Hamiltonian describing interaction with an external field

1 Hamiltonian describing interaction with an external field 

A charged particle (of charge q and mass m positioned at r) in a potential V(r) interacting with a linearly polarized external field is described by the Hamiltonian, 

Invoking the weak-field approximation (neglecting the term involving A A) and the long wavelength approximation (e 1), we arrive at the familiar dipole approximation expression, 

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