Uncertainty Principle.... Intent Behind It?

[Madi Araly]

I've been pre-occupied with Heisenberg's uncertainty principle for around four years now, and I've come to fabricate a lot of questions.

The most pressing one, however, is as follows:
To me, the uncertainty principle seems to reference our (relatively) poorly controlled methods to measure a particle's momentum and position rather than being some special quantum phenomenon. Is this how it was intended?

If I measure a coffee mug's position using a crowbar, I change the coffee mug's momentum by measuring it. I do not, however, change its momentum simply by having knowledge of the coffee mug's position. This is how I think of the uncertainty principle, but was it meant this way? If it was, then doesn't that screw up multiple other concepts such as entanglement and electron configuration around nuclei?

Or did Heisenberg believe in some phenomenon that changed one of the particle's traits merely because we observed it?

 

[phinds]

To me, the uncertainty principle seems to reference our (relatively) poorly controlled methods to measure a particle's momentum and position rather than being some special quantum phenomenon. Is this how it was intended?

Absolutely not. The POINT of the HUP is that it is not at all a measurement problem but rather a fundamental fact of nature.

 

[fresh_42]

See what a search here on it results in:

https://www.physicsforums.com/search/3199649/?q=Heisenberg+uncertainty+principle&o=relevance

 

[Nugatory]

To me, the uncertainty principle seems to reference our (relatively) poorly controlled methods to measure a particle's momentum and position rather than being some special quantum phenomenon.

You have been misled by superficial treatments of the subject. The uncertainty principle is not a consequence of the way that we measure position and momentum, nor does it only apply to those two properties; that's a misconception dating back to the early days of quantum mechanics, before we had developed a complete understanding of the theory.

Instead, it is an inescapable mathematical consequence of the laws of QM: If the operators corresponding to two observables have a particular mathematical relationship (they "do not commute" in the lingo) then any quantum state in which one of them has a definite value is necessarily also a state in which the other one does not. Position and momentum are everbody's favorite example of such a pair of incompatible operators, but there are many more.

There are many more threads here, both on what the uncertainty principle is and the history behind it.

 

[Feeble Wonk]

Position and momentum are everbody's favorite example of such a pair of incompatible operators, but there are many more.

Just for curiosity sake... could you list a few other such pairs of operators?

 

[Nugatory]

Just for curiosity sake... could you list a few other such pairs of operators?

The x, y, and z components of angular momentum. I cannot prepare a quantum system in a state such that more than one of the three is definitely known. (Although I can prepare a system in which the magnitude of the angular momentum vector and anyone of its three components is are both definite).

 

[bhobba]

To me, the uncertainty principle seems to reference our (relatively) poorly controlled methods to measure a particle's momentum and position rather than being some special quantum phenomenon. Is this how it was intended?

To understand the uncertainty principle you really need to see a proper statement of it.

The correct statement is the following. Suppose you have a large number of similarly prepared systems ie all are in the same quantum state. Divide them into two equal lots. In the first lot measure position to a high degree of accuracy. QM places no limit on that accuracy - its a misunderstanding of the uncertainty principle thinking it does. The result you get will have a statistical spread. In the second lot measure momentum to a high degree of accuracy - again QM places no limit on that. It will also have a statistical spread. The variances of those spreads will be as per the Heisenberg Uncertainty principle.

Note:
1. You can measure momentum and position to any accuracy you like. QM places no limit on that.
2. Its simply a statistical statement about a fundamental property of QM - if you observe a system in a quantum state the result can only be predicted statistically.

If you want to read up on it a very careful and correct analysis can be found in Ballentine:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

Unfortunately in physics generally, and its particularly true in QM, correct understanding is not always given in beginning texts or popularizations. Its a big problem - Feynman for example, being the great educator he was, worried about it a lot but saw no way around it. Its simply not possible to give the full detail to start with - you must build up to it which means you need to unlearn and relearn things as you progress.

That's why you will usually only find correct treatments of the uncertainty principle in advanced books like Ballentine.

Thanks
Bill

 

[Madi Araly]

You have been misled by superficial treatments of the subject. The uncertainty principle is not a consequence of the way that we measure position and momentum, nor does it only apply to those two properties; that's a misconception dating back to the early days of quantum mechanics, before we had developed a complete understanding of the theory.

Instead, it is an inescapable mathematical consequence of the laws of QM: If the operators corresponding to two observables have a particular mathematical relationship (they "do not commute" in the lingo) then any quantum state in which one of them has a definite value is necessarily also a state in which the other one does not. Position and momentum are everbody's favorite example of such a pair of incompatible operators, but there are many more.

There are many more threads here, both on what the uncertainty principle is and the history behind it.

Well, this is frustrating, but I'd rather be displeased and know the truth than live with an incorrect understanding. Thank you. :)

Why is it that humans are so certain of this, though? Was it purely because of the double slit experiment and the calculations following it? The double slit experiment is less controlled than I would like to take that as our fundamental proof... Given what you said, however, electron configurations do make more sense to me now that I'm not viewing this as a classical situation. Don't get me wrong, I take no pleasure in trying to argue with what physicists have discovered thus far, but my view is that it should try to be explained with classical physics, and if it can, it isn't unique nor is it quantum mechanics at work.

 

[Madi Araly]

The x, y, and z components of angular momentum. I cannot prepare a quantum system in a state such that more than one of the three is definitely known. (Although I can prepare a system in which the magnitude of the angular momentum vector and anyone of its three components is are both definite).

So theoretically, if I had all three axis components of a particle's angular momentum, I would know/be able to calculate its position at any given time and "break", for lack of a better term, the uncertainty principle. Correct?

 

[phinds]

Why is it that humans are so certain of this, though? Was it purely because of the double slit experiment and the calculations following it?

The single slit experiment is more persuasive regarding the HUP. The double slit experiment is more about showing how light can have the characteristics of both particles and waves.

 

[Madi Araly]

The single slit experiment is more persuasive regarding the HUP. The double slit experiment is more about showing how light can have the characteristics of both particles and waves.

But was the double slit experiment really the only reason this was theorized? Or was there something else that made physicists think particles could also behave as waves?

 

[DrChinese]

So theoretically, if I had all three axis components of a particle's angular momentum, I would know/be able to calculate its position at any given time and "break", for lack of a better term, the uncertainty principle. Correct?

You can't know the non-commuting components simultaneously. Position and momentum are non-commuting as well, so you can't know those either. You can't cheat the uncertainty principle, many have tried! EPR being a great example of that.

 

[phinds]

But was the double slit experiment really the only reason this was theorized? Or was there something else that made physicists think particles could also behave as waves?

When you start talking about two entirely different phenomenon in the same thread, it gets confusing. If you want to know about the double slit experiment and the fact that light can act like both a wave and a particle, please read some of the thousands of threads on that subject and if you still have questions, start a new thread.

 

[Madi Araly]

You can't know the non-commuting components simultaneously. Position and momentum are non-commuting as well, so you can't know those either. You can't cheat the uncertainty principle, many have tried! EPR being a great example of that.

No haha that's not what I'm saying. I know it can't be done, but what I was trying to clarify was that if I had all 3 axis components at the same time, I would in turn be able to calculate its position. Now that I retype it, it becomes clear to me that this is the case.

 

[Madi Araly]

When you start talking about two entirely different phenomenon in the same thread, it gets confusing. If you want to know about the double slit experiment and the fact that light can act like both a wave and a particle, please read some of the thousands of threads on that subject and if you still have questions, start a new thread.

That's not necessarily what I'm referring to. I'm just wondering what the first thing was that made physicists think "huh, these particles are behaving as if they're in two positions at once". I need to verify that the double slit experiment, or something of a very similar nature, isn't the reason Heisenberg even came up with the uncertainty principle.

 

[DrChinese]

If you knew the momentum, you could not calculate the ending position unless you knew the starting position. But any previous position information would be invalidated by learning the momentum. So you still could not calculate the ending position.

 

[Madi Araly]

If you knew the momentum, you could not calculate the ending position unless you knew the starting position. But any previous position information would be invalidated by learning the momentum. So you still could not calculate the ending position.

So then why are we incapable of measuring all three parts of the momentum? It's not possible to measure it multiple times and account for the variations in those measurements to gather an x, y, and z vector?

 

[DrChinese]

So then why are we incapable of measuring all three parts of the momentum? It's not possible to measure it multiple times and account for the variations in those measurements to gather an x, y, and z vector?

If you measure any fundamental observable very precisely - say position - any non-commuting partner observable moves into a superposition of states (momentum, for example). When you later measure that, you will find that the new outcome (for momentum) is random and uncorrelated to any prior measurement of that observable.

Keep in mind that for all practical purposes, particles do not have simultaneous (well-defined) values for both position and momentum. Experiments on entangled particle pairs demonstrate this very convincingly.

 

[Madi Araly]

If you measure any fundamental observable very precisely - say position - any non-commuting partner observable moves into a superposition of states (momentum, for example). When you later measure that, you will find that the new outcome (for momentum) is random and uncorrelated to any prior measurement of that observable.

Keep in mind that for all practical purposes, particles do not have simultaneous (well-defined) values for both position and momentum. Experiments on entangled particle pairs demonstrate this very convincingly.

Forgot about relativity... Looks like I'll be doing a lot more reading and contemplation. Thank you!

 

[phinds]

Forgot about relativity...

Relativity has nothing to do with what we have been discussing, so if you think it does then that's some more reading you need to take on.

 

[Madi Araly]

Relativity has nothing to do with what we have been discussing, so if you think it does then that's some more reading you need to take on.

I'll continue reading. I read the last response too quickly, saw commuting and somehow jumped to relativity.

 

[Nugatory]

Why is it that humans are so certain of this, though? Was it purely because of the double slit experiment and the calculations following it? The double slit experiment is less controlled than I would like to take that as our fundamental proof...

You've been misled again, probably because you are relying on popularizations and other dumbed-down treatments of QM. The double slit experiment gets a lot of press because it is a simple setup that (in its one particle at a time two slit version) starkly demonstrates behavior that cannot be explained classically. This makes it a good starting point for authors trying to get the flavor of QM across without asking their readers to do any intellectual heavy lifting; but there's a lot more to understanding QM than that.

Why is it that humans are so certain of this, though?

Why are we so confident in the uncertainty principle? The uncertainty principle is derived from the principles of QM by using logic as unassailable as (for example) that which we use to conclude from the laws of arithmetic that no odd number is the sum of two even numbers. So if we are confident in the principles of quantum mechanics, we're confident in the uncertainty principle (correctly stated, that is - the stuff you've been reading about measuring position changing momentum and vice versa is NOT the uncertainty principle).

Of course this just invites the next question: Why are we so confident in the principles of quantum mechanics? The answer is that QM has more observational confirmation than any other science in human history. Quantum mechanics correctly predicts the behavior of valence electrons and chemical bonds, and therefore explains all of chemistry, and all of biochemistry. It correctly predicts the interactions between atoms in solids and liquids, and therefore explains all of materials science. Every electronic device in the world depends on QM to function properly - the screen you're reading this on right now is another successful experimental test of the principles of QM. So there are a lot of reasons to believe that QM is correct - the double-slit experiment only comes in because it's a good starting point for explaining how QM is different from classical physics.

my view is that it should try to be explained with classical physics, and if it can, it isn't unique nor is it quantum mechanics at work.

Classical physics emerges from quantum mechanics, in the sense that every problem that can be correctly solved with classical physics can also be solved using the methods of quantum mechanics (another of the reasons that we trust QM is that it agrees with classical physics in any situation in which classical physics produces the right answer). We generally try using classical methods first because the math is less demanding (I know someone whose PhD thesis was the quantum mechanical treatment of a bouncing rubber ball - but this is a first-year undergraduate problem in classical mechanics). So you're right about trying to explain things with classical physics first, but mistaken to think that succeeding means that quantum mechanics is not still at work.

Furthermore, there is a huge area where classical physics just doesn't work. For example, classical physics predicts that all atoms are unstable (with a half-life measured in fractions of a second) and won't chemically bond to form molecules. Stuff like this is a very strong argument that QM is a more complete theory - it does everything that classical physics does, and much more.

 

[Nugatory]

So theoretically, if I had all three axis components of a particle's angular momentum, I would know/be able to calculate its position at any given time and "break", for lack of a better term, the uncertainty principle. Correct?

Saying that you have all three components of a particle's angular momentum is equivalent to saying that you have a wave function that is an eigenfunction of $L_x$, $L_y$, and $L_z$. That's like saying that you have a triangle with four sides, or the factors of a prime number, or an odd number that is divisible by 2... There is no such thing.

 

[anorlunda]

Just for curiosity sake... could you list a few other such pairs of operators?

I think the list is very interesting. From https://en.wikipedia.org/wiki/Conjugate_variables

1. The energy of a particle at a certain event is the negative of the derivative of the action along a trajectory of that particle ending at that event with respect to the time of the event.
2. The linear momentum of a particle is the derivative of its action with respect to its position.
3. The angular momentum of a particle is the derivative of its action with respect to its orientation (angular position).
4. The electric potential (φ, voltage) at an event is the negative of the derivative of the action of the electromagnetic field with respect to the density of (free)electric charge at that event.
5. The magnetic potential (A) at an event is the derivative of the action of the electromagnetic field with respect to the density of (free) electric current at that event.
6. The electric field (E) at an event is the derivative of the action of the electromagnetic field with respect to the electric polarization density at that event.
7. The magnetic induction (B) at an event is the derivative of the action of the electromagnetic field with respect to the magnetization at that event.
8. The Newtonian gravitational potential at an event is the negative of the derivative of the action of the Newtonian gravitation field with respect to the mass density at that event.

By the way, the same wiki article offers this answer to the OPs question, and gives a hint on how to derive HUP.

Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals of one another, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty in physics called the Heisenberg uncertainty principle relation between them.

[mentor's note: Interesting followup question has been moved into its own thread]

 

[bhobba]

Why are we so confident in the principles of quantum mechanics?

At rock bottom QM is simply the most reasonable extension of probability for modelling physical systems:
http://www.scottaaronson.com/democritus/lec9.html

Thanks
Bill

 

[Madi Araly]

You've been misled again, probably because you are relying on popularizations and other dumbed-down treatments of QM. The double slit experiment gets a lot of press because it is a simple setup that (in its one particle at a time two slit version) starkly demonstrates behavior that cannot be explained classically. That makes it a good starting point for authors trying to get the flavor of QM across without asking their readers to do any intellectual heavy lifting; but there's a lot more to understanding QM than that.

Why are we so confident in the uncertainty principle? The uncertainty principle is derived from the principles of QM by using logic as unassailable as that which (for example) we use to conclude from the laws of arithmetic that no odd number is the sum of two even numbers. So if we are confident in the principles of quantum mechanics, we're confident in the uncertainty principle (correctly stated, of course - the stuff you've been reading about measuring position changing momentum and vice versa is NOT the uncertainty principle).

Of course this just invites the next question: Why are we so confident in the principles of quantum mechanics? The answer is that QM has more observational confirmation than any science in human history. Quantum mechanics correctly predicts the behavior of valence electrons and chemical bonds, and therefore explains all of chemistry, and all of biochemistry. It correctly predicts the interactions between atoms in solids and liquids, and therefore explains all of materials science. Every electronic device in the world depends on QM to function properly - the screen you're reading this on right now is another successful experimental test of the principles of QM. So there are a lot of reasons to believe that QM is correct - the double-slit experiment only comes in because it's a good starting point for explaining how QM is different from classical physics.Classical physics emerges from quantum mechanics, in the sense that every problem that can be correctly solved with classical physics can also be solved using the methods of quantum mechanics (another of the reasons that we trust QM is that it agrees with classical physics in any situation in which classical physics produces the right answer). We generally try using classical methods first because the math is less demanding (I know someone whose PhD thesis was the quantum mechanical treatment of a bouncing rubber ball - but this is a first-year undergraduate problem in classical mechanics). So you're right about trying to explain things with classical physics first, but mistaken to think that succeeding means that quantum mechanics is not still at work.

Furthermore, there is a huge area where classical physics just doesn't work. For example, classical physics predicts that all atoms are unstable (with a half-life measured in fractions of a second) and won't chemically bond to form molecules. Stuff like this is a very strong argument that QM is a more complete theory - it does everything that classical physics does, and much more.

Thank you for explaining this in a very clear, non-condescending manner. The chemistry aspect was something I pushed to the side and must have forgotten about, and I agree that it certainly doesn't make sense if explained from a classical perspective.

So you're right about trying to explain things with classical physics first, but mistaken to think that succeeding means that quantum mechanics is not still at work.

I'm a little disappointed I overlooked this, but then again I don't expect to draw perfect conclusions with only a few years of study on this. Hopefully my further research makes it clearer for me. I'll try to push aside any instinctive doubt in quantum mechanics and just deal with some temporary confusion, haha.

 

[phinds]

I'll try to push aside any instinctive doubt in quantum mechanics ...

That's the best advice you could give yourself when looking at both QM (the very small) and cosmology (the very large). Neither are in the domain of our direct experience and we did not evolve with any survival value being based on understanding them. Consequently, "common sense" and "intuition" are not only often wrong, they are often a direct impediment to learning these things.

 

[phinds]

Why is it that humans are so certain of this, though?

Feynman was fond of saying that the results of QM has been shown to be true to the same accuracy as measuring the width of the United States to within the diameter of one human hair. And that was about 50 years ago.

 

[Madi Araly]

That's the best advice you could give yourself when looking at both QM (the very small) and cosmology (the very large). Neither are in the domain of our direct experience and we did not evolve with any survival value being based on understanding them. Consequently, "common sense" and "intuition" are not only often wrong, they are often a direct impediment to learning these things.

Will do. Yet I assume you can understand my hesitance... To compare, the reason I'm not religious is that I dislike simply having "faith" in matters, I prefer to observe it directly.

I suppose I would compare it to how many (religious) people point towards events such as the survival of a person with severe injuries as being "the work of a <insert deity here>", yet offer no proof to support that it is indeed their deity causing the survival rather than anything else.

In QM, obviously there is a lot more proof, yet my brain automatically tests to see if there could be an alternative explanation... It's very difficult to resist this.

 

[phinds]

Sure, I understand that POV and there's nothing wrong with it but you want to be careful about going to the point where basically you are assuming that all the thousands of very smart physicists who have worked on this stuff are idiots who got it all wrong.

 

[Madi Araly]

Sure, I understand that POV and there's nothing wrong with it but you want to be careful about going to the point where basically you are assuming that all the thousands of very smart physicists who have worked on this stuff are idiots who got it all wrong.

As I said in a previous comment, that is clearly not my intent. I look up to these physicists more than anyone else, but I refuse to put clouded or blind faith in anything. After all, I wouldn't want to repeat Ancient Greece.

 

[anorlunda]

As I said in a previous comment, that is clearly not my intent. I look up to these physicists more than anyone else, but I refuse to put clouded or blind faith in anything. After all, I wouldn't want to repeat Ancient Greece.

Well then, don't be blind by choice. You should study QM yourself. May I recommend Leonard Susskind's video course on quantum mechanics? He is an excellent teacher. Here is a link to the first lecture.

 

[Madi Araly]

Well then, don't be blind by choice. You should study QM yourself. May I recommend Leonard Susskind's video course on quantum mechanics? He is an excellent teacher. Here is a link to the first lecture.

I recognize him, perhaps I've seen one of his videos before. I'll watch his lectures, thanks for the link!

 

[malawi_glenn]

The uncertainty property follows from Cauchy–Schwarz inequality

As some people have pointed out, the Heisenberg Uncertainty is a statistical relation, one can not thus ask what happens on an event-by-event basis

The standard deviation concerned in the Heisenberg uncertainty relation is INTRINSIC to the system/object itself, no matter how good detectors and devices we will invent - there will always be a non-zero standard deviation in our measurements

 

[Traruh Synred]

In a more fundamental you can think about this w/o explicit reference to commutation rules. A 'particle' is an quantized excitation of a field. It is more wave than particle. The particle like-behavior arises when the excitation is in the form of a wave-packet which is localized, but to get a localized excitation/wave you need to superimpose many wave lengths (inversely related to momenta via Planck's constant). The more localized the packet/'particle' the more different momenta (wavelengths) it needs to be constructed from, hence, the uncertainty principle. The inverse is also true. If you want to get an excitation that has a small range of momentum it need to will have to occupy a lot of space. You might look at wave packet on Wikipedia to see how it works.

-Traruh