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Quantum Mechanics I
The Fundamentals
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Book Description
Quantum Mechanics I: The Fundamentals provides a graduatelevel account of the behavior of matter and energy at the molecular, atomic, nuclear, and subnuclear levels. It covers basic concepts, mathematical formalism, and applications to physically important 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 reflectionless potential
 Modified Heisenberg relation
 Wave packet revival and its dynamics
 Hydrogen atom in Ddimension
 Alternate perturbation theories
 An asymptotic method for slowly varying potentials
 Klein paradox, EinsteinPodolskyRosen (EPR) paradox, and Bell’s theorem
 Numerical methods for quantum systems
A collection of problems at the end of each chapter develops students’ understanding of both basic concepts and the application of theory to various physically important systems. This book, along with the authors’ followup Quantum Mechanics II: Advanced Topics, provides students with a broad, uptodate introduction to quantum mechanics.
Print Versions of this book also include access to the ebook version.
Table of Contents
Why Was Quantum Mechanics Developed?
INTRODUCTION
BLACK BODY RADIATION
PHOTOELECTRIC EFFECT
HYDROGEN SPECTRUM
FRANCK–HERTZ EXPERIMENT
STERN–GERLACH EXPERIMENT
CORRESPONDENCE PRINCIPLE
COMPTON EFFECT
SPECIFIC HEAT CAPACITY
DE BROGLIE WAVES
PARTICLE DIFFRACTION
WAVEPARTICLE DUALITY
Schrödinger Equation and Wave Function
INTRODUCTION
CONSTRUCTION OF SCHRÖDINGER EQUATION
SOLUTION OF TIMEDEPENDENT EQUATION
PHYSICAL INTERPRETATION OF ψ∗ψ
CONDITIONS ON ALLOWED WAVE FUNCTIONS
BOX NORMALIZATION
CONSERVATION OF PROBABILITY
EXPECTATION VALUE
EHRENFEST’S THEOREM
BASIC POSTULATES
TIME EVOLUTION OF STATIONARY STATES
CONDITIONS FOR ALLOWED TRANSITIONS
ORTHOGONALITY OF TWO STATES
PHASE OF THE WAVE FUNCTION
CLASSICAL LIMIT OF QUANTUM MECHANICS
Operators, Eigenvalues, and Eigenfunctions
INTRODUCTION
LINEAR OPERATORS
COMMUTING AND NONCOMMUTING OPERATORS
SELFADJOINT AND HERMITIAN OPERATORS
DISCRETE AND CONTINUOUS EIGENVALUES
MEANING OF EIGENVALUES AND EIGENFUNCTIONS
PARITY OPERATOR
ALL HERMITIAN HAMILTONIANS HAVE PARITY
SOME OTHER USEFUL OPERATORS
Exactly Solvable Systems I: Bound States
INTRODUCTION
CLASSICAL PROBABILITY DISTRIBUTION
FREE PARTICLE
HARMONIC OSCILLATOR
PARTICLE IN THE POTENTIAL V (x) = x2k, k = 1, 2, · · ·
PARTICLE IN A BOX
PÖSCHL–TELLER POTENTIALS
QUANTUM PENDULUM
CRITERIA FOR THE EXISTENCE OF A BOUND STATE
TIMEDEPENDENT HARMONIC OSCILLATOR
RIGID ROTATOR
Exactly Solvable Systems II: Scattering States
INTRODUCTION
POTENTIAL BARRIER: TUNNEL EFFECT
FINITE SQUAREWELL POTENTIAL
POTENTIAL STEP
LOCALLY PERIODIC POTENTIAL
REFLECTIONLESS POTENTIALS
DYNAMICAL TUNNELING
Matrix Mechanics
INTRODUCTION
LINEAR VECTOR SPACE
MATRIX REPRESENTATION OF OPERATORS AND WAVE FUNCTION
UNITARY TRANSFORMATION
TENSOR PRODUCTS
SCHRÖDINGER EQUATION AND OTHER QUANTITIES IN MATRIX FORM
APPLICATION TO CERTAIN SYSTEMS
DIRAC’S BRA AND KET NOTATIONS
EXAMPLES OF BASIS IN QUANTUM THEORY
PROPERTIES OF KET AND BRA VECTORS
HILBERT SPACE
PROJECTION AND DISPLACEMENT OPERATORS
Various Pictures and Density Matrix
INTRODUCTION
SCHRÖDINGER PICTURE
HEISENBERG PICTURE
INTERACTION PICTURE
COMPARISON OF THREE REPRESENTATIONS
DENSITY MATRIX FOR A SINGLE SYSTEM
DENSITY MATRIX FOR AN ENSEMBLE
TIME EVOLUTION OF DENSITY OPERATOR
A SPIN1/2 SYSTEM
Heisenberg Uncertainty Principle
INTRODUCTION
THE CLASSICAL UNCERTAINTY RELATION
HEISENBERG UNCERTAINTY RELATION
IMPLICATIONS OF UNCERTAINTY RELATION
ILLUSTRATION OF UNCERTAINTY RELATION
THE MODIFIED HEISENBERG RELATION
Momentum Representation
INTRODUCTION
MOMENTUM EIGENFUNCTIONS
SCHRÖDINGER EQUATION
EXPRESSIONS FOR hXi AND hpi
TRANSFORMATION BETWEEN MOMENTUM AND COORDINATE REPRESENTATIONS
OPERATORS IN MOMENTUM REPRESENTATION
MOMENTUM FUNCTION OF SOME SYSTEMS
Wave Packet
INTRODUCTION
PHASE AND GROUP VELOCITIES
WAVE PACKETS AND UNCERTAINTY PRINCIPLE
GAUSSIAN WAVE PACKET
WAVE PACKET REVIVAL
ALMOST PERIODIC WAVE PACKETS
Theory of Angular Momentum
INTRODUCTION
SCALAR WAVE FUNCTION UNDER ROTATIONS
ORBITAL ANGULAR MOMENTUM
EIGENPAIRS OF L2 AND Lz
PROPERTIES OF COMPONENTS OF L AND L2
EIGENSPECTRA THROUGH COMMUTATION RELATIONS
MATRIX REPRESENTATION OF L2, Lz AND L±
WHAT IS SPIN?
SPIN STATES OF AN ELECTRON
SPINORBIT COUPLING
ROTATIONAL TRANSFORMATION
ADDITION OF ANGULAR MOMENTA
ROTATIONAL PROPERTIES OF OPERATORS
TENSOR OPERATORS
THE WIGNER–ECKART THEROEM
Hydrogen Atom
INTRODUCTION
HYDROGEN ATOM IN THREEDIMENSION
HYDROGEN ATOM IN DDIMENSION
FIELD PRODUCED BY A HYDROGEN ATOM
SYSTEM IN PARABOLIC COORDINATES
Approximation Methods I: TimeIndependent Perturbation Theory
INTRODUCTION
THEORY FOR NONDEGENERATE CASE
APPLICATIONS TO NONDEGENERATE LEVELS
THEORY FOR DEGENERATE LEVELS
FIRSTORDER STARK EFFECT IN HYDROGEN
ALTERNATE PERTURBATION THEORIES
Approximation Methods II: TimeDependent Perturbation Theory
INTRODUCTION
TRANSITION PROBABILITY
CONSTANT PERTURBATION
HARMONIC PERTURBATION
ADIABATIC PERTURBATION
SUDDEN APPROXIMATION
THE SEMICLASSICAL THEORY OF RADIATION
CALCULATION OF EINSTEIN COEFFICIENTS
Approximation Methods III: WKB and Asymptotic Methods
INTRODUCTION
PRINCIPLE OF WKB METHOD
APPLICATIONS OF WKB METHOD
WKB QUANTIZATION WITH PERTURBATION
AN ASYMPTOTIC METHOD
Approximation Methods IV: Variational Approach
INTRODUCTION
CALCULATION OF GROUND STATE ENERGY
TRIAL EIGENFUNCTIONS FOR EXCITED STATES
APPLICATION TO HYDROGEN MOLECULE
HYDROGEN MOLECULE ION
Scattering Theory
INTRODUCTION
CLASSICAL SCATTERING CROSSSECTION
CENTRE OF MASS AND LABORATORY COORDINATES SYSTEMS
SCATTERING AMPLITUDE
GREEN’S FUNCTION APPROACH
BORN APPROXIMATION
PARTIAL WAVE ANALYSIS
SCATTERING FROM A SQUAREWELL SYSTEM
PHASESHIFT OF ONEDIMENSIONAL CASE
INELASTIC SCATTERING
Identical Particles
INTRODUCTION
PERMUTATION SYMMETRY
SYMMETRIC AND ANTISYMMETRIC WAVE FUNCTIONS
THE EXCLUSION PRINCIPLE
SPIN EIGENFUNCTIONS OF TWO ELECTRONS
EXCHANGE INTERACTION
EXCITED STATES OF THE HELIUM ATOM
COLLISIONS BETWEEN IDENTICAL PARTICLES
Relativistic Quantum Theory
INTRODUCTION
KLEIN–GORDON EQUATION
DIRAC EQUATION FOR A FREE PARTICLE
NEGATIVE ENERGY STATES
JITTERY MOTION OF A FREE PARTICLE
SPIN OF A DIRAC PARTICLE
PARTICLE IN A POTENTIAL
KLEIN PARADOX
RELATIVISTIC PARTICLE IN A BOX
RELATIVISTIC HYDROGEN ATOM
THE ELECTRON IN A FIELD
SPINORBIT ENERGY
Mysteries in Quantum Mechanics
INTRODUCTION
THE COLLAPSE OF THE WAVE FUNCTION
EINSTEIN–PODOLSKY–ROSEN (EPR) PARADOX
HIDDEN VARIABLES
THE PARADOX OF SCHRÖDINGER’S CAT
BELL’S THEOREM
VIOLATION OF BELL’S THEOREM
RESOLVING EPR PARADOX
Numerical Methods for Quantum Mechanics
INTRODUCTION
MATRIX METHOD FOR COMPUTING STATIONARY STATE SOLUTIONS
FINITEDIFFERENCE TIMEDOMAIN METHOD
TIMEDEPENDENT SCHRÖDINGER EQUATION
QUANTUM SCATTERING
ELECTRONIC DISTRIBUTION OF HYDROGEN ATOM
SCHRÖDINGER EQUATION WITH AN EXTERNAL FIELD
Appendix A: Calculation of Numerical Values of h and kB
Appendix B: A Derivation of the Factor h_/(eh_/kBT − 1)
Appendix C: Bose’s Derivation of Planck’s Law
Appendix D: Distinction between SelfAdjoint and Hermitian Operators
Appendix E: Proof of Schwarz’s Inequality
Appendix F: Eigenvalues of a Symmetric Tridiagonal Matrix—QL Method
Appendix G: Random Number Generators for Desired Distributions
Solutions to Selected Exercises
Index
Concluding Remarks, Bibliography, and Exercises appear at the end of each chapter.
Author(s)
Biography
S. Rajasekar received his B.Sc. and M.Sc. in physics both from the St. Joseph’s College, Tiruchirapalli. In 1987, he received his M.Phil. in physics from Bharathidasan University, Tiruchirapalli. He was awarded a Ph.D. in physics (nonlinear dynamics) from Bharathidasan University in 1992. In 2005, he became a professor at the School of Physics, Bharathidasan University. His recent research focuses on nonlinear dynamics with a special emphasis on nonlinear resonances. He has coauthored a book, and authored or coauthored more than 80 research papers in nonlinear dynamics.
R. Velusamy received his B.Sc. in physics from the Ayya Nadar Janaki Ammal College, Sivakasi in 1972 and M.Sc. in physics from the P.S.G. Arts and Science College, Coimbatore in 1974. He received an M.S. in electrical engineering at the Indian Institute of Technology, Chennai in the year 1981. In the same year, he joined in the Ayya Nadar Janaki Ammal College as an assistant professor in physics. He was awarded an M.Phil. in physics in 1988. He retired in 2010. His research topics are quantum confined systems and wave packet dynamics.
Reviews
"The first volume of this course of quantum mechanics contains basic concepts of quantum mechanics, mathematical formalism, and a wide range of applications to physically important systems. The problems concerning the considered subject are included at the end of each chapter. The textbook is intended for graduate students and also as a reference book. Doubtless advantage of this tutorial is the discussion of such mysteries in quantum mechanics as the collapse of the wave function, EinsteinPodolskyRosen paradox, hidden variables, the paradox of Schrödinger cat, and Bell's theorem. It should be noted the presence of numerical methods in quantum mechanics."
—Zentralblatt MATH 1318"… excellent, uptodate … can be used as either a twotothreesemester graduate text or as a standalone reference book. Quantum Mechanics I: The Fundamentals covers the canonical basics and Quantum Mechanics II: Advanced Topics covers a range of modern developments from introductory quantum field theory through quantum information theory and other quantum technologies, such as quantum metrology and imaging, that are not discussed in other sources … I recommend this set highly."
—Dr. Jonathan P. Dowling, Hearne Professor of Theoretical Physics and CoDirector, Hearne Institute for Theoretical Physics, Louisiana State University, and Author of Schrödinger's Killer App: Race to Build the World's First Quantum Computer"Be assured … these two books by Rajasekar and Velusamy will definitely tell you how to do quantum mechanics."
—Dr. K.P.N. Murthy, Professor, School of Physics, and Director, Centre for Integrated Studies, University of HyderabadVol I quote: "The real strength of this book lies in its scope. Each individual chapter covers the fundamentals of a topic and acts as an excellent reference for quantum researchers….I will certainly ensure that a copy remains on my bookshelf, for when I have a query on any fundamental aspect of quantum mechanics."
—Contemporary Physics (Nov 2017), review by Prof. Thomas Collier, University of ExeterVol II quote: "I found digesting the latter part of this book a most enjoyable experience. Indeed, the final chapters on broad applications of advanced quantum mechanics are entertaining to read and refrain from delving too deeply into the intricacies. The reader is given an enticing glimpse of the research directions one might apply the knowledge of quantum mechanics presented over the two volumes. As with the first volume, I will keep this book close at hand for when I require a concise and mathematically rich guide to these advanced topics of quantum mechanics."
—Contemporary Physics (Nov 2017), review by Prof. Thomas Collier, University of Exeter
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