Fourier Transform, and the uncertainty principle

[YaelPerkal]

Hello,

Recently I've learned about Fourier Transform, and the uncertainty principle that is arose from it.

According to Fourier Transform, if there is only one pulse in a signal, then it is composed from a lot more frequencies, compared to the number of frequencies that are building a repeating pulses in a signal.

But I'd like to talk about a logical paradox- if we believe that the Fourier components, the harmonic waves, are building our signals in nature, and the particles, and etc, so let's look on a repeating pulse in a signal - when we operate the first pulse, it is composed from many frequencies, and when we operate the second and the following pulses of the signals, then all the pulses including the first one in the signal are composed from less frequencies. Can the second pulse change the number of frequencies that built the first pulse?!

Also, when we make a noise in certain dt [sec] it is composed of more frequencies, then if we continue the pulse to: 2dt[sec]. Where the many frequencies gone to? Could the first part of the pulse know that we will continue the pulse later, so it will effect it to use less frequencies to build it?

This paradox, can be explained, as I understand it, by observing the Fourier Transform as only a mathematical algorithm that helps as to analyze the signal, only after we know the whole shape of it. Namely, only after we see how many pulses are in it, and how long it is.

And for the famous uncertainty principle that says: dt x dw ~1. It is only the limitation of this Fourier algorithm, but is not a limitation of nature. This uncertainty principle is more an uncertainty limitation when we try to analyze the nature in harmonic functions as building blocks.

But if we won't assume nature is built on sin, and cos waves, so we won't suffer from this uncertainty principle everywhere. Only when we try to measure things with our instruments ,and analyze it with Fourier algorithm, then we will have the uncertainty limitation of our measurements.

Because I'm only starting to learn with physics, so maybe those things are wrong or already known, so I'd like to hear your opinion on this matter.

Yael Perkal, Israel.

 

[phinds]

It is only the limitation of this Fourier algorithm, but is not a limitation of nature.
...
Only when we try to measure things with our instruments ,and analyze it with Fourier algorithm, then we will have the uncertainty limitation of our measurements.

You either have been misinformed or you misunderstand. The Uncertainty Principle has NOTHING to do with our ability to measure things, it IS a fundamental limitation of nature.

 

[Astrum]

You either have been misinformed or you misunderstand. The Uncertainty Principle has NOTHING to do with our ability to measure things, it IS a fundamental limitation of nature.

I'll put this out there in risk of sounding like an idiot. I'm not very well versed in the philosophy of QM, but couldn't you say that the uncertainty principle is caused by measurement, rather than something that's intrinsic to nature?

 

[phinds]

I'll put this out there in risk of sounding like an idiot. I'm not very well versed in the philosophy of QM, but couldn't you say that the uncertainty principle is caused by measurement, rather than something that's intrinsic to nature?

NO ... I just said no in post #2. Do you need me to say it again? NO

 

[Astrum]

NO ... I just said no in post #2. Do you need me to say it again? NO

One more time, please?

I understand the explanation before for why it conceptually is the way it is, as well as the mathematical formulation of the general UP, but I was curious if perhaps there existed some interpretation that took a different view to this being "definite".

I did some checking, and it seems that the UP is something that's fundamental.

 

[YaelPerkal]

Can someone give us a link to a proof of the uncertainty principle, which is not based on Fourier transforms at all?

 

[Astrum]

Can someone give us a link to a proof of the uncertainty principle, which is not based on Fourier transforms at all?

Heisenberg Uncertainty Principle

I couldn't find a proof of the generalized uncertainty principle online though, and I don't have enough time to write one out here.

NB: The generalized uncertainty principle is proven through use of matrices, which is what it seems you're looking for. The link I posted uses the Fourier transform to derive the momentum, position uncertainty relation.

 

[YaelPerkal]

Thanks, but I want to make it clearer by a simple example-

I'm making a noise in a known frequency of: 50[Hz], three times.

1. In the first time, I make this noise for 0.1 [sec]
2. In the second time, I make this noise for 1 [sec]
3. In the third time, I make this noise for 10 [sec]

According to the uncertainty principle, the frequency uncertainty in those three cases is:

1. 10 [Hz]
2. 1 [Hz]
3. 0.1[Hz]

Now please tell me, was anything different in the frequency that I've made, in all those cases? Or it only was harder to the one who heard the noise to identify it, the more shorter the signel was?

That's why this uncertainty limitation is true only to the one who try to identify a signal by using Fourier algorithm.

Let's, for example, try to use another way to identify a signal: Suppose the one who "hear" the signal, can get the signal and compare it to many signals that he has in his stock. When he gets the best fit, he declares on the frequency that he heard. I believe in this way he can know the exact frequency, but in this way it will take him much longer time. But there won't be any uncertainty related to the length of the signal.

 

[jtbell]

So you're not discussing quantum mechanics, but instead, classical signal processing?

 

[YaelPerkal]

I think it's the basic of quantum mechanics, as the probability waves are at the base of building all the particles and the signals there. And what I'm imply on, is that maybe many paradoxes that we get in quantum mechanics is related to this Fourier algorithm, like the simple paradox that I've shown here. For conclusion, I suggest to reconsider using harmonic functions as the basic to quantum mechanics, because of the limitation that Transform Fourier has - can't depict phenomenon while it is being built, but only to tell us things after it is being done. Only about the history, and with uncertainty limitation.

 

[George Jones]

I'll put this out there in risk of sounding like an idiot. I'm not very well versed in the philosophy of QM, but couldn't you say that the uncertainty principle is caused by measurement, rather than something that's intrinsic to nature?

Maintaining this viewpoint in quantum theory is very difficult.

What about the Kochen-Specker theorem? This says, roughly, that quantum systems do not, in general, possesses properties.

Consider the measurement of S_z of a spin-1 particle. If the state is psi = |1> just before measurement, then, with certainty, the result of the measurement is S_z = 1, and it seems to make sense that S_z = 1 was a property of the particle before measurement. Now suppose that the state is psi = (sqrt(3)|-1> + |1>)/2 just before the measurement, and that the result of the measurement is S_z = 1. Is it possible, within the framework of standard quantum theory, to say that the particle had the property S_z = 1 before the measurent? In other words, is possible that to say that the observable S_z possessed the value 1 before the measurent?

Now generalize this. Let A be an observable for a quantum system. Is it possible that a value function V that takes as input an observable and a state of the system, and spits out, say V(A ,psi), for the value of the observable A when the system is in state psi?

No one has ever thought of a way of doing this. Is this because no one has been able to hit on the right idea, or is it because it is impossible? If it were possible to do this, many of the ontological "problems" associated with quantum theory would disappear, and realists, like Patrick (I think) and me, would be happier campers :-). The Kochen-Specker theorem says that for quantum systems having state spaces of dimension 3 or higher, no value function (with reasonable properties) exists. :-(

This one of the things that makes a "disturbance" interpretation of the uncertainty principle difficult. If a system doesn't possesses a particular property, how is it possible to disturb that property.

Nice discussions of the Kocken-Specker theorem can be found in Lectures on Quantum Theory: Mathematical and Structure Foundations by Chris Isham, and in The Structure and Interpretation of Quantum Mechanics by R.I.G.Hughes. Hughes is a philosopher, and some philosophy he espouses in his preface is particularly interesting: "Having thus outlined my program and declared my allegiances, I leave the reader to decide whether to proceed further, or to open another beer, or both."

The Kochen-Specker theorem makes life somewhat difficult for realists, but positivists probably have no problem with it. Isham says "... very effective professionally without needing to lose any sleep over the implications of the Kochen-Specker theorem or the Bell inequalities. However, in the last twenty years there has been a growing belief among physicists from many different specialisations that even if modern quantum theory works well at the pragmatic level, it simply cannot be the last word on the matter."

Patrick, this is essentially your hope, isn't it? Maybe a subtle 4).

Isham says further "... there are now any any physicists who feel that any significant advance will entail a radical revision in our understanding in the meaning and significance of the categories of space, time and substance."

To me, this seems to point towards quantum gravity, which is no surprise coming from Isham. Many physicists would disagree this, but I am inclined to believe that a theory of quantum gravity might have interesting things to say about these issues. Unfortunately, despite claims by various camps to the contrary, I see no viable candidate for a quantum theory of gravity. Also, such a theory could bring forth new interpretational issues just as thorny as any it alleviates.

 

[Jilang]

I'll put this out there in risk of sounding like an idiot. I'm not very well versed in the philosophy of QM, but couldn't you say that the uncertainty principle is caused by measurement, rather than something that's intrinsic to nature?

No, my take on it is that's intrinsic. If you want something to move in a straight line you have to give it enough space (or leg-room) to do so. Think of it as a random walk... If you don't give it the chance to come back to the mean it won't. The more you restrict the space the more chance you have of it going off at an angle.

 

[bahamagreen]

Just thinking out loud, but might both ideas be fundamentally correct (intrinsic vs caused by measurement)?

Isn't it true that the Fourier typically uses sines and cosines because those were how it was discovered, because they are simple; but Fourier works with other waveforms as building blocks as well - all possible waveforms - any waveform may be used to compose or decompose any other waveform... one has to chose the building block waveform (historically sine / cosine), but it could be the impulse, the triangle, the ramp, or some meandering wiggle or any shape...?

If so, then it seems perhaps the choice of waveform to be used as the building blocks of the analysis corresponds to the choice of what attribute one wants to measure, which does seem to imply that the object has no intrinsic attributes until the measurement is defined and applied, or it may be the same thing to imagine that the object has all possible potential attributes and one's choice of waveform for the analysis determines which attribute is selected to provide the measured value.

If so, this might have bearing on both uncertainty and complementarity (as pairs of attributes would be like opposite surface points of a diameter through a sphere of which all possible waveforms are represented on the surface).

If the simple waveforms correspond to the more familiar attributes, the more complicated waveforms would be corresponding to increasingly complex attributes and increasingly more unlikely and difficult experimental measurement arraignments, virtually all of which are technically impossible.

 

[Naty1]

Hi Yael:
Let me see if I can help clear up one area of misunderstanding. It takes time and for most of us effort to figure out how these all fit together...it is not obvious, so stick with it.

Recently I've learned about Fourier Transform {FT}, and the uncertainty principle that is arose from it.

No, the uncertainty principle [HUP] has no source from the FT. It does not stem from the DETERMINISTIC Schrodinger wave equation, either.

The FT reflects that the expression for describing particles and states in quantum mechnaics, the Schrodinger wave equation, is a linear DETERMINISTIC equation and therefore subject to superposition principles. Those superpositions are conveniently expressed via the FT.

[It turns out real numbers within the Schrodinger wave equation are found to represent real particles, imaginary numbers anti particles, and complex numbers virtual particles!]


Two examples of HUP from prior discussions in these forums on HUP...these helped me clarify my thinking:

...Bouncing a single photon off an atom tells us nothing about any [Heisenberg] uncertainties. We must bounce many ‘identically’ prepared photons off like atoms in order to get the statistical distributions of atomic position measurements and atomic momentum measurements. What we call "uncertainty" is a property of a statistical distribution. You cannot determine an uncertainty from a single measurement.

Regarding the double slit experiment [if you are not familiar with it, no problem, the explanation that follows is I think clear enough]:

... a single measurement uncertainty is still the same as in the classical case. If I shoot the particle one at a time, I still see a distinct, accurate "dot" on the screen to tell me that this is where the particle hits the detector. However, unlike the classical case, my ability to predict where the NEXT one is going to hit becomes worse as I make the slit smaller. As the slit becomes smaller and smaller, I know less and less where the particle is going to hit the screen...

Note that HUP has nothing to do with a single measurement! Remember that!

Over time, a definite pattern always emerges, but not like the purely classical case, and not predictable, dot by dot by dot in any specific orderly location of events [detection dots on a screen].


From Roger Penrose, the mathematical physicist:

at Stephen Hawking’s 60th birthday in 1993 at Cambridge England...he was addressing the elite of the physics world...this description offered me a new perspective into quantum/classical relationships:

..Either we do physics on a large scale, in which case we use classical level physics; the equations of Newton, Maxwell or Einstein and these equations are deterministic, time symmetric and local. Or we may do quantum theory, if we are looking at small things; then we tend to use a different framework where time evolution is described... by what is called unitary evolution...which in one of the most familiar descriptions is the evolution according to the Schrodinger equation: deterministic, time symmetric and local. These are exactly the same words I used to describe classical physics.

However this is not the entire story... In addition we require what is called the "reduction of the state vector" or "collapse" of the wave function to describe the procedure that is adopted when an effect is magnified from the quantum to the classical level...quantum state reduction is non deterministic, time-asymmetric and non local...The way we do quantum mechanics is to adopt a strange procedure which always seems to work...the superposition of alternative probabilities involving w, z, complex numbers...an essential ingredient of the Schrodinger equation. When you magnify to the classical level you take the squared modulii (of w, z) and these do give you the alternative probabilities of the two alternatives to happen...it is a completely different process from the quantum (realm) where the complex numbers w and z remain as constants "just sitting there"..
So key differences between the classical and quantum world are superposition and complex numbers in our models. This 'strange procedure' describes what we observe!

Hope that helps...
If not, read it five or ten more times...that's what I had to do!

 

[atyy]

Can someone give us a link to a proof of the uncertainty principle, which is not based on Fourier transforms at all?

Thanks, but I want to make it clearer by a simple example-

I'm making a noise in a known frequency of: 50[Hz], three times.

1. In the first time, I make this noise for 0.1 [sec]
2. In the second time, I make this noise for 1 [sec]
3. In the third time, I make this noise for 10 [sec]

According to the uncertainty principle, the frequency uncertainty in those three cases is:

1. 10 [Hz]
2. 1 [Hz]
3. 0.1[Hz]

Now please tell me, was anything different in the frequency that I've made, in all those cases? Or it only was harder to the one who heard the noise to identify it, the more shorter the signel was?

That's why this uncertainty limitation is true only to the one who try to identify a signal by using Fourier algorithm.

Let's, for example, try to use another way to identify a signal: Suppose the one who "hear" the signal, can get the signal and compare it to many signals that he has in his stock. When he gets the best fit, he declares on the frequency that he heard. I believe in this way he can know the exact frequency, but in this way it will take him much longer time. But there won't be any uncertainty related to the length of the signal.

Once you say frequency, you refer to something which is periodic or repeats itself. If something repeats itself, it must take up some time, so it cannot be completely localized in time. The Fourier transform is used because the specific meaning of periodicity we usually use is the periodicity of sinusoids.

 

[Naty1]

Isn't it true that the Fourier typically uses sines and cosines because those were how it was discovered, because they are simple; but Fourier works with other waveforms as building blocks as well - all possible waveforms - any waveform may be used to compose or decompose any other waveform... one has to chose the building block waveform (historically sine / cosine), but it could be the impulse, the triangle, the ramp, or some meandering wiggle or any shape..

.?

generally, yes.

If so, then it seems perhaps the choice of waveform to be used as the building blocks of the analysis corresponds to the choice of what attribute one wants to measure, which does seem to imply that the object has no intrinsic attributes until the measurement is defined and applied, or it may be the same thing to imagine that the object has all possible potential attributes and one's choice of waveform for the analysis determines which attribute is selected to provide the measured value.

It's not the 'waveform', but mathematical operators that represent 'observables' in QM. Beyond that is philosophy...like creation and annihilation operators ...what we think all the math represents...

Brain Greene says this:

The telltale difference between quantum and classical notions of probability is that
the former is subject to interference and the latter is not. Brian Greene.

And interference results from superposition in linear systems as I described above.

 

[YaelPerkal]

Let's, for example, try to use another way to identify a signal: Suppose the one who "hear" the signal, can get the signal and compare it to many signals that he has in his stock. When he gets the best fit, he declares on the frequency that he heard. I believe in this way he can know the exact frequency, but in this way it will take him much longer time. But there won't be any uncertainty related to the length of the signal.

Isn't this example contradicts the uncertainty principle?

 

[Naty1]

Isn't this example contradicts the uncertainty principle?

No. I can also guess you did not read my prior caution carefully enough:

Note that HUP has nothing to do with a single measurement! Remember that!

I'm making a noise in a known frequency of: 50[Hz], three times.

1. In the first time, I make this noise for 0.1 [sec]
2. In the second time, I make this noise for 1 [sec]
3. In the third time, I make this noise for 10 [sec]

from a prior post:

According to the uncertainty principle, the frequency uncertainty in those three cases is:

1. 10 [Hz]
2. 1 [Hz]
3. 0.1[Hz]

That is NOT an accurate application of HUP. You are applying a formula intended for discrete [h type] quantum interactions...subatomic interactions. We have frequency measuring equipment that far,far exceed the accuracy you imply. If we did not, radio, TV,radar and GPS could not work...we could not process signals properly,accurately.

It's analogous to trying to apply classical velocity addition V[total] = V₁ + V₂ near relativistic speeds...experimental Observations refute such simple calculations, they don't apply. [Special relativity applies at high speeds.]



Here is one of the best source explanations of HUP I have come across...
from one of the experts of these forums:

From Zapper: Misconception of the Heisenberg Uncertainty Principle. http://physicsandphysicists.blogspot.com/2006/11/misconception-of-heisenberg-uncertainty.html

One of the common misconceptions about the Heisenberg Uncertainty Principle (HUP) is that it is the fault of our measurement accuracy. A description that is often used is the fact that ….a very short wavelength photon has a very high energy, and thus, the act of position measurement will simply destroy the accurate information of that electron's momentum.

While this is true, (about measurement limitations of equipment) it isn't really a manifestation of the HUP. The HUP isn't about a single measurement and what can be obtained out of that single measurement. It is about how well we can predict subsequent measurements given the ‘identical’ {initial, state preparation} conditions. In classical mechanics, if you are given a set of identical conditions, the dynamics of a particle will be well defined. The more you know the initial position, the better you will be able to predict it's momentum, and vice versa…...

 

[atyy]

Thanks, but I want to make it clearer by a simple example-

I'm making a noise in a known frequency of: 50[Hz], three times.

1. In the first time, I make this noise for 0.1 [sec]
2. In the second time, I make this noise for 1 [sec]
3. In the third time, I make this noise for 10 [sec]

According to the uncertainty principle, the frequency uncertainty in those three cases is:

1. 10 [Hz]
2. 1 [Hz]
3. 0.1[Hz]

Now please tell me, was anything different in the frequency that I've made, in all those cases?

Here is perhaps a more precise way of stating the uncertainty principle.

By definition, frequency refers to something periodic, and hence infinite in duration. A sine wave is periodic and infinite in duration. While a sine wave may fit some part of a short sound, it will not capture its short duration. It is the superposition of sine waves of different frequencies and phases that can make a short sound.

 

[atyy]

From Zapper: Misconception of the Heisenberg Uncertainty Principle. http://physicsandphysicists.blogspot.com/2006/11/misconception-of-heisenberg-uncertainty.html

I believe ZapperZ's post is wrong, drawing from the error in Ballentine's 1970 article.

 

[Naty1]

I believe ZapperZ's post is wrong, drawing from the error in Ballentine's 1970 article.

I still like it, but claim no special expertise here. [Edit: I thought Zapper's explanation consistent with conclusions here: I seem to recall a general agreement that Ballentine's paper was in error.]
For those interested in more detail than is likely appropriate for the current discussion, here is a very long, sometimes antagonistic discussion, sometimes a bit confusing [for me] which was even closed by a moderator for a period:

what is it about position and momentum that forbids knowing both quantities at once?
https://www.physicsforums.com/showthread.php?t=516224

 

[atyy]

I still like it, but claim no special expertise here. For those interested in more detail than is likely appropriate for the current discussion, here is a very long, sometimes antagonistic discussion, sometimes a bit confusing [for me] which was even closed by a moderator for a period:

what is it about position and momentum that forbids knowing both quantities at once?
https://www.physicsforums.com/showthread.php?t=516224

Let me try to explain what I think the error is, since I think it is basic and important.

In QM, it is canonically conjugate position and momentum that cannot be precisely and simultaneously attributed to a particle. In ZapperZ's and Ballentine's example in which the position and momentum of a particle are claimed to be precisely and simultaneously measured, the position and momentum are not canonically conjugate. A good exposition of the uncertainty principle in QM is given here http://www.youtube.com/watch?v=TcmGYe39XG0&list=PL5A6DBFFBEFF3A92E.

 

[Jilang]

Let me try to explain what I think the error is, since I think it is basic and important.

In QM, it is canonically conjugate position and momentum that cannot be precisely and simultaneously attributed to a particle. In ZapperZ's and Ballentine's example in which the position and momentum of a particle are claimed to be precisely and simultaneously measured, the position and momentum are not canonically conjugate. A good exposition of the uncertainty principle in QM is given here http://www.youtube.com/watch?v=TcmGYe39XG0&list=PL5A6DBFFBEFF3A92E.

Zapper Z's example shows really well how uncertainty has nothing to do with measurement! However whilst the final position is determined by the dot on the screen, the momentum at that point in time is not determined. The best that can be arrived at is the average momentum along the path from the slit. It's still sufficient though to demonstrate the point well. I don't think it was ever claimed that position and momentum are simultaneously measured. Please correct me if I'm wrong!

 

[Naty1]

Zapper Z's example shows really well how uncertainty has nothing to do with measurement!

That's what I think...[edit] if you mean an individual, single measurement. It seems in QM the state preparation is what results in uncertainty rather than an individual measurement.

I don't think it was ever claimed that position and momentum are simultaneously measured. Please correct me if I'm wrong!

I took it to mean exactly that...a simultaneous measurement...see PAllen's description below which I so far find consistent with this interpretation of Zapper's comments.

The two illustrations post #14 I think correcty illustrate HUP based on opinions of experts in these forums...not my own 'mumbo jumbo'. I'm not qualified to opine on what is and what is not a 'canonically conjugate' measurement.

But I have collected other similar descriptions which I think consistent:

...attempts at identical state preparation procedures do not result in identical SYSTEMS, but systems with a statistical distribution.

"canonical conjugates" [a function and its Fourier transform] of the wavefunction representing an ensemble of similarly prepared particles;

The commutativity and non commutivity of operators applies to the distribution of multiple results, not an individual measurement of an individual particle. The uncertainty principle restricts the degree of statistical homogeneity of measurements and predictions which it is possible to achieve in an ensemble of similarly prepared systems.

Born’s postulate is that the square of the wave function, psi, represents a probability density function. The width of, say, a measured momentum distribution for many particles is what is limited by the HUP.

A quantum mechanical state vector thus refers not to properties of, say, single neutron spins, but to ensembles of neutrons with equally prepared spins…

I believe the above comport with Zapper and Fredrik's understanding...In addition PAllen of these forums has posted the following:

If you are measuring position and momentum of the 'same thing' at two different times, the measurements are necessarily timelike. The measurements occur at two times on the world line of the thing measured. This order will never change, no matter what the motion of the observer is. If, instead, they occur for the same time on the "thing's" world line, they are simultaneous for the purposes of the uncertainty principle. To measure a particle's momentum, we need to interact with it via a detector, which localizes the particle. So we actually do a position measurement (to arbitrary precision). Then we calculate the momentum, which requires that we know something else about the position of the particle at an earlier time (perhaps we passed it through a narrow slit). Both of those position measurements, and the measurement of the time interval, can be done to arbitrary precision, so we can calculate the momentum to arbitrary precision. From this you can see that in principle, there is no limitation on how precisely we can measure the momentum and position of a single particle.

Where the HUP comes into play is that if you then repeat the same sequence of arbitrarily precise measurements on a large numbers of identically prepared particles (i.e. particles with the same wave function, or equivalently particles sampled from the same probability distribution), you will find that your momentum measurements are not all identical, but rather form a probability distribution of possible values for the momentum. The width of this measured momentum distribution for many particles {that are measured} is what is limited by the HUP. In other words, the HUP says that the product of the widths of your measured momentum probability distribution, and the position probability distribution associated with your initial wave function, can be no smaller than Planck's constant divided by 4 times pi.

Somebody also posted the following in a HUP discussion which I thought consistent with the above interpretational examples:

Course Lecture Notes, Dr. Donald Luttermoser, East Tennessee State University:

.. The HUP strikes at the heart of classical physics: the trajectory. Obviously, if we cannot know the position and momentum of a particle at t[0] we cannot specify the initial conditions of the particle and hence cannot calculate the trajectory...Due to quantum mechanics probabilistic nature, only statistical information about aggregates of identical systems can be obtained. QM can tell us nothing about the behavior of individual systems...

But I'll bet we can conjure up different interpretations as well.

 

[atyy]

I don't think it was ever claimed that position and momentum are simultaneously measured. Please correct me if I'm wrong!

I was thinking of this specific statement in http://physicsandphysicists.blogspot.com/2006/11/misconception-of-heisenberg-uncertainty.html: "I have shown that there's nothing to prevent anyone from knowing both the position and momentum of a particle in a single measurement with arbitrary accuracy that is limited only by our technology." In my understanding, this is an error from Ballentine's 1970 article.

 

[Naty1]

I want to agree with atyy that the video he posted,

A good exposition of the uncertainty principle in QM is given here http://www.youtube.com/watch?v=TcmGY...6DBFFBEFF3A92E.

is really,really good. It's an hour overall. The professor makes some qualifications which are really interesting...just the sort of background assumptions that seem required to interpret the math.

I will not make any interpretation til I finish watching the video today or tomorrow [because there may always be further refinements during the lecture] but for those whose time is short, I think if you tune in at minute 14 and watch for a minute or so you can make your own call reagrding the issue under discussion.

 

[Naty1]

I was thinking of this specific statement in http://physicsandphysicists.blogspot...certainty.html:

"I have shown that there's nothing to prevent anyone from knowing both the position and momentum of a particle in a single measurement with arbitrary accuracy that is limited only by our technology."

yes, that is a clear statement of interpretation. So I am interested if and how your youtube video addresses that.

 

[atyy]

yes, that is a clear statement of interpretation. So I am interested if and how your youtube video addresses that.

I take back my criticism of ZapperZ's post. His argument is different from Ballentine's, and does refer to conjugate position and momentum.

In the Bohmian interpretation, the wave function is not the complete description of a particle, and a particle always has a definite position and momentum, so his statement may make sense in that interpretation. However, in that case, the trajectory of the particle is not governed by Newtonian mechanics, but by the Bohmian equation for the trajectory.

I was thinking of an interpretation in which the wave function is the complete description of a particle. In those interpretations, a wave function that yields a definite position (all particles with that wave function give the same result when their position is measured) does not yield have a definite momentum (all particles with the same wave function give different results when their momentum is measured).

 

[Naty1]

I watched the first Youtube lecture. Good stuff as atyy implied...

Depending on one's interpretation, I think the professor confirms that HUP applies to probability distributions, not an individual measurement. Others may disagree.

Prior to minute 14: [These are close to quotes but not exact]

First a general question; Anybody know what this means:

...Quantum mechanics with friction, where Hamilton formalism does not apply, is unclear. Open systems which exchange energy remains an open question: for systems essentially at absolute zero and with no dissipation…that is where the uncertainty principle applies...

And onto our issue of single measurements versus statistical distributions of multiple measurements:

To me, clearly a statistical explanation:

14.40

‘delta x’ in the HUP uncertainty statement is a standard deviation. No matter how accurate your measuring device you will still get a standard deviation of measurement results. The whole purpose of QM is to determine the distribution, which probability distribution, which can be calculated from the wavefunction…… QM uncertainty is intrinsic, in addition to any noise or other perturbations affecting the system. Can derive the HUP inequality [position, momentum] from the postulates of QM and the Schrödinger equation. Poisson brackets of canonical variables are replaced in QM by operators….and we have expectation values, or average values, of these operators

Perhaps a single measurement description:

min 30:

...You cannot even {theoretically} specify x,p to arbitrary simultaneous accuracy...has nothing to do with equipment sensitivity….States of a system can no longer be represented by POINTS in phase space as can be done classically. A POINT implies we know both parameters to arbitrary accuracy. There is an intrinsic fuzziness and we need some other representation. So an electron does not move in an orbit about a nucleus because ‘orbit’ implies a precise trajectory.

Again, what seems clearly a statistical explanation:
Min 36

The Bohr radius can be calculated as a QM statistical average; it is not a fixed classical circular orbit of a point prticle. The electron does not have a precise classical position and momentum of a point particle….

And a general insight of interest:

..A state vector is a generalization of the wavefunction used to describe such a particle. We should not extrapolate 'particle' and 'wave' macroscopic concepts to quantum world…how we probe determines what the observational results we extract. Quantum mechanics is like learning another language. A particle is a state vector.

I'm going to watch another lecture...If the prof. explains the assumptions behind the mathematics, and how he interprets the subsequent math, that sure would be of interest.

 

[Len M]

I want to agree with atyy that the video he posted,

A good exposition of the uncertainty principle in QM is given here http://www.youtube.com/watch?v=TcmGY...6DBFFBEFF3A92E.

is really,really good. It's an hour overall. The professor makes some qualifications which are really interesting...just the sort of background assumptions that seem required to interpret the math.

I will not make any interpretation til I finish watching the video today or tomorrow [because there may always be further refinements during the lecture] but for those whose time is short, I think if you tune in at minute 14 and watch for a minute or so you can make your own call reagrding the issue under discussion.

I thought the video was very good as well, I very much enjoyed it. Your link wouldn't work for me though, so just for completeness this link does work:

http://www.youtube.com/watch?v=TcmGYe39XG0&list=PL5A6DBFFBEFF3A92E