Algebraic treatments of the problems of the spin-1/2 particles in the one and two-dimensional geometry: a systematic study

We consider solutions of the 2x2 matrix Hamiltonians of the physical systems within the context of the su(2) and su(1,1) Lie algebra. Our technique is relatively simple when compared with the others and treats those Hamiltonians which can be treated in a unified framework of the $% Sp(4,R)$ algebra. The systematic study presented here reproduces a number of earlier results in a natural way as well as leads to a novel findings. Possible generalizations of the method are also suggested.