[HN Gopher] The Fourier Uncertainty Principles [pdf] (2021)
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The Fourier Uncertainty Principles [pdf] (2021)
Author : mindcrime
Score : 110 points
Date : 2024-09-03 17:23 UTC (1 days ago)
(HTM) web link (math.uchicago.edu)
(TXT) w3m dump (math.uchicago.edu)
| gballan wrote:
| There is an approachable explanation in [1], chapter 16
| ("Duration-bandwidth relationships and the uncertainty
| principle"), that says that the product of rise-time and
| bandwidth of a signal must be greater than some minimum.
|
| [1] Siebert, W. M. (1986). Circuits, Signals, and Systems.
| McGraw-Hill.
| peter_d_sherman wrote:
| >"The most popular use of Fourier uncertainty principles is as a
| description of the natural _tradeoff_ between the _stability_ and
| _measurability_ of a system "
|
| Related:
|
| https://en.wikipedia.org/wiki/Complementarity_(physics)
| wyager wrote:
| I have read a lot of documents on ncatlab and other places to try
| to pin down a coherent model of the physical role of planck's
| constant in fourier transforms on physical systems. I understand
| that it often serves as the scale factor for embedding the
| integers into the reals, but it's not totally clear to me what
| its role is in physical pontryagin duality/fourier transforms.
| It's some kind of volume in phase space, but where does that
| volume come from? For a constant like c, we have the narrative "c
| is the ratio of unit lengths in time and space", but I have not
| yet found a good narrative about the meaning of h that works for
| fourier transforms. Would appreciate any articles on the matter.
| CarpaDorada wrote:
| I hope this answers your question. Let me preface this by
| saying that there are (probably) no satisfying answers for
| these questions, and that I'm not an expert. There is a
| classical limit <https://en.wikipedia.org/wiki/Classical_limit>
| that will recover classical equations from quantum equations
| from the limit -> 0. Such a thing is a heuristic, which means
| that we just know some equations/models where it works, but
| have not discovered a general truth. There are also situations
| where you may take c -> +[?] for example, and that would be
| called the non-relativistic limit. Why do we take these limits?
| Because when we did, the answer was not complete nonsense. We
| don't know what to make of them, i.e. we don't have complete
| theories. Also, what these limits mean is not a simple matter
| of calculus, they are not point-wise limits.
|
| In one such instance I've been studying for years, the WKB
| approximation, I've realized two things: 1) the approximations
| are not well understood and 2) the mathematics are quite
| complicated, but these points notwithstanding the equations are
| used in experiments. You can read the few-page introduction in
| "Lectures on the Geometry of Quantization" by Bates & Weinstein
| <https://math.berkeley.edu/~alanw/GofQ.pdf> to see some of
| this, in particular the subsection "Quantization and the
| classical limit". I'll just quote the relevant paragraph:
|
| > Although there remain some unsettled issues connected with
| the question, "How can become small?" the answer is
| essentially the following. For any particular mechanical
| system, there are usually characteristic distances, masses,
| velocities, . . . from which a unit of action appropriate to
| the system can be derived, and the classical limit is
| applicable when divided by this unit is much less than 1.
|
| But remember, this is just one approach to the subject. Another
| heuristic is this: h has dimensions energy x time, which means
| it converts frequency into energy, e.g. E = hf. In the Fourier
| transform, the character is exp(2pihx*x), where x is the
| frequency. The effect of h -> 0 would be to dampen high-energy
| waves. Irregularity comes from high frequencies (think of it
| like this: a sum of sines of large periods would not have many
| kinks.) When you "iron out" the irregularity of the quantum
| solution, you end up with a classical one.
|
| Again disclaimer: not a physicist, nor an expert.
| adrian_b wrote:
| Like the speed of light in vacuum provides a natural unit for
| velocity, i.e. for the ratio between length and time, Planck's
| constant (computed by him in 1899, before any quantum theory,
| because it is a consequence of the laws of blackbody radiation)
| provides a natural unit for angular momentum, i.e. for the
| ratio between kinetic energy and frequency (a.k.a. angular
| velocity).
|
| (Many books claim that Planck's constant is a quantum of
| "action". This is extremely wrong, because action is not a
| quantized physical quantity, so it cannot have quanta, and
| moreover the ratio between energy and frequency is not an
| action, but an angular momentum. This ridiculous mistake is
| caused by the failure to understand that the unit of plane
| angle is a base unit that cannot be derived from any of the
| units of the other physical quantities and forgetting to write
| the plane angle in the dimensional relationships between
| physical quantities leads to very serious errors. Angular
| momentum is the ratio between action and plane angle, while
| energy is the ratio between action and time and the linear
| momentum is the ratio between action and length. Frequency is
| not the inverse of time, but the ratio between plane angle and
| time, as the old name of "Hertz", i.e. "cycles per second" was
| making this obvious.)
|
| Dividing or multiplying by combinations of Planck's constant
| with the speed of light and the elementary charge only changes
| the system of units, between the traditional units and natural
| units. There are several variants of "natural" systems of units
| and the difference between using them and using SI is that when
| using natural systems of units there are much less "universal"
| constants in the relationships between physical quantities.
|
| Like electric charge, angular momentum is one of the quantities
| that are discrete, not continuous (though angular momentum is
| not necessarily discrete like electric charge; depending on the
| physical system it may be either discrete or continuous).
|
| Like the elementary charge is the quantum of electric charge
| (when all quarks are bound in hadrons), a half of Planck's
| constant is the quantum of angular momentum (there are multiple
| possible definitions of the angular momentum, depending on the
| unit chosen for plane angle, which lead to multiple possible
| numeric values for Planck's constant).
|
| The importance of the _c_ , _e_ and _h_ constants is due to
| them being respectively the limit value for velocity and the
| quanta of two fundamental discrete quantities (electric charge
| and angular momentum), which makes them appear in many
| relationships between physical quantities, unless the system of
| units is changed to a "natural" system of units, when the
| "universal" constants become "1", so they disappear from the
| formulae.
|
| When a natural system of units is not used in quantum
| mechanics, then all the equations that contain both energy or
| momentum and time or length, like the equation of Schroedinger,
| will contain Planck's constant, possibly combined with other
| universal constants. This will have as a consequence the
| appearance of those constants also in the expressions that give
| the pairs of quantities related by Fourier transforms.
| wyager wrote:
| I don't think your explanation hits the bottom of the ontic
| causal hierarchy.
|
| The fact that hbar is the quantum of angular momentum can be
| derived from the boundary conditions of a wavefunction with a
| closed dimension (such as an angle), combined with the fact
| that one h of action phases a wavefunction by 360o.
|
| In particular, for a wavefunction with only angular
| dependence:
|
| boson: \psi(2pi) = \psi(0) fermion: \psi(4pi) = \psi(0)
|
| Because the value has to be the same when you get back to
| where you started.
|
| Therefore, (for a boson) psi must decompose into
| eigenfunctions of the form \psi(\theta) = exp(i * n * \theta)
| where n is an integer.
|
| These eigenfunctions satisfy the boundary condition stated
| above and are a complete basis
|
| We can take the fourier transform of this (closed) function
| and we get a dirac delta at n. I.e. the fourier transforms of
| this basis are just dirac deltas at integers in the fourier
| domain.
|
| But to convert this to physical units, we need to use the
| embedding from integers to reals, which is done by
| multiplication by planck's constant.
|
| So, some related questions that I think are further down the
| causal hierarchy:
|
| * Why does one h of action phase a wavefunction by 360o
|
| * Why does h necessarily show up in the fourier transform
| jinwoo68 wrote:
| When I learned the Uncertainty Principle, I noticed its
| similarity with the characteristics of the Fourier transform, but
| couldn't understand why. This paper says:
|
| > In quantum mechanics, the wave function of position is the
| Fourier transform of the wave function of momentum.
|
| That explains it!
| tines wrote:
| So the Fourier transform on wave functions is like integrating
| a normal function, like the velocity function, in Newtonian
| mechanics?
| tomyedwab wrote:
| That sounds more complicated than what is actually happening
| here. The way wave functions evolve over time, their velocity
| of movement is proportional to the frequency of oscillation.
| That is why measuring the frequency component of the wave
| function gives you the momentum function.
|
| You have to keep in mind that the wave function represents
| the many places the particle can be with some probability, as
| well as the many frequencies it could have, so what
| uncertainty means in this case is that if you constrain the
| function to a small area in space (with zero probability
| outside it) you necessarily end up with a momentum function
| that spreads across many different velocities.
| jinwoo68 wrote:
| The similarity I noticed was the fact that in the Fourier
| transform, when the time domain graph is focused in a small
| time range, its frequency domain graph is spread out in a
| wide frequency range, and vice versa, like the position and
| the momentum in the uncertainty principle. That makes sense
| if the wave function of position is the Fourier transform of
| the wave function of momentum.
| throwawaymaths wrote:
| It's also basically the Nyquist Shannon sampling theorem
| xeonmc wrote:
| not quite, sampling theorem specifically refers to aliasing,
| whereas uncertainty is the general reciprocal correspondence.
| throwawaymaths wrote:
| Yeah I should have specified they are related by an inversion
| AtomicOrbital wrote:
| https://m.youtube.com/watch?v=D1WfID6kk90
|
| take a time series dataset like an audio file or stock ticker
| price over time ... give your self a healthy period of time ...
| for example a second of broadcast quality audio gives you 44,100
| data points spread across that time period stored as information
| ... importantly this time series audio curve wobbles up and down
| as it's recorded over time ... in order to justify taking 44,100
| audio samples per second (on the X axis) you must balance that by
| breaking up the granularity of your measurement of the up and
| down wobble (Y axis) by devoting two bytes (a bit depth of 16
| bits) of memory storage per data point which gives you 2 raised
| to the 16 power distinct gradations of resolution
|
| above defines the time domain representation of the one second of
| audio data ... now feed this dataset into a Fourier transform
| which will output the same information you started with but now
| in the frequency domain ... it will give you not 44,100 points in
| time but instead 44,100 distinct frequencies ... super cool side
| note you can feed this new frequency domain representation of the
| dataset into an inverse Fourier transform to rescue back the
| original time series audio
|
| If instead of a second of audio we start with a fraction of that
| number this reduction of recording duration will compromise the
| frequency resolution of the data in the frequency domain giving
| it less granularity hence larger increments to the next frequency
| quantadev wrote:
| First paragraph is a description of single channel redbook CD-
| AUDIO format right? Second paragraph is basically describing
| FFT and inverse FFT. Third paragraph is basically how MP3s work
| right? Wavlets that are essentially FFTs (list of frequencies)
| over a finite range of time, with enough of them removed to
| compress the data.
| kragen wrote:
| 'wavelets' means something different
| quantadev wrote:
| Both FFT and FWT beak a signal down into frequency
| components. I used the word 'essentially' to make it clear
| I'm not equating the two but saying they both output
| frequency components, but that the FWT coefficients apply
| to specific points in time, whereas FFT doesn't
| kragen wrote:
| yes, there are important similarities
| TheOtherHobbes wrote:
| No, you're confusing two things. The uncertainty in a Fourier
| transform applies whether it's continuous or discrete. It does
| not require sampling. It still appears in the sampled DFT, but
| that's an extra wrinkle. It's a feature of the transform
| itself, not the sampling process.
|
| You're also confusing horizontal and vertical resolution.
| Sampling bit depth sets the maximum possible dynamic range
| resolution of both pre-transformed samples and post-transformed
| frequency components.
|
| The number of samples defines how many frequency components
| there are. The number of bits define how accurate their levels
| are.
|
| The uncertainty trade off is in the number of samples. You can
| do an FFT on multi-second chunks of music. You get superb
| frequency resolution, and it will transform back to the
| original. But you can't use the spectrum to see fine detail in
| individual notes, because the frequency domain view is just a
| bar graph with the same number of samples, and shorter features
| - like individual notes - are smeared out across the entire
| frame.
| a-dub wrote:
| "if i want a good look at big things, i need a big window so i
| can see as much of them as possible, but if i use a big window,
| then i don't know where exactly things are in that big window."
| freework wrote:
| Here is my question to those who understand this "paper":
|
| How does the discovery described in this paper help engineer
| something the world has never seen before?
|
| As an engineer, I'm always looking for some new thing to make.
| What does this paper make more possible to make that was less
| possible to make before?
| quantadev wrote:
| The way I interpreted it they're claiming their mathematical
| approach to relating the wave uncertainty in FFTs to
| uncertainty formula in Quantum Mechanics is a novel one. I
| don't think there's any actual new discoveries however, because
| there's an infinite number of ways to show that all of
| mathematics is internally consistent. However I have great
| respect for all their math, if it's all correct, and it may be
| useful to someone just like when Einstein "found" Lorentz
| formulas and Minkowski space which were done before him and
| ready for him to recognize the pattern that fit into his own
| tinkerings that we now call relativity.
| eigenket wrote:
| Nothing in this paper is actually new. Its a review. In general
| understanding various uncertainty principles is pretty
| foundational in engineering quantum things, for example
| transistors. They're also an essential part of how we
| understand electromagnetic waves from radio through WiFi and
| xrays.
|
| In terms of direct engineering implications I think there are
| essentially none, but this is in the background of a lot of
| important stuff.
| Cadwhisker wrote:
| The clearest, most intuitive explanation of the Fourier
| Uncertainty Principle that I have come across is by Grant
| Sanderson of 3Blue1Brown.
|
| https://www.youtube.com/watch?v=MBnnXbOM5S4
| a-dub wrote:
| as per usual. such beautifully simplifying explainers. true
| mastery.
| pixelpoet wrote:
| *Lebesgue, bottom of page 1. And it's easy to tell where someone
| is from when they refer to dates using seasons :P
| KvanteKat wrote:
| The variable n comes out of nowhere in theorem 3.3, and they do
| not refer to it in the proof itself as far as I can tell. Is this
| just an editing error (I think the formula 3.4 needs the variable
| n if f is multidimensional and we are integrating over R^n, but
| since f is in L^1(R) I'm not sure what it signifies. I am however
| worried that there's something I'm missing).
| eigenket wrote:
| Yeah it's an editing error. Looks like originally they wrote
| the theorem down for R^n and later decided to just stick to R.
| Pinegulf wrote:
| >Uncertainty principles are not formally defined
|
| In physics it seem to be pretty straight forward. For instance
| https://www.wolframalpha.com/input/?i=uncertainty+principle
| diffeomorphism wrote:
| Not really. You defined exactly one, the Heisenberg uncertainty
| principle. There are many, many more.
|
| For instance, a function cannot both be compactly supported and
| have a compactly supported Fourier transform. Or you can have
| other transforms, other operators than position and momentum
| etc.
| ballooney wrote:
| I remember an undergraduate homework question that was just
| asking you to calculate the fourier transform of a gaussian of
| mean 0 and variance a. You get out a gaussian of mean 0 and
| variance 1/a.
|
| I missed the significance of this, until we went over the
| homework with the TA and pointed out the implications of this
| result, heisenberg, etc.
|
| It was very enjoyable that something I had previously taken as a
| sort of spooky truth of the quantum universe (Heisenberg's
| uncertainty principle) was actually just a pretty mechanically
| apparent consequence of some basic algebra on an EE homework.
| Astro-Domine wrote:
| Everything I'm interested in is expressed in a language I don't
| understand. I need to get a grounding in mathematics.
| shannifin wrote:
| At least in some contexts, I never really agreed with calling it
| "uncertainty"; a frequency cannot exist in less time than the
| time needed to measure it. You're not really uncertain about it,
| it does not exist at all. Like looking at a single pixel's color
| and saying you're uncertain about the picture.
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