Observables in Quantum Mechanics and the Importance of Self-Adjointness

Abstract

We are focused on the idea that observables in quantum physics are a bit more then 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.

Anomalous Bootstrap on the half line

“…But there is now an extra term A ≡ (H † − H)A ψ . This modification to the Ehrenfest theorem has been noticed before in the literature [13,14]. It is dubbed an 'anomaly', which is an appropriate term for a number of reasons.…”

“…This state is created by the boundary conditions. It is also exactly the same as the energy of a state bound in an inverted delta function potential on R (see Appendix A, also [13]).…”

“…The required restriction of operator domains is just the familiar task of supplying boundary conditions to eigenvalue problems like (13). By supplying boundary conditions for the solutions, we eliminate some functions in the Hilbert space from the domain of consideration for the operator.…”

“…shows that there is no suitable self-adjoint momentum operator on the half line. Trying to define such an operator can lead to various paradoxes [13]. Recently, the authors in [15] also considered these issues.…”

“…To illustrate the physical consequences [13], consider the free particle on R + ; −ψ = Eψ with ψ(0) + aψ (0) = 0. The solutions with E = k 2 are forward and backward traveling plane waves:…”