In the present thesis, we consider variants of the stationary one-dimensional quantum drift-diffusion model and the stationary one-dimensional viscous quantum hydrodynamic model with a non-smooth barrier potential. For the latter, existence results are established by a reformulation in terms of a viscosity-adjusted Fermi-level, variational methods and fixed point arguments. The dependence of solutions on the quantum and viscous parameters is examined and the combined viscous semi-classical limit is considered. The existence of fluid-dynamical boundary layers is verified by the construction of asymptotic expansions of the solutions
