A Textbook on Modern Quantum Mechanics: A Textbook on Modern Quantum Mechanics

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Acknowledgments
About the Author
1. Introduction to Quantum Mechanics
	1.1. Blackbody Radiation and Planck’s Hypothesis
	1.2. Photoelectric Effect
	1.3. Bohr’s Atomic Model and the Hydrogen Atom
	1.4. Compton Scattering of Photons
	1.5. De Broglie Hypothesis
	1.6. Pauli Exclusion Principle
	1.7. Schrödinger Wave Equation
	1.8. Born Interpretation of Wave Function
	1.9. Heisenberg Uncertainty Principle
	1.10. Davisson and Germer Wave Properties of Electrons
	1.11. Bohr-Sommerfeld Quantization Condition
	1.12. Correspondence Principle
	1.13. Heisenberg Quantum Mechanics
	1.14. Dirac Theory of Quantum Mechanics
	1.15. Important Quantum Mechanical Parameters in SI Units
	1.16. Solved Examples
	1.17. Exercises
2. Wave Mechanics and Its Simple Applications
	2.1. Schrödinger Equation
	2.2. Bound States and Scattering States
	2.3. Probability Density, Probability Current, and Expectation Value
	2.4. Simple Applications of Time-Independent Schrödinger Equation
		2.4.1. Free Particle Motion
		2.4.2. Infinite Potential Well (Particle in a Box)
		2.4.3. The Finite Potential Well
		2.4.4. Step Potential
		2.4.5. Finite Potential Barrier and Tunneling
		2.4.6. Relevance of Free Particle, Potential Wells, and Potential Barriers
	2.5. Periodic Solids and their Band Structures
		2.5.1. The Kronig-Penney Model
		2.5.2. Confined States in Quantum Wells, Wires, and Dots
	2.6. Solved Examples
	2.7. Exercises
3. Matrix Formulation of Quantum Mechanics
	3.1. Matrices and their Basic Algebra
	3.2. Bra and Ket Notations
	3.3. Vectors and Vector Space
		3.3.1. Linearly Independent Vectors
		3.3.2. Orthogonal and Orthonormal Vectors
		3.3.3. Abstract Representation of a Vector
		3.3.4. Outer Product of Vectors
	3.4. Gram-Schmidt Method for Orthogonalization of Vectors
	3.5. Schwarz Inequality
	3.6. Linear Transformation of Vectors
		3.6.1. Eigenvalues and Eigenvectors of a Matrix
		3.6.2. Numerical Method to Find Eigenvalue and Eigenvector
	3.7. Inverse Matrix
	3.8. Orthogonal Matrix
	3.9. Hermitian Matrix
	3.10. Unitary Matrix
	3.11. Diagonalization of a Matrix
	3.12. Cayley-Hamilton Theorem
	3.13. Bilinear, Quadratic, and Hermitian Forms
	3.14. Change of Basis
		3.14.1. Unitary Transformations
	3.15. Infinite-dimensional Space
	3.16. Hilbert Space
		3.16.1. Basis Vectors in Hilbert Space
		3.16.2. Quantum States and Operators in Hilbert Space
		3.16.3. Schrödinger Equation in Matrix Form
	3.17. Statement of Assumptions of Quantum Mechanics
	3.18. General Uncertainty Principle
	3.19. One Dimensional Harmonic Oscillator
	3.20. Solved Examples
	3.21. Exercises
4. Transformations, Conservation Laws, and Symmetries
	4.1. Translation in Space
	4.2. Translation in Time
	4.3. Rotation in Space
	4.4. Quantum Generalization of the Rotation Operator
	4.5. Invariance and Conservation Laws
		4.5.1. Infinitesimal Space Translation
		4.5.2. Infinitesimal Time Translation
		4.5.3. Infinitesimal Rotation
		4.5.4. Conservation of Charge
	4.6. Parity and Space Inversion
		4.6.1. Parity Operator
	4.7. Time-Reversal Operator
		4.7.1. Properties of Antilinear Operator
		4.7.2. Time Reversal Operator for Non-zero Spin Particles
	4.8. Solved Examples
	4.9. Exercises
5. Angular Momentum
	5.1. Orbital Angular Momentum
	5.2. Eigenvalues of Angular Momentum
	5.3. Eigenfunctions of Orbital Angular Momentum
	5.4. General Angular Momentum
	5.5. Spin Angular Momentum
		5.5.1. Pauli Theory of Spin One-half Systems
	5.6. Addition of Angular Momentum
		5.6.1. Clebsch-Gordon Coefficients and their Properties
		5.6.2. Recursion Relations for Clebsch-Gordon Coefficients
		5.6.3. Computation of Clebsch-Gordon Coefficients
	5.7. Solved Examples
	5.8. Exercises
6. Schrödinger Equation for Central Potentials and 3D System
	6.1. Motion in a Central Field
	6.2. Energy Eigenvalues of the Hydrogen Atom
	6.3. Wave Functions of the Hydrogen Atom
	6.4. Radial Probability Density
	6.5. Free Particle Motion
	6.6. Spherically Symmetric Potential Well
	6.7. Electron Confined to a 3D Box
	6.8. Solved Examples
	6.9. Exercises
7. Approximation Methods
	7.1. Perturbation Theory
		7.1.1. Perturbation Theory for Nondegenerate States
			7.1.1.1. First Order Corrections to Energy and Wave Function Ket
			7.1.1.2. Second Order Corrections to Energy and Wave Function Ket
			7.1.1.3. kth Order Corrections to Energy and Wave Function Ket
			7.1.1.4. Anharmonic Oscillator
		7.1.2. Perturbation Theory for Degenerate States
			7.1.2.1. Effect of an Electric Field on the First Excited State in a Hydrogen Atom (Linear Stark Effect)
	7.2. Variation Method
		7.2.1. The Ground State of the Helium Atom
		7.2.2. Rayleigh-Ritz Variational Method
		7.2.3. The Hydrogen Molecule Ion
		7.2.4. Variational Method for Excited States
		7.2.5. Application of Variational Method to the Excited State of a 1D Harmonic Oscillator
	7.3. The W K B Approximations
		7.3.1. The Classical Region
		7.3.2. Alternative Derivation of the WKB Formula
		7.3.3. Non-classical or Tunneling Region
		7.3.4. Connecting Formulae
		7.3.5. Quantum Condition for Bound State
	7.4. Solved Examples
	7.5. Exercises
8. Quantum Theory of Scattering
	8.1. Scattering Cross-Section and Frame of Reference
	8.2. Asymptotic Expansion and Scattering Amplitude
	8.3. Partial Wave Analysis
		8.3.1. Free Particle and Asymptotic Solutions
		8.3.2. Scattering Amplitude and Phase Shift
		8.3.3. Optical Theorem
		8.3.4. Scattering Length
		8.3.5. Scattering by a Square Well Potential
		8.3.6. Scattering by a Hard Sphere Potential
		8.3.7. Interpretation of the Phase Shift
	8.4. Expression for Phase Shift
	8.5. Integral Equation
	8.6. The Born Approximation
		8.6.1. Scattering by Screened Coulomb Potential
		8.6.2. Validity of the Born Approximation
	8.7. Transformation from the Center of Mass Coordinate System to the Laboratory Coordinate System
	8.8. Solved Examples
	8.9. Exercises
9. Quantum Theory of Many Particle Systems
	9.1. System of Indistinguishable Particles
		9.1.1. Non-interacting System of Particles
		9.1.2. Space and Spin Parts of Wave Function
	9.2. The Helium Atom
		9.2.1. Ground State of Helium
		9.2.2. Excited State of Helium
	9.3. Systems of N-Electrons
		9.3.1. Hartree Approximation
		9.3.2. Hartree-Fock Approximation
		9.3.3. Thomas-Fermi Theory
		9.3.4. Thomas-Fermi Model of Atom
		9.3.5. Density Functional Theory
	9.4. Solved Examples
	9.5. Exercises
10. Time-dependent Perturbations and Semi-classical Treatment of Interaction of Field with Matter
	10.1. Time-dependent Potentials
	10.2. Exactly Solvable Time-dependent Two-state Systems
	10.3. Time-dependent Perturbation Theory
		10.3.1. First Order Perturbation
	10.4. Harmonic Perturbation
		10.4.1. Transition Probability
		10.4.2. Fermi’s Golden Rule
	10.5. Constant Perturbation
		10.5.1. Fermi’s Golden Rule
	10.6. Semi-classical Treatment of Interaction of a Field with Matter
		10.6.1. Absorption and Stimulated Emission
		10.6.2. Electric Dipole Approximation
	10.7. Spontaneous Emission and Einstein Coefficients
	10.8. Dipole Selection Rules
	10.9. Solved Examples
	10.10. Exercises
11. Relativistic Quantum Mechanics
	11.1. The Klein-Gordon Equation
		11.1.1. Probability Density and Probability Current
		11.1.2. The Klein-Gordon Equation in a Coulombic Field
	11.2. The Dirac Equation
		11.2.1. Derivation of the Dirac Equation
		11.2.2. Covariant Form of the Dirac Equation
		11.2.3. Probability Density and Probability Current
	11.3. Free Particle Solutions of the Dirac Equation
		11.3.1. Positive and Negative Energy Eigenvalues
	11.4. The Dirac Equation and the Constants of Motion
	11.5. Spin Magnetic Moment (the Dirac Electron  in an Electromagnetic Field)
	11.6. Spin-Orbit Interaction Energy
	11.7. Solution of the Dirac Equation for Central Potential
		11.7.1. Hydrogen-like Atom
	11.8. Solved Examples
	11.9. Exercises
12. Quantization of Fields and Second Quantization
	12.1. Quantization of an Electromagnetic Field
		12.1.1. Field Operators
	12.2. Second Quantization
		12.2.1. Second Quantization of the Schrödinger Equation for Bosons
		12.2.2. Second Quantization of the Schrödinger Equation for Fermions
		12.2.3. Matrix Representation of Fermionic Operators
		12.2.4. Number Operator
	12.3. System of Weakly Interacting Bosons
	12.4. Free Electron System
	12.5. Solved Examples
	12.6. Exercises
Annexure A: Useful Formulae
	I. Table of Integrals
	II. Series and Expansions
	III. Basic Functional Relations
	IV. Coordinate Systems
Annexure B: Dirac Delta Function
	I. Properties of Delta Function
	II. Representation of Delta Function
Answers to Exercises
	Chapter 1
	Chapter 2
	Chapter 3
	Chapter 5
	Chapter 6
	Chapter 7
	Chapter 8
	Chapter 9
	Chapter 10
	Chapter 11
	Chapter 12
Bibliography
Index