Question

Consider the Lagrangian density ih Construct the equations of motion for the field from the Euler-Lagrange equations, and show that it leads to the Schrödinger equation in dt2m and its complex conjugate.

Answer 1

Lagrangian Density L = h^2/2m nabla psi middot nabla psi^* + v psi psi^* - ih/2 (psi^* psi^dot - psi psi^*^dot) Given Euler-lagrange equation of motion for psi^*. partial differential L/partial differential psi^* - partial differential mu partial differential h/partial differential(partial differential_mu middot psi^*) = 0 equation (1) partial differential L/partial differential psi^* - partial differential/partial differential t partial differential L/partial differential(partial differential psi^*^dot) - nabla^vector partial differential L/partial differential (nabla psi^*) = 0 equation (2) since partial differential_mu = (partial differential_0, nabla^vector) = (partial differential_t, nabla^vector) Now, for Given Lagragain- partial differential L/partial differential psi^*^dot = + ih/2 psi, partial differential R/partial differential (nabla psi^*) = h^2/2m nabla psi partial differential L/partial differential psi^* = -ih/2 psi^dot + v psi equation (2) using these in (2) \r\n v psi + -i k/2 psi^dot - partial differential/partial differential t (ih/2 psi) - h^2/2m middot nabla middot nable psi = 0 -h^2/2m nabla^2 psi + v psi - ih