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Symmetric operator

For physical consistency, the mean values of any dynamical variable $\; \mathcal{A}$ must be real numbers: this implies that the mean values of the operator $\; A$ which describes $\; \mathcal{A}$ must be real, i.e. that

$\langle \psi,A\psi \rangle= \langle \psi,A\psi\rangle^* = \langle A\psi,\psi\rangle , \quad \psi \in \mathcal{D}(A),$

where $\; \mathcal{D}(A)$ is the domain of the operator $\; A$ .

It can be proved that this condition is equivalent to the following:

$\langle \psi,A\chi \rangle= \langle A\psi,\chi\rangle , \quad \psi, \chi \in \mathcal{D}(A)$ .

Having defined the linear operator $\; A^\dagger$ in $\; \mathcal{H}$ , which is called the adjoint of $\; A$ , such that

$\langle A^\dagger\psi,\chi \rangle= \langle \psi,A\chi\rangle , \quad \psi, \chi \in \mathcal{H}$ ,

it follows that $\; A$ must be Hermitian, i.e. its domain must be such that $\; \mathcal{D}(A) \subseteq \mathcal{D}(A^\dagger)$ and $\; A$ must coincide with $\; A^\dagger$ in $\; \mathcal{D}(A)$ .