Geometrical representation of the equations for solving quantum beat problems

Equations for solving a quantum beat problem in a three-level system, which determine two complex variables of time dependence, are rewritten in terms of four real functions constructed from the two complex variables. The Minkowski space is reasonably introduced in order to represent the time evolution of the four real functions as the motion of a four-dimensional vector, though the equations are irrelevant to the special theory of relativity. It is found that the four-dimensional vector precesses around the zeroth axis on the cone which is constructed from all of the points whose norms are zero in the Minkowski space, and that the Euclidean norm of the vector decreases with the increase of time. Though the visualized motion of the vector is similar to those in the well-known magnetic resonance precession model, the picture obtained from the equations for quantum beats cannot be connected with such a phenomenon as photon echo.