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 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.
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.
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.
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.
>"The most popular use of Fourier uncertainty principles is as a description of the natural tradeoff between the stability and measurability of a system"
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.
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.
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 ℏ -> 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
ℏ 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 ℏ 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(2πihx·ξ), where ξ 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.
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.
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:
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
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).
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.
"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."
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.
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.
I've had this feeling before. Even now I read that doc and feel I need to study it for some time.
I think that's actually the point of dense math formulas/papers like this, but I want to share a resource that helped me start from "ground zero" per-se.
Starting with Mathematical Thinking [1], and adding in practice books for Algebra [2] and Calculus [3] to grok what the different parts of the formulas are trying to capture.
Once I did some basic problems, I found the what and why became much clearer. At this point I tend to read it more as programming code than as archaic formulae.
Thank you for taking the time to write this.
I have sourced the two books off eBay and will start the coursera course.
I’ve just finished Robert Pool and Anders Ericsson’s book “Peak” - which has convinced me to stop comparing my unpractised, lack of understanding to the practiced expertise of others.
So the two practice books you recommended have come at a time where I am especially receptive to the idea!
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
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.
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.
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
> In quantum mechanics, the wave function of position is the Fourier transform of the wave function of momentum.
That explains it!