Why Is the Probability of Finding a Particle in a Position Eigenstate Zero?

[bluecap]

Unfortunately I'm still not sure you are. See below.
I did not say they were. In principle, one can define observables for multi-particle systems such that the multi-particle system will be in one particular eigenstate of that observable when the observable is measured. However, in practice, it is often very difficult to find such observables that can actually be measured, and it gets more and more difficult the more particles a multi-particle system has. For a macroscopic object it is practically impossible. Which means that most of the time, when we talk about observables for multi-particle systems, we are talking about observables like the center of mass position observable, that don't put the system into a single particular eigenstate when they are measured. There might not even be any definable eigenstates for such an observable at all.

This leads to a question: why are you so interested in eigenstates? It seems to me that you would be better served by taking a step back and looking at the way QM actually models multi-particle systems mathematically. Your current understanding seems to be based on misconceptions.

I'm interested in eigenstates because I want to know what observables were chosen in Decoherence Einselection (Environment Induced Superselection). Zurek wrote:

"Interactions that
depend on a certain observable correlate it with the environment,
so its eigenstates are singled out, and phase
relations between such pointer states are lost".

In an apple, what eigenstates were singled out by the environment in Decoherence, Einselection which cause "phase relations between such pointer states are lost"??

 

[PeterDonis]

I'm interested in eigenstates because I want to know what observables were chosen in Decoherence Einselection

Then I think you need much, much more background than you currently appear to have. As I said before, Zurek is trying to deal with a very complex and advanced subject in QM. Also, the way he is using the word "eigenstate" looks nonstandard to me. It might be appropriate for the particular advanced subject he is dealing with, but it certainly doesn't lead to a simple answer to this question of yours:

In an apple, what eigenstates were singled out by the environment in Decoherence, Einselection which cause "phase relations between such pointer states are lost"??

As far as I can tell, the short answer to this question is that there are no such eigenstates if we use "eigenstate" in the way that term is usually used. He is basically using "eigenstate" to mean "pointer state", but his "pointer states" are really what I was calling "subspaces" earlier; they aren't single states, they're huge sets of states that are all consistent with a particular measured value for some macroscopic observable. All he is saying about decoherence and einselection is that there is no way to measure quantum interference effects between the different possible "pointer states". None of this has anything to do with the questions you have been asking about eigenstates as they appear in simpler measurements, like measuring the position of a single particle or the double slit experiment.

 

[bluecap]

Then I think you need much, much more background than you currently appear to have. As I said before, Zurek is trying to deal with a very complex and advanced subject in QM. Also, the way he is using the word "eigenstate" looks nonstandard to me. It might be appropriate for the particular advanced subject he is dealing with, but it certainly doesn't lead to a simple answer to this question of yours:
As far as I can tell, the short answer to this question is that there are no such eigenstates if we use "eigenstate" in the way that term is usually used. He is basically using "eigenstate" to mean "pointer state", but his "pointer states" are really what I was calling "subspaces" earlier; they aren't single states, they're huge sets of states that are all consistent with a particular measured value for some macroscopic observable. All he is saying about decoherence and einselection is that there is no way to measure quantum interference effects between the different possible "pointer states". None of this has anything to do with the questions you have been asking about eigenstates as they appear in simpler measurements, like measuring the position of a single particle or the double slit experiment.

Peter. I have a last question that I appeal you to answer before you tell me to spend 4 years undergraduate coarse in QM and Ballentine. I need to know the answer to the following that is so basic. Here's my puzzle. You said in old archive that:

However, once again, the size of the disturbance relative to the size of the object matters. If you are measuring an object that has only one quantum building block, like an electron, any measurement you make is going to disturb it significantly--heuristically, because the measurement itself has a minimum size which is basically one quantum building block. (For example, if we try to measure the electron by bouncing photons off of it, the minimum measurement we can make is to use one photon.) But if you are measuring an object with 1025'>10251025 10^{25} building blocks, like a piece of wood, there are lots of ways to measure it without significantly affecting its state, simply because of the huge number of building blocks. In fact, measuring an object of that size is really no different from what its environment is continually doing to it anyway--which is part of Zurek's point.

But Zurek own Pointer States being Subspace can be re-prepared. See https://arxiv.org/pdf/0707.2832v1.pdf "This (iii) quantum Darwinism allows observers to use environment as a witness to acquire information about pointer states indirectly, leaving system of interest
untouched and its state unperturbed." Zurek has to create the complicated concepts of fragments just so we can't directly perturb the pointer states. And the pointer states are macroscopic. Can you give example of what it means to directly perturb the pointer states if there is no fragments.. like can you make apples change shape or something (just one example). And lastly. In conventional QM.. how come people don't talk about perturbing the subspaces and effecting macroscopic quantum effect? Please answer these last questions before you lock this thread again and censor any discussions about this. Thank you!

 

[PeterDonis]

Zurek own Pointer States being Subspace can be re-prepared. See https://arxiv.org/pdf/0707.2832v1.pdf "This (iii) quantum Darwinism allows observers to use environment as a witness to acquire information about pointer states indirectly, leaving system of interest
untouched and its state unperturbed."

What you quote from Zurek's paper does not say that pointer states can be re-prepared. It says that observers can acquire information about pointer states (which is not the same as "re-preparing" them) from the environment.

Also, where does Zurek say that pointer states are subspaces? That word does not appear at all in the paper linked to in the quote above. If you are basing this on what I've said previously about subspaces, don't; you need much, much more background than you currently have before you will be able to use that concept correctly.

Zurek has to create the complicated concepts of fragments just so we can't directly perturb the pointer states.

He uses the concept of fragments to explain how we can repeatedly make observations that give us information about the same pointer state, without directly perturbing the pointer state. It's not that we can't perturb the pointer state; in principle it should be physically possible to do that (see below). But that can't be what we are actually doing when we make ordinary observations; if we did, if that's what "observing" the pointer state was, then we would not be able to make repeated observations of the same pointer state that gave the same results. The only way that can be possible is if our observations don't actually interact with the pointer state at all, but instead interact with something else (the fragments in the environment) that has information about the pointer state.

Can you give example of what it means to directly perturb the pointer states if there is no fragments.

Not based on anything in Zurek's paper; he doesn't include any theoretical model of how to do that. Nor does anyone else that I'm aware of. In principle it should be physically possible (though not necessarily practical) to perturb any quantum state; but saying that's possible in principle is not at all the same as having a specific model for how to do it in a particular case.

In conventional QM.. how come people don't talk about perturbing the subspaces and effecting macroscopic quantum effect?

First, please see my comment above about the term "subspaces". You are not using it correctly here.

Second, nobody knows how to make a "macroscopic quantum effect" in practice (see above), so nobody talks about doing it. But experimentalists have been gradually increasing the size of systems that can be shown to have quantum interference effects under the right conditions. The double slit experiment, for example, has now been done with molecules having 810 atoms each and massing more than 10,000 amu. See here:

https://arxiv.org/abs/1310.8343

Please answer these last questions before you lock this thread again

I have done so; and now I am locking the thread after answering them.