[QUE/QFT-02002] QFT-PROBLEM

Compute inifinitesimal variations of the Lagrangian density for the Schrodimmger field under the Galiniean 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} Verfiy that the the change in Lagrangian is a total time derivative. Find the corresponding constant of motion.