Superconductivity is an example of an 
intrinsically many bodied quantum method, 
which doesn't really have a single 
particle counterpart. 
So it cannot be really understood based 
on just a single particle Schrodinger 
equation. 
So in the context of superconductors, of 
course, these many particles are 
electrons. 
So that play together in some sense to 
create this amazing quantum coherent 
state. 
Now, to understand the very complicated 
array of many, Body quantum phenomena. 
The first step is to understand the 
concepts of quantum statistics. 
Which is an increasing quantum 
interaction between identical particles. 
It doesn't really have a classical 
counterpart. 
So unlike let's say the electrostatic 
let's say [UNKNOWN] which depends on 
distance between the two charges. 
So this statistical interaction between 
identical particles is quite different. 
It appears to never have identical 
particles and they feel each others 
presence. 
independently of the distance separating 
them, so this is very interesting. 
And so we're going to learn about it and 
also we are going to learn about the two 
fundamental classes of particles, with 
respect to their quantum statistics, that 
are called bosons and fermions. 
And those guys they lose energy space, 
[INAUDIBLE] was very much drastically 
different. 
And we will see that it's actually a 
transition in some sense from fermion 
like behavior, to boson like behavior. 
That is exactly what explains transition 
from a regular metal into a quantum 
superconductor. 
The first step in defining quantum 
statistics. 
Let us consider two identical quantum 
particles. 
This blue represented by these blue dots 
here. 
And let's assume that all intrinsic 
properties of these particles are exactly 
the same. 
And so the question I'm going to be 
asking is what happens if we braid two 
particles? 
So, in other words, if these two 
particles exchange places. 
So the first particle goes to the place 
of the second one and vice-versa. 
So what it means is that the wave 
functions. 
In this case, the wave function 
describing this two, this two particle 
system depends well, on the coordinates 
of, of these two particles, r1 and r2, in 
principle, some other quantum numbers 
that I'm not writing. 
And so the question we're asking is what 
happens with this wave function if we 
replace r1 with r2, and otherwise errors. 
So it turns out that upon this braiding 
the wave function requires quantum 
mechanical phase. 
So this is statistical phase phi. 
And well we can sort of get an idea of 
why it might happen. 
Because the wave function really is a 
tool for us to understand the observable 
phenomenon, most observable answer. 
Through the let's say the absolute value 
of the wave function or the various 
metric elements wave function phase is 
not something we worry about. 
And so we may imagine that even though 
from our perspective nothing has changed 
since two particles are identical which 
one is at which point doesn't really 
matter. 
So we still main principle get a quantum 
mechanical phase here, and indeed this 
happens, okay? 
So and so the question that we can ask, 
and that was asked a long time ago what 
are the possible values of this quantum 
statistical phase? 
And the answer was provided in this 
seminal paper entitled The Connection 
Between Spin and Statistics, by Wolfgang 
Pauly, who proved there exists a 
one-to-one correspondence between the 
internal angular momentum of a particle, 
so called Spin that you're going to learn 
about a little later in the course and 
it's quantum statistics so I want to give 
you a lecture on spin because there will 
be a separate one on that but that will 
just mention that their are just two 
types of particles allowed in nature 
which have either integer or half integer 
spin. 
And those differ by how their wave 
functions transform under rotation. 
So if we rotate our [INAUDIBLE] 
coordinate system by a certain angle of 
this rotation [INAUDIBLE] that we would 
encounter would look differently 
depending on what kind of spin of the 
particle has. 
And, interestingly enough let's say spin 
one half particles an electron is an 
example of such a particle. 
is going to be described by simple 
representation of this rotation. 
Which upon repeating it by 360 degrees, 
actually picks up a minus sign. 
As opposed to integer spin particles, or 
particles with no spins that don't really 
change. 
Whose wave function doesn't really change 
upon this rotation by 360 degrees. 
Now I'm just sort of giving you a flavor, 
but the outcome of this so-called spin 
statistics theory can be formulated in a 
very simple way. 
So if a particle with integer a internal 
angle and momentum including zero the 
phase here is really zero, so there is no 
change in the wave function. 
If we replace r one bl-, by r2, nothing 
changes. 
Well, for particles with a half integer 
spin this statistical [INAUDIBLE], phi is 
equal to pi. 
Or, in other words, this psi of r1 and r2 
is equal to minus of psi of r2 and r1. 
And these guys with half integer spins 
are half integer spin are called 
fermions. 
So, we'll just make one comment that it 
turns out that this classification is 
specific to a three dimensional 
relativistic quantum theory to the world 
that which we actually leave but if we 
consider two dimensional systems which 
appear as a matter of fact in, let's say 
in condensed measure context. 
So we have two dimensional metals and two 
dimensional super conductors. 
So there it turns out that the 
possibilities for phi are much more 
interesting. 
So apart from zero and pi we may have an 
extra arbitrary phase of pi, which are 
called in the particles like that, so the 
fictitious immersion particles like that 
are called anions this pi can be a matrix 
in which case they are called [UNKNOWN] 
in news but this is the only thing I'm 
going to mention about it. 
But it just, I want to want you to know 
that the parts of these well known 
possibilities for fundamental particles. 
So there are other situations which arise 
in a [UNKNOWN] manner of physics. 
But now let me focus on this case. 
So well, so we for irregular particles we 
have I, either a zero or pi. 
So let's assume we're dealing with 
fermions, let's say we're dealing with 
electrons. 
So what are the consequences of this 
minus sign that we pick up in front of 
the wave function. 
Does it really matter at all, so should 
we worry about it? 
So let us think about it. 
So what we're really saying again is that 
psi of r 1 and r 2 is equal to minus of 
psi of r 2 and r 1. 
And it immediately leads to a conclusion 
that for an important case r 1 equals r 
2. 
So psi of r 1 and r 1 is equal to minus 
of psi of r 1 and r 1. 
Which means that the size is equal to 0. 
So what does it mean? 
It means that the probability of find two 
identical fermions in the same point is 
0, because the size is 0. 
So, well and It means that no two 
particles can be like no two fermions. 
Identical fermions cannot occupy the same 
point. 
And more generally we can actually 
generalize this for an arbitrary 
representation of the way function. 
We can prove and what this is about 
exclusion principle is that no two 
identical fermions cannot occupy the same 
quantum state. 
They cannot have exactly the same quantum 
numbers. 
But there is no such constraint for 
bosons okay. 
So, this has very important consequences 
so this statement again is called the 
Pauli exclusion principle and for bosons 
it doesn't apply. 
So, and that the consequences are that if 
we have certain quantum state, let's see 
an [UNKNOWN] quantum state i and we want 
to populate this state with fermions, 
there are only two possibilities for the 
occupation number of this state. 
So the occupation number can be either 0 
or 1. 
Well for bosons, it can be anything, it 
can be any 0, 1, 2, can put any, any 
number of bosons we want in this quantum 
[INAUDIBLE] . 
Now what does it mean in the context of 
many body systems, when we have actually 
more than one and two particles, when we 
have billions and billions of identical 
particles as we usually do in quantum 
systems such as, let's say, metals and 
superconductors? 
So let us discuss this question first on 
for many bosons system. 
And what we're actually asking is what 
happens let's say if we have many energy 
levels that are initially empty and we 
want to know what happens if we sort of 
pour identical bosons into this 
Prescribe, landscape of quantum states, 
with some states I with some energy E, 
energy E sub I. 
And more specifically what we're asking 
is, what we're interested in is what is 
the lowest energy state, the so called 
ground state, that the system would want 
to, Form at low temperatures. 
So at low temperatures the excitations 
are going to sort of calm down and the 
system would want to form the lowest 
energy state. 
Now since any number of bosons can occupy 
any state, so clearly the lowest energy 
state would correspond to essentially a 
state in which all bosons, 
non-interacting bosons that is, are 
sitting just at the lowest possible 
level. 
Now for a free particle with dispersion, 
just having the kinetic energy dispersion 
p squared over 2 m. 
Some sort of parabolic dispersion here, 
I'm just plotting it in sort of an 
example of a two-dimensional system px, 
py and this is the energy basically all 
bosons would want to drop into the zero 
momentum state. 
Okay, and this kind of state is called 
Bose-Einstein condensate. 
The state in which a single quantum 
mechanical level in this case the state 
was zero momentum, zero velocity is 
microscopically occupied. 
So this kind of phenomenon has been known 
for a long time, since Einstein. 
But it was sort of, very observed in the 
clear way, only very recently in 1995 
first. 
Now it's observed routinely in various 
laboratories including our own joint 
quantum institute. 
But back in 1995, it was new. 
And, this discovery was actually awarded 
2001 Nobel Prize in physics. 
So here, I'm showing the [INAUDIBLE] . 
So, basically, what the experimentalists 
did. 
They trapped atoms, bosonic atoms in a 
confined geometry in this certain 
trapping potential. 
And cooled them down to ultra cool 
temperatures. 
And then let them remove the trap. 
And let them expand. 
And they just sort of took photographs of 
these atoms as time went by. 
And they saw this pic. 
So this is basically the so called time 
of flight extension. 
And this pic is shows the population. 
The density of particles in real space. 
Large number of particles sort of don't, 
move too much, as time goes by. 
And this speaks sort of [COUGH] in real 
space is [UNKNOWN] of condensation and 
[UNKNOWN] space. 
So this corresponds to particles with 
zero velocity so the just sort of stay 
still. 
So now I would like to mention a very 
important circumstance namely that. 
This Bose-Einstein condensates may host a 
very interesting phenomenon of 
superfluidity where a condensate of 
interacting bosons form a superfluid 
which has exact 0 viscocity and which can 
flow therefore without resistance. 
So, here for example I'm showing a photo 
of superfluid. 
Helium in a cup, the helium which can 
actually escape the cup climbing up the 
walls. 
So let me mention that superfluidity, the 
phenomenon of superfluidity has actually 
been known for a very very long time. 
Since 1937 it was discovered then in 
Helium by Pyotr Kapitsa. 
And, and Kapitsa got his Nobel prize for 
this discovery of 1978. 
Well, it would be too late, 40 years 
after the initial discovery. 
So the mathematical theory of 
superfluidty was put together by Lev 
Landau a famous Russian theorist who got 
his Nobel prize for this theory in 1962. 
So actually before Kapitsa and hey, let 
me just mention that the You know this 
noble prize was awarded to him after Lev 
Landau got into a very serious accident. 
Everybody thought he wouldn't survive but 
by some miracle he lived for a few more 
years. 
And this allowed the Nobel community to 
award him long overdue Nobel prize. 
So, in any case, I'm not going to 
mention. 
I'm not going to talk too much about the 
theory of superfluidity now because it 
actually requires understanding of the 
interactions in Bose-Einstein 
condensates, and this is a non-trivial 
business interacting many-body states. 
But I will just that, superfluidity sort 
of rings a bell in the context of 
superconductivity because. 
If we want to explain the main property 
of a superconductor, namely the zero 
resistance state of the superconductor, 
we would want essentially a superfluid of 
electrons, which would move with zero 
viscosity and without resistance. 
This is exactly what the doctor had as 
prescribed, but there is one little 
problem with this explanation of super, 
superconductivity, namely well electrons 
in the metal are not [INAUDIBLE], they 
have half of half of the [INAUDIBLE], so 
they spin one half, so they're 
[INAUDIBLE], so they do not form a 
superfluid. 
So, what do they do? 
And, to answer, to answer this question, 
I need to consider the ground state of 
many-fermion system now. 
We talked about the ground state of 
bosons. 
Now let me talk about what happens with 
fermions if I pour them into a given 
energy landscape. 
Let's say I have, Has some available 
levels and they have a many fermion 
system that would want to occupy those 
levels and the question is, How will I 
minimize energy? 
So due to the power of exclusion 
principle I cannot allow my fermions to 
be sitting on the same lowest energy 
state. 
So I can only at minimized energy, can 
only sort of stack them up one by one to 
the lowest energy states as so until they 
reach the the certain level called the 
fermion level or fermion energy up here 
and basically this is the picture of 
fermion ground state, which is quite 
different from bosonic ground state. 
Now in the context of a metal and three 
dimensional, three dimensional metal, so 
what we have Is well our energy landscape 
is given by the usual disbursement 
relation. 
The energy versus momentum is equal to p 
squared over 2m. 
So what I need to do in order to sort of 
satisfy the same picture with this energy 
is that I will occupy all momentum stays. 
So here I have k x, k y,k z ,or I could 
have written p x, p y, and p z. 
So below certain momentum is called 
fermion momentum I will have all states 
occupied just like I have here all these 
levels occupied and while these levels 
are all empty. 
So and the threshold momentum it is just 
ignore this red and white dots for the 
moment. 
So below this threshold momentum. 
I have all states occupied above this 
threshold momentum I have all states 
empty and this is exactly the picture 
which describes electrons in, in metals. 
Actually it works amazingly well for 
metals. 
And it's also known that the typical 
Fermi temperature in metal, the Fermi 
temperature being the threshold momentum 
here The p, pF square divided by 2m this 
is the Fermi energy divided by the Bols, 
Bolsman constants or constant which 
converts energy into the temperature. 
So if we do this calculation the typical 
Fermi temperature in the metal would be 
around 10000k, which is huge so even if 
the room temperature left alone at 4 
kelvin so this energy scale is normal as 
compared to all other energy. 
[UNKNOWN] So and what the temperature, 
well Fermi temperature usually does is it 
excited electrons from below the thermo 
surface. 
For instance I can not excite electron 
from here to here. 
I can not move, so let me just draw this 
paralysis and then I will say it's not 
allowed. 
So have, let's say, an electron here, I 
can move it to this point, it's not a 
lot. 
Because both the stays are occupied. 
So this stay is already occupied. 
So power exclusion wouldn't allow me to 
do this thing. 
So, but what I can do if I have 
temperature, I can, I have some energy to 
spend, so I can excite an electron let's 
say from here to here. 
And so the y point here is an empty space 
that the electron leaves behind. 
It's called a hole and the red one is the 
actual electronic excitation. 
And this is exactly the picture of a 
metal and so this guys, basically this 
red guys and this white guys. 
these guys form basically they are the 
linked leaders in a typical metal. 
But they do not condense, they don't form 
a super fluid. 
They just move around and experience 
scattering you know from one point to a 
different point and this result in a 
finite resist. 
So how do you reconcile this picture of a 
metal with the presence of super 
conductor is that is observed at low 
temperatures. 
So this was again a major mystery for 
many many years, was not at all obvious 
how to do this. 
But of course there is a hint about it 
due to the presence of superfood so sort 
of in retrospect it's pretty natural to 
ask whether or not it's possible to 
somehow convert to electronic liquid 
[UNKNOWN] liquid Into a bosonic 
superfluid. 
So this guys are fermions, so they're not 
bosons. 
So can we make a bosonic superfluid out 
of a fermionic gas such as here? 
Well, it's difficult unless, so unless I 
allow my electrons to form composite 
objects. 
Of two electrons states whose total spin 
is either one or a zero. 
Remember the speed of a single electron 
is one half but if I combine the two of 
them together, so it's going to be an 
integer. 
So they are going to be boson. 
And so if I somehow find a way to say 
force objects well I could have actually 
drawn it like this but well knowing the 
results, I'm drawing these objects this 
week. 
But this station doesn't matter. 
So if I allow the formation of to 
electron states, this, these 2 electrons 
states are going to be bosons. 
And these bosons would be able to 
condense. 
And these guys would be able to form a 
super-fluid now, a charge super-fluid. 
And this is great because this state will 
superconduct. 
And this is exactly what happens and this 
is bound states of two electrons called 
Cooper burst. 
And from this perspective superconductor 
is a Bose-Einstein condensate of cooper 
burst. 
But the main question remains is why 
would this Cooper burst would form? 
Why would such states appear? 
And this is the question that we're 
actually going to answer in the remaining 
two segments. 
And we're going to solve the cooper 
pairing problem. 
And this was also established connection 
between this lecture and the previous 
lecture. 
When we discussed, [INAUDIBLE] states and 
weak potentials.