In the double slit experiment, how the uncertainty principle contribute to verify the complementary principle?

[Axel Togawa]

With the double slit, experiment we show the double nature of light and matter as wave and particle. In particular, the so called "which way" thought experiment illustrate the complementary principle. In my book, this experiment is analyzed putting a series of particles in front of one of the two slit, so when the electron pass through the slit it scatter with the particles, changing the component of the momentum perpendicular to the direction of motion Δpy. Using the uncertainty principle, it's said that the uncertainty on the position of the electron is now Δy<<D with D the distance between the two slits, and consequently, Δpy is so large that the interference pattern is destroyed. But since the uncertainty principle represents an intrinsic property of the electron, independently of the measurement (correct me if I'm wrong), even if I don't alter the state of the electron but can still determine in which slit the electron crosses (I don't know if it's possible), the interference pattern should be destroyed. Doesn't that mean that considering the scattering I should take into consideration the uncertainty principle and the uncertainty due to the scattering?

[joigus]

7 hours ago, Axel Togawa said:

it's said that the uncertainty on the position of the electron is now Δy<<D with D the distance between the two slits,

I think you mean \( \triangle y\geq D \) or \( \triangle y\sim D \).

They're gonna tell you about the observer effect vs HUP. One thing is 1st-kind measurements or preparations, and quite a different thing are 2nd-kind measurements, or "interventions" --I would call them.

I'll be back.

Edited November 12, 2020 by joigus

[Axel Togawa]

I mean Δy as the uncertainty on the position of my electron when i know which slit it passed through, so i think it should be right Δy<<D.

Is therefore correct add the uncertainty due to HUP to the one caused by the observer effect?

[joigus]

No sorry. I made a mistake. \( \triangle y\geq D \) is not correct. For your example, the uncertainty (dispersion) in the \( y \) position is of order \( \triangle y\sim D \), so by HUP,

\[ \triangle p_{y}\geq\frac{\hbar}{2D} \]

which means that the dispersion in \( y \) is of order \( D \). That holds just after the particles go through the slit.

What you wrote is,

\[\triangle y\ll D\]

That means that the dispersion in \( y  \) is much less than \( D \), and that's not necessarily correct either. It depends on what regime of scattering you have after the electrons go through the slit, and it's certainly not true just after the electrons cross it, and before they scatter. The fact that you scatter particles after the passing of the electron through the slit of width \( D \) further changes the dispersion. So \( D \) no longer plays much of a role if you scatter the particles after they go through the slits. If you scatter the electrons with very energetic particles, you get better precision in \( y \) than \( D \), but you lose precision in \( p_y \).

Dispersions are not added. But a detailed calculation requires scattering theory and is outside the scope of what we can deal with here.

Whatever you do to the electrons to determine which way they go does destroy the interference pattern. Even if you set a detector in one of two alternative paths after they go through the slits and consider only the electrons that go through the other path --counterfactual measurement or so-called "interaction-free measurement."

I do have a feeling that I'm not being very clear and I apologize for that. Maybe tomorrow.

 

Edited November 13, 2020 by joigus