The "size" of entangled particles

[Souma]

Hello everyone, I hope you are having a nice day,

I was reading [The principles of Quantum Mechanics by P.A.M Dirac], and I was attracted to the definition of size. The book says: "If the object under observation is such that the unavoidable limiting disturbance is negligible, then the object is big in the absolute sense and we may apply classical mechanics to it. If, on the other hand, the limiting disturbance is not negligible, then the object is small in the absolute sense and we require a new theory for dealing with it." This definition makes me wonder about the "size" of entangled particles.
If I am correct, we can observe the state of one of the entangled particles (in case of 2 particles) by observing the state of the other particle. If this implies that the particle is observed with very little disturbance (maybe no disturbance at all), then
the particle should be really, really big (if not infinite) in "size", right? Does that mean entangled particles can be divided into two halves, a quantum mechanical half (the particle we observe) & a classical mechanical half (the particle we observe by relying on the observation of the other particle)?
Is the problem with our definition of size? or is it that I am missing more information?

 

[anuttarasammyak]

Since the days of the text about 1930, QM has progressed. Disturbance is not an essential concept as it used to be and entanglement is more fully analyzed now. As a classics formalism of Dirac is still worth learning but I think you do not have to go into details of his literature.

 

[DrClaude]

"If the object under observation is such that the unavoidable limiting disturbance is negligible, then the object is big in the absolute sense and we may apply classical mechanics to it. If, on the other hand, the limiting disturbance is not negligible, then the object is small in the absolute sense and we require a new theory for dealing with it."

That's simply a heuristic. You shouldn't take it too literally.

And as for entangled particles, the system total is being disturbed. You can't take the particles individually in that context.

 

[Souma]

Since the days of the text about 1930, QM has progressed. Disturbance is not an essential concept as it used to be and entanglement is more fully analyzed now. As a classics formalism of Dirac is still worth learning but I think you do not have to go into details of his literature.

I will keep that in mind. But do you think there's any better way of defining the size of things?

 

[vanhees71]

Before you define an expression, you have to ask yourself, what do you want to achieve with this definition, i.e., what do you want to describe by "size"?

In quantum theory, it's not at all very clear what "size" should mean. As an example take a hydrogen atom. It consists of a proton and an electron bound together by the electromagnetic force. What you can define as "size" is something like the average distance between the proton and the electron. In the ground state you get the Bohr radius. If you have excited states this size becomes somewhat larger.

 

[Souma]

That's simply a heuristic. You shouldn't take it too literally.

And as for entangled particles, the system total is being disturbed. You can't take the particles individually in that context.

I see. But what if we considered the degree of disturbance? even if the whole system is disturbed, do you think the particles of the system will be disturbed equally?

 

[anuttarasammyak]

But do you think there's any better way of defining the size of things?

Dirac said in the very beginning of his book QM is more suitable (than classical physics) for the description of phenomena on the atomic scale. I think in other words the phenomena that ##\hbar## matters.

 

[Souma]

Before you define an expression, you have to ask yourself, what do you want to achieve with this definition, i.e., what do you want to describe by "size"?

In quantum theory, it's not at all very clear what "size" should mean. As an example take a hydrogen atom. It consists of a proton and an electron bound together by the electromagnetic force. What you can define as "size" is something like the average distance between the proton and the electron. In the ground state you get the Bohr radius. If you have excited states this size becomes somewhat larger.

It is actually not clear what size really means, but I think its meaning lies in solving the problem of "getting deeper" in the atom. We can't keep saying there's something inside something else, otherwise no ground will be reached. Dirac gave that definition of size in an attempt to reach the ground. If we considered this approach, is there any better definition of size?

 

[vanhees71]

Physics is about observables, and a quantity is defined by a procedure how to measure it. Often it's not easy to clearly define in that sense what "size" means and how to get the precise value of it. A recent famous example is the "charge radius" of the proton, which has been an enigma for quite some time since it seemed to depend on how it's meausured. One way is to define the charge radius by electron-proton scattering. Then the charge radius can be extracted from what's called the electromagnetic form factor of the proton. Another way to get the same quantity is to measure very accurately the bound-state energies of hydrogen atoms (i.e., a usual hydrogen atom with an electron around the proton) or of muonic hydrogen atoms (i.e., with a muon bound to the proton instead of an electron). All these measurements should lead to the same value for this quantity, "charge radius", but for quite some time there was a discrepancy between the different ways to measure it. Only recently it turned out that one has to carefully take into account all kinds of effects to precisely define the quantity from the measured data.

 

[DrChinese]

If I am correct, we can observe the state of one of the entangled particles (in case of 2 particles) by observing the state of the other particle. If this implies that the particle is observed with very little disturbance (maybe no disturbance at all), then
the particle should be really, really big (if not infinite) in "size", right? Does that mean entangled particles can be divided into two halves, a quantum mechanical half (the particle we observe) & a classical mechanical half (the particle we observe by relying on the observation of the other particle)?

The EPR paradox, 1935, attempted to use your logic on a pair of entangled particles. You should definitely familiarize yourself with that paper. The paper is now known to come to an incorrect conclusion due to the theoretical work of Bell and others.

Specifically: An entangled system of 2 particles cannot be considered to be a system of 2 separate independent particles. It is not "separable". Consequently, attempts to model it that way fall apart almost immediately. The only thing you can say is that certain conservation rules apply to the system as a whole. Total charge, total momentum, total spin, etc.

So no, it cannot be divided into 2 halves as you imagine.

 

[vanhees71]

The EPR paper is very hard to understand, and Einstein himself didn't like it very much, because for him the main problem with QM has not been well formulated in this paper, as he put it: The argument is "buried under erudition". In 1948 he wrote a much better understandable paper about his issue concerning the problem of an entangled system consisting of two parts which are observed at two very far distant places. His main quibble indeed was the implied "inseparability", i.e., the existence of long-ranged correlations, described by entanglement.

The 1948 paper is unfortunately in German, but there's is a translation here. I have not carefully checked the entire document, how well this translation has been done, but from reading the first paragraphs it seems to be very good:

https://www.informationphilosopher.com/solutions/scientists/einstein/dialectica.html

Forget most of the remarks in the gray boxes though, which are partially very misleading from a physics point of view!

 

[Souma]

Dirac said in the very beginning of his book QM is more suitable (than classical physics) for the description of phenomena on the atomic scale. I think in other words the phenomena that ##\hbar## matters.

The EPR paper is very hard to understand, and Einstein himself didn't like it very much, because for him the main problem with QM has not been well formulated in this paper, as he put it: The argument is "buried under erudition". In 1948 he wrote a much better understandable paper about his issue concerning the problem of an entangled system consisting of two parts which are observed at two very far distant places. His main quibble indeed was the implied "inseparability", i.e., the existence of long-ranged correlations, described by entanglement.

The 1948 paper is unfortunately in German, but there's is a translation here. I have not carefully checked the entire document, how well this translation has been done, but from reading the first paragraphs it seems to be very good:

https://www.informationphilosopher.com/solutions/scientists/einstein/dialectica.html

Forget most of the remarks in the gray boxes though, which are partially very misleading from a physics point of view!

Thank you very much for this, I will definitely check it out.

 

[DrChinese]

The 1948 paper is unfortunately in German, but there's is a translation here. I have not carefully checked the entire document, how well this translation has been done, but from reading the first paragraphs it seems to be very good:

https://www.informationphilosopher.com/solutions/scientists/einstein/dialectica.html

Forget most of the remarks in the gray boxes though, which are partially very misleading from a physics point of view!

I can't comment on the translation, but the argument seems to be a well presented version of the original EPR paper. Admittedly the 1935 paper was unnecessarily long and convoluted. This is sharper, still same basic logic. That being that according to QM, entangled systems evidence a subjective (observer dependent) reality that is not bounded by c (i.e. in different parts of space). It was natural that Einstein would want to assume that what happens to a distant particle cannot affect a local one. He acknowledged the opposing position:

"There seems to me no doubt that those physicists who regard the descriptive methods of quantum mechanics as definitive in principle would react to this line of thought in the following way: they would drop the requirement II for the independent existence of the physical reality present in different parts of space; they would be justified in pointing out that the quantum theory nowhere makes explicit use of this requirement."

And in fact this is the consensus opinion of physicists today: requirement II is dropped. That requirement is often called "local realism", and was later found to be experimentally incompatible with the predictions of QM.

 

[vanhees71]

Well, the facts however are that these long-ranged correlations are there. That's due to the experimental work following the theoretical breakthrough by Bell in the last 3-4 decades. I also do not think that there is any tension between causality and inseparability as Einstein thought, because with the formulation of local (microcausal) relativistic QFT there are no actions at a distance. The correlations are there due to the preparation of the system in the corresponding entangled state. What one has to give up are local deterministic hidden-variable theory and stick to a strictly probabilistic interpretation of the quantum state, which is sometimes dubbed "non-realistic" though ironically QT is, according to all empirical evidence, the most realistic description of nature in the sense that it is most accurately describing quantitatively what's observed, but from there on we usually get into endless discussions about interpretation, which usually do not lead to any conclusion.

 

[Souma]

QT is, according to all empirical evidence, the most realistic description of nature in the sense that it is most accurately describing quantitatively what's observed, but from there on we usually get into endless discussions about interpretation, which usually do not lead to any conclusion.

This is what makes quantum mechanics unsatisfactory. How can the most realistic description of nature give only numbers without meaning? I mean, yes we can predict events to great accuracy by using QM, but as long as the "why" question is not answered, QM will remain unsatisfying.

 

[vanhees71]

QT gives numbers with meaning. It explains all phenomena known today. How do you come to the conclusion that QT is unsatisfactory? There's no answer to any "why" question in the natural sciences. They rather describe "how" nature behaves and tries to reduce the description to some fundamental laws, and amazingly are very successful in doing so.

 

[Souma]

QT gives numbers with meaning. It explains all phenomena known today. How do you come to the conclusion that QT is unsatisfactory? There's no answer to any "why" question in the natural sciences. They rather describe "how" nature behaves and tries to reduce the description to some fundamental laws, and amazingly are very successful in doing so.

Your right. Natural sciences don't have the obligation to answer the "why" questions. I am more into the philosophical aspects of natural sciences, that is why I feel knowing why is as important as knowing how things work in our universe. It's as if you have a machine you don't know anything about, you may understand how it works, but after understanding that, what is the next thing you are going to look for? For me, I would like to know why it works the way it works. There are a lot of unanswered "why" questions about our universe, and as long as they are not fully answered, science won't know the nature of reality at all.

 

[vanhees71]

That's not science but philosophy or even rather theology. So it's off-topic here!

 

[Souma]

That's not science but philosophy or even rather theology. So it's off-topic here!

You are right. We drifted away from our main topic. Thank you very much for your time.