Until now we have been studying mostly
quantum theories and physical phenomena
which were discovered relatively a long
time ago.
So I think maybe the latest theoretical
development we talked about, the path
integral itself goes back to the 40's,
1940's.
Today in this video, towards the end of
the video I'm going to tell you a, about a
more recent theoretical development.
A so-called theory of quantum localization
or more precisely weak localization.
And that will do so using the main ideas
of the Feyman path integral.
So I should mention that the 4 theory of
quantum localization is an extremely
complicated mathematical theory and it
could be a subject though for separate
course.
So I don't have time for that and
obviously, we don't have the mathematical
sort of background to talk about it.
But it turns out that some basic ideas
behind this phenomenon can be understood
using the basic concepts behind the
Feynman path integral, which comes handy
here.
So just in a few words, what I'm going to
be taking about here is the following.
So we know that in solid state physics,
materials sort of can be divided into
roughly two categories.
Insulators and metals.
They're also semiconductors but from this
perspective, they're much like insulators.
So and if you, what insulators are, if you
apply a weak electric field the electrons
which exist inside the material don't move
and so there is no current really in
response to electric field.
So, If you think about a piece of food or
something like that.
So we obviously understand it's an
insulator, it's not going to be a very
useful electric component.
On the other hand, if you have a metallic
system, let's say a piece of copper, and
if we apply electric field or voltage
there will be current in response to this
electric field.
And so this characterizes a metal, so in
[unknown] there is some intersections
disorder, impurities , dislocation all
kind of things and these imperfections.
They lead to a finite resistance.
Okay, and so, the next slide will tell you
about how this resistance comes about.
But later on we will see how essentially
the classical theory of transport that
applies to most metal at high temperature
get more defined as you go to lower and
low temperatures quantum effects come into
play, the way of nature for instance, the
play, and this is very interesting and it
turns out that there under certain
circumstances you can get complete
completely different behavior at low
temperatures.
Metal actually would become insulators and
this process of trapping electrons due to
quantum mechanical effects is called
mobile localization.
Of course, it's a very mathematical
theory, as I told you, but we will try to
get some idea about where it comes from.
But before discussing a complicated
quantum theory, let me first talk about a
basic classical, very simple, theory of
how we can understand electrical transport
in a metal.
So here this black dot labels an electron
which can move around in, in a piece of
metal.
And these red dots here, large dots, they
represent some impurities, both of which
the electron can scatter.
And that's exactly what it does, as it
moves around so it gets scattered by these
random impurities and soon the trajectory
of this electron is this diffused
trajectory, so it's random walk.
Of course it's not just one, there are
many, many electrons in the metal and they
all experience this random walk.
Which in the absence of an electric field
just averages out to zero.
There is no net current, there is no
preferential direction, there is nothing
which breaks the symmetry.
But now, if we let's say apply electric
field in this direction the electrons
which have a negative charge are going to
move in the opposite direction
preferentially.
They are still going to move around in all
directions, but there will be sort of a
drift in one particular direction, and
this will give rise to an a current.
Now it turns out that the first theory of
this conductivity of this current in a
metal was put together back in 1900 and
this [unknown] theory is the so called
Drude theory which is a very simplistic
theory that I am going to explain now.
So, basically the Drude model assume that
we can just view the [unknown] as a
classical particle which there [inaudible]
was the classical equations of motion, the
Newton equation.
And in the presence of electric field so
that the charge, let's see if the charge
is q, so it experiences this force due to
the electric field, and these impurities
all these imperfections, he just replaced
with the friction force.
And this friction force was assumed to be
proportional to the velocity.
Lets say, it sort of makes sense if you
have a viscous fluid for instance, and if
you put.
An object is going to move in this fluid
so its going to experience some force, a
friction which is proportional eh
proportional to the velocity and with the
minus sign of course and there is some
coefficient.
So velocity is momentum divided by the
mass.
So we can write instead of writing the
friction force this way we can write it as
minus the p over t of some tau which is
the time scale which describes.
When we have the [unknown] in this prose.
So in some sense you may think about the
stall as a typical time between collision
for the electron.
But in the drude model it was just a sort
of in the logical parameter of the
described friction, or momentum relaxation
of this electron.
And now what he did, he simply wrote down
this model, and so of course we can write
the left-hand side of the Newton equation
as dp/dt, and so the right-hand side just
putting this together is going to be the
electric force and this friction force.
And in equilibrium, we demand that there
is no acceleration of electrons.
So if we, sort of, if we, if we take let's
say, a piece of metal, apply some electric
field, eventually it's going to come to
some sort of state of equilibrium.
There's going to be a current, but there's
not going to be indefinite acceleration of
electrons.
So they're going to come to a certain sort
of steady state, and this steady state
will determine the current.
Now and from here what we can do we can
just solve for, for the velocity so
momentum is equal from this equation is
going to be equal q times E times tal and
velocity therefore is going to be just
momentum divided by the map.
And at this stage what we can do, so we
can write the current.
So, the current is simply the charge of,
of the carriers in this case electrons,
times the density of these carriers how
many of those participate in transport and
v is the velocity, which we have just
determined.
And so, if we put everything together,
we're going to get this expression.
So, it's going to be this [unknown] of
friction, which is called conductivity and
times the electric field.
So, by definition actually the
conductivity is the coefficient of
proportionality between the current and
the electric field.
So this expression, this nq squared tau
over m is the so-called drude
conductivity, which as you can see is a
purely classical entities.
So there is no quantum mechanics here
whatsoever.
Now, embarrassing thing that I have to
admit at this stage, is that almost all
sort of useful theories of metals, of
transporting metals rely on this drude
equation up to this date.
So since, for more than 100 years we've
been using this equation and in most cases
actually it works fine.
But fortunately well for us scientists
otherwise it would have been really
boring.
So if you go to lower temperatures, the
oxidation becomes a little more tricky.
And this is because quantum mechanics
become, becomes important.
So quantum mechanics come into play.
And what it means is that, in this
picture, for instance, what, what, how we
can envision the appearance of quantum
effects is that instead of just straight
trajectories of classical particles, we're
going to have waves.
Bouncing ground of, of this incurities and
so this waves are going to interfere with
one at, with each other in some sense.
So its going to be the particles going to
interfere with itself and this quantum
interference [inaudible] are going to,
going to modify this picture a very
significantly.
So now let us look at the mo, electron
motion in methalanic disorder, a metal
from a slightly different perspective,
actually from a completely different
perspective because now we want to
exclusively to chernon quantum mechanics.
And we will use quantum mechanics
[unknown] of quantum mechanics due to
Feynman and ask the question of what is
the probability of for an electron to
diffuse from an initial point into the
final point F for this forest of impurity?
And according to the Feynman this
probability can be written as the absolute
value squared of a sum of quantum
mechanical amplitudes associated with all
possible classical paths.
So this diffusion trajectory here, I'm
just showing two such possible
trajectories.
You know this blue one and the black one
and this guys are going to interfere with
one another and this interference suppose
instead if I call this trajectory,
trajectory 1, and this is going to be
trajectory 2 so, the interference term
which is which is going to appear due to
this from this expression is going to
involve a product of this black
exponential [unknown] h bar times the Lew
exponential which is complex conjugate
[unknown] divided h bar plus its complex
conjugate.
So, oh no, the interference term coming
from these two particular trajectories,
again this is just two examples.
There are infinitely many such
trajectories and infinitely many such
term.
But in any case, each of these terms is
going to produce this sort of cosine of s1
minus s2 divided by h bar.
So this is basically the quantum
interference terms.
And the terms which don't involve the
cross product of different trajectories I
classical terms.
So in some sense, these are the terms.
Which are responsible for the judic
conductivity, the classical judic
conductivity that we obtained in the
previous discussion, okay and so now the
question of what want what is, what is
quantum mechanics doing, what are the
quantum directions goes down to figuring
out what this terms are.
Now in the beginning of this video I
mentioned something about temperature,
said well at high temperature so this
terms don't really matter at low
temperatures.
They start to measure and the reason this
is the case because at high temperatures
what the temperature really is, it
involves lot of motion which occurs around
these electrons which are sort of doing
their thing moving around, diffusing
around.
But there is also let's say lattice
vibrations, all kinds of things and this
sort of shaking of the electron disturbs
the phases, so in some sense this is what
is called sometimes the quantum mechanical
literature, this is called the phasing.
So basically here in this expression
whatever theory we're going to be doing
we're going to determine these actions
We're going to assume that phase one of
mechanical phase is going to remain
constant during the , or unperturbed more
to say during the motion of election or
along these trajectories, but it turns out
that even if we assume that there is no
disturbance of the phases and no new
phasing.
So classical physics in some sense
protects itself in that this quantum
interference theorems tends to cancel each
other out and completely disappear from
the picture.
And we'll actually have to search really
hard in order to find the [inaudible]
where they start being essential.
In order to see all this let me estimate
the typical action that appears in this
equation.
So remember that.
Well, at this stage, we're just talking
about particles bouncing, bouncing around
from these impurities.
But otherwise, in between each scattering
they are free particles.
So their energy is essentially just mv
squared over 2.
The kinetic energy of a free particle.
And therefore, in order to estimate the
action.
We can just say that well of the, by, by
the at the projector.
So now using this estimate we can also
estimate the typical a quantum
interference terms which are which I
described by the sum of over all pairs of
non-equivalent different trajectories L1
and L2 and now instead of the action S we
write exclusively at the typical momentum
which by the way is called Feyman momentum
divided by.
H bar times the difference in length
between the two trajectories under
consideration.
Or we can write the [inaudible] to be as
the cosine sum over l one not equal to l
two cosine of typical momentum u times
delta l, divided by h bar.
So it turns out that at these argumental,
the cosine in the typical metal is
actually very large and this is because
the typical velocity of electron in a
metal is less than the percent of the
speed of light which I can't[INAUDIBLE]
the speed of light which still is pretty
fast and therefore the typical firmly to
the typical wavelength turns out to be
less than a.
[unknown] minus nine meters, which is
usually much smaller than the distance
between these red dots here.
So I'm just giving you facts, it's not by
any means obvious, it doesn't follow from
the previous discussion and I'm just
giving you certain information and the
bottom line here is that the typical wave
length, in some sense you might think
about action's really being a wave, and
this wavelength is much, much smaller than
the distance between these red dots.
And so the interference terms are not very
important.
How it plays out here is that?
So, this typical [inaudible] of delta l
divided by essentially, this lambda is
very large.
And so, you have different trajectories
contributing to this sum.
And so you have cosines of different
phases, completely random phases.
And cosine as we know is the function
which can be a positive or it can be
negative depending so if this our cosine
of pi, this is pi.
So it can be positive or negative.
So if we completely pick this phase is
completely in random.
So the average value of the cosine is
going to be 0.
So the terms with different signs of
cosine basically cancel each other out and
this is exactly.
What happens with quantum interference
terms?
And so, seemingly classical physics
survives, and we can just completely
forget about this quantum interference
phenom.
So fortunately, it's not quite the case.
It's almost the case, actually in
[inaudible] disorder [inaudible] you see
there is a leap year when an electron can
actually travel in the clockwise
direction, and this is going to be our
path number one.
Or it can decide to travel in the opposite
direction, counterclockwise, in which case
it's going to be a path number two.
And the two paths, they're going to be
different paths but they're going to have
exactly the same way.
And therefore the quantum interference
terms responding to this pass are going to
survive the this phase difference will be
0 and this terms are going to play
[inaudible] significantly and so this is
where we going to task and this second
part of this of this video.
Now going back to little bit to physics in
order just to get give you an intuition of
what, what, what it has to do with
conductivity.
So we can sort of can individually
understand that the easier it is for an
electron to move from an initial point to
some just and final point battery, this
material conducts.
Okay the battery diffuses the battery it
conducts and the classical level it is
just [inaudible] by this [inaudible].
On the other hand the easier for the
electron to return to the same point the,
the more often it just sort of goes back
and back.
So the worst conductor it is.
So in some sense the probability to have
this self crossing trajectories, should in
general, by general arguments reduce the
conductivity.
And this is exactly what happens, and
we'll most cases.
And therefore, it is called [inaudible].
Basically tries to bring electron back to
where it came from.