Self-adjointness in quantum mechanics: a pedagogical path

Observables in quantum mechanics are represented by self-adjoint operators on Hilbert space. Such ubiquitous, well-known, and very foundational fact, however, is traditionally subtle to be explained typical first classes mechanics, as well senior physicists who have grown up with the lesson that self-adjointness “just technical”. The usual difficulties clarify connection between demand for certain physical features theory corresponding mathematical requirement of self-adjointness, distinguish hermitian operator not just at level definition but most importantly from perspective mere hermiticity, without does ensure desired requirements leaves inconsistent. In this work, we organise an amount standard facts role into a coherent pedagogical path aimed making observables emerge necessarily self-adjoint, merely operators. Next central core our line reasoning—the necessity non-trivial declaration domain associate formal action observable, emergence consequence fundamental requirements—we include some complementary materials consisting few instructive proofs short retrospective, ranging past decades current research agenda, problem Hamiltonians relevance applications.