How is the uncertainty relation preserved in this experiment

[weezy]

For an electron can I not do the following to determine both the position and momentum? I take a screen with a small hole and I eventually make the hole smaller and smaller. Cathode rays emitted will hence get diffracted after passing through the hole making momentum more and more uncertain. What if I weaken my cathode ray to such an extent that only one electron passes at an instant. I could calculate the electron's velocity by measuring precisely how much time it took from gun to detector(placed behind the wall with the hole). In the most ideal condition at least one electron passes through the hole and when it does I calculate the velocity(hence momentum) and I could keep doing this to arbitrary precision. I'm aware this won't work. What I want to know is the limit where this experiment fails to violate the principle.

 

[mfb]

If you measure the time at gun and detector precise enough, you won't get the usual interference pattern any more as you get a large velocity uncertainty. Also, what exactly would you measure? The velocity at which point in time? Not at the time the electron passes the hole (where your position measurement is).

 

[Vanadium 50]

What I want to know is the limit where this experiment fails to violate the principle.

The second electron. It's state will not perfectly match the first's.

 

[drvrm]

actually that type of experiment has been done as reported in the book by Anderson- what happens that the feeble beam of electrons comes out from the hole and gives impressions on the screen and if one electron is coming say in one second after a good lapse of time one gets a jumble of spreading marks on the screen and no diffraction pattern emerges.

 

[jfizzix]

In a way, you can measure both the position and momentum of an electron, but you can't violate the uncertainty principle (sounds crazy, I know). The thing is that in the current setup, the position and momentum are being measured at different times, with different measurements.

By forcing the electrons to exit a pinhole, you know their initial position to high precision.
By using where and when the final detection takes place, you know the final position to high precision, and the time interval with high precision.

From these, you can calculate the average momentum (mass times total displacement over total time), and know it to a high precision, but this does not tell you very much at all about what the initial momentum or final momentum are.
Many random paths can have the same initial and final points, and we can't even assume the electrons travel in straight lines (since electron diffraction is the very thing we're looking at here).

What the uncertainty principle would say here, is that there are no set of measurements that will tell you both the position and momentum of a particle at a single point in time with unlimited precision.

 

[zonde]

From these, you can calculate the average momentum (mass times total displacement over total time), and know it to a high precision, but this does not tell you very much at all about what the initial momentum or final momentum are.
Many random paths can have the same initial and final points, and we can't even assume the electrons travel in straight lines (since electron diffraction is the very thing we're looking at here).

What the uncertainty principle would say here, is that there are no set of measurements that will tell you both the position and momentum of a particle at a single point in time with unlimited precision.

This is Copenhagen interpretation not a generic QM, right? Given idea that uncertainty principle is applicable to a single particle at a single point in time.

 

[jfizzix]

The general statement of the uncertainty principle is that it is impossible to prepare a quantum system in such a state that the outcome of both a position measurement and a momentum measurement may be predicted with unlimited precision. Whether the quantum state applies to a single particle, or only to an ensemble of particles is a matter of the interpretation of quantum mechanics, but the consequence remains the same in all interpretations.