Exact solution of few multistate problems in quantum and statistical mechanics (PhD)

Abstract

The thesis entitled "Exact solution of few multistate problems in quantum and statistical mechanics" basically consists of two parts. Out of these parts is devoted to multistate problems in quantum mechanics while another is based upon multistate problems in statistical mechanics. Multistate problems in quantum mechanics is an interdisciplinary topic which covers a wide range of fields like physics, chemistry and biology and economics as well. Nonadiabatic transitions is a part of multistate problems and such transitions has a long history starting from 1932 onwards when pioneering works in the area were published by Landau, Zener, Stuckelberg and Rosen. Nonadiabatic transitions due to crossing of the potential energy curves is one of the most probable mechanism responsible for electronic transitions. Various spectroscopic, collision processes and reactions are governed by such kind of transitions. Some of the examples involving such kind of transitions may include radationless transitions in condensed matter physics, laser assisted collisions reactions, Zener transitions in flux driven metallic rings, super conducting Josephson junctions, reactions in nuclear physics and electron proton transfer processes in biological systems. Neutrino conversion in the sun, dissociation of molecules on metal surfaces are some other examples which explain the importance of nonadiabatic transitions. The pioneering work by Landau, Zener, Stuckelberg and Rosen opens a pathway for solutions to the problems including such kind of transitions. The approach used by them is purely analytical which can be mapped to problems like electron detachment, ionization in slow atomic and ionic collisions and electronic transitions in crystals where one state of a system is interacting with a group of states of different nature and many more. From1932 onwards we have numerous citations in literature based on nonadiabatic transitions which involves analytical as well as computational approach. The work presented in this thesis pays attention to the use of analytical methods for problems involving nonadiabatic transitions where one state of a system is interacting with a number of states of different nature through Dirac Delta interactions and we provide a simple analytical formula for calculation of transition probabilities between different interacting states. The latter part of the thesis is devoted to multistate problems in statistical mechanics. It will make use of the Smoluchowski equation along with a coupling term represented by a Dirac Delta function. Models of such type are useful to study a variety of dynamical processes and in diffusion controlled reactions. Such a model can be used to study electron transfer reactions in polar solvents, barrierless electronic relaxation in solutions, multichannel and electro chemical electron transfer cases. Our work in this area is devoted to exact analytical solution of the Smoluchowski equation with time dependence in case of a flat, linear and parabolic potential. Further exact solution is also provided in different cases where the strength of the coupling term has varied time dependence. More than one potential case in statistical mechanics is also solved exactly. The present thesis is divided into 6 chapters. Chapter 1 is the introductory overview of the work carried out in this thesis. Chapter 2 includes different analytical properties used to study time independent multistate problems in quantum mechanics. Chapter 3 is devoted to the exact solution of time dependent multistate problems in quantum mechanics. Chapter 4 includes the exact solution of time dependent multistate problems in statistical mechanics. Chapter 5 is devoted to multistate problems solved using the computational package MOLPRO. Chapter 6 concludes this thesis by providing the summary of all five chapters and future prospects in this area.