So in this final part of the lecture 
we'll solve the Cooper problem, which is 
a problem of weakly attractive electrons 
near the thermos surface that form pairs, 
so-called Cooper pairs bound states of 2 
electrons. 
And as we discussed these Cooper pairs, 
the, at low temperatures, they both 
condense into [INAUDIBLE] condensate 
should become superfluid, and this is 
this amazing state that we know as 
superconductor. 
I should mention that we will see 
actually the solution to the Cooper 
problem will involve essentially the same 
method we used in the first lecture this 
week in the context of single particles 
quantum mechanics, so it's going to be 
actually very simple. 
Technically, a very simple solution for a 
Nobel prize winning work anyway. 
So, it's not trivial but, you know, this 
is a very important work. 
So, here I'm actually showing the the 
title of the original paper by Leon 
Cooper. 
So and this work of Cooper eventually has 
progressed, has developed into a much 
more sophisticated theory, which is the 
Bardeen-Cooper-Schrieffer theory of 
superconductivity, also called the BCS 
theory. 
So, and this BCS theory has been the 
cornerstone of the theory of 
superconductivity for many, many years. 
And frankly, there is nothing better on 
the market in some sense. 
So, this BCS theory describes very well 
most conventional superconductors. 
And it provides some understanding also, 
in the physics of unconventional 
superconductors. 
Although they're sensuously a high 
temperatures super conductors for example 
although their understanding is not 
complete. 
now as I well I shown many times already 
excitations for various Nobel prizes and 
and that just caused briefly a number of 
Nobel prize winning works and they should 
mid, admit that. 
In most of these cases I had to sort of 
simplify things enormously when I was 
providing explanations, sort of dumb it 
down to some degree. 
So here, well I'm not going to discuss 
the BCS-theory, this is a very 
complicated theory, but the paper of 
Cooper is actually very, very simple. 
So, the original paper which started it 
all is very simple. 
And we will be able to understand 
essentially its main message and the 
derivation. 
Now but before going into this derivation 
let me discuss the origin of this 
attractive interaction. 
So I basically, w,hat I, what I want to 
solve again is a problem of electrons 
interacting via an attractive potential. 
But where is it exactly coming from? 
So this is actually very complicated and 
intriguing question and well, it 
sometimes the mystery of let's say high 
temperature of super conductivity is in 
that, we don't really know where it is 
sort glue this attractive interaction 
might be coming from. 
So the origin of electron interaction and 
conventional super conductors is also far 
from obvious actually. 
So, but there was a breakthrough paper 
back in the beginning of the 50s. 
so I'm showing you here the title of this 
paper, Superconductivity in Iso, of 
Isotopes of Mercury. 
And to appreciate the importance of this 
paper, let me just remind you of the 
basic picture of, what a metal is. 
So, well, if you have a metallic system, 
well, assorted system, so the reason it's 
assorted is because, the ions form a 
crystal lattice. 
So they are positioned in space in a 
regular fashion. 
So I'm just drawing here for the sake of 
simplicity, as a square a lot as 2D but 
in relatives, it could be more 
complicated. 
Three crystal lattice And, you should 
think about having positive ions sitting 
here in these slides, and these line are 
slides. 
And these lattice is sort of elastic, 
well it's a recent object but it can have 
a oscillations, waves running through it, 
and we're going to discuss a little bit 
in the few, in a few weeks, and so these 
oscillations of the lattice are called 
phonons. 
Now electrons in the metal are moving 
moving around, they're free to move 
around, but they move around in the on 
the background of this lattice. 
So this red guys here are electrons. 
Okay? 
So this is pretty much what, well it's 
very [UNKNOWN] model, simplified, over 
simplified model of, metallic soil. 
Now, so this paper, what this paper has 
achieved is that it looked at different 
isotopes of mercury which is a 
superconductor. 
And let me remind you that the isotopes 
are essentially different versions of the 
same chemical elements. 
and basically everything is the same for 
different isotopes apart from the number 
of neutrons in each ion sitting on the 
lattice size, but basically. 
So different isotopes means that 
everything is the same, but the only 
difference is the mass of these objects 
on the lattice size. 
Which well now really shouldn't affect 
too much superconductivity because 
clearly superconductivity's coming from 
whatever electron, electrons are doing 
and it shouldn't have much to do with the 
lattice. 
Right? 
So because these guys are actually, well 
they can oscillate a little bit near 
their equilibrium positions, but they 
cannot conduct electricity. 
So as it turns out, and this is really 
the main message of the paper, that the 
transition temperature, in the 
superconductor, the temperature at which 
the resistivity drops exactly to zero, 
depended very strongly on the mass of 
these guys. 
So and the behavior the ions the lower 
was the transition temperatures. 
So here maybe it is hard to see so there 
are a few points so this has average mass 
number and this has transition 
temperature. 
So so this is basically let me write a tc 
and this is the mass of this ions on 
lattice size. 
Okay and so this was a very clear-cut 
trend and it was a smoking gun of that 
superconductivity had something to do 
With interactions between the electrons, 
and decrease the lattice. 
So a little later, people realized that 
what actually happens is that this these 
electrons, moving around, they whenever 
the electron passes through a region in a 
lattice, essentially because electron's 
negatively charged. 
And the ions sitting on lattice size are 
positively charged, so this electron 
polarizes the lattice, okay, so locally. 
And it takes some time for the lattice 
sort of to relax back. 
And then if a second electron comes in, 
so, let's say I have here now this guy 
goes away, and there's a second electron 
which comes in into the same, region. 
So, this, electron, this second electron 
is attracted in some sense to, to this 
region and, this, results in an effective 
correlation, effective, phonon-mediated 
attraction between the electrons. 
It's a very tricky business. 
So, I, if you don't understand it, well, 
don't be surprised. 
I don't understand it either. 
Attraction but that's how it works so, 
we'll have to you know just accept this 
fact. 
So that what happens is that electrons in 
some sense they exchange waves elastic 
waves which are called phonons and this 
leads to effective attraction. 
Well, and by the way for the model for 
the actual solution, I'm going to present 
the origin of the attraction does matter 
that much it will be just some 
attraction, some constant minus v not 
but, you know, it's good to know the 
physics behind it. 
For the actual compilation we'll need a 
specific mathematical model of this 
electron, electron attraction. 
And the true attraction the true 
interaction is actually quite 
complicated. 
But here I will present what I call a 
spherical cow model of the phonon 
mediated attraction between electrons. 
which I should say actually is used 
commonly even in research papers. 
So it works perfectly well. 
And even though we know how to write the 
true model which would involve all the 
complications of the theory. 
So the results, the outcomes, of the 
simplified model and the I'm not going to 
give a little more, or pretty close to 
each other, so there is really no reason 
for us to complicate things. 
In any case, so what we were talking 
about is that again, electrons moving 
around in a presence of a crystal 
lattice, they exchange, waves drawing 
through the lattice are these phonons. 
And so this diagram is an example of a 
fine metallic like diagram which shows 
this kind of a change. 
And in any process like this we must 
satisfy basic conservation laws 
particularly when must ensure that both 
energy and momentum are conserved. 
So and therefore what's important here in 
understanding the structure of this 
interaction is the typical energies 
involved in such processes. 
So what we know, what we've already 
discussed is the the typical energy of 
the electrons in the in the middle. 
So and this typical energy, I denoted as 
E sub F, the Fermi energy divided by the 
Boseman constant, sort of to convert into 
a more, familiar temperature units so the 
corresponding typical sort of temperature 
of electrons is going to be about 10,000 
Kelvin. 
So this guys which are actually the main 
players. 
Of the theory have the energy, which you 
shall see with this temperature scale. 
Now it turns out that the phonons, it's 
not obvious from the previous discussion. 
It's not something we can get at, I'm 
just giving you an experimental fact, is 
that if the typical energy of the 
phonons, which I will call s h times I 
mean divide, so called divide frequency 
divided by the Boseman constant. 
So when convert it into temperatures, 
it's about 400 kelvin. 
A few hundred kelvin, so, which is two 
orders of magnitude smaller than the 
energy of the electron. 
So what we should think about is that, 
this phonon actually being exchanged by 
the electrons have energy which is much, 
much more than the electron energy 
itself. 
And, this, puts certain constrains on the 
possible, momentum and energies of the 
electrons that may experience sort of a 
process. 
And the simplest, sort of, version of the 
interaction which takes this into account 
is, presented here. 
So this is an equation for an 
interaction, momentum independent 
interaction between the two electrons V 
of p and what it tells us that the two 
electrons attract each other in this 
language if they are located in the 
narrow shell near the Fermi surface and 
the width of the shell is of the order of 
this phonon energy. 
So basically if, two electrons can 
exchange phonons, they do so and this 
results in the attraction which is 
manifested here through this negative 
coefficient. 
And V naught is some constant. 
And if two electrons are located well, 
beyond this narrow region So then they 
don't have a mechanism to attract each 
other. 
And their interaction, therefore, is 0. 
So if we look just at the upper part. 
So v of p is equal to minus v, 0. 
So this would correspond to, essentially, 
a local attraction real speed. 
So this is something that we actually 
saw. 
In, lecture number five for the delta 
potential. 
So this is the kind of problem we, 
discussed already. 
And, but, however, then, we were talking 
about the, one particle in a, in an 
quantum well. 
So now we're talking about two particles 
two excitations interacting with each 
other by this potential. 
But per, well put the results of the 
previous segment in this lecture we know 
that the two particle problem and quantum 
mechanics is to a large degree equivalent 
to a single particle quantumic angle 
problems and as we discussed in the 
previous video the only difference 
between the. 
single particle problem in potential v of 
r and 2 particle problem of 2 particles 
interacting by the same potential is that 
in the lecture we will get the reduced 
mass instead of the mass of eac, of each 
individual particle. 
And in the case of two identical 
electrons the reduced mass is equal to m 
r 2. 
So we, we have this simply this 
Schrodinger equation. 
And here I write it first in real space 
sort of but keeping in mind that there 
are in fact additional constraints on the 
interaction. 
So if we do if we follow the same route 
that we did in lecture number five and do 
Fourier transform. 
So, we can write this equation in this 
form, so delta of r emptying on psi of r 
picks up psi of of 0. 
And so here I'm going to have this 
integral over all momentum. 
But basically a way to introduce 
constraints on the potential which we 
just discussed would be to limit the 
integration or momentum here by only 
momenta that appear in the vicinity of 
the Fermi surface, you know, so that the 
possibility of phonon exchange exists. 
And so this equation essentially, these 2 
equations with the appropriate sort of 
caveats. 
About where this reaction is possible is 
exactly the Cooper pairing problem, and 
as you can see it's essentially identical 
mathematically to the kinds of problems 
we saw in lecture number five in the 
single particle of quantum mechanics. 
And so, well we can we can follow the 
same route but at at this stage we may be 
a little bit surprised by this 
resemblance. 
Because, I mentioned that this, 
eventually the result of this calculation 
is going to be the appearance of a bound 
state between the two electrons. 
But, as we discussed, weak attraction in, 
three dimensions, and this is a 
3-dimensional problem, does not result in 
a bound state. 
So delta function potentially 3D does not 
have a bound state. 
Uh,however, an interesting thing that 
happens here is that because electrons 
really play a role in, in this, pairing 
and this interaction, with for-nodes, 
they exist in the vicinity of this firmer 
surface. 
And this surface is two dimensional. 
This, gives rise, in a sense, to 
reduction of dimensionality. 
Although, in real space, we started with 
a three dimensional, system, so this 
effectively what you are dealing with, 
with here has used dimension. 
And this gives rise to the appearance of 
a bond state in a very unexpected way. 
So, to see this we essentially have to 
repeat exactly the same steps as we did 
in parts 3 and 4 of lecture number 5. 
So, so the only difference here is due to 
the fact that this energy in the left 
hand, in the right-hand side, if we want 
to find, really consists of two parts. 
So this is the energy of the electrons 
that already have a finite energy at the 
at the Fermi surface. 
So remember is that we're talking here 
about these excitations that have the 
energy of two E-Fermi, and then there is 
an energy due to the interaction which we 
want to negative. 
So if this guy delta is negative, it 
means that we can have former bonds 
stayed, and by doing so lowering, lower 
the energy. 
So and well this fact actually makes all 
the difference. 
Actually this is how mathematically the 
fact that we're dealing with this 
manifest itself in this equation. 
Since the remaining calculation follows 
almost 1 to 1, that in lecture number 5, 
and I also present this particular 
calculation, all the details in 
additional materials, so you can read 
through this. 
So I'm not going to repeat this again in 
this video. 
So what I'm going to do, I'm going to 
just present the final result for this 
Our key parameter delta which again is 
the energy of the bounced state of two 
electrons that form sort of a large 
molecule, if you want, which is called 
Gutterberg here. 
So and this delta is is going to be of 
the water of minus h of energy so this is 
phonon energy, typical phonon energy that 
we discussed in the previous slide. 
But the key thing here is that there is 
multiplied by an. 
Exponential of minus one, some constant, 
I'll call it n naught, and define it in a 
second, times v naught. 
So where n naught is equal to m, p fermi, 
so this is this threshold momentum, 
divided by 4 pi squared, hq. 
So this is just constant, but what I want 
to emphasize, well, first of all, there 
is a solution, which is great. 
So the electrons indeed,can find a way to 
lower their energy by forming these 
[UNKNOWN] that eventually form the 
super-conductor. 
but what is not so great is that these 
small interaction energy appears in the 
exponential in, minus 1 over d naught. 
So if you notice, small then, well the, 
energy of the bound state, so 
called,[UNKNOWN], is exponentially small 
and this is exactly what happens. 
And those of you followed, all segments 
in lecture number five can recognize. 
This result is a result that we have 
seen, actually, in the context of 
two-dimensional, single particle problem. 
And this reflects this reduction in 
dimensionality that I mentioned before. 
So, the last two things I'm going to 
mention in the end is. 
First of all, this result. 
For delta is a function of not is a very 
unusual function which we've already seen 
but, this function is special because it 
doesn't have a Taylor expansion. 
So there is no way one can approach this 
result by doing so called perturbation 
theory. 
And this may be one of the reasons why it 
took so long for people to figure out the 
heat of super conductivity, because there 
was no way to approach, it sort of in a 
conservative way. 
And, the last thing I am going to 
mention, is that well, if we did not have 
this exponential, if we were to image 
that this delta was the order of this 
phonon energy. 
This would have actually implied that we 
can get super conductivity at very high 
temperatures up to room temperatures. 
So, this would have been great we would 
have the ability to transport electricity 
with no losses. 
So, this is really the grand challenge in 
the field of how to go beyond this 
coupling this so called weak coupling. 
Bearing model, so how to first of all 
describe this situation where the 
interaction is so strong and how to get 
rid of this exponentially this is a sort 
of practical problem that there's no 
solution at the moment.