Symmetric operator

For physical consistency, the mean values of any dynamical variable $\backslash ;\; \backslash mathcal\{A\}$ must be real numbers: this implies that the mean values of the operator $\backslash ;\; A$ which describes $\backslash ;\; \backslash mathcal\{A\}$ must be real, i.e. that :$\backslash langle\; \backslash psi,A\backslash psi\; \backslash rangle=\; \backslash langle\; \backslash psi,A\backslash psi\backslash rangle^*\; =\; \backslash langle\; A\backslash psi,\backslash psi\backslash rangle\; ,\; \backslash quad\; \backslash psi\; \backslash in\; \backslash mathcal\{D\}(A),$ where $\backslash ;\; \backslash mathcal\{D\}(A)$ is the domain of the operator $\backslash ;\; A$. It can be proved that this condition is equivalent to the following: :$\backslash langle\; \backslash psi,A\backslash chi\; \backslash rangle=\; \backslash langle\; A\backslash psi,\backslash chi\backslash rangle\; ,\; \backslash quad\; \backslash psi,\; \backslash chi\; \backslash in\; \backslash mathcal\{D\}(A)$. Having defined the linear operator $\backslash ;\; A^\backslash dagger$ in $\backslash ;\; \backslash mathcal\{H\}$, which is called the ''adjoint'' of $\backslash ;\; A$, such that :$\backslash langle\; A^\backslash dagger\backslash psi,\backslash chi\; \backslash rangle=\; \backslash langle\; \backslash psi,A\backslash chi\backslash rangle\; ,\; \backslash quad\; \backslash psi,\; \backslash chi\; \backslash in\; \backslash mathcal\{H\}$, it follows that $\backslash ;\; A$ must be ''Hermitian'', i.e. its domain must be such that $\backslash ;\; \backslash mathcal\{D\}(A)\; \backslash subseteq\; \backslash mathcal\{D\}(A^\backslash dagger)$ and $\backslash ;\; A$ must coincide with $\backslash ;\; A^\backslash dagger$ in $\backslash ;\; \backslash mathcal\{D\}(A)$. '''Definition: Hermitian [[operators]] whose domain is dense in $\backslash ;\; \backslash mathcal\{H\}$ are called '''symmetric'''.''' In particular, if a ''bounded'' linear operator is symmetric, it is also a Hermitian and [[self-adjoint operator]]. {{stub}} [[Category:Mathematical Structure]] [[Category:Handbook of Quantum Information]]