The Heisenberg Uncertainty Principle Is Pure Mathematics(cantorsparadise.com)
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The Heisenberg Uncertainty Principle Is Pure Mathematics
https://www.cantorsparadise.com/heisenbergs-uncertainty-principle-is-pure-mathematics-e14c840659e1
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The Fourier transform of the position wave for a particle yields the space frequency wave of the particle in the same way that the Fourier transform of a picture gives you the spatial frequency of that picture.
We can then relate the spatial frequency of a photon to its momentum by the formula p = h * f / c, where h and c are the Planck constant and the speed of light in a vacuum, respectively. From this we see that the momentum of a photon is a function of frequency, which, from the properties of the Fourier transform, we know to be the conjugate pair of position.
We can then relate the spatial frequency of a photon to its momentum by the formula p = h * f / c, where h and c are the Planck constant and the speed of light in a vacuum, respectively. From this we see that the momentum of a photon is a function of frequency, which, from the properties of the Fourier transform, we know to be the conjugate pair of position.
First part makes sense to me. But second part doesn't. I remember learning and successfully using the p=h* f/c formula in high school physics, but what is the justification for this?
And if the formula only holds for photons, why can we say that frequncy = constant * momentum for other particles?
And if the formula only holds for photons, why can we say that frequncy = constant * momentum for other particles?
> Why is the fourier transform of position equal to momentum?
That comes out of the way you get the probability distribution of the position and the probability distribution of the momentum out of the wave function.
If you do the math, then one turns out to be the fourier transformation of the either. (And of course the article skipped over this, because it needs quite a bit of math).
Of course this then begs the question why there are wave functions, and why you get the position and momentum from the wave function in that particular way. And I guess the only answer I have to that is "that's how quantum mechanics works, and it matches what we observe".
> More generally, why would the fourier transform of an observable be another observable?
Again, that comes out of how you calculate particular observables from the wave function. Like above, if you do this for the energy probability distribution and the time probability distribution, they turn out to be related.
And it's not the case for any observables you can compute. For example, the energy and position are NOT fourier transforms of each other.
That comes out of the way you get the probability distribution of the position and the probability distribution of the momentum out of the wave function.
If you do the math, then one turns out to be the fourier transformation of the either. (And of course the article skipped over this, because it needs quite a bit of math).
Of course this then begs the question why there are wave functions, and why you get the position and momentum from the wave function in that particular way. And I guess the only answer I have to that is "that's how quantum mechanics works, and it matches what we observe".
> More generally, why would the fourier transform of an observable be another observable?
Again, that comes out of how you calculate particular observables from the wave function. Like above, if you do this for the energy probability distribution and the time probability distribution, they turn out to be related.
And it's not the case for any observables you can compute. For example, the energy and position are NOT fourier transforms of each other.
Energy and time are conjugate variables, not energy and position. And as we would expect, energy is the fourier transform of phase. This is why E=hf. Frequency just measures the rate of phase change akin to how momentum measures the rate of position change. As far as current knowledge goes, phase is not an observable that we can measure. Phase shift is actually a gauge symmetry in electromagnetism.
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This is an incredible explanation! I'd love a deeper explanation of the uncertainty principle applied to the Doppler effect for radar that is mentioned in passing.
Here's a decent explanation:
https://www.radartutorial.eu/01.basics/Doppler%20Dilemma.en....
https://www.radartutorial.eu/01.basics/Doppler%20Dilemma.en....
Cool. I confess I don't have this 100% straight, but it must be related to the CRB for range estimation. For time (equivalent to range) tau, [2, example 3.13] gives it as var(tau) > 1/(Eb/N0 * MSB), where MSB is the mean-square bandwidth of the pulse.
[2] Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory.
[2] Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory.
Thank you!
Isn’t this being taught in quantum mechanics and Fourier transform courses at the advanced undergraduate level nowadays? The basic idea is discussed in Wikipedia pages for both Uncertainty Principle and Fourier Analysis. Since Fourier Analysis is so fundamental to Information Technology, I assumed FA would be covered in the first year or two, so this sort of discussion would pop up in a third year or fourth year QM or signals processing course.
It’s much more insightful than the uncertainty principle discussions in the Paleolithic era when I went to school, albeit the first FA derivation of the uncertainty principle dates back to the 1950s and 1960s.
It’s much more insightful than the uncertainty principle discussions in the Paleolithic era when I went to school, albeit the first FA derivation of the uncertainty principle dates back to the 1950s and 1960s.
That is, it's a property of waves.
>"Waves
In the end, it all comes down to something pretty simple. All types of signals or functions, no matter how complicated, are really superpositions of sine waves. That is pure waves that have a well-defined constant wavelength and amplitude.
A superposition simply means that all the waves interact and the sum of all the waves (called interference) is then the superposition that makes up the more complicated signal."
I agree!
But, here's a related question...
All waves require some medium of oscillation/vibration.
In Classical Physics, you might think of this as a rope which has tension which is tied between two poles, or an instrument string, tied between two points on the instrument, and also under tension...
Well... here's my question then...
Let's suppose that
a) That the rope or string that connect the two points under tension -- is so strong that it cannot break under any circumstances
And that
b) More and more (increasing) tension is applied to the rope or string between the two points -- until the tension becomes INFINITE... or close to it...
Now, here's the question...
When the medium of the transmission of a wave (or wave packet, i.e., superposition of waves) reaches (or approaches) INFINITE tension -- then what happens?
That's question #1.
Question #2...
How does approaching INFINITE tension in a medium -- affect the SPEED of waves using that medium to travel in?
Do they move faster, slower, or at the same speed they always moved at? And WHY?
?
I agree!
But, here's a related question...
All waves require some medium of oscillation/vibration.
In Classical Physics, you might think of this as a rope which has tension which is tied between two poles, or an instrument string, tied between two points on the instrument, and also under tension...
Well... here's my question then...
Let's suppose that
a) That the rope or string that connect the two points under tension -- is so strong that it cannot break under any circumstances
And that
b) More and more (increasing) tension is applied to the rope or string between the two points -- until the tension becomes INFINITE... or close to it...
Now, here's the question...
When the medium of the transmission of a wave (or wave packet, i.e., superposition of waves) reaches (or approaches) INFINITE tension -- then what happens?
That's question #1.
Question #2...
How does approaching INFINITE tension in a medium -- affect the SPEED of waves using that medium to travel in?
Do they move faster, slower, or at the same speed they always moved at? And WHY?
?
Uncertain about what is being added here … a straw man seemed to be set up and hit … It is not surprising one cannot take any pop science description of science too seriously … like the so-called weird nature of a quantum thing “being at two places at the same time” is not accurate and misleading …
The article seems to promise to explain the fundamental uncertainty of physics in simple terms, but it accepts axiomatically that physicists describe physical properties with probability distributions. Too bad.
The heisenberg uncertainty principle is a different type of uncertainty than the uncertainty modeled by probability distributions.
The assertion “Is Pure Mathematics” is wrong and misleading …
How about “a logical consequence of the wave nature…”? (Personally, I make no distinction between “logical” and “mathematical.”)
Isn't all of theoretical physics pure mathematics? That's the point.
Well, maybe not "pure", but HUP isn't particularly "pure" either.
Well, maybe not "pure", but HUP isn't particularly "pure" either.
What the article means is that the Heisenberg Uncertainty Principle isn't just some problem of measurement that could possibly be overcome by better tools.
Rather, HUP is a mathematical consequence of the axioms of quantum mechanics, and you won't get around it by clever technology. Only blowing up QM entirely would get around HUP -- and the results would almost certainly be less familiar, rather than more like classical mechanics. (Indeed, this is already the case: it's called Quantum Field Theory.)
Rather, HUP is a mathematical consequence of the axioms of quantum mechanics, and you won't get around it by clever technology. Only blowing up QM entirely would get around HUP -- and the results would almost certainly be less familiar, rather than more like classical mechanics. (Indeed, this is already the case: it's called Quantum Field Theory.)
hey, since you might be familiar with this: so quantum field theory is actually a departure from QM instead of its continuity ? I'm confused, i thought QM was the best thing we had regarding quantum behaviors...
Also, i sometimes saw theoretical physicists talk about "quantum gravity" as if it was a solid theory (i tried to look at some presentation on the topic, but it was way beyond my level)... That made me even more confused, because i was under the impression that chord theory was the only thing we had to unify quantum mechanic and general relativity, and that it was far from accomplished...
Could you (or anyone reading this) help me out of my confusion ?
Also, i sometimes saw theoretical physicists talk about "quantum gravity" as if it was a solid theory (i tried to look at some presentation on the topic, but it was way beyond my level)... That made me even more confused, because i was under the impression that chord theory was the only thing we had to unify quantum mechanic and general relativity, and that it was far from accomplished...
Could you (or anyone reading this) help me out of my confusion ?
Quantum mechanics is the low-energy case of quantum field theory. They have the same kind of relationship as Newtonian mechanics have to general relativity: a very, very different basis that just happens to have a convenient low-energy formulation, where the differences between truth and experiment are too small to be observed.
Quantum mechanics can't even begin to talk about quarks. The weak and strong forces just don't exist. And they're very, very different kinds of forces from gravity and electromagnetism. It requires a completely different kind of basis, one based on fields rather than particles.
As for "quantum gravity", there's a lot to unpack. There are very sound and useful theories of Quantum Field Theory in curved spacetime, such as semiclassical gravity. That's what Stephen Hawking was working on when he produced his black hole results.
But those theories have limitations and aren't a full integration of quantum mechanics with gravity. Unfortunately, that seems to require yet another complete rewrite of the fundamental basis of things, such as strings or loop quantum gravity. And more unfortunately, the differences only occur at stupid high energy levels, so it's practically impossible for us to do experiments to help figure out which one is right.
Quantum mechanics can't even begin to talk about quarks. The weak and strong forces just don't exist. And they're very, very different kinds of forces from gravity and electromagnetism. It requires a completely different kind of basis, one based on fields rather than particles.
As for "quantum gravity", there's a lot to unpack. There are very sound and useful theories of Quantum Field Theory in curved spacetime, such as semiclassical gravity. That's what Stephen Hawking was working on when he produced his black hole results.
But those theories have limitations and aren't a full integration of quantum mechanics with gravity. Unfortunately, that seems to require yet another complete rewrite of the fundamental basis of things, such as strings or loop quantum gravity. And more unfortunately, the differences only occur at stupid high energy levels, so it's practically impossible for us to do experiments to help figure out which one is right.
QM alone fails to account for certain facts. QFT does this by using somewhat “weird” methods in order to bring Special Relativity and QM together. It has been working pretty well so far, and it is indeed the best we have. String Theory, on the other hand, while being a consistent theory of quantum gravity, uses a model (or models) that we may never be able to verify experimentally.
HUP is may be a physical phenomena, but the same kind of uncertainty is part of Fourier transforms which are entirely mathematical constructs.
So even more intriguing than HUP, is the thing that purely mathematical constructs exibit the same sort of uncertainty as HUP.
Gibbs phenomenon was initially thought by physicits to be due to defects in instruments making the measurements, but later it was unferstood to be a mathematical rather than physical characteristic.
So even more intriguing than HUP, is the thing that purely mathematical constructs exibit the same sort of uncertainty as HUP.
Gibbs phenomenon was initially thought by physicits to be due to defects in instruments making the measurements, but later it was unferstood to be a mathematical rather than physical characteristic.
The difference between theoretical physics and mathematics is 1) the subject and 2) the criterion of truth.
We must have "The Talk".
https://www.smbc-comics.com/comic/the-talk-3
https://www.smbc-comics.com/comic/the-talk-3
Note that interestingly, momentum and position in orthogonal directions do commute and don't have uncertainty.
You can know both a particle position in the x direction and it's momentum in the y direction with 100% certainty. Some intuitive based explanations would give you the wrong impression here.
Another way of looking at it is that momentum operator is equivalent to derivation, and derivation and multiplying by x don't commute: d/dx x - x d/dx = 1. So, depending on the order in which you measured (first momentum then position, or first position then momentum) you get a different result.
You can know both a particle position in the x direction and it's momentum in the y direction with 100% certainty. Some intuitive based explanations would give you the wrong impression here.
Another way of looking at it is that momentum operator is equivalent to derivation, and derivation and multiplying by x don't commute: d/dx x - x d/dx = 1. So, depending on the order in which you measured (first momentum then position, or first position then momentum) you get a different result.
Why is the fourier transform of position equal to momentum?
I.e why is position conjugate to momentum?
More generally, why would the fourier transform of an observable be another observable?