  Although the spectral lines of hydrogen contain valuable information on the
physical properties of a variety of astrophysical plasmas, including the upper
solar chromosphere, relatively little is known about their scattering
polarization signals whose modification via the Hanle effect may be exploited
for magnetic field diagnostics. Here we report on a basic theoretical
investigation of the linear polarization produced by scattering processes and
the Hanle effect in Ly-a, Ly-b and H-a taking into account multilevel radiative
transfer effects in an isothermal stellar atmosphere model, the fine-structure
of the hydrogen levels, as well as the impact of collisions with electrons and
protons. The main aim of this first paper is to elucidate the physical
mechanisms that control the linear polarization in the three lines, as well as
its sensitivity to the perturbers density and to the strength and structure of
micro-structured and deterministic magnetic fields. To this end, we apply an
efficient radiative transfer code we have developed for performing numerical
simulations of the Hanle effect in multilevel systems with overlapping line
transitions. For low density plasmas such as that of the upper solar
chromosphere collisional depolarization is caused mainly by collisional
transitions between the fine-structure levels of n=3, so that it is virtually
insignificant for Ly-a but important for Ly-b and H-a. We show the impact of
the Hanle effect on the three lines taking into account the radiative transfer
coupling between the different hydrogen line transitions. For example, we
demonstrate that the linear polarization profile of the H-a line is sensitive
to the presence of magnetic field gradients in the line core formation region
and that in solar-like chromospheres selective absorption of polarization
components does not play any significant role on the emergent scattering
polarization.
