Algebraic Realizations of the Solutions for Three Quantum Mechanical Systems

Several quantum mechanical problems are studied all of which can be approached using algebraic means. The first problem introduces a large class of Hamiltonian operators which can be related to elements of su(2) or su(1, 1) Lie algebras. A Casimir operator can be obtained and the model can be solved in general by introducing an appropriate basis. The second system involves a collection of lattice spin Hamiltonian models. It is shown how a matrix representation can be determined for these types of models with respect to a specific basis. Using these matrices secular polynomials as functions of the energy eigenvalues and their corresponding eigenvectors are calculated. The last system is similar to the first, but is suited to a particular application The eigenvectors of the model are used used to calculate the Berry phase for these states