Graduate Double-Slit Experiment: Momentum & Position

[Ad VanderVen]

TL;DR

In the double-slit experiment with two open slits, is there a fixed relationship between the momentum (p) of the particle immediately after passing through the slit and the position (q) of the impact on the screen?

In the double-slit experiment with two open slits, is there a fixed relationship between the momentum (p) of the particle immediately after passing through the slit and the position (q) of the impact on the screen?

 

[hutchphd]

Yes. The relationship is called Schrodinger's Equation for a non-relativistc massive particle.
If you wish a more specific answer you will need to formulate a more specific question.

 

[Ad VanderVen]

My specific question is, that I have a formula for the probability density function of the momentum (p) of the particle immediately after it passes through the slit in the double-slit experiment with two open slits and I want to derive from that formula a formula for the probability density function of the position (q) of the impact on the screen?

 

[hutchphd]

Then you need to solve Schrodinger to match your specific boundary conditions in time and space. This will likely be very difficult.
There are many approximation techniques that are useful for particular situations. What does your formula look like?

 

[Ad VanderVen]

It is formula (11) from Uffink and Hilgevoord. (1985):

$$\begin{align}
\phi \left(p \right) \, = \, \frac{\sqrt{2}~\sqrt{\frac{a }{\pi }}~\cos \left(A ~p \right)~\sin \left(a ~p \right)}{a ~p }
\end{align}$$

Reference

Uffink, J.B.M. and Hilgevoord, J. (1985). Uncertainty Principle and Uncertainty Relations. Foundations of Physics, Vol. 15, No. 9,