Concepts in Quantum Mechanics

Taking a conceptual approach to the subject, Concepts in Quantum Mechanics provides complete coverage of both basic and advanced topics. Following in the footsteps of Dirac’s classic work Principles of Quantum Mechanics, it explains all themes from first principles. The authors present alternative ways of representing the state of a physical system, outline the mathematical connection between the representatives of the same state in different representations, and highlight the connection between Dirac brackets and their integral forms in the coordinate and momentum representations. They also logically develop the equations of motion in Schrödinger and Heisenberg pictures. In addition, the book covers motion in the presence of potential steps and wells, bound state problems, symmetries and their consequences, the role of angular momentum in quantum mechanics, approximation methods, time-dependent perturbation methods, and second quantization. Written by authoritative professors who have taught quantum mechanics at the graduate level for a combined forty years, this textbook provides students with a strong foundation in quantum mechanics. After reading the book, students will be ready to take on quantum field theory.

NEED FOR QUANTUM MECHANICS AND ITS PHYSICAL BASIS Inadequacy of Classical Description for Small Systems Basis of Quantum Mechanics Representation of States Dual Vectors: Bra and Ket Vectors Linear Operators Adjoint of a Linear Operator Eigenvalues and Eigenvectors of a Linear Operator Physical Interpretation Observables and Completeness Criterion Commutativity and Compatibility of Observables Position and Momentum Commutation Relations Commutation Relation and the Uncertainty Product Appendix: Basic Concepts in Classical Mechanics REPRESENTATION THEORY Meaning of Representation How to Set up a Representation Representatives of a Linear Operator Change of Representation Coordinate Representation Replacement of Momentum Observable p by -ih d/dq Integral Representation of Dirac Bracket The Momentum Representation Dirac Delta Function Relation between the Coordinate and Momentum Representations EQUATIONS OF MOTION Schrödinger Equation of Motion Schrödinger Equation in the Coordinate Representation Equation of Continuity Stationary States Time-Independent Schrödinger Equation in the Coordinate Representation Time-Independent Schrödinger Equation in the Momentum Representation Time-Independent Schrödinger Equation in Matrix Form The Heisenberg Picture The Interaction Picture Appendix: Matrices PROBLEMS OF ONE-DIMENSIONAL POTENTIAL BARRIERS Motion of a Particle across a Potential Step Passage of a Particle through a Potential Barrier of Finite Extent Tunneling of a Particle through a Potential Barrier Bound States in a One-Dimensional Square Potential Well Motion of a Particle in a Periodic Potential BOUND STATES OF SIMPLE SYSTEMS Introduction Motion of a Particle in a Box Simple Harmonic Oscillator Operator Formulation of the Simple Harmonic Oscillator Problem Bound State of a Two-Particle System with Central Interaction Bound States of Hydrogen (or Hydrogen-Like) Atoms The Deuteron Problem Energy Levels in a Three-Dimensional Square Well: General Case Energy Levels in an Isotropic Harmonic Potential Well Appendix 1: Special Functions Appendix 2: Orthogonal Curvilinear Coordinate Systems SYMMETRIES AND CONSERVATION LAWS Symmetries and Their Group Properties Symmetries in a Quantum Mechanical System Basic Symmetry Groups of the Hamiltonian and Conservation Laws Lie Groups and Their Generators Examples of Lie Group Appendix 1: Groups and Representations ANGULAR MOMENTUM IN QUANTUM MECHANICS Introduction Raising and Lowering Operators Matrix Representation of Angular Momentum Operators Matrix Representation of Eigenstates of Angular Momentum Coordinate Representation of Orbital Angular Momentum Operators and States General Rotation Group and Rotation Matrices Coupling of Two Angular Momenta Properties of Clebsch–Gordan Coefficients Coupling of Three Angular Momenta Coupling of Four Angular Momenta (L - S and j - j Coupling) APPROXIMATION METHODS Introduction Nondegenerate Time-Independent Perturbation Theory Time-Independent Degenerate Perturbation Theory The Zeeman Effect WKBJ Approximation Particle in a Potential Well Application of WKBJ Approximation to a-decay The Variational Method The Problem of the Hydrogen Molecule System of n Identical Particles: Symmetric and Antisymmetric States Excited States of the Helium Atom Statistical (Thomas–Fermi) Model of the Atom Hartree’s Self-consistent Field Method for Multi-Electron Atoms Hartree–Fock Equations Occupation Number Representation QUANTUM THEORY OF SCATTERING Introduction Laboratory and Center-of-Mass (CM) Reference Frames Scattering Equation and the Scattering Amplitude Partial Waves and Phase Shifts Calculation of Phase Shift Phase Shifts for Some Simple Potential Forms Scattering due to Coulomb Potential The Integral Form of Scattering Equation Lippmann–Schwinger Equation and the Transition Operator Born Expansion Appendix: The Calculus of Residues TIME-DEPENDENT PERTURBATION METHODS Introduction Perturbation Constant over an Interval of Time Harmonic Perturbation: Semiclassical Theory of Radiation Einstein Coeffcients Multipole Transitions Electric Dipole Transitions in Atoms and Selection Rules Photo-Electric Effect Sudden and Adiabatic Approximations Second-Order Effects THE THREE-BODY PROBLEM Introduction Eyges Approach Mitra’s Approach Faddeev’s Approach Faddeev Equations in Momentum Representation Faddeev Equations for a Three-Body Bound System Alt, Grassberger, and Sandhas (AGS) Equations RELATIVISTIC QUANTUM MECHANICS Introduction Dirac Equation Spin of the Electron Free Particle (Plane Wave) Solutions of Dirac Equation Dirac Equation for a Zero Mass Particle Zitterbewegung and Negative Energy Solutions Dirac Equation for an Electron in an Electromagnetic Field Invariance of Dirac Equation Dirac Bilinear Covariants Dirac Electron in a Spherically Symmetric Potential Charge Conjugation, Parity, and Time-Reversal Invariance Appendix: Theory of Special Relativity QUANTIZATION OF RADIATION FIELD Introduction Radiation Field as a Swarm of Oscillators Quantization of Radiation Field Interaction of Matter with Quantized Radiation Field Applications Bethe’s Treatment of Atomic Level Shift Due to the Self Energy of the Electron: (Lamb–Retherford Shift) Compton Scattering Appendix: Electromagnetic Field in Coulomb Gauge SECOND QUANTIZATION Introduction Classical Concept of Field Analogy of Field and Particle Mechanics Field Equations from Lagrangian Density Quantization of a Real Scalar (KG) Field Quantization of Complex Scalar (KG) Field Dirac Field and Its Quantization Positron Operators and Spinors Interacting Fields and the Covariant Perturbation Theory Second-Order Processes in Electrodynamics Amplitude for Compton Scattering Feynman Graphs Calculation of the Cross-Section of Compton Scattering Cross-Sections for Other Electromagnetic Processes Appendix 1: Calculus of Variation and Euler–Lagrange Equations Appendix 2: Functionals and Functional Derivatives Appendix 3: Interaction of the Electron and Radiation Fields Appendix 4: On the Convergence of Iterative Expansion of the S Operator EPILOGUE Introduction Einstein–Podolsky–Rosen Gedanken Experiment Einstein–Podolsky–Rosen–Bohm Gedanken Experiment Theory of Hidden Variables and Bell’s Inequality Clauser–Horne Form of Bell’s Inequality and Its Violation in Two-Photon Correlation Experiments GENERAL REFERENCES INDEX