneling 151 4.9 Tunneling of α-particles 152 4.10 Carbon atoms shown by scanning tunneling microscopy 153 4.11 Potential and energy levels of the harmonic oscillator 154 4.12 Eigenfunctions for the one-dimensional harmonic oscillator 162 4.13 Potential energy corresponding to a particle in a uniform field 165 4.14 Triangular well 167 4.15 A quantum particle with energy E encounters a potential step of height V0 < E 171 4.16 Rectangular potential barrier with finite width a 171 5.1 Representation of pure and mixed states on a sphere 187 6.1 Angular momentum of a classical particle 194 6.2 Levi–Civita tensor 195 6.3 Relationship between rectangular and spherical coordinates 200 6.4 s- and p-states 206 6.5 Rigid rotator 207 6.6 Energy levels and transition frequencies for a rigid rotator 209 6.7 Cylindrical coordinates 212 6.8 Energy levels of the three-dimensional harmonic oscillator 215 6.9 Levels in the spectrum of hydrogen atom 218 6.10 An electric dipole with charges +e and −e in an electric field gradient 218 6.11 Scheme of spin superposition in single-crystal neutron interferometry 223 6.12 Landé vectorial model for angular momentum 227