Momentum-space detection of an Electron (EWP)

[logic smogic]

[SOLVED] Momentum-space detection of an Electron (EWP)

I know there have been plenty of questions about electrons, momentum, and wave packets recently - but my question is distinct (and comes with a picture!).

Question: If you ionize an atom and detect the resulting free electron wave packet, how should it appear in momentum space?

My understanding is that the spacially-localized electron wave packet (EWP) that is born during ionization will be of a Gaussian shape, due to:

$$\Psi(x,t) = \frac{1}{\sqrt{2 \pi \hbar}} \int^{+\infty}_{-\infty}{\phi(p)e^{i(px - Et)/\hbar} dp$$

and that it will spread as it moves away from the ion and towards the detector. But won't the detector still process it as either a delta function or a Gaussian (in position space)? And if so, why isn't the resulting detection in momentum space a Gaussian (the Fourier Transform)?

The picture below is of an EWP that was single photon ionized by an XUV attosecond pulse. It looks to me like a "Sinc" function, which means that in position space it would be a rectangular function. This is without scattering (off of the ion core). My question really is why does this picture look like this? Thanks!

 

[logic smogic]

Oh, nevermind!

I found a paper that derives the final (drift) momentum as

$$p(t_{0}) = \frac{e E}{\omega} [cos(\omega t_{0}) + \gamma]$$

which is effectively a sinc function.

 

[denian]

I can provide an explanation for the momentum-space detection of an electron (EWP).

Firstly, it is important to understand that the concept of momentum space is a mathematical representation of the momentum states of a particle. In this case, we are dealing with an electron, which has both particle and wave-like properties. The momentum of an electron can be described by its wave function, which is a mathematical description of the probability amplitude of finding the electron at a particular momentum.

When an atom is ionized and a free electron is produced, it will have a wave function that is a superposition of different momentum states. This wave function can be described by a Gaussian shape, as mentioned in the question. However, as the electron moves away from the ion and towards the detector, it will spread out in space and therefore also in momentum space.

The detector will process the electron's wave function, and the resulting detection in momentum space will not necessarily be a Gaussian. This is because the detector will measure the electron's momentum at a specific point in time, and the wave function at that point in time will be a superposition of different momentum states. This can result in a non-Gaussian shape in momentum space.

Additionally, the scattering of the electron off of the ion core can also affect its momentum and therefore its detection in momentum space. This can result in a different shape than what is expected from the initial Gaussian wave function.

In the given picture of the EWP, the rectangular shape in position space can be explained by the fact that the electron is localized in a small region of space. This can also affect its momentum and result in a non-Gaussian shape in momentum space.

In conclusion, the momentum-space detection of an electron (EWP) can vary depending on the specific experimental setup and conditions. The resulting shape in momentum space may not always be a Gaussian, and this can be influenced by factors such as the electron's initial wave function, its movement, and any interactions with its surroundings.