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gulsen

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Euler-Lagrange equations for the Lagrangian density [tex]\mathcal{L} = V\psi \psi^* + \frac{\hbar^2}{2m}\frac{\partial \psi}{\partial x}\frac{\partial \psi^*}{\partial x} + \frac{1}{2}\left(i\hbar \frac{\partial \psi^*}{\partial t} \psi- i\hbar \frac{\partial \psi}{\partial t} \psi^*\right)[/tex] gives (complex conjuagate of) Schördinger equation, when it's considered as the minumum of [tex]\int \int \mathcal{L} dx dt[/tex]. Is this of any use?

edit: corrected Lagrangian density. This LD results in [tex]\left(-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + V\psi - i\hbar \frac{\partial \psi}{\partial t}\right) + \left(-\frac{\hbar^2}{2m}\frac{\partial^2 \psi^*}{\partial x^2} + V\psi^* + i\hbar \frac{\partial \psi^*}{\partial t}\right) = 0[/tex]

edit: corrected Lagrangian density. This LD results in [tex]\left(-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + V\psi - i\hbar \frac{\partial \psi}{\partial t}\right) + \left(-\frac{\hbar^2}{2m}\frac{\partial^2 \psi^*}{\partial x^2} + V\psi^* + i\hbar \frac{\partial \psi^*}{\partial t}\right) = 0[/tex]

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