Quantum mechanical evolution operator in the presence of a scalar linear potential: discussion on the evolved state

We discuss the form of the wave-function of a state subjected to a scalar linear potential, paying special attention to quantum tunneling. We analyze the phases acquired by the evolved state and show that some of them have a pure quantum mechanical origin. In order to measure one of these phases, we propose a simple experimental scenario. We finally apply the evolution equations to re-analyze the Stern\&Gerlach experiment and to show how to manipulate spin by employing constant electric fields.