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Today I'm going to explain the Heisenberg uncertainty principle.
In case you haven't heard it the most common version of the Heisenberg Uncertainty basically
says that you can't know a particle's position and its momentum both perfectly, instead,
the more know about one, the less you can know about the other.
Today we're going to figure out what that actually means.
If you remember way back to the first episode, I introduced you to the wavefunction, which
is the wave that tells you where the particle is most likely to turn up.
We said that a particle is most likely to turn up n areas around where the wave is most
intense.
So from the wavefunction you find out the place that on average, most particles will
turn up.
But of course, the average by itself isn't that useful, we also want to know the range
around the average that the particles are most likely to fall into.
Well for a peak like this, it looks like the particles will mostly turn up in this region.
To calculate this precisely we use what's called the standard deviation.
This number also gets called the uncertainty because the bigger it is, the more uncertain
we are about where a particle will be.
Now let's talk about momentum.
We know that while we're not looking, a particle must take all possible momentums at once.
If we plot all the different momentums a particle has, it also looks like a wave.
We'll call this wave the momentum wavefunction.
Just like with the other wavefunction, the momentums where the wave is most intense are
the more likely to be measured momentums.
At the moment we don't know how to find this wave, but we'll get to that.
But meanwhile, we define uncertainty for momentum as well, in the exact same way we did with
position.
Again this uncertainty tells you that if you do a measurement of the particle's momentum,
you're most likely to find it in that region.
So now the Heisenberg Uncertainty is the claim that some how, these two uncertainties are
connected.
In fact, if you multiply the two, the number you get is always bigger than this tiny number.
So what does this actually mean?
Imagine we manage to get a particle into a state like this.
Now we're pretty sure about where the particle will end up.
However as a result, we don't know very well what momentum we'd get if we measured it.
So you see the uncertainty principle is about prediction.
It restricts how well we can predict the particle's momentum and position at the same time.
Let's look at a really interesting example.
Say we go ahead and measure the particle's position.
What happens if we decide to measure the particle's position again straight afterwards?
Well, remember that before we do our first measurement, the particle's in a superposition
of being in many places at once, but when we measure it, we break the superposition,
and now the particle really is in just one place.
So if we measure it again quickly, it will in fact be in the exact same spot.
Ok, but what about if we are going to measure the particle's momentum after the position?
Now your uncertainty about position is practically zero so your uncertainty about must be momentum
is huge.
That means it will be really hard to guess the particle's momentum.
Still, we go ahead and measure the momentum, and we get some number.
But wait.
Now it seems like we know the position and the momentum of the particle perfectly well,
and doesn't the uncertainty principle forbid that?
Well are we sure we know the particle's position?
What happens if we measure it again and check.
The overwhelming chances are it will be somewhere other than where it was the first time.
Why?
Because by measuring the momentum, we actually changed the particle's position wavefunction.
The particle went from being in pretty much in just one spot to be in many different spots.
So you see, we can never pin down both the momentum and the position of a particle.
Why is that?
The uncertainty principle isn't just something that we add into quantum in an ad hoc way.
The reason it happens is because of a special relationship between the position wavefunction,
and the momentum wavefunction.
Well, firstly it's weird that there should be any relationship between them at all.
Usually we think and object's velocity is independent of its position.
By that I mean, an object can be moving at all kinds of speeds regardless of where it
is.
In quantum mechanics that's not at all the case.
The position wavefunction actually contains all the information about the momentum wavefunction.
In fact it contains all the information about a particle's state.
Say we're interested in measuring something else about a particle like its energy, angular
momentum or in the double slit experiment case, which slit it went through.
You can calculate what values are most likely just from the position wavefunction.
In other words, in quantum mechanics, the wavefunction tells us everything about the
particle that is possible to know.
Still there is a particularly special relationship between momentum and position that makes the
uncertainty principle happen.
I'd like to explain it to you but it's a bit of a detour so I'm not sure all of you would
want to see it.
That's why I've decided to make two short supplementary episodes that explain it.
If you're interested, you can check them out, they'll be up soon.
Alright, see you then!