The Quantum Formulation of Polarizability by Bethe Rules

Starting from the consecrated second order perturbation energy (see Volume I/Section 3.7 of the present five-volume set) 

going to evaluate the atomic polarizability in terms of the quantum basic information contained within the atomic quantum numbers (e.g., n, k), one starts recognizing the general opeiatorial identity over the complete set of quantum (eigen) states (Putz, 2010c). 

Equation (4.415) is eventually known as the sum rule of Bethe and Jackiw (Bethe Jackiw, 1968 Jackiw, 1967), while its simplest dipole matrix element stun rule casts as 

On the other hand, recalling the basic quantum commutation rule of momentum with space coordinate 

Quantum Nanochemistry—Volume II Quantum Atoms and Periodicity 

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