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I heard that the Lagrangian is defined in the path integral formulation of quantum mechanics. How would the Lagrangian in this formulation be used to recover the Schrodinger equation that we normally use?

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I am not sure if you are looking for this, but you can define a Lagrangian in such a way that the L-EOM (equation of motion) is the Schrödinger equation.

$\cal{L}=\Psi^{t}(i\frac{\partial}{\partial t}+\nabla^2/2m)\Psi$

$\frac{\partial\cal{L}}{\partial\Psi^t}=0$

The second term of the Lagrange-equation (derivative with respect to $\partial_{\mu}\Psi^t$) is zero since no derivative of $\Psi^t$ occurs in our field Lagrangian density .

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To get from a given Lagrangian to the Schrödinger equation you have to realize that the latter is actually given in terms of the Hamiltonian of the theory. Lagrangian and Hamiltonian are related by a Legendre transformation.

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