QUANTUM MECHANICS SECOND EDITION G. ARULDHAS

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2 QUANTUM MECHANICS SECOND EDITION G. ARULDHAS Formerly, Professor and Head of Physics and Dean, Faculty of Science University of Kerala New Delhi

3 QUANTUM MECHANICS, 2nd Ed. G. Aruldhas 2009 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN The export rights of this book are vested solely with the publisher. Eighth Printing (Second Edition) November, 2008 Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi and Printed by Rajkamal Electric Press, B-35/9, G.T. Karnal Road Industrial Area, Delhi

4 To My Parents Gnanasigamoni & Rajammal who taught me the merits of discipline and the rewards of education

5 Contents Preface Preface to the First Edition xiii xv 1 Origin of the Quantum Theory Limitations of Classical Physics Planck s Quantum Hypothesis Einstein s Theory of Photoelectric Effect Compton Effect Quantum Theory of Specific Heat Bohr Model of Hydrogen Atom Existence of Stationary States Wilson Sommerfeld Quantization Rule Elliptic Orbits of Hydrogen Atom The Harmonic Oscillator The Rigid Rotator Particle in a Box The Correspondence Principle The Stern-Gerlach Experiment Inadequacy of Quantum Theory 18 Worked Examples 19 Review Questions 22 Problems 23 2 Wave Mechanical Concepts Wave Nature of Particles The Uncertainty Principle 27 v

6 vi Contents 2.3 The Principle of Superposition Wave Packet Time-Dependent Schrödinger Equation Interpretation of the Wave Function Ehrenfest s Theorem Time-Independent Schrödinger Equation Stationary States Admissibility Conditions on the Wave Function 45 Worked Examples 45 Review Questions 54 Problems 55 3 General Formalism of Quantum Mechanics Linear Vector Space Linear Operator Eigenfunctions and Eigenvalues Hermitian Operator Postulates of Quantum Mechanics Simultaneous Measurability of Observables General Uncertainty Relation Dirac s Notation Equations of Motion Momentum Representation 79 Worked Examples 83 Review Questions 95 Problems 96 4 One-Dimensional Energy Eigenvalue Problems 4.1 Square-Well Potential with Rigid Walls Square-Well Potential with Finite Walls Square Potential Barrier Alpha Emission Bloch Waves in a Periodic Potential Kronig-Penney Square-Well Periodic Potential Linear Harmonic Oscillator: Schrödinger Method Linear Harmonic Oscillator: Operator Method The Free Particle 123 Worked Examples 124 Review Questions 138 Problems 139

7 Contents vii 5 Three-Dimensional Energy Eigenvalue Problems 5.1 Particle Moving in a Spherically Symmetric Potential System of Two Interacting Particles Rigid Rotator Hydrogen Atom Hydrogenic Orbitals The Free Particle Three-Dimensional Square-Well Potential The Deuteron 162 Worked Examples 163 Review Questions 174 Problems Heisenberg Method The Heisenberg Method Matrix Representation of Wave Function Matrix Representation of Operator Properties of Matrix Elements Schrödinger Equation in Matrix Form Eigenvalue Problems Unitary Transformations Linear Harmonic Oscillator: Matrix Method 185 Worked Examples 189 Review Questions 191 Problems Symmetry and Conservation Laws Symmetry Transformations Translation in Space: Conservation of Linear Momentum Translation in Time: Conservation of Energy Rotation in Space: Conservation of Angular Momentum Space Inversion: Parity Conservation Time Reversal 201 Worked Examples 203 Review Questions 205 Problems 205

8 viii Contents 8 Angular Momentum The Angular Momentum Operators Angular Momentum Commutation Relations Eigenvalues and Eigenfunctions of L 2 and L z General Angular Momentum Eigenvalues of J 2 and J z Angular Momentum Matrices Spin Angular Momentum Spin Vectors for Spin-(1/2) System Addition of Angular Momenta 218 Worked Examples 224 Review Questions 234 Problems Time-Independent Perturbation Theory Basic Concepts Nondegenerate Energy Levels Anharmonic Oscillator: First-order Correction The Ground State of Helium Effect of Electric Field on the Ground State of Hydrogen Degenerate Energy Levels Effect of Electric Field on the n = 2 State of Hydrogen Spin-Orbit Interaction 250 Worked Examples 252 Review Questions 261 Problems The Variation Method The Variational Principle Rayleigh Ritz Method Variation Method for Excited States The Hellmann Feynman Theorem The Ground State of Helium The Ground State of the Deuteron 270 Worked Examples 272 Review Questions 278 Problems 279

9 Contents ix 11 WKB Approximation The WKB Method The Connection Formulas Validity of WKB Method Barrier Penetration Alpha Emission Bound States in a Potential Well 288 Worked Examples 290 Review Questions 292 Problems Time-Dependent Perturbation Theory Introduction First-Order Perturbation Harmonic Perturbation Transitions to Continuum States Absorption and Emission of Radiation Einstein s A and B Coefficients Selection Rules Rayleigh Scattering Raman Scattering 310 Worked Examples 313 Review Questions 318 Problems Many Electron Atoms Indistinguishable Particles Pauli Principle Inclusion of Spin Spin Functions for Two-Electrons Spin Functions for Three-Electrons The Helium Atom Central Field Approximation Thomas Fermi Model of the Atom Hartree Equation Hartree Fock Equation 335 Worked Examples 337 Review Questions 346 Problems 346

10 x Contents 14 Scattering Scattering Cross-Section Scattering Amplitude Partial Waves Scattering by a Central Potential: Partial Wave Analysis Significant Number of Partial Waves Scattering by an Attractive Square-well Potential Breit-Wigner Formula Scattering Length Expression for Phase Shifts Integral Equation The Born Approximation Scattering by Screened Coulomb Potential Validity of Born Approximation Laboratory and Centre of Mass Coordinate Systems 367 Worked Examples 370 Review Questions 376 Problems Relativistic Wave Equations Klein Gordon Equation Interpretation of the Klein Gordon Equation Particle in a Coulomb Field Dirac s Equation for a Free Particle Dirac Matrices Covariant Form of Dirac Equation Probability Density Plane Wave Solution Negative Energy States Spin of the Dirac Particle Magnetic Moment of the Electron Spin Orbit Interaction Radial Equation for an Electron in a Central Potential The Hydrogen Atom Lamb Shift 404 Worked Examples 404 Review Questions 408 Problems 409

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