# How would Lagrangian be used tor recover Schrodinger equation?

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

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Googling "path integrals in quantum mechanics" will give you many sources, for example en.wikipedia.org/wiki/Path_integral_formulation If you have difficulty following the standard presentation please ask another question :) – Michael Brown Mar 2 at 9:06
The derivation in the link shows how you get from a solution of the Schrödinger equation to the path intergral formalism, but not the other way around. In this sense, the question is not really answered by it. – Frederic Brünner Mar 2 at 9:35
This is probably what you're looking for: users.physik.fu-berlin.de/~kleinert/kleiner_re242/node2.html – elfmotat Mar 2 at 9:59

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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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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