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So in classical mechanics, we can generally convert from the Hamiltonian to the Lagrangian and back using Legendre transformations. However, I have found this to be a bit troublesome in QM. Specifically, I am working on problems in quantum chemistry and I was wondering if there was a way to get from a wavefunction to a Lagrangian. For example, the s-orbitals in hydrogen are very well understood; can we write a Lagrangian for this orbital? I ask about obtaining Lagrangians from wavefunctions instead of Hamiltonians because the wavefunction has more similarity to a "trajectory" with a unique action than it is similar to an energy operator.

I apologize if this question is native, my understanding of Feynman's sum-over-history approach to QM is limited at best.

EDIT/ADDITION: my question is essentially how can we derive a Lagrangian from a wave function or a propagator? Is this analytically possible?

Stoby
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    Perhaps look at this question/answer https://physics.stackexchange.com/q/390299/157704 – user1379857 Mar 01 '21 at 23:45
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    While it is possible to analyze the hydrogen atom with the Feynman Path Integral is is very complicated. Solving for the wave functions is far far simpler and there is really no reason to stray from that formalism. – user1379857 Mar 01 '21 at 23:47
  • @user1379857 yes, I know that the full Feynman path integral formulation will be difficult, my question is really more about writing Lagrangians, I appreciate the practical input, but I really only reference the path integral approach to show familiarity with QM Lagrangians, not b/c I want to solve its entirety. – Stoby Mar 02 '21 at 00:19

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Yes, for what it's worth, the TDSE has a Lagrangian formulation, cf. e.g. this & this Phys.SE posts.

Qmechanic
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