Hello, welcome to this sixteenth class, in
this physics of materials lecture series.
So, in
the past few classes, we have developed the
Drude model for materials metallic systems
in particular and we looked at how well it
was able to explain thermal conductivity,
thermal properties, electronic properties
and also what it did with respect to the
Wiedemann Franz law. So, these are the things
that we did. We recognized that it had
some limitations that in terms of explaining
a material properties. It could only go so
far,
but was not able to successfully explain all
the major facets of material properties that
we
are able to measure.
As I mentioned in right at the beginning,
the way we proceed on this courses that and
in
fact the way we should proceed and in the
search in general is that, the experimental
information that we are able to obtain. So
far as we are able to obtain it properly,
where
we have eliminated or significantly reduced
or control the errors involved. Once you do
all that and we obtain some experimental information
that is the supreme piece of
information. So, any theory that you put together
no matter how suttle it seems, no
matter how interesting it seems, when you
write it down or when you explain it to
somebody. At the end of it or at the end of
the whole analysis, it should actually match
very well with the experimental data. Your
success depends on how well it matches with
that experimental data.
So therefore, all the work that we did with
respect to the Drude model and the analysis
that we did is partially successful. After
all it does explain some of the experimental
data
that we have seen, but we recognize that we
have to move forward, we have to look at
alternative to it. So that, we can explain
the properties more in a more complete manner
with less of the kinds of a loopholes or holes
that are there in the ability of Drude model
to explain the experimental data. So, in this
context we digress, we basically said that
you know Drude model treats the particles
as classical particles and we over the past
100
years or so, we have come to recognize that
there are quantum mechanical effects that
there are effects that we call, we now recognize
as a as or designate a quantum
mechanical effects. And that they are very
relevant to the kinds of system that we are
dealing with, in which, in this case bunch
of electrons in a solid.
So, in this context we digressed a bit, we
looked at the history of quantum mechanics
in
the last couple of classes and we did this
particularly to understand or to pull together
all
the major concepts of quantum mechanics that
we will use through this course. And also
to understand; where they came from? How they
related to each other? And perhaps also
the kinds of difficulties we face, while utilizing
those concept for trying to understand
those concepts. And so, that is where we are
now and we will proceed forward from
here.
I am right now showing you the in summary
the major concepts of quantum mechanics,
in the single slide all that we discussed
to the last of last couple of classes. So,
the very
first and foremost equation of quantum mechanics
that we will that we saw was that the
Planck has given to us, Max Planck. And that
is simply that energy is h times nu for for
a
radiation of frequency given. In other words,
it is in that quantum of h nu that you can
exchange energy with with a system that is
a giving out energy with that frequency nu.
And specifically the whole idea of quantum
mechanics comes down to this one equation.
In the sense that, if this h had turned out
to be 0, then we would be existing in a world
where quantum mechanical effects did not actually
exist.
So, if h had turned out to be 0, it would
be in that a particular source of energy or
or a
electromagnetic radiation of frequency nu,
could give out could exchange energy with
any of its surroundings in or in any manner
in in any amounts. So, in in extremely small
incremental amounts to very large amounts
that we know real restriction in how it would
go about in this transaction of energy. The
fact that the h is actually non 0, even though
it
is small, implies that now the transaction
can occur only in steps of h nu and this is
what
Planck discovered. I mean he assumed that
there are may be step size, which he was not
aware of and he treated that to be h nu and
he ran through the calculations with respect
to
black body radiation and came up with the
value for h.
And he found that it was non 0. Now that,
he has found he found that it was non 0, the
implication was that the transaction could
occurs could occur only in the in step sizes
of
h nu. So, it could be h nu, 2 h nu, 3 h nu
and so on. Which which of course, meant that
if
the frequency was very high that h nu value
would also be high and it was quite possible
that the source of energy did not have the
amount of energy required to be equal to that
step size. And therefore, if we went to higher
and higher frequencies that source of
energy would not be able to transmit energy
at those higher frequencies or absorb energy
at those higher frequencies. So, this is the
basic idea that Planck discovered and that
has
formed the basis of quantum mechanics.
And as I mentioned in those last 2 classes
that, he himself was quite uncomfortable with
his idea that there was a step size involved
in this process and several of the scientists,
leading scientist; who worked on quantum mechanics,
who have developed all of the
theory that we today accept as quantum mechanics
have all been uncomfortable at one
point or the other with this idea that there
was a step size and that nature had a step
size.
So that is important thing. That that we are
finding that nature has a step size and that
step size for these transactions is h nu and
this is something that lot of people felt
was not
natural for from an intuitive prospective,
but it turns out that if you look at all the
examination that people have done over the
years seems to indicate that this basic
concept is right. I mean that all of the there
is so much of data that had not been
explainable in about in hundred years ago,
out of which seem to fall into place the
minute we accepted that there was this h size.
So, in even though as a from our own
intuitive feel of nature perhaps it comes
across as something that is not natural, in
that
there is a step size apparently nature does
have this step size and and therefore, we
have
to accept is as is.
So, this is the first equation, foremost equation
that brought us to quantum mechanics
and it was put together around in the year
1900, Albert Einstein extended this. And he
explained the other phenomenon, which is photoelectric
effect. Planck had explained
black body radiation, Albert Einstein explained
photoelectric effect. Where he took
exactly the same idea and see he said that
light would now be able to initiate would
would then be absorbed from an electromagnetic
radiation would be absorbed by a body
which could then give out photoelectrons,
only in step sizes of h nu. Therefore, he
wrote
down this equation that we see here, for the
photoelectric effect. So, this is the equation
he put down. The idea also came about that
people began to think that if light could
be
thought of as particles, which would then,
which were then being called as photons and
then it was quite possible that may be particles
also displayed wave like behavior.
And this was demonstrated using techniques
which enabled us to see electron diffraction;
for example. And then the de Broglie equation
came about which basically said that, if
there was a any particle that had a momentum
p, then we could associate it with it a
wavelength lambda and that they would be related
then by lambda equals h by p. So, this
was the de Broglie relationship. So, these
sort of formulize the idea that all matter
could
now be associated with a wavelength. So in
fact, even macroscopic objects could be
associated with a, could be assume to have
a wave like behavior and using this equation
you could come up with the wavelength corresponding
to those objects.
You would find then in fact if you did this
analysis, you would find that for macroscopic
objects, the kinds of numbers would come up
with would put you in a situation, where it
is more than enough to treat it simply as
a particle and not bother about the wave nature
of that object.
So therefore, it is not that the quantum mechanics
principle, these principles that have
put forth become irrelevant or are inconsistent
with the macroscopic life as we see it. It
is
just that the effects become lesser and lesser
significant, as you get a larger and larger
objects. And so at at, some point it is very
very reasonable for you to assume that we
can
neglect the effect and you will not know any
difference in the calculation in in all that
we
experience and so on. Therefore, quantum mechanics
pervades all aspects of life of of
nature from very small scale to larger scales,
but largely it is not of enough significance
at larger size scales. So, having then discovered
that, you know having now reach the
situation where we recognize that matter can
also show wave like behavior.
All of this information of quantum mechanics
was then captured together elegantly by a
Schrodinger, when you put together this Schrodinger
wave equation. So the Schrodinger
wave equation is here. And so, he basically
captured the idea that since everything had
this wave like behavior, we could now associate
with any particle or any system. A wave
functions psi and that wave function psi would
then capture the most important aspects
of that system, the attributes of the system.
And then the way function psi could then
could actually be obtained by solving the
Schrodinger wave equation, which would
actually put together the constrains that
the system is facing. So, that that is how
the
Schrodinger wave equation is. The Schrodinger
wave equation actually pulls together all
the constrains that the system faces and if
you pull all those constrains together and
place
it as the terms of the Schrodinger wave equation,
what you will get out of the
Schrodinger wave equation is the wave function
psi. Once you get the way function psi
that then represents the system. It captures
all the details of the system. So, he came
up
with this and it seem to actually, nicely
meet the requirements of the quantum
mechanical description of a systems, but there
was some problem in interpreting what psi
was? In I am trying to understand; what the
psi represent?
And therefore, the other major contribution
was that from Born Max, Born he basically
said that psi psi star d x would then represent
the probability of an electron existing in
the
location x and x plus d x. So, that is the
contribution of Max Born, which would also
be
the modulus of psi square. So, that was the
contribution there. And then finally, there
was Heisenberg, who recognized that when once
once you have things, once you have
particles and systems being described using
way functions, then by the very nature of
how the description comes about, there there
is an uncertainty principle that is present.
So, this is the very again, a very non intuitive
kind of an or what should I say?
Relationship which does not immediately become
convenient for us to accept because of
our experience with large scale objects, where
we apparently do not have any uncertainty
in trying to identify the velocity or position
of a ball; for example. So, of a macroscopic
ball; for example. So he does something that
again, he shows to you that the effects of
quantum mechanics are not that significant
when you get a large scales, but when you
go
down to small scales, this is an the impact
of that particular kind of a behavior quantum
mechanical behavior are becomes very significant.
So, this is something that we also
discuss, I also indicated you that especially
with respect to the Heisenberg uncertainty
principle that there is this general based
on the descriptions you hear of it. There
is a, it is
likely that you may get the impression, that
it has simply a experimental limitation and
I
specifically emphasize to you that it is not
simply a matter of experimental limitation.
It
is not simply a case of if you try to observe
something you may disturb it and and so on.
Although, descriptively that seems to convey
in a simple sense, what the uncertainty
uncertainty principle is, but more so it is
it is something more fundamental than that.
It is
a fundamental requirement that if you treat
particles have has a waves distributed across
space, then the more wave functions that you
need to add, to get it to get localized would
increase the uncertainty, would all enable
the particle to have possessive wide range
of a
momenta; for example. So at a given point
in time the more precisely you get its
position, the less precisely you can specify
it is a momentum and this is got to do with
the fact that they are mathematically related
each to other as conjugate variables.
So so, these are all major concepts that we
saw for quantum mechanics in the past few
classes and the reason we saw it was was of
course, that we realized that the classical
description of a particles was not good enough
to serve our purposes. Now, we will move
forward from here, we will try to develop
our next model or improve model. So, to speak
for understanding the behavior of a electrons
in a solid. And therefore, explaining the
properties of a the solid. So, before we do
that I will briefly introduce to 4 personalities,
historical personalities, of great importance
in the history of science. A couple of whom
I
have already mentioned, but we will still
go over then and then we will see, where it
takes us.
In chronological order in terms of their accomplishments,
the first person I will introduce
you to is Heisenberg, whom we have discussed
in great detail including the very last the
very last equation that we saw of the uncertainty
principle. So, Heisenberg received his
Nobel prize in 1932 and as you can see, it
is for the his contribution to quantum
mechanics and such. So, he was it was a Nobel
prize in physics, in the year 1932 and as
I
indicated his contributions were very significant
to the quantum mechanical description.
Then we come 4 years down, we we meet Peter
Debye, who received a Nobel prize in
1936, in chemistry not in physics in chemistry.
And he he had made major contributions
to molecular structure. So, molecular structure
and through his investigations on dipole
moments and also to the diffraction of X rays.
So, X ray diffraction was another aspect
of a Debye’s contributions. So, there are
cameras which are Debye Scherer cameras and
so on. And so, he had made very significant
contributions in all these aspects and for
this
he received a Nobel Prize in 1936. And Peter
Debye is of course, a Debye is a name that
a people become very familiar with when you
study science.
The third person I introduce you to is Wolfgang
Pauli, again Nobel laureate, Nobel Prize
in physics, in the year 1945. And of course,
his contribution is something that we
mentioned earlier on which is that he put,
he postulated this Pauli’s exclusion principle.
So, he basically said that you know, if you
have this quantum mechanical description of
systems existing then you cannot have 2 particles
which have exactly the same quantum
numbers. They cannot they cannot have all
of the quantum numbers being exactly
identical. So, that is the Pauli’s exclusion
principle. It is a very important contribution;
in
fact we will immediately be using it in this
later in this discussion and even in the
discussion that will come up in the next class.
So, Pauli’s exclusion principle is a very
integral part of how we are going to look
at the behavior of electrons in solids. So
he got
a Nobel Prize in physics, in the year 1945
so.
Much later 20 years down the road, we meet
Hans Bathe, again a Nobel laureate in
physics. So, his contributions are to the
theory of nuclear reactions and especially
his
discoveries concerning energy production in
stars. So, we have 4 Nobel laureates spaced
out in time over a period of about 35 years.
So, Heisenberg for uncertainty principle,
Debye for contributions in chemistry and in
X ray diffraction. Then Pauli’s exclusion
principle and then now we have Hans bathe,
who has looks at the generation of a energy
in stars and such. So, these are 4 personalities,
who apparently are you know spread out
across the wide areas of science have and
not at first glance may be expect for a for
Pauli
and Heisenberg perhaps not immediately relatable
to each other.
And spread out in time, spread out in areas
and science, but perhaps the thing that is
common to them, the immediate obvious thing
that is common to them is that there are
Nobel laureates. And so, their highly accomplished
in in their areas of work. What is of
interest to us, is to is to recognize that
there is something more common to this 4 people.
Which which is more relevant to our immediate
discussion. What is common to this 4
people and relevant to our discussion, is
the fact that all of them did their P H D
or
doctoral thesis under the same person. There
was one teacher for who was common to all
of these people, they are all they are all
students for the same person and who is this
person?
This person is goes by the name; I have the
name Arnold Sommerfeld. So, he is
considered a very eminent scientist and several
of his students, you saw now 4 of his
students won Nobel prizes. Which is perhaps
perhaps quite unique, I am not aware of
any other persons, whose whom who had 4 students
earning up us Nobel laureates. So,
he has this distinction that he had 4 students
who became Nobel laureates. He had
several students, who who went on to win several
awards themselves and are or also
extremely distinguish. The specifically the
4 that we discussed went on to win the Nobel
Prize. Sommerfeld, himself was nominated for
a Nobel Prize several times, although he
did not win it.
And he has made major contributions in the
fields of in the fields of mathematics, in
the
fields of physics and so on. So, he is an
highly accomplished person, who did not who
of
course, who apparently did not win the Nobel
Prize though, but has won a number of
other awards and was the doctoral adviser
for 4 people, who won the Nobel Prize. So,
his
contribution is many fold as I mentioned,
the specific contribution of a Sommerfeld,
which is interest to us is the Drude Sommerfeld
model. So, what and that is the model
that we are going to examine in some detail,
the in the immediate time that we are going
to discuss this.
And what he has actually done is he looked
at the Drude model and looked at what he
said that was good about the model, what was
said that apparently seem to be lacking in
the model and then try to make improvements
on it. So, he came up with a new model
which is a modified Drude model, as you made
you may want to call it and in fact as
called a Drude Sommerfeld model or this is
considered very important contribution.
Because it really helped move forward the
theory of the solids, to understand why he
said this solids have the properties that
they have and what can we do, what can we
say
about the fundamental properties of the constituents
of the solids and how they add up to
give us the property of the solid. So, the
Drude Sommerfeld model is what we will look
at in the immediate.
So, the Drude Sommerfeld model is what we
will see, it is actually in continuation with
the Drude model as I said, it is going to
take some features of the Drude model and
it is
going to improve on that. So, we will see
the features that it takes up and features
that it
improves in our discussion right now.
So, the Drude Sommerfeld model, I said the
Drude Sommerfeld model is first of all a
free electron model. So in this sense, in
the sense that we call it a free electron
model. In
this sense it is borrowing the same idea or
it is starting out with the same idea that
the
original Drude model did.
So, in in the sense that, when we say it is
a free electron model, what we mean is that
you
would think of a solid as containing those
ionic cores, which are present within the
solid.
And that there are electrons, which are free
to run across the extent of that solid and
that
largely those electrons are not really impacted
by those ionic cores. So, they are they are
free, that is what we mean. There is no specific
preference for them to be at any one
location, they can freely run across, all
of those electrons can freely run across the
extent
to the solid. This was a primary assumption
and requirement in the original Drude model
and it is it remains an assumption and a requirement
in the Drude Sommerfeld model.
So therefore, in this sense it is the same
as the original Drude model. Then, the
immediate thing that, the Sommerfeld model,
the Drude Sommerfeld model does is that
it applies, it takes quantum mechanical principles.
And we have discussed all the major
quantum mechanical principles that we will
atheist immediately use and it takes quantum
apply mechanical principles and it applies
them to the Drude model. So, what that
means, I mean it is put down as a sentence
here, that it applies quantum mechanical
principles to the Drude model, exactly what
that means we will we will see in in just
a
few moments.
So, but the idea is that that in the original
Drude model, the fundamental idea that the
electrons are classical particles is being
utilized. And we have discussed that, we will
again touch up on it at least, but here we
specifically move away from that idea. We
recognize that, we cannot merely treat electrons
as classical particles; there is something
more to it. So, we recognize this idea and
in the Drude Sommerfeld model that is
basically done. It is formally incorporated
into the model, formally we incorporate the
fact that the electrons are not classical
particles, they are quantum mechanical particles,
they show quantum mechanical behavior.
And therefore, the way in which we handle
the electrons in our analysis has to change,
to
accommodate for the fact that they are showing
displaying quantum mechanical
behavior. Specifically, the model also incorporates.
So, specifically the model also
incorporates the Pauli’s exclusion principle.
So, we already have the quantum
mechanical behavior that is being incorporated,
in that specifically we also incorporate
the Pauli’s exclusion principle. So, that
is what the Drude Sommerfeld model is doing.
Incorporating the Pauli’s exclusion principle
into the Drude model, the Drude
Sommerfeld model does that additional thing.
It also makes the assumption that; so, this
is an assumption that again existed already
in
the Drude model. And so, this is simply continuing
with that assumption that potential is
constant within the solid, this ties with
this idea that it is a free electron model.
So the
fact that the potential is constant within
the solid, simply emphasizes the idea that
the
electrons have no specific preference, that
they need to be at one that all the electrons
need to be in any one preferred location,
all the free electrons. So, it recognizes
that there
are electrons, which are tightly bound, closely
bound to the ionic cores and neglects
them. And then treats the whole of the solid
as being of some kind of a uniform potential.
So that, the electrons that have escaped from
each of those electron atomic or ionic cores.
And therefore, are the free electrons that
fill this solid, those electrons are actually
free to
run through the entire extent to the solid,
there is no preferred location in terms of
potential that they would have to gather or
where they would have, which they would
they have to avoid. So, this is being assumed
in this model. So, as these are the major
ideas that the Drude Sommerfeld model employs
and the manner in which we the
significance of these ideas we will explore
a little more.
And then we will see that, there is a formal
way in which we can incorporate these ideas
into the model from a mathematical prospective.
So, these are the major things for the
Drude Sommerfeld model.
So when we say that the 
the ideas that I have specifically indicated
as the Drude
Sommerfeld ideas. When those ideas are now
incorporated into the model, there is
specific implication to incorporating those
ideas. So, what we will look at briefly or
the
the implications of those ideas. Excuse me.
The first is that a electrons are identical
and indistinguishable. The first is that electrons
are identical and indistinguishable. So, this
is the basic idea that is being that this
is the
basic point, where in we are incorporating
the fact that the Drude Sommerfeld model is
employing quantum mechanical principles in
the model, where originally the Drude
model did not look at quantum mechanical principles
treat at the electrons is classical
particles. So, the electrons were treated
as identical, but distinguishable particles.
Now
because we are using quantum mechanical description,
they are giving treated as
identical and indistinguishable particles.
So, this is where the quantum mechanical
principle comes in.
So so, we will see the significance of that
in in just a moment. And therefore, the fact
that they are identical and indistinguishable
impacts directly impacts the way in which
we do the statistics of the set of particles.
So, once again as we did with the original
Drude model, we will now have to develop a
statistical distribution for how the electrons
behave within the solid and except that that
will now incorporate the fact that the
particles are identical and indistinguishable.
And therefore, that impacts the mathematics
the way in which we put in, put the equations
together to develop that statics. And
therefore, the result will will also change.
So, when you do that the original work that
that actually did this. That looked at identical
and indistinguishable particles. And therefore,
particles that were demonstrating quantum
mechanical behavior and also incorporated
the fact that they were obeying Pauli’s
exclusion principle. So, those that combination
of identical and indistinguishable
particles obeying Pauli’s exclusion principle,
that combination was a a examined and the
statistic corresponding to that combination
was first put together and demonstrated or
indicated by Fermi and Dirac. And so, it is
named after them.
So so, there is this statistics, which are
referred to as Fermi Dirac statistics or Fermi
Dirac statistical behavior. So, Fermi Dirac
statistics actually examines the, excuse me,
the behavior of a set of particles, which
are showing us the quantum mechanical behavior
in that they are identical, but indistinguishable,
but also that they are actually following
Pauli’s exclusion principle. Now the thing
is the for it to follow Fermi Dirac statistics,
the we are saying that Pauli’s exclusion
principle is followed and which normally means
that it applies to basically particles that
have half integer spin. So, one of the criteria
is
that you should have half integer spin.
And and therefore, the particles that are
having this half integer spin and are identical
and indistinguishable and follow Fermi Dirac
statistics. So, this combination is then,
this
a particle that does all of this is referred
to as a Fermions. So Fermions, these particles
that follow the Fermi Dirac statistics and
therefore, actually also have a half integer
spin.
They are referred to as Fermions. This is
to be distinguished from our original set
of
particles that we explore explore explore
or investigated under the original Drude model.
So, the Drude Sommerfeld model uses the Fermi
Dirac statistics to describe electrons in
a solid.
And therefore, incorporates all of those ideas
that we are seeing, that the Drude
Sommerfeld model is trying to put into the
picture, which is that the electrons or
quantum mechanical particles and they are
following Pauli’s exclusion principle. So,
the
Drude Sommerfeld model treats the electrons
as Fermions. Now that you understand
what we mean by Fermions, I can simply say
this that the Drude Sommerfeld model
treats the electrons in a solid as Fermions.
And therefore, they meet all these criteria.
They also; therefore, we we will put that
down the original Drude model, treats the
particles as classical particles.
Drude Sommerfeld model 
treats the electrons as Fermions. So, this
is the difference.
Drude model treats the particles as classical
particles; the Drude Sommerfeld model
treats the particles as Fermions. So, this
is the difference between the Drude model
and
the Drude sommerfeld model. So, and I mentioned
that we are talking of, when we say
classical particle versus a quantum mechanical
particle. We are basically saying that in
classical particles, the particles are identical,
but distinguishable and in quantum
mechanical particles, we are talking of identical,
but indistinguishable particles. So,
when we build this statistic, we were we are
actually going to build this Fermi Dirac
statistic. We are actually going through,
going to do this calculation, which is the
Fermi
Dirac statistic.
So, the Drude model, which applies classical
particle behavior is what we saw and for
this we actually went ahead and we we explore
this in greater detail. We try to
understand, what we said that, when we say
classical particle, what is the statistical
behavior that we need to impose on the system
or or what is that behavior that we are
assuming about that system. When we did that
the statistical distribution that we found
was of relevant to this kind of behavior was
the Maxwell Boltzmann statistics. So, when
we used classical behavior for the particles,
when we assume that the electrons behave
like classical particles, effectively we had
impose the Maxwell Boltzmann statistics on
those particles. So, all the results that
we got were consistent with the fact that
those
particles were assumed to obey the Maxwell
Boltzmann statistics. So, an and we went
ahead and derive this statistics. We we looked
at the mathematical process by which, if
you mathematically incorporate all those descriptive
things that we say about those
particles, we will end up with set set with
a certain set of equations, it would then
give us
how the particles are distributed across all
the energy levels available in that system.
And
so that, is how we came up with the Maxwell
Boltzmann statistics.
Now, we are saying that, we have modified
the model and it is now the Drude
Sommerfeld model, the Drude Sommerfeld model
has changed the kinds of assumptions
we have made about the particles. And therefore,
as I mentioned of the beginning that set
of assumptions then represents the model that
we are talking of for the particle. So, the
Drude Sommerfeld model changes those assumptions,
it is basically says that now the
particles are identical and indistinguishable.
Therefore, they and therefore, they are no
longer classical particles and and therefore,
the statistical distibution way in which they
are distributed changes. Or they are following
Pauli’s exclusion principle. So, all of
that
is captured in the set of particles that are
called Fermions using the Fermi Dirac statistics.
So, this Fermi Dirac statistical behavior
or the Fermi Dirac statistical distribution
is the is
the thing that we are going to derive. So,
we we will derive this Fermi Dirac statistics
because it is of immediate relevant to what
we are doing, in particular it will also tell
us
specifically what is it? We can say about
electrons present within the solid, when we
understand, when we pull this derivation together,
we will understand in in the the results
of the derivation will show us how we have
changed the picture of the electrons in the
solid.
When when I say we have changed, all I am
saying is we have, how we have changed
from a theoretical prospective, the electrons
in the solid already already have whatever
it
is that they have. So we in that sense, in
the in that fundamental sense, we are not
altering anything about the solid. We are
only trying to look at our ability to describe
it
and so our ability to describe it using Fermi
Dirac statistics changes the kinds of things
we could, we will say about those electrons.
And and on the basis of those change
statements are we are now able to make of
those electrons.
We will see, if some of the properties that
we have a difficulty explaining; for example,
this specific heat. That we have a difficulty
explaining using Maxwell Boltzmann
statistics, we will see if by using Fermi
Dirac statistics that anomaly that existed
on the
on the a prediction of the specific heat of
the electronic contribution to specific heat.
That anomaly that we face, we will see if
by simply incorporating Fermi Dirac statistics
are we have been able to overcome that problem.
Therefore, that is very important
contribution, here we we over estimated the
electronic contribution to specific heat by
a
factor of 100. So, that is quite a significant
over estimate. So, two orders of magnitude
we we over estimated, we will see if by using
Fermi Dirac statistics we are able to
actually make make the correction.
So so, these are, this is the direction in
which we will proceed, that we will look at
the
contributions in that way. Even as so, as
we go ahead and make the development, we will
I will just high light here, some of the key
aspects in which these 2 statistics are going
to
differ. So, and that will set the base that
will enable us to then properly relate whatever
it
is that we are deriving at this point with
what we have derived earlier. So, I will high
light those specific aspects of those derivations
that are different, that are going to be
different. Now relative to what we have derived
earlier on for the Maxwell Boltzmann
statistics.
So, the first thing that we first manner in
which, these 2 differ differences. The first
thing
is of course, as I said descriptively Fermions
are identical and indistinguishable.
Whereas, Maxwell Boltzmann classical particles
are identical and distinguishable, what
is the difference from a mathematical prospective?
For a mathematical prospective, what
it means is that, when we try and write this
equations for the manner in which the
particles are distributed at various energy
levels. If you consider a situation, where
you
have a certain number of particles in one
energy level and so let us say, n 1 particle
sitting in one energy level and n 2 particle
sitting in a other energy level. If you have
this
situation, if you simply swap 2 particles,
if you move one particle from the higher energy
level to the lower energy level and at the
same time move another particle from the lower
energy level to higher energy level. What
have you done? You have not changed the
number of particles at each of those energy
level. So, you still have n 1 particle sitting
at
the higher energy level, you still have a
n 2 particles sitting at the lower energy
level. So,
in terms of the number of particles those
2 energy levels, you are not changed anything.
Now if you have this situation, but they are
classical particles, since the each particle
is
distinguishable from the other, simply swapping
these particles would now be treated as
another arrangement.
So, even though the number of particles at
the higher energy level remains the same,
number of particles in the lower energy level
remains the same and everything else about
the system remain the same. The fact that
you swapped 2 particles, you moved 1 particle
up and simultaneously you moved another particle
down. This situation, this step, it will
now result in the system being treated as
though it had attained a new state. So, it
would
be counted as a new state, in a classical
system, in a in a classical way of counting
the
statistics of the state of the system.
So, when we looking at micro states of the
system, macro state of the system and so on,
this would be one another way in which the
same micro state is being attained. In in
the
quantum mechanical description of the system,
when you when you say that the particles
are identical, but indistinguishable. When
they are identical and indistinguishable,
if you
swap a particle you have n 1 sitting at higher
energy level, n 1 particle sitting at a higher
energy level, n 2 particle sitting at a lower
energy level, when you just take 1 particle
from there move it down and take 1 particle
from below and move it up in energy level.
Since the particles are anyway indistinguishable,
what this means is that this swap cannot
be treated as a new state, it cannot be treated
as a as a new implementation of that micro
state. It cannot be treated as that and why
is it, why is this issue coming? It is coming
because of what I already described for you
regarding this idea of indistinguishability.
It
is simply that, when you say it is a quantum
mechanical particle, it we no longer
uniquely think of it is a hard object, we
think of it has being distributed in space,
as
something that has being distributed in space
there is a probability of it is existence,
which is distributed in space. So, when when
when you have 2 particles have with
certain, I mean to identical particles 2 electrons
for this in this case, having specific
attributes, the fact that they are in this
indistinguishable and the fact that they are
actually
just probability distributions across a particular
regional space. Simply, implies that there
is always an inherent chance that, they may
swap with each other.
So, more specifically we use the example that
when they colloid and they move up apart.
If they were classical the fact that, you
they collided and moved apart, would still
enable
you to say what was the ball that started
on the what does the particle that started
on the
left side and where it ended up? What was
the particle that started on the right side
and
where did it end up? The minute it is quantum
mechanical and these are only probability
distributions, when you go through this process,
there is as a chance that they would have
swapped, there is a non 0 chance that they
would have swapped. And therefore, you
cannot say for a fact, that the particle does
started with on the left side is the is the
particle that is sitting here, the particle
that started on the is the particle that is
sitting
here. They might have anyway switched. So,
when you have such a situation, when you
have 2 energy levels and you switch, you do
everything else is the same and you simply
switch the particles, you cannot with confidence
call it as a new state. Because for all
you know it might have occur at anyway and
even if you had not intended for it occur.
So, between the Drude model using Maxwell
Boltzmann statistics, so more specifically
between the Maxwell Boltzmann statistics and
the Fermi Dirac statistics. The first and
most important thing that changes is the manner
in which you count the number of micro
states. And that is very integral, I mean,
when I say that I have change the manner in
which I am counting the number of micro states
or the number of ways in which that
micro state can exist, if the the minute I
change that manner in which I do that. At
that
very significantly alters the result that
I am going to end up with. Because that is
the
basic idea that is there in that whole process.
It is the it is the most it is the core of
that entire statistical distribution process.
So,
whatever result we get, very critically depends
on the manner in which we count the
number of ways and which the micro states
can exist. So, a fundamentally they differ
and
I and I just mentioned, they differ simply
because of the character of the particle that
we
have made an assumption, the assumption that
we making about the character of the
particle. So, that is the fundamental manner
in which they differ. So, this is one very
important difference between Maxwell Boltzmann
statistics and Fermi Dirac statistics.
There is a second important difference and
that is got to do with the fact that as
Fermions, the particles are assumed to obey
the Pauli’s exclusion principle. So, the
Pauli’s exclusion principle basically tells
us that when you have quantum numbers
assigned to all the particles, then you have
a situation where all the quantum numbers
including the spin quantum number, all of
the quantum numbers cannot be exactly the
same for 2 electrons. So, at least one there
has to be a change, in at least one quantum
number, so at least one quantum number has
to differ. Therefore, when you look at it
that
way, the fact that there are quantum numbers
and so on, it also means that at a given
energy level, we will have now have to specify
the number of states that are available.
So, if so in for each energy level, we will
now have to say that quantum mechanics
allows us to have so many energy levels and
then we will look and see how we are going
to fill those numbers of states using particles.
So, the Pauli’s exclusion principle creates
situation, with respect to differences where
you are talking of Fermions, we are basically
saying that there is a fixed number of states
at each energy level. And that is where; we
are able to say that with respect to the states,
we have to ensure that you cannot have more
than , if those states also incorporate all
the quantum numbers. And that is how you are
actually indicating the number of states at
that particular energy level. We ensure that
that puts a limit on the number of particles
you can place that energy level. So, this
is a very important difference, when we did
the
Maxwell Boltzmann statistics, when we derived
the Maxwell Boltzmann statistics, at at
at one of the early stages of the derivation,
we simply said that we will have let there
be
n 0 particles at E 0 energy level, n 1 particles
at E 1 energy level and so on. So, that is
how we did it, we came down and said we have
n r particles at energy level E r. We
when we did this description for the Maxwell
Boltzmann statistics, at no stage did we
place any restriction or what is the upper
limit for n 0, we did not place any restriction
for the upper limit of n 0, no restriction
for n 1, no restriction for n 2, n 3, n 4
and so on.
Up to n r they was no restriction at all and
that is fundamental to the idea that these
are
classical particles and and there is no question
of no 2 particles being in in exactly the
same state, all those issues are do not arise
if you are talking treating them as classical
particles. And therefore, we simply had some
number at some at a given energy level; we
did not care of what that number was, now
the minute we instead of treating them as
classical particles, we now start it treating
them as Fermions.
Which are quantum mechanical particles also
following the Pauli’s exclusion principle?
Once we do that, we cannot just in addition
for the system itself, in addition to the
energy
levels that are available in the system, we
are also up front placing some restriction
that
at a given energy level. There are only so
many states available. So, we are saying that
in
in our system now in the description for our
system, once we talk of Fermi Dirac
statistics, so this much would be valid for
Maxwell Boltzmann statistics. In addition
we
would now incorporate as 0 states, S 1 states,
S r. So, only if if you have only n 0 at E
0,
n 0 particles at E 0, n 1 particles at E 1
and so on. And n r particles at E r and this
is all
the restriction you place on the system and
your entire description for the system, your
mathematical description that you build for
the system is based only on this much
information, which is present within the first
box, then that would then that would lead
us to the Maxwell Boltzmann statistics. If
on top of it we also place the restriction
that
there is at E 0, you cannot an arbitrary number
of particles because there is a fixed
number of states available at E 0. And therefore,
there is a certain upper limit on the
number of particles you can place in those
this number of states because Pauli’s
exclusion principle exists or is valid for
our system.
So then n 0 is not some arbitrary number,
it is in some way restricted by S 0, n 1 is
restricted by S 1 and so on. n r is restricted
by S r, if you take this entire body of
information that there is certain number of
energy levels, there are the which are
indicated here. There are certain number of
states at each of those energy levels, which
are indicated here and the fact that there
are the particles will now have to populate
the
states within those restrictions. The fact
that there are so many limited states at each
energy level and Pauli’s exclusion principle
prevents them prevents you from putting any
number of particles within a limited number
of states because if you if you cross some
number at you will be forced to ensure that
you would be forced to put 2 particles into
exactly the same state.
And Pauli’s exclusion principle prevents
you from doing so. If you assume Pauli’s
exclusion principle is valid, we can we cannot
do that. So, that that places an upper limit
on the number of particles at each energy
level. So, this combined picture now. The
energy levels the number of states at those
energy levels and then the number of particles
that you can place in those states. That combined
picture will now lead us to the Fermi
Dirac statics. So, if you step back here,
the 2 major differences are that the manner
in
which we count the states simply because we
are they are identical and indistinguishable
under Fermions, but they are identical and
distinguishable as classical particles. In
that
fundamental way, the the 2 statistics will
differ plus the fact that at each energy level,
we
now have, we are also specifying controlling
the number of states available at each
energy level. And therefore, you cannot put
an arbitrary number of particles at at a given
state, at a given energy level.
So, what we will do in our next class is,
we will actually derive the statistics and
all of
these ideas that I have now described to you
and where I have shown you the difference
between these 2 systems between what is that
we have done? And where it is that we are
going, we are headed? All of these ideas will
be incorporated in our derivation of the
Fermi Dirac statistics. And we will come up
with the actual final result for the Fermi
Dirac statistics; we will then see that having
got that result, what does that imply in terms
of material properties. And how successful
is this new description in taking care of
the
Anomaly’s that the Drude model had a problem
dealing with and so in in other words we
will see in in what ways is the Drude Sommerfeld
superior to the original Drude model
and then we will see if there is if there
is room for even further improvement before
and
beyond that. So, with this we will halt for
today, we will pick it up in the next class.
Thank you.
