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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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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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