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

[PeterDonis]

So what really are the pointer states of the apples?

I don't think Zurek ever writes one down. And Zurek's paper is an attempt to tackle a very complex issue; I don't think you should even be trying to understand what Zurek is saying until you are crystal clear about much simpler cases. See below.

For each particle or for the entire apple (but you said in last message it's not possible for entire apple at once).

I still don't think you understand what I've been saying. Once again, before even trying to understand the case of an apple, you should try to understand the case of a two-particle quantum system, which I've posted about several times now.

First: for the two particle system, there is no such thing as "the state of particle 1" in isolation. There is only the state of the system. (Here we are using "state" to mean "wave function" or "state vector in Hilbert space".) If the system is such that we can specify a state for each individual particle, then it is not a quantum two-particle system; it is just two individual quantum particles that do not interact with each other at all, and no quantum interference is possible between them. (More precisely: the state of the two-particle system is separable, and therefore no quantum interference effects can be observed between the two particles.)

Second: if we construct an observable on the two-particle system like the "center of mass" observable, this observable does not pick out a single state of the two-particle system. It only picks out a subspace of the full Hilbert space of all possible states--the subspace that satisfies the constraint that the center of mass position (the average position of the two particles) is equal to the measured value of the center of mass observable.

Do you understand what the above statements mean? And do you understand how they make the things you have been saying meaningless?

 

[bluecap]

I don't think Zurek ever writes one down. And Zurek's paper is an attempt to tackle a very complex issue; I don't think you should even be trying to understand what Zurek is saying until you are crystal clear about much simpler cases. See below.
I still don't think you understand what I've been saying. Once again, before even trying to understand the case of an apple, you should try to understand the case of a two-particle quantum system, which I've posted about several times now.

First: for the two particle system, there is no such thing as "the state of particle 1" in isolation. There is only the state of the system. (Here we are using "state" to mean "wave function" or "state vector in Hilbert space".) If the system is such that we can specify a state for each individual particle, then it is not a quantum two-particle system; it is just two individual quantum particles that do not interact with each other at all, and no quantum interference is possible between them. (More precisely: the state of the two-particle system is separable, and therefore no quantum interference effects can be observed between the two particles.)

Second: if we construct an observable on the two-particle system like the "center of mass" observable, this observable does not pick out a single state of the two-particle system. It only picks out a subspace of the full Hilbert space of all possible states--the subspace that satisfies the constraint that the center of mass position (the average position of the two particles) is equal to the measured value of the center of mass observable.

Do you understand what the above statements mean? And do you understand how they make the things you have been saying meaningless?

So for 2-particle system, it is entangled and in pure state.. but it's component is not in pure state. But the entangled state is a superposition. So I understand your first case where you stated "for the two particle system, there is no such thing as "the state of particle 1" in isolation".. because it is entangled. In the second case. Are you saying that we can only get observable for each component and not the entire pure state? The component being same meaning as "subspace" of the full Hilbert space of all possible states? So we can only get observable by tracing the rest in the density matrix (only focusing on a subspace or component)?

How do you apply this concept to one particle that is in superposition of taking left or right slit. Going to the double slit experiments. The single particle is in superposition taking account left and right slits. And the detections in the screens are its eigenstates (?) Here what you mean the observable is only defined for each component or subspace.. you are like saying the screen detection is only for each slit.. but isn't it for both slits (hence pick out a single state of the two-slit paths)?

Thanks a while lot! Appreciated so much for all the help.

 

[PeterDonis]

So for 2-particle system, it is entangled and in pure state..

If the state is not separable, which is what we are discussing, yes.

but it's component is not in pure state.

Remember how I defined "state": a wave function/state vector. The components (the individual particles--note that this is one of several possible usages of the term "component", see below) do not have such a state at all; only the two-particle system does. If one uses the density matrix formalism, one can assign a "state" (a density matrix) to one of the particles individually, but I am not talking about that definition of "state", because we first need to get clear about the simpler case.

But the entangled state is a superposition.

That depends on the observable. For any pure state, it is always possible in principle to find an observable for which that state is an eigenstate, not a superposition. Since this is true of any pure state for any quantum system, it is true for any pure state of the two-particle system under discussion.

You need to take a step back and rethink your understanding in the light of what I have just said; I think your confusion between an entangled state and a superposition is underlying much of your confusion in this thread.

Are you saying that we can only get observable for each component and not the entire pure state? The component being same meaning as "subspace" of the full Hilbert space of all possible states?

It appears that you don't understand what a "subspace" is. Let me illustrate with a particular example. Suppose we have measured the center of mass position of a two-particle system to be $X = 0$; i.e., we have measured the center of mass of the two-particle system to be at the spatial origin. The full Hilbert space of the two-particle system is the set of all wave functions $\Psi(x_1, x_2)$ of two positions. Our measurement of the center of mass position restricts the state of the two-particle system to the subset $\Psi_0(x_1, x_2)$ for which $x_2 = - x_1$ (because the average position must be $0$), i.e., to the set of functions $\Psi(x_1, x_2)$ which only have nonzero values when $x_2 = - x_1$. This set of functions is a subspace of the full Hilbert space. But a "pure state" is a single function $\Psi(x_1, x_2)$, not a whole set of them. (Strictly speaking, this is only true if we have defined the wave functions appropriately, so that each one corresponds to a distinct ray in the Hilbert space; but we'll ignore that complication here.)

For another usage of the term "component", different from how I used it above, see below.

Going to the double slit experiments. The single particle is in superposition taking account left and right slits.

More precisely: the wave function at the detector can be expressed as a superposition of two "components", one for each slit; each component can be thought of, heuristically, as expressing the probability amplitude for the particle to arrive at a given position on the detector after passing through one of the slits. Here the "components" are parts of a single particle wave function, whereas earlier in this post, the "components" were individual particles in a multi-particle system. These are two different concepts and you need to be very careful not to confuse them.

And the detections in the screens are its eigenstates (?)

No. As I have said several times now, eigenstates depend on the observable. You continue to be very sloppy about how you are expressing these things, and I think this is contributing to your confusion. Also, you are confusing the double slit case with the two-particle case; they are not the same. See below.

Here what you mean the observable is only defined for each component or subspace.. you are like saying the screen detection is only for each slit..

No, that's not what I am saying at all. In the double slit experiment, we do not have a two-particle system. We have a single-particle system. So none of the things I've been saying about two-particle systems even apply to this case.

In the double slit experiment, if we are sloppy and talk about "position eigenstates" as if they were actual states, then after the particle hits the detector and is detected, it is in a position eigenstate. If we are not sloppy, then we have to say that after the particle hits the detector and is detected, it is in a state with a very narrow spread of amplitude as a function of position--heuristically, a state in which the particle's position is somewhere within a very small "box" (the size of the "box" corresponds to the spatial resolution of the detector). But these are states of a single-particle system, not a multi-particle system.

Also, you are still confused about what I'm saying about the two-particle system. I am not saying an observable like the "center of mass position" is "only defined for each component or subspace". Components and subspaces are different things (see above). The observable is defined on the entire Hilbert space; but for each possible measured value of the observable, there is a corresponding subspace of the Hilbert space that contains all the possible states that are consistent with that measured value, as I described earlier in this post.

 

[bluecap]

If the state is not separable, which is what we are discussing, yes.
Remember how I defined "state": a wave function/state vector. The components (the individual particles--note that this is one of several possible usages of the term "component", see below) do not have such a state at all; only the two-particle system does. If one uses the density matrix formalism, one can assign a "state" (a density matrix) to one of the particles individually, but I am not talking about that definition of "state", because we first need to get clear about the simpler case.
That depends on the observable. For any pure state, it is always possible in principle to find an observable for which that state is an eigenstate, not a superposition. Since this is true of any pure state for any quantum system, it is true for any pure state of the two-particle system under discussion.

You need to take a step back and rethink your understanding in the light of what I have just said; I think your confusion between an entangled state and a superposition is underlying much of your confusion in this thread.
It appears that you don't understand what a "subspace" is. Let me illustrate with a particular example. Suppose we have measured the center of mass position of a two-particle system to be $X = 0$; i.e., we have measured the center of mass of the two-particle system to be at the spatial origin. The full Hilbert space of the two-particle system is the set of all wave functions $\Psi(x_1, x_2)$ of two positions. Our measurement of the center of mass position restricts the state of the two-particle system to the subset $\Psi_0(x_1, x_2)$ for which $x_2 = - x_1$ (because the average position must be $0$), i.e., to the set of functions $\Psi(x_1, x_2)$ which only have nonzero values when $x_2 = - x_1$. This set of functions is a subspace of the full Hilbert space. But a "pure state" is a single function $\Psi(x_1, x_2)$, not a whole set of them. (Strictly speaking, this is only true if we have defined the wave functions appropriately, so that each one corresponds to a distinct ray in the Hilbert space; but we'll ignore that complication here.)

For another usage of the term "component", different from how I used it above, see below.
More precisely: the wave function at the detector can be expressed as a superposition of two "components", one for each slit; each component can be thought of, heuristically, as expressing the probability amplitude for the particle to arrive at a given position on the detector after passing through one of the slits. Here the "components" are parts of a single particle wave function, whereas earlier in this post, the "components" were individual particles in a multi-particle system. These are two different concepts and you need to be very careful not to confuse them.
No. As I have said several times now, eigenstates depend on the observable. You continue to be very sloppy about how you are expressing these things, and I think this is contributing to your confusion. Also, you are confusing the double slit case with the two-particle case; they are not the same. See below.
No, that's not what I am saying at all. In the double slit experiment, we do not have a two-particle system. We have a single-particle system. So none of the things I've been saying about two-particle systems even apply to this case.

In the double slit experiment, if we are sloppy and talk about "position eigenstates" as if they were actual states, then after the particle hits the detector and is detected, it is in a position eigenstate. If we are not sloppy, then we have to say that after the particle hits the detector and is detected, it is in a state with a very narrow spread of amplitude as a function of position--heuristically, a state in which the particle's position is somewhere within a very small "box" (the size of the "box" corresponds to the spatial resolution of the detector). But these are states of a single-particle system, not a multi-particle system.

Also, you are still confused about what I'm saying about the two-particle system. I am not saying an observable like the "center of mass position" is "only defined for each component or subspace". Components and subspaces are different things (see above). The observable is defined on the entire Hilbert space; but for each possible measured value of the observable, there is a corresponding subspace of the Hilbert space that contains all the possible states that are consistent with that measured value, as I described earlier in this post.

I tried to understand everything and I'm getting it thanks.
But if eigenstates are good only for one particle.. how is it useful for practical applications where you have more than one particle? (such as quantum chemistry)
What is the common applications of eigenstates when the system has more than one particle? Can you give an actual example of more than one particle system and they happily apply the concept of eigenstates? Thank you.

 

[PeterDonis]

I'm getting it

Unfortunately I'm still not sure you are. See below.

if eigenstates are good only for one particle..

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.

 

[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.