You've probably heard of the Heisenberg uncertainty principle from quantum mechanics
that the more you know about a particles position the less certain you can
be sure of its momentum and vice-versa
My goal here is for you to come away from this video feeling like this is utterly reasonable
It'll take some time
But I think you'll agree that digging deep is well worth it
You see the uncertainty principle is actually one specific example of a much more general trade-off that shows up in a lot of everyday
totally non-quantum circumstances involving waves
The plan here is to see what this means in the context of sound waves which should feel reasonable, and then Doppler radar
which should again feel reasonable and a little bit closer to the quantum case and
Then for particles, which if you're willing to accept one or two premises of quantum mechanics. Hopefully feels just as reasonable as the first two
The core idea here has to do with the interplay between
frequency and duration
And I bet you already have an intuitive idea of this principle before we even get into the math or the quantum
If you were to pull up behind a car at a red light and your turn signals were flashing together for a few seconds
You might kind of think that they have the same frequency
But at that point for all you know they could fall out of sync as more time passes
revealing that they actually had different frequencies, so
An observation over a short period of time gave you low confidence over what their frequencies are
But if you were to sit at that red light for a full minute and the signals continued to click in sync
You would be a lot more confident that the frequencies are actually the same so that certainty about the frequency information
required an observation spread out over time and
And this trade-off right here between how short your observation can be and how confident you can feel about the frequency
Is an example of the general uncertainty principle.
Similarly think of a musical note the shorter it lasts in time the less certain you can be about what its exact frequency is.
In the extreme I could ask you what the pitch of a clap or a shock wave is and
Even someone with perfect pitch would be unable to answer and on the flip side a more definite frequency
requires a longer duration signal or
Rather than talking about definiteness or certainty it would be little more accurate here to say that the short signal
correlates highly with a wider range of frequency
And at the signal correlating strongly with only a narrow range of frequencies must last for a longer time
Here that's the kind of phrase that's made a little bit clearer when we bring in the actual math
So let's turn now to talking about the Fourier transform, which is the relevant construct for analyzing frequencies the last video?
I put out was a visual intuition for this transform, and yes it probably would be helpful if you've seen it
But I'm gonna go ahead and give a quick recap here. Just to remind ourselves of how it went
Let's say you have a signal and it plays five beats per second over the course of two seconds
The Fourier transform gives a way to view any signal not in terms of the intensity at each point in time
But instead in terms of the strength of various frequencies within it
the main idea was to take this signal and to kind of wind it around a circle as
In imagine some rotating vector whose length is determined by the height of the graph at each point in time
Right now this little vector is rotating at 0.3 cycles per second
that's the frequency with which were winding the graph around the circle and
For most frequencies the signal is kind of just averaged out over the circle
This was the fun part of last video
Don't you think just seeing the different patterns that come up as you wind a pure cosine around a circle like this?
But the key point is what happens when that winding frequency matches the signal frequency in this case five cycles per second as
Our little vector is rotating around and it draws all of the peaks line up on one side and all of the valleys on another
Side so the whole weight of the graph is kind of off-center so to speak
The idea behind the Fourier transform, is that if you follow the center of mass of the wound-up graph with frequency F
the position of that center of mass
Encodes the strength of that frequency in the original signal
the distance between that center of mass and the origin captures the strength of that frequency and
This is something I didn't really talk about in the main video
But the angle of that center of mass the horizontal corresponds to the phase of the given frequency
Now one way to think of this whole winding mechanism
Is that it's a way to measure how well your signal correlates with a given pure frequency?
So remember when we say the Fourier transform were referring to this new function whose
Input is that winding frequency and whose output is the center of mass thought of as a complex number or more technically
It's a certain multiple of that center of mass
but whatever the overall shape remains the same and the graph that I'm drawing is just going to be the
x-coordinate of that center of mass the real part of its output
If you wanted you could also plot the distance between the center of mass and the origin and maybe that better conveys how strongly each
Possible frequency correlates with the signal the downside is that you lose some of the nice linearity properties that I talked about last video
anyway point is this spike you're looking at here above the winding frequency of five is the Fourier transforms way of telling us that the
dominant frequency of the signal is five beats per second and
equally importantly the fact that it's a little bit spread out around that five is an indication that pure sine waves
near five beats per second also correlate pretty well with the signal
And that last idea is key for the uncertainty principle what I want you to do is think about how this spread
Changes as the signal persists longer or shorter over time
You've already seen this at an intuitive level all we're doing right now is just illustrating it in the language of Fourier transforms
if the signal persists over a long period of time
Then when the winding frequency is even slightly different from five
The signal goes on long enough to wrap itself around the circle and balance out
So looking at the Fourier plot over here
That corresponds to a super sharp drop-off in the magnitude of the transform as your frequency shifts away from that five beats per second
On
The other hand if your signal was really localized to a short period of time
Then as you adjust the frequency away from five beats per second
The signal doesn't really have as much time to balance itself out around the circle you have to change the winding frequency to be
Meaningfully different from five before that signal starts to balance out again
over on the frequency plot that corresponds to a much broader peak around the five beats per second and
that's the uncertainty principle just phrased a little bit more mathematically a
signal concentrated in time must have a spread out fornia transform meaning it correlates with a wide range of frequencies and a signal with a
concentrated Fourier transform has to be spread out in time and
One other place where this comes up in a really tangible way is Doppler radar
so with radar the idea is you send out some radio wave pulse and the pulse might reflect off of
objects and the time that it takes for this echo signal to return to you lets you deduce how far away those objects are and
You can actually take this one step further and make deductions about the velocities of those objects using the Doppler effect
Think about sending out a pulse with some frequency if this gets reflected off an object
Moving towards you then the beats of that wave get kind of smushed together
So the echo you hear back is going to be a slightly higher frequency
Fourier transforms give a neat way to view this the Fourier transform of your original signal tells you the frequencies that go into it and
For simplicity let's think of that as being
dominated by a single pure frequency
Though as you know if it's a short pulse that means that our Fourier transform has to be spread out a little bit
And now think about the Doppler shifted echo
By coming back at a higher frequency
It means that the Fourier transform will just look like a similar plot shifted up a bit
Moreover if you look at the size of that shift you can deduce how quickly the object was moving
By the way, there is an important technical point that I'm choosing to gloss over here
And I've outlined it a little more in the video description
What follows is meant to be a distilled if somewhat oversimplified description of the Fourier trade-off in this setta?
The ceiling in fact is that time and frequency of that echo signal?
Correspond respectively to the position and the velocity of the object being measure
Which is what makes this example much more closely analogous to the quantum mechanical Heisenberg uncertainty principle
You see there is a very real way in which a radar operator
Faces a dilemma where the more certain you can be about the positions of things the less certain you are about their velocities
Here imagine sending out a pulse that persists over a long period of time
Then that means the echo from some object is also spread out over time and on its own that might not seem like an issue
But in practice there's all sorts of different objects in the field
So these echoes are all gonna start to get overlapped with each other
Combine that with other noise and imperfections and this can make the locations of multiple objects extremely ambiguous
Instead a more precise understanding of how far away all these things are would require having a very quick little pulse
Confined to a small amount of time
But think about the frequency
Representations of such a short echo as you saw with the sound example the Fourier transform of a quick pulse is
necessarily more spread out
So for many objects with various velocities that would mean that the Doppler shifted echoes
Despite having been nicely separated in time are more likely to overlap in frequency space
So since what you're looking at is the sum of all of these it can be really ambiguous
How you break it down?
if you wanted a nice clean sharp view of the
Velocities you would need to have an echo that only occupies a very small amount of frequency space
But for a signal to be concentrated in frequency space it necessarily has to be spread out in time
this is the Fourier trade-off you cannot have crisp delineation for both and
This brings us to the quantum case
Do you know who else spent some time immersed in the pragmatic world of radio wave transmissions? A young otherwise
philosophically inclined history major in World War one France. Louis de Broglie
And this was a strangely fitting post given his predispositions to philosophizing about the nature of waves
Because after the war as de Broglie switched from the humanities to physics in his 1924
PhD thesis he proposed that all matter has wave-like properties
and
More than that he concluded that the momentum of any moving particle is going to be proportional to the spatial frequency of that wave
How many times that wave cycles per unit distance.
Okay now
That's the kind of phrase that can easily fly into one year and out the other
Because as soon as you say matter is a wave
It's easier to just throw up your hands and say physics is just weird
But really think about this even if you're willing to grant that particles behave like waves in some way whatever that means
Why on earth should the momentum of those particles the thing we classically think of as mass times velocity?
Have anything to do with the spatial frequency of that wave
Now being more of a math than a physics guy
I asked a number of people with deeper backgrounds in physics about helpful intuitions here
And one thing that became clear is that there is a surprising variety of viewpoints
Now personally one thing I found to be interesting
Was just going back to the source and seeing how debris frame things in his seminal paper on the topic
You see there is a sense in which it's not all that different from the Doppler effect where relative movement corresponds to shifts in frequency
It has a slightly different flavor since we're not talking about frequency over time instead
We're talking about frequency over space and special relativity is going to come into play, but I still think it's an interesting analogy
In his thesis dubrow lays out what is in his own words a crude comparison for the kind of wave phenomenon he has in mind
Imagine many weights hanging from Springs with all of these weights oscillating up and down in sync and with most of the mass
concentrated towards a single point
The energy of these oscillating weights is meant to be a metaphor for the energy of a particle
Specifically the e equals mc-squared style energy residing in its mass and debris emphasized how the conception he had in mind involves the particle being
dispersed across all of space
The whole premise he was exploring here
Is that the energy of a particle might have with something that oscillates over time?
since this was known to be the case for photons and
These oscillating weights are just meant to be a metaphor for whatever that something might be
With Einsteins relatively new theory of relativity in mind he pointed out that if you view this whole setup while moving
Relative to it all of the weights are going to appear to fall out of phase
That's not obvious
And I'm certainly exaggerating the effect in this animation
It has to do with a core fact from special relativity that what you consider to be
Simultaneous events in one reference frame may not be simultaneous in a different reference frame
So even though from one point of view you might see two of these weights as reaching their peaks and their valleys at the same
Instant from a different moving point of view those events might actually be happening at different times
Understanding this more fully requires some knowledge of special relativity
So we'll all just have to wait for Henry Rice's series on that topic to come out
Right here our only goal is to get an inkling for y momentum that thing
we usually think of as mass times velocity should have anything to do with spatial frequency and
The basic line of reasoning here is if mass is the same as energy via e equals mc-squared
And if that energy was carried as some kind of oscillating phenomenon similar to how it is for photons then this sort of
Relativistic Doppler effect means changes to how that mass moves corresponds to changes in the spatial frequency
So what does our general 4ei trade-off tell us in this case?
Well if a particle is described as a little wave packet over space then the Fourier transform
Where we're thinking of this as a function over space naught over time tells us how much various pure frequencies correspond with this top wave
So if the momentum is the spatial frequency up to a constant multiple the momentum is also a kind of wave
Namely some multiple of the Fourier transform of the original wave
So if that original wave was very concentrated around a single point as
we have seen multiple times now that means that its Fourier transform must necessarily be more spread out and
Hence the wave describing its momentum must be more spread out and vice versa
Notice unlike the Doppler radar case where the ambiguity arose because waves were being used to measure an object with a definite distance and speed
What we're saying here is that the particle is the wave?
So the spread out over space and over momentum is not some artifact of imperfect measurement techniques
It's a spread fundamental to what the particle is
Analogous to how a musical note being spread out over time is fundamental to what it even means to be a musical note
One pet peeve I have in mainstream references to quantum is that they often treat Heisenberg's uncertainty principle as some?
Fundamental example of things being unknowable in the quantum realm as if it is a core nugget of the universe's indeterminacy
But really it's just a trade-off between how concentrated a wave
and it's frequency representation can be
Applied to the premise that matter is some kind of wave and hence spread out
All of the stuff about randomness and unknowability is still there
But it comes one level deeper in the way that these waves have come to be interpreted
You see when you measure these particles say trying to detect if it's in a given region
Whether or not you find it there appears to be probabilistic where the probability of finding
It is proportional to the strength of the wave in that region, so
When one of these waves is concentrated near a point what that actually means is that we have a higher?
probability of finding it near that point that we are more certain of its location and
Just to beat this drum one more time since that concentration implies a more spread-out Fourier transform
Then the wave describing its momentum would also be more spread out
so you wouldn't be able to find a narrow range of momenta that the particle has a high probability of occupying I
Quite like how if you look at the German word for this principle it might be more directly translated as the unsharp ness relation
Which I think more faithfully captures the Fourier trade-off at play here without imposing on questions of no ability
When I think of the Heisenberg uncertainty principle what makes it fascinating is not so much that it's a statement about randomness
I mean yes
That randomness is very thought-provoking and contentious and just plain weird but to me equally fascinating is
underpinning Heisenberg's conclusion is that position and momentum have the same relationship as sound and frequency as
If a particles momentum is somehow the sheet music describing how it moves through space
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