Interference pattern after a particle is detected?

[Nickyv2423]

Say we have a particle, like an electron in a double slit experiment. And say that we have a measuring apparatus to detect it right after its fired out of the electron gun. And after that it goes through the double slit and hits the detector plate. Would we see an interference pattern? If the electron is measured right after it gets shot out of the gun it's wave function would collapse. So when it keeps traveling through the double slit would it create an interference pattern, even though it already collapsed before?

 

[Drakkith]

I believe so. If the detector has no way of knowing which slit the electron will go through, I think the electron will still interfere with itself and show an interference pattern.

 

[mikeyork]

This is about the fifth or sixth thread on this same subject in the last week or so. Each variation makes no essential difference as to how we discuss what is going on. Here is IMO a way to analyze the situation in all variations:

1. Electrons are prepared by shooting them from an electron gun (usually they are coherent i.e. have well-defined momentum).
2. They pass through multiple stages (slits, detectors, quantum erasers, being eaten and excreted by unicorns , etc..). Each of these stages affects the prepared state in a different way, but the result is still a prepared state of one or more electrons.
3. Each electron is detected at a point on a screen (or not). The frequency of impact of electrons at any small local area of the screen is determined by the final "prepared" state information that arrives at the screen and its projection onto the spatial context of the screen. All that the final "observer" (the screen) does is force the projection onto the screen context (a spatial co-ordinate basis in QM parlance).

In the case of detection at the gun it depends on what is detected. We'll assume that it does not significantly alter the electron state. A. If the detector at the gun determines the trajectory, then it will also determine through which slit, if any, the electron passes. The effect will be the same as placing a detector at the slit. Absent a quantum eraser the result will be multiple overlapping single slit diffraction patterns. B. If the detector at the gun does not determine through which slit the electron will pass, if any then, absent any other apparatus that can detect through which slit the electron passes (such as a detector at the slit) the result will be multi-slit diffraction pattern (a complex pattern of fringes produced by the single-slit patterns "interfering" with each other).

 

[PeroK]

Say we have a particle, like an electron in a double slit experiment. And say that we have a measuring apparatus to detect it right after its fired out of the electron gun. And after that it goes through the double slit and hits the detector plate. Would we see an interference pattern? If the electron is measured right after it gets shot out of the gun it's wave function would collapse. So when it keeps traveling through the double slit would it create an interference pattern, even though it already collapsed before?

I think that "collapse" is a misleading term for the change to a wavefuction. This creates the impression that something terminal and irreversible has happened to the particle.

I guess "collapse" is used to distinguish the change from the continuous time evolution.

But, all collapse really means is a sudden change caused by the measurement of some observable, such as position.

And, in something like a double-slit experiment an electron's wavefuction may "collapse" several times during the experiment: preparation, firing, detection along the way and finally detection at the screen.

 

[PeterDonis]

say that we have a measuring apparatus to detect it right after its fired out of the electron gun.

What exactly is the apparatus detecting?

If the apparatus is detecting the electron's momentum, that won't change the electron's state significantly, since it basically comes out of the gun in an eigenstate of momentum (at least to a good approximation). So this won't change anything much after that--you will still see interference at the detector.

If the apparatus is detecting the electron's position, then that will change the electron's state drastically, since it comes out of the gun in an eigenstate of momentum and the detector puts it into an eigenstate of position (again, to a good approximation at least). This will drastically change the overall experiment, because now most of the electrons that come out of the gun won't even reach the slits, let alone the detector! Most of them will fly off in some other direction. Of those few electrons that still end up going through the slits and reaching the detector, they will still form an interference pattern, but it will take many, many more runs to get a pattern of the same intensity as the original apparatus without the detector at the gun.

 

[mikeyork]

If the apparatus is detecting the electron's momentum, that won't change the electron's state significantly, since it basically comes out of the gun in an eigenstate of momentum (at least to a good approximation). So this won't change anything much after that--you will still see interference at the detector.

I disagree. The electron may be in a momentum eigenstate to a good approximation and so its position will be uncertain to a good approximation assuming we know the location of the detector. But Planck's constant is tiny and the quantum context is microscopic. The slit separation, however, is macroscopic and huge by comparison. So it seems to me that the momentum will effectively determine if the electron passes through a slit and, if so, which slit. The prepared state after passing through the slits would then be effectively the same as that obtained by putting detectors at the slits. Yes, there would still be a slight ambiguity and a bit of slit interference but it would be negligible in the overall distribution on the screen unless the slit width were comparable to the slit separation..

 

[PeterDonis]

it seems to me that the momentum will effectively determine if the electron passes through a slit and, if so, which slit

I agree that whether a momentum measurement determines which slit the electron will go through depends on the spread in momentum vs. the distance between the slits. By "momentum eigenstate" I meant a plane wave, or at least a plane wave to a good enough approximation as compared with the slit spacing, which would mean being in a momentum eigenstate would not determine which slit the electron would go through. But there is certainly a lot of room to specify momentum measurements which are uncertain, but not uncertain enough to make it uncertain which slit the electron will go through. Which just makes it even more important to specify exactly what measurement is being made.

 

[mikeyork]

By "momentum eigenstate" I meant a plane wave, or at least a plane wave to a good enough approximation as compared with the slit spacing

That would typically require the slits to be a great distance from the gun. Although a plane wave is usually assumed as an approximation in order to calculate the fringe separation a plane wave isn't necessary for there to be slit interference in a standard double slit experiment. The full details of the pattern depend on the distance of slits from the gun, the distance of screen from slits, the slit separation and the slit width as well as the momentum range. For instance, if the screen were sufficiently close to the slits, then the screen would itself act as a slit detector!

 

[PeterDonis]

That would typically require the slits to be a great distance from the gun.

A great distance compared to the wavelength, yes. For an electron that's not very far. AFAIK this condition is typically met quite easily in actual experiments.

Although a plane wave is usually assumed as an approximation in order to calculate the fringe separation a plane wave isn't necessary for there to be slit interference in a standard double slit experiment.

Yes, agreed.

 

[mikeyork]

A great distance compared to the wavelength, yes. For an electron that's not very far. AFAIK this condition is typically met quite easily in actual experiments.

Sure. But if there is a momentum detector at the gun, it has to be a lot further since it has to lose the trajectory separation information as well,depending on the precision of the trajectory determination.

 

[PeterDonis]

if there is a momentum detector at the gun, it has to be a lot further

I think this depends on how close the eigenstates of the detector are to the states prepared by the gun. If they're close enough I don't think the detector distance matters.

 

[mikeyork]

I think this depends on how close the eigenstates of the detector are to the states prepared by the gun. If they're close enough I don't think the detector distance matters.

The detector is essentially at the gun. I was referring to the distance from detector to slits.

It's the difference in momentum component parallel to the plane of the slits and perpendicular to the slit orientation that matters. I think it is only if the detector is able to measure that difference that the question of overlapping single slit or a double-slit pattern arises.