Self-adjointness in Quantum Mechanics: a pedagogical path

TL;DR

This paper provides a pedagogical approach to understanding the importance of self-adjoint operators in quantum mechanics, clarifying their physical necessity over mere hermitian operators through mathematical and conceptual explanations.

Contribution

It organizes standard facts into a coherent pedagogical path emphasizing the physical and mathematical necessity of self-adjointness for quantum observables.

Findings

Self-adjointness ensures physically meaningful quantum observables.

Mere hermiticity does not guarantee the physical consistency of operators.

A domain declaration is crucial for associating operators with physical observables.

Abstract

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.