Time dependent scattering theory - cross section

[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?

 

[eljose79]

Thank you for your question about the double-slit experiment and its interpretation in terms of wave packets. The double-slit experiment is a classic experiment in quantum mechanics that has been studied and debated for many years. The experiment involves shining a beam of particles, such as electrons or photons, through two parallel slits and observing the resulting pattern on a screen behind the slits.

One of the key observations of the double-slit experiment is the interference pattern that is produced on the screen. This pattern can be explained by the wave nature of the particles, where the waves from the two slits interfere with each other to produce areas of constructive and destructive interference. This is known as the wave theory of light.

However, as you mentioned, there is also a wave packet interpretation of the double-slit experiment. In this interpretation, the particles are described as localized wave packets, rather than extended waves. This means that the particles have a well-defined position and momentum, but their wave function is not spread out over space like a traditional wave.

To explain the interference pattern in terms of wave packets, we can use the concept of scattering cross section. The scattering cross section is a measure of the probability of a particle being scattered in a particular direction. In the double-slit experiment, the particles are scattered by the two slits and the resulting interference pattern on the screen can be calculated by integrating the scattering cross section over all possible scattering directions.

In order to calculate the scattering cross section for a wave packet, we need to use time-dependent scattering theory. This theory takes into account the time evolution of the wave packet as it travels through the slits and interacts with the screen. By using this theory, we can calculate the probability current, which is the product of the wave function and its complex conjugate. This is similar to the expression you mentioned in your forum post.

In summary, the double-slit experiment can be explained in terms of wave packets by using time-dependent scattering theory and calculating the probability current. This allows us to calculate the interference pattern on the screen and understand the behavior of the particles in a more rigorous way. I hope this explanation helps to clarify the wave packet interpretation of the double-slit experiment.