Differential realizations of the various Lie superalgebra and their applications to the physical problems

Lie algebras have been studied for many years and their roles in solving quantumechanical problems are well known. In particular, exactly and quasi-exactly solvable quantumechanical problems are classifed according to their Lie algebraic properties. They provide thelgebraic basis for a unified language for physics and mathematics which offers many advantagesover current techniques. In this thesis, the solutions of the quantum optical problems in theframework of the Lie algebraic technique have been mainly studied and this work consists ofthree parts.One of the principle aims of this thesis is the development of a number of new algebraictechniques which serve to broaden the field of applicability of Lie (super) algebra. The spectra ofthe many quantum optical Hamiltonians have been obtained by introducing a novel similaritytransformation technique which transforms the multi-boson systems in the form of the singleboson systems. This transformation leads to the determination of the solvability of the associatedHamiltonian.The other contention of this thesis is that systems of the spin ½ particles can be solved byreformulating the Hamiltonians. To support this contention a diagonalization method isdeveloped. In the solution of a general Hamiltonian for spin ½ particles, it is argued that thediagonalization method is easily applicable to solve a wide range of quantum mechanicalproblems and it provides many new insights.Both methods have been applied to the solution of the quantum optical Hamiltonians aswell as the solution of the Dirac equation. In this context second harmonic generation problem,interacting electrons in a quantum dot, Jaynes-Cumings model, quantum dot including spin-orbitcoupling and Dirac equation have been solved. It has been shown that these physical systemsassociated with the algebras of the groups SU(2), SU(1,1), OSP(2,1) and OSP(2,2).This thesis also includes the solution of Eckart potential in the s-wave Klein-Gordonequation by using super-symmetric quantum mechanical method.The ultimate goal of the methods developed in this thesis is that they can be extended tosolve the other physical Hamiltonians which include multi-boson or multi-fermion-bosonsystems.Key words: Lie (super)algebra, (Quasi)exactly solvable potentials, Dirac equation, Klein-GordonEquation