Abstract
Most quantum mechanics textbooks approximate the nonrelativistic but high-energy continuum-state wave function by a plane wave in the evaluation of photoelectric effect matrix element. This approximation leads to very wrong results if the gauge is chosen so that the EM interaction Hamiltonian is −(e/m)A⋅p and the matrix element is evaluated directly for the momentum operator. On the other hand, the same approximation is a fair one if the interaction Hamiltonian is written in the −eE⋅r gauge, or when p is replaced by the commutator −im[H,r], where H is the atomic Hamiltonian. This is explicitly demonstrated for the photoelectric effect for the hydrogen atom in the ground state.
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