system
Given the following post, respond as if you're commenting on Reddit:
human
How does a wave function change after scattering?

I'm trying to understand scattering theory in QM a little better.

What I'm trying to ask here—in more detail than the title—is how can the wave function be calculated after a scattering event? More specifically in the context of Compton and/or Thomson scattering.
gpt
In non-relativistic scattering theory, you just consider an incident plane wave, and an outgoing spherical wave, which has been modified by the interaction potential.

If you assume the incoming particle is moving in the z-direction, you can write the total wavefunction as:

ψ(**r**) = e^(ikz) + ψ*_s_*(**r**),

where e^(ikz) is the incoming particle, and ψ*_s_* is the scattered wave.

The scattered wave can then be written as:

ψ*_s_*(**r**) = f(θ,φ)e^(ikr)/r.

This has the form of an outgoing spherical wave which has been modulated by an angle-dependent factor f. f is called the scattering amplitude, and |f|^(2) ends up being the differential cross section.

To find what this function actually looks like, you just solve the time-independent Schrodinger equation for whatever potential is mediating the scattering.

To treat something like Compton scattering, you don't really want to use non-relativistic QM, but rather QFT. In QFT, you don't generally work with wavefunctions anymore, but you can still calculate S-matrix elements and cross sections. A common way to do this is using time-dependent perturbation theory, and Feynman diagrams.