In this study, expectation value dynamics of the quantum harmonic oscillator under the influence of the external dipol effects is discussed in some details. The dipole function of the system has been taken as an odd cubic spatial polynomial and the external field amplitude has been assumed to be known. Instead of the wave function of the system, expectation values of the position and momentum operators are considered as unknowns and certain matrix ordinary differential equations are obtained by using the fluctuation free matrix representation concept. The resulting equations are in the category of Quantum Mechanical Matrix Ordinary Differential Equations (QMMODE) and can be solved by a recently developed scheme called "Characteristic Evolutions Method (CEM)". We mention also how to implement this approach. The purpose of this work can be considered as a preparation to quantum optimal control problem solutions via expectation value dynamics.