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From what I have read, the evolution of a quantum state is determined by the Hamiltonian (Schrodinger equation). However, I'm trying to understand if the Hamiltonian itself can be fully derived from the quantum state, or if it needs to be defined externally. From my understanding, the Hamiltonian includes information about the potential energy (when particles are interacting, etc), and that the laws of physics are actually "embedded" in the Hamiltonian, and not in the actual state vector. Is this correct, or does the state vector itself contain information about all the laws of physics? I hope my question is clear... Thanks!

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No. The state is determined by the preparation procedure, which is quite distinct and independent from the Hamiltonian.

As additional reading on this I recommend this excellent article by a master in foundational issues:

Peres, Asher. "What is a state vector?." American Journal of Physics 52.7 (1984): 644-650.

The abstract is by itself enlightening:

“ A state vector is not a property of a physical system (nor of an ensemble of systems). It does not evolve continuously between measurements, nor suddenly ‘‘collapse’’ into a new state vector whenever a measurement is performed. Rather, a state vector represents a procedure for preparing or testing one or more physical systems. No ‘‘quantum paradoxes’’ ever appear in this interpretation. The formulation of dynamical laws may involve path integrals and/or S‐matrix theory.”

Edit: I understand the “state” question as meaning the initial state $\vert\Psi(0)\rangle$.

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  • $\begingroup$ However, could one determine (at least something about) the Hamiltonian if we could perform many experiments where we could track the change in the state over time? Certainly this doesn't determine the Hamiltonian, but could this be used to "find it"? This is what I thought when I read in the question "I'm trying to understand if the Hamiltonian itself can be fully derived from the quantum state..." (I didn't down vote). $\endgroup$ – BioPhysicist May 19 '20 at 20:15
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    $\begingroup$ @BioPhysicist I think the OP is asking if the Hamiltonian can somehow be read directly from the initial state. If you want to determine the Hamiltonian doing some experiments you can do it probably much more simply if you know how to measure energy (as long as you can find a tomographically complete set of observables). $\endgroup$ – Dvij D.C. May 19 '20 at 20:46
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    $\begingroup$ @BioPhysicist If you only have one state to work with, the answer is no. To see why, suppose that the state you happen to have is a stationary state of the Hamiltonian. The only thing you can find out about the Hamiltonian in this case is that the state you have is one of its eigenstates; you learn nothing else about the structure. $\endgroup$ – probably_someone May 19 '20 at 20:50
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    $\begingroup$ @BioPhysicist Yes, I agree. It's a bit unclear what OP means by "quantum state". $\endgroup$ – Dvij D.C. May 19 '20 at 20:53
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    $\begingroup$ @hyportnex so I guess it's unclear as to "which state" we are talking about. $\endgroup$ – ZeroTheHero May 19 '20 at 22:55

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