The Schrodinger equation for an electron and a single-mode photon field with interactions is solved by a direct method. A unique feature of these solutions is the inclusion of retardation effects in the photon field. Some interesting physical questions arising from the solutions are discussed. The Keldysh-Faisal-Reiss formula for the transition rate of multiphoton ionization modified by the inclusion of retardation effects is simplified by averaging the degenerate initial states. The result shows that the retardation effects can be calculated in terms of the radial part of the momentum wavefunction of the initial state. The physical significance of the inclusion is analysed in the near-threshold case of multiphoton ionization. The result shows that in the near-threshold case, retardation effects depend exponentially on the orbital angular momentum of the initial state. The effect vanishes for s-states, but is significant for states with high orbital angular momentum.
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