The moving boundary truncated grid method is developed to significantly reduce the number of grid points required for wave packet propagation. The time-dependent Schrodinger equation (TDSE) and the imaginary time Schrodinger equation (ITSE) are integrated using an adaptive algorithm which economizes the number of grid points. This method employs a variable number of grid points in the Eulerian frame (grid points fixed in space) and adaptively defines the boundaries of the truncated grid. The truncated grid method is first applied to the time integration of the TDSE for the photodissociation dynamics of NOCl and a three-dimensional quantum barrier scattering problem. The time-dependent truncated grid precisely captures the wave packet evolution for the, photodissociation of NOCl and finely adjusts according to the process of the wave packet bifurcation into reflected and transmitted components for the barrier scattering problem. The truncated grid method is also applied to the tithe integration of the ITSE for the eigenstates of quantum systems. Compared to the full grid calculations, the truncated grid method requires fewer grid points to achieve high accuracy for the stationary state energies and wave functions for a two-dimensional double well potential and the At trimer. Therefore, the truncated' grid method demonstrates a significant reduction in the number of grid points needed to perform accurate wave packet propagation governed, by the TDSE or the ITSE.