I’m working on a paper discussing whether quantum mechanics can be said to be deterministic, and while I believe I’ve already established that it can’t, I’d like to spice it up a bit with a rather extreme example, like the butterfly effect apparently found in cancer, nuclear bombs and the like (eg, very small differences in particle position and/or movement can have very real and very measurable results, like whether or not a potentially historically important person dies).
I read the argument somewhere on this board (looked, but can’t find), but I don’t really think citing a message board would be appropriate, and as physics isn’t really my field, I wouldn’t know where else to look. So any suggestions would be greatly appreciated, and I really hope this isn’t considered cheating 
Did you check Wiki?
Forgive me if I’m wrong, but is the “Butterfly Effect” related to chaos theory, not quantum theory. Heisenberg tells us with certainty that Quantum effects can’t be deterministic right ?
The Butterfly Effect is related to Chaos Theory and is commonly misunderstood.
The media gets it wrong (often) by saying something like, “A butterfly flapping its wings in Brazil causes a tornado in Iowa.” That is not the case. All it means is when you try to predict a complex system very, very small variables can grow into large changes in the prediction in the future. So, if you account for everything in your climate model but forget that lone butterfly in time your model will totally miss something like that tornado.
I think you are referring to chaos theory and not quantum theory. They aren’t the same at all.
Assuming you still want an example or the Butterfly Effect as it relates to chaos theory, then reproduction is a good example. Human reproduction consists of lots of things but one of the key things is that millions of sperm race towards the egg and the specific sperm that wins determines the child that will be born. Almost anything can influence the “winner” both before or during the act. Slight changes in position or deciding when to take the trash out would do it. Once that specific child is born, he or she has a lifelong influence on countless people ranging from large to microscopic effects and the cycle continues. Another example is traffic. Every driver changes the traffic pattern in large or small ways and this will determine what accidents happen. Leaving you house 5 minutes late could mean that a child that was destined to be a world leader is caught in an accident on the way to school and killed. That small event will influence world history from then on.
I have never really understood why people say this. My understanding of his theories is that the very act of measuring something changes what you are measuring, thereby limiting what you can know about a system. Thus, even if it is deterministic, it is effectively non-deterministic as you can never know enough to accurately predict how the system will evolve. However, I am unaware of any theory on his part that says that even if you did know, in complete exactitude, the initial state, you would still not be able to accurately predict what would happen in the system. Am I missing something?
Well, I do not think the Uncertainty Principle says you could not make perfect predictions IF you knew all there was to know about the system. However, the Uncertainty Principle says you cannot know in complete exactitude the state of a system. Period. So speculating on the possibility of perfect predictions is a moot point.
The uncertainty principle arises as a consequence of the basic assumptions in the mathematical model of QM:
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Every physical system has a state function φ(x,t) associated with it (x and t are generalized space/time coordinates).
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Every physical observable of the system (position, momentum, angular momentum, etc.) is associated with a mathematical operator. Here’s a list of QM operators; the momentum of a system in the x-direction, for example, is found using the associated operator -(ih)d/dx (“d/dx” here is the partial derivative in the x direction).
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The only possible values an observable can have are the eigenvalues λ of the equation Pφ = λφ, where P is the operator associated with the observable.
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The expected value (i.e. measurement) of any observable is given by the integral over space of φPφ (φ is the complex conjugate of φ ), often written <P>. This is something akin to the usual statistical average of possible measurements (i.e. eigenvalues), but it takes into account the possibly complex nature of φ.
The uncertainty principle arises when you take the expected value <P> of some operator P, and think about the operator P-<P>. If you calculate <P-<P>>, you get a value somehat akin to the variance/standard deviation of the expected value <P>. Similarly, the variance of a different observable Q’s measurement <Q> is given by <Q-<Q>>.
Now multiply the two variances together: <P-<P>><Q-<Q>>. Remember that <> is shorthand here for an integral, so one can apply the Schwartz inequality, and say the product of the variance must be at least as great as <(P-<P>)(Q-<Q>)> = <PQ-<P>Q-P<Q>+<P><Q>>.
If you do enough math (a pretty good derivation can be found here), you can reduce the entire expression to some multiple of <PQ-QP>, the expected value of a new operator PQ-QP. Despite it’s apparent form, this operator may not be zero, because some operators do not commute (a simple example: if P is the operator “add one” and Q is the operator “multiply by 2”, the answer after applying these operators to some number x will depend upon the order in which these operations are performed, i.e. they are not commutative). The operators for momentum and position do not commute, so PQ-QP for these is non-zero.
This idea of operator commutivity at the heart of the Heisenberg Uncertainty Principle’s proof is why, I think, many folks interpret it as a measurement problem: If the order of measurement is important, then the taking of one measurement must somehow affect the other. But the proof itself says nothing about how measurements are made; it is simply a consequence of the mathematical model used to describe QM.
I’m aware that the butterfly effect is related to chaos theory, not quantum mechanics. However, I wanted an example on quantum systems behaving chaotically, as I understand that due to the microscopic nature of particle physics, the uncertainty normally “cancels out” at the macroscopic level, and for most cases you get a pretty good aproximation. However, if I’ve understood things correctly, this is not the case with cancer and nuclear bombs, as small variations on the microscopic level can have profound effects on the macroscopic level.
The paper is on quantum physics, so I’ll need a cite for something that says that even though we can calculate particle behaviour to a high degree of precision, there are systems (such as nuclear bombs) that behave so chaotically, even at the sub-atomic level, that these uncertainties amount to serious variations in output, such as whether or not a historically important person dies (or better yet, an added blast radius of, say, a kilometre, due to the properties of a few particles)
The position of a free quantum particle is represented by a wave packet which sort of looks like a bell curve. The width of the packet represents the uncertainty in the particle’s position.
In order to know the position exactly the width of this packet must be reduced to zero. (A Dirac Delta function) In order to accomplish this an infinite range of frequencies must be superposed.
However, the momentum of the particle is proportional to the frequency and since we have an infinite range of these we can say nothing whatsoever about the momentum. (Other than it’s completely unknown)
So the more we know about the position the less we know about the momentum and vice versa. This uncertainty in conjugate variables is an intrinsic part of reality and has nothing to do with measurement.
ETA sorry Yelim I didn’t see your intervening posts.
My uneducated two cents:
If you want to talk about nondeterminism, I think what you really need to be talking about is Bell’s theorem rather than just Heisenberg’s Uncertainty Principle. Without Bell’s theorem, it’s always possible to say things like “The values are well-defined, we just don’t know what they are.” But with Bell’s theorem, you are forced to abandon any explanation dependent upon local hidden variables.
Indistinguishable: I know, and I’m scared stiff, just like Al. However, I want something that says not only “QM is indeterministic, but we get good results”, but “QM is indeterministic, and sometimes, we haven’t even a clue as to what is going to happen”. For artistic effect only, but I think it suits the flow of the paper.
(I’m a determinist myself and really hate these arguments, by the way.)
Ah, that’s interesting. Not to force you into something you hate, but given that you are a determinist who considers as established that quantum mechanics cannot be said to be deterministic, should I take it, then, that you reject quantum mechanics?
Au contraire! My paper is concerned with an instrumental vs a realistic interpretation of QM, and all I’ve read suggest that those who consider QM indeterministic do so solely as instrumentalists, suggesting that as a theory, QM is not able to predict the future, and also (according to some) it seems principally impossible to do so, and so it is meaningless to talk of determinism, at least as a physicist. However, I believe in philosophical arguments for determinism, altough I accept (as a layman to physics, of course) that it is not QM’s task to even consider determinism.
Anyway, the difference between saying it is meaningless to talk about determinism and saying that the world is not determined should be noted. And from a philosophical point of view, it doesn’t seem meaningless to talk about it at all, and even Hume, the famous doubter of causal realations, was at heart a determinist. (Quick cite found in Schopenhauer’s essay on the freedom of will, I could look it up if anyone’s interested)