# Thread: Heisenberg uncertainty

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## Heisenberg uncertainty

I'm trying to understand the HU formula where the product of sigma_p and sigma_x is always greater than or equal to hbar over two. I have some understanding of the meaning of the left hand side of the equation and appreciate p and x are not the only possible complimentary variables, but I totally don't understand why hbar (Planck's reduced constant) over two is the benchmark. Indeed, I neither get why it's hbar, nor why it's over two. Any advice appreciated.

Thanks!

Steve

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Let's just do why it's hbar, the 1/2 is technical and really depends on how one defines the uncertainty in x and p. Also, note there is no explanation for why hbar has the value it does, that's a matter of pure observation. But we can understand why there is such a thing as hbar, and why it shows up in that relation. It all begins with something called "wave/particle duality." You cannot ask why matter exhibits wave/particle duality, because nobody knows that either, but you can understand that given that it does, we're going to have a HUP.

Many confusing things get said about wave/particle duality, like that it means that matter sometimes acts like a wave and sometimes like a particle, depending on what environment it's in or what questions it is answering. I don't think that's the best way to think about it, because we can't feel like we understand nature by making nature herself sound confused. Instead, just recognize something that was always true that people just never really noticed: waves are capable of every type of motion that particles exhibit. So wave/particle duality is not about accepting that waves and particles are two different types of motion and the object sometimes does one and sometimes the other, it is about noticing a type of unification that was always there: all motions we see are motions that waves do. All that differs between "wavelike" and "particlelike" is the size of the wavelength involved-- everything we mean about "wavelike motion" (like doing diffraction and being spread out) corresponds to having a large wavelength, (you get diffraction when the wavelength is not much smaller than the size of the obstacles encountered, and you always have to have at least as much nonlocality as the size of the wavelength), and everything we mean about "particlelike motion" (like following trajectories and well-defined locations) corresponds to having a small wavelength (you don't get much diffraction when the wavelength is much smaller than the obstacles, and if the wavelength is small the wave can be well localized). The "aha" moment is when you realize that all motion has always been the motion of a wave, we just never needed to see that until our measurements were precise enough to see down to the wavelength scale of the particles.

So if all motion is wave, then where does the particle part come in? It appears when you ask "what is moving." The answer to that is always "discrete particles, that's what," we just never realized this before because sometimes there are so many tiny particles involved that all we observe is their collective behavior. But we've been down that road many times before-- such as when people were first trying to understand air, and what is moving when there's a wind. Early philosophers sometimes concluded that air, or water or any matter, was a continuous substance that could move around. Others thought the only thing that could move had to be tiny discrete particles. Then came one of the great ironies of physics: it was discovered that the mathematics of waves could explain the behavior of lots and lots of particles, while still holding that individual particles moved differently from waves. Thus was born the fundamental distinction between wave motion and particle motion, and obscured for centuries the fact that wave motion by itself can do particle motion also, the only thing it cannot do is the requirement that everything be composed of discrete elements (discrete numbers of charge, discrete numbers of elemental mass, discrete energy levels, etc.). So the best way to think of wave/particle duality, I argue, is particle handles "what it is", and wave handles "how it moves."

Now we are at the doorstep of the HUP, because if you want to have waves tell particles how to move, you need to connect the momentum of the particle (the fundamental characteristic of a moving particle) to the appropriate wavelength (the fundamental characteristic of a wave). That connection is called the "deBroglie wavelength," where wavelength times momentum equals h, a constant Planck had encountered with light and deBroglie extended to a fundamental relation for all particles. So the wave/particle
duality that had recently been discovered for light (culminating in Einstein's Nobel prize for the photoelectric effect) was extended to all particles, and wave/particle duality was born.

So we see where h gets into the picture, it's in the connection between momentum and associated wavelength for a particle. How does that give us a HUP? It's a simple consequence of how precisely one can associate a wavelength with a wave. The quintessential wavelength is the distance between peaks in a sine wave, but notice that a sine wave extends over an infinite domain, and that's what lets you be very clear what it's wavelength is-- no matter where you look, you see that same distance between peaks. But if the wave had only, say, 3 peaks, and then trailed off in both directions, you only see that distance over part of the wave-- over the rest of it, you see something different, you see a different way that the wave is changing with distance. So you are uncertain what wavelength means when there are only 3 peaks. To see this more quantitatively, perform what is called "Fourier transform" of the wave-- if it's an infinite sine wave, you get a narrow sharp peak at the exact correct wavelength, but if it only has 3 peaks and then trails off, you get a much broader Fourier transform. The ratio of the spread of the Fourier transform (which we can take as the "uncertainty" in the value of the wavelength) to the central value of the wavelength is characterized by one over the number of peaks in the sine wave before it trails off. Another way to say one over the number of peaks is the ratio of the spread of the wave in physical space, to its wavelength, right?

Now we have all the ingredients of the HUP. You simply say that for any wave, its spread in wavelength space, divided by its wavelength, is about the same as its wavelength divided by its spread in physical space, because that's how Fourier transforms work. Then you tack on the concept of the deBroglie wavelength, to replace the wavelength with h over the momentum. Write out everything I just said as an equation, and it looks like delta lambda / lambda ~ lambda / delta x ~ h / (p * delta x). Since delta lambda / lambda is, by the deBroglie relation, the same thing as delta p / p, this gives us the HUP, to within factors like 4pi that come from the more technical aspects of Fourier transforms and what we are using to quantify the concept of uncertainty. It's all a consequence of wave/particle duality, and the mathematical properties of Fourier transforms-- which physically mean that there is always a connection between how localized a wave is, and the uncertainty in its wavelength.

3. Originally Posted by Ken G
Let's just do why it's hbar, the 1/2 is technical and really depends on how one defines the uncertainty in x and p. Also, note there is no explanation for why hbar has the value it does, that's a matter of pure observation. But we can understand why there is such a thing as hbar, and why it shows up in that relation. It all begins with something called "wave/particle duality." You cannot ask why matter exhibits wave/particle duality, because nobody knows that either, but you can understand that given that it does, we're going to have a HUP.

Many confusing things get said about wave/particle duality, like that it means that matter sometimes acts like a wave and sometimes like a particle, depending on what environment it's in or what questions it is answering. I don't think that's the best way to think about it, because we can't feel like we understand nature by making nature herself sound confused. Instead, just recognize something that was always true that people just never really noticed: waves are capable of every type of motion that particles exhibit. So wave/particle duality is not about accepting that waves and particles are two different types of motion and the object sometimes does one and sometimes the other, it is about noticing a type of unification that was always there: all motions we see are motions that waves do. All that differs between "wavelike" and "particlelike" is the size of the wavelength involved-- everything we mean about "wavelike motion" (like doing diffraction and being spread out) corresponds to having a large wavelength, (you get diffraction when the wavelength is not much smaller than the size of the obstacles encountered, and you always have to have at least as much nonlocality as the size of the wavelength), and everything we mean about "particlelike motion" (like following trajectories and well-defined locations) corresponds to having a small wavelength (you don't get much diffraction when the wavelength is much smaller than the obstacles, and if the wavelength is small the wave can be well localized). The "aha" moment is when you realize that all motion has always been the motion of a wave, we just never needed to see that until our measurements were precise enough to see down to the wavelength scale of the particles.

So if all motion is wave, then where does the particle part come in? It appears when you ask "what is moving." The answer to that is always "discrete particles, that's what," we just never realized this before because sometimes there are so many tiny particles involved that all we observe is their collective behavior. But we've been down that road many times before-- such as when people were first trying to understand air, and what is moving when there's a wind. Early philosophers sometimes concluded that air, or water or any matter, was a continuous substance that could move around. Others thought the only thing that could move had to be tiny discrete particles. Then came one of the great ironies of physics: it was discovered that the mathematics of waves could explain the behavior of lots and lots of particles, while still holding that individual particles moved differently from waves. Thus was born the fundamental distinction between wave motion and particle motion, and obscured for centuries the fact that wave motion by itself can do particle motion also, the only thing it cannot do is the requirement that everything be composed of discrete elements (discrete numbers of charge, discrete numbers of elemental mass, discrete energy levels, etc.). So the best way to think of wave/particle duality, I argue, is particle handles "what it is", and wave handles "how it moves."

Now we are at the doorstep of the HUP, because if you want to have waves tell particles how to move, you need to connect the momentum of the particle (the fundamental characteristic of a moving particle) to the appropriate wavelength (the fundamental characteristic of a wave). That connection is called the "deBroglie wavelength," where wavelength times momentum equals h, a constant Planck had encountered with light and deBroglie extended to a fundamental relation for all particles. So the wave/particle
duality that had recently been discovered for light (culminating in Einstein's Nobel prize for the photoelectric effect) was extended to all particles, and wave/particle duality was born.

So we see where h gets into the picture, it's in the connection between momentum and associated wavelength for a particle. How does that give us a HUP? It's a simple consequence of how precisely one can associate a wavelength with a wave. The quintessential wavelength is the distance between peaks in a sine wave, but notice that a sine wave extends over an infinite domain, and that's what lets you be very clear what it's wavelength is-- no matter where you look, you see that same distance between peaks. But if the wave had only, say, 3 peaks, and then trailed off in both directions, you only see that distance over part of the wave-- over the rest of it, you see something different, you see a different way that the wave is changing with distance. So you are uncertain what wavelength means when there are only 3 peaks. To see this more quantitatively, perform what is called "Fourier transform" of the wave-- if it's an infinite sine wave, you get a narrow sharp peak at the exact correct wavelength, but if it only has 3 peaks and then trails off, you get a much broader Fourier transform. The ratio of the spread of the Fourier transform (which we can take as the "uncertainty" in the value of the wavelength) to the central value of the wavelength is characterized by one over the number of peaks in the sine wave before it trails off. Another way to say one over the number of peaks is the ratio of the spread of the wave in physical space, to its wavelength, right?

Now we have all the ingredients of the HUP. You simply say that for any wave, its spread in wavelength space, divided by its wavelength, is about the same as its wavelength divided by its spread in physical space, because that's how Fourier transforms work. Then you tack on the concept of the deBroglie wavelength, to replace the wavelength with h over the momentum. Write out everything I just said as an equation, and it looks like delta lambda / lambda ~ lambda / delta x ~ h / (p * delta x). Since delta lambda / lambda is, by the deBroglie relation, the same thing as delta p / p, this gives us the HUP, to within factors like 4pi that come from the more technical aspects of Fourier transforms and what we are using to quantify the concept of uncertainty. It's all a consequence of wave/particle duality, and the mathematical properties of Fourier transforms-- which physically mean that there is always a connection between how localized a wave is, and the uncertainty in its wavelength.
what a beautiful explanation, it is a story of discovery and interpretation.

4. Originally Posted by profloater
what a beautiful explanation, it is a story of discovery and interpretation.
No kidding! Thanks, Ken!

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Thank you!

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Originally Posted by Ken G
Let's just do why it's hbar, the 1/2 is technical and really depends on how one defines the uncertainty in x and p. Also, note there is no explanation for why hbar has the value it does, that's a matter of pure observation. But we can understand why there is such a thing as hbar, and why it shows up in that relation. It all begins with something called "wave/particle duality." You cannot ask why matter exhibits wave/particle duality, because nobody knows that either, but you can understand that given that it does, we're going to have a HUP.

Many confusing things get said about wave/particle duality, like that it means that matter sometimes acts like a wave and sometimes like a particle, depending on what environment it's in or what questions it is answering. I don't think that's the best way to think about it, because we can't feel like we understand nature by making nature herself sound confused. Instead, just recognize something that was always true that people just never really noticed: waves are capable of every type of motion that particles exhibit. So wave/particle duality is not about accepting that waves and particles are two different types of motion and the object sometimes does one and sometimes the other, it is about noticing a type of unification that was always there: all motions we see are motions that waves do. All that differs between "wavelike" and "particlelike" is the size of the wavelength involved-- everything we mean about "wavelike motion" (like doing diffraction and being spread out) corresponds to having a large wavelength, (you get diffraction when the wavelength is not much smaller than the size of the obstacles encountered, and you always have to have at least as much nonlocality as the size of the wavelength), and everything we mean about "particlelike motion" (like following trajectories and well-defined locations) corresponds to having a small wavelength (you don't get much diffraction when the wavelength is much smaller than the obstacles, and if the wavelength is small the wave can be well localized). The "aha" moment is when you realize that all motion has always been the motion of a wave, we just never needed to see that until our measurements were precise enough to see down to the wavelength scale of the particles.

So if all motion is wave, then where does the particle part come in? It appears when you ask "what is moving." The answer to that is always "discrete particles, that's what," we just never realized this before because sometimes there are so many tiny particles involved that all we observe is their collective behavior. But we've been down that road many times before-- such as when people were first trying to understand air, and what is moving when there's a wind. Early philosophers sometimes concluded that air, or water or any matter, was a continuous substance that could move around. Others thought the only thing that could move had to be tiny discrete particles. Then came one of the great ironies of physics: it was discovered that the mathematics of waves could explain the behavior of lots and lots of particles, while still holding that individual particles moved differently from waves. Thus was born the fundamental distinction between wave motion and particle motion, and obscured for centuries the fact that wave motion by itself can do particle motion also, the only thing it cannot do is the requirement that everything be composed of discrete elements (discrete numbers of charge, discrete numbers of elemental mass, discrete energy levels, etc.). So the best way to think of wave/particle duality, I argue, is particle handles "what it is", and wave handles "how it moves."

Now we are at the doorstep of the HUP, because if you want to have waves tell particles how to move, you need to connect the momentum of the particle (the fundamental characteristic of a moving particle) to the appropriate wavelength (the fundamental characteristic of a wave). That connection is called the "deBroglie wavelength," where wavelength times momentum equals h, a constant Planck had encountered with light and deBroglie extended to a fundamental relation for all particles. So the wave/particle
duality that had recently been discovered for light (culminating in Einstein's Nobel prize for the photoelectric effect) was extended to all particles, and wave/particle duality was born.

So we see where h gets into the picture, it's in the connection between momentum and associated wavelength for a particle. How does that give us a HUP? It's a simple consequence of how precisely one can associate a wavelength with a wave. The quintessential wavelength is the distance between peaks in a sine wave, but notice that a sine wave extends over an infinite domain, and that's what lets you be very clear what it's wavelength is-- no matter where you look, you see that same distance between peaks. But if the wave had only, say, 3 peaks, and then trailed off in both directions, you only see that distance over part of the wave-- over the rest of it, you see something different, you see a different way that the wave is changing with distance. So you are uncertain what wavelength means when there are only 3 peaks. To see this more quantitatively, perform what is called "Fourier transform" of the wave-- if it's an infinite sine wave, you get a narrow sharp peak at the exact correct wavelength, but if it only has 3 peaks and then trails off, you get a much broader Fourier transform. The ratio of the spread of the Fourier transform (which we can take as the "uncertainty" in the value of the wavelength) to the central value of the wavelength is characterized by one over the number of peaks in the sine wave before it trails off. Another way to say one over the number of peaks is the ratio of the spread of the wave in physical space, to its wavelength, right?

Now we have all the ingredients of the HUP. You simply say that for any wave, its spread in wavelength space, divided by its wavelength, is about the same as its wavelength divided by its spread in physical space, because that's how Fourier transforms work. Then you tack on the concept of the deBroglie wavelength, to replace the wavelength with h over the momentum. Write out everything I just said as an equation, and it looks like delta lambda / lambda ~ lambda / delta x ~ h / (p * delta x). Since delta lambda / lambda is, by the deBroglie relation, the same thing as delta p / p, this gives us the HUP, to within factors like 4pi that come from the more technical aspects of Fourier transforms and what we are using to quantify the concept of uncertainty. It's all a consequence of wave/particle duality, and the mathematical properties of Fourier transforms-- which physically mean that there is always a connection between how localized a wave is, and the uncertainty in its wavelength.
Thanks - an elegant explanation.

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Originally Posted by Ken G
But if the wave had only, say, 3 peaks, and then trailed off in
both directions, you only see that distance over part of the wave--
over the rest of it, you see something different, you see a different
way that the wave is changing with distance. So you are uncertain
what wavelength means when there are only 3 peaks.
Does this description need to be modified at all when just
three peaks are measured, and nothing before or after is
detected? Does Fourier analysis give the same result?

-- Jeff, in Minneapolis

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Originally Posted by Jeff Root
Does this description need to be modified at all when just
three peaks are measured, and nothing before or after is
detected? Does Fourier analysis give the same result?
Although it is possible to use clever "weak measurements" to get correlations between observations that look a lot like a wavefunction, in general you don't actually measure or detect the wave function itself, you infer it based on what you know about the "preparation" of the system. So you can think of it as a mathematical tool for quantifying your information about the preparation of a system. Even if you use clever weak measurements to get the wavefunction as a measurable output, you still don't know the system "really is" in that state, in the sense that there can still be aspects of that system's history that you don't know about and are not taking into account. So I think it's best to think of a wavefunction as a quantification of the information you have about a system, and it will give you statistical results consistent with that information. Someone else with different information about the exact same system can use a different wavefunction and also make successful statistical predictions. Information itself is classical, so you can think of people playing cards who only see their own hands-- if they are expert players they can all be making successful statistical predictions about how the game might play out, but their knowledge and predictions are different from each other. The main difference quantum mechanically is that it seems to be impossible to "see the entire deck"-- the wave nature of the deck implies fundamental constraints like the Fourier transform issues.

So once you infer the presence of a wavefunction, that's what you apply the Fourier transform to, and generate a concept of uncertainty consistent with the HUP. Different people could have different knowledge of the exact same system, and use different wavefunctions, and get perfectly correct predictions-- statistically. But to test that uncertainty, you need an ensemble of many particles that you have the same information about. It's that information that gives you the wavefunction, just like looking at your own hand gives you the information about a given game of cards you are playing. What makes it possible for different predictions to all be correct is that the tests of those predictions will involve binning the experimental trials differently-- you bin all the outcomes based on similar information going in, so people with different information will bin the experimental trials differently. Everyone's statistical expectations check out, if they make no errors.

So to answer your question, if we have a huge ensemble of systems, and one person has incomplete information about them, they will bin all the cases with three peaks in a given incomplete window, and get the spread of outcomes their Fourier transform predicts. If someone else can see a wider window, then some of the data the first person was binning as "three peak wavefunctions", they might bin as "six peak wavefunctions", and get a narrower spread of outcomes for that subset of the data. We get out what is consistent with the information that is available to us. This is why there is the debate in quantum mechanics about whether the wavefunction is "the information that nature has", or if it contains "hidden variables" (Bohm's approach) that represent information we are simply not privy to. Then there's the "many worlds" approach, which says all the information is out there, but we only inhabit a tiny subset that our perceptions have access to. So we don't know if the HUP is just a limit on what we can know, or a limit on what is, because science is never about what is, it is always about what we can know (and therefore test) about what is. We never get to know the price we pay for requiring that we can test it.
Last edited by Ken G; 2019-Feb-10 at 03:27 PM.

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This answer is much more complex than I expected.

You introduced several new terms that did not previously
appear in the thread:

- wavefunction
- system
- preparation of a system
- information about the preparation of a system
- history of a system
- ensemble of systems
- predictions

It isn't apparent why you needed to introduce these ideas
to answer my question.

I don't know what a wavefunction is. I belive I have a
pretty good understanding of what waves are and how they
behave. Likewise I think I understand what mathematical
functions are, what they do, and how they can be used.

But I suspect that a wavefunction is neither a wave nor a
mathematical function. And I'm skeptical that you can
explain it in a way that I could understand.

From what I've seen elsewhere, my impression is that a
wavefunction is a description of something physical, in
terms of quantum-mechanical quantities. Analogous to how
a cardboard box could be partially described in terms of
the physical quantities of height, width, and depth.

Except that a wavefunction seems to be considered a
complete description, somehow-- not partial. So either the
wavefunction needs to describe something very, very simple,
or the wavefunction needs to be very, very complex, which
does not at all fit my understanding of what a mathematical
function is. Yet wavefunctions apparently are supposed to
be able to describe anything and everything. No limits on
complexity.

I don't see how any of that helps to answer my question.

If my question involved a system, I do not know what system
that is. I asked about your example of measuring a wave in
which only three peaks are measured. You introduced the wave
as being a sine wave, but I presume that if only three peaks
are measured, the actual waveform is unknown, but Fourier
analysis compares it to a sine wave and describes it as an
approximation of the wave as if it were a sine wave.

You described the measurement as trailing off before and after
the part of the wave that is measured. The only change I made
was to assert nothing about anything before or after the three
peaks. The wave might be infinite in length, or the three
peaks that are measured might be the entire wave. Since you
specified peaks rather than null points, this wave would begin
with an instantaneous rise and end with an instantaneous fall
in amplitude. My impression is that Fourier analysis doesn't
handle that, or doesn't handle it well, but I don't know.

My question is essentially, if nothing is known about the
wave before or after the section that is measured, including
whether or not the wave extends beyond the region measured,
then in the example case you suggested in which only three
peaks are measured, does Fourier analysis give the same
result as when the wave "trails off" in both directions?

What happens when you analyze a portion of a sound wave of
constant pitch, versus a portion of a sound wave of varying
pitch, when there is no variation over the length of the
sample because the sample is so short? I would expect the
results of the analyses to be identical. But if Fourier
analysis really does assume a sine wave of infinite length,
then that seems questionable.

You also introduced the idea of predictions. What is being
predicted in the scenario I described? I thought I was just
describing a measurement, not a prediction.

-- Jeff, in Minneapolis

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Originally Posted by Jeff Root
I asked about your example of measuring a wave in
which only three peaks are measured.
But it is very rare that anyone ever "measures" a wavefunction. Instead, you normally measure some observable. If you measure pressure in a sound wave, that's a classical measurement of many many quantum systems, it's not what is meant by a quantum measurement to which we would be applying the HUP. However, the way Fourier transforms appear is very analogous, so that's why Fourier transforms are insightful to think about.
You introduced the wave
as being a sine wave, but I presume that if only three peaks
are measured, the actual waveform is unknown, but Fourier
analysis compares it to a sine wave and describes it as an
approximation of the wave as if it were a sine wave.
I wouldn't say "three peaks are measured," I would say the mathematical abstraction that is the wavefunction would have three peaks. That mathematical abstraction incorporates all the information that some physicist has about some system, and they arrive at that abstraction via their knowledge of the history of the system. For example, you could prepare a particle in a state of definite momentum, and then the "wavefunction" expressed in terms of position measurements (which means, expanded on a "position basis"-- you don't really need this jargon, it's just there to show you that a lot of choices are being made behind using this language) would be a plane wave over position (as we might often picture a wave). Then if the preparation also includes a weak position measurement that only tells us the particles are not outside some bounded region that includes three peaks of the original plane wave, that's when we'd say the wavefunction has only three peaks. Maybe a bright light was shined outside that region, and only particles that did not reflect any of that light are included in the ensemble in question. Now you have a wavefunction with three peaks, and you can Fourier transform it to learn about the possible momenta in the particles in that ensemble, and you will find the HUP at play-- the uncertainty in the position that comes from knowing only that the particle is within that region corresponds to the spread in momenta you can get from doing subsequent momentum measurements on that ensemble. Notice that at no time are you ever measuring a wavefunction, it's always just a mathematical tool that is in your mind. (But there are clever ways of using weak measurements that will give you the wavefunction as an output of your measuring device, it's just not the usual way we think about a wavefunction.)
You described the measurement as trailing off before and after
the part of the wave that is measured. The only change I made
was to assert nothing about anything before or after the three
peaks. The wave might be infinite in length, or the three
peaks that are measured might be the entire wave.
The point is, you have to have some prior knowledge of the system, or else you cannot even say you have a wavefunction there at all (or, you have no idea what wavefunction to put in your head). Then you do measurements, with apply additional constraints, and tell you how to modify that wavefunction. If you ever get a definite result of a measurement (say, a definite momentum), then you have a specific wavefunction that goes along with that definite outcome, but often you don't have definite outcomes (as in the example I gave above), and then you need to know the prior wavefunction to know how to change it based on the new information. So that's the kind of additional information needed to decide the answer to your question as to what is meant by "three peaks are measured."
Since you
specified peaks rather than null points, this wave would begin
with an instantaneous rise and end with an instantaneous fall
in amplitude.
Yes, that would be like the example I just gave, with the bright light outside some region.
My impression is that Fourier analysis doesn't
handle that, or doesn't handle it well, but I don't know.
Fourier transforms work fine there, as then the integrals are over finite regions.
My question is essentially, if nothing is known about the
wave before or after the section that is measured, including
whether or not the wave extends beyond the region measured,
then in the example case you suggested in which only three
peaks are measured, does Fourier analysis give the same
result as when the wave "trails off" in both directions?
You have to use the wavefunction (about position) that you are applying the Fourier transform to (to understand the momentum implications) as a means of describing your current knowledge of the system. If you know nothing, you have to decide what that means for the wavefunction. In a situation like that, you would have to combine essentially classical information (like a person playing a card game) with quantum information (which you could put in a single wavefunction). You'd end up with some weird combination of quantum and classical forms of information, the situation is not the simplest case we'd like to apply quantum mechanics to. But you can think about what it would mean to be playing your favorite card game, and knowing nothing other than the cards in your own hand. What expectations do you have about the other players? That all goes into your predictions for that game and how you should play it. You have to know something, like that they aren't cheaters, or that there isn't some cards missing from the deck, but if you make the normal assumptions you just average with equal weight over every possible other configuration. You could tack on that kind of thinking on top of a quantum mechanical expectation as well, it would work just the same as the card game after you are done the quantum part.

Technically, when you mix classical and quantum information, you no longer call it a "wavefunction," you call it a "mixed state." So it's not that different information leads to different wavefunctions for the same ensemble, it is that different information causes you to bundle or bin the ensemble data differently, and the information you do not possess you analyze classically and generate a different "mixed state" from what someone else is using, but still get correct statistical predictions for that same ensemble. Mixing the classical information we already know how to do, it's like playing cards, where experts can play each other and form different expectations because they have different information, not because any are in error. A way that different information can lead to different wavefunctions without any classical mixing is when you have entanglement in complicated systems, but that's getting quite far afield to the issue.
What happens when you analyze a portion of a sound wave of
constant pitch, versus a portion of a sound wave of varying
pitch, when there is no variation over the length of the
sample because the sample is so short?
That's an example of mixing classical knowledge with the Fourier analysis part. You'd have to know something about your expectations outside the region you have analyzed or else you can't get anywhere. Do you treat what is outside what you've seen as "white noise", or do you expect that if the part you looked at had a tight frequency, then so should the rest of it because it's all the same source? This requires additional knowledge about the situation. My point is only that the job of physics is to tell you what to do with the information you have, not to tell you information that you don't have and have no access to. Nothing can do that.

You also introduced the idea of predictions. What is being
predicted in the scenario I described? I thought I was just
describing a measurement, not a prediction.
The HUP is about uncertainties in predicted outcomes, so it is about the range of outcomes you expect to see if you do the same experiment over and over on a particular similarly prepared ensemble. This aspect is classical, like buying stocks and analyzing the outcomes you expect. What would you mean by the uncertainty in a stock price? That's classical information, the quantum mechanics is over by then. At the end of the day, all of physics is about making testable predictions, then testing them.
Last edited by Ken G; 2019-Feb-11 at 06:37 PM.

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