Physics “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.” WITH VITALSOURCE ® EBOOK Quantum Mechanics I Dr. Dowling, Hearne Professor of Theoretical Physics, Louisiana State University The Fundamentals Rajasekar • Velusamy “… these two books by Rajasekar and Velusamy will definitely tell you how to do quan- tum mechanics. 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 The Fundamentals Quantum Mechanics I • 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. • Access online or download to your smartphone, tablet or PC/Mac • Search the full text of this and other titles you own • Make and share notes and highlights • Copy and paste text and figures for use in your own documents • Customize your view by changing font size and layout K24364 S.indd 1 11/5/14 1:23 PM www.com Quantum Mechanics I www.com Quantum Mechanics I The Fundamentals S.com CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U. Government works Version Date: 20141118 International Standard Book Number-13: 978-1-4822-6338-1 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources.
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Visit the Taylor & Francis Web site at http://www.com and the CRC Press Web site at http://www.com To our wives.com Contents Preface xvii About the Authors xxi Chapter 1 Why Was Quantum Mechanics Developed? 1 1.2 BLACK BODY RADIATION 2 1.5 FRANCK–HERTZ EXPERIMENT 12 1.6 STERN–GERLACH EXPERIMENT 14 1.9 SPECIFIC HEAT CAPACITY 21 1.10 DE BROGLIE WAVES 22 1.12 WAVE-PARTICLE DUALITY 24 1.15 EXERCISES 29 Chapter 2 Schrödinger Equation and Wave Function 31 2.2 CONSTRUCTION OF SCHRÖDINGER EQUATION 32 2.3 SOLUTION OF TIME-DEPENDENT EQUATION 34 ∗ 2.4 PHYSICAL INTERPRETATION OF ψ ψ 35 2.5 CONDITIONS ON ALLOWED WAVE FUNCTIONS 36 2.6 BOX NORMALIZATION 38 vii www.com viii Contents 2.7 CONSERVATION OF PROBABILITY 39 2.11 TIME EVOLUTION OF STATIONARY STATES 51 2.12 CONDITIONS FOR ALLOWED TRANSITIONS 52 2.13 ORTHOGONALITY OF TWO STATES 53 2.14 PHASE OF THE WAVE FUNCTION 55 2.15 CLASSICAL LIMIT OF QUANTUM MECHANICS 56 2.18 EXERCISES 60 Chapter 3 Operators, Eigenvalues and Eigenfunctions 65 3.3 COMMUTING AND NONCOMMUTING OPERATORS 69 3.4 SELF-ADJOINT AND HERMITIAN OPERATORS 73 3.5 DISCRETE AND CONTINUOUS EIGENVALUES 76 3.6 MEANING OF EIGENVALUES AND EIGENFUNCTIONS 78 3.8 ALL HERMITIAN HAMILTONIANS HAVE PARITY 83 3.9 SOME OTHER USEFUL OPERATORS 84 3.12 EXERCISES 87 Chapter 4 Exactly Solvable Systems I: Bound States 91 4.2 CLASSICAL PROBABILITY DISTRIBUTION 92 4.5 PARTICLE IN THE POTENTIAL V (x) = x , k = 1, 2, · · · 105 4.6 PARTICLE IN A BOX 107 www.com Contents ix 4.7 PÖSCHL–TELLER POTENTIALS 114 4.9 CRITERIA FOR THE EXISTENCE OF A BOUND STATE 117 4.10 TIME-DEPENDENT HARMONIC OSCILLATOR 120 4.14 EXERCISES 129 Chapter 5 Exactly Solvable Systems II: Scattering States 135 5.2 POTENTIAL BARRIER: TUNNEL EFFECT 136 5.3 FINITE SQUARE-WELL POTENTIAL 143 5.5 LOCALLY PERIODIC POTENTIAL 153 5.10 EXERCISES 164 Chapter 6 Matrix Mechanics 167 6.2 LINEAR VECTOR SPACE 168 6.3 MATRIX REPRESENTATION OF OPERATORS AND WAVE FUNCTION 170 6.6 SCHRÖDINGER EQUATION AND OTHER QUANTITIES IN MATRIX FORM 174 6.7 APPLICATION TO CERTAIN SYSTEMS 176 6.8 DIRAC’S BRA AND KET NOTATIONS 179 6.9 EXAMPLES OF BASIS IN QUANTUM THEORY 180 6.10 PROPERTIES OF KET AND BRA VECTORS 181 6.11 HILBERT SPACE 183 www.12 PROJECTION AND DISPLACEMENT OPERATORS 184 6.15 EXERCISES 189 Chapter 7 Various Pictures and Density Matrix 191 7.2 SCHRÖDINGER PICTURE 191 7.5 COMPARISON OF THREE REPRESENTATIONS 199 7.6 DENSITY MATRIX FOR A SINGLE SYSTEM 199 7.7 DENSITY MATRIX FOR AN ENSEMBLE 202 7.8 TIME EVOLUTION OF DENSITY OPERATOR 204 7.12 EXERCISES 207 Chapter 8 Heisenberg Uncertainty Principle 209 8.2 THE CLASSICAL UNCERTAINTY RELATION 210 8.3 HEISENBERG UNCERTAINTY RELATION 211 8.4 IMPLICATIONS OF UNCERTAINTY RELATION 215 8.5 ILLUSTRATION OF UNCERTAINTY RELATION 217 8.6 THE MODIFIED HEISENBERG RELATION 220 8.9 EXERCISES 224 Chapter 9 Momentum Representation 227 9.3 SCHRÖDINGER EQUATION 230 9.4 EXPRESSIONS FOR hXi AND hpi 231 www.com Contents xi 9.5 TRANSFORMATION BETWEEN MOMENTUM AND CO- ORDINATE REPRESENTATIONS 233 9.6 OPERATORS IN MOMENTUM REPRESENTATION 234 9.7 MOMENTUM FUNCTION OF SOME SYSTEMS 235 9.10 EXERCISES 239 Chapter 10 Wave Packet 241 10.2 PHASE AND GROUP VELOCITIES 241 10.3 WAVE PACKETS AND UNCERTAINTY PRINCIPLE 245 10.4 GAUSSIAN WAVE PACKET 247 10.5 WAVE PACKET REVIVAL 254 10.6 ALMOST PERIODIC WAVE PACKETS 255 10.9 EXERCISES 259 Chapter 11 Theory of Angular Momentum 261 11.2 SCALAR WAVE FUNCTION UNDER ROTATIONS 262 11.3 ORBITAL ANGULAR MOMENTUM 266 2 11.4 EIGENPAIRS OF L AND Lz 267 2 11.5 PROPERTIES OF COMPONENTS OF L AND L 268 11.6 EIGENSPECTRA THROUGH COMMUTATION RELATIONS 272 2 11.7 MATRIX REPRESENTATION OF L , Lz AND L± 276 11.8 WHAT IS SPIN? 278 11.9 SPIN STATES OF AN ELECTRON 283 11.10 SPIN-ORBIT COUPLING 285 11.12 ADDITION OF ANGULAR MOMENTA 288 11.13 ROTATIONAL PROPERTIES OF OPERATORS 294 11.14 TENSOR OPERATORS 295 www.com xii Contents 11.15 THE WIGNER–ECKART THEOREM 298 11.18 EXERCISES 301 Chapter 12 Hydrogen Atom 305 12.2 HYDROGEN ATOM IN THREE-DIMENSION 305 12.3 HYDROGEN ATOM IN D -DIMENSION 320 12.4 FIELD PRODUCED BY A HYDROGEN ATOM 321 12.5 SYSTEM IN PARABOLIC COORDINATES 324 12.8 EXERCISES 328 Chapter 13 Approximation Methods I: Time-Independent Perturbation Theory 331 13.2 THEORY FOR NONDEGENERATE CASE 332 13.3 APPLICATIONS TO NONDEGENERATE LEVELS 339 13.4 THEORY FOR DEGENERATE LEVELS 342 13.5 FIRST-ORDER STARK EFFECT IN HYDROGEN 345 13.6 ALTERNATE PERTURBATION THEORIES 350 13.9 EXERCISES 354 Chapter 14 Approximation Methods II: Time-Dependent Perturbation Theory 359 14.6 SUDDEN APPROXIMATION 371 www.com Contents xiii 14.7 THE SEMICLASSICAL THEORY OF RADIATION 373 14.8 CALCULATION OF EINSTEIN COEFFICIENTS 375 14.11 EXERCISES 377 Chapter 15 Approximation Methods III: WKB and Asymptotic Methods 381 15.2 PRINCIPLE OF WKB METHOD 381 15.3 APPLICATIONS OF WKB METHOD 385 15.4 WKB QUANTIZATION WITH PERTURBATION 392 15.5 AN ASYMPTOTIC METHOD 394 15.8 EXERCISES 397 Chapter 16 Approximation Methods IV: Variational Approach 399 16.2 CALCULATION OF GROUND STATE ENERGY 399 16.3 TRIAL EIGENFUNCTIONS FOR EXCITED STATES 404 16.4 APPLICATION TO HYDROGEN MOLECULE 407 16.5 HYDROGEN MOLECULE ION 410 16.7 EXERCISES 413 Chapter 17 Scattering Theory 415 17.2 CLASSICAL SCATTERING CROSS-SECTION 417 17.3 CENTRE OF MASS AND LABORATORY COORDINATES SYSTEMS 418 17.5 GREEN’S FUNCTION APPROACH 424 17.6 BORN APPROXIMATION 427 www.com xiv Contents 17.7 PARTIAL WAVE ANALYSIS 431 17.8 SCATTERING FROM A SQUARE-WELL SYSTEM 437 17.9 PHASE-SHIFT OF ONE-DIMENSIONAL CASE 440 17.13 EXERCISES 443 Chapter 18 Identical Particles 445 18.3 SYMMETRIC AND ANTISYMMETRIC WAVE FUNCTIONS 449 18.4 THE EXCLUSION PRINCIPLE 451 18.5 SPIN EIGENFUNCTIONS OF TWO ELECTRONS 452 18.7 EXCITED STATES OF THE HELIUM ATOM 456 18.8 COLLISIONS BETWEEN IDENTICAL PARTICLES 458 18.11 EXERCISES 461 Chapter 19 Relativistic Quantum Theory 463 19.2 KLEIN–GORDON EQUATION 464 19.3 DIRAC EQUATION FOR A FREE PARTICLE 468 19.4 NEGATIVE ENERGY STATES 476 19.5 JITTERY MOTION OF A FREE PARTICLE 477 19.6 SPIN OF A DIRAC PARTICLE 480 19.7 PARTICLE IN A POTENTIAL 482 19.9 RELATIVISTIC PARTICLE IN A BOX 487 19.10 RELATIVISTIC HYDROGEN ATOM 490 19.11 THE ELECTRON IN A FIELD 492 19.12 SPIN-ORBIT ENERGY 493 www.com Contents xv 19.15 EXERCISES 497 Chapter 20 Mysteries in Quantum Mechanics 499 20.2 THE COLLAPSE OF THE WAVE FUNCTION 500 20.3 EINSTEIN–PODOLSKY–ROSEN (EPR) PARADOX 501 20.5 THE PARADOX OF SCHRÖDINGER’S CAT 507 20.7 VIOLATION OF BELL’S THEOREM 510 20.8 RESOLVING EPR PARADOX 515 20.11 EXERCISES 518 Chapter 21 Numerical Methods for Quantum Mechanics 519 21.2 MATRIX METHOD FOR COMPUTING STATIONARY STATE SOLUTIONS 521 21.3 FINITE-DIFFERENCE TIME-DOMAIN METHOD 529 21.4 TIME-DEPENDENT SCHRÖDINGER EQUATION 535 21.6 ELECTRONIC DISTRIBUTION OF HYDROGEN ATOM 548 21.7 SCHRÖDINGER EQUATION WITH AN EXTERNAL FIELD 551 21.10 EXERCISES 558 Appendix A Calculation of Numerical Values of h and kB 561 Appendix B A Derivation of the Factor hν/(ehν/kB T − 1) 563 www.com xvi Contents Appendix C Bose’s Derivation of Planck’s Law 565 Appendix D Distinction Between Self-Adjoint and Hermitian Operators 567 Appendix E Proof of Schwarz’s Inequality 569 Appendix F Eigenvalues of a Symmetric Tridiagonal Matrix—QL Method 571 Appendix G Random Number Generators for Desired Distributions 575 Solutions to Selected Exercises 579 Index 587 www.com 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.