[QUE/QFT-02]

Compute infinitesimal variations of the Lagrangian density for the Schrodinger field under the Galilean transformation \begin{equation}  \vec{x} \longrightarrow \vec{x}\,^\prime = \vec{x} + \vec{v} t \end{equation} and \begin{equation}  \psi(\vec{x}) \longrightarrow \psi\,^\prime(\vec{x}\,^\prime)
= e^{-im\vec{v}\,^{\prime\,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.