Partial waves method in scattering problems

Question

One way of deal with scattering problems is to use partial wave analysis. This is the procedure I've found in my notes: It's assumed that the observer is in the center-of-mass reference frame and the system is treated using the reduced mass.

When the interaction potential in the origin is zero the wave function is a plane wave $$\psi_{{\rm incident}}(x)=e^{i\vec{k}\cdot \vec{x}} $$ The plane wave can be rewritten as a sum of partial waves $$\psi_{{\rm incident}}=\sum_l i^l(2l+1)\left[e^{-i\left(kr-\frac {\pi}{2} l\right)}-e^{i\left(kr-\frac {\pi}{2} l\right)}\right]P_l(\cos\theta).$$

When the interaction within the two particles is present, it's assumed that the wave function becames: $$\psi=\sum_l i^l(2l+1)\left[e^{-i\left(kr-\frac {\pi}2 l\right)}-\eta(k)e^{i\left(kr-\frac {\pi}2 l\right)}\right]P_l(\cos\theta)$$

Finally using these wave function we derive an expression for the probability currents $j_{incident}$ and $j_{scattered}$ and doing some consideration we obtain an expression for the cross section.

Scattering between two particles is a process that evolves in time, so I don't understand why we treat it as it were a static phenomena (using stationary states). Can you explain me in words what is the main idea behind this method?

Answer 1

I would formulate the main idea as follows:
You consider a system with a scatterer at the origin. The interaction is modelled by some potential with a finite range/support around the origin. Your incident wave approaches this scatterer from 'infinity', in the sense that the incident wave is a free state far from the scatterer. Getting closer to the origin it starts interacting with the scatterer and eventually leaves it range. When it is far from the origin again, it can essentially be considered as a free wave. Therefore, the only change that could've happened should be encoded in its amplitude and should take the following form \begin{equation} \psi_\text{scattered} = f(\vec{k},\theta,\phi)e^{ikz} \end{equation}
where I assumed the $z$-direction to be the direction the wave is travelling in. A very similar philosophy holds in order to define the $S$-matrix.

Your questions about it being 'static' is that we don't use an equation like the Schrödinger equation to evolve the quantum state while it is passing through the origin, which is the region of interaction. In a sense, what you're interested in is the phase shift which is the result of the total interaction, a sort of time-averaged effect. This will allow you to learn something about the potential (attractive vs repulsive, etc.).

The Born approximation is usually used in this approach, it gives an adequate model potential for the interaction but what the interaction actually looks like is different. For example, the Coulomb and Yukawa potentials can be extracted from its corresponding QFT via this approximation.

Answer 2

If there is no scatterer present the solution will be $\psi_{\rm inc}$. When the there is a scattering center present the solution will change. The change in $\psi$ due to the scatterer is, by definition, $\psi_{\rm scat}$.