 Observable matrices in an energy basis

Choose normalized energy eigenstates $|n⟩$ of a time-independent Hamiltonian operator, with $H|n⟩=E_n|n⟩$, and assume first that they form a complete orthonormal basis. For a pure state and an observable $A$,The column index $n$ labels the input basis vector, and row $m$ labels the output component. Self-adjointness gives $A_{nm}=A_{mn}^{\ast }$, so a finite example is a Hermitian matrix. A matrix element can be negative or complex; by itself it is not generally a probability. The Born rule assigns measurement probabilities using the eigenstates of $A$, which need not be energy eigenstates. For an unbounded observable, these expressions require a suitable operator domain; a system with continuous spectrum also needs the corresponding spectral integrals.