# Time dependent scattering theory - cross section

1. Nov 10, 2014

### tom.stoer

I am looking for a realistic explanation of the double-slit experiment in terms of wave packets (instead of stationary waves). First of all this results in using the scattering cross section, i.e. the probability current (not the density). Then, I guess, there is a kind of time average. So one should end up with something like

$$j_\text{scatt}(x,t) \sim \text{Im}\psi^\ast\nabla\psi$$

calculated for a scattered wave packet

$$|\psi,t\rangle = U(t,t_0)\,|\psi,t_0\rangle$$

and an integration like

$$N(\Omega) \sim \int_{-T}^{+T}dt\,\int_\Omega d\Omega \, j_\text{scatt}(x,t)$$

to calculate the number of particles N detected in Omega on a spherical screen.

Is there a rigorous derivation of such an expression for wave packets using e.g. time-dependent scattering theory?

Last edited: Nov 10, 2014
2. Nov 15, 2014