In the double slit experiment, what is the position and momentum of a particle?

Question

1. Is the position of an electron/photon which slit it went through?
2. And the momentum of the electron/photon the wave-like interference pattern detected on the detector screen?

Answer 1

In the double slit experiment, what is the position and momentum of a particle?

The experiment , even with single particle at a time , extends in time. The direction of the ingoing particle is known, and the hit on a screen after the slits is known.

For a classical experiment with billiard balls This is mathematically fitted by a trajectory. The mathematics of throwing billiard balls at slits gives a trajectory up to the slit and then straight through , or if the billiard ball scatters off an edge of the slits a distribution will appear at the screen, and an image of two blurred lines. Even in the classical case at the region between the two slits images one cannot say which slit the particle went through (unless extra detection is made at the slits)

1. Is the position of an electron/photon which slit it went through?

Here is a double slit experiment with a build up of single electrons at a time, starting from the top.

The only position measurements we have are the (x,y) of the electron on the screen and the (x,y) positions of the slits. The mathematical trajectories would not differ from the classical ones. As more and more electrons accumulate an interference pattern emerges in space.

This interference pattern has led to a lot of confusion. Since the electrons are found one electron at a time they obviously are not split. The pattern is a probability distribution, i.e. answers :how probable is it that I will find an electron at the (x,y) of the screen. This probability distribution fits the quantum mechanical solutions of "an electron scattering through two slits" . Generally detectors on slits destroy the interference accumulation. unless care is taken . In this experiment the detectors are subtle , the quantum mechanical problem changes but still some interference is seen, since each measurement changes the boundary conditions , and a changed pattern appears in the accumulation.

This experiment displays the quantum mechanical character of elementary particles, their point particle behavior on the screen, their quantum mechanical nature going through the screen.

2. And the momentum of the electron/photon the wave-like interference pattern detected on the detector screen?

The momentum of the electron is decided by the experimenter and chosen in the range where the Heisenberg uncertainty principle, HUP will allow to display the wave probability pattern.

The delta(x) in this case is the size and distance between slits chosen so that delta(x)*delta(p) is within the bounds of the HUP.

From the wavelength of the interference pattern using the known mass of the electron and the de Broglie relation one can get the momentum chosen for the incoming beam.

It is important to stress that the wave nature displayed in the experiments on elementary particles are neither energy waves nor matter waves as far as individual particles go. The wave nature of the particles appears into the distributions of probability of detection/interaction. The double slit experiment answers the quantum mechanical boundary value problem "one electron interacting with two slits " and the answer is a probability density distribution when the boundaries are within the HUP.

Answer 2

Is the position of an electron/photon which slit it went through?

No. The position is the result of an interaction. You have a slit, and electrons are free to go through both slits and travel until they reach some kind of particle detector such as a screen. The screen makes a record of the position based on which part of the screen reacts to the electron. So each electron leaves a mark based on where it lands.

There is no information about which slit you went through unless some other particle changes its state based on whether the electron goes through.

And the momentum of the electron/photon the wave-like interference pattern detected on the detector screen?

No. The momentum is what it had before it went through the slits. You get position by interacting and you get momentum by interacting. So first you interact so you have a momentum but no position, this allows you to go through both slits in a nice coherent way. Then later you acquire a position instead of a momentum.

The wave-like interference pattern is built up over time. When you send many electrons through one at a time you see that some places get a larger share of hits than others. Some places are more popular destinations for position to be determined and other places are less popular.