614 Pages 107 B/W Illustrations
    by Chapman & Hall

    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 <A2|F|A1>

    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

    Biography

    Vishnu S. Mathur, Surendra Singh

    … this book has much to recommend it. The impression is that it is written for students who may not have a deep grounding in the required mathematics. Each required mathematical point is explained clearly but concisely … The breadth of coverage is such that the book would be suitable as a general text for students embarking on advanced work in most fields of physics …
    Contemporary Physics, Vol. 51, No. 2, March 2010

    … This text is intended for graduate students studying quantum mechanics or for someone very interested in learning about the details of quantum mechanics. It is has a high level of technical depth with complex mathematical expressions used to describe the topics being discussed.
    IEEE Electrical Insulation Magazine, March/April 2010, Vol. 26, No. 2