Quantum Mechanics

Description

Designed for a two-semester advanced undergraduate or graduate level course, this distinctive and modern textbook provides students with the physical intuition and mathematical skills to tackle even complex problems in quantum mechanics with ease and fluency. Beginning with a detailed introduction to quantum states and Dirac notation, the book then develops the overarching theoretical framework of quantum mechanics, before explaining physical quantum mechanical properties such as angular momentum and spin. Symmetries and groups in quantum mechanics, important components of current research, are covered at length. The second part of the text focuses on applications, and includes a detailed chapter on quantum entanglement, one of the most exciting modern applications of quantum mechanics, and of key importance in quantum information and computation. Numerous exercises are interspersed throughout the text, expanding upon key concepts and further developing students' understanding. A fully worked solutions manual and lecture slides are available for instructors. Cover Half-title Endorsement Title page Copyright information Contents Preface Book Organisation Acknowledgements Part I Basics 1 Stories and Thoughts about Quantum Mechanics Further Reading 2 Quantum States 2.1 States of a Quantum System 2.1.1 Kets 2.1.2 Scalar Product 2.1.3 Bras 2.1.4 Norm 2.1.5 Orthogonality 2.1.6 Operators 2.1.7 Bases 2.1.8 Tensor Product 2.2 Two-State Systems 2.3 The Wave Function of a One-Dimensional System 2.3.1 Discretised System 2.3.2 Continuum System Summary Problems 3 Observables 3.1 Introducing Observables 3.1.1 A First Example: Position Operator 3.1.2 Generic Observables 3.2 Observing Observables 3.3 Hermitian Operators 3.3.1 Hermitian Conjugate 3.3.2 Hermitian Operators 3.4 Properties of Hermitian Operators 3.5 Spectral Decomposition 3.6 Collapse of the State Vector 3.7 Compatible Observables 3.8 Complete Sets of Commuting Observables 3.9 Continuous Spectrum 3.9.1 Eigenvalue Equation 3.9.2 Orthonormality 3.9.3 Spectral Decomposition 3.10 Momentum Operator 3.10.1 Momentum as Generator of Translations 3.10.2 Position and Momentum Representations 3.10.3 Wave Packets 3.11 The Uncertainty Principle Summary Problems 4 Dynamics 4.1 Schrödinger Equation 4.1.1 Equation of Motion 4.2 Eigenstates of the Hamiltonian 4.3 Evolution of a Generic State 4.4 One-Dimensional System 4.5 Some Properties of One-Dimensional Potentials 4.5.1 General Restrictions on Ψ(x,t) 4.5.2 Energy Expectation Value 4.5.3 Ehrenfest’s Theorem 4.5.4 Degeneracy 4.5.5 Nodes 4.6 Probability Current Summary Problems 5 Potentials 5.1 Potential Step 5.2 Tunnelling 5.3 Infinite Potential Well 5.4 Symmetry under Parity 5.5 Finite Potential Well Summary Problems 6 Harmonic Oscillator 6.1 The Harmonic Oscillator in Classical Mechanics 6.2 The Quantum Harmonic Oscillator 6.3 Factorising the Hamiltonian 6.4 Creation and Annihilation 6.5 Eigensystem 6.5.1 Eigenvalues 6.5.2 Normalisation of Eigenstates 6.5.3 Wave Functions 6.6 Brute Force Solution Summary Problems 7 Systems in Three Spatial Dimensions 7.1 Quantum States 7.2 Observables 7.2.1 General Considerations 7.2.2 Position 7.2.3 Momentum 7.3 Dynamics 7.4 Separation of Variables: Three-Dimensional Harmonic Oscillator 7.5 Degeneracy Summary Problems 8 Angular Momentum 8.1 Angular Momentum Operator 8.2 Squared Norm of the Angular Momentum 8.3 Eigenfunctions 8.4 Physical Interpretation 8.5 Algebraic Solution of the Eigenvalue Equations 8.6 Nomenclature 8.7 Normalisation 8.8 Matrix Representations 8.9 Wave Functions Summary Problems 9 Spin 9.1 The Stern–Gerlach Experiment 9.2 Physical States 9.3 Matrix Representation 9.4 Eigenvectors 9.5 Scalar Products 9.6 Eigenvectors of [sub(x)] 9.7 The Stern–Gerlach Experiment Reloaded Summary Problems 10 Addition of Angular Momenta 10.1 Total Angular Momentum Operator 10.2 Addition Theorem 10.3 Example: Two Spin-1/2 Particles Summary Problems 11 Central Potentials 11.1 Stationary States 11.2 Physical Interpretation 11.3 Quantum Rotator 11.4 Central Square Well Summary Problems 12 Hydrogen Atom 12.1 Stationary States 12.2 Solution of the Radial Equation 12.3 Physical Interpretation Summary Problems 13 Identical Particles 13.1 Permutation Symmetry 13.2 A First Look at Helium 13.3 Two-Electron Wave Function 13.4 More on the Helium Atom 13.5 Pauli Exclusion Principle Summary Problems 14 Symmetries in Quantum Mechanics 14.1 Classical Symmetry 14.2 Quantum Symmetry 14.3 Symmetry Groups in Physics 14.4 Group of Translations 14.5 Group of Rotations 14.5.1 Rotations in Two Dimensions 14.5.2 Rotations in Three Dimensions 14.6 Rotations in the Space of Quantum States 14.7 Rotations Acting on Operators 14.8 Commutation Relations for a Generic Non-Abelian Group Summary Problems Part II Applications 15 Quantum Entanglement 15.1 Hidden Variables and the Einstein–Podolsky–Rosen Paradox 15.1.1 The Einstein–Podolsky–Rosen Paradox 15.2 Bell’s Inequality 15.2.1 Spin Correlations 15.2.2 Experimental Tests of Bell’s Inequality 15.3 Characterising Entanglement 15.3.1 Bit and Qubit 15.3.2 Information Theory Essentials – Shannon Entropy 15.3.3 Von Neumann Entropy 15.3.4 Entanglement Entropy 15.3.5 Other Measures of Information 15.4 Quantum Communication 15.4.1 No-Cloning Theorem 15.4.2 Quantum Teleportation of a Qubit 15.4.3 Superdense Coding 15.5 Quantum Computing 15.5.1 Quantum Register and Logic Gates 15.5.2 Deutsch’s Algorithm 15.5.3 Grover’s Algorithm 15.A Appendix: Klein’s Inequality Summary Reference Further Reading Problems 16 Time-Independent Perturbation Theory 16.1 Nondegenerate Time-Independent Perturbation Theory 16.1.1 First-Order Correction 16.1.2 Second-Order Energy Correction 16.1.3 Normalisation and Orthogonality of the Eigenstates to First Order 16.1.4 Higher-Order Corrections 16.1.5 Properties of the Perturbation Expansion 16.2 Degenerate Perturbation Theory 16.2.1 First-Order Degenerate Perturbation Correction 16.2.2 Second-Order Degenerate Perturbation Energy Correction 16.3 Applications 16.3.1 Hydrogen Fine Structure 16.3.2 Helium Atom Perturbation Treatment of Electron Repulsion Term 16.A Appendix: Derivation of the Fine-Structure Terms from Relativistic Theory Summary References Further Reading Problems 17 Calculation Methods Beyond Perturbation Theory 17.1 Rayleigh–Ritz Variational Method 17.1.1 Ground State of Hydrogen 17.1.2 Ground State of Helium 17.1.3 Excited States 17.2 Atomic Systems 17.2.1 First Excited States of Helium 17.2.2 Multi-electron Atoms 17.3 Born–Oppenheimer Approximation 17.3.1 The H[sub(2)sup(+)] Ion and Bonding 17.4 Hellmann–Feynman Method 17.5 Wenzel–Kramers–Brillouin–Jeffreys Approximation 17.5.1 Potential Barrier 17.5.2 Potential Well 17.5.3 Symmetric Double Well Summary Further Reading Problems 18 Time-Dependent Perturbation Theory 18.1 Time-Dependent Hamiltonians 18.1.1 Time-Dependent Perturbation Theory 18.2 Time-Dependent Perturbations 18.2.1 Time-Independent Perturbation Switched on at a Given Time 18.2.2 Fermi’s Golden Rule 18.2.3 Harmonic Perturbations 18.3 Interaction of Radiation with Quantum Systems 18.3.1 Semiclassical Treatment of Electromagnetic Radiation 18.3.2 Interaction with a One-Electron Atom 18.3.3 Selection Rules 18.4 Time-Dependent Perturbation Theory at Higher Order 18.4.1 Time Dependence in Quantum Mechanics 18.4.2 Time Evolution Operator in the Interaction Picture Summary Problems 19 Quantum Scattering Theory 19.1 Scattering Kinematics 19.1.1 The Differential Cross-Section 19.2 Evolution of a Wave Packet 19.3 Time-Dependent Approach to Scattering 19.3.1 The Born Approximation 19.3.2 Two-Body Scattering 19.4 Time-Independent Approach to Scattering 19.4.1 Relation of Wave Packet to Time-Independent Approach 19.4.2 The Scattering Amplitude 19.4.3 Green Function Method for Calculating f(θ, [phi]) 19.4.4 Partial Wave Analysis 19.A Appendix: Solutions of the Radial Equation for a Free Particle Summary Further Reading Problems Appendix: Gaussian/SI Electromagnetic Units Index