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If you were comfortable with the, the level of physics and

Â math in the last lecture that's great, we're going to keep going.

Â If you were uncomfortable with it, you can do one of a couple things.

Â You can look up some basic physics differential equations

Â on, on many different sites, will provide some of them.

Â Or you can just follow along in a sort of higher conceptual level, don't sweat the

Â details of the physics and the math so much, but just still try to get the point.

Â The point last time was Hydrostatic equilibrium, you have a balance between

Â compressing a gas, and how much it wants to re-expand, and that balance is

Â between the pressure of that gas and all of the weight on top

Â of it, and the other important point to remember is the equation of state.

Â That's such an important point we're going to work on it again, today.

Â Equation of state, really is a huge part of the name of the game

Â for understanding what's going on inside of

Â a giant planet, really, inside any planet.

Â I'll remind you equation of state simply means what is

Â the connection between the pressure and the density, and can we

Â do it a little better than our simple PV equals nRT

Â that we learned in high school, for the ideal gas law?

Â We're going to make an approximation today that's the

Â opposite end of the spectrum from PV equals nRT.

Â We're going to do something called a Fermi gas.

Â Now it's going to look a little bit crazy, a little bit complicated, but

Â in the end, it's relatively straightforward

Â to at least conceptually understand going on.

Â So, bear with me because we are now delving into quantum mechanics.

Â And we're delving into quantum mechanics because in

Â the limit of high density, the reason that

Â things don't just collapse on top of themselves,

Â is because of quantum mechanics, pretty much anything.

Â If I take a solid thing like this pen, why

Â can I not push my fingers together on this pen?

Â It's because, essentially, the electrons inside of the atoms, inside

Â of this pen, do not want to be pushed together.

Â 2:01

It's not quite as simple as that, but it's almost as simple as that.

Â Why don't the electrons want to be pushed together?

Â Has nothing to do with electrostatics like negatively charged

Â electrons like to repel each other because for every electron

Â inside of here, there is a proton inside of here,

Â so overall this is neutral, so it's not electrostatic repulsion.

Â It's purely quantum mechanical repulsion, and the reason it happens is because

Â electrons are a fundamental particle of a type called fermions.

Â Fermions have a critically important property that is the

Â reason for this lack of ability to compress this pen.

Â Fermions, no two Fermions can occupy the

Â same quantum mechanical state at the same time.

Â Usually when we think about quantum mechanical states, if you ever think

Â about quantum mechanical states, you think about an electron going around a nucleus.

Â And it has an energy level, and an electron can

Â then jump up to a different energy level or down.

Â And you can only have a certain number of electrons in each energy level.

Â Well, actually I'm not even going to talk about electrons and energy levels.

Â We're just going to now make a, a simplifying approximation that

Â we have electrons, a sea of electrons in free space.

Â And in that sea of electrons, we magically put enough

Â positive charges, so we overall have a, a net neutral charge.

Â Even in this sea of electrons, the electrons

Â don't want to be in the same state.

Â Now, I keep saying that word, don't want to be in the same state.

Â Let me draw you a picture of what that means quantum mechanically.

Â If, if I had a box, and I put an electron in a

Â box, and I asked myself the question, where is the electron in that box?

Â Well, one of the weird things about quantum mechanics

Â is that there's no answer to where the electron is.

Â Really there's a, a probability that it's in any particular place.

Â And that probability inside of a box goes something like this.

Â There's a high probability it's, it's close to the center, at any point

Â in time there's a low probability that it's at its, it's at the end.

Â That's for one electron.

Â If we put in two electrons, well, electrons have this

Â funny property that they, they can either be electrons that

Â are spinning up, spinning in this direction, or they can

Â be electrons that are spinning down, and spinning in that direction.

Â And those two electrons can actually be in the same state at the same time.

Â So, the second electron will be found in the same range of places.

Â Put a third electron there.

Â Where does it go?

Â Well, it can't have the same probability distribution as those first two electrons.

Â This is what I mean when I say it's in a different state.

Â In fact, the sec, the third electron will be humped.

Â 5:05

Seventh and eighth electrons, you guessed it.

Â It's a very strange thing, why you might ask, why because quantum mechanics.

Â Quantum mechanics does this.

Â This is when I say that electrons can't occupy the same state.

Â Every time I put new electrons into the box, those new electrons have

Â to be in a new state that has, has not been occupied before.

Â We're going to label these states by the number of peaks they have.

Â This one has one, two, three, four peaks.

Â We're going to name it as k equals 4 for this one.

Â This one had a k equals 3, and we're going to try to figure

Â out how much energy each electron has in each one of its states.

Â And I'm going to do this by using a very, very poor analogy.

Â That really is a pretty bad representation of how you really calculate

Â the energy of the state, and yet it, it works moderately well.

Â So, so if you know your quantum mechanics better, I apologize.

Â If you don't know quantum mechanics, think of it

Â this way, it's a good way to think about.

Â The first and second electrons, which are somewhere in

Â the box with, with a probability something like this.

Â Moving around with some velocity, we don't know what it is, we'll call it v-not.

Â So for k equals 1, the velocity is v-not.

Â So the electrons are moving around here

Â with some effective velocity, something like that.

Â Now the third and fourth electrons, think of it this way, either here or here.

Â There's some probability that they're over here, and

Â some probability that they're over here, and they

Â are, in some sense, having to move faster

Â to jump back and forth between these, these possibilities.

Â They have something like twice as much region to cover,

Â they have to be moving something like twice as fast.

Â I know, again, if you know your quantum

Â mechanics, you're probably about to revolt at this point.

Â But, but go with me.

Â Three humps, one, two, three.

Â going to make the same argument, if three times as

Â many peaks to have to jump around between to

Â get around all those three peaks with equal probability,

Â they have to move something like three times the speed.

Â [BLANK_AUDIO]

Â And you can see how this continues on.

Â All right what good does this do me?

Â I'm actually interested in energy, and I'm interested in the kinetic energy.

Â As you remember, kinetic energy is one half mv square so the energy of

Â these electrons, I'm going to ignore things like

Â one halves, I'm even going to ignore things like

Â ms, I'm going to even ignore things like V naughts, I'm going to say that the energy

Â of this electron is proportional to the

Â velocity squared, which is equal to k squared.

Â 7:39

So that first electron, that only has one hump

Â in it, is the lowest energy electron that is.

Â Next, lowest energy has two humps, next lowest energy has three humps.

Â By the time you put in, a million electrons

Â into one box, and you have a million humps,

Â you have an energy that, last electron that you

Â put in there, has to have a very high energy.

Â It'd like to have a lower energy, everything likes to have a lower energy.

Â But there are no low energy states available to it.

Â So if I now just count electrons and I wanted

Â to say what the energy of what, of that last

Â electron I put in, well now I have to admit

Â that this is, I was just doing one dimensional box here.

Â Boxes are really three dimensional so, you can

Â put electrons with humps in these, this direction.

Â You could put electrons with humps in this direction, and you could put

Â electrons with humps in the direction in and out of the, the screen here.

Â Each one of those counts as a different state.

Â So think of it this way.

Â If I have a certain number of electrons that I'm

Â filling inside this box, what's my highest level of k?

Â Well, I can do fill it up k in

Â this direction, k in this direction, k in this direction.

Â So the total number of electrons is

Â proportional to that highest value of k cubed.

Â 8:49

And it's nice to think of this not in

Â terms of total number of electrons, but in terms

Â of something like distance between electrons, or maybe volume

Â that each electron gets to occupy on its own.

Â Even though that's not a real concept, it's a nice easy concept to do.

Â So the number of electrons is proportional to one over

Â the volume that each electron occupies, which of course is

Â equal to, proportional to one over the radius cubed of

Â the distance to the next closest electron of each electron.

Â So we have the k is proportional to 1 over r.

Â Kind of makes sense, if that number of humps is huge, that means

Â that that distance to the nearest electron, 1 over r, is kind of small.

Â That's kind of what this is saying.

Â This is great, because now we know the energy is

Â proportional to 1 over r squared, and we know that

Â the pressure, a, one of the definitions of pressure, is

Â that it is the change in energy with respect to volume.

Â That makes sense if you compress it, make the volume go down,

Â the energy inside there goes up, and that's the pressure that you feel.

Â So dE dV, energy is proportional to 1 over r squared, change in energy

Â with respect to volume is going to be proportional to 1 over r to the 5th.

Â The density of the material, of course, is

Â going to be proportional to 1 over r cubed.

Â The closer you put those electrons together, the denser the thing is

Â going to be, and so you're left with a very simple equation of state.

Â I just made up an equation of state.

Â Pressure is proportional to density to the five thirds.

Â Let's think where all this came from, one more time.

Â It's the simple fact that the electrons cannot occupy the same

Â state inside of a box, and by a box I mean anything.

Â This pen is a box and so every time we add one more electron into this

Â box, we have to add another hump in

Â this distribution of where that electron must be.

Â What I'm really saying is that every time I add another

Â electron in there, it has to be an ever higher energy electron.

Â The energy of the electron that I add

Â is proportional to the number of humps squared.

Â That's because energy is one half Mv squared, and that number

Â of humps is also related to the average distance between electrons.

Â So, when you compress the material, you make

Â R smaller, you make Rho larger, but you're

Â making the pressure even larger, because you're increasing

Â the energy of all those electrons inside there.

Â 11:15

Okay, this is probably the last time we're going to do this much level of

Â detail of quantum mechanics, but I want you to think of it this way.

Â I want you to think of walking down the road.

Â Every time your foot goes on the street, you are pressing

Â down on the road and you are pressing down on those electrons.

Â You're forcing those electrons a little bit close together and they're resisting.

Â They increase their density a little bit.

Â They increase their pressure quite a bit and they resist

Â being deformed by the weight of your foot coming down.

Â Drive down the road in your pickup truck, and it's slightly

Â different, you have more weight pushing down, but again, not too bad.

Â Put yourself inside of Jupiter, however, and you have enough pressure

Â pushing you down that the pressure can finally increase by dramatic amounts.

Â More importantly, though, we have an equation of state.

Â 12:07

That is appropriate, close to appropriate for high pressure material.

Â It's not perfect, the inside of Jupiter is not a sea

Â of electrons inside of a bigger sea of positively charged thing.

Â It is not a fermi gas like this one is, but it is pretty close.

Â And this gets approximately the right form of behavior

Â over some of the ranges that we'll care about.

Â We'll use an equational state like this as an experimental equational

Â of state, and we will explore what's going on inside of Jupiter.

Â