[Slide 1] Welcome everyone to the second lecture
of atoms to materials. Today we're going to
start talking about quantum mechanics and hopefully start
developing an intuition for how quantum mechanics works.
[Slide 2] And we're going to start by
not debating why we need quantum mechanics and
we're going to instead example, we're going to
start with the simplest possible atom, okay, and
we're going to start the road of connecting
atoms to materials. So the simplest atom, of
course, hydrogen. Hydrogen is made- it consists of
a proton and an electron, okay. The proton
has charge plus e and the electron has
charged negative e. So, of course, the atom
is neutral and the proton is significantly more
massive than the electron, okay, the proton, the
mass of the proton is about 2000 times
larger than that of the electron, so we're
going to assume the proton is standing still
at the origin- is not moving and the
electron is around.. I'm going to worry about
the position of the electron. What we're going
to do is describe the hydrogen atom, try
to describe it using classical mechanics. Okay so,
in classic mechanics the state of the system
is neither at a given time by its
position and its velocity, okay, so if I
know the position and velocity of the electron
then I know everything about my system. Remember
we said that the problem is the proton
is standing still. So what I'm going to
do is I'm going to write an expression
for the total energy of the system and
as you all know the equilibrium configuration of
the atom would be the configuration with the
lowest energy, okay? So we know that in
materials and in nature in general, systems try
to minimize their energy right, that's why if
I drop this pen it's going to fall
back to the table, that's minimizing its energy
pulled by gravity. So let's do that, there's
the terms of the total energy of a
system, there is a potential energy and in
this case, in the case of hydrogen, the
potential energy is dominated by the by electrostatic
interactions, by coulomb interactions, and the total energy
depends on the separation distance between the proton
and the electron and it's in this form,
this is Coulomb's law. The total energy is
the charge of one particle times the charge
of the other particle divided by the separation
distance, so the energy goes down. The magnitude
of the energy decreases as you move away
from each other. Now in our case, the
charge of the proton is plus e, the
charge of the electron is negative e, so
the energy has this form, negative e square
over r, over the separation distance. If we
plot this potential r as a function of
distance between these two what we see if
that the potential energy becomes more and more
negative as I bring these two atoms together,
okay, that means that the electron is pulled
by the proton, okay, the energy goes down
as we bring them together. So that so
that's potential energy, Coulomb's law of interaction. You
may remember that the Coulomb force- it is
one distance squared, okay, the force is the
gradient of the energy- of negative gradient of
the energy- so energy goes with one over
r. Force, the derivative, goes with one over
r square. That's one component of the energy,
potential energy; this has to do with the
interaction between the two particles. The other component
is kinetic energy, is the energy associated with
the motion of the electron and all of
you would know that that kinetic energy of
a moving particle is one half times its
mass, times v square, okay, a one half
m v square, and we're going to write
that in a slightly different way in terms
of the linear momentum, okay, the linear momentum
is mass times velocity, it's typically called p.
Okay and so the kinetic energy can be
written as p square over 2m, okay, it's
the same expression. Alright, that's great we have
an expression for the potential of the total
of the kinetic energy, if we add them
up would get the total energy. [Slide 3]
And now what I'm going to do is
try to minimize this energy, okay so let's
call it p; it's going to be the kinetic
energy is going to be k, so minimizing
K. Of course they could minimize kinetic energy,
of course it has to be positive- zero
cannot be negative so the smallest possible kinetic
energy is p equals zero, okay so zero
velocity- that means electron is not moving at
all. That's correct, the second term; potential energy,
we're going to call it V, so minimizing
V the lowest possible volume that this expression
can have is negative infinity when the position
is zero. That means that the classic solution
predicts that the electron is going to collapse
into the proton and it started to be
moving there. That's very bad news, that tells
us that with classical mechanics would predict that
the atom collapses into point, it has negative
infinite energy, it's not a good description. Of
course we know that the atoms of finite
size and they have finite energies and so
this clearly shows the shortcoming of classical mechanics,
that it's just unable to describe these electron-
electronic atomical physics. [Slide 4] So let's switch
to quantum mechanics. And we are going to
try to solve the hydrogen atom using quantum
mechanics as well as the basic principles of
quantum mechanics along the way. So quantum mechanics
tells you well, forget about knowing the position
and the velocity of a particle independently. These
two quantities are not completely independent or related
to one another and quantum mechanics tells you
the state of the system at a given
time is fully determined by its wave function
and so the wave function is a function
of space, Psi of r, and I'm going
to just sketch it here- sketch a possible
shape of the function as a function of
position, the wave is a function of position
and we're going to see in a minute
that the wave function more or less tells
you where the electron is okay? So if
the wave function is zero that the electron
is not there and we probably have some
familiarity with quantum mechanics- we all know that
the electrons describe us to cloud of probability,
and the wave function tells you it is
associated with that probability, so again you cannot
know the position and velocity independently. All the
information as we will see in a minute
is in the wave function. Great. So the
second thing that we need to learn about
quantum mechanics is that any physical observable, anything
that you can measure physically is associated with
an operator, a mathematical operator. So the operators
are mathematical objects that act on functions, they
are not functions, they are not numbers, they
are objects that act on functions and specifically
in quantum mechanics we're going to- the operators
are going to act on wave functions. And
there's two operators- only two that we'll have
to learn. One is the operator position and
the operator position is very simple. It says
multiply times r, okay? Take the wave function
and multiply it times r. Simple enough. In
a minute we're going to do some examples
of how we use operators to make predictions.
The second operator that we need to worry
about is the operator for momentum, okay, and
the operator for momentum is proportional to the
gradient of the wave function, okay, so it
has the prefactors H-Bar here at the top.
H-Bar is Planck's constant and i here is
the imaginary unit, okay, so the square root
of negative one. So it's those two, just
the constant prefactor times the gradient operator. The
gradient operator what, you see on the right
it's described by that called Lambda. This operator
is a vector. It has components along X,
Y and Z. The component along X is
the derivative of X, the component along Y
the derivative of component Y, the component along
Z is the derivative with respect to Z.
And as we said, it's not a function;
it's something that can act on a function.
We put a function there; we take derivatives
of that particular function. So two things that
are important to remember momentum like velocities of
vectors so the operator is also a vector
and interestingly you can see here how that
the-the same wave function will be able to
tell us about position and momentum, it has
information about position and velocity. Because we're going
to use an operator acting on the similar
function to have a position in a moment.
[Slide 5] So the last piece of background
information before we go back to hydrogen. When
we measure things in the lab, okay, what
comes out is a number. It is not
a function, is not an operator. If I
measure an energy it could not be 13.6
electoral volts, if I measure distance I get
something in nanometers or in action and in
quantum mechanics takes up to measuring in the
lab r, associated with a mathematical operation called
the expectation value of an operator. So operators
are associated with physical observables, things that I
can measure. It is the expectation value of
that operator what I measure in the lab.
So what you see here in this equation
is the definition of the expectation value of
a generic operator, O. And what it's done
is it's an integral over all of space
of the wave function outside. The operator acting
on the wave function, it hasn't arrived- in
operator acting on the function, and the result
multiplied again by the function. And we want
to be mathematically more rigorous. The second function
is such a complex conjugate; we're not going
to worry about those details for the most
part. So again what I measure in the
lab is the expectation value of the operator
that I'm interested in which is an average
over all space, involving the wave function and
the operator, okay. As we go, I know
this is a lot if it is your
first time learning about quantum mechanics, this is
going to look counterintuitive. Now we'll do it
work on various examples throughout the lectures and
I think these would make a lot of
sense very soon. Okay so let's do an
example. Let's say I want to calculate the
expectation value of the operator position. I to
go to the lab, I measure the the
position of the electron and I want to
do a mathematical calculation to show me the
position. So as we said- it's an expectation
value that I need to calculate. I have
the wave function multiplied the operator acting on
the wave function, the upper position simply multiply
times r, and then the wave function again.
So in this particular case I can move
things around and I have an integral of
r, position, times the wave function squared integrated
over all of space, okay, and in nomenclature,
I'm going to use this d3r to indicate
dx times dy times dz, integrated over all
of space. Alright, so this expectation value of
position or the average position of the electron
as a way to think about it is
an integral over all of space of a
possible position times the wave functions squared. And
if you remember the definition of probabilities, what
you will realize is that the wave function
squared is the probability density of finding the
election at position r. What that means is
that the average position is sum up over
all possible positions of the value of that
possible position times the probability of that position
actually being the location of the electron. So
the wave function squared- this is very important-
gives us an interpretation for the wave function-
of the wave function squared- which will guarantee
that it's positive is the probability density of
finding the electron around position of r.
[Slide 6] Now we're going to go back to
hydrogen, okay, and I'm going to do the
same thing in quantum mechanics that I did
classically as in classical mechanics but I'm going to write the
expectation body for the energy and I'm going
to try to minimize the energy with respect
to the state of the system, in this
case try to think about what wave functions
will minimize the energy in the same way
we classically did, we found what position and
velocity would minimize the energy. So for energy
I'm going to do- to know what the
expectation value for the energy will be, I'm
going to do this expression that you see
at the top. It's the expectation of a
different operator and this operator for total energy
of the system and this operator is called
Hamiltonian. And we're going to learn quite a
bit more about the Hamiltonian in the coming
lectures, but as in classical mechanics it has
two components. It has a potential energy, e
squared over r; same as in classical mechanics
and it has a kinetic energy, that's as
in classical mechanics, v square over r. And
remember- the operator, p, momentum had to do
with the gradient so the kinetic energy has
to do with the gradient squared which is
this operator here, that's not a square. And
what I'm doing here on the right-hand side
of the equation is just writing out the
kinetic energy, the expectation value for the kinetic
energy and the expectation value for the potential
energy. And so as before, as we did
in classical mechanics we're going to try to
minimize this expression, so what type of wave
function will give me low kinetic energy and
low potential energy. So let's start with minimizing
the kinetic energy, okay and then we're going
to think about potential energy. So what do
I want to do to minimize the kinetic
energy, we look at the expression, we see
that the kinetic energy has to do with
gradients of the wave function with how fast
or how slow the wave function is changing
so to minimize the wave function- the kinetic
energy- what I want is a smooth wave
function, smooth wave functions. Small gradients, okay, so
if the wave function has small gradient, it
is going to have smaller, lower momentum and
smaller kinetic energy. For the potential energy, what
I need to do is look at the
expression over here. What I need to do
is the expression is wave function squared over
r. So I want wave function squared to
be large for small rs. And this means
I want the wave function squared- that is
the probability of finding the electron's position at
position r to be very large, for small
r's, that means that the electron spends a
lot of time, probability defining the electrons, very
close to the proton, very high, that's exactly
like in classical mechanics. [Slide 7] So that
keeping these two things in mind, we're going
to explore different shapes of wave functions and
try to find what would be the best
possible wave function for hydrogen. Let's start here
on the right. What you see here, well
in all of the three plots, you see
the potential energy at the bottom, is one
over r. We see coulomb's energy r in
black, and 3 possible wave functions in red.
The wave function on the right is is
a very compact wave function, where the electron
is confined to spending a lot of time
near the proton and it cannot wander away,
far away from the problem very much, okay?
So this wave function- we have a very
low potential energy, that's very good, the potential
energy, remember, to minimize the potential energy, we
wanted the electron to spend a lot of
time next to the proton. However, the kinetic
energy is going to be very high because
kinetic energy has to do with with gradients
of the wave function and so this functional
have gigantic gradients, so the kinetic energy is
really, really bad. Now let's go to the
other extreme on the left. Here I have
a very diffused wave function, so the electron
will spend a lot of time moving away
from the proton, being far away from the
proton. So here the kinetic energy will be
great, will be very low because I have
very small gradients. However, the potential energy will
be high and it will be high because
the electron spends a lot of time away
from the proton, where the potential is not
so great. The potential energy here is not
very good. Over here the potential energy is
very high- er, very low, so very good.
So between these two extremes there's a happy
medium, okay. There's an optimal solution and this
optimal solution that we're going to work in
the next few lectures in detail tells us
that quantum mechanics predicts a finite sized atom,
quantum mechanics is able to stabilize the atom
as opposed to collapsing it like classical mechanics
does and we've had success, okay? And so
with a few concepts that that we talked
about today and will continue developing the next
few lectures, we have been able to explain
the finite size of finite energy of the
hydrogen atom, and hopefully start to develop an
intuition as to how quantum mechanics works. Let
me make one and one more comment which
is that it's really the kinetic energy that
saves the day, say if my hands are
the wave function, okay, as I try to
make the wave function more and more compact,
the kinetic energy is going up because the
gradients of the wave functions are increasing and
that's bad, right so at one point it's
going to find the potential energy wants to
make the wave function very compact, the kinetic
energy wants to keep its spread out and
there is a balance between these two and
the compromise is reached for a finite size.
So we can relate these to a principle
that we all heard about which is called
the Heisenberg uncertainty principle. And this will hopefully
give us an intuitive understanding of kinetic energy
in quantum mechanics. Heisenberg says that you cannot
know both position and velocity with infinite accuracy,
that's a fundamental aspect of quantum mechanics and
say this is the principle that it goes
a bit further and it tells you that
the more you know about the position of
a particle, the less you know about its
velocity. So as I try to make the
hydrogen atom more and more compact, as I
try to collapse the wave function into a
point, I am knowing more and more about
the position of where that electron is and
confining the electron more and more. And as
I find out more and more and more
about its position and other uncertainties about this
velocity, so that means that it's probability of
having a very large velocity increases and the
kinetic energy will also increase. It's really the
Heisenberg uncertainty principal and the kinetic energy that
stops the hydrogen atom from collapsing with stability
and allows us to- will allow us to
start from hydrogen and build our way up
to molecules and materials and crystals. [Slide 9]
So let's sum up in a minute what
we learned in this lecture. In classical mechanics
the state of the system is position and
momentum. They're independently known variables for the kinetic
energy and the potential energy we found that
the minimum energy was negative infinity and that
the atom really collapsed into a point, which
is not up to the very good result.
Fortunately, quantum mechanics saves the day and it
tells us, well, forget about position of velocity.
The wave function of the system has all
the information you know, both about position and
velocity. We have been able to write an
expression for the total energy, that's kinetic, that's
potential, it's called Hamiltonian operator, and we found
that the ground state or the lowest energy
state of hydrogen has a finite size, it
doesn't collapse, and really quantum mechanics saves the
day. And this all thanks to the kinetic
energy, so we have to thank the kinetic
energy and Heisenberg's uncertainty principle for stabilizing atoms
from which we can build molecules, materials and
people. Thank you very much; I'll see you
in lecture three. [Slide 10]
