[QUE/QFT-04013] QFT-PROBLEM

$\newcommand{\Lsc}{\mathscr L}$ 
$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
$\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$
Using the Heisenberg equation of motion for the Schrodinger field with Lagrangian density \[ \Lsc = i\hbar\psi^\dagger(\mathbf x,t)\pp[\psi(\mathbf x,t)]{t} - \frac{\hbar^2}{2m} (\nabla\psi^\dagger(\mathbf x))(\nabla\psi(\mathbf x)) - \psi^\dagger(\mathbf x,t)V(\mathbf x)\psi(\mathbf x,t) \] compute \(\dd[\rho]{t}\), where \(\rho=\psi^\dagger(\mathbf x)\psi(\mathbf x)\). Hence prove the equation of continuity \[\dd[\rho]{t} + \nabla\cdot \mathbf J =0\] where \(J\) is the probability current density \[\mathbf J = \frac{\hbar}{2im}[\psi^\dagger(\mathbf x)\big(\nabla\psi(\mathbf x)\big)- \big(\nabla\psi^\dagger(\mathbf x)\big)\psi(\mathbf x)].\] How does the interpretation of \(\rho\) and \(\mathbf J\) differ in the second quantized theory from that in Schrodinger quantum mechanics?