Abstract
We discuss some of the geometric properties of the state spaces in Quantum Mechanics with emphasis in the two-level systems case. Then the dynamics of a two-level atom affected by radiation fields is revisited in both the semiclassical and quantum approaches. In the calculations we use Hubbard operators to reduce the matrix operations into simple relations of subscripts. The time-evolution of the energy states of the atom is associated to trajectories either on the 2-sphere or in the three-dimensional unit ball according to the classical or quantum approach we are using.
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