Analytic representations of finite quantum systems based on $SU(2)$ coherent states in $C\cup\{\infty\}$ and potential quantum applications to energy works

This study considers the finite-dimensional Hilbert space with the quantum systems. The $SU(2)$ coherent states in the extended complex plane, which is stereo graphically equivalent to a sphere, serve as the foundation for analytical representation functions of the states in this space. The analytical representation function is practically implemented using the numerical style for both the periodic and non-periodic systems. This study shows in detail the zeroes trajectories of these analytic representation functions which apply the chronological evolution of the system and derive vital findings regarding to the behaviour of the quantum system. More potentially, there are several crucial applications of quantum mechanics to advance energy works for more in depth investigation in future pathways.