How can we form Euler-Lagrange equations for Time-Independent Schrodinger Equations?

Question

Is it possible to form a lagrangian of the TISE using the concept of Lagrange Multipliers? I am new to this topic so any help would be much appreciated.

Answer 1

Finding the stationary points of $$ \chi[\psi, \psi^*]= \int \left\{\psi^*(-\nabla^2 \psi)+V(x)\right\}\psi\, dx $$ subject to $\int |\psi|^2 dx=1$ leads to the Euler-Lagrange equations $$ -\nabla^2 \psi +V\psi=E\psi\\ -\nabla^2 \psi^* +V\psi^*=E\psi^* $$ Here $E$ is the Lagrange multiplier enforcing the constraint and the equations are the Schroedinger equation and its conjugate.