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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1$\begingroup$ 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 :) $\endgroup$– MichaelMar 2, 2013 at 9:06
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2$\begingroup$ 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. $\endgroup$– Frederic BrünnerMar 2, 2013 at 9:35
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3$\begingroup$ This is probably what you're looking for: users.physik.fu-berlin.de/~kleinert/kleiner_re242/node2.html $\endgroup$– JoldMar 2, 2013 at 9:59
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$\begingroup$ Duplicate: physics.stackexchange.com/q/69982 $\endgroup$– Abhimanyu Pallavi SudhirJul 27, 2013 at 9:26
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2 Answers
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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.
answered Mar 2, 2013 at 9:44