The goal of this article is to provide backward uniqueness results for several models of parabolic equations set on the half line, namely the heat equation, and the heat equation with quadratic potential and with purely imaginary quadratic potentials, with non-homogeneous boundary conditions. Such result can thus also be interpreted as a strong lack of controllability on the half line, as it shows that only the trivial initial datum can be steered to zero. Our results are based on the explicit knowledge of the kernel of each equation, and standard arguments from complex analysis, namely the Phragmén Lindelöf principle.