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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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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