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Sometimes intuitive large-scale phenomena can give us incredible insights into the extremely unintuitive world of quantum mechanics
Today, the humble sound wave is going to open the door to really understanding
Heisenberg's uncertainty principle and,
ultimately, quantum fields and Hawking radiation.
One of the most difficult ideas to swallow in quantum mechanics is Werner
Heisenberg's famous uncertainty principle. It expresses the fundamental limit on the knowability of our universe.
We've discussed it in earlier videos on quantum mechanics
But it's time we looked a little deeper. See the apparent weirdness of the uncertainty principle
hints at an even weirder underlying reality that gives rise to it. The universe we experience seems to be
constructed of singular particles with well-defined properties, but this intuitive
mechanical reality is emergent from an underlying reality in which the particles that form matter
arise from the combination of an infinity of possible properties. And
forget matter - the vacuum itself
can be thought of as constructed from the sum of infinite possible particles. If we fully unravel this idea,
we'll be on the verge of tackling things like Hawking radiation.
But as you'll see today in that unraveling, we are led
unavoidably to Heisenberg's uncertainty principle.
The uncertainty principle is most often expressed in terms of position and momentum
We cannot simultaneously know both position and momentum for a quantum system with absolute precision
Try to perfectly nail down a particles position, and we have complete uncertainty about its momentum
And it's not just because our measurement of position requires us to interact with the particle therefore changing its momentum
No, the uncertainty principle exists alongside this observer effect.
It's instead a statement about how information we are ever able to extract from a quantum system
To understand the origin of the uncertainty principle
We don't need to know any quantum mechanics
At least not to start with, see, quantum mechanics is a type of wave mechanics - a very weird type.
However, it turns out that something like the uncertainty principle
Arises in any way of mechanics, so let's choose a type of wave that's a little more intuitive -
sound waves. You can describe a sound wave just as the intensity of the wave as it passes by, so
intensity changing over time. It can take really any shape, that shape determines what the wave sounds like to our ears.
The sound wave for a simple pure tone, like a middle C, is a sinusoidal wave
with the frequency determining the pitch of the ton. The sound wave from, say, an orchestra is extremely complex,
but,
amazingly, it can always be broken down into a combination of many simple sine waves of different frequencies.
This is Fourier's theorem, after French mathematician Jean-Baptiste Joseph Fourier.
It states that any complex sound wave can be decomposed into a number of sine waves of different frequencies,
each with a different strength, stacked on top of each other or
superposed. In fact instead of representing a sound wave in terms of intensity changing with time,
you can also represent it in terms of its frequency components: each with its own weighting or strength.
When you switch between a time and a frequency representation you're doing a Fourier transform. In fact
digital audio equipment stores, manipulates, and transmits sound in its frequency
representation. In the physics of sound, time and frequency have a special relationship,
because any sound wave can be represented in terms of one or the other: we call them Fourier pairs.
Also, sometimes, conjugate variables. Okay,
so we can make any shape sound wave with a series of sine waves of different frequencies.
For example, you can build a wave packet by adding frequency components with the right phases to destructively interfere
everywhere, except within a small region. The tighter you want to make that time window for the wave packet,
the more frequency components you need to use. In fact, to get those steep edges of the wave packet
you need to add higher and higher frequencies, because the high frequency components are the ones that give you rapid changes in intensity.
So, what if you try to
compress the wave packet to a single spike, a blip of sound, that exists for only one instant in time ?
Is it even possible to make an instantaneous spike at one point in time, out of a bunch of sine waves that themselves extend
infinitely through time. In fact, it is. However, to get a spike at one point in time you need to use
infinitely many different frequency sine waves, each of which exists at all points in time.
So then, if we make a sound that is
perfectly located in time it doesn't have a frequency or it has all frequencies. At the same time, a sound wave with a
perfectly known frequency is a simple traveling sine wave that extends infinitely in time,
so the time of its existence is undefined.
That sounds an awful lot like a frequency time uncertainty principle for sound waves.
Now, it's not really a statement about the fundamental knowability of a sound wave as is Heisenberg's uncertainty principle.
It's more a statement about the sampling of frequencies needed to produce a given wave packet.
But the underlying idea is the same. So, how does this relate to the quantum world?
Well, before we get back to quantum fields
let's think about the wave function - the solution to the Schrodinger equation that contains all of the information
about a quantum system.
Like the sound wave, it oscillates through space at a particular frequency.
To keep things simple
we're just going to consider a wave function that doesn't vary in time, it only changes with position in space.
This is more like a standing sound wave inside an organ pipe, rather than the traveling sound wave from earlier. So, position,
rather than time, becomes the first of our Fourier pair. The second of this pair is
momentum, not frequency. See,
momentum is sort of the generalization of frequency for what we call a matter wave. In the early days of quantum mechanics
it was realized that photons are electromagnetic wave packets, whose momentum is given by their frequency.
Louis de Broglie extended this idea to particles, and his de Broglie relation
generalizes the relationship between frequency and momentum of a matter wave.
We now call matter waves wave functions, and we can describe them in terms of position or momentum,
just as a traveling sound wave can be expressed in terms of time or frequency.
So, any particle, any wave function can be represented as a combination of many locations in space with accompanying intensities.
Think of it as the particle being smeared of possible positions,
or as a combination of many
momenta with accompanying intensities, in which case the particle would be smeared in momentum space, and,
of course, this means that position and momentum
have the same kind of uncertainty relation that time and frequency had in the sound wave.
But, what does it even mean for a particle to be comprised of waves of many different positions or momenta ?
To answer this we need one more bit of physics - the
interpretation of the wave function itself, known as the Born rule.
The magnitude of the wave function squared is the probability distribution
for the particle. If we're expressing the wave function in terms of position, then applying the born rule
tells us how likely we are to find the particle at any given point when we make a measurement. Or,
put another way, the range of positions in which the particle is likely to be located
were we to look. If we apply the born rule to the momentum wave function, then we learn the range of momenta
the particle is likely to have. So, if we measure a particles position then from our point of view its wave function is
highly localized in space: we know where the particle is.
The resulting particle wave packet, now constrained in position, can only be described as a
superposition of waves with a very large range of different momenta, via a Fourier transform.
The result is a very fat momentum wave function that gives a wide range of possible momenta. The more precisely we try to measure the position,
the narrower we make its position wave function and, so, the less certain we become about its momentum as that momentum wavefunction
gets wider. This is all super abstract,
but a concrete example is single slit diffraction. If we
increase our certainty of the position of a particle by narrowing the slit we also increase the uncertainty of its momentum,
as it passes the slit. This results in an increasing spread in final locations. Check out Veritasium's
excellent video to see this in action.
So, that's exactly the uncertainty principle.
It's a statement, about how much of a quantum systems information is accessible at a fundamental level.
It's an unavoidable outcome of describing particles as the superposition of waves,
waves that can be represented in terms of either position or momentum. The fact that both can't be known
simultaneously with perfect precision is a property of the nature of the wave function itself.
Precision in one is actually constructed by the uncertainty in the other.
OK, so what does this old-school quantum mechanics have to do with quantum field theory and Hawking radiation?
Well, the key to understanding these things is to be able to switch between thinking about quantum fields in terms of position versus momentum.
See, a single particle, a quantum field vibration perfectly localized at one spot in space can also be described as
infinite oscillations in momentum space, spanning all possible momenta.
But each of these oscillations in momentum space are equivalent to particles with highly specific momenta.
The uncertainty principle therefore tells us, that they must be completely unconstrained in position.
So, a perfectly specially localized particle is equally an infinite number of
momentum particles that themselves occupy all locations in the universe.
It's only by manipulating quantum fields in this strange momentum space, by adding and removing these
spatially infinite particles, that we can describe how the quantum vacuum changes to give us a phenomena,
like Unruh and Hawking radiation,
which you'll soon understand as some of the weirdest behaviors of space-time.
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OK, so, I traveled a bit over the past two weeks until we missed one of our comment responses. Today
we're gonna cover our episode on robots that sacrifice themselves for science, as well as our episode on citizen science.
First, thanks to everyone who pointed out the editing error at the end of the suicide robots episode,
where we accidentally included both of the two takes of the last line.
Totally unintentional
But, hey, it was a pretty good line right. Also, thanks to everyone who noticed my
mispronunciation of Enceladus in that episode. You know those words you've never heard said that have been pronouncing wrong in your head forever.
Yeah, this isn't one of them. I think I was just recording through lunchtime and really craving enchiladas. A few of you
who've been involved in citizen science projects that we didn't cover. One of my favorites is the exoplanet search program embedded in the EVE Online game.
Players killed time during warp journeys by scanning light curves of distant stars for the characteristic dips in brightness, due to
transiting alien planets.
sakurasleight suggested, that SETI@home uses computing cycles to mine Bitcoin instead of look for alien signals.
A couple of you pointed out a better alternative,
the cryptocurrency,
GridCoin, which you mine by dividing computing cycles to BOINC research programs
Seems a bit more useful than Bitcoin's useless cryptographic calculations.
Daniel Soltesz reminded me of the coolest object found in Galaxy Zoo,
discovered by a Dutch school teacher Hanny van Arkel. Hanny's Voorwerp
Is a weird blob of light right next to a spiral galaxy. It's hypothesized to be the light echo from a dead quasar,
that was once in that galaxy, so the cloud of gas
ionized by the last burp of energy from an active supermassive black hole in the middle of the spiral galaxy
just before it ran out of food. It is the first of its kind discovered.
Regarding our
zero-point challenge answer
Jaden Andrews asked how to prevent the geckos tail falling off when you harness them for wall climbing.
Well, Jaden, the trick is to assume geckos with infinitely high tensile strength. As a couple of you pointed out, we already
assumed the geckos of zero mass, so who's to say those rare zero mass geckos don't have infinite tensile strength.
Perhaps you just haven't tried enough geckos.