Quantum Mechanics

Description

Gives a fresh and modern approach to the field. It is a textbook on the principles of the theory, its mathematical framework and its first applications. It constantly refers to modern and practical developments, tunneling microscopy, quantum information, Bell inequalities, quantum cryptography, Bose-Einstein condensation and quantum astrophysics. The book also contains 92 exercises with their solutions. Cover page Title page Copyright page Preface Contents Physical Constants 1. Quantum Phenomena 1.1 The Franck and Hertz Experiment 1.2 Interference of Matter Waves 1.2.1 The Young Double-Slit Experiment 1.2.2 Interference of Atoms in a Double-Slit Experiment 1.2.3 Probabilistic Aspect of Quantum Interference 1.3 The Experiment of Davisson and Germer 1.3.1 Diffraction of X Rays by a Crystal 1.3.2 Electron Diffraction 1.4 Summary of a Few Important Ideas Further Reading Exercises 2. The Wave Function and the Schr¨odinger Equation 2.1 The Wave Function 2.1.1 Description of the State of a Particle 2.1.2 Position Measurement of the Particle 2.2 Interference and the Superposition Principle 2.2.1 De Broglie Waves 2.2.2 The Superposition Principle 2.2.3 The Wave Equation in Vacuum 2.3 Free Wave Packets 2.3.1 Definition of a Wave Packet 2.3.2 Fourier Transformation 2.3.3 Structure of the Wave Packet 2.3.4 Propagation of a Wave Packet: the Group Velocity 2.3.5 Propagation of a Wave Packet 2.4 Momentum Measurements and Uncertainty Relations 2.4.1 The Momentum Probability Distribution 2.4.2 Heisenberg Uncertainty Relations 2.5 The Schr¨odinger Equation 2.5.1 Equation of Motion 2.5.2 Particle in a Potential: Uncertainty Relations 2.5.3 Stability of Matter 2.6 Momentum Measurement in a Time-of-Flight Experiment Further Reading Exercises 3. Physical Quantities and Measurements 3.1 Measurements in Quantum Mechanics 3.1.1 The Measurement Procedure 3.1.2 Experimental Facts 3.1.3 Reinterpretation of Position and Momentum Measurements 3.2 Physical Quantities and Observables 3.2.1 Expectation Value of a Physical Quantity 3.2.2 Position and Momentum Observables 3.2.3 Other Observables: the Correspondence Principle 3.2.4 Commutation of Observables 3.3 Possible Results of a Measurement 3.3.1 Eigenfunctions and Eigenvalues of an Observable 3.3.2 Results of a Measurement and Reduction of the Wave Packet 3.3.3 Individual Versus Multiple Measurements 3.3.4 Relation to Heisenberg Uncertainty Relations 3.3.5 Measurement and Coherence of Quantum Mechanics 3.4 Energy Eigenfunctions and Stationary States 3.4.1 Isolated Systems: Stationary States 3.4.2 Energy Eigenstates and Time Evolution 3.5 The Probability Current 3.6 Crossing Potential Barriers 3.6.1 The Eigenstates of the Hamiltonian 3.6.2 Boundary Conditions at the Discontinuities of the Potential 3.6.3 Reflection and Transmission on a Potential Step 3.6.4 Potential Barrier and Tunnel Effect 3.7 Summary of Chapters 2 and 3 Further Reading Exercises 4. Quantization of Energy in Simple Systems 4.1 Bound States and Scattering States 4.1.1 Stationary States of the Schrodinger Equation 4.1.2 Bound States 4.1.3 Scattering States 4.2 The One Dimensional Harmonic Oscillator 4.2.1 Definition and Classical Motion 4.2.2 The Quantum Harmonic Oscillator 4.2.3 Examples 4.3 Square-Well Potentials 4.3.1 Relevance of Square Potentials 4.3.2 Bound States in a One-Dimensional Square-Well Potential 4.3.4 Particle in a Three-Dimensional Box 4.4 Periodic Boundary Conditions 4.4.1 A One-Dimensional Example 4.4.2 Extension to Three Dimensions 4.4.3 Introduction of Phase Space 4.5 The Double Well Problem and the Ammonia Molecule 4.5.1 Model of the NH3 Molecule 4.5.2 Wave Functions 4.5.3 Energy Levels 4.5.4 The Tunnel Effect and the Inversion Phenomenon 4.6 Other Applications of the Double Well Further Reading Exercises 5. Principles of Quantum Mechanics 5.1 Hilbert Space 5.1.1 The State Vector 5.1.2 Scalar Products and the Dirac Notations 5.1.3 Examples 5.1.4 Bras and Kets, Brackets 5.2 Operators in Hilbert Space 5.2.1 Matrix Elements of an Operator 5.2.2 Adjoint Operators and Hermitian Operators 5.2.3 Eigenvectors and Eigenvalues 5.2.4 Summary: Syntax Rules in Dirac’s Formalism 5.3 The Spectral Theorem 5.3.1 Hilbertian Bases 5.3.2 Projectors and Closure Relation 5.3.3 The Spectral Decomposition of an Operator 5.3.4 Matrix Representations 5.4 Measurement of Physical Quantities 5.5 The Principles of Quantum Mechanics 5.6 Structure of Hilbert Space 5.6.1 Tensor Products of Spaces 5.6.2 The Appropriate Hilbert Space 5.6.3 Properties of Tensor Products 5.6.4 Operators in a Tensor Product Space 5.6.5 Simple Examples 5.7 Reversible Evolution and the Measurement Process Further Reading Exercises 6. Two-State Systems, Principle of the Maser 6.1 Two-Dimensional Hilbert Space 6.2 A Familiar Example: the Polarization of Light 6.2.1 Polarization States of a Photon 6.2.2 Measurement of Photon Polarizations 6.2.3 Successive Measurements and “Quantum Logic” 6.3 The Model of the Ammonia Molecule 6.3.1 Restriction to a Two-Dimensional Hilbert Space 6.3.2 The Basis {|S, |A} 6.3.3 The Basis {|R, |L} 6.4 The Ammonia Molecule in an Electric Field 6.4.1 The Coupling of NH3 to an Electric Field 6.4.2 Energy Levels in a Fixed Electric Field 6.4.3 Force Exerted on the Molecule by an Inhomogeneous Field 6.5 Oscillating Fields and Stimulated Emission 6.6 Principle and Applications of Masers 6.6.1 Amplifier 6.6.2 Oscillator 6.6.3 Atomic Clocks Further Reading Exercises 7. Commutation of Observables 7.1 Commutation Relations 7.2 Uncertainty Relations 7.3 Ehrenfest’s Theorem 7.3.1 Evolution of the Expectation Value of an Observable 7.3.2 Particle in a Potential V (r) 7.3.3 Constants of Motion 7.4 Commuting Observables 7.4.1 Existence of a Common Eigenbasis for Commuting Observables 7.4.2 Complete Set of Commuting Observables (CSCO) 7.4.3 Completely Prepared Quantum State 7.4.4 Symmetries of the Hamiltonian and Search of Its Eigenstates 7.5 Algebraic Solution of the Harmonic-Oscillator Problem 7.5.1 Reduced Variables 7.5.2 Annihilation and Creation Operators ˆa and ˆa† 7.5.3 Eigenvalues of the Number Operator ˆ N 7.5.4 Eigenstates Further Reading Exercises 8. The Stern–Gerlach Experiment 8.1 Principle of the Experiment 8.1.1 Classical Analysis 8.1.2 Experimental Results 8.2 The Quantum Description of the Problem 8.3 The Observables ˆµx and ˆµy 8.4 Discussion 8.4.1 Incompatibility of Measurements Along Di?erent Axes 8.4.2 Classical Versus Quantum Analysis 8.4.3 Measurement Along an Arbitrary Axis 8.5 Complete Description of the Atom 8.5.1 Hilbert Space 8.5.2 Representation of States and Observables 8.5.3 Energy of the Atom in a Magnetic Field 8.6 Evolution of the Atom in a Magnetic Field 8.6.1 Schr¨odinger Equation 8.6.2 Evolution in a Uniform Magnetic Field 8.6.3 Explanation of the Stern–Gerlach Experiment 8.7 Conclusion Further Reading Exercises 9. Approximation Methods 9.1 Perturbation Theory 9.1.1 Definition of the Problem 9.1.2 Power Expansion of Energies and Eigenstates 9.1.3 First-Order Perturbation in the Nondegenerate Case 9.1.4 First-Order Perturbation in the Degenerate Case 9.1.5 First-Order Perturbation to the Eigenstates 9.1.6 Second-Order Perturbation to the Energy Levels 9.1.7 Examples 9.1.8 Remarks on the Convergence of Perturbation Theory 9.2 The Variational Method 9.2.1 The Ground State 9.2.2 Other Levels 9.2.3 Examples of Applications of the Variational Method Exercises 10. Angular Momentum 10.1 Orbital Angular Momentum and the Commutation Relations 10.2 Eigenvalues of Angular Momentum 10.2.1 The Observables ˆ J2 and ˆ Jz and the Basis States |j,m 10.2.2 The Operators ˆ J± 10.2.3 Action of ˆ J± on the States |j,m 10.2.4 Quantization of j and m 10.2.5 Measurement of ˆ Jx and ˆ Jy 10.3 Orbital Angular Momentum 10.3.1 The Quantum Numbers m and are Integers 10.3.2 Spherical Coordinates 10.3.3 Eigenfunctions of ˆL2 and ˆLz: the Spherical Harmonics 10.3.4 Examples of Spherical Harmonics 10.3.5 Example: Rotational Energy of a Diatomic Molecule 10.4 Angular Momentum and Magnetic Moment 10.4.1 Orbital Angular Momentum and Magnetic Moment 10.4.2 Generalization to Other Angular Momenta 10.4.3 What Should we Think about Half-Integer Values of j and m ? Further Reading Exercises 11. Initial Description of Atoms 11.1 The Two-Body Problem; Relative Motion 11.2 Motion in a Central Potential 11.2.1 Spherical Coordinates 11.2.2 Eigenfunctions Common to ˆ H, ˆL2 and ˆLz 11.3 The Hydrogen Atom 11.3.1 Orders of Magnitude: Appropriate Units in Atomic Physics 11.3.2 The Dimensionless Radial Equation 11.3.3 Spectrum of Hydrogen 11.3.4 Stationary States of the Hydrogen Atom 11.3.5 Dimensions and Orders of Magnitude 11.3.6 Time Evolution of States of Low Energies 11.4 Hydrogen-Like Atoms 11.5 Muonic Atoms 11.6 Spectra of Alkali Atoms Further Reading Exercises 12. Spin 1/2 and Magnetic Resonance 12.1 The Hilbert Space of Spin 1/2 12.1.1 Spin Observables 12.1.2 Representation in a Particular Basis 12.1.3 Matrix Representation 12.1.4 Arbitrary Spin State 12.2 Complete Description of a Spin-1/2 Particle 12.2.1 Hilbert Space 12.2.2 Representation of States and Observables 12.3 Spin Magnetic Moment 12.3.1 The Stern–Gerlach Experiment 12.3.2 Anomalous Zeeman Effect 12.3.3 Magnetic Moment of Elementary Particles 12.4 Uncorrelated Space and Spin Variables 12.5 Magnetic Resonance 12.5.1 Larmor Precession in a Fixed Magnetic Field B0 12.5.2 Superposition of a Fixed Field and a Rotating Field 12.5.3 Rabi’s Experiment 12.5.4 Applications of Magnetic Resonance 12.5.5 Rotation of a Spin 1/2 Particle by 2ð Further Reading Exercises 13. Addition of Angular Momenta, Fine and Hyperfine Structure of Atomic Spectra 13.1 Addition of Angular Momenta 13.1.1 The Total-Angular Momentum Operator 13.1.2 Factorized and Coupled Bases 13.1.3 A Simple Case: the Addition of Two Spins of 1/2 13.1.4 Addition of Two Arbitrary Angular Momenta 13.1.5 One-Electron Atoms, Spectroscopic Notations 13.2 Fine Structure of Monovalent Atoms 13.3 Hyper.ne Structure; the 21 cm Line of Hydrogen 13.3.1 Interaction Energy 13.3.2 Perturbation Theory 13.3.3 Diagonalization of ˆ H1 13.3.4 The Effect of an External Magnetic Field 13.3.5 The 21 cm Line in Astrophysics Further Reading Exercises 14. Entangled States, EPR Paradox and Bell’s Inequality 14.1 The EPR Paradox and Bell’s Inequality 14.1.1 “God Does not Play Dice” 14.1.2 The EPR Argument 14.1.3 Bell’s Inequality 14.1.4 Experimental Tests 14.2 Quantum Cryptography 14.2.1 The Communication Between Alice and Bob 14.2.2 The Quantum Noncloning Theorem 14.2.3 Present Experimental Setups 14.3 The Quantum Computer 14.3.1 The Quantum Bits, or “Q-Bits” 14.3.2 The Algorithm of Peter Shor 14.3.3 Principle of a Quantum Computer 14.3.4 Decoherence Further Reading Exercises 15. The Lagrangian and Hamiltonian Formalisms, Lorentz Force in Quantum Mechanics 15.1 Lagrangian Formalism and the Least-Action Principle 15.1.1 Least Action Principle 15.1.2 Lagrange Equations 15.1.3 Energy 15.2 Canonical Formalism of Hamilton 15.2.1 Conjugate Momenta 15.2.2 Canonical Equations 15.2.3 Poisson Brackets 15.3 Analytical Mechanics and Quantum Mechanics 15.4 Classical Charged Particles in an Electromagnetic Field 15.5 Lorentz Force in Quantum Mechanics 15.5.1 Hamiltonian 15.5.2 Gauge Invariance 15.5.3 The Hydrogen Atom Without Spin in a Uniform Magnetic Field 15.5.4 Spin-1/2 Particle in an Electromagnetic Field Further Reading Exercises 16. Identical Particles and the Pauli Principle 16.1 Indistinguishability of Two Identical Particles 16.1.1 Identical Particles in Classical Physics 16.1.2 The Quantum Problem 16.2 Two-Particle Systems; the Exchange Operator 16.2.1 The Hilbert Space for the Two Particle System 16.2.2 The Exchange Operator Between Two Identical Particles 16.2.3 Symmetry of the States 16.3 The Pauli Principle 16.3.1 The Case of Two Particles 16.3.2 Independent Fermions and Exclusion Principle 16.3.3 The Case of N Identical Particles 16.3.4 Time Evolution 16.4 Physical Consequences of the Pauli Principle 16.4.1 Exchange Force Between Two Fermions 16.4.2 The Ground State of N Identical Independent Particles 16.4.3 Behavior of Fermion and Boson Systems at Low Temperature 16.4.4 Stimulated Emission and the Laser Effect 16.4.5 Uncertainty Relations for a System of N Fermions 16.4.6 Complex Atoms and Atomic Shells Further Reading Exercises 17. The Evolution of Systems 17.1 Time-Dependent Perturbation Theory 17.1.1 Transition Probabilities 17.1.2 Evolution Equations 17.1.3 Perturbative Solution 17.1.4 First-Order Solution: the Born Approximation 17.1.5 Particular Cases 17.1.6 Perturbative and Exact Solutions 17.2 Interaction of an Atom with an Electromagnetic Wave 17.2.1 The Electric-Dipole Approximation 17.2.2 Justification of the Electric Dipole Interaction 17.2.3 Absorption of Energy by an Atom 17.2.4 Selection Rules 17.2.5 Spontaneous Emission 17.2.6 Control of Atomic Motion by Light 17.3 Decay of a System 17.3.1 The Radioactivity of 57Fe 17.3.2 The Fermi Golden Rule 17.3.3 Orders of Magnitude 17.3.4 Behavior for Long Times 17.4 The Time-Energy Uncertainty Relation 17.4.1 Isolated Systems and Intrinsic Interpretations 17.4.2 Interpretation of Landau and Peierls 17.4.3 The Einstein–Bohr Controversy Further Reading Exercises 18. Scattering Processes 18.1 Concept of Cross Section 18.1.1 Definition of Cross Section 18.1.2 Classical Calculation 18.1.3 Examples 18.2 Quantum Calculation in the Born Approximation 18.2.1 Asymptotic States 18.2.2 Transition Probability 18.2.3 Scattering Cross Section 18.2.4 Validity of the Born Approximation 18.2.5 Example: the Yukawa Potential 18.2.6 Range of a Potential in Quantum Mechanics 18.3 Exploration of Composite Systems 18.3.1 Scattering Off a Bound State and the Form Factor 18.3.2 Scattering by a Charge Distribution 18.4 General Scattering Theory 18.4.1 Scattering States 18.4.2 The Scattering Amplitude 18.4.3 The Integral Equation for Scattering 18.5 Scattering at Low Energy 18.5.1 The Scattering Length 18.5.2 Explicit Calculation of a Scattering Length 18.5.3 The Case of Identical Particles Further Reading Exercises 19. Qualitative Physics on a Macroscopic Scale 19.1 Con.ned Particles and Ground State Energy 19.1.1 The Quantum Pressure 19.1.2 Hydrogen Atom 19.1.3 N-Fermion Systems and Complex Atoms 19.1.4 Molecules, Liquids and Solids 19.1.5 Hardness of a Solid 19.2 Gravitational Versus Electrostatic Forces 19.2.1 Screening of Electrostatic Interactions 19.2.2 Additivity of Gravitational Interactions 19.2.3 Ground State of a Gravity-Dominated Object 19.2.4 Liquefaction of a Solid and the Height of Mountains 19.3 White Dwarfs, Neutron Stars and the Gravitational Catastrophe 19.3.1 White Dwarfs and the Chandrasekhar Mass 19.3.2 Neutron Stars Further Reading 20. Early History of Quantum Mechanics 20.1 The Origin of Quantum Concepts 20.1.1 Planck’s Radiation Law 20.1.2 Photons 20.2 The Atomic Spectrum 20.2.1 Empirical Regularities of Atomic Spectra 20.2.2 The Structure of Atoms 20.2.3 The Bohr Atom 20.2.4 The Old Theory of Quanta 20.3 Spin 20.4 Heisenberg’s Matrices 20.5 Wave Mechanics 20.6 The Mathematical Formalization 20.7 Some Important Steps in More Recent Years Further Reading Appendix A. Concepts of Probability Theory 1 Fundamental Concepts 2 Examples of Probability Laws 2.1 Discrete Laws 2.2 Continuous Probability Laws in One or Several Variables 3 Random Variables 3.1 Definition 3.2 Conditional Probabilities 3.3 Independent Random Variables 3.4 Binomial Law and the Gaussian Approximation 4 Moments of Probability Distributions 4.1 Mean Value or Expectation Value 4.2 Variance and Mean Square Deviation 4.3 Bienaym´e–Tchebyche. Inequality 4.4 Experimental Verification of a Probability Law Exercises Appendix B. Dirac Distribution, Fourier Transformation 1 Dirac Distribution, or ä “Function” 1.1 Definition of d(x) 1.2 Examples of Functions Which Tend to ä(x) 1.3 Properties of ä(x) 2 Distributions 2.1 The Space S 2.2 Linear Functionals 2.3 Derivative of a Distribution 2.4 Convolution Product 3 Fourier Transformation 3.1 Definition 3.2 Fourier Transform of a Gaussian 3.3 Inversion of the Fourier Transformation 3.4 Parseval–Plancherel Theorem 3.5 Fourier Transform of a Distribution 3.6 Uncertainty Relation Exercises Appendix C. Operators in Infinite-Dimensional Spaces 1 Matrix Elements of an Operator 2 Continuous Bases Appendix D. The Density Operator 1 Pure States 1.1 A Mathematical Tool: the Trace of an Operator 1.2 The Density Operator of Pure States 1.3 Alternative Formulation of Quantum Mechanics for Pure States 2 Statistical Mixtures 2.1 A Particular Case: an Unpolarized Spin-1/2 System 2.2 The Density Operator for Statistical Mixtures 3 Examples of Density Operators 3.1 The Micro-Canonical and Canonical Ensembles 3.2 The Wigner Distribution of a Spinless Point Particle 4 Entangled Systems 4.1 Reduced Density Operator 4.2 Evolution of a Reduced Density Operator 4.3 Entanglement and Measurement Further Reading Exercises Solutions to the Exercises Index