Hi again physics fans, and especially to Toon van K, who I heard say “Hi” back during
the last episode.
How is everyone doing?
This series of videos is great fun for me, both the first part and the questions.
Several questions in previous episodes would have been much easier to answer if I had made
a video about the Heisenberg Uncertainty Principle.
So, I decided…that’s it…I’m gonna do it.
That’s what I’m going to talk about in this episode of Subatomic Stories.
The simplest description of the Heisenberg Uncertainty Principle says that it is impossible
to simultaneously know an object’s location and velocity.
The more accurately you know the object’s location, the less accurately you know how
fast it is moving, and vice versa.
It’s like a teeter-totter, where when one side goes up, the other side goes down.
This is just true.
Mathematically, the uncertainty principle can be written as the uncertainty in position
times the uncertainty in momentum has to be bigger than this constant which is called
hbar divided by two.
Those delta symbols mean uncertainty.
P means momentum and x means position.
Hbar is a super small number called the reduced Planck constant.
What it means really doesn’t matter unless you’re trying to be an expert.
All you really need to know is that it is small and constant.
Because it’s small, the Heisenberg Uncertainty principle only matters for things the size
of atoms or smaller.
At low velocities, momentum is just mass times velocity, so you can explicitly write Heisenberg’s
equation to show the relationship between uncertainty in velocity and uncertainty in
position.
The meaning of this equation is often explained incorrectly.
Many people claim that it is a measurement thing.
For instance, suppose you have an electron and you want to precisely know its position.
The way you would find its position it to shine the light on it to see where it is.
Light comes in many different wavelengths.
The shorter the wavelength, the higher the energy.
And, if you want to find the position of the electron with great precision, you need to
use very short wavelength light.
That’s because you can’t see anything smaller than the wavelength of light you’re
using.
So, the story goes, if you use very short wavelength light, you are hitting the electron
with a photon with a lot of energy, so you hit it hard and now you don’t know its velocity
because you knocked it off in some direction or another.
So, this is a totally wrong explanation of Heisenberg’s principle, although it was
the first one I was taught back in high school.
Sorry to out you Mr. G.
This explanation is an example of what is called the Observer’s effect.
It’s definitely not Heisenberg.
Kind of like this guy.
In fact, the Heisenberg Uncertainty principle is inextricably linked to the wave function
of matter, like electrons and photons.
Let me explain.
First you need to know that the location of a subatomic particle is determined by what
are called a wave function.
The wave function gives the probability of finding a particle at any particular location.
Actually, it’s the square of the wave function, but I’m glossing over that because it’s
not crucial here.
Where the wave function is big, it’s likely that you’ll find a particle there and where
it’s small, it’s rare to find it.
I made a long form video on the wave function and the URL is in the description
The second important point is that subatomic particles are both particle and waves.
When you’re looking at them as waves, they have a wavelength and the wavelength is related
to the momentum.
High momentum particles have a short wavelength and low momentum ones have long wavelengths.
And, since in this case we know the momentum, the Heisenberg Uncertainty Principle says
that we don’t know the position, so the waves stretch off to infinity in both directions.
Now if you want to have some knowledge of both the position and momentum of the particle,
you need to change the wave function from an infinite sine wave to something more localized.
And this is where it gets interesting.
To do that, you can simply start adding up the wave functions of particles whose momentum
is well known, but for whom the position is not known.
You start with a single wave, but then you add a series of waves that have a wavelength
that are slightly different than the initial one.
The cumulative wave function, which is the sum of more and more different wavelengths
slowly morphs from being a sine wave to being a wave function that is more localized.
You can see here how adding a bunch of similar waves of different wavelengths can change
the wave function.
The mathematics of this is called a Fourier Transform by the way.
If you like calculus, it’s worth looking up.
But even if you aren’t much into the math, you can see the key points.
Any particularly shaped wave function can be created by mixing together a bunch of different
wavelengths of different amplitudes.
We can best illustrate what I’m talking about by showing what happens when you start
with a single wavelength.
On a plot of wavelengths, everything is zero, and on a plot of position, you see a sine
wave.
If you then mix several different wavelengths, each with the same amplitude, you get a wiggly
wave function that is more localized than a single wavelength.
If you mix even more wavelengths, you get a wave function that is more localized.
And, finally, if you mix every wavelength with equal amplitude, you get a wave function
that is perfectly localized.
So, this is the core reason for the Heisenberg Uncertainty Principle.
If you have a single wavelength, which I remind you is related to the velocity or momentum,
then you have no information about the position of the wave function.
It’s everywhere.
And in order to localize the wave function, you need to add all the wavelengths, which
means you have no information about the wavelength and therefore the velocity.
And that’s just how it works.
Because wavefunctions are built of mixes of wavelengths, the more you focus the position,
the broader range of wavelengths is needed.
The more you restrict the wavelength, the less information you have about the position.
That’s the real reason for the Heisenberg Uncertainty principle.
Of course, to use it, you just need the equation and don’t need to ding into the deeper cause.
Now, there is another version of the uncertainty principle, which relates the uncertainty in
energy, which is E and time, which is t.
This is Delta E times Delta t is greater than hbar over 2, which is the same constant as
was used in the position/momentum version.
Everything I said before still matters, but this version is helpful when talking about
the masses and lifetimes of subatomic particles.
Particles which have a very long lifetime, which is to say that Delta t is very big must,
by the Heisenberg Uncertainty principle, have a very small uncertainty in the energy, which
is to say mass.
Turning it around, particles that live for a very short amount of time, which is to say
that the delta t is very short, must have a correspondingly large uncertainty in the
energy or mass.Thus short-lived particles have a large range of allowed masses around
their average, while long-lived particles have no range at all.
That’s why subatomic particles like the top quark or W or Z bosons can have any number
of different masses, while a stable particle like the electron have basically one and only
one observed mass.So that’s the Heisenberg Uncertainty Principle in a nutshell.
It takes a while to fully appreciate all of its implications, but I hope you have a deeper
understanding of just why it is impossible to simultaneously know a subatomic object’s
position and velocity, or energy and lifetime.
How about we take a look at a few viewer’s questions?
It’s question time and time for me to pay off a debt.
In a previous episode, HL65536 asked if electrons can have different masses than the accepted
electron mass of 0.511 MeV.
So I hope that this video on the Heisenberg Uncertainty Principle answers that.
Given that the electron is stable, that means that its lifetime is infinite.
A delta T of infinity means a delta E of zero, which means that the electron has a single
and unique mass.
Now the answer changes when we consider the virtual particles we talked about in episode
nine.
In the case of virtual particles, the electrons exist for only a very short time.
That means that both the energy and momentum could differ from the average and the result
is that you could have electrons with a range of masses – even cases where the square
of the mass is negative.
Virtual particles break a lot of the rules and are super confusing when you first encounter
them.
George GDL asks where neutrinos get their mass from.
Hi George!
The short answer is that we don’t know.
They could get their mass from interactions with the Higgs field, but we aren’t sure
about that.
Neutrinos are so much less massive than other particles, that it’s possible that they
get their mass from another mechanism.
Theorists have invented other possibilities, but the truthful answer is “I dunno.”
TheBibo Sez asks if a neutrino could pass through a neutron star.
The answer is basically no, or at least mostly so.
The probability that a neutrino will interact depends on energy and the density of the material
it is travelling through.
For a high energy neutrino of a hundred billion electron volts of energy, the neutrino will
travel less than a tenth of a millimeter before interacting.
So, it doesn’t get far.
For low energy neutrinos, they can penetrate much farther, but we’re still talking a
distance of about a meter or so.
Very low energy muon neutrinos don’t have enough energy to make muons, so they bounce
around inside the star essentially forever and eventually find their way out.
The neutrino that would penetrate a neutron star most easily would have to an be electron
neutrino with a few million electron volts of energy.
In principle, they could encounter a proton and make a neutron.
The density of protons inside neutron stars is very poorly known, but if you take reasonable
numbers, then a very low energy electron neutrino might travel as far as a kilometer before
finding a rare proton and turning it into a neutron.
Peli Mies asks after leptogenesis, how long did it take for antimatter and matter to destroy
one another?
Hi Peli.
First, you should know that leptogenesis isn’t known to be true.
Maybe it is and maybe it isn’t.
For those of you who don’t know what leptogenesis is, I made a long form video about it.
The URL is in the description.
However, whatever mechanism is responsible for the matter/antimatter imbalance in the
universe, it took only a tiny fraction of a second to complete.
Oisnowy asks if there have been estimates of the masses of neutrinos.
The answer is yes, although to cover all of the answers is a huge undertaking.
I guess I’m going to issue another IOU on this one.
I think that would be a great topic for an upcoming video.
Nibbler correctly points out that the Schrodinger cat example isn’t what most people think.
Schrodinger came up with the scenario to show that the Copenhagen interpretation of quantum
mechanics was silly.
And that’s totally true.
But it’s also true that the example, if it wasn’t absurd, is a quick way to convey
a key quantum concept when you have to explain something as complex as neutrino oscillations
in five minutes.
I’ll let you know if Schrodinger’s ghost haunts me and, if he does, what he says.
Schlynn properly points out the correct pronunciation of the thirteenth letter of the Greek alphabet
is nee.
Well that’s news to me.
I crack myself up.
Schlynn is right, but it’s certainly pronounced “nu” by scientists in this context – even
the Greek ones I know.
Hey, I don’t make the rules…
Russell Subedi asks for a function that characterizes neutrino oscillation and what parameters are
relevant.
Hi Russell – that’s a big question.
The Wikipedia is a pretty good reference, but if you want to make plots, I recommend
the package I used.
It’s a Wolfram Alpha demo.
I’ll put the links in the description.
And the good thing is that the demon is totally free.
It’s worth your time.
Mark Sakowski asks if the quarks and antiquarks annihilated after the big bang, where did
the energy go?
Hi Mark, that’s a good question.
You’re right.
After the annihilation, there was a tremendous amount of gamma radiation.
But that was when the visible universe was tiny.
It has expanded a bunch since then.
The expansion also stretched the wavelength of the gamma rays.
Nowadays, those gamma rays have a wavelength of about a millimeter, or what is called microwaves.
In fact, that one millimeter wavelength radiation is called the cosmic microwave background
radiation or CMB.
The CMB is one of the strongest bits of evidence we have that the Big Bang actually occurred.
We see it everywhere we look in the universe.
I’ll talk more about it in a subsequent episode.
There’s another very interesting bit of information we can use the CMB to determine.
First, we can measure the energy held in the CMB.
Second, we can calculate how many photons that means.
Finally, we can calculate the number of the protons in the universe and compare it to
the number of CMB photons.
We find that the proton to photon ratio of about three billion to one.
that's how scientists can quantify the amount of matter/antimatter imbalance in the early
universe.
The universe is just full of interesting data if you know how to look.
OK, that’s all the time we have for questions today.
Are you liking the series?
I hope you’re liking the series.
If you are, please be sure to like, subscribe and share on social media.
As long as you’re watching, I’ll keep making videos.
And since you’re watching, I hope that you’re now a firm believer that, even at home, physics
is everything.