where m e is the mass of the electron. This means when (E α -E β ) is larger (for well-separated energy levels), |x αβ | will be smaller in order to satisfy the TRK sum rule. This can be clearly seen in the middle panels of Fig. , where the largest matrix elements for x αβ only show up for nearest neighbor energy levels. Thus, in the energy representation, x is "local" in the sense an electron in a 1-D square potential is strongly coupled to a cavity whose frequency is resonant to the electronic transition from the ground state to the first excited state. Electronic matrix element magnitudes shown for the coordinate x and its conjugate momentum, p. Note that the coordinate matrix is significantly more diagonal than the momentum matrix. The bottom panel shows the eigenspectra of the Coulomb (H C Rabi ) and dipole (H D Rabi ) gauges truncated to two levels compared to the full basis limit. The stark disagreement between to two gauges demonstrates the gauge ambiguities. For this model, the two level approximation is not a terribly good approximation. (b) This repeats the analysis for panel (a) for a double well potential. For this model, the two-level approximation is valid. This shows how for a valid level of truncation the dipole gauge results match very well with the full space results. (c) Demonstration of non-local potentials, V (x, x ′ ), that form upon a finite n-level truncation of the electronic Hilbert space. In the infinite basis limit, V (x, x ′ ) -→ V (x) and is completely local.