For analyzing the time evolution of quantum mechanically driven harmonic oscillator systems, we first establish the eigenvalue equation for the time-evolved wave function, and then we dynamically decompose the Hamiltonian into two parts. One part is the operator which does not change the state, and the other part does. Through this decomposition, the time evolution of the state can be solved more easily than in the familiar way. We illustrate this method by exactly solving the system of the driven harmonic oscillator. We show that nonspreading wave packets can also exist in this system in addition to the historically known paradigms, the profiles of which are the same as those in the harmonic oscillator system.