Concepts in Quantum Mechanics

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





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