[QUE/QFT-02001] QFT-PROBLEM

Compute infinitesimal variations of the Lagrangian density for the Schrodinger field under the Galilean transformation \begin{equation} \vec{x} \longrightarrow \vec{x}{'} = \vec{x} + \vec{v} t \end{equation} and \begin{equation} \psi(\vec{x}) \longrightarrow \psi{'}(\vec{x}\,{'}) = e^{-im\vec{v}\,^{{'}\,2} t/(2\hbar)} e^{im\vec{v}\cdot\vec{x}/\hbar} \psi(\vec{x}). \end{equation} Verify that the the change in Lagrangian is a total time derivative. Find the corresponding constant of motion.