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“Excellent, up-to-date… Quantum Mechanics I: The Fundamentals covers the ca-

nonical basics and Quantum Mechanics II: Advanced Topics covers a range of mod-

ern developments…I recommend this set highly.”

Dr. Jonathan P. Dowling, Hearne Professor of Theoretical Physics, Louisiana State

University

“… these two books by Rajasekar and Velusamy will definitely tell you how to do quan-

tum mechanics.”

—Dr. K.P.N. Murthy, Professor, School of Physics, and Director, Centre for Integrated

Studies, University of Hyderabad

Quantum Mechanics I: The Fundamentals provides a graduate-level account of the be-

havior of matter and energy at the molecular, atomic, nuclear, and sub-nuclear levels. It

covers basic concepts, mathematical formalism, and applications to physically impor-

tant systems.

The text addresses many topics not typically found in books at this level, including:

• Bound state solutions of quantum pendulum

• Pöschl–Teller potential

• Solutions of classical counterpart of quantum mechanical systems

• A criterion for bound state

• Scattering from a locally periodic potential and reflection-less potential

• Modified Heisenberg relation

• Wave packet revival and its dynamics

• Hydrogen atom in D-dimension

• Alternate perturbation theories

• An asymptotic method for slowly varying potentials

• Klein paradox, Einstein-Podolsky-Rosen (EPR) paradox, and Bell’s theorem

• Numerical methods for quantum systems

A collection of problems at the end of each chapter develops readers’ understanding

of both basic concepts and the application of theory to various physically important

systems. This book, along with the authors’ follow-up Quantum Mechanics II: Advanced

Topics, provides readers with a broad, up-to-date introduction to quantum mechanics.

Physics

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EBOOK

Quantum Mechanics I

The Fundamentals

Rajasekar • Velusamy

Quantum Mechanics I

The Fundamentals

S. Rajasekar

R. Velusamy

K24364_cover.indd 1 11/5/14 1:23 PM

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Quantum Mechanics I

The Fundamentals

S. Rajasekar

R. Velusamy

Quantum Mechanics I

The Fundamentals

S. Rajasekar

R. Velusamy

To our wives.

Contents xi

Chapter 1 Why Was Quantum Mechanics Developed? About the Authors xxi
- 1.1 INTRODUCTION
- 1.2 BLACK BODY RADIATION
- 1.3 PHOTOELECTRIC EFFECT
- 1.4 HYDROGEN SPECTRUM
- 1.5 FRANCK–HERTZ EXPERIMENT
- 1.6 STERN–GERLACH EXPERIMENT
- 1.7 CORRESPONDENCE PRINCIPLE
- 1.8 COMPTON EFFECT
- 1.9 SPECIFIC HEAT CAPACITY
- 1.10 DE BROGLIE WAVES
- 1.11 PARTICLE DIFFRACTION
- 1.12 WAVE-PARTICLE DUALITY
- 1.13 CONCLUDING REMARKS
- 1.14 BIBLIOGRAPHY
- 1.15 EXERCISES
Chapter 2 Schrödinger Equation and Wave Function
- 2.1 INTRODUCTION
- 2.2 CONSTRUCTION OF SCHRÖDINGER EQUATION
- 2.3 SOLUTION OF TIME-DEPENDENT EQUATION
- 2.4 PHYSICAL INTERPRETATION OF ψ∗ψ
- 2.5 CONDITIONS ON ALLOWED WAVE FUNCTIONS
- 2.6 BOX NORMALIZATION
- 2.7 CONSERVATION OF PROBABILITY viii Contents
- 2.8 EXPECTATION VALUE
- 2.9 EHRENFEST’S THEOREM
- 2.10 BASIC POSTULATES
- 2.11 TIME EVOLUTION OF STATIONARY STATES
- 2.12 CONDITIONS FOR ALLOWED TRANSITIONS
- 2.13 ORTHOGONALITY OF TWO STATES
- 2.14 PHASE OF THE WAVE FUNCTION
- 2.15 CLASSICAL LIMIT OF QUANTUM MECHANICS
- 2.16 CONCLUDING REMARKS
- 2.17 BIBLIOGRAPHY
- 2.18 EXERCISES
Chapter 3 Operators, Eigenvalues and Eigenfunctions
- 3.1 INTRODUCTION
- 3.2 LINEAR OPERATORS
- 3.3 COMMUTING AND NONCOMMUTING OPERATORS
- 3.4 SELF-ADJOINT AND HERMITIAN OPERATORS
- 3.5 DISCRETE AND CONTINUOUS EIGENVALUES
- 3.6 MEANING OF EIGENVALUES AND EIGENFUNCTIONS
- 3.7 PARITY OPERATOR
- 3.8 ALL HERMITIAN HAMILTONIANS HAVE PARITY
- 3.9 SOME OTHER USEFUL OPERATORS
- 3.10 CONCLUDING REMARKS
- 3.11 BIBLIOGRAPHY
- 3.12 EXERCISES
Chapter 4 Exactly Solvable Systems I: Bound States
- 4.1 INTRODUCTION
- 4.2 CLASSICAL PROBABILITY DISTRIBUTION
- 4.3 FREE PARTICLE
- 4.4 HARMONIC OSCILLATOR
- 4.5 PARTICLE IN THE POTENTIAL V (x) = x^2 k, k = 1, 2 , · · ·
- 4.6 PARTICLE IN A BOX
- 4.7 PÖSCHL–TELLER POTENTIALS Contents ix
- 4.8 QUANTUM PENDULUM
- 4.9 CRITERIA FOR THE EXISTENCE OF A BOUND STATE
- 4.10 TIME-DEPENDENT HARMONIC OSCILLATOR
- 4.11 RIGID ROTATOR
- 4.12 CONCLUDING REMARKS
- 4.13 BIBLIOGRAPHY
- 4.14 EXERCISES
Chapter 5 Exactly Solvable Systems II: Scattering States
- 5.1 INTRODUCTION
- 5.2 POTENTIAL BARRIER: TUNNEL EFFECT
- 5.3 FINITE SQUARE-WELL POTENTIAL
- 5.4 POTENTIAL STEP
- 5.5 LOCALLY PERIODIC POTENTIAL
- 5.6 REFLECTIONLESS POTENTIALS
- 5.7 DYNAMICAL TUNNELING
- 5.8 CONCLUDING REMARKS
- 5.9 BIBLIOGRAPHY
- 5.10 EXERCISES
Chapter 6 Matrix Mechanics
- 6.1 INTRODUCTION
- 6.2 LINEAR VECTOR SPACE
  - WAVE FUNCTION 6.3 MATRIX REPRESENTATION OF OPERATORS AND
- 6.4 UNITARY TRANSFORMATION
- 6.5 TENSOR PRODUCTS
  - IN MATRIX FORM 6.6 SCHRÖDINGER EQUATION AND OTHER QUANTITIES
- 6.7 APPLICATION TO CERTAIN SYSTEMS
- 6.8 DIRAC’S BRA AND KET NOTATIONS
- 6.9 EXAMPLES OF BASIS IN QUANTUM THEORY
- 6.10 PROPERTIES OF KET AND BRA VECTORS
- 6.11 HILBERT SPACE
- 6.12 PROJECTION AND DISPLACEMENT OPERATORS x Contents
- 6.13 CONCLUDING REMARKS
- 6.14 BIBLIOGRAPHY
- 6.15 EXERCISES
Chapter 7 Various Pictures and Density Matrix
- 7.1 INTRODUCTION
- 7.2 SCHRÖDINGER PICTURE
- 7.3 HEISENBERG PICTURE
- 7.4 INTERACTION PICTURE
- 7.5 COMPARISON OF THREE REPRESENTATIONS
- 7.6 DENSITY MATRIX FOR A SINGLE SYSTEM
- 7.7 DENSITY MATRIX FOR AN ENSEMBLE
- 7.8 TIME EVOLUTION OF DENSITY OPERATOR
- 7.9 A SPIN- 1 / 2 SYSTEM
- 7.10 CONCLUDING REMARKS
- 7.11 BIBLIOGRAPHY
- 7.12 EXERCISES
Chapter 8 Heisenberg Uncertainty Principle
- 8.1 INTRODUCTION
- 8.2 THE CLASSICAL UNCERTAINTY RELATION
- 8.3 HEISENBERG UNCERTAINTY RELATION
- 8.4 IMPLICATIONS OF UNCERTAINTY RELATION
- 8.5 ILLUSTRATION OF UNCERTAINTY RELATION
- 8.6 THE MODIFIED HEISENBERG RELATION
- 8.7 CONCLUDING REMARKS
- 8.8 BIBLIOGRAPHY
- 8.9 EXERCISES
Chapter 9 Momentum Representation
- 9.1 INTRODUCTION
- 9.2 MOMENTUM EIGENFUNCTIONS
- 9.3 SCHRÖDINGER EQUATION
- 9.4 EXPRESSIONS FOR 〈X〉 AND 〈p〉
  - ORDINATE REPRESENTATIONS 9.5 TRANSFORMATION BETWEEN MOMENTUM AND CO-
- 9.6 OPERATORS IN MOMENTUM REPRESENTATION
- 9.7 MOMENTUM FUNCTION OF SOME SYSTEMS
- 9.8 CONCLUDING REMARKS
- 9.9 BIBLIOGRAPHY
- 9.10 EXERCISES
Chapter 10 Wave Packet
- 10.1 INTRODUCTION
- 10.2 PHASE AND GROUP VELOCITIES
- 10.3 WAVE PACKETS AND UNCERTAINTY PRINCIPLE
- 10.4 GAUSSIAN WAVE PACKET
- 10.5 WAVE PACKET REVIVAL
- 10.6 ALMOST PERIODIC WAVE PACKETS
- 10.7 CONCLUDING REMARKS
- 10.8 BIBLIOGRAPHY
- 10.9 EXERCISES
Chapter 11 Theory of Angular Momentum
- 11.1 INTRODUCTION
- 11.2 SCALAR WAVE FUNCTION UNDER ROTATIONS
- 11.3 ORBITAL ANGULAR MOMENTUM
- 11.4 EIGENPAIRS OF L^2 AND Lz
- 11.5 PROPERTIES OF COMPONENTS OF L AND L^2
  - RELATIONS 11.6 EIGENSPECTRA THROUGH COMMUTATION
- 11.7 MATRIX REPRESENTATION OF L^2 , Lz AND L±
- 11.8 WHAT IS SPIN?
- 11.9 SPIN STATES OF AN ELECTRON
- 11.10 SPIN-ORBIT COUPLING
- 11.11 ROTATIONAL TRANSFORMATION
- 11.12 ADDITION OF ANGULAR MOMENTA
- 11.13 ROTATIONAL PROPERTIES OF OPERATORS
- 11.14 TENSOR OPERATORS
- 11.15 THE WIGNER–ECKART THEOREM xii Contents
- 11.16 CONCLUDING REMARKS
- 11.17 BIBLIOGRAPHY
- 11.18 EXERCISES
Chapter 12 Hydrogen Atom
- 12.1 INTRODUCTION
- 12.2 HYDROGEN ATOM IN THREE-DIMENSION
- 12.3 HYDROGEN ATOM IN D-DIMENSION
- 12.4 FIELD PRODUCED BY A HYDROGEN ATOM
- 12.5 SYSTEM IN PARABOLIC COORDINATES
- 12.6 CONCLUDING REMARKS
- 12.7 BIBLIOGRAPHY
- 12.8 EXERCISES
  - Perturbation Theory Chapter 13 Approximation Methods I: Time-Independent
- 13.1 INTRODUCTION
- 13.2 THEORY FOR NONDEGENERATE CASE
- 13.3 APPLICATIONS TO NONDEGENERATE LEVELS
- 13.4 THEORY FOR DEGENERATE LEVELS
- 13.5 FIRST-ORDER STARK EFFECT IN HYDROGEN
- 13.6 ALTERNATE PERTURBATION THEORIES
- 13.7 CONCLUDING REMARKS
- 13.8 BIBLIOGRAPHY
- 13.9 EXERCISES
  - Perturbation Theory Chapter 14 Approximation Methods II: Time-Dependent
- 14.1 INTRODUCTION
- 14.2 TRANSITION PROBABILITY
- 14.3 CONSTANT PERTURBATION
- 14.4 HARMONIC PERTURBATION
- 14.5 ADIABATIC PERTURBATION
- 14.6 SUDDEN APPROXIMATION
- 14.7 THE SEMICLASSICAL THEORY OF RADIATION Contents xiii
- 14.8 CALCULATION OF EINSTEIN COEFFICIENTS
- 14.9 CONCLUDING REMARKS
- 14.10 BIBLIOGRAPHY
- 14.11 EXERCISES - Asymptotic Methods Chapter 15 Approximation Methods III: WKB and
- 15.1 INTRODUCTION
- 15.2 PRINCIPLE OF WKB METHOD
- 15.3 APPLICATIONS OF WKB METHOD
- 15.4 WKB QUANTIZATION WITH PERTURBATION
- 15.5 AN ASYMPTOTIC METHOD
- 15.6 CONCLUDING REMARKS
- 15.7 BIBLIOGRAPHY
- 15.8 EXERCISES - Approach Chapter 16 Approximation Methods IV: Variational
- 16.1 INTRODUCTION
- 16.2 CALCULATION OF GROUND STATE ENERGY
- 16.3 TRIAL EIGENFUNCTIONS FOR EXCITED STATES
- 16.4 APPLICATION TO HYDROGEN MOLECULE
- 16.5 HYDROGEN MOLECULE ION
- 16.6 CONCLUDING REMARKS
- 16.7 EXERCISES
Chapter 17 Scattering Theory
- 17.1 INTRODUCTION
- 17.2 CLASSICAL SCATTERING CROSS-SECTION
  - COORDINATES SYSTEMS 17.3 CENTRE OF MASS AND LABORATORY
- 17.4 SCATTERING AMPLITUDE
- 17.5 GREEN’S FUNCTION APPROACH
- 17.6 BORN APPROXIMATION
- 17.7 PARTIAL WAVE ANALYSIS xiv Contents
- 17.8 SCATTERING FROM A SQUARE-WELL SYSTEM
- 17.9 PHASE-SHIFT OF ONE-DIMENSIONAL CASE
- 17.10 INELASTIC SCATTERING
- 17.11 CONCLUDING REMARKS
- 17.12 BIBLIOGRAPHY
- 17.13 EXERCISES
Chapter 18 Identical Particles
- 18.1 INTRODUCTION
- 18.2 PERMUTATION SYMMETRY
  - FUNCTIONS 18.3 SYMMETRIC AND ANTISYMMETRIC WAVE
- 18.4 THE EXCLUSION PRINCIPLE
- 18.5 SPIN EIGENFUNCTIONS OF TWO ELECTRONS
- 18.6 EXCHANGE INTERACTION
- 18.7 EXCITED STATES OF THE HELIUM ATOM
- 18.8 COLLISIONS BETWEEN IDENTICAL PARTICLES
- 18.9 CONCLUDING REMARKS
- 18.10 BIBLIOGRAPHY
- 18.11 EXERCISES
Chapter 19 Relativistic Quantum Theory
- 19.1 INTRODUCTION
- 19.2 KLEIN–GORDON EQUATION
- 19.3 DIRAC EQUATION FOR A FREE PARTICLE
- 19.4 NEGATIVE ENERGY STATES
- 19.5 JITTERY MOTION OF A FREE PARTICLE
- 19.6 SPIN OF A DIRAC PARTICLE
- 19.7 PARTICLE IN A POTENTIAL
- 19.8 KLEIN PARADOX
- 19.9 RELATIVISTIC PARTICLE IN A BOX
- 19.10 RELATIVISTIC HYDROGEN ATOM
- 19.11 THE ELECTRON IN A FIELD
- 19.12 SPIN-ORBIT ENERGY
- 19.13 CONCLUDING REMARKS Contents xv
- 19.14 BIBLIOGRAPHY
- 19.15 EXERCISES
Chapter 20 Mysteries in Quantum Mechanics
- 20.1 INTRODUCTION
- 20.2 THE COLLAPSE OF THE WAVE FUNCTION
- 20.3 EINSTEIN–PODOLSKY–ROSEN (EPR) PARADOX
- 20.4 HIDDEN VARIABLES
- 20.5 THE PARADOX OF SCHRÖDINGER’S CAT
- 20.6 BELL’S THEOREM
- 20.7 VIOLATION OF BELL’S THEOREM
- 20.8 RESOLVING EPR PARADOX
- 20.9 CONCLUDING REMARKS
- 20.10 BIBLIOGRAPHY
- 20.11 EXERCISES
Chapter 21 Numerical Methods for Quantum Mechanics
- 21.1 INTRODUCTION
  - STATE SOLUTIONS 21.2 MATRIX METHOD FOR COMPUTING STATIONARY
- 21.3 FINITE-DIFFERENCE TIME-DOMAIN METHOD
- 21.4 TIME-DEPENDENT SCHRÖDINGER EQUATION
- 21.5 QUANTUM SCATTERING
- 21.6 ELECTRONIC DISTRIBUTION OF HYDROGEN ATOM
  - FIELD 21.7 SCHRÖDINGER EQUATION WITH AN EXTERNAL
- 21.8 CONCLUDING REMARKS
- 21.9 BIBLIOGRAPHY
- 21.10 EXERCISES
Appendix A Calculation of Numerical Values ofh and kB
Appendix B A Derivation of the Factor hν/(ehν/kBT − 1)

Preface

Quantum mechanics is the study of the behaviour of matter and energy at the molecular, atomic, nuclear levels and even at sub-nuclear level. This book is intended to provide a broad introduction to fundamental and advanced topics of quantum mechanics. Volume I is devoted to basic concepts, mathematical formalism and application to physically important systems. Volume II covers most of the advanced topics of current research interest in quantum mechan- ics. Both the volumes are primarily developed as texts at the graduate level and also as reference books. In addition to worked-out examples, numerous collection of problems are included at the end of each chapter. Solutions are available to confirmed instructors upon request to the publisher. Some of the problems serve as a mode of understanding and highlighting the significances of basic concepts while others form application of theory to various physically important systems/problems. Developments made in recent years on various mathematical treatments, theoretical methods, their applications and exper- imental observations are pointed out wherever necessary and possible and moreover they are quoted with references so that readers can refer them for more details. Volume I consists of 21 chapters and 7 appendices. Chapter 1 summa- rizes the needs for the quantum theory and its early development (old quan- tum theory). Chapters 2 and 3 provide the basic mathematical framework of quantum mechanics. Schrödinger wave mechanics and operator formalism are introduced in these chapters. Chapters 4 and 5 are concerned with the ana- lytical solutions of bound states and scattering states respectively of certain physically important microscopic systems. The basics of matrix mechanics, Dirac’s notation of state vectors and Hilbert space are elucidated in chapter

The next chapter gives the Schrödinger, Heisenberg and interaction pic- tures of time evolution of quantum mechanical systems. Description of time evolution of ensembles by means of density matrix is also described. Chap- ter 8 is concerned with Heisenberg’s uncertainty principle. A brief account of wave function in momentum space and wave packet dynamics are presented in chapters 9 and 10 , respectively. Theory of angular momentum is covered in chapter 11. Chapter 12 is devoted exclusively to the theory of hydrogen atom. Chapters 13 through 16 are mainly concerned with approximation methods such as time-independent and time-dependent perturbation theories, WKB method and variational method. The elementary theory of elastic scattering is presented in chapter 17. Identical particles are treated in chapter 18. The

xvii

xviii Preface

next chapter presents quantum theory of relativistic particles with specific emphasize on Klein–Gordon equation, Dirac equation and its solution for a free particle, particle in a box (Klein paradox) and hydrogen atom. Chapter 20 examines the strange consequences of role of measurement through the paradoxes of EPR and a thought experiment of Schrödinger. A brief sketch of Bell’s inequality and the quantum mechanical examples violating it are given. Considering the rapid growth of numerical techniques in solving phys- ical problems and significances of simulation studies in describing complex phenomena, the final chapter is devoted for a detailed description of numeri- cal computation of bound state eigenvalues and eigenfunctions, transmission and reflection probabilities of scattering potentials, transition probabilities of quantum systems in the presence of external fields and electronic distribution of atoms. Some supplementary and background materials are presented in the appendices. The pedagogic features volume I of the book, which are not usually found in textbooks at this level, are the presentation of bound state solutions of quan- tum pendulum, Pöschl–Teller potential, solutions of classical counter part of quantum mechanical systems considered, criterion for bound state, scattering from a locally periodic potential and reflectionless potential, modified Heisen- berg relation, wave packet revival and and its dynamics, hydrogen atom in D- dimension, alternate perturbation theories, an asymptotic method for slowly varying potentials, Klein paradox, EPR paradox, Bell’s theorem and numeri- cal methods for quantum systems. The volume II consists of 10 chapters. Chapter 1 describes the basic ideas of both classical and quantum field theories. Quantization of Klein–Gordon equation and Dirac field are given. The formulation of quantum mechanics in terms of path integrals is presented in chapter 2. Application of it to free particle and linear harmonic oscillator are considered. In chapter 3 some il- lustrations and interpretation of supersymmetric potentials and partners are presented. A simple general procedure to construct all the supersymmetric partners of a given quantum mechanical systems with bound states is de- scribed. The method is then applied to a few interesting systems. The next chapter is concerned with coherent and squeezed states. Construction of these state and their characteristic properties are enumerated. Chapter 5 is devoted to Berry’s phase, Aharonov–Bohm and Sagnac effects. Their origin, properties, effects and experimental demonstration are presented. The features of Wigner distribution function are elucidated in chapter 6. In a few decades time, it is possible to realize a computer built in terms of real quantum systems that operate in quantum mechanical regime. There is a growing interest on quan- tum computing. Basic aspects of quantum computing is presented in chapter

Deutsch–Jozsa algorithm of finding whether a function is constant or not, Grover’s search algorithm and Shor’s efficient quantum algorithm for inte- ger factorization and evaluation of discrete logarithms are described. Chapter 8 deals with quantum cryptography. Basic principles of classical cryptogra- phy and quantum cryptography and features of a few quantum cryptographic

quantum mechanics for partical physics, Thesis of Physics

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Partial preview of the text

Download quantum mechanics for partical physics and more Thesis Physics in PDF only on Docsity!

Quantum Mechanics I

The Fundamentals

S. Rajasekar

R. Velusamy

The Fundamentals

S. Rajasekar

R. Velusamy

To our wives.