Time-Dependent Quantum Mechanics and Spectroscopy

These notes are meant as a resource for chemists that study the time-dependent quantum mechanics, dynamics, and spectroscopy of molecular systems. The notes are derived from my lectures in graduate quantum mechanics that focus on condensed phase spectroscopy, dynamics, and relaxation. As with any notes, there will be mistakes. I am happy to hear from those that find them, and I regularly post revised versions with corrections.
Overview of Time-Independent Quantum Mechanics

DOWNLOAD (v. 3/2015)
1. Describing a System Quantum Mechanically
2. Matrix Mechanics
3. Basic Quantum Mechanical Models
4. Exponential Operators
5. Numerically Solving the Schrödinger Equation

1. Introduction

DOWNLOAD (v. 12/2014)
1. Time-Evolution with a Time-Independent Hamiltonian
2. Exponential Operators
3. Two-Level System

2. Time-Evolution Operator

DOWNLOAD (v. 8/2022)
1. Time-Evolution Operator
2. Integrating the TDSE Directly
3. Transitions Induced by Time-Dependent Potential
4. Resonant Driving of a Two-Level System
5. Schrödinger and Heisenberg Representations
6. Interaction Picture
7. Time-Dependent Perturbation Theory
8. Fermi’s Golden Rule

3. Irreversible Relaxation
4. The Density Matrix
5. Adiabatic Approximation

DOWNLOAD (v. 11/2014)
1. Born–Oppenheimer Approximation
2. Nonadiabatic Effects
3. Diabatic and Adiabatic States
4. Adiabatic and Nonadiabatic Dynamics
5. Landau–Zener Transition Probability

6. Interaction of Light and Matter

DOWNLOAD (v. 3/2015)
1. Introduction
2. Classical Light–Matter Interactions
3. Quantum Mechanical Electric Dipole Hamiltonian
4. Relaxation and Line-broadening
5. Absorption Cross Section
6. Appendix: Review of Free Electromagnetic Field
7. Appendix: Magnetic Dipole and Electric Quadrupole Transitions

7. Mixed States and the Density Matrix
8. Irreversible and Random Processes

DOWNLOAD (v. 11/2014)
1. Concepts and Definitions
2. Thermal Equilibrium
3. Fluctuations

9. Time-Correlation Functions

DOWNLOAD (v. 11/2014)
1. Definitions, Properties, and Examples
2. Correlation Function from a Discrete Trajectory
3. Quantum Time-Correlation Functions
4. Transition Rates from Correlation Functions

10. Linear Response Theory

DOWNLOAD (v. 11/2014)
1. Classical Linear Response Theory
2. Quantum Linear Response Functions
3. The Response Function and Energy Absorption
4. Relaxation of a Prepared State

11. Time-Domain Description of Spectroscopy

DOWNLOAD (v. 11/2014)
1. Classical Description of Spectroscopy
2. Time-Correlation Function Description of Absorption Lineshape
3. Different Types of Spectroscopy Emerge from the Dipole Operator
4. Ensemble Averaging and Line-Broadening

12. Coupling of Electronic and Nuclear Motion

DOWNLOAD (v. 11/2014)
1. The Displaced Harmonic Oscillator Model
2. Coupling to a Harmonic Bath
3. Semiclassical Approximation to the Dipole Correlation Function

13. Fluctuations in Spectroscopy

DOWNLOAD (v. 12/2014)
1. Fluctuations and Randomness: Some Definitions
2. Line-Broadening and Spectral Diffusion
3. Gaussian-Stochastic Model for Spectral Diffusion
4. The Energy Gap Hamiltonian
5. Correspondence of Harmonic Bath and Stochastic Equations of Motion

14. Energy and Charge Transfer

DOWNLOAD (v. 12/2014)
1. Electronic Interactions
2. Förster Resonance Energy Transfer
3. Excitons in Molecular Aggregates)
4. Multiple Excitations and Second Quantization
5. Marcus Theory for Electron Transfer 

15. Quantum Relaxation Processes

DOWNLOAD (v. 11/2014)
1. Vibrational Relaxation
2. A Density Matrix Description of Quantum Relaxation