Solvable One-Dimensional Multi-State Models for Statistical and Quantum Mechanics

✍ Scribed by Rajendran Saravanan, Aniruddha Chakraborty

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 186
• Category: Library

✦ Synopsis

This book highlights the need for studying multi-state models analytically for understanding the physics of molecular processes. An intuitive picture about recently solved models of statistical and quantum mechanics is drawn along with presenting the methods developed to solve them. The models are relevant in the context of molecular processes taking place in gaseous phases and condensed phases, emphasized in the introduction. Chapter 1 derives the arisal of multi-state models for molecular processes from the full Hamiltonian description. The model equations are introduced and the literature review presented in short. In Chapter 2, the time-domain methods to solve Smoluchowski-based reaction-diffusion systems with single-state and two-state descriptions are discussed. Their corresponding analytical results derive new equilibrium concepts in reversible reactions and studies the effect of system and molecular parameters in condensed-phase chemical dynamics. In Chapter 3, time-domain methods to solve quantum scattering problems are developed. Along side introducing a brand new solvable model in quantum scattering, it discusses transient features of quantum two-state models. In interest with electronic transitions, a new solvable two-state model with localized non-adiabatic coupling is also presented. The book concludes by proposing the future scope of the model, thereby inviting new research in this fundamentally important and rich applicable field.​

✦ Table of Contents

Foreword
Preface
Acknowledgements
Contents
Acronyms
1 Theoretical Background and Motivation
1.1 Conservative Versus Dissipative Dynamics in Molecular Studies
1.1.1 Introduction to Non-equilibrium Systems
1.1.2 Fokker–Planck Equation Description of Random Walk Approach to Non-equilibrium Systems
1.2 Arisal of Multi-state Problems for Representing Electron …
1.2.1 Born–Oppenheimer Surfaces of the Molecules in Solution
1.2.2 Derivation of a Multi-state Hamiltonian Involving Reactant and Product States of the System
1.2.3 Libby's Theory on Electron Transfer: Concepts, Applicability and Limitations
1.2.4 Marcus' Theory on the Electron Transfer
1.2.5 Extensions to the Marcus' Description
1.2.6 Formulation of Multi-state Problems in Electron Transfer
1.3 Existing Analytical Methods for Solving Multi-state Reaction-Diffusion Models
1.3.1 The Survey of Already Existing Models Solvable in Time-Domain
1.3.2 Survey of Laplace-Domain Solvable Models
1.4 Models Considered in the Monograph
1.4.1 Effective Single State Models
1.4.2 Multi-state Reaction Diffusion Systems
1.4.3 Few More Applications of Reaction-Diffusion Models
1.5 Introduction to Quantum Multi-level Systems
1.5.1 Few Examples of Multi-State Processes
1.5.2 Equation of Motion Governing the Multi-level Systems
1.5.3 Survey of Existing Analytical Methods to Solve Quantum Multi-state Models
1.5.4 Models Considered for Quantum Mechanical Case in This Monograph
1.6 Brief Summary About the Organization of This Monograph
2 Mathematical Methods for Solving Multi-state Smoluchowski Equations
2.1 Prelude
2.2 Solution of Single-State Problems in Statistical Physics
2.2.1 Introduction to the Kernel Method
2.2.2 Diffusion Dynamics of a Distribution in Flat Potential with a Dirac Delta-Function Sink
2.2.3 Exact Diffusion Dynamics of a Distribution in the Presence of Two Competing Sinks: Oster–Nishijima Model
2.2.4 A General Method to Solve Diffusion in Piece-Wise Linear Potentials in the Time Domain
2.2.5 Understanding Condensed-Phase Dynamics Using Parabolic Potential Models in Presence of Delta-Function Reactive Terms
2.3 Exact Dynamics of Coupled Problems
2.3.1 Exact Diffusion Dynamics of a Distribution in a Coupled System: A Simple Open System
2.3.2 Deriving General Characteristics About Open and Closed Systems Using a Simple Model
2.3.3 Exact Diffusion Dynamics of a Distribution in a Closed System
3 Investigation of Wave Packet Dynamics Using the Presented Time-Domain Method
3.1 Prelude
3.2 Exact Time-Domain Solution of the Schrödinger Equation …
3.2.1 The Mathematical Methodology to Evaluate the Wave Packet Dynamics
3.2.2 Results and Discussions
3.2.3 Conclusion
3.3 Exact Wave Packet Dynamics of Gaussian Wave Packets in Two-Flat …
3.3.1 Methodology: Kernel Method to Calculate Wave Packet Dynamics
3.3.2 Asymptotic and Exact Solutions, Results
3.4 Deriving General Characteristics of a Two-State …
3.4.1 Application to Set of Flat Diabatic Curves
3.4.2 Application to Linear Diabatic Curves
3.4.3 Application to Harmonic Diabatic Potentials
3.4.4 Summary of Results, Discussions
3.5 Exact Solution of a Delta-Function Coupled Two-State …
3.5.1 Conclusion
4 Summary and Future Scope
Appendix Miscellaneous Discussions of Sect.2.2.5摥映數爠eflinksubsec:harmonic2.2.52
Appendix References
Index