# Fourier Transform, and the uncertainty principle

1. Nov 16, 2013

### YaelPerkal

Hello,

Recently I've learnt 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 belive 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.

2. Nov 16, 2013

### phinds

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.

3. Nov 16, 2013

### Astrum

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?

4. Nov 17, 2013

### phinds

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

Last edited: Nov 17, 2013
5. Nov 17, 2013

### Astrum

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.

6. Nov 17, 2013

### YaelPerkal

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

7. Nov 17, 2013

### Astrum

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.

8. Nov 17, 2013

### 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 belive 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.

9. Nov 17, 2013

### Staff: Mentor

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

10. Nov 17, 2013

### 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.

11. Nov 17, 2013

### George Jones

Staff Emeritus
Maintaining this viewpoint in quantum theory is very difficult.

12. Nov 17, 2013

### Jilang

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.

13. Nov 17, 2013

### 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.

14. Nov 17, 2013

### 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.

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:

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

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:

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

15. Nov 17, 2013

### atyy

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.

Last edited: Nov 17, 2013
16. Nov 17, 2013

### Naty1

.?

generally, yes.

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:

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

17. Nov 18, 2013

### YaelPerkal

Isn't this example contradicts the uncertainty principle?

18. Nov 18, 2013

### Naty1

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

19. Nov 18, 2013

### atyy

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.

Last edited: Nov 18, 2013
20. Nov 18, 2013

### atyy

21. Nov 18, 2013

### Naty1

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?

22. Nov 18, 2013

### atyy

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.

23. Nov 18, 2013

### Jilang

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!

24. Nov 18, 2013

### Naty1

That's what I think.... 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 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:

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

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:

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

Last edited: Nov 18, 2013
25. Nov 18, 2013

### atyy

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.