PAUL: So how can quantum mechanics help explain a white dwarf?
Well, let's think about the problem.
We've got this small star with absolutely, incredibly intense gravity
but no nuclear fusion going on.
Now, let's zoom in on the surface.
Here's a little surface layer.
That weighs a lot, so there's very strong
gravitational pull down towards the center.
Because it's not falling in, it's just sitting there,
there must be an equal and opposite force outwards
to balance the gravity downwards.
And remember because these things are so dense and so small
that gravity is very intense, so it must be a very strong force outwards
to balance the gravity.
What is this force?
Well, it's caused by pressure.
But what is pressure?
Well, pressure is just a cumulative effect
of being hit by atoms and molecules.
So what's happening here is the inside edge,
and you got lots of atoms, electrons, protons, whatever.
They're flying in, and they bounce off the insides in very large numbers.
You don't get sore by getting hit by one atom.
Ow!
That atom hurt me.
Oh, no.
It doesn't matter.
But there are so many billions upon trillions of atoms
hitting that their cumulative effect can be quite strong,
and that's what pressure is.
So the idea would be that in a normal star, the pressure inside is very high,
so vast numbers of atoms are bouncing off the inside of the surface layers.
And their impacts apply enough net force to balance the downward gravity pull.
But there's a problem.
These items must be moving really fast to supply enough force,
which means their temperature is high.
But if they're very hot, the heat will leak out.
The heat will radiate out into space somehow.
It might be conducted through there and then radiated out into space.
So they will get cooler and therefore slower.
And it they're slower, there will be fewer impacts.
The impact will have less oomph behind them, and so the pressure will go down.
So no longer balanced, gravity and the star will shrink.
As it shrinks, it compresses what is inside.
It makes it hotter again.
Once again that heat will radiate out, and so the whole star
will shrink smaller and smaller and smaller until it's no size at all.
So that doesn't work.
What we need is something that stops these moving particles from slowing
down below a certain speed, like a minimum possible speed.
And that's what quantum mechanics, in the form of degeneracy pressure,
gives us.
What's the idea?
Well, here we've got our star just shrinking,
and it's got a whole bunch of electrons in it.
Now, from the Pauli exclusion principle, we know that the electrons cannot all
be-- no two electrons can be in the same state.
So one way to think about that would be to divide a white dwarf
into little cubes.
And we'll put one electron in each cube.
Now, the electrons could move back and forth inside the cube,
but they can't trespass into another cube
because that would be overlapping in a state with another electron.
But now let's look at one of these cubes,
and apply what we know about quantum mechanics.
The electron, remember, it behaves like a probability wave.
So let's, for the moment, just to imagine
it's going to be a wave along one dimension.
It could be, like a guitar string, in the ground
state, which is something like this.
Or, it could be in the first excited state.
Or, it could be in the second excited state and so on.
But it's always going to have an integer number of waves across the length.
Now of any wave, whether it be a light wave or electron,
the shorter the wavelengths, the more the energy.
So the lowest energy will be this blue state.
And the green and yellow lines have more energy, and so a state like this
would be a very energetic state.
Normally, if you're on something like the Sun,
most electrons are in states like this.
They're in their little cube.
They've got very wound up in terms of their wavelength,
so that they have way more than the minimum energy.
But what quantum mechanics tells us is the wavelength
can get longer and longer and longer, but the longest it can be
is when you only got onto wavelength covering the entire length of the box.
And you can't be longer than that because then it
would have to not be 0 at one end, so it wouldn't be a proper standing wave.
And that is what the Uncertainty Principle encapsulates.
It tells us delta p delta x is greater than h-bar over 2.
So this is the momentum related to the energy,
and what that's telling us is as the box becomes smaller,
the longest wavelength you can have also becomes
smaller, which means the momentum becomes larger.
So there's a lower limit to the momentum corresponding to the size.
I'm going to do a very approximate, very rough calculation
to show how a white dwarf could work.
The full calculation is too complicated for this course,
but we can get the basic physics, a roughly right answer
out by making some big approximations.
So let's start off by thinking about electrons stuck in a box.
This box could be 1 cubic meter of white dwarf material
or 1 cubic meter of anything really.
And let's assume we have a number of electrons per unit volume,
so a number density of electrons of ne.
So that's the number of electrons per unit volume.
Now, all the electrons can't be in the same space because of the Pauli
Exclusion Principle.
They all have to have their own discrete identity,
and they can't all sit in the same state.
So each electron-- so we've got an electron
there-- has to have its own unique volume.
So assume each has a cube like this.
As the electron is trapped in that volume,
it's going to-- the size of this volume, the size of each side here,
is the uncertainty in its position, delta x.
And what is delta x?
Well, each electron has a volume, delta x cubed,
that being the volume of a square of each side delta x.
And there are ne electrons in one unit volume.
So what that is telling us is that delta x is
equal to 1 over the cube root of ne.
So that's ne to the minus a third.
Now, if we have that uncertainty in its position,
that means there's going to be uncertainty in momentum.
So the uncertainty in momentum times uncertainty in the position
is going to h-bar.
That's Heisenberg uncertainty principle.
So the uncertainty and the momentum is going to be h-bar over delta x.
So that's going to be h-bar ne to the plus 1/3.
So that tells us that the electrons are not still.
They have momentum.
There's an uncertainty in it.
And that's going to be very roughly equal to momentum.
Some are going to be going forward.
Some are going to be going backwards.
This is momentum in one direction.
So roughly speaking, the uncertainty is going
to be about the size of the momentum.
So roughly speaking, again, not a very good approximation,
but close enough for this work, the momentum
is going to be about the size of the uncertainty.
And it's given by this.
