View Full Version : The uncertainty principal: measuring things with uncertain instruments.

WaxRubiks

2019-Feb-13, 02:53 AM

It occurred to me the other day, that although there is uncertainty in the thing you are measuring, there will also be uncertainty in the thing you are measuring with.

I'm sure this is well understood, and it must make the uncertainty principal beautifully complex.

This is how it goes I guess...?

Ken G

2019-Feb-13, 01:46 PM

The key point is, you can make the instrument better and reduce the uncertainty in the thing you are measuring with. But you can't reduce the minimum uncertainty in what you are measuring, that doesn't improve with a better instrument. So to study the latter, you just make sure the instrument has the smaller uncertainty of the two.

I'm not sure if this is really related to the uncertainty principle? That is only significant for sub-atomic particles, not macroscopic properties.

For most things the uncertainty principle is negligible compared to the measurement uncertainty.

For example, the speed of a vehicle is just that. Intrinsically it has hardly any uncertainty. When you try and measure the speed, there is an uncertainty in that speed measurement, and that uncertainty certainly derives in part from the devices used in the measurement.

George

2019-Feb-14, 02:56 PM

Attempting another georgeeze analogy, in the subatomic world it's as if there is a fork in the road at every infinitesimal point for a waveicle. If you take the left path, the farther you get from the right (and the less you know about it), and vice versa.

I happened to be looking at Bell's Theorem (why is it called a theorem?) and HUP seems to be a big factor to it, right? I don't understand the hidden variables claim (or refutation of it) but the statistical probability part in comparing QM with normal statistical probability is not that hard to understand, though amazing in its own right. Given a testing of three angles for, say, the polarity of a pair of entangled photons the probability should be better than 33% (perhaps closer to 50%) that any choice for the angles of both would match. Yet QM demands only 25% (using 120o angles...cos(120o)^2). Is this correct?

Grey

2019-Feb-14, 09:16 PM

... Bell's Theorem (why is it called a theorem?) ...The word "theorem" is usually used for proven statements of mathematics (given some particular choice of axioms). This is distinct from the word "theory" which is generally a unifying understanding of some scientific principle. Bell's Theorem is the former: a statement about probabilities of events that can be proven to satisfy a certain inequality, given some basic assumptions. It has some interesting implications for quantum theory, but that doesn't change the fact that it's essentially a mathematical statement.

George

2019-Feb-14, 09:36 PM

The word "theorem" is usually used for proven statements of mathematics (given some particular choice of axioms). This is distinct from the word "theory" which is generally a unifying understanding of some scientific principle. Bell's Theorem is the former: a statement about probabilities of events that can be proven to satisfy a certain inequality, given some basic assumptions. It has some interesting implications for quantum theory, but that doesn't change the fact that it's essentially a mathematical statement.So it's a mathematical construct (from QM math), and experimental results are secondary to the theorem. Most if not all scientific hypotheses use the language of math to define them, so it's an interesting distinction, if not a nuance, worth noting. Thanks.

Ken G

2019-Feb-14, 10:02 PM

One way to state Bell's theorem is that the predictions made by quantum mechanics about entangled particles have to violate the inequalities you can derive from making the basic assumptions called "local realism." So Bell's theorem is not purely a mathematical statement (that part would be called "Bell's inequality", but it was around before Bell), it brings in physics via the recognition that the predictions of quantum mechanics violate that inequality. Experimental data comes in when it is shown that quantum mechanics was right, but that's not part of Bell's theorem-- it's part of why Bell's theorem matters. The theorem is that if quantum mechanics is right, local realism isn't (as long as you buff up local realism with a basic physics tool called "counterfactual definiteness", which basically means that physics will take the perspective that experimenters get to decide what questions they want to answer and are not ruled by the same deterministic outcomes they are studying). The experimental verification is that quantum mechanics is indeed right.

George

2019-Feb-14, 10:16 PM

One way to state Bell's theorem is that the predictions made by quantum mechanics about entangled particles have to violate the inequalities you can derive from making the basic assumptions called "local realism." So Bell's theorem is not purely a mathematical statement (that part would be called "Bell's inequality", but it was around before Bell), it brings in physics via the recognition that the predictions of quantum mechanics violate that inequality. I guess it's the inequality term that is tricky for me, though this probably should be another thread if you want the incredible challenge of trying to convert this to georgeeze. On my level, albeit cursory, the difference in predicted probabilities (Classic vs. QM) seems more like an inequality than a violation of it.

Experimental data comes in when it is shown that quantum mechanics was right, but that's not part of Bell's theorem-- it's part of why Bell's theorem matters. Of course; if i'm raising it up then it must already be important. :)

Ken G

2019-Feb-15, 02:22 AM

I guess it's the inequality term that is tricky for me, though this probably should be another thread if you want the incredible challenge of trying to convert this to georgeeze. On my level, albeit cursory, the difference in predicted probabilities (Classic vs. QM) seems more like an inequality than a violation of it.

It probably should be a different thread, but the basic idea is that if any system can broken down into parts, and the parts "carry with them" the information that determines the likelihood of various outcomes of measurements on them, then there is a limit on the degree of correlation you can find in those outcomes when you compare how the measurements came out for the different parts. That limit is "Bell's inequality." Quantum mechanics does not treat systems as if they can be broken down into parts, it treats them holistically, whereby even if the parts are separated by light years, the quantum mechanical formalism says they are still part of the same whole system, and that whole system has attributes that controls correlations between the parts in ways that violate Bell's inequality. Experiments show quantum mechanics is right, and physically separating the parts of a system (even by light years) does not mean they carry with them all the information that would go into any correlations they show with the other parts (unlike the way a right glove and a left glove work, they carry with them their identity as opposite gloves). In any classical description, that's what would need to hold, the parts carry the information and the whole is naught but the sum of those parts. The mathematics of quantum mechanics says that a system is more than the sum of its parts, and remarkably, it's true.

(And actually, it would be more correct to say quantum mechanics says a system is less than the sum of its parts, in the sense that high degrees of correlation between seemingly independent measurements actually means the system contains less independent information. Indeed, any pure state in quantum mechanics has zero entropy, even though observations on that system are often unpredictable.)

Shaula

2019-Feb-15, 09:11 AM

Attempting another georgeeze analogy, in the subatomic world it's as if there is a fork in the road at every infinitesimal point for a waveicle. If you take the left path, the farther you get from the right (and the less you know about it), and vice versa.

Or looking at the original question, about measurement versus quantum uncertainty...

There is a fundamental limit to how well you can understand a scientific model about quantum physics using analogies involving everyday objects. The more familiar the situations used in the analogy the less matched to the model the resulting understanding usually is. That is how quantum uncertainty behaves. This is independent of how good the person making up the analogy is at explaining things using analogies. Which is instrumental error in this case. So while both result in the same sorts of confusion they are independent and behave quite differently - plus the person making up the analogies (instrument measuring the system) can usually be improved whereas the gap between reasoning using analogies and using the model (quantum uncertainty) represents a fundamental limit to understanding.

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