Calculating Electric Transition Dipoles with the Gradient Operator

8 Calculating Electric Transition Dipoles with the Gradient Operator 

10 The Relationship Between Matrix Elements of the Electric Dipole and Gradient Operators 

Equation (4.27) can be derived by relating the gradient operator V to the commutator of the Hamiltonian and dipole operators ([H, p]). (See Box 2.2 for an introduction to commutators.) The Hamiltonian operator H includes terms for both potential energy (V) and kinetic energy (T) however, we only need to consider T because V commutes with the position operator ([V,r] = O) and fi is simply ex. For a one-dimensional system, in which V is just d/dx, the commutator of H and ji is 

Generalizing to three dimensions, and treating the commutator as an operator gives 

We can relate the matrix element on the left side of Eq. (B4.10.2) to the transition dipole by expanding and formally in the basis of all the 

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