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Schrödinger cat

derekc153
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Posted 08/16/08 - 11:37 PM:	#11
ragnus wrote: There's something I don't understand about the Schrödinger cat experiment. The starting point is a live cat. When, where and how does the cat go into a superposition of states? In what ways would the superposed cat be different from a live cat? I think it really depends on how you interpret the wave-function. If you see it as a real physical entity, like (I think) in the many-worlds interpretation, then maybe you'll start talking about multiple universes and crazy stuff like that. However, if you see the wave-function as a predictive tool, then it's easier to answer your question. After a while, there is a chance the cat is in state A (alive) and a chance it is in state B (dead). The modulus-squared of the wave-function is the probability density, which in this case will simply be two delta functions (basically just an infinitely tall, infinitesimally wide spike with a defined area underneath the curve), one at point A and one at point B. When you integrate the probability density over some part of the space you're working in (in this case, the alive/dead axis), you get the probability of measuring the system to be in the state represented by that part of the space (say, measuring a particle to be in a certain range of positions or have a momentum within a certain range). Thus, the wave-function of the cat can be said to be a superposition of the two delta functions, representing state A and state B, indicating that there is a chance of measuring the system to be in either state. The key here is that probability is an epistemic notion, meaning that if we look inside the box we change the probability from a 50/50 chance that the cat is in either state to a 100% chance that the cat is in whatever state we observe. In other words, the cat goes into a superposition of states as soon as we stop directly observing what state it's in (and thus stop knowing what state it's in). From our knowledge of the half-life of the radioactive material, we know there is a 50/50 chance that the cat is alive or dead at time x, and the superposition of the states reflects this--both delta-functions have an area of 1/2. The "superimposed cat" is not really comparable to a live cat because, unlike the live cat, it's not a real thing. That is, it isn't a real state that the cat can be in. It's simply a predictive tool that reflects the fact that we don't know what state the cat is in, but we know there is a chance it is in either state A or state B. When we observe the cat again, we "collapse its wave-function" by discovering exactly which state it's in (and thus discarding the possibility of other states). I hope I wasn't too unclear and that I didn't say anything incorrect, but both are possible as it is rather late. Edit: It just occurred to me that I might be presuming hidden variables, which demonstrably do not exist. I'll take a look at this tomorrow and possibly reevaluate my position. Edited by derekc153 on 08/16/08 - 11:43 PM
"There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy." -Shakespeare's Hamlet "If God does not exist, then all things are permitted." -Fyodor Dostoyevsky

abba
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Posted 08/17/08 - 07:50 AM:	#12
Excellent.

Kwalish Kid
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Posted 08/17/08 - 09:36 AM:	#13
swstephe wrote: A photon acts like a particle if you measure it like a particle and it is like a wave when you measure it like a wave. It is more accurate to say that it acts like neither a classical particle nor a classical wave. A photon always delivers a quantum of energy whenever it delivers energy and its position and momentum are determined according to wave propagation. This doesn't change no matter how one makes a measurement on a photon. So how do you treat it if you want the best of both worlds? Half particle and half wave. This is demonstrably incorrect. To accurately present a photon, one has to use quantum theory, not treat the photon at some times as a classical particle and at other times as a classical wave.
"Scientific truth is always paradox, if judged by everyday experience, which catches only the delusive nature of things." - KM, V, P and P "A fishnet is made up of a lot more holes than strings, but you can't therefore argue that the net doesn't exist. Just ask the fish." - Jeffrey Kluger "…Love of God and compassion and empathy leads you to a very glorious place, and science leads you to killing people." -Ben Stein [This is included for the irony.]

Pete
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Posted 08/17/08 - 12:19 PM:	#14
derekc153 wrote: Thus, the wave-function of the cat can be said to be a superposition of the two delta functions, representing state A and state B, indicating that there is a chance of measuring the system to be in either state. The key here is that probability is an epistemic notion, meaning that if we look inside the box we change the probability from a 50/50 chance that the cat is in either state to a 100% chance that the cat is in whatever state we observe. I don't know anything about physics, but the impression I'm getting from your comment and the Wiki article is that Schrodinger came up with the cat example to show that physicists were wrong to reify these wave-functions. What I mean is, he thought that, even if it is not intuitively obvious that a radioactive element must be in one state or another, it is intuitively obvious that a cat must be. By linking the fate of a cat with the fate of a radioactive element, we can give a cat a wave-function. Since we're inclined to think that there still must be a fact of the matter, at any moment, about whether the cat is alive or dead, we should similarly conclude that there is a fact of the matter about the state of radioactive elements--i.e. that these wave-functions represent only our uncertainty, not reality. Is this a fair representation?

derekc153
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Posted 08/17/08 - 10:33 PM:	#15
Pete wrote: Since we're inclined to think that there still must be a fact of the matter, at any moment, about whether the cat is alive or dead, we should similarly conclude that there is a fact of the matter about the state of radioactive elements--i.e. that these wave-functions represent only our uncertainty, not reality. That is exactly what I was getting at, but actually I don't think it is correct. Bell's Theorem shows that the idea that there is a local reality (some definite state of the system, such as an exact momentum and position of a particle, which exists independently of any measurement of it) which we simply cannot observe due to practical problems of measurement is in fact a testable hypothesis. Moreover, that hypothesis has been refuted. I don't know all the details, but supposedly the probabilities work out differently if there are local "hidden variables" (a definite state of the system before it is measured). There is a wikipedia article on the subject, though I got lost in some of the math last time I looked at it.
"There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy." -Shakespeare's Hamlet "If God does not exist, then all things are permitted." -Fyodor Dostoyevsky

Benkei
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Posted 08/18/08 - 07:39 AM:	#16
Edit: It just occurred to me that I might be presuming hidden variables, which demonstrably do not exist. I'll take a look at this tomorrow and possibly reevaluate my position. What "demonstrated" that hidden variables do not exist? Furthermore, Bell's Theorem shows that QM is inherently non-local and that any theory, with or without hidden variables, has to take this into account.
- How are you doing? - I'm doing good. - No, Superman is doing Good, you're doing well. You need to brush up on your grammar.

Kwalish Kid
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Posted 08/18/08 - 07:52 AM:	#17
Bell's theorem shows that two systems that are quantum entangled are not separable. That is, one cannot simply treat them as two separate systems, each with their own independent description. It also shows that if you want to account for this non-separability with hidden variables, then one must use a non-local theory. Non-separability is compatible with locality. In any given description (of the kind of thing we normally think of in this context), there is one event in the system that precedes another. This isn't the same event in each description, but there is nothing in locality that requires that each description have the same event ordering.
"Scientific truth is always paradox, if judged by everyday experience, which catches only the delusive nature of things." - KM, V, P and P "A fishnet is made up of a lot more holes than strings, but you can't therefore argue that the net doesn't exist. Just ask the fish." - Jeffrey Kluger "…Love of God and compassion and empathy leads you to a very glorious place, and science leads you to killing people." -Ben Stein [This is included for the irony.]

Pete
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Posted 08/18/08 - 11:55 AM:	#18
derekc153 wrote: Moreover, that hypothesis has been refuted. Wow, so much for common sense I guess.

Makarismos
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Posted 08/18/08 - 01:48 PM:	#19
Pete wrote: Wow, so much for common sense I guess. What is common about sense?

Benkei
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Posted 08/18/08 - 10:00 PM:	#20
Kwalish Kid wrote: Bell's theorem shows that two systems that are quantum entangled are not separable. That is, one cannot simply treat them as two separate systems, each with their own independent description. It also shows that if you want to account for this non-separability with hidden variables, then one must use a non-local theory. Non-separability is compatible with locality. In any given description (of the kind of thing we normally think of in this context), there is one event in the system that precedes another. This isn't the same event in each description, but there is nothing in locality that requires that each description have the same event ordering. Kwalish, I'm in the end only a lawyer so you could be right but everything I've read about it, states that Bell proves that QM is inherently non-local. Perhaps a part of his Theorem also proves non-separability while being silent on non-locality? I'll give you my sources: http://plato.stanford.edu/entries/bell-theorem/ http://plato.stanford.edu/entries/qm-bohm/ (paragraph 2) and just found this one http://plato.stanford.edu/entries/physics-holism/ this seems to suggest that physical relational holism could lead to explainable violation of Bell's inequalities, which allows for possible QM interpretations that are local. Interesting. I had the impression most physicists thought Bell's Theorem was conclusive. The three articles have been written by different people, so it does not reflect a single person's opinion.
- How are you doing? - I'm doing good. - No, Superman is doing Good, you're doing well. You need to brush up on your grammar.

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