Symmetric operator

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

ψ, Aψ⟩ = ⟨ψ, Aψ *  = ⟨Aψ, ψ⟩,  ψ ∈ D(A),
where  D(A) is the domain of the operator A.

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

ψ, Aχ⟩ = ⟨Aψ, χ⟩,  ψ, χ ∈ D(A)
.

Having defined the linear operator A †  in  H, which is called the adjoint of A, such that

A † ψ, χ⟩ = ⟨ψ, Aχ⟩,  ψ, χ ∈ H
, it follows that A must be Hermitian, i.e. its domain must be such that  D(A) ⊆ D(A † ) and A must coincide with A †  in  D(A).

'Definition: Hermitian operators whose domain is dense in  H are called symmetric.'

In particular, if a bounded linear operator is symmetric, it is also a Hermitian and self-adjoint operator.