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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.

Qmechanic
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Anirudh Kalla
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  • What do you mean by "a lagrangian of the TISE"? Euler-Lagrange equations are equations for something evolving in time, but the time-independent Schrödinger equation does not evolve in time, that's its entire point. – ACuriousMind Sep 20 '20 at 16:38
  • Related: https://physics.stackexchange.com/q/579929/2451 and links therein. – Qmechanic Sep 20 '20 at 17:16
  • @ACuriousMind it is very much possible to find E-L equations that do not depend explicitly on time. The time dependence is built into the function $\psi$ as $\psi$ depends on time. Same goes for $\psi^{*}$. Please correct me if I am wrong. – Anirudh Kalla Sep 21 '20 at 07:44

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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.

AccidentalTaylorExpansion
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mike stone
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