[QUE/QFT-04007] QFT-PROBLEM

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• The Lagrangian density for the Schrodinger equation is given to be \[\Lsc = i\hbar\psi^*(x,t)\pp[\psi(x,t)]{t} - \frac{\hbar^2}{2m} |\nabla \psi|^2 - \psi^*(x,t)V(x)\psi(x,t)\] Verify that the Euler Lagrange equations for the Schrodinger field coincide with the Schrodinger equation.
• Find the Hamiltonian of the system. Use Poisson brackets to obtain the Hamiltonian equations of motion.
• Verify that the Hamilton's equations imply the Euler Lagrange equation of motion.