[MUSIC PLAYING] 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.

[MUSIC PLAYING] 13 00:00:33,750 --> 00:00:36,990 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 particle's 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 much 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 wave 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 tone.

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.

OK.

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 familiar.

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.

Luis 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 over 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 function, then we learn the range of momenta the particle is likely to have.

So if we measure a particle's position, then from our point of view, it's 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 position, the narrower we make its position wave function, and so the less certain we become about its momentum, as that momentum wave function 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 system's 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 so 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 phenomena like Unruh and Hawking radiation, which you will soon understand as some of the weirdest behaviors of space time.