Several forms of the dipole matrix element

Within the dipole approximation, one can have different forms for the dipole matrix element (see [BSa57]). The form presented so far is called the momentum form (or the velocity form) because the relevant operator contains the momentum p  

The dipole matrix element on the right-hand side of equ. (8.19b) is called the length form of the matrix element, because the vector r acts as the photon operator (see the discussion of equ. (1.28a) in which the name dipole approximation is also explained). Equ. (8.16) can then be replaced by 

Finally, a third form can be derived by applying equ. (8.20) to the time derivative of the momentum p, which gives the acceleration form of the dipole matrix element  

All three forms of the dipole matrix element are equivalent because they can be transformed into each other. However, this equivalence is valid only for exact initial- and final-state wavefunctions. Since the Coulomb interaction between the electrons is responsible for many-body effects (except in the hydrogen atom), and the many-body problem can only be solved approximately, the three different forms of the matrix element will, in general, yield different results. The reason for this can be seen by comparing for the individual matrix elements how the transition operator weights the radial parts R r) and Rf(r) of the single-particle wavefunction differently  

As an example, the Is photoionization cross section in helium calculated for the three forms of the dipole matrix element is shown in Fig. 8.1, and the deviations can be clearly seen. 

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