Time-Dependent Quantum Mechanics and Spectroscopy Andrei Tokmakoff
5 800107 417351 Contents
TIME‐DEPENDENT QUANTUM MECHANICS 1. INTRODUCTION 1. Time‐Evolution with a Time‐Independent Hamiltonian 2. Exponential Operators 3. Two‐Level System
THE TIME‐DEPENDENT HAMILTONIAN 2. TIME‐EVOLUTION OPERATOR 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 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 1. Electric Dipole Hamiltonian 2. Supplement: Review of Free Electromagnetic Field 3. Absorption Cross Section
CONCEPTS AND TOOLS FOR CONDENSED PHASE DYNAMICS 7. MIXED STATES AND THE DENSITY MATRIX 8. IRREVERSIBLE AND RANDOM PROCESSES 1. Concepts and Definitions 2. Thermal Equilibrium 3. Fluctuations 9. TIME‐CORRELATION FUNCTIONS 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 1. Classical Linear Response Theory 2. Quantum Linear Response Functions 3. The Response Function and Energy Absorption 4. Relaxation of a Prepared State
SPECTROSCOPY 11. TIME‐DOMAIN DESCRIPTION OF SPECTROSCOPY 1. A 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 1. The Displaced Harmonic Oscillator Model 2. Coupling to a Harmonic Bath 3. Semiclassical Approximation to the Dipole Correlation Function 13. FLUCTUATIONS IN SPECTROSCOPY 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
APPLICATIONS 14. ENERGY AND CHARGE TRANSFER 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 1. Vibrational Relaxation 2. A Density Matrix Description of Quantum Relaxation
1. INTRODUCTION
1.1. Time Evolution with a Time-Independent Hamiltonian The time evolution of the state of a quantum system is described by the time-dependent Schrödinger equation (TDSE): irtHrtrt ,,, ˆ (1.1) t Hˆ is the Hamiltonian operator which describes all interactions between particles and fields, and determines the state of the system in time and space. Hˆ is the sum of the kinetic and potential energy. For one particle under the influence of a potential 2 HVrtˆˆˆ 2 , (1.2) 2m The state of the system is expressed through the wavefunction rt, . The wavefunction is complex and cannot be observed itself, but through it we obtain the probability density 2 Prt , which characterizes the spatial probability distribution for the particles described by Hˆ at time t. Also, it is used to calculate the expectation value of an operator Aˆ Atˆˆ * r,, t A r tdr t A ˆ t (1.3)
Physical observables must be real, and therefore will correspond to the expectation values of Hermitian operators AAˆˆ † . Our first exposure to time-dependence in quantum mechanics is often for the specific case in which the Hamiltonian Hˆ is assumed to be independent of time: HHrˆˆ . We then assume a solution with a form in which the spatial and temporal variables in the wavefunction are separable: