Uncertainty Principle and Compatible Observables

As we know, observables are associated to Hermitian operators. Given one such operator A we can use it to measure some property of the physical system, as represented by a state Ψ. If the state is in an eigenstate of the operator A, we have no uncertainty in the value of the observable, which coincides with the eigenvalue corresponding to the eigenstate. We only have uncertainty in the value of the observable if the physical state is not an eigenstate of A, but rather a superposition of various eigenstates with different eigenvalues. We want to define the uncertainty ΔA(Ψ) of the Hermitian operator A on the state Ψ. This uncertainty should vanish if and only if the state is an eigenstate of A. The uncertainty, moreover, should be a real number. In order to define such uncertainty we first recall that the expectation value of A on the state Ψ, assumed to be normalized, is given by (A) = (Ψ|A|Ψ) = (Ψ, AΨ) . (1.1) The expectation (A) is guaranteed to be real since A is Hermitian. We then define the uncertainty as the norm of the vector obtained by acting with (A − (A)I) on the physical state (I is the identity operator):