Classicality and the Correspondence Principle

[jlcd]

Correspondence Principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations."

But is it applied for only one system (say a single atom or single system like the quantum oscillator), or is it also correct to say that the human body is not quantum because of the correspondence principle? Or a bicycle is not quantum because of the correspondence principle?

 

[PeterDonis]

Correspondence Principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers.

Basically, yes. But note that this is not the same as saying that macroscopic objects (objects with large quantum numbers) are not quantum objects.

is it applied for only one system (say a single atom or single system like the quantum oscillator)

It can't be applied to such a system, because such a system does not have large quantum numbers.

is it also correct to say that the human body is not quantum because of the correspondence principle? Or a bicycle is not quantum because of the correspondence principle?

No. See above.

 

[jlcd]

Basically, yes. But note that this is not the same as saying that macroscopic objects (objects with large quantum numbers) are not quantum objects.
It can't be applied to such a system, because such a system does not have large quantum numbers.
No. See above.

Human body has zillions and zillions of atoms and molecules. Can't you considered it as having large quantum numbers?

I mean does correspondence principle apply to the human body or not?

 

[jlcd]

Basically, yes. But note that this is not the same as saying that macroscopic objects (objects with large quantum numbers) are not quantum objects.
It can't be applied to such a system, because such a system does not have large quantum numbers.
No. See above.

To reword it. The human body is ruled by classical physics and also quantum? But if it is ruled by classical physics. It shouldn't have quantum effects. Or maybe you are saying macroscopic body is quantum yet doesn't have quantum effects due to the correspondence principle?

Also the human body atomic numbers were scattered in different atoms. They are not continuously. So technically. why does the correspondence principle apply in the human body? It's not like this closely spaced oscillator here http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc6.html which produces classical physics. What parts of the human body atoms have large quantum numbers or identical to it?

 

[PeterDonis]

Human body has zillions and zillions of atoms and molecules. Can't you considered it as having large quantum numbers?

Yes. But that doesn't mean the human body is not a quantum object.

does correspondence principle apply to the human body or not?

It does.

The human body is ruled by classical physics and also quantum?

No. The human body is a quantum object, which, because it has large quantum numbers, exhibits classical-like behavior (i.e., behavior similar to that which would be predicted by classical physics) due to the correspondence principle. In other words, the correspondence principle says, as you say in your OP:

for large orbits and for large energies, quantum calculations must agree with classical calculations.

Again, that is not the same as saying objects stop being quantum when they have large quantum numbers.

the human body atomic numbers were scattered in different atoms

So what? The correspondence principle doesn't say the large quantum numbers have to be in a single atom.

If you're bothered by the term "large quantum numbers", just use "large number of degrees of freedom" instead. That's really a better way of describing what differentiates macroscopic objects which act "classically" from microscopic ones which act "quantum".

 

[jlcd]

Yes. But that doesn't mean the human body is not a quantum object.
It does.
No. The human body is a quantum object, which, because it has large quantum numbers, exhibits classical-like behavior (i.e., behavior similar to that which would be predicted by classical physics) due to the correspondence principle. In other words, the correspondence principle says, as you say in your OP:
Again, that is not the same as saying objects stop being quantum when they have large quantum numbers.
So what? The correspondence principle doesn't say the large quantum numbers have to be in a single atom.

If you're bothered by the term "large quantum numbers", just use "large number of degrees of freedom" instead. That's really a better way of describing what differentiates macroscopic objects which act "classically" from microscopic ones which act "quantum".

Thank you.

Say, does correspondence principle exist because of the existence of the born rule? In the oscillator with close spacing, it produces the classical one because of born rule. Can you give one example where there is no born rule but the correspondence principle still apply?

 

[PeterDonis]

does correspondence principle exist because of the existence of the born rule?

Meaning, do objects with large numbers of degrees of freedom behave classically because of the Born rule? I'm not sure, because the Born rule is not clear about what a "measurement" is, so it's not clear about when to apply it.

In the oscillator with close spacing, it produces the classical one because of born rule.

How so? There is no mention of the Born rule in the article you linked to. Nor does the argument it makes depend on the Born rule; the argument describes the shape of the quantum wave function and notes that it is similar to the shape of the classical probability curve for the particle's location. None of that requires applying the Born rule.

 

[jlcd]

Meaning, do objects with large numbers of degrees of freedom behave classically because of the Born rule? I'm not sure, because the Born rule is not clear about what a "measurement" is, so it's not clear about when to apply it.
How so? There is no mention of the Born rule in the article you linked to. Nor does the argument it makes depend on the Born rule; the argument describes the shape of the quantum wave function and notes that it is similar to the shape of the classical probability curve for the particle's location. None of that requires applying the Born rule.

If you will look at the hyperphysics link again. There is the psi^2 besides the graphics. It is the born rule. Born rule states that the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point. So psi^2 is direclty related to born rule. Unless you can state psi^2 is valid even without born rule? If you can't find the psi^2 in the graphics. I'll share the graphics later.

 

[PeterDonis]

There is the psi^2 besides the graphics.

Ah, yes, sorry, I missed that.

 

[A. Neumaier]

But is it applied for only one system (say a single atom or single system like the quantum oscillator)

It can't be applied to such a system, because such a system does not have large quantum numbers.

If you're bothered by the term "large quantum numbers", just use "large number of degrees of freedom" instead.

This is not the same.

The correspondence principle dates back to Niels Bohr 1920 (i.e., even before the advent of modern quantum mechanics 1925). He applied it to single electrons (of hydrogen atoms) with high angular momentum quantum numbers.

 

[PeterDonis]

This is not the same.

Not for single electrons in hydrogen atoms, no. But nobody has ever observed a hydrogen atom with an electron that has quantum numbers large enough for Bohr's argument to apply to them. So if you limit yourself to Bohr's original argument, you're making the correspondence principle irrelevant in any practical sense.

If, OTOH, you want to apply the correspondence principle to macroscopic objects, then you have to think about large numbers of degrees of freedom, not just large quantum numbers for individual quantum systems (since, as you note, there generally will not be any individual quantum systems in such an object that have large quantum numbers by themselves). That is, as I understand it, implicitly what is being done in any discussion of the correspondence principle that claims to apply it to macroscopic objects.

In a practical sense, I'm not sure how much of a role the correspondence principle actually plays in modern QM. In the early days of QM it was a useful guide to finding quantum models of systems, but all of the systems for which it was a useful guide back then have been well understood for decades and nobody actually appeals to the correspondence principle to justify the quantum models of them, since those models have extensive experimental confirmation by now.

 

[jlcd]

Ah, yes, sorry, I missed that.

About the born rule. For simple system or even molecules where it is only the system and apparatus. The probability amplitudes or density can range a lot like you see in normal quantum chemistry (where the wave is largest or highest, you get more probability by squaring it).

However, when the apparatus has entangled with the environment causing decoherence. The mixed states don't form probabily densities that have large amplitude differences akin to the Gerlach experiment with only Up and Down branches? I mean. Let us say the position preferred basis is chosen in the apparatus-environment decoherence. It is just positions and doesn't form any probability amplitude with huge differences in the probability? Does it make sense one location has higher probability than another in zillions of apparatus-environment superpositions? Although in energy basis, it can happen. But not in position? (I'm aware that technically position eigenstates aren't actually states since they aren't normalizable, and one has to do a fair bit of mathematical work to make all this rigorous, but we'll ignore that here)

Can you please give example where the probability density of different position mixed states (apparatus-environement, not system-apparaturs where it is more obvious) have large differences to one another (like top of the wave function compare to near the zero axis?)

This is related to this thread becuse it's about how classicality is derived from the quantum. Correspondence principle seems to involve system and apparatus per Bohr specification. But to model apparatus and environmental decoherence and emergence of classicality. Here I want to understand the behavior of wave function amplitudes in such apparatus-environment setup (let's use this in place of Born Rule to just focus on the math of it and not the interpretations).

 

[PeterDonis]

when the apparatus has entangled with the environment causing decoherence. The mixed states don't form probabily densities that have large amplitude differences akin to the Gerlach experiment with only Up and Down branches?

I'm sorry, I don't understand what you're trying to say here or asking in the rest of your post. Decoherence just means that there is no interference between the different branches. It doesn't place any limitations on the relative probabilities of the different branches.

I also don't understand the distinction you're trying to draw between "system and apparatus" and "apparatus and environment". You seem to be using the Stern-Gerlach experiment as an example of "system and apparatus", as contrasted with "apparatus and environment"; but the way you detect the result of the Stern-Gerlach experiment is by seeing where a spot appears on the detector, which involves an "environment" since the detector has many degrees of freedom which are not kept track of.

 

[PeterDonis]

nobody has ever observed a hydrogen atom with an electron that has quantum numbers large enough for Bohr's argument to apply to them.

Actually, this isn't quite true:

https://en.wikipedia.org/wiki/Rydberg_atom
But these cases are still rare, and require specially controlled methods to produce. They certainly are nothing like the states of ordinary atoms in macroscopic objects.

 

[jlcd]

I'm sorry, I don't understand what you're trying to say here or asking in the rest of your post. Decoherence just means that there is no interference between the different branches. It doesn't place any limitations on the relative probabilities of the different branches.

I also don't understand the distinction you're trying to draw between "system and apparatus" and "apparatus and environment". You seem to be using the Stern-Gerlach experiment as an example of "system and apparatus", as contrasted with "apparatus and environment"; but the way you detect the result of the Stern-Gerlach experiment is by seeing where a spot appears on the detector, which involves an "environment" since the detector has many degrees of freedom which are not kept track of.

For simple quantum system. I can understand how there can be say 15% probability, 60% probability, 25% probability (that equals 100%). But in decoherence. I can't think of a scenerio where there is 30% probability, 70% probability in the different branches. Can you give an example?

About the second paragraph. But if the Stern-Gerlach apparatus is also entangled with the environment. What produced the up and down when the apparatus is entangled in all sorts of ways with the environment. It should also produce Up-down-up states and all combinations. This is why I thought the apparatus is isolated from environmental decoherence.

 

[jlcd]

For simple quantum system. I can understand how there can be say 15% probability, 60% probability, 25% probability (that equals 100%). But in decoherence. I can't think of a scenerio where there is 30% probability, 70% probability in the different branches. Can you give an example?

About the second paragraph. But if the Stern-Gerlach apparatus is also entangled with the environment. What produced the up and down when the apparatus is entangled in all sorts of ways with the environment. It should also produce Up-down-up states and all combinations. This is why I thought the apparatus is isolated from environmental decoherence.

Here are more details of the latter.

In a controlled measurement in the lab, where we can write down the explicit Hamiltonian and its eigenstates of say the Stern-Gerlach apparatus for measuring spin. The basis up and down are clear.

However, in apparatus-environmental decoherence, we don't control the interaction between the apparatus and the environment that determines which states of the apparatus they are in.

So if the Stern-Gerlach experiment is exposed to the environment. Why doesn't it produce Up and Down basis ambiguity of the pointer states we commonly find in macroscopic apparatus environment decoherence?

 

[PeterDonis]

It should also produce Up-down-up states and all combinations.

No, that's what decoherence eliminates. You seem to have decoherence backwards. It doesn't cause interference between pointer states like "up" and "down". It eliminates it.

ambiguity of the pointer states we commonly find in macroscopic apparatus environment decoherence?

I have no idea what you're talking about here. Decoherence eliminates ambiguity of the pointer states.

 

[charters]

Can you please give example where the probability density of different position mixed states (apparatus-environement, not system-apparaturs where it is more obvious) have large differences to one another (like top of the wave function compare to near the zero axis?)

Take a well localized macroscopic object A and isolate it in a large vacuum chamber but where its center of mass is prepared in a very small finite region R, where the volume/extension of A >>> R. Now let A evolve freely for a trillion years. The center of mass wavefunction will now have spread to have, say, 99% probability of still being in R, 1% outside it. (I am ignoring issues with permanent tails, but the same point works for Newton-Wigner-esque effectively localized states). Now, pump the chamber full of air. The air will quickly decohere A such that it is now an improper mixture, weighted 99% in R and 1% outside R.

The same thing happens for realistic macro objects but instead of a trillion years, its only for the briefest of moments between collisions/photon absorbtions&emissions/any other decohering interactions, so you have even less spreading off the initial state, and possibly Zeno type suppression on top.

So if the Stern-Gerlach experiment is exposed to the environment. Why doesn't it produce Up and Down basis ambiguity of the pointer states we commonly find in macroscopic apparatus environment decoherence?

The preferred/stable basis for the SG device is essentially the position basis. The up/down basis of its spin measurement is the relative position of its exit ports, versus whatever filters you used to preselect your particle's spin axis. So, the SG's (stable against decoherence) position eigenstate fixes the basis for the spin measurement.

 

[PeterDonis]

The preferred/stable basis for the SG device is essentially the position basis.

No, actually it's the momentum basis; the SG magnetic field entangles the electron's momentum with its spin. The detector is what converts the momentum to a position (because where on the detector the spot appears depends on the direction in which the electron's momentum points when it exits the SG magnetic field).

 

[charters]

No, actually it's the momentum basis; the SG magnetic field entangles the electron's momentum with its spin. The detector is what converts the momentum to a position (because where on the detector the spot appears depends on the direction in which the electron's momentum points when it exits the SG magnetic field).

Isn't this what I said? By device I meant detector.

 

[jlcd]

No, that's what decoherence eliminates. You seem to have decoherence backwards. It doesn't cause interference between pointer states like "up" and "down". It eliminates it.
I have no idea what you're talking about here. Decoherence eliminates ambiguity of the pointer states.

I mean. Right now. In the case of the cat, we don't know what pick out the alive/dead basis as the one that is physically relevant. We don't know how the quantum interaction between the cat and its environment picks out the alive/dead basis as the one that gets decohered, so that all observers will agree that the cat is either alive (in one branch) or dead (in the other branch).

But in the case of spin up and down in the Stern Gerlach experiment. Why do we know how to do it?

 

[A. Neumaier]

Here I want to understand the behavior of wave function amplitudes in such apparatus-environment setup

The apparatus is usually described by a density matrix (as it is decohered by its environmnt). Thus no wave function picture applies.

But in decoherence. I can't think of a scenerio where there is 30% probability, 70% probability in the different branches. Can you give an example?

The reduced density matrix may have quite arbitrary diagonal entries; decoherence only let's the off-diagonal part decay to zero. Thus uou can easily have diagonal enties 0.7 and 0.3.

 

[PeterDonis]

By device I meant detector.

But you talked about the position basis being defined by the "exit ports", which would be exit ports of the magnetic field source, not the detector. You say it's the "relative position" of those ports, but really that's not the case; it's the orientation of the magnetic field. There don't even have to be "ports" in the apparatus: the magnetic field could be in empty space, and it would still entangle the electron's momentum with its spin and split one input beam into two output beams. None of this involves any decoherence or any selection of a preferred position basis; as I said, if there's a preferred basis after the electron-magnet interaction, it's momentum, not position, since that's what gets entangled with the spin.

 

[charters]

But you talked about the position basis being defined by the "exit ports", which would be exit ports of the magnetic field source, not the detector

I am just thinking of the whole SG as one piece, as in https://ds055uzetaobb.cloudfront.net/brioche/uploads/wPEXhoGgui-q2p2.svg?width=300

I wasn't dividing it into the "field source" and "detector", just treating it as one contraption.

There don't even have to be "ports" in the apparatus: the magnetic field could be in empty space, and it would still entangle the electron's momentum with its spin and split one input beam into two output beams

But this is part of the process is not even a decohering interaction anyway. You can still recombine the electron paths after the magnet, and they will interfere/unitarily restore the initial state. This portion of the SG is akin to the MZI after the beamsplitters, which is not decoherence either. There's no decoherence until the electron is detected at the exit ports or whatever you want to call them. So I don't see why we are talking about this. If there are no exit ports, there's no decoherence, and actually not even an experiment.

 

[PeterDonis]

am just thinking of the whole SG as one piece

If there are no exit ports, there's no decoherence

The detector is not "exit ports", it's a screen of some kind (in the original SG experiment it was photographic film) that shows spots when electrons (or whatever particles you're using; in the original SG experiment it was silver atoms with one unpaired electron) hit it. The screen has a large number of degrees of freedom that can't be kept track of, which is why it produces decoherence. Just exiting the magnetic field area, whether it's through "exit ports" or just traveling through empty space, does not, as you point out, decohere anything or even constitute a measurement.

 

[charters]

The detector is not "exit ports", it's a screen of some kind (in the original SG experiment it was photographic film) that shows spots when electrons (or whatever particles you're using; in the original SG experiment it was silver atoms with one unpaired electron) hit it. The screen has a large number of degrees of freedom that can't be kept track of, which is why it produces decoherence. Just exiting the magnetic field area, whether it's through "exit ports" or just traveling through empty space, does not, as you point out, decohere anything or even constitute a measurement.

Ok, when I said exit ports I was picturing them having two little screens that capture the electron like in the image I just linked, so I think we agree and this is just semantics. I can see what you mean from other images.

 

[jlcd]

Peterdonis. Neumaier and i sent replies at exactly the same time so you might have missed my reply. It happened to me. This is a question of vital import.

Right now. In the case of the cat, we don't know what pick out the alive/dead basis as the one that is physically relevant. We don't know how the quantum interaction between the cat and its environment picks out the alive/dead basis as the one that gets decohered, so that all observers will agree that the cat is either alive (in one branch) or dead (in the other branch). But in the case of spin up and down in the Stern Gerlach experiment. Why do we know how to do it?

 

[A. Neumaier]

In the case of the cat, we don't know what pick out the alive/dead basis as the one that is physically relevant [...] But in the case of spin up and down in the Stern Gerlach experiment. Why do we know how to do it?

In both cases, Nature picks and we describe the statistics of what it picks. Nobody knows how.

 

[PeterDonis]

We don't know how the quantum interaction between the cat and its environment picks out the alive/dead basis as the one that gets decohered

There doesn't even have to be any interaction between the cat and its environment; the cat is its own "environment". The cat, all by itself, has a huge number of degrees of freedom which will decohere each other. So the cat's own internal structure is what picks out the alive/dead basis. We can't write down in detail how that works because there are way too many degrees of freedom.

In fact, even stating the cat's "preferred basis" as alive/dead is misleading. The cat's state at the classical level, i.e., after decoherence, can be specified much more precisely than just "alive" or "dead", and whether it is in fact alive or dead is assessed based on observations of multiple pieces of information at the classical level. Otherwise nobody would have been able to tell whether a cat was alive or dead until QM was discovered. In other words, "alive" and "dead" are not really orthogonal quantum states of the cat; they're names for two disjoint subspaces of the set of classical cat states, where the classical state space has many more than two degrees of freedom (but many fewer than the quantum state space).

in the case of spin up and down in the Stern Gerlach experiment. Why do we know how to do it?

We only know how to do part of it for the SG experiment, because for that part of it the number of degrees of freedom is so small. The SG magnetic field entangles spin with momentum; it's easy to write down and easy to understand. But if you ask why the detector (photographic film or whatever) decoheres in the position basis, we don't have a good answer, because, just as with the cat above, the detector has way too many degrees of freedom and we can't write down in detail how it works.

 

[jlcd]

There doesn't even have to be any interaction between the cat and its environment; the cat is its own "environment". The cat, all by itself, has a huge number of degrees of freedom which will decohere each other. So the cat's own internal structure is what picks out the alive/dead basis. We can't write down in detail how that works because there are way too many degrees of freedom.

In fact, even stating the cat's "preferred basis" as alive/dead is misleading. The cat's state at the classical level, i.e., after decoherence, can be specified much more precisely than just "alive" or "dead", and whether it is in fact alive or dead is assessed based on observations of multiple pieces of information at the classical level. Otherwise nobody would have been able to tell whether a cat was alive or dead until QM was discovered. In other words, "alive" and "dead" are not really orthogonal quantum states of the cat; they're names for two disjoint subspaces of the set of classical cat states, where the classical state space has many more than two degrees of freedom (but many fewer than the quantum state space).
We only know how to do part of it for the SG experiment, because for that part of it the number of degrees of freedom is so small. The SG magnetic field entangles spin with momentum; it's easy to write down and easy to understand. But if you ask why the detector (photographic film or whatever) decoheres in the position basis, we don't have a good answer, because, just as with the cat above, the detector has way too many degrees of freedom and we can't write down in detail how it works.

But if you express|LIVE⟩ and |DEAD⟩ as products of a state of the cat ("live" or "dead") and a state of the cat's environment ("observed to be live, etc." or "observed to be dead, etc."). So now the state of the overall system is a sum of products of states of two subsystems, i.e., an entangled state.
But those definitions depend on being able to separate the system into the two disjoint subsystems "cat" and "cat's environment". That is what we don't know how to do in the general case using just the information in the overall system's wave function.

By the arguments above. Isn't "alive" and "dead" really orthogonal quantum states of the cat. And not as you described "they're names for two disjoint subspaces of the set of classical cat states, where the classical state space has many more than two degrees of freedom (but many fewer than the quantum state space)."

Or is the truth we really don't know if alive and dead are orthogonal quantum states of the cat or classical state space due to lack of information?