Quantum wires continue to be a subject of novel applications in the fields of electronics and optoelectronics. In this work, we revisit the problem of determining the electron states in semiconductor quantum wires in a self-consistent way. For that purpose, we numerically solve the 2D system of coupled Schrödinger and Poisson equations within the envelope function and effective mass approximations. The calculation method uses the finite-element approach. Circle, square, triangle and pentagon geometries are considered for the wire cross-sectional shape. The features of self-consistent band profiles and confined electron state spectra are discussed, in the latter case, as functions of the transverse wire size and temperature. Particular attention is paid to elucidate the origin of Friedel-like oscillations in the density of carriers at low temperatures.