Instantaneous Or Gradual?

From the previous post "$\psi_{max}$ Is New And Important",

$F=\psi_{n}-\psi_o=m\cfrac{dv}{dt}$

assumes that the photon $(\psi_{n}-\psi_o$) is absorbed by the particle instantaneously.  This may not be true.  In this case, the transit particle gain non zero momentum at $\psi(x=x_a/2)=\psi_{max}$, as it was accelerated.  It is then decelerated beyond $(x_a/2,\,\psi_{max})$.  The particle will oscillate about $(x_a/2,\,\psi_{max})$, until it emits the packet of energy gained and make a downward energy transition.

Is the photon absorbed by the particle instantaneous, on direct collision, or is the transfer of energy from a passing photon in helical motion to the particle gradual?