## Violation of Heisenberg's Uncertainty Principle 155

mbone writes

*"A very interesting paper (PDF) has just hit the streets (or, at least,**Physics Review Letters*) about the Heisenberg uncertainty relationship as it was originally formulated about measurements. The researchers find that they can exceed the uncertainty limit in measurements (although the uncertainty limit in quantum states is still followed, so the foundations of quantum mechanics still appear to be sound.) This is really an attack on quantum entanglement (the correlations imposed between two related particles), and so may have immediate applications in cracking quantum cryptography systems. It may also be easier to read quantum communications without being detected than people originally thought."
## Insert Breaking Bad Joke Here (Score:5, Funny)

Let's just get all the Walter White jokes out of the way...

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I came here for this. Come on people, where are they?

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Following Hank's "Holy Shit" moment, Heisenberg's future is decidedly uncertain.

## I have the fix (Score:5, Funny)

I learned about it on the factual science TV show (currently honored on Google.com), Star Trek. They need a Heisenberg compensator.

## Re:I have the fix (Score:4, Funny)

## Re:I have the fix (Score:5, Informative)

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## Here we go again (Score:5, Funny)

## Walter White (Score:1, Funny)

He's not only a fantastic meth cook, but a stellar physicist as well

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If you can find him.

## Re: (Score:2)

Besides, if guns kill people, then meth will ruin your teeth... no... ehhhh... never mind...

Just dont focus on his speed ok!?!

## Magic (Score:2, Insightful)

## Re:Magic (Score:5, Interesting)

This is exactly how I feel when it comes to quantum-

anything. Especially quantum-computing, which leaves me looking at papers on it the way my cat looks at me when I ask him to do my taxes. It's one of the best examples I've encountered of anything sufficiently advanced enough being indistinguishable from magic.## Re:Magic (Score:5, Funny)

Back in the day we didn't have Quantum Computers, but we did have Quantum hard drives. You were never certain when they were going to fail

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## Re:Magic (Score:4, Funny)

I have a lemon 20MB Quantum hard drive in an ancient box. It's a lemon because it still reads and writes!

## Re:Magic (Score:4, Funny)

I have a lemon 20MB Quantum hard drive in an ancient box. It's a lemon because it still reads and writes!

You won't know whether it still works unless you open the box.

Wait...

## Not magic (Score:4, Interesting)

Most people won't consider quantum physics magic simply because it involves things that aren't experienced in everyday life. If I see a chair float in the air, I'd say it's magic because a chair suddenly floating up is contrary to my everyday experience of chairs. Familiar things behaving in unfamiliar ways, that's magic. A person being cut up and put back together is a magic trick. A medieval person might consider the Amazon Kindle magic because it resembles a book or at least a biblical tablet and yet contains the contents of thousands of books.

I'd consider quantum states magical only in so far as they produce macroscopic effects, a real-life cat that's both alive and dead. Quantum entanglement would be magical if it would allow us to develop instantaneous communication devices or, even more magical, Star Trek-style teleportation.

## Re:Not magic (Score:4, Informative)

## Re: (Score:2)

f(x) = C[n](X[n]^n) + C[n-1](X[n-1]^(n-1))+....C[1]X[1]

Now, let N run from 0 to infinity. The trick, however, if determine the correct values of the constant C and the var

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Needless to say

pfffft.

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A person being cut up and put back together is a magic trick.

You confused me until I realized how this fit with the rest of your your post (with which I agree BTW.) I thought at first you were talking about a stage-magician sawing someone in half. But you're actually talking about

surgery, aren't you? A medieval person might think a Kindle is magic, but I doubt they'd see surgery that way. Surgery has been practiced for longer than recorded history.Familiar things behaving in unfamiliar ways, that's magic.

Yes, exactly! But whether a magical event is treated by the observer as

parnormalor asexplainablewill be determi## Re: (Score:2)

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Yet you type that in a magic box (only explained by quantum physics) that puts your text on a magic screen (built witth he help of quantum mechanics).

## Re:the way my cat looks at me (Score:2)

Is your cat named Schrodinger? And are you quite certain of how he was looking at you? (Ba-dump-duush!)

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## Re:Magic (Score:5, Informative)

Bearing in mind that it's generally an error to try to summarise anything about quantum mechanics in a paragraph or two:

Actually, it's the equivalent of finding socks in the dark. If two photons are produced by an interaction of spin zero then the two photons will have spin up and spin down, although you can't know which is which without measuring one. .

I'm sorry,. but the way you write that makes it seem that they have spin up and spin down, and then you measure them to find out which is which. If that's indeed what you meant, I'm afraid that's fundamentally incorrect.

The whole point about the weirdness of quantum entanglement is that the quanta are NOT in a state where one is up and one is down prior to the measurement. Only when you make the measurement does this happen. Prior to the measurement, quantum mechanics says that they are both in a state that is BOTH up and down at the same time.

In other words, quanta are not like socks. We can be reasonably sure that socks' measurable properties are fixed before we actually look at them. Not so with quanta.

You can think of this in this way: when you make a measurement on one of the quanta, it flips a coin that tells it whether to be up or down. Its twin quantum is then bound to give the opposite result. But prior to the coin toss, neither quantum knows how it will respond to a measurement. The most that can be said is that whatever the result of measuring one, the other will give the opposite result.

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Only when you make the measurement does this happen.

How do we know that? Is there any way to figure out if a quanta has bean measured or not? Don't think so, as otherwise we could use it to transmit information via quantum teleportation, which we can't.

## Re:Magic (Score:5, Informative)

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By definition it can't be tested experimentally. You don't know until you measure, and the only way to know is therefore to measure. So it's not science as it can't be tested.

It's the old "if a tree falls in the forest and there's nobody around to hear it, does it make a sound" question in another form. I would assert they do have a spin, now disprove this assertion.

## Re: (Score:2)

By definition it can't be tested experimentally. You don't know until you measure, and the only way to know is therefore to measure. So it's not science as it can't be tested.

You have it backwards. Science is about predicting the results of measurements. If a theory correctly predicts the results of experiments then we consider the theory to be correct. The meaning of words like "truth", "knowledge" and "reality" in this context is best left to the philosophers.

## Re: (Score:2)

No. It seems that you're too sexy for science.

What Bell did was give us a way to check to see if those properties existed before the measurement. Experiments show Bell's inequality to be violated. You're still thinking in classical terms -- science moved on nearly 100 years ago.

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There's no way to measure whether a measurement has been performed.

Howeverthere's a way to determine whether the measurement result has been predetermined by the state before the measurement. The most striking one is the Mermin paradox: There you measure a certain state (called GHZ state), and get a complete contradiction to predetermined values, no probabilities involved!Here's how it goes:

You have a system composed of three subsystems, and prepared in a certain way (namely the way which gives that GHZ s

## Re: (Score:2)

There's no way to measure whether a measurement has been performed. However there's a way to determine whether the measurement result has been predetermined by the state before the measurement.

But aren't those the same thing? Say you have two physicists. One does his little quantum teleportation experiment and writes down the states of the photons. Then he hands of the photons to another physicist, but doesn't tell him that the photons come from a teleportation experiment. The second guy now does all those fancy other experiments to check if they have a predefined state. So how can the second guy come out negative, but the first guy can have all the states written down on a piece of paper?

## Re: (Score:2)

There's no way to measure whether a measurement has been performed. However there's a way to determine whether the measurement result has been predetermined by the state before the measurement.

But aren't those the same thing? Say you have two physicists. One does his little quantum teleportation experiment and writes down the states of the photons. Then he hands of the photons to another physicist, but doesn't tell him that the photons come from a teleportation experiment. The second guy now does all those fancy other experiments to check if they have a predefined state. So how can the second guy come out negative, but the first guy can have all the states written down on a piece of paper?

No, those are not the same thing. Maybe I should clarify that first sentence, because if interpreted in a too broad way, it's actually wrong; for example, if we know that we prepared the spin of a set of spin-1/2-particles in positive x direction, and someone

mighthave measured them in z direction, then of course we can find out whether that happened by measuring the spins: Just measure them in x direction again, and if the z measurement had been performed, and only then, half of them will be found to have## Re: (Score:2)

Basically in certain circumstances the probability spread of results is different depending on whether the states are predetermined but unknown (the hidden-variable interpretation) or actually in a superposition of states. Various experiments have been performed under such circumstances, and the measured results are reliably consistent with superposition rather than hidden variables.

Not quite as easy or satisfying as dropping rocks in a vacuum to measure gravity, but it can be done. It's just that rather

## Re: (Score:3)

Does this experiment have any bearing on Bell's Inequality? (And on that thread, would Bell's Theorem be satisfied by an infinite number of hid

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To prove Bell's Theorem you simply assume a single other hidden variable exists, perhaps signifying if the particle is actually spin up or spin down before you measure it. This assumption contradicts quantum mechanics and therefore cannot be true, so there is nothing else you can know about the system if quantum mechanics is the correct description.

If simply one more variable produced this result, I do not see how adding infinitely many more variables would help, or be of any practical use as a theory of n

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## Re:Magic (Score:4, Interesting)

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"Prior to the measurement, quantum mechanics says that they are both in a state that is BOTH up and down at the same time"

Prior to measurement by whom or what? - You, me, some physical process somewhere that no-one is aware of? Surely it's just a status of lack of knowledge - not an actual physical status.

Say someone elsewhere in the universe does the measurement - what state are they in for you? Can they have both a defined state and an undefined state simultaneously?

Clearly they are always in a definite s

## Re: (Score:3)

Prior to measurement by whom or what? - You, me, some physical process somewhere that no-one is aware of? Surely it's just a status of lack of knowledge - not an actual physical status.

This has been troubling philosophers for the last 100 years or so, but the majority viewpoint now is that when something "measures" a system, what's happening is that the measurer interacts with the system, and they become entangled together. The result of this entanglement turns out to be that "me-seeing-down + it-being-down" and "me-seeing-up + it-being-up" dominate the possible outcomes.

Look up 'interpretation of quantum mechanics' on wikipedia for much more detailed info

The 'lack of knowledge' theory is

## Re: (Score:2)

The whole point about the weirdness of quantum entanglement is that the quanta are NOT in a state where one is up and one is down prior to the measurement. Only when you make the measurement does this happen. Prior to the measurement, quantum mechanics says that they are both in a state that is BOTH up and down at the same time

It's even cooler than that. If you measure the spin of one of the particles in *any direction* -- say, northwest, then the other one will be found to be spinning the opposite way , southeast in this example.

(In the language of linear algebra, the space of possible spin states is a two-dimensional complex vector space. Opposite directions are considered orthogonal, and since any pair of two orthogonal unit vectors forms a basis that spans a two-dimensional space; the state can be represented in any of these

## Re: (Score:2)

Actually, it's the equivalent of finding socks in the dark.Actually, it's not at all like finding socks in the dark. What you are suggesting here is hidden variable theory. The state of the sock(or quantum particle) is determined at the beginning of the experiment and hidden until the observation is made.

This is a convenient way to think about it, but inaccurate. It's a bit much to go into here, but Bell's theorem [wikipedia.org] prohibits this possibility. Basically, if you angle the detectors you get an observed co

## Re: (Score:1)

Help me to understand what entanglement really means. As it's explained above, I don't see how it's different than this scenario:

Take two playing cards, the king of hearts and the king of spades, and place them face down on a table. Mix them up until you don't know which is which. Have a friend pick one card without looking at it and drive away with it in his car. When he's gone 100 miles have him call you up. Tell him you will now perform magic and tell him what card he has. Look at the card that's remain

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Isn't having his friend state that the card is the king of spades a measurement?

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## Re:Magic (Score:5, Insightful)

That is a good description of classical entanglement - what, in this context, would be called a hidden variable theory (the cards have a certain face value, even if you can't see them).

Let's see if I can expand this analogy. Suppose you had two decks of cards, each with only two cards - say the king of hearts and the king of spades. Off-stage, I shuffle them, so that there is either one deck of 2 hearts, and one of two spades, or one deck of both, and another of both. Say that the chances of either shuffle are the same.

Now, repeat your experiment, except you and your friend only get to pull 1 card each, each from your own deck. Classically, the chances are

- 50%, you pull from 1 spade and 1 heart

- 25%, you pull from 2 spades

- 25%, you pull from 2 hearts.

And, of course, ditto for your friend.

Now, if you pull a spade, then the classical chances are

2/3 the other card is a heart

1/3 the other card is a spade

and the classical chances for your friend are thus

2/3 he has a spade and a heart

1/3 he has 2 hearts

so his (classical) chances on his card are

2/3 he pulls a heart

1/3 he pulls a spade.

(If you pull a spade, you CANNOT have two hearts, while he can.)

So, if you pull a Spade, you can tell your friend he is likely to have a heart. Do this a lot of times, and you should be correct 2/3 of the time. The cards are indeed entangled, but classically. Experimental error (maybe you can't always see your cards well) will lower this, but (for a long enough term average) cannot raise this.

In Quantum Mechanics, however, you can get correlations that you cannot get in classical physics, i.e., greater than 2/3 in this case. That is the essence of Bell's Theorem - you have correlations that you just can't "get there from here," classically. This is a consequence of having a complex amplitude. Again, it's not just having a correlation, it's that you can get correlations you just can't classically.

I saw a lecture from Dick Feynman once where he showed that you could explain all of this by allowing for negative probabilities for intermediate results, and that this was mathematically the same as the normal (i.e., complex) formulation of QM. (Since you cannot actually measure the intermediate results, you never actually measure a negative probability.) In some ways, I find that helps to grasp the weirdness. YMMV.

## Re: (Score:2)

Help me to understand what entanglement really means. As it's explained above, I don't see how it's different than this scenario:

Take two playing cards, the king of hearts and the king of spades, and place them face down on a table. Mix them up until you don't know which is which. Have a friend pick one card without looking at it and drive away with it in his car. When he's gone 100 miles have him call you up. Tell him you will now perform magic and tell him what card he has. Look at the card that's remained with you. If it's the king of hearts, tell him he has the king of spades. If it's the king of spades, tell him he has the king of hearts.

OK, but imagine that you can also rotate your card by 90 degrees, and when you turn it over you get the jack of diamonds. If you tell your friend to rotate his card by 90 degrees before turning it over, then he must always find the jack of clubs.

Further, if you don't rotate your card (and you see the king of spades), but your friend does rotate his card by 90 degrees. What happens then? The theory predicts that there's a 50% chance he sees the king of spades, and 50% he sees the king of hearts.

Now, try to

## Re: (Score:2)

Heh, I munged up my own example. He has 50% chance of seeing jack of clubs and 50% of seeing jack of diamonds.

## Like I even tried to resist... (Score:3, Funny)

"...so the foundations of quantum mechanics still appear to be sound..."Are they sure about that? I think they fe-line to us.

## So this means (Score:3, Funny)

## Uncertain uncertainty limit (Score:1)

So basically this "uncertainty limit" is itself uncertain.

I don't know much about quantum physics but isn't that how it's supposed to work?

Is there more truth to recursive opensource software algorithims than we previously thought?

(-1 Completely Ignorant)

## Re: (Score:2)

>(-1 Completely Ignorant)

"The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt." --Bertrand Russell

--

BMO

## Re: (Score:1)

"Quoting dead people all the time is the trouble of the world" - Abraham Lincoln (The Vampire Slayer )

## Re: (Score:2)

"All quotes on the internet are true" -- Andrew Jackson

--

BMO

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Ahhh interesting, I suspected those two opposite quotes were entangled but I had to measure it by posting it to make sure, it seems to work

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I know, the results could just be dead cat bounce.

## Nobody with a clue is surprised (Score:5, Informative)

Quantum "encryption" was never that. It is only quantum "modulation" and its "security" is pure conjecture, not anything actually provable in the mathematical sense as you get with real encryption. That does not hinder a log of gullible fools to hail it as the new thing. (It does have a lot of other fundamental and unsolved problems, even if it should be secure.)

## Re: (Score:2)

The security of "real" encryption hasn't been proved mathematically.

## Re:Nobody with a clue is surprised (Score:4, Informative)

## Re: (Score:2)

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is.## Re: (Score:2)

There is quite a bit more. Some proofs need assumptions and an attacker model is always required. But your knowledge is outdated.

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The whole basis of quantum encryption was mathematical and the error checking routines have been prove mathematically. It's only used to exchange keys.

No. The whole basis was physical. The mathematics came only in via a physical theory. That that is "theory" in the sense that it may well be wrong.

As to key-exchange: If that is compromised, it basically does not matter what you do afterwards.

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Exactly.

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Then again mathematical proofs likewise only hold true if the theory there is correct. That the foundational theories are based on logical precepts rather than physical properties does not make them inherently immune to falseness, it may be that our logical framework contains fundamental inconsistencies. Take the Banach–Tarski paradox as an example - it states (roughly) that there exists a way to subdivide a sphere into pieces which can then be reassembled into

twospheres identical to the original,## Re: (Score:2)

should be: ... if the theories they're based on are correct.

## Re: (Score:1)

...not anything actually provable in the mathematical sense as you get with real encryption. That does not hinder a log of gullible fools to hail it as the new thing.

Almost every technological breakthrough that has made life better and some people quite rich was based on things not actually "provable in the mathematical sense" (which is nearly everything we think we know, including the whole of empirical science).

## Re: (Score:2)

You seem to be unaware of how encryption works and is made secure. It does not follow the rules you quote. As in "not at all".

Hint: If no clue, maybe shut up?

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## Maybe yes, maybe no (Score:2)

They thought they found a violation of Heisenberg's Uncertainty Principle but they weren't sure.

## Re:Maybe yes, maybe no (Score:4, Funny)

I'm waiting for the undead cat.

## Interesting but... (Score:1)

I'm not really sure how to feel about this.

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## Uncertainty (Score:2)

## Obligatory (Score:3)

Q: "So, how do your Heisenberg compensators work?"

The researchers: "They work just fine, thank you."

## I can only hope (Score:3)

that they checked heir cables before publishing this.

## Argh science journalism. (Score:5, Insightful)

This article is horrible.

"The Heisenberg uncertainty principle is in part an embodiment of the idea that in the quantum world, the mere act of observing an event changes it."

That's not the Heisenberg uncertainty principle. That's just the observer effect, and it's not something peculiar to quantum mechanics. You want to measure the temperature of a system, so you stick a thermometer in there. Okay, the mercury in the thermometer absorbs a bit of heat from the system, providing you with a temperature measurement at the same time it changes the temperature of the system. If you want to measure the parameters of a particle, you stick a bubble chamber in the way, and as the particle flies through the chamber it smacks into hydrogen molecules, showing you what it's doing but also taking a different path than it would have if none of those hydrogen molecules were in the way. Big fat hairy deal.

The HUP doesn't just say that you can't simultaneously measure the position and momentum of a particle, it says that a particle *does not simultaneously possess* a well-defined position and momentum. If the particle's doing something in a system and is interacting in such a way that you can define its position to arbitrary precision, then it *does not have* a well-defined momentum for you to measure, and vice versa. Position and momentum are what are called quantum conjugate variables, and the HUP says that when you have a pair of those variables, then the product of their uncertainties is greater than or equal to a constant. There is *no state* in which that particle is even *allowed* to exist in which it possesses both a well-defined position and well-defined momentum.

A signal processing analogy, for any analog people. A particle's wavefunction carries information about its position and its momentum. Where the wave exists is where the particle actually is, and the wavelength is the particle's momentum. Take a particle whose momentum you know to the utmost precision, and graph that. Range of momentums on the x axis, probability of the particle having that momentum on the y axis. You'll get a graph that looks like a Dirac function, a value of 0 everywhere except for a single spike corresponding to the particle momentum, area under the curve of 1.

Now switch domains, change from the momentum to the position domain, this is mathmatically the same thing as changing from a time domain to a frequency domain, which means you can use your old friend the Fourier Transform.

What do you get when you do an FT of a Dirac function? You get a constant value everywhere, from -infinity to +infinity. If you know exactly where that particle is, you have no idea *where* it is, and it's not because you disturbed it in measuring it, it's because *it* has no idea where it is, a well-defined position does not exist; since the uncertainty in the momentum measurement approaches zero than the uncertainty in the position measurement has to approach infinity so that the product of those uncertainties remains greater than a constant.

The "you change the system by measuring it" is an analogy, and it's one that Heisenberg himself used to explain the HUP, but *that is not what it says*. The HUP is not a statement about the process of measuring things, it is a statement about the nature of the universe, and finding a way to improve a measuring system to reduce the disturbance it creates in the system it's measuring has nothing to do with the HUP.

## Re:Argh science journalism. (Score:4, Insightful)

## Re: (Score:3, Interesting)

Only on Slashdot can you find a comment better than the article. Someone give him a modpoint.

With the proviso that the comment would be utterly incomprehensible to the target audience of the original article. "Better" is thus a relative term, and an assessment the BBC would rightfully disagree with in this case.

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Only on Slashdot? You must not look at anything produced by any news outlet. (Not even talking about Fox 'News')

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The 'you change the system by measuring it' is called the observer effect, and it has to do with wave function collapse and not with uncertainty.

Regarding the HUP: you can derive that without referring to measurement apparatuses, by just looking at the momentum and position operators. Let position x and momentum p each have standard deviation dx and dp, respectively, then by the Cauchy-Schwarz inequality

dx dp >= |cov(x,p)|.

The covariance of x and p is

## Re:Argh science journalism. (Score:5, Informative)

While the article is terrible, the actual paper is very clear about this. There are two different things that are commonly referred to as "the Heisenberg uncertainty principle". One refers to the intrinsic properties of a wavefunction and the impossibility of being in an eigenstate of two noncommuting observables. The other - which is what Heisenberg originally proposed - refers to the fact that performing a measurement alters the state of the thing being measured. Many people, including the authors of quantum mechanics textbooks, frequently talk about these as if they were equivalent, but they aren't.

Here's the first paragraph of the paper, which lays all this out very clearly:

The Heisenberg Uncertainty Principle is one of the cornerstones of quantum mechanics. In his original paper on the subject, Heisenberg wrote “At the instant of time when the position is determined, that is, at the instant when the photon is scattered by the electron, the electron undergoes a discontinuous change in momentum. This change is the greater the smaller the wavelength of the light employed, i.e., the more exact the determination of the position” [1]. Here Heisenberg was following Einstein’s example and attempting to base a new physical theory only on observable quantities, that is, on the results of measurements. The modern version of the uncertainty principle proved in our textbooks today, however, deals not with the precision of a measurement and the disturbance it introduces, but with the intrinsic uncertainty any quantum state must possess, regardless of what measurement (if any) is performed [2–4]. These two readings of the uncertainty principle are typically taught side-by-side, although only the modern one is given rigorous proof. It has been shown that the original formulation is not only less general than the modern one – it is in fact mathematically incorrect [5]. Recently, Ozawa proved a revised, universally valid, relationship between precision and disturbance [6], which was indirectly validated in [7]. Here, using tools developed for linear-optical quantum computing to implement a proposal due to Lund and Wiseman [8], we provide the first direct experimental characterization of the precision and disturbance arising from a measurement, violating Heisenberg’s original relationship.

## No (Score:1)

It's always seemed self-evident to me - if a particle is changing it's position, then it has a momentum but no fixed position

No. If it was that simple then this issue would arise already in Newtonian mechanics. A Newtonian particle with a well-defined momentum is constantly changing its position, but at any given instant in time it does have a particular position. This is just not the case in quantum mechanics; one has

onlya probability of finding a particle at a given point, and if it has a definite momentum then that probability isuniform over space, so it's position is completelyindeterminate(in a 1D example, anyway).if it has no momentum then it is not changing it's position, so it has a fixed position. In other words, the quality of each depends on a changing value for the other.

Ag

## Re: (Score:2)

That absolute finite Plank scale is so small as to be functionally irrelevant when speaking of classical Newtonian mechanics. The implications that are attempting to be explained are that within quantum mechanics, there is no distinct, solid particle that could be applied to Newtonian mechanics. The particle is just a probabilistic field, not because we simply don't know where it is, and as such are assigning a field where it probably is, but because that actually is its true nature.

Consider the original,

## Uncertainty (Score:5, Funny)

## Quantum Entangelment/Mechanics Lectures (Score:1)

http://www.youtube.com/watch?v=0Eeuqh9QfNI : Quantum Entanglement Lectures from Leonard Susskind. It really isn't that complicated, there are a lot of people here making statements that should instead be asking questions. This series along with his series on Quantum Mechanics should help answer those questions.

## Fantasy Physics still confuses (Score:2)

The Black Swann, clearly demolishes the Uncertainty Principle and too few people still don't understand Bell's Theorem and therefore are confused on "quantum entanglement" and the EPR.(Read that fellow who's a prof in quantum mechanics at MIT, Seth Lloyd.)

It's about universal balance --- too many people are still unfamiliar with GFB Riemann, most unfortunately. In the present we are saddled with Fantasy Finance and Fantasy Physics, I fear.....

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Well, the pdf link goes to arXiv, which is accessible by anyone. For quantitative results, see esp. figure 4.