Quantum Mechanics I The Fundamentals

Quantum Mechanics I: The Fundamentals provides a graduate-level account of the behavior of matter and energy at the molecular, atomic, nuclear, and sub-nuclear 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 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 students’ 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 students with a broad, up-to-date introduction to quantum mechanics. Print Versions of this book also include access to the ebook version.

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 WAVE-PARTICLE DUALITY Schrödinger Equation and Wave Function INTRODUCTION CONSTRUCTION OF SCHRÖDINGER EQUATION SOLUTION OF TIME-DEPENDENT EQUATION PHYSICAL INTERPRETATION OF ?*? CONDITIONS ON ALLOWED WAVE FUNCTIONS BOX NORMALIZATION CONSERVATION OF PROBABILITY EXPECTATION VALUEEHRENFEST’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 SELF-ADJOINT 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 TIME-DEPENDENT HARMONIC OSCILLATOR RIGID ROTATOR Exactly Solvable Systems II: Scattering States INTRODUCTION POTENTIAL BARRIER: TUNNEL EFFECT FINITE SQUARE-WELL 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 SPIN-1/2 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 SPIN-ORBIT COUPLINGROTATIONAL TRANSFORMATIONADDITION OF ANGULAR MOMENTA ROTATIONAL PROPERTIES OF OPERATORS TENSOR OPERATORS THE WIGNER–ECKART THEROEM Hydrogen Atom INTRODUCTION HYDROGEN ATOM IN THREE-DIMENSION HYDROGEN ATOM IN D-DIMENSION FIELD PRODUCED BY A HYDROGEN ATOM SYSTEM IN PARABOLIC COORDINATES Approximation Methods I: Time-Independent Perturbation TheoryINTRODUCTION THEORY FOR NONDEGENERATE CASE APPLICATIONS TO NONDEGENERATE LEVELS THEORY FOR DEGENERATE LEVELS FIRST-ORDER STARK EFFECT IN HYDROGEN ALTERNATE PERTURBATION THEORIES Approximation Methods II: Time-Dependent 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 CROSS-SECTION CENTRE OF MASS AND LABORATORY COORDINATES SYSTEMS SCATTERING AMPLITUDE GREEN’S FUNCTION APPROACHBORN APPROXIMATION PARTIAL WAVE ANALYSISSCATTERING FROM A SQUARE-WELL SYSTEM PHASE-SHIFT OF ONE-DIMENSIONAL 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 SPIN-ORBIT 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 FINITE-DIFFERENCE TIME-DOMAIN METHOD TIME-DEPENDENT 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 LawAppendix D: Distinction between Self-Adjoint 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.

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.