>> Welcome back to Chem131A.
Where we had last left
our hero, we had decided
that it was possible for
a light particle to tunnel
through a forbidden region,
much like a high jumper going
over a bar, but not going
over it, just appearing
on the other side,
collecting the trophy,
without having enough energy to
actually go over, a phenomenon
that we call tunneling to
indicate that we went through
but we did not go over.
Today what we're going to talk
about is we're going to talk
about tunneling microscopy and
that's an application that turns
out to be very, very interesting
for a lot of reasons.
And then we're going
to introduce a little more
complex problem on vibrations.
The reason why vibrations
are a more complex problem is
that the potential energy
for a vibrational
problem is not square well
or something that's
mathematically so easy.
It's trickier because we
get x squared in there.
We're going to see how
we have to handle that.
It certainly seems at first
blush that this phenomenon
of quantum mechanical tunneling
is just a small niche field
for experts and people
in ivory towers to study,
but just like a lot of basic
research, it oftentimes leads
to killer applications and this
is very, very true in this case.
We saw, for example,
that the phenomenon
that when you measure
something you cause it
to change allowed us to
do quantum cryptography
so that we could have this
key that we could tell
if somebody was spying on us
and we could establish an
unbreakable code between us.
Likewise, this phenomenon
of tunneling,
because when the barrier is
big, it depends exponentially
on the distance and that
means it's very sensitive
to it in some sense.
That means that something
that's close tends
to dominant everything and
that lets us have a trick
to make very sharp
looking point.
And one of these
applications then
of tunneling is called
scanning tunneling,
the scanning tunneling
microscope or the STM,
sometimes called scanning probe.
It was Gerd Binnig and Heinrich
Rohrer, when they were working
at IBM, they were
granted a patent
on the scanning tunneling
microscope
in 1982 while they were at IBM
and I've given you a
reference here on Google.
They have a list
basically of all the patents
that have ever been granted.
They're public knowledge.
You can search them.
You can find out and, boy,
are there are a lot of them.
And this patent was 4,343,993
and that was back in 1982.
If you take a sharpened metal
tip and I mean really sharp,
as sharp as you can make
it, but, as we'll see,
it may not matter
how sharp you make it
because when you look
closely it's going
to be extremely sharp no matter
what if you're fairly lucky.
And you bring it up to a clean
surface, like a gold surface,
and you put voltage on the
tip, then there's nothing
between the tip and the
surface except the vacuum
and the way we interpret that is
that the electron can't come off
from the gold atom too
far because its energy,
its potential is
getting too high
as it goes away from the atom.
The atom has a big positive
nuclear charge that's pulling
the atom down, that's holding
it down, and, therefore,
crossing through this region
of space is like jumping
through that region v in
our tunneling problem.
Classically, it should
not happen.
We need to have a conductor to
have a current with electrons.
But if the electron's a wave,
then if we get near the surface
of the gold or something
else, then the fact
that the wave function can
sneak out a little bit means
if it can sneak into the
tip, then there's some chance
that the electron
materializes in the tip
and that gives us a current.
And this current is going
to be some noisy thing
because it occurs because
of tunneling but it's going
to be a current and
it's going to depend
if we move the tip closer,
it's going to get much,
much bigger because as
the barrier gets thinner,
the tunneling gets
exponentially more likely.
And what that means is that the
position of this tip floating
over this surface is extremely
sensitive to the distance.
How can you use that
to do something?
Well, this is not a microscope
in the conventional sense
of a light microscope
where you might look
at a hair or look at cells.
This is only looking at
the topography of a surface
but it can be fantastic because
you can put pieces of DNA
on a surface or something like
that and you can use variance,
the atomic force microscope for
example, to look at these things
and see all kinds of things.
The amount of current
that you get is a measure
of the local density of states,
in other words it has to do
with how many electrons
can be there
and how their wave functions are
on the surface and the distance
of the tip to the actual
surface itself and this works
with a conducting surface.
If we raster the tip over
the surface by moving it back
and forth and we keep
track of where we are
with a little system of
piezoelectrics that's just
like a GPS for your
car, it knows exactly
where the thing is,
then what we can do is
if the current gets too big,
we assume we're too close
to the surface and
we pull the tip up.
And we try to keep the
current at a certain value
and what we keep track of
as a function of the value
of the current trying
to keep it locked
on to a certain value
that's convenient,
that means we're close
enough to get something,
we know we are too far away
we aren't getting anything
but we're far enough away
that we can move reasonably.
The problem is if we're
too close and we come along
with the tip and there's a mesa
or something on the surface
and we come along, then it'll
increase, it'll increase,
it'll increase, but then will
crash the tip into the surface
and then we put a
notch in the surface,
sort of like scratching
an LP in the old days
if you were careless,
and we may change the tip
because the tip has atoms on
it and they may get knocked off
and they get knocked around.
We have to strike a compromise.
We want to get a current that we
can tell that we're in contact,
we aren't actually touching
but we're in tunneling contact
with the surface, but we
don't want it to be so high
that if we move too quickly that
we're likely to crash the tip.
In the early days of
doing these experiments,
people crashed the tip into
the surface all the time.
Now there are commercial
kinds of machines
that even people have made STMs
on their own, avid hobbyists,
you can actually
make this device,
so it's not so difficult
to make.
Because of the exponential
dependence then,
let's just imagine
the tip, the tip,
whatever it is, has
been sharpened.
It's very sharp.
It's as sharp as you
can make a sharp thing.
And why do you want
it to be so sharp?
Because you want to be able to
see little egg carton things
of atoms and things
on the surface
and so your tip should
be very sharp.
If your tip is this big
wide thing, then you kind
of blur everything together
and you can't see anything.
But it turns out because of
the exponential dependence,
if there's a cluster of grapes
hanging off the tip of atoms
that are piled up there
and it could be anywhere,
it doesn't have to be in
the middle of the tip,
but whatever's closest, all
the current is going to go
through there and so that's
perfect because it means
that you just get
lucky with the tip
and then all the current goes
through there and the tip sort
of makes itself sharper
by the way the current
depends on the proximity.
And so unless you're
very unlucky,
unlucky would be two tips coming
down, two bunches of grapes,
both the same size, that would
give you a very confusing image.
That doesn't happen very likely.
The other ones contribute
exponentially less current
and that means that they
don't bother you much.
You can experiment
with different tips
and people did a lot of that
until you find a good one.
And then you conduct
experiments with this good tip.
You try to get your PhD with it
if you can until you're unlucky
and you're too aggressive
or something
and you crash the
tip into the surface.
And, of course, most of the time
when you're looking at things
with the STM, you're looking
at things that you kind
of assume are pretty smooth.
You're trying to look at details
of flat surface of atoms.
You're trying to see, for
example, if you have a material,
whether some of the
atoms might--
If you have two different
kinds of atoms,
some of the atoms might
like to be on the surface
of the material in a different
amount than in the bulk
and you can see that kind
of behavior with this.
And you can also use the STM
to do chemical reactions.
If you have a surface of
some molecules and you bring
down your tip and now
you can influence,
you've got current
going through,
what you can do is
you can do a pulse,
sort of like having a
meteor come in and hit
and you can make a small
chemical reaction at that point
and you can write a little
dot there, for example,
and then you can move the
tip and pulse it again
and you can modify the surface
by doing this over and over,
and there was interest
in that at one point.
Here's a figure from the patent,
you can see back in 1982,
you drew things by hand and you
drew a lot of things by hand.
In fact, you used something
called a Leroy set back them
to get the numbers to
look nice and so forth
and you drew them in India ink.
This is from the original
scan of the original document.
Here, they're showing this tip.
They're showing a flat surface.
They're showing z, which
is the up and down.
They're showing x and y,
which they're controlling
with these piezoelectric
controllers.
They're showing a
plot of what they get.
And they're showing a
screen which is going
to show the topography, what the
current did as a function of z.
And here's figure 3
from the same patent.
What you can see here is that
they're showing the current
as a function of the distance
from the tip from the surface
and they're drawing
an exponential,
which is exactly what we
derived for tunneling,
and they're showing that
the current should go like e
to the minus something
and then you can see
that there's a distance
in the exponential.
That's the exponential distance
dependence that we derived
for a very unlikely event and
you want this to be unlikely.
You don't want it
to be too likely.
And here is a schematic again
figure 4 from their patent.
There's a surface.
There's a tip.
The tip is very near
the surface.
The tip has some shape.
And all the current is
coming from the part
of the tip that's closest
to the surface, so that part
of the tip is a super-sharp part
that's giving all the current
and the other part is giving a
little current, which is kind
of a noisy background
but it's not a big deal.
And then you're going to move
the tip along and you're going
to keep the current constant and
by keeping track of how you have
to adjust the height of the tip
to keep the current constant,
you get a picture of
the lay of the land.
The tip can also be
used to actually pick
up atoms and move them around.
And here's a spectacular
example of that,
which Don Eigler published
in "Nature" in 1990
and again working at IBM,
here you have a very,
very flat atomic
precision nickel surface
and on it are scattered
some xenon atoms.
Xenon atoms have a lot of
electron density and they show
up there as these
round things just
like you might imagine a xenon
atom should look and you can see
in frame A, they're all
randomly positioned.
And what they were able to do is
first use the microscope to see
where everything is,
be quite careful,
and then go to an exact position
where you know there is
an atom and, of course,
this has to be extremely cold.
This is basically at
liquid helium temperature
because if this is
at room temperature
or even liquid nitrogen
temperature, it's going to be
like drops of water
on a fry pan.
They'll be moving all over
the place and it won't matter
that you scan through
and see where they are
because the next time you
come through, they're going
to be somewhere else and half
the time they'll just pop off
the surface entirely.
There won't be enough
sticking for them to stay on.
So this is extremely, extremely
cold, which is also a challenge
because you have to
cool your microscope.
You have to cool the surface.
You have to cool
everything down.
You have to be extremely
careful.
And then you come down
with the tip on top.
You know it's still there
because it's so cold.
And you actually push on it
and then you drag it somewhere.
And with your x, y magical
GPS system there, you park it
and then you go get
another atom and you drag it
and another one and you drag it.
And then in between times,
you then image the
surface very gently
so that you are not
dragging any atoms
and then you can
see your progress.
And what they show here is they
can write out IBM in xenon atoms
on a nickel surface
using the STM,
and this was really just
a spectacular example
of how you can manipulate
the very smallest things
with such exquisite
detail using this device.
Unfortunately, I think
nobody has figured out how
to use the device to make
extremely small things
like computer chips or other
things that might be ultra,
ultra, ultra miniaturized
because it's too slow.
It takes too long and
it has limited ability
to make any kind of 3D shapes.
Here's another image.
This is from a group at Carnegie
Mellon in the physics department
and what this is, is this
is a picture of the surface
of silicon and the
111 just means
that it's a certain crystal
plane and so if I take silicon
that has a certain
crystalline structure,
I can cut at certain angles,
just like cutting a diamond,
and I cut at certain
angles and I would expect
to have certain kinds of
atomic patterns, but what tends
to happen once I cut is that
the atoms are very unhappy
if they're sticking out too far.
They're very unhappy because
they don't have enough bonding
neighbors, so-called dangling
bonds, bonds that are going
out into space, doing nothing,
and what they may decide to do
if they're unhappy enough,
it's sort of like a lonely
person going to a bar,
they may pull in and try to make
extra bonds with other atoms
which has nothing to do
with the original structure
that you would expect and
that's called reconstruction.
And here you can see this
so-called 5 by 5 reconstruction
of silicon and you can even see
there's one defect in the middle
of the picture, where there's an
atom that's kind of dislocated,
that's out of position but
most of them seem very perfect.
So it's very interesting.
Of course, the color
here is false.
The color here is
just to guide the eye.
All right, now let's
go on to vibrations.
Vibrations are important because
when we do a chemical reaction,
we take a chemical bond
and we usually break it.
We break it and we
make a new bond.
And the whole business of
chemistry is to take stuff
where things are organized,
they're the same atoms
but they're worthless, just
junk, manure, and then we make
and break bonds, a
little witchcraft,
and out comes some very,
very important antibiotic,
which is worth a lot more money.
And in order to understand
how that works in detail,
we need to have a very good idea
of how strong the bonds are.
We need to be able to predict
if we're going to make something
if it's going to have strong
bonds or weak bonds and we need
to understand also if
it's going to absorb light
so that we can do
an assay to see
if we've made what
we think we've made,
like IR spectroscopy.
Here what I have shown
in potential energy curve
for the hydrogen molecule,
H2, the simplest molecule.
The proton-proton distance is
along the x axis in picometers.
And the energy, the
electronic energy,
so we calculate this curve by,
moving the protons together
at different distances
and then we freeze them,
even though we know
they can't be frozen
by the uncertainty principle,
and we calculate the
electronic energy.
And if the protons
are too close,
they tend to repel each
other, plus the electrons,
the orbitals are too close
together, it's not optimum.
If they're at the
right distance,
then the electrons
can be in between.
Each proton sees both electrons
as part of the principle
of bonding, as we'll see is
that they share each proton,
each hydrogen thinks
it's a helium atom,
and that's a very
stable configuration.
And then as we tend to
pull the protons apart,
the electron clouds
can't overlap.
This proton cannot see anything
to do with this electron
and so the strength of
interaction decreases
and finally when
they're far apart,
they're just two hydrogen atoms.
And that's shown then in this
so-called potential energy
curve, which is just the
electronic energy plotted
as a function of the frozen
distance of the two protons.
When they're too close, you see
that the electronic
energy is above zero.
That means that when
they're that close
that they're more unstable than
just two hydrogen atoms apart.
But there is a well,
there is a position
where the two hydrogen atoms
working together are much more
stable than two hydrogen
atoms apart
and that's the stable
H2 molecule
and then the potential
curve goes back to zero
as they go back toward just
two isolated hydrogen atoms.
Now near the bottom of the well
at the equilibrium distance,
which was 74 picometers
in that previous figure,
the potential has a minimum
and where the potential has
a minimum, calculus tells us
that the slope must be zero.
And that means that if we
expand the potential V of r,
that curve, whatever the form
of it is, in a Taylor series,
which we do by taking
the function value
and then the derivative,
second derivative and so on,
around the equilibrium position,
we can write V of r is equal
to V of re, which
is the equilibrium,
the lowest point,
plus r minus re.
So again, a trick of
rearranging something
by making it seem more
complicated and we can write
that as V of re plus delta r,
where delta r is how much
the bond is stretched
or compressed from
the equilibrium.
What we get is in the Taylor
Series, we get V of r evaluated
at re, which we'll just call V
of re, plus the derivative of V
of r evaluated at
re, times delta r,
plus 1 over 2 factorial times
the second derivative of V
of r evaluated at re times delta
r squared plus blah, blah, blah.
It keeps on going,
the same pattern.
If we look at the
bottom of the well,
the derivative is
zero and, therefore,
it simplifies quite a bit.
Near the bottom of the well, the
derivative is zero right there,
so we throw that term
away because we evaluate
that term right at r equals re.
If we're near the bottom of the
well, then r is close to re,
so what we will assume then is
that r minus re squared is
something but r minus re cubed
and all the higher ones are too
small because r is very close
to re and so, therefore,
they're much smaller.
And if we do that, we end
up with this following very
simple form for the potential,
which is what we're going to use
when we do the Schrodinger
equation because we don't want
to use the real potential or
we'll never get out alive.
It'll be far too
difficult for us to do.
We get V of re plus 1/2 k times
r minus re, quantity squared,
where k is the second derivative
of V with respect to r,
evaluated at r equals re, and
k here is the force constant,
not the-- So I apologize
for using k again,
but k is conventionally the
force constant of the spring
and before k was the
wave vector e to the ikx,
this is a different k
and there's another k,
Boltzmann constant, which you
might put k sub b to try to keep
that one separate, but we'll use
k when we talk about vibrations
to mean the force
constant of a spring.
And for small displacements,
then we have
that the motion should be
harmonic unless k is zero.
If k happens to be zero,
then that term goes away
and then the whole
motion is described
by something very funny,
whichever terms are left over,
but usually k is not zero
because the thing comes down
and goes back up and
so it has some part
of it that's quadratic.
And around the bottom, that's
going to be the main part
of the actual potential.
Therefore, we can
model a chemical bond
as a one-dimensional
harmonic oscillator.
We totally ignore any kinds
of other displacements
in other directions or anything
funny and we just say, look,
these are two things on a
line here, there's a distance
between them, we know the
energy, what we want to figure
out is what's the wave function
as a function of this distance
between them given the form
of the potential energy.
We can always adjust the energy
zero, so we can call the bottom
of the well zero,
even though it's not,
even though for hydrogen,
it's minus 4.5 eV down,
we can call it zero and then we
can just add it later if we want
to get the real energy, so we
don't have to worry about that
in the math, and so
we'll call it zero
when the displacement is zero.
And just to keep in keeping with
what we've done for consistency,
rather than using r minus re,
I'll just introduce a variable x
and psi will be a function
of x and so if x is zero,
then they're at the equilibrium
and if x is something else plus
or minus, then it's
away from equilibrium.
We get the same tired old time
independent Schrodinger equation
to solve, minus h
bar squared over 2m,
d squared psi dx squared,
now plus 1/2 k x squared,
psi is equal to E psi.
And given k and m,
our task is to figure
out what the allowed
values of E are
and what the functional
form of psi is.
For the particle in the box,
the allowed values of E went
like n squared and
psi was a sine wave.
Now we've got a different
potential, completely.
It's got this x squared, so it
keeps continuously changing,
so we could guess it's going to
be quite a bit harder to do it
and that would be a very
good guess, as we'll see.
If we have the two
nuclei connected together,
let me just remark that the
mass m here is the mass,
the reduced mass of
the oscillator or m1,
m2 over m1 plus m2
but we won't worry.
For what we're doing, we just
want to get a qualitative field,
so we'll just keep m and
m is some mass associated
with the oscillator.
Now before we solve the
differential equation,
it's a good idea to take
a second and try to figure
out what it is we would
predict that we should see.
That way if we get
something ridiculous
because we make a
mistake, we'll know it.
So the question is what
properties should the wave
function have and that's
pretty easy to [inaudible] out.
First, the energy
will be quantized.
Why? Because the
potential is going like that
and something that's
going like that is tending
to confine the particle.
The particle cannot
just go anywhere
because the potential gets
bigger and bigger and bigger
and bigger, so it's
going to be trapped.
If it's trapped, it's
got to be quantized.
It's got to go way out there
and the energy, it has to fit
into the space and, therefore,
it's going to be quantized.
We don't know what shape
it's going to have.
Secondly, the lowest energy
eigenstate can't be zero energy.
It wasn't zero energy for
the particle in a box either.
The problem with that is if we
pick zero there, it went away.
And we have a similar
problem here.
If we want to have a
real wave function,
it's going to have
to be in there.
It's going to have to satisfy
the uncertainty principle and,
therefore, it's going to have
to have non-zero p squared
and it has non-zero x squared
because it's not an infinitely
narrow box and, therefore,
it's going to have
non-zero energy.
And thirdly, the wave function
has to die away somehow
as x gets far from zero.
The reason why there is that the
potential keeps getting larger
and also it has to
die away to zero
because it has to
be normalizable.
And finally, the ground
state should have no nodes.
And the reason for that is that
we by analogy with the particle
in a box, when we did
the particle in a box,
it was zero at the edge
because it had to be
but it wasn't zero
anywhere in between.
It was just a lump and if
we change this to this,
we expect this lump to change
but we don't expect it to change
into two lumps; therefore, we
expect something like a turtle
in there somehow sitting
there, not very exciting
but just sitting there.
And if we have an excited state,
if there are excited states
in this potential, we would
expect them to have nodes just
like the higher excited states
of the particle in a box.
But, well, how should we solve
the differential equation?
Remember I said the
most powerful method
to solve differential
equations is often guessing
and I'm going to try to guess.
I have something times the
derivative, the derivative
of the wave function twice,
plus some other stuff is equal
to the wave function
times a number.
I've got to get the other
stuff to go away; therefore,
I need to have a function
that generates itself again
so I can get this
part and I need it
to generate a little
other garbage times itself
so that I can get rid of the
1/2 kx squared, which I want
to go away, because
there's no 1/2 kx squared
on the right-hand side.
And I couldn't use just e to
the x to do that because e
to the x will give itself times
a number but I could use e
to the something else and I
would expect that I'm going
to have to use an
exponential function
because there are
always the solutions
of these differential equations.
Furthermore, I could guess that
look, this thing has symmetry.
If I draw the potential,
at the bottom x is zero
and then it's going up and it's
symmetrical and, therefore,
the wave function has
to be symmetrical, too.
And that means that we
can't have anything like e
to the minus alpha x
or something like that
because that's not
symmetrical around zero.
We could have that plus
e to the plus alpha x
but we can see right
away that neither
of those would be any good.
Since those don't look
good, try the next power up
and if you try the
next power up,
which is a Gaussian function,
you get very lucky and, in fact,
it seems to work,
so let's guess.
Let's guess psi of x is equal to
A times the exponential function
of minus a times x squared,
little a. Big A is the
normalization constant.
We won't worry about what
that is at this time.
And little a is something that
we're going to have to pick
to make it work and we'll
see what the condition is.
You wouldn't necessarily
know that this would work
but you can easily work it out,
so let's go ahead
and work it out.
If we take-- We want to take the
second derivative and multiply
by h bar squared over 2m.
The first derivative
of that function is
that function again
times the derivative,
remember the derivative of e
to the u is e to the u du/dx.
We have minus ax squared
as u. The derivative
of that is minus 2ax,
so the first derivative is
A times exponential minus ax
squared minus 2ax,
that whole thing.
The second derivative, now we've
got a product of two things.
We've got the derivative
of u times v
and that is the derivative of
u times v plus the derivative
of v times u. And the first
one we've done the derivative
before, so therefore
the derivative,
the second derivative
is capital A e
to the minus ax squared
times minus 2a plus a times e
to the minus x squared times
minus 2ax times minus 2ax again.
The first 2ax comes from du/dx;
the second minus 2ax comes
from the fact that that
second one is there.
And we can then put
these together and we see
that we got what we want.
We got A times e to
the minus ax squared,
so that gave the same
thing reproduced,
and then we have this
term and it has two parts.
It has a 4 a squared x squared,
which if we pick a right
it's going to cancel
out the 1/2 kx squared, and
then it's got the other part 2a,
which is going to have
something to do with the energy.
Let's have a look.
If we put everything in then
into the Schrodinger equation,
we come to the following
conclusion, minus h bar squared
over 2m capital A e to the minus
ax squared times 4 a squared,
x squared minus 2a, plus 1/2k
squared, again capital A e
to the minus ax squared is
equal to e times the same thing.
And so now we can divide
both sides by capital A e
to the minus ax squared and get
a relationship between little a
and k. If we want
those terms to cancel,
because there's no x squared
term on the right-hand side,
it's just e, the terms
in x squared cancel,
that means that minus
h bar squared
over 2m times 4a squared x
squared plus 1/2kx squared is
equal to 0.
And for that to be equal to
0 for all values of x, a,
little a, has to equal to the
square root of mk upon 2h bar.
And a, therefore, the
exponential argument has to do
with the mass and
the spring constant
and Planck's constant
and, boy, is that sweet
because this is quantum
mechanics
and that's exactly
the kind of behavior
that we would have
expected to see.
And we can make a connection
with a classical oscillator
if you've done the
classical oscillator.
If you haven't, then you should
but the angular frequency
of the classical oscillator,
what's the angular frequency?
Well, if I see this thing
going back and forth like that,
I can interpret it as
something as a projection
of something going
around because
if something's going
around, it's going like this,
and that angular frequency is
the square root of k over m
and since omega, the angular
frequency, is the square root
of k over m, the square root of
mk is equal to m times omega.
And using that, we can
now figure out the energy.
The energy is minus h bar
squared over 2m times minus 2a;
that's the only part that's
left over because the x is gone
by our choice of a. And
that's h bar squared over m,
square root of 2 mk
over 2h and that's h bar
over 2m times m omega,
the m's cancel,
and we get h bar omega over 2.
That is the ground state energy
of the oscillator,
which is not zero.
The oscillator is always
a little bit excited.
It can't be zero because of
the uncertainty principle.
And we can make a connection
between what we found with light
by saying , look, omega, the
angular frequency, is 2 pi nu;
nu is the regular
frequency, so h bar omega
over 2 is also equal
to h nu over 2.
And h nu was the quantization
of a photon, so this is similar
to the quantization of light
except that now there's a factor
of 2 in the denominator for the
energy, but other than that,
it's very closely related.
Knowing the value of a now,
little a, then we can calculate
and normalize the entire wave
function and determine the value
of big A because now we know how
wide this Gaussian function is.
We know what area it's going
to have when we integrate it.
And so we know how big
this big A has to be
to make the probability
of finding the displacement
somewhere equal to unity.
And our normalization
condition now is
that we should take the
square integral of this thing
from minus infinity to infinity,
as usual, and the integral of e
to minus 2ax squared dx,
that's a standard integral,
which you can look up,
that's the square root of pi
over 2 divided by
the square root of a,
and capital A squared times
that should be equal to 1 and,
therefore, if we take the
square root of both sides,
we find that a squared
should be 2a divided by pi
to the 1/4 power and that's
capital A. We've picked A
as usual to have real
phase because we always
like A having real phase;
we're just biased toward that.
Just as we did with the particle
in a box, wave functions,
we had that 2iA and we said,
well, we'll get rid of the i
because it doesn't matter.
This, therefore, gives
us our very final form
for the ground state of
the harmonic oscillator.
And that is 2a over pi to the
1/4, e to the minus ax squared
and then I can put in all the
things with m and omega for a
and I get this very
nice formula.
Now the problem with this
approach of guessing is
that whatever you don't
guess doesn't turn up
and we guessed one thing
and we found one thing
and it made sense.
Why? Because it's a Gaussian,
it's like a big turtle,
it has no nodes.
It has a low energy,
which satisfies the
uncertainty principle
but it doesn't have any
more energy than that,
but now what we would have
to do is we would have to try
to guess some higher thing.
It's not like the particle
in a box where we had n pi.
We don't have any n yet
here in this problem.
We just have this one wave
function and this one solution.
We suspect and we're right that
because it's like a particle
in a box, just slightly
different,
that there should be a ladder
of states and, in fact,
the ladder is actually
equally spaced
at n plus 1/2 times h bar omega,
which makes this
potential really unique
because it's the only form where
you get an equally spaced ladder
of states all the
way up to infinity.
And there are some very,
very nice mathematical ways
of attacking that problem
that are very beautiful
but they take us a little bit
too far field for our course
and they don't have that
much to do with chemistry.
They have more to do
with operator algebra
and quantum physics
than chemistry.
And so we're not
going to explore those
but we'll just quote that
the general solution is some
polynomial in x that you have
to pick carefully times e
to the minus ax squared.
So that part of it
is always the same.
Now what is the interpretation
of this?
It's completely different
than a classical oscillator
because the classical oscillator
has the highest probability
if we take a film of it, it goes
out, it stretches, it stops,
it turns around, comes back
through, that's where x is zero,
that's where it's
perfect, compresses, stops,
back through again, back
through again, and so on.
And the chance of finding
it if it's oscillating right
at the equilibrium
position is not likely.
If you just grab
it at some point
and measure its distance
apart, it's much more likely
to be either fully stretched
because it has to turn
around there, that's why it's
called the turning point,
or fully compressed.
But when we look
at the ground state
of the harmonic oscillator,
what we find is something
completely different.
The highest chance is that
the thing is in the middle
in the perfect position,
where it shouldn't be,
so it seems as if it's
trying to stay in the middle
but it's actually
spreading out a little bit,
not because it wants to but
because it has to because
of the uncertainty
principle and, therefore,
this ground state
of the harmonic oscillator
looks completely different
than a classical oscillator.
And that caused a lot
of consternation I think
in the very early days
because it looked so different
than how do you interpret
this thing,
you want to make sure you
haven't made a mistake
or something's not
right in the equations.
But it turns out this agrees
exactly with what we observe
and there are many experiments
where we take a molecule
and we excite an electron
and it really looks
like it comes mostly from
the equilibrium position.
We very rarely find something
coming from extended position,
so this interpretation
of this is correct.
Like I said, the
oscillator can't sit still.
It's like a small kid.
It has to squirm around
to satisfy the uncertainty
principle and the
zero-point energy depends
on the square root of
the spring constant k
and on the inverse
square root of the mass.
And this gives rise to something
called the isotope effect.
Let's have a look at
the isotope effect.
Suppose we have two isotopes,
for example hydrogen
has a single proton;
deuterium is a single proton
plus a neutron, it's heavier
but the charge is the same
and deuterium behaves much
the same way as hydrogen does.
There's D2O you can use it to
make NMR samples, for example.
The neutrons don't have any
charge, no electric charge,
so the electrons don't
really care too much
about the neutrons and,
therefore, what we expect
to a first approximation
is that the force constant,
which has to do with
the electronic orbitals,
and the repulsion
and the repulsion
of the two protons
doesn't depend
on whether there are two
protons or two deuterons,
it depends on the charge
and the separation.
And the electrons
pretty much don't care.
There are small effects
because the electron is sneaky;
it can see the neutron.
There are some small
effects but basically,
the form of the potential
energy surface is the same.
But what's different then
is the mass and, therefore,
this whole field of isotope
effects is a great field to get
into if you're interested
in theoretical effects
of zero-point energy
and tunneling
and all these subtle things
because you have a perfect thing
where everything's the
same except the mass,
and so you're calculation even
doesn't have to be quite so good
because even the things
in your calculation
that are slightly wrong are
the same except for the mass
and so unless you're
very unlucky,
you get a pretty
good result anyway.
If we have then a CD bond,
a carbon-deuterium bond,
versus a carbon-hydrogen bond,
the carbon-deuterium bond is
stronger because it in the sense
that the amount of
energy it takes to go
from the ground vibrational
state, which is as close
to the bottom of the
well as you can get,
to where the bond is broken,
which is the same place
on the curve, is
higher for the CD.
And here's then--
We'll close with this;
here's how it might look.
On the left, we have a molecule
that could either have a
hydrogen or a deuterium
and let's say it's carbon,
and then in the chemical
reaction this bond breaks
and as it's breaking, the
force constant gets less,
so in the transition state,
it's almost off and, therefore,
the potential is very wide
because the force constant
for something that is a weak
bond is not a very stiff spring,
but when it is actually the
reactant, it is stiff and,
therefore, there
is a big difference
when it's the reactant and
there's a small difference
when it's the transition
state, and that then translates
into a different
rate of reaction.
That means that if we have
molecules that could be CD or CH
and we react them somehow,
we expect the CH ones
to react more quickly
than the CD ones
and this is called the
kinetic isotope effect,
and I have adapted this little
figure here from Wikipedia
to just show how this works.
In fact, this again seems
like a very esoteric thing
but you would be amazed if you
go to the enzymology literature
and you look, how do I know
if this enzyme is breaking
this bond or that bond
or what is the rate-limiting
step for the synthesis
of cholesterol or the removal
of something from the body,
you put in a deuterium
as a spy and you look
for the deuterated product
versus the protonated
product and, boy,
do you find a ton
of information.
So this is now translated
all the way
from this esoteric effect
of the harmonic oscillator,
the uncertainty principle and
zero-point energy all the way
to modern medicine,
where it's used to figure
out what's going on in the body.
Next time we'll continue
on with some
of these one-dimensional model
problems and then we'll begin
to actually do some
multi-dimensional quantum
mechanical problems.
I'll close it there.
------------------------------dce2c7b047c3--
