The nonrelativistic electron driven by a classical homogeneous time-varying field is given an exact wave-mechanical treatment. The problem is formulated and solved in two separate reference frames. One is a frame that accelerates like a classical electron that is in the same field, and here the wave mechanics is formally that of a free electron. The second frame is an inertial frame, and here the time-development operator is given two explicit forms. States of sharp but time-dependent momentum are found. Wave functions in both momentum and configuration spaces are given. Packets that move in the manner of a classical electron are produced conveniently. The physical interpretation of the wave-mechanical description is given graphical illustration in a phase-space plot. The circumstances where a purely classical description suffices are elucidated. It is shown that the wave-mechanical analysis based on a homogeneous electric field can be suitably applied to several physical situations as a good approximation. One situation is the electron in a propagating classical electromagnetic wave of arbitrary frequency. Another situation is an electron in transit through the fields in a microwave cavity. The wave mechanics of an electron plasma in a homogeneous time-dependent electric field is treated separately and yields some tangible information.