[QUE/QFT-04014] QFT-PROBLEM

$\newcommand{\Hca}{\mathcal H}$
$\newcommand{\Pca}{\mathcal P}$

For the classical Schrodinger field\(\psi(x)\) calculate the following Poisson brackets.

• \(\{N, \psi\}_\text{PB}\) 
• \(\{ \psi, \Hca\}_\text{PB}\)
• \(\{\Pca, \psi\}_\text{PB}\) 
• \(\{\Pca, \Hca\}_\text{PB}\)

where \(N,\Pca\) and \(\Hca\), respectively, are the momentum and the Hamiltonian of the Schrodinger field. \begin{eqnarray}\nonumber N &=&\int \psi^*(x)\psi(x)\, dx; \qquad \Pca = -i\hbar\int\, dx \psi^*(x) \nabla \psi(x)\\\nonumber \Hca &=& \int \Big\{ \frac{\hbar^2}{2m}(\nabla\psi(x))^*(\nabla \psi(x)) + \psi^*(x)V(x)\psi(x)\Big\}. \end{eqnarray}