# Concepts in Quantum Mechanics

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## Book Description

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.

## Table of Contents

**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 <*A*_{2}|*F*|*A*_{1}>

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**

## Reviews

… 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