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We analytically demonstrate the emergence of surface non-Hermitian boundary contributions that appear in an extended form of the quantum Ehrenfest theorem and are crucial (although so far overlooked) in the calculation of optical matrix elements that govern the optical transitions in semiconductors, e.g. solar cells. Their inevitable existence, strongly related to the boundary conditions of a given quantum mechanical problem, is far-reaching in the sense that they play a crucial role in the dynamics of solar absorption and the corresponding optical transitions that follow. Processes like optical transitions in localized and delocalized states and probabilities of intermolecular transitions can be investigated through this generalized off-diagonal Ehrenfest theorem, employed in the present work for the first time. An explicit demonstration of bulk-boundary correspondence is shown, as the extended Ehrenfest theorem can be separated into bulk and surface contributions, each behaving separately from the other, but at the end collaborating to give the correct time-derivative of the desired optical element; this paves the way for future application of the extended theorem to optical transitions in topologically nontrivial quantum systems. It is also demonstrated through two examples in the literature as well as through a new example (of a system exhibiting the Integer Quantum Hall Effect) that non-Hermitian boundary terms (that have been designated "topological anomalies" in the mathematical literature) may be expected to be quantized, especially in topologically nontrivial quantum systems but also in certain conventional ones.

Topological anomalies in the off-diagonal Ehrenfest theorem and their role on optical transitions in solar cells