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!
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 $\vert\Psi(0)\rangle$.