A time-dependent approach for the calculation of eigenenergies, eigenfunctions and optical spectra, within the effective-mass and single-hole-band approximations, of low-dimensional semiconductor systems is presented. In this method, energy and optical spectra are obtained by Fourier analysis of a time-autocorrelation function constructed from an appropriately chosen propagated wavepacket. Eigenfunctions are determined by a direct Fourier analysis of such a propagated wavepacket. The time evolution of the wavefunction is obtained by numerical integration of the time-dependent Schrödinger equation. This approach shares the virtues of the more common numerical methods (accuracy, flexibility and automatic enforcement of boundary conditions) with the additional advantage that it can yield an entire eigen or optical spectrum in a single calculation. The methodology is applied to the calculation of all the eigenenergies and eigenfunctions of the conduction band, and the intensities of the transitions from the ground state of the valence band to all the states of the conduction band, in spherical GaAs-(Ga,Al)As quantum dots with finite confinement and a wide range of radii, without and with an on-centre shallow donor impurity.