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Let's take the Schrodinger's cat as an example.

  • In what sense is the cat different from the isotope ?

We say that the isotope is "microscopic/quantum" and the cat is "macroscopic/classical" object however they are both quantum objects/systems, because the cat is made up of atoms and each atom is a quantum system/object.

  • In other words, Given a (quantum) system/object, how can I tell if it is macroscopic/classical object or if it is a quantum object?

  • In other words, What does it mean that something is a macroscopic object from the quantum mechanics point of view?

If we treat the isotope and cat systems independent then there are two independent Hilbert spaces $H_{I}$ and $H_{Cat}$ that describe them. The cat in $H_{Cat}$ is described as a state $|Cat>$ . Similarly the Isoptope is described as a state $|Isotope>$ in the $H_{I}$ space.

  • In other words, How can we tell if $|Cat>$ is a "classical" state or a "quantum" state? It is the same kind of state in a Hilbert space as the $|Isotope>$ state, I don't see how they are different.

  • In other words, What is the difference between $|Cat>$ and $|Isotope>$ that makes one "quantum" and the other "classical"? They are both "quantum". So why/what is the difference?

Why is this interesting? Without defining the meaning of these words, the Schrodinger cat's paradox is meaningless.

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    $\begingroup$ There is no fundamental difference between a "classical" object and a "quantum" object. As objects get larger, and acquire more internal degrees of freedom, they slowly lose their quantum properties. $\endgroup$ Mar 22, 2017 at 12:33
  • $\begingroup$ In the first week of your QM course, presumably, you learned that QM objects exhibit interference but classical ones don't. Cats and we interfere in principle, but their interference is washed out and we appear not to. $\endgroup$ Mar 22, 2017 at 12:34
  • $\begingroup$ Hmm, "lose their quantum properties". I need to think about this. What quantum properties for example? I mean, is it possible to quantify the degree to which a quantum object lost it's quantum properties? Some kind of an "order parameter". For example if the Cat would go from a Cat into independent separated atoms , then it would go from "classical" to "quantum". Can this be considered as some kind of phase transition (as the interaction between particles becomes weeker) ? Is there a way to look at a quantum system and tell that this quantum system has lost it's quantum properties ? $\endgroup$
    – jhegedus
    Mar 22, 2017 at 14:03

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The point of the paradox is precisely that macroscopic does not seem to imply classical, contrary to our intuition that objects we see with our own eyes naively seem to always be in states of definite position and momentum instead of states actually allowed by quantum mechanics.

A "macroscopic" object is simply one that consists of many atoms and exists at our usual scales. A cat is clearly macroscopic, and in our everyday experience where it is not tormented by mad scientists it can be well-described by classical mechanics. So the crux of the paradox is simply that it creates a situation where the classical description of the cat does not apply, which you are supposed to find counterintuitive, if not impossible. If you don't, then there is no paradox.

There cannot be a proper definition of "classical object" because nothing is classical. It is generally believed that quantum mechanics is the proper description of the world and that everything is ultimately a quantum object - "classicla objects" are simply those where we can get away with describing them by classical physics and still get the relevant points right. What the "relevant points" are changes from application to application, so there's no unique idea of what objects qualify as "classical" - it depends on the specific situation and your goal in describing it physically.

In particular your idea that a classical state could somehow be a state in Hilbert space is not correct. Classical states are states in a classical theory, not in a quantum theory, there is no Hilbert space in a classical theory unless you insist on the awkward Koopman-von Neumann formulation. You need to take a proper classical limit of the quantum theory (heuristically, $\hbar\to 0$) and see which classical states the quantum states are mapped to, but this will highly depend on the system and the precise general procedure for this classical limit is, to my knowledge, unknown, though actively researched.

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    $\begingroup$ What I mean is that there is a function g: (H, a Hilbert space) -> True/False such that, for any state s in that space H ,g can tell if s is "classical" or "quantum". Why is it so interesting to find that function ? Because the whole notion of wave function collapse and Born rule depends on the existence of such a function because g defines what is "classical" and what is "quantum". Measurement takes quantum states and maps them to "classical" states. Now what is a classical state? $\endgroup$
    – jhegedus
    Mar 22, 2017 at 14:12
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    $\begingroup$ @jhegedus There is no such function. All states in your Hilbert space are quantum (or all are classical, in Koopman-von Neumann, but that's beside the point). There is no notion of "classical state" inside quantum mechanics, as I've tried to explain in the answer. $\endgroup$
    – ACuriousMind
    Mar 22, 2017 at 14:13
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    $\begingroup$ But if there is no such function then isn't the concept of wave function collapse and Born rule ill-defined? Like talking about pink unicorns? So if I send a paper about pink unicorns to Nature, why don't they publish it? $\endgroup$
    – jhegedus
    Mar 22, 2017 at 14:17
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    $\begingroup$ @jhegedus Collapse is a notion that is inherently linked with the field of quantum interpretations - it exists in some interpretations and not in others, but it is no objective property of the physical theory. I'm afraid I have no interest to talk about collapse at all, and it is not relevant to the question of what people mean by a "classical object", which is what I've answered. $\endgroup$
    – ACuriousMind
    Mar 22, 2017 at 14:18
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    $\begingroup$ Well, the whole point of classical vs quantum is the collapse, otherwise why would I care about the difference ? Without collapse there is no such thing as "classical" whatever that is. The definition of "classical" is the definition of collapse. $\endgroup$
    – jhegedus
    Mar 22, 2017 at 21:28

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