= Observable matrices in an energy basis
{title2=$A_{mn}=\langle m|A|n\rangle$}
Choose normalized <energy eigenstates> $|n\rangle$ of a time-independent <Hamiltonian operator>, with $H|n\rangle=E_n|n\rangle$, and assume first that they form a complete <orthonormal basis>. For a <pure state> and an <observable> $A$,
$$
|\psi\rangle=\sum_n c_n|n\rangle,\qquad A_{mn}=\langle m|A|n\rangle,\qquad (A\psi)_m=\sum_n A_{mn}c_n,\qquad \langle A\rangle=\sum_{m,n}c_m^*A_{mn}c_n.
$$
The column index $n$ labels the input <basis vector>, and row $m$ labels the output component. <Self-adjointness> gives $A_{nm}=A_{mn}^*$, 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.
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