Question re: Compton Scattering

Many QM texts seem to use Spherical Waves (e^{i k r} / r) to describe scattered Reflection Waves ([math]\Psi_R[/math]). Does this imply, that, in Compton Scattering, the photon's Wave Function is composed, of an incident plane-wave-like Wave Packet (e^{i k z}), plus an outgoing, reflected, Spherical Wave, with an angular dependent Wave Vector ([math]\vec{k}( \theta )[/math]), which varies according to the Compton Formula ? Presumably, the electron would be, similarly, scattered out, into a spherically spreading Wave Packet (whose momentum distribution was Quantum Entangled, with the momentums of the reflected photon Wave Function, according to the conservation laws, for total linear momentum) ?? As a rule, the "Classical Distributions", found from high-intensity beams, which map out many individual wave functions, are the simple sum of all those individual wave functions, so that the "Classical Patterns" accurately map out the individual Wave Functions (Greenstein & Zajonc. Quantum Challenge, ch.2,8) ???