Editing Self-adjoint operator (section)

=== Non-self-adjoint operators in quantum mechanics===
{{see also|Non-Hermitian quantum mechanics}}
In quantum mechanics, observables correspond to self-adjoint operators. By [[Stone's theorem on one-parameter unitary groups]], self-adjoint operators are precisely the infinitesimal generators of unitary groups of [[time evolution]] operators. However, many physical problems are formulated as a time-evolution equation involving differential operators for which the Hamiltonian is only symmetric. In such cases, either the Hamiltonian is essentially self-adjoint, in which case the physical problem has unique solutions or one attempts to find self-adjoint extensions of the Hamiltonian corresponding to different types of boundary conditions or conditions at infinity.

'''Example.''' The one-dimensional Schrödinger operator with the potential <math>V(x) = -(1 + |x|)^\alpha</math>, defined initially on smooth compactly supported functions, is essentially self-adjoint for {{math|0 < ''α'' ≤ 2}} but not for {{math|''α'' > 2}}.{{sfn|Berezin|Shubin|1991|pp=55,86|ps=}}{{sfn|Hall|2013|pp=193-196|ps=}}

The failure of essential self-adjointness for <math>\alpha > 2</math> has a counterpart in the classical dynamics of a particle with potential <math>V(x)</math>: The classical particle escapes to infinity in finite time.<ref>{{harvnb|Hall|2013}} Chapter 2, Exercise 4</ref>

'''Example.''' There is no self-adjoint momentum operator <math>p</math> for a particle moving on a half-line. Nevertheless, the Hamiltonian <math>p^2</math> of a "free" particle on a half-line has several self-adjoint extensions corresponding to different types of boundary conditions. Physically, these boundary conditions are related to reflections of the particle at the origin.{{sfn | Bonneau | Faraut | Valent | 2001 |ps=}}