This interpretation explains for example the linear relationship of the oscillator strength of the lowest excited state with system size in the case of some conjugated organic polymers (see e.g. this paper): If there are more electrons available to oscillate, then the transition strength increases.
The oscillator strength fij between two non-degenerate states i and j is defined (in atomic units) as two thirds of the squared transition dipole moment multiplied by the energy gap
where the vector r contains all 3N spatial coordinates of the N electrons
The Thomas-Reiche-Kuhn sum rule now states that the sum over the oscillator strengths from one state i to all possible other states is equal to the number of electrons in the system, i.e.
The derivation of this sum rule starts by realizing that the momentum operator with respect to any spatial coordinate x of any particle (e.g. x=y2) is given as the commutator of the Hamiltonian with this coordinate
The remaining proof follows what is shown here (sorry that I am switching the notation, but I copy-and-pasted a little bit ...). First one realizes that the commutator of x and px is equal to i
Then one expands the commutators and inserts a resolution of the identity over the eigenstates of the Hamiltonian
The commutators are evaluated by letting H act either on the bra or the ket, which results in a multiplication with the respective eigenvalue. And after summing together the equivalent terms one obtains
The actual r vector was composed of 3N individual electron coordinates. The above equation holds for each of these coordinates. Thus, in summary:
which is just what we wanted to show.