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

Abstract

We consider solutions of the 2 · 2 matrix Hamiltonians of the physical systems within the context of the su (2) and su (1, 1) Lie algebras. Our technique is relatively simple when compared with those of 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 finding. Possible generalizations of the method are also suggested.

Sp(4, R) algebraic approach of the most general Hamiltonian of a two-level system in two-dimensional geometry

“…The most general form of the Hamiltonian that describe a two-level system for a single particle with spin-1/2, in two dimensional geometry is given by [27]…”

“…From the energy spectrum of equation (107), we obtain that the eigenvalues of the interaction JC Hamiltonian are given by [27]…”

“…√ mω , and ω 0 = mc 2 / . With these definitions the Hamiltonian of equation ( 67) is reduced to [27,28]…”

“…In addition to the Jaynes-Cummings model, there are other equally important models in quantum optics like the Rabi model [19,20], the Dicke model [21] (also called Tavis-Cummings model [22]), the E ǫ Jahn-Teller Hamiltonian [23,24], the modified Jaynes-Cummings model [25], among others [26]. Some of these models are particular cases of a general Hamiltonian which is related to the Sp(4, R) group, as can be seen in reference [27]. The aim of the present work is to solve exactly the most general Hamiltonian of a two-level system in two-dimensional geometry, by using the Sp(4, R) group theory.…”

Berry phase of the Tavis-Cummings model with three modes of oscillation

“…As a result, it has been found that a wide variety of these models are expressed in terms of bosons and fermions or matrixdifferential equations. By choosing an appropriate realization, many of these models can be put in the context of the su(1, 1) and su(2) Lie algebras, as it is shown in the references [35][36][37]. In this Section, as an simple but useful application of the theory developed earlier, we are going to diagonalize two Hamiltonians with a simple structure given in terms of the su(1, 1) and su(2) Lie algebras.…”