Quantum Mechanics I: The Fundamentals

Description

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. This fully updated new edition addresses many topics not typically found in books at this level, including: Bound state solutions of quantum pendulum Morse oscillator 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 An asymptotic method for slowly varying potentials Klein paradox, Einstein-Podolsky-Rosen (EPR) paradox, and Bell’s theorem Delayed-choice experiments Fractional quantum mechanics 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. Cover Half Title Title Page Copyright Page Dedication Contents Preface About the Authors 1. Why Was Quantum Mechanics Developed? 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 2. Schrodinger Equation and Wave Function 2.1. Introduction 2.2. Construction of Schrodinger 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. A Special Feature of Occurrence of i in the Schrodinger Equation 2.8. Conservation of Probability 2.9. Expectation Value 2.10. Ehrenfest’s Theorem 2.11. Basic Postulates 2.12. Time Evolution of Stationary States 2.13. Conditions for Allowed Transitions 2.14. Orthogonality of Two States 2.15. Phase of the Wave Function 2.16. Classical Limit of Quantum Mechanics 2.17. Concluding Remarks 2.18. Bibliography 2.19. Exercises 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. Some Other Useful Operators 3.9. Concluding Remarks 3.10. Bibliography 3.11. Exercises 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) = x2k, k = 1, 2, . . . 4.6. Particle in a Box 4.7. Morse Oscillator 4.8. Poschl–Teller Potentials 4.9. Quantum Pendulum 4.10. Criteria for the Existence of a Bound State 4.11. Time-Dependent Harmonic Oscillator 4.12. Damped and Forced Linear Harmonic Oscillator 4.13. Two-Dimensional Systems 4.14. Rigid Rotator 4.15. Concluding Remarks 4.16. Bibliography 4.17. Exercises 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 Tunnelling 5.8. Concluding Remarks 5.9. Bibliography 5.10. Exercises 6. Matrix Mechanics 6.1. Introduction 6.2. Linear Vector Space and Tensor Products 6.3. Matrix Representation of Operators and Wave Function 6.4. Unitary Transformation 6.5. Schrodinger Equation and Other Quantities in Matrix Form 6.6. Application to Certain Systems 6.7. Dirac’s Bra and Ket Notations 6.8. Dimensions of Kets and Bras 6.9. Hilbert Space 6.10. Symmetry Operators in Hilbert Space 6.11. Projection and Displacement Operators 6.12. Quaternionic Quantum Mechanics 6.13. Concluding Remarks 6.14. Bibliography 6.15. Exercises 7. Various Pictures and Density Matrix 7.1. Introduction 7.2. Schrodinger 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. Concluding Remarks 7.10. Bibliography 7.11. Exercises 8. Heisenberg Uncertainty Principle 8.1. Introduction 8.2. The Classical Uncertainty Relation 8.3. Heisenberg Uncertainty Relation 8.4. Condition for Minimum Uncertainty Product 8.5. Implications of Uncertainty Relation 8.6. Illustration of Uncertainty Relation 8.7. Some Extensions of Uncertainty Relation 8.8. The Modified Heisenberg Relation 8.9. Concluding Remarks 8.10. Bibliography 8.11. Exercises 9. Momentum Representation 9.1. Introduction 9.2. Momentum Eigenfunctions 9.3. Schrodinger Equation 9.4. Expressions for and 9.5. Transformation Between Momentum and Coordinate Representations 9.6. Operators in Momentum Representation 9.7. Momentum Function of Some Systems 9.8. Concluding Remarks 9.9. Bibliography 9.10. Exercises 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 11. Theory of Angular Momentum 11.1. Introduction 11.2. Scalar Wave Function Under Rotations 11.3. Orbital Angular Momentum 11.4. Spin Angular Momentum 11.5. Spin-Orbit Coupling 11.6. Addition of Angular Momenta 11.7. Rotational Transformation of a Vector Wave Function and Spin 11.8. Rotational Properties of Vector Operators 11.9. Tensor Operators and the Wigner–Eckart Theorem 11.10. Concluding Remarks 11.11. Bibliography 11.12. Exercises 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 13. Approximation Methods I: Time-Independent Perturbation Theory 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 14. Approximation Methods II: Time-Dependent Perturbation Theory 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 14.8. Concluding Remarks 14.9. Bibliography 14.10. Exercises 15. Approximation Methods III: WKB and Asymptotic Methods 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 16. Approximation Methods IV: Variational Approach 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 17. Scattering Theory 17.1. Introduction 17.2. Classical Scattering Cross-Section 17.3. Centre of Mass and Laboratory Coordinates Systems 17.4. Scattering Amplitude 17.5. Green’s Function Approach 17.6. Born Approximation 17.7. Partial Wave Analysis 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 18. Identical Particles 18.1. Introduction 18.2. Permutation Symmetry 18.3. Symmetric and Antisymmetric Wave Functions 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. Uncertainty Principle for a System of Identical Particles 18.10. Concluding Remarks 18.11. Bibliography 18.12. Exercises 19. Relativistic Quantum Theory 19.1. Introduction 19.2. Klein–Gordon Equation 19.3. Dirac Equation for a Free Particle 19.4. Minimum Uncertainty Wave Packet 19.5. Spin of a Dirac Particle 19.6. Particle in a Potential 19.7. Klein Paradox 19.8. Relativistic Particle in a Box 19.9. Relativistic Hydrogen Atom 19.10. The Electron in a Field 19.11. Spin-Orbit Energy 19.12. Relativistic Quaternionic Quantum Mechanics 19.13. Concluding Remarks 19.14. Bibliography 19.15. Exercises 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 Schrodinger’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 21. Delayed-Choice Experiments 21.1. Introduction 21.2. Single Slit and Double Slit Experiments 21.3. Quantum Mechanical Explanation 21.4. Experiment With Mach–Zehnder Interferometer 21.5. Delayed-Choice Experiment 21.6. Delayed-Choice Quantum Eraser 21.7. Concluding Remarks 21.8. Bibliography 21.9. Exercises 22. Fractional Quantum Mechanics 22.1. Introduction 22.2. Integer and Fractional Diffusion Equations 22.3. Wave Function and Kernel 22.4. Space Fractional Schrodinger Equation 22.5. Solutions of Certain Space Fractional Schrodinger Equations 22.6. Fractional Schrodinger Equation in Relativistic Quantum Mechanics 22.7. Time Fractional Schrodinger Equation 22.8. Solutions of Certain Time Fractional Schrodinger Equations 22.9. Space-Time Fractional Schrodinger Equation 22.10. Concluding Remarks 22.11. Bibliography 22.12. Exercises 23. Numerical Methods for Quantum Mechanics 23.1. Introduction 23.2. Matrix Method for Computing Stationary State Solutions 23.3. Finite-Difference Time-Domain Method 23.4. Time-Dependent Schrodinger Equation 23.5. Quantum Scattering 23.6. Electronic Distribution of Hydrogen Atom 23.7. Schrodinger Equation with an External Field 23.8. Concluding Remarks 23.9. Bibliography 23.10. Exercises A. Calculation of Numerical Values of h and kB B. A Derivation of the Factor hν/(ehν/kBT − 1) C. Bose’s Derivation of Planck’s Law D. Distinction Between Self-Adjoint and Hermitian Operators E. Proof of Schwarz’s Inequality F. Calculation of Eigenvalues of a Symmetric Tridiagonal Matrix–QL Method G. Random Number Generators for Desired Distributions Solutions to Selected Exercises Index