We obtain the eigenfunctions, eigenvalues, and typical wave-packet dynamics of a ``harmonic oscillator'' defined on a lattice, when subjected to an added constant force field \ensuremath{\alpha}. The eigenfunctions of this displaced harmonic oscillator are given in terms of Mathieu functions. In the momentum representation, the Mathieu eigenfunctions have period \ensuremath{\pi} (or 2\ensuremath{\pi}) when the bottom of the parabolic potential well corresponds to a lattice point (or is halfway between two lattice points). This occurs when \ensuremath{\alpha} is an integer (or half-integer) in units of the force needed to stretch the oscillator by one lattice constant. For an incommensurate displacement of the well (i.e., for \ensuremath{\alpha} irrational in the above units) the eigenfunctions are not periodic. In real space, the eigenfunctions are displaced zero-field eigenfunctions centered on the new equilibrium position only for the \ensuremath{\alpha} integer. The eigenvalues are obtained as the sum of a periodic function of the force \ensuremath{\alpha}, plus an overall quadratic shift in \ensuremath{\alpha}. While this quadratic term is common to the continuum limit of the theory, the first periodic one is not. As \ensuremath{\alpha} is turned on, the periodically varying relative position of the parabolic well with respect to the underlying lattice gives rise to the oscillating term for the energy eigenvalues. The time evolution of a Gaussian wave packet is followed, and, in contrast with the continuum limit, its noncoherency is displayed. Damped oscillations of this wave packet occur between the spatial limits imposed by the parabolic well, or between those given by a Bloch oscillation. The ``periods'' for these two types of motion are quite different, and each depends on both the force \ensuremath{\alpha} and on the spring constant of the oscillator. As time increases, Bloch oscillations are eventually destroyed by the inhomogeneous restoring force of the harmonic oscillator, but not by the constant force field. Finally, we compare and contrast our results with the predictions of the semiclassical solid-state model, especially those involving effects due to Bloch oscillations.