I'm having a gap in understanding the relation between them, and resolving my confusion is really appreciated. For example, the Hamiltonian operator, why do we call its eigenvalues energies? how do we actually measure it in the laboratory, quantum mechanically? And maybe I need a better sense in which I understand energy? What is it, in quantum mechanics? More generally, can I write down any hermitian operator and claim that it can be measured? How do I know how it is measured?