Saydanzad Group Direct Method

Course Notes

Table of Contents

Chapter 1. Mathematical Tools of Quantum Mechanics

  1. 1.1 Introduction
  2. 1.2 Postulate I: States and Hilbert Space
  3. 1.3 Postulate II: Observables as Operators
  4. 1.4 Commutators and the Heisenberg Uncertainty Principle
  5. 1.5 Postulate III and IV: Measurement and State Reduction
  6. 1.6 Postulate V: Time Evolution
  7. 1.7 Postulate VI: Composite Systems
  8. 1.8 Homework

Chapter 2. Simple Model Systems: Solving the Schrödinger Equation

  1. 2.1 Introduction
  2. 2.2 Stationary States
  3. 2.3 The Free Particle
  4. 2.4 The Infinite Square Well
  5. 2.5 The Finite Square Well
  6. 2.6 The Potential Barrier and Tunnelling
  7. 2.7 The Harmonic Oscillator
  8. 2.8 Summary
  9. 2.9 Exercises
  10. 2.10 Project: Solving One-Dimensional Problems Numerically

Chapter 3. Angular Momentum and Central Potentials

  1. 3.1 Introduction
  2. 3.2 The Angular Momentum Operators
  3. 3.3 Ladder Operators
  4. 3.4 Matrix Representations
  5. 3.5 Orbital Angular Momentum in Spherical Coordinates
  6. 3.6 Parity and Selection Rules
  7. 3.7 Spin
  8. 3.8 Central Potentials
  9. 3.9 Summary
  10. 3.10 Exercises
  11. 3.11 Project: The Centrifugal Barrier

Chapter 4. The Hydrogen Atom

  1. 4.1 Introduction
  2. 4.2 The Two-Body Problem and the Reduced Mass
  3. 4.3 The Time-Independent Schrödinger Equation
  4. 4.4 Separation of Variables
  5. 4.5 Solving the Angular Equation
  6. 4.6 Solving the Radial Equation
  7. 4.7 The Complete Solution
  8. 4.8 Atomic Units
  9. 4.9 Rydberg States
  10. 4.10 Summary
  11. 4.11 Exercises
  12. 4.12 Project: Hydrogen, Numerically and Exactly

Chapter 5. Many-Electron Atoms

  1. 5.1 Introduction
  2. 5.2 Identical Particles and Antisymmetry
  3. 5.3 Helium: The Hamiltonian
  4. 5.4 First Approximation: Independent Electrons
  5. 5.5 Second Approximation: First-Order Perturbation Theory
  6. 5.6 Third Approximation: The Variational Method
  7. 5.7 Exchange: Singlet and Triplet Helium
  8. 5.8 Heavier Atoms: Screening and the Quantum Defect
  9. 5.9 Term Symbols
  10. 5.10 States Inside a Continuum
  11. 5.11 Summary
  12. 5.12 Exercises
  13. 5.13 Project: Screening and the Quantum Defect

Chapter 6. Time-Dependent Potentials and Light–Matter Interaction

  1. 6.1 Introduction
  2. 6.2 The Coupled Amplitude Equations
  3. 6.3 How Light Couples to an Atom
  4. 6.4 Two Levels, Solved Exactly
  5. 6.5 Perturbation Theory and the Golden Rule
  6. 6.6 Summary
  7. 6.7 Exercises
  8. 6.8 Project: Driving a Two-Level Atom

Chapter 7. Molecular Structure, Vibration and Rotation

  1. 7.1 Introduction
  2. 7.2 The Molecular Hamiltonian
  3. 7.3 The Born–Oppenheimer Separation
  4. 7.4 Molecular Orbitals: the LCAO Method
  5. 7.5 Molecular Orbital Labels
  6. 7.6 Vibration
  7. 7.7 Rotation
  8. 7.8 Polyatomic Molecules: Normal Modes
  9. 7.9 Symmetry Workflow for Molecular Spectroscopy
  10. 7.10 The Three Energy Scales
  11. 7.11 Summary
  12. 7.12 Exercises
  13. 7.13 Project: A Potential Energy Curve and Its Levels

Chapter 8. Rotational and Vibrational Spectroscopy

  1. 8.1 Introduction
  2. 8.2 Radiative Processes and Einstein Coefficients
  3. 8.3 Rotational Spectroscopy
  4. 8.4 Vibrational Spectroscopy
  5. 8.5 Normal Modes and CO2
  6. 8.6 Summary
  7. 8.7 Exercises
  8. 8.8 Project: The Normal Modes of CO2

Chapter 9. Electronic Spectroscopy

  1. 9.1 Introduction
  2. 9.2 What an Electronic Spectrum Measures
  3. 9.3 Factorising the Transition Dipole
  4. 9.4 The Franck–Condon Principle
  5. 9.5 Excited-State Relaxation Pathways
  6. 9.6 Photoelectron Spectroscopy
  7. 9.7 Molecular Rydberg Series
  8. 9.8 Summary
  9. 9.9 Exercises
  10. 9.10 Project: Franck–Condon Factors

Chapter 10. Rydberg and Continuum States; Fano Resonances

  1. 10.1 Introduction
  2. 10.2 Continuum Normalization
  3. 10.3 A Discrete State Coupled to a Continuum
  4. 10.4 Fano's Diagonalisation
  5. 10.5 The Fano Profile
  6. 10.6 A Rydberg Series of Autoionizing States
  7. 10.7 Two Series, and Why They Differ
  8. 10.8 Summary
  9. 10.9 Exercises
  10. 10.10 Project: Fano Profiles from a Measured Series

Chapter 11. Time-Dependent Rydberg–Fano Dynamics in CO\(_2\)

  1. 11.1 Introduction
  2. 11.2 The Physical Setup
  3. 11.3 The Channel Basis
  4. 11.4 Deriving the Coupled Equations
  5. 11.5 The Approximations
  6. 11.6 What the Model Predicts
  7. 11.7 Summary
  8. 11.8 Exercises
  9. 11.9 Project: Time-Resolved Fano Buildup