The Heisenberg Uncertainty Principle Is Pure Mathematics
cantorsparadise.com
cantorsparadise.com
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?
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.
And if the formula only holds for photons, why can we say that frequncy = constant * momentum for other particles?
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.
[1] Siebert, William McC.. Circuits, Signals, and Systems. MIT Press, 1986.
https://www.radartutorial.eu/01.basics/Doppler%20Dilemma.en....
[2] Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory.
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.
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?
?
Well, maybe not "pure", but HUP isn't particularly "pure" either.
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.)
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 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.
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.
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.