= Transition frequencies and matrix multiplication
{title2=$\omega_{mn}=(E_m-E_n)/\hbar$}
In the <Heisenberg picture>, an <observable> without explicit time dependence has
$$
A_H(t)=e^{iHt/\hbar}Ae^{-iHt/\hbar},\qquad (A_H)_{mn}(t)=A_{mn}e^{i\omega_{mn}t},\qquad \omega_{mn}=\frac{E_m-E_n}{\hbar}.
$$
Here $\hbar$ is the reduced <Planck constant>. An emitted <photon> on a transition $n\to m$ with $E_n>E_m$ has positive <angular frequency> $(E_n-E_m)/\hbar=|\omega_{mn}|$. The sign of the array's <quantum phase> follows its index convention; it is not a negative measured <photon> <frequency>. The frequencies involve differences of <energy levels>, rather than the absolute values $E_n/\hbar$. Adding a constant to $H$ leaves these frequencies unchanged.
Insert the complete <orthonormal basis> between two <linear operators>:
$$
(AB)_{mn}=\sum_k A_{mk}B_{kn},\qquad \omega_{mk}+\omega_{kn}=\omega_{mn}.
$$
The sum runs over intermediate states. The second identity follows by cancellation of $E_k$; it makes every term in the product acquire the same time factor $e^{i\omega_{mn}t}$. This is a useful reason for <matrix multiplication>: it composes the two indexed arrays consistently with the addition of transition frequencies. It does not multiply matching entries independently. Swapping $A$ and $B$ generally changes the intermediate-state sum and hence the result.
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