How Do We Model Wave Function Collapse After Measuring Particle Location?

[Sciencemaster]

Hello!
Let's say we have a wave function. Maybe it's in a potential well, maybe not, I think it's arbitrary here. This wave function is one-dimensional for now to keep things simple. Then, we use a device, maybe a photon emitter and detector system where the photon crosses paths with the wave function of our other particle at some point. If the photon does not reach the detector, that's where are particle is. As such, we know where it is to some finite region. There is going to be some uncertainty in our measurement, so I don't imagine it's simply a delta function. Additionally, if our detector does not measure the particle, what happens to the wave function? I would imagine that the new wave function is the same as the pre-collapse wave function, except for in the region with the detector, where the probability drops to zero in a piecewise fashion.
Essentially, my question is the same as the tagline. Once we collapse a particle's wave function to a finite region, or measure it to not be in a finite region, how do we mathematically model its wave function?

 

[anuttarasammyak]

I assume at least you need to change and renormalize wavefunction with the information of area where particle does not exist by your observation.

In the special setting that particle go though gates A or B and you observe no passing at B we may deduce that wavefunction collapse to pass gate A though no direct observation is done there. This example suggests to me that wavefunction is not more than information we have for the special setting we prepare. It does not seem physical entity to me. I know it is a controversial issue.

 

[Sciencemaster]

I assume at least you need to change and renormalize wavefunction with the information of area where particle does not exist by your observation.

In the special setting that particle go though gates A or B and you observe no passing at B we may deduce that wavefunction collapse to pass gate A though no direct observation is done there. This example suggests to me that wavefunction is not more than information we have for the special setting we prepare. It does not seem physical entity to me. I know it is a controversial issue.

Regardless of what is happening behind the scenes, our mathematical models should be able to predict reality. As such, we should be able to model a wave function post-collapse that can time-evolve and describe measurements. I'm hoping to learn how to model this given a measurement done on a given particle's wave function.

 

[StevieTNZ]

I thought you could predict the probability of an event happening (at t=0) regardless at what time it happens in the future or what happened beforehand. As such, superposition evolves into superposition, entanglement may occur etc.

 

[anuttarasammyak]

we should be able to model a wave function post-collapse that can time-evolve and describe measurements.

You may be interested in learning how particle wave function evolves after passing slits, i.e. position observation allowing width, and goes to screen for full position observation in Young double or other slit experiments.

 

[Demystifier]

As such, we know where it is to some finite region. There is going to be some uncertainty in our measurement, so I don't imagine it's simply a delta function. Additionally, if our detector does not measure the particle, what happens to the wave function? I would imagine that the new wave function is the same as the pre-collapse wave function, except for in the region with the detector, where the probability drops to zero in a piecewise fashion.

Yes, this is described by projectors to the region to which the wave function collapses. A projection is needed even if the detector does not click, because it also brings new information about position of the particle. For mathematical details see e.g. my recent paper https://arxiv.org/abs/2107.08777.

 

[Sciencemaster]

You may be interested in learning how particle wave function evolves after passing slits, i.e. position observation allowing width, and goes to screen for full position observation in Young double or other slit experiments.

I mean, this seems at the very least similar to what I was looking for, I think that would help!

 

[Sciencemaster]

Yes, this is described by projectors to the region to which the wave function collapses. A projection is needed even if the detector does not click, because it also brings new information about position of the particle. For mathematical details see e.g. my recent paper https://arxiv.org/abs/2107.08777.

Alright, so a projector can be used to model a new wave function post-collapse. However, I'm talking about spatial collapse, not arrival time like in your paper. So, for a particle whose position is being measured, would it still use projectors? How would that work?

 

[Demystifier]

However, I'm talking about spatial collapse, not arrival time like in your paper.

My paper also talks about spatial collapse, you just need to read more than title and abstract. See in particular Sec. 3.