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