By performing a anitary transformation, we transform the Hamiltonian of the
trapped ion in any position of standing wave to that of the normal Jaynes-
Cummings model in ionic bare basis and we obtain a general evolution operat
or of the trapped ion system. We study the quantum entropic dynamics of the
phonons and trapped ion. Our results show that, when the trapped ion is lo
cated at the node of standing wave, the quantum entropic dynamics of phonon
s and trapped ion are the same as the one of the field in the Jaynes-Cummin
gs model. When the trapped ion deviates from the node of standing wave, the
entropies of the phonons and ion keep their maximum value except at the in
itial stage, and the phonons and trapped ion extremely entangled.