  The formation of a polaron quasiparticle from a bare electron is studied in
the framework of the Holstein model of electron-phonon coupling. Using
Schr\"{o}dinger's formalism, we calculate the time evolution of the
distribution of the electron and phonon density, lattice deformation, and the
electron-phonon (el-ph) correlation functions in real space. The quantum
dynamical nature of the phonons is preserved. The polaron formation time is
related to the dephasing time of the continuum of unbound phonon excited states
in the spectral function, which depends on the phonon frequency and the el-ph
coupling strength. As the el-ph coupling increases, qualitative changes in
polaron formation occur when the one-phonon polaron bound state forms. In the
adiabatic regime, we find that a potential barrier between the quasi-free and
heavy polaron states exists in both 2D and 3D, which is crucial for polaron
formation dynamics. We compare to recent experiments.
