Observables in Quantum Mechanics and the Importance of Self-adjointness

TL;DR

This paper explores the mathematical subtleties of quantum observables, emphasizing the crucial role of self-adjointness over mere hermiticity, with examples illustrating physical implications.

Contribution

It clarifies the importance of self-adjoint operators in quantum mechanics and discusses their extensions through physical examples, highlighting their significance in the theory.

Findings

Self-adjointness is essential for well-defined quantum observables.

The theory of self-adjoint extensions has practical physical applications.

Mathematical distinctions impact the physical interpretation of quantum measurements.

Abstract

We are focused on the idea that observables in quantum physics are a bit more than just hermitian operators and that this is, in general, a "tricky business". The origin of this idea comes from the fact that there is a subtle difference between symmetric, hermitian, and self-adjoint operators which are of immense importance in formulating Quantum Mechanics. The theory of self-adjoint extensions is presented through several physical examples and some emphasis is given on the physical implications and applications.