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CATHERINE DRENNAN: OK.
We're going to take
10 more seconds.
OK.
Does someone want to explain
how they got the right answer?
We have a Faculty of 1,000
research bag for them.
Do you want to hand that
up and the bag, too?
We're just--
AUDIENCE: So the theme
of light gives enough
energy for the
electrons to be ejected.
And the amount of energy
for that is 4.3 eV.
And then it also has
kinetic energy of 7.9 eV.
So you just add the two.
So 4.3 plus 7.9 is 12.2 eV.
CATHERINE DRENNAN: Thanks.
Let's bring it back.
OK.
Thank you.
All right.
We'll have lots more
practice with this today,
and we'll get the hang
of doing these problems.
So let's just jump
in and get started.
We're still continuing to think
about the photoelectric effect,
and think about
light as a particle.
So we're going to finish up
with the photoelectric effect,
and we're going to have a little
demo on that in a few minutes.
Then we're going to go on.
If light is, in fact, quantized,
and you have these photons,
then photons should
have momentum.
And so we'll talk about that.
Then we've talked about
white as a particle.
And most of you are probably
pretty OK with matter
being a particle.
But what about
matter being a wave?
So we're going to talk
about matter being a wave.
And if we have time
at the end, we're
going to start on the
Schrodinger equation, which
we're going to continue
with on Friday.
So I'll just say that sometimes,
I am a little overly ambitious,
and I put things on the
handout that I'm not really
sure I'm going to get to.
Just because I've
never gotten into it
before doesn't mean that I
won't get into it this time.
So if I don't finish
everything on a handout,
bring your handout
to the next class,
and we'll just
continue from there.
And there'll be a new
handout then as well, so
just a heads up on that.
All right.
So let's continue with
the photoelectric effect
and get good at doing
these kinds of problems.
So let's look at these
particular examples.
We have three different
examples here.
We have the energy
of an incoming photon
must be equal or greater to that
threshold energy or that work
function, in order for an
electron to be ejected.
So in this case, the energy is
greater than the work function.
So tell me whether an
electron will be ejected
or will not be ejected.
And you can just yell it out.
What do you think?
AUDIENCE: Will.
CATHERINE DRENNAN: Yes.
So an electron is ejected.
It will be ejected.
What about this
scenario over here,
where the energy is less
than that threshold energy?
Is or is not ejected?
AUDIENCE: Is not.
CATHERINE DRENNAN: Yes.
OK now we have another scenario.
We have three
photons, each of which
have half of the energy needed,
half of that threshold energy.
But you have three of them.
So will an electron
is or is not ejected?
AUDIENCE: Is not.
CATHERINE DRENNAN: Is not.
OK.
So three photons each
that have half the energy
does not add up.
You cannot add it.
It will not eject an electron.
So let's just think
about it for a minute.
Suppose the threshold
knowledge for passing
an exam is answering three
specific questions correctly.
Suppose over here, we have the
answer to one of the questions,
but not to the other two.
Over here, we have an
answer to the middle one,
but not the first or the second.
And over here, we have
the answer to the third,
but not the first or the second.
So everyone knows the answer
to a different question.
Will there be the threshold
energy, a threshold knowledge,
to pass this test?
No.
Everyone needs to have
the threshold knowledge
themselves to be able to pass.
Everyone has to overcome that
critical amount of knowledge
to be able to pass the test.
So that's the same thing here.
You can't add it up.
Now, with the test here at MIT,
if everyone has that threshold
knowledge, and a really
high level of the threshold
knowledge, everyone can get
an A. So the more people
with the threshold knowledge,
the more tests that are passed,
and the more the course
is passed by people.
So the more photons coming
in with that threshold
energy, the more
electrons being ejected.
But you can't add up
if you have photons
that don't have enough,
if they're not greater
than, the threshold energy.
You won't eject an electron.
So everyone needs to meet
that threshold criteria.
You can't add things up.
OK.
So here's just some
useful terminology
for solving problems
on this problem set.
And there will also be
problems on problems set two
related to this topic.
So photons-- also
called light, also
called electric
magnetic radiation--
may be described by their
energy, by their wavelength,
or by their frequency.
Whereas electrons, which
are sometimes also called
photoelectrons, may be described
by their kinetic energy,
their velocity, and, as you'll
see later, by their wavelength.
So you'll be given
problems where,
given different
pieces of information,
you have to think about how
you're going to convert it.
You've got to think about
am I talking about a photon?
Am I talking about an electron?
And you also want to
think about units.
You'll sometimes be
told about energy in eVs
and sometimes be told
about energies in joules.
So this is a conversion factor.
All conversion factors are
given to you on the exam.
You do not need to memorize
any kind of conversion factor.
But you need to be
aware, when someone
said joules, what's that a unit
for, or eV, what's that a unit
for.
All right.
So now we're going to do
and in-class demonstration
of the photoelectric effect.
But before we actually
do the experiment,
we're going to predict what
the experiment will show.
Always dangerous to
do that, so we'll
hope it works after
we do the prediction.
All right.
So we're going to be
looking at whether we're
going to get an injection of an
electron from a zinc surface.
And we're given the threshold
energy, or the work function,
of zinc.
Every metal-- this is
a property of metals.
They're different,
as we saw last time.
So this is 6.9 times 10
to the minus 19th joules.
And we're going to use
two different light
sources that are going to
have different wavelengths.
And we'll predict whether
they have enough energy
to meet this threshold
to go over the threshold
and inject an electron.
So the two different
sources, we have a UV lamp
with a wavelength
of 254 nanometers
and a red laser pointer
with a wavelength
of about 700 nanometers.
OK.
So before we do the
experiment, let's
do some calculations
to see what we expect.
So first, we want to
see what the energy,
or calculate what the
energy, of the photon
will be that's a
emitted by the UV lamp.
And I will write this down.
So what do we know?
We know a bunch
of things already.
We know that energy is equal
to Planck's constant times
the frequency.
We also know that the frequency
is related to wavelength
by c, the speed of light.
And then we can put
those two things together
to say the energy, then, is
also the Planck's constant times
the speed of light
divided by the wavelength.
So we can use that last
equation to do a calculation,
and figure out the
energy that's associated
with that particular
wavelength of light.
So here we have energy.
We're going to write in Planck's
constant, 6.626 times 10
to the minus 34.
And the units are
joules times seconds
and the speed of
light, 2.998 times 10
to the 8 meters per second.
And we want to divide this,
then, by the wavelength.
So we have the wavelength
here that we're
using first is a 254 times
10 to the minus 19 meters.
Oh sorry-- 9 meters.
Thank you.
I wrote down 19.
I'm like, wait a minute.
That's not right.
OK.
OK.
So then we can do
the calculation out.
And here is where I
got excited about 19.
We have 7.82 times 10
to the minus 19 joules.
And if we look at
the equation, we'll
see that the meters
are going to cancel.
The seconds cancel,
and we're left
with joules, which is good,
because we want an energy.
So joules is a
good thing to have.
So there, we can do
a simple calculation.
And we can look and say, OK,
if the energy, then, associated
with that wavelength
is 7.82 times 10
to the minus 19th joules, then
we ask, is this greater or less
than the threshold energy?
And it's greater than that.
So it does have enough energy.
It should eject an electron.
So we can try that out and see.
Now we can look at what happens
with the red laser pointer
and see whether that should
have the energy that's needed.
And so I will just write
these things down here
instead of writing it again.
So that was our UV.
So now our red light, we have
700 times 10 to the minus 9
meters, or 700 nanometers.
And so here is our
answer for the UV.
And our answer for the red
light should be 2.84 times 10
to the minus 19 joules.
And I'll move this up a
little so people can see that.
So does that have enough
energy to eject an electron?
AUDIENCE: [INAUDIBLE]
CATHERINE DRENNAN:
No, that should not
work, because that's less than
the threshold energy that's
needed.
All right.
So we'll do one more
calculation just for fun.
And then we'll do
the experiment.
So the last
calculation we'll do is
we'll think about
the number of photons
that are emitted by
a laser in 60 seconds
if you have an intensity
of one milliwatt.
And a milliwatt is equal
to 10 to the minus 3 joules
per second.
So we can just do that
calculation over here.
So we have 1.00 times 10 to
the minus 3 joules per second,
1 photon, and here, this
is for the red laser.
So we'll use the number that
we just calculated over here.
So we have 2.84 times 10
to the minus 19th joules
for the red laser
and times 60 seconds.
And we should get 2.1
times 10 to the 17 photons.
So that's how much
photons, if we hold it
for 60 seconds, that were going
to be shooting at our metal's
surface.
So these are the
kind of calculations
that you'll be doing on
these kind of problems.
And now let's see how
well the experiment works.
So we're going to bring out
our demo TAs, who are going
to tell you about this demo.
And we're going to try
to do some fancy stuff
with this document camera
to project it on the screen.
So this is all very exciting.
Oh, I guess I should
put that down,
the number, in case
you couldn't see
it-- 2.1 times 10 to the 17.
All right.
So let's bring--
you've got the mic.
GUEST SPEAKER 1: OK.
So we've got our
metal plate here
that Eric's got in his hand.
And what he's doing
right now is he's
rubbing it with a little bit
of-- what is that, actually?
ERIC: It's just steel wool.
CATHERINE DRENNAN: Steel wool.
GUEST SPEAKER 1: OK.
So that's just going
to get the aluminum
oxide, because sometimes--
you guys will get to it.
But sometimes you can get
a reaction of aluminum
with the moisture in
the air, and that's
going to cause aluminum oxide.
So he's getting get rid of that.
And now we've put this
on a-- what is this?
ERIC: [INAUDIBLE].
GUEST SPEAKER 1:
What do you call it?
A detector of some kind.
So basically, when he charges
this, what's going to happen
is that you have this plate,
and you have this joint.
And they're both going to
be electrically negative,
because you've introduced
some electrons.
And they're going
to repel each other,
because they're both negative.
Two negative charges
repel each other.
So you're going to see some
space develop as Eric's done.
Now, what he's doing is
he's got a plastic rod here
that he's charging with the fur.
And he's introducing those
electrons onto the plate.
So now we've got a
negatively charged plate,
and you can see that
by the fact that you
see some repulsion between
that rod and the rest
of the detector, which
is actually working out
pretty nicely.
So once--
CATHERINE DRENNAN: So
say this experiment
is very weather-dependent.
If it's really humid or too dry,
it doesn't work nearly as well.
But today, today's good weather.
Today's good weather
for this experiment,
not so much good for
sunbathing outside, but good
weather for this experiment.
GUEST SPEAKER 1: Although
we have UV lamps, so maybe.
CATHERINE DRENNAN: That's true.
GUEST SPEAKER 1: OK.
So now we've got a charge.
CATHERINE DRENNAN: That's
the green laser pointer.
Let's get the red.
GUEST SPEAKER 1: It's
underneath here, I think.
CATHERINE DRENNAN: Oh, yeah.
GUEST SPEAKER 1: OK.
So now--
CATHERINE DRENNAN: We could do
the calculation for the green.
If you want to do the
calculation for the green,
we can try it later.
GUEST SPEAKER 1: Eric's got a
red laser pointer in his hand.
He's going to shine it.
And we're going to see
that nothing happens,
because as we calculated,
the energy of these photons
is not enough to get
over the threshold
of this particular metal.
CATHERINE DRENNAN: So if
electrons were being ejected,
you should see it move.
GUEST SPEAKER 1: And we'll
do that one more time.
Maybe the green one will work.
It doesn't.
CATHERINE DRENNAN: All right.
Well, now we have to see
if the UV-- we built it up.
The UV should--
GUEST SPEAKER 1: So
hopefully this works.
CATHERINE DRENNAN: --work.
Let's see.
GUEST SPEAKER 2: [INAUDIBLE]
GUEST SPEAKER 1: OK.
So oh-- maybe--
AUDIENCE: [INAUDIBLE]
CATHERINE DRENNAN: Oh.
GUEST SPEAKER 1: Oh,
well, I guess it worked.
CATHERINE DRENNAN: It did work.
You could sort of see that.
GUEST SPEAKER 1: So
maybe we can charge it
up again while I talk about it.
CATHERINE DRENNAN:
Yeah, sometimes
the charge [INAUDIBLE].
GUEST SPEAKER 1: The
UV lamp, obviously,
has enough energy in
each of these photons.
So when you shine that
light at the metal,
you have the electrons
on the surface,
which are being ejected.
And if those
electrons get ejected,
then the whole system
becomes neutral.
If the systems become neutral,
then that rod can go back
and is no longer
feels a repulsion,
because the two parts
are no longer negative.
So once we charge this
up again, maybe we
can go to the other side
and-- I think it's good.
It's good.
CATHERINE DRENNAN:
Yeah, that's good.
Oh--
GUEST SPEAKER 1:
It will be fine.
GUEST SPEAKER 2: Wavering.
CATHERINE DRENNAN: OK.
GUEST SPEAKER 1: OK.
Now we're just going
to try it again.
And yay.
CATHERINE DRENNAN: Yay.
GUEST SPEAKER 1: We got it.
[APPLAUSE]
CATHERINE DRENNAN: OK.
Great.
We can just leave this here.
All right.
And I think he held it
for 60 seconds, so you
know how many photons
were coming off,
too, if you want to
do that calculation.
So again, the
photoelectric effect
was really important
at this time
in understanding the properties
that were being observed,
to help us understand about this
quantized energy of particles,
that light had this
particle-like property.
It had this quantized energy.
And you needed a
certain amount of it
to eject an electron
from a metal surface.
So we all know that
light is a wave.
But now there's
this evidence that,
even though it's pretty
much this massless particle,
that it still has
particle-like properties.
So light is a really
amazing thing.
This doesn't really
show up very well.
It's a view of the Stata Center.
Stata Center always has some
really spectacular sunlight
coming around it sometimes.
All right.
So now, if this is
true, that means
that photons that have
this quantized energy
should have momentum as well.
And so Einstein was
thinking about that.
And so he reasoned that
this had to be true.
There had to be some kind of
momentum associated with them.
And so momentum, or p,
here is equal to Planck's
constant times the frequency
divided by the speed of light,
c.
And since the speed of light
is equal to the frequency
times the wavelength
of the light,
then the momentum should be
equal to Planck's constant
divided by the wavelength.
So this is really-- we're
talking about momentum
in terms of wavelength, this
inverse relationship here.
This was just a
kind of a crazy idea
to be thinking
about momentum, when
you're talking about light.
And this really came out of
the photoelectric effect.
And also, there were
some experiments
done by Arthur Compton that
also showed that you could
sort of transfer this momentum.
And so that's again the
particle-like property.
So it's a really exciting time.
OK.
So we're going to
now move to matter.
So we've been
talking about light
and how light has this dual,
particle, wavelike properties.
But what about matter?
So we accept that matter has
particle-like properties.
But what about as a wave?
So enter de Broglie
into this area.
And so he was following what
Einstein was thinking about.
And he said, OK, so
that's pretty cool.
If you have momentum is equal
to Planck's constant divided
by wavelength, if
you could think
of things that have
wavelengths as having momentum.
And he said, or I can
rewrite this equation,
that wavelength equals Planck's
constant divided by momentum.
And we know something
about momentum.
We know that momentum is often
associated with something's
mass times its velocity.
So therefore, I should be able
to rewrite this equation again
in terms of wavelength
being equal to Planck's
constant divided by a
mass and a velocity.
And here, we are expressing
wavelengths in terms of masses.
So this was really something.
And this was basically
his PhD thesis.
I think it maybe had
more pages than that,
but this would have
probably been enough,
this sort of cover page.
This is my PhD thesis.
And Einstein said
that he had lifted
the corner of a great veil
with really just manipulating
what was known at the time and
rearranging these equations
and presenting relationships
that people hadn't really
put together before.
So he ended up
winning a Nobel Prize,
basically, for his PhD thesis,
which is a fairly rare thing
to have happen.
But this was really
an incredible time.
OK.
So if this is true,
if you have equations
that relate wavelengths
to mass, and particles
have wavelike properties,
how come we don't see this?
How come this isn't
part-- how come
no one noticed the
particle going by
and this wavelength
associated with it?
So why don't we observe
this wavelike behavior
if, in fact, it is
associated with particles?
So let's think
about this a minute.
And we can consider why,
when you go to Fenway Park--
and you should,
because it's fun--
and you watch someone
throw a fastball,
why you don't see a wave
associated with that fastball.
So we can consider a fastball
and that the mass of a baseball
is about 5 ounces,
or 0.142 kilograms.
And the velocity of a fastball
is around 94 miles per hour,
or 42 meters per second.
And so we can do a
little calculation
and figure out what the
wavelength associated
with that ball should be.
So wavelength should be
Planck's constant over the mass
times the velocity of the ball.
And we can plug in these values.
And here's Planck's
constant again.
And now you'll note I did
something with the units.
So instead of joule
seconds, I substituted
joules with kilograms meters
squared seconds to the minus 2.
And that's what's
equal to a joule.
And I'm going to do that so
I can cancel out my units.
And again, all of this will be
provided on an equation sheet.
You do not need to remember
all of these conversions.
And so over the mass of the
baseball and the velocity
of the baseball--
and we're going
to put the velocity
in meters per second
so our units can cancel out.
And so I'll just
cancel units out.
So we're canceling
our kilograms.
We're canceling
one of the meters,
and canceling all
of the seconds.
And we have one
meter left, which
is good, because we're
talking about wavelengths.
So that's the unit
we should have.
And the wavelength is 1.1 times
10 to the minus 34 meters.
That is a really small
number times 10 to the 34.
And it is, in fact,
undetectably small.
OK.
So now why don't you
try your hand at this,
and we'll try a
clicker question.
Yeah, it's very tiny.
All right.
Let's take just 10 more seconds.
Oh, or five seconds.
OK.
Awesome.
It went away.
That's OK.
So they're in-- 97%.
I like 97%.
That's a good number.
So again, you want
to think about this
and just realize the
relationship, the equation,
involved.
And so thinking about--
oops, I switched pointers.
I like the green better.
So think about the relationship
between the velocity
of the ball and the wavelength.
And so Wakefield, who
was an knuckleballer,
is the winner here, with
the longest wavelength.
But still, the number
for this is 1.4 times 10
to the minus 34.
And so this is still
undetectably small.
So of course, no one had
noticed this property before.
But it still, it still exists.
So when you're talking
about a baseball,
the wavelength is really
not very, relevant to you,
because it is this incredibly
small, undetectable number.
But if you're talking
about an electron,
it's entirely different.
So now, if we think about a
gaseous electron traveling
at 4 times 10 to the 6
meters per second, and so
that's associated with
an eV of about 54.
So we have this electron
traveling with this velocity.
And now, if we do
this calculation,
so if we use Planck's
constant divided
by the mass of the
electron-- and that's
known, in another great
experiment-- and its velocity,
now we can calculate
out the wavelength.
And it's 2 times 10 to the minus
10, or about two angstroms.
Now, 2 angstroms is
a relevant number,
when you're talking
about an electron,
because an electron
is in an atom.
And atoms tend to be-- you have
diameters 0.5 to 4 angstroms.
So now the wavelength
is on the same scale
as the size of the object
you're talking about.
And so when that's true, all
of a sudden, the wavelength--
the wavelike property becomes
super important to thinking
about this.
So for an electron
that is a particle,
it's really important to think
about its wavelike properties.
And so people were saying, OK,
if electrons are waves, then
maybe we should see other
wavelike properties,
such as diffraction.
Diffraction, we talked
about last time,
is an important
wavelike property
of constructive interference,
destructive interference.
So people looked to
see whether there
were diffraction-like
properties,
and in fact, there are.
So we had observed,
then, the first
was observing
diffraction of electrons
from a nickel crystal.
And then JP Thomson
showed that electrons
that pass through
gold foil again
produced a diffraction pattern.
So again, this was
a wavelike property.
So you might think Thomson, that
sounds a little familiar to me.
Didn't she just talk
about that last week?
And yes, here there are
two important Thomsons
in this story.
And this is a
father and son team.
And so JJ Thomson won
a Nobel Prize in 1906
for showing that an
electron is a particle.
He discovered an electron.
And then in 1937, his son wins
a Nobel Prize for showing-- son
just had to be like, Dad, I'm
going to show you're wrong.
An electron is, in fact, a wave.
But I think they
were both happy.
I think they both got along,
no father-son rivalry.
I think this is one of the
cooler stories in science,
how this father,
son both had kind
of opposite discoveries, which
both ended up being true,
and really changed the way
we thought about matter.
All right.
So we have light as a
particle and as a wave.
We have matter,
particularly electrons,
as particles and waves.
And now we are ready
for a way to think
about how to put this together.
So before we move on and talk
about the Schrodinger equation,
I just want to take a break
from history for a minute,
because some of you are
like, OK, well, this
is really cool for the
father and son team,
but what about today?
What's happening today?
So let's take a break
from history for a second
and talk about why you should
care about small particles.
Small particles of
special properties,
if they're on the
subatomic scale,
their properties are different.
If you have very, very few
atoms, versus many atoms,
the things with very few
atoms have special properties.
So why should you
care about that?
Why should you care
about the energies
that we can get out of
the Schrodinger equation?
So why should we care about the
Schrodinger equation or quantum
mechanics?
So there are many reasons,
but I will share one with you.
And this is a segment
in their own words.
So you're going to hear
from Darcy, who was actually
a former TA for 5.111.
So she is associated
with this class.
She actually just got her
PhD in the spring from MIT,
and she now works at Google.
But in this short,
she's going to tell you
about research in
Moungi Bawendi's Lab,
and why you should care
about quantum dots, which
are small collections of atoms.
So I'm going to
try to switch over
now and hope that
our demo before
didn't screw up the sound.
But we'll see what we can do.
And I think it should be good.
[VIDEO PLAYBACK]
- My name is Darcy Wanger, and
I work as a graduate student
in the Bawendi Lab at MIT.
I work with quantum
dots in my research.
Quantum dots are really,
really tiny particles
of a semiconductor.
So we're talking like 4
nanometers in diameter.
In a particle that small, there
are only 10,000 or so atoms,
which seems like a
lot of atoms if you're
comparing to
something like water,
which only has 3 atoms in it.
But if you compare it to
something you can actually
hold in your hand, which
has a lot of atoms in it,
10,000 is actually a
pretty small number.
So a particle this small has
really strange properties.
Different things start to matter
when you get really small.
And just like an
atom, a quantum dot,
or semiconductor nanocrystal,
has discrete energy levels.
So if an electron is sitting
at this energy level,
and it absorbs
light, an electron
can get excited to a
higher energy level.
And then, when that electron
relaxes back down to the ground
state, it emits light.
And the energy of
that light is exactly
the difference between
these two energy levels.
The difference between
the energy levels
is related to the
size of the dot.
So in a really
small quantum dot,
the energy levels are far apart.
So the light it emits
is higher energy,
because there's a large energy
difference between the energy
levels.
If we use a larger
quantum dot, the distance
between the energy levels is
smaller, so the light it emits
is lower energy, or redder.
People in our lab are working
to make quantum dots bind
to a tumor.
So when a doctor goes
in to remove a tumor,
they can see the shining
of the UV light on it,
and see whether
it's all gone when
they've taken out the tumor.
They can also use quantum
dots to label other things
other than tumors, like pH
or oxygen level or antibodies
or the other drugs that are
treating the cancer tumor.
Each of those can
be different colors.
So if you shine a light
on that whole area,
you can see, oh, that orange
spot, that's some cancer cells.
Oh, and that green tells me
that the pH is above 7.4.
So it's pretty cool that we can
use the idea of energy levels
in something so applicable like
surgery, where it can actually
be used to track things and
make it easy for doctors
to see what's going on while
they're doing a surgery.
[END PLAYBACK]
CATHERINE DRENNAN: OK.
So that's an example
for course 5 research.
[APPLAUSE]
And you can see all
these credits online.
I will mention that some
of those nice animations
were done by a former graduate
student in the chemistry
department.
So these videos,
even the art was
done by chemists,
which is a lot of fun.
OK.
So let's introduce the
Schrodinger equation.
And we'll spend some
more time on this
as we go along, on Friday.
So we needed now--
we had learned a lot
about wave particle
duality and about
these subatomic particles.
And we needed a way
to think about it.
We needed a theory to
describe their behavior.
And classical mechanics had
some flaws in with respect.
So we needed a new
kind of mechanism.
We needed quantum mechanics.
So here, if we're
thinking about particles
that are really
small like electrons,
we need to consider the
wavelike properties.
It's really important when
you have a wavelength that
is so similar to the
size of the object
that you're thinking about.
So the Schrodinger
equation really
became to quantum mechanics
like Newton's equations
were to classical mechanics.
So what is the
Schrodinger equation?
So here's a picture
of Schrodinger.
And he looks so happy.
I would be happy, too, if I
had come up with this equation,
I think.
So here's the simplest
form of the equation
that you will probably ever see.
And so we'll just define
some of these terms.
So we have wave function, psi.
And over here is
the binding energy,
and that's the energy of binding
an electron to a nucleus.
And then an H with a hat, we
have our Hamiltonian operator.
And in this course, you will
not be solving this equation.
We're just going to be talking
about what sorts of things
came out of this equation.
So I'm going to give
you a little bit longer
version of the equation now.
And so again, you're
thinking about the electron.
It has these
wavelike properties.
And it's somewhere in the atom,
not crashing into the nucleus.
And it needs to be defined
in three dimensions.
And it has momentum,
so it's moving.
So we need to think about
this as an equation of motion
in a three-dimensional space.
And the equation
is going to change.
The math will change,
depending on where
the electron is located,
which you won't know exactly.
So this is a very hard problem.
But it's not totally
without anything
to do with classical mechanics.
And if we write
the longest version
you'll see, at least in this
course, for the hydrogen atom,
I just want to show
this to point out
that there are some terms from
classical mechanics in here.
This is Coulomb's energy,
also sometimes called
potential energy.
So we saw Coulomb's
force before.
Here is Coulomb's energy.
So some of the
classical mechanics
is contained within
this, but it expands
from classical
mechanics to consider
the wavelike properties
of the electrons.
So whenever I talk
about this, I always
feel like I want to
have something better
to say about really what this
is doing and where it came from.
In terms of what it's doing,
how is solving this helping you?
What are you learning
from solving this?
So one thing you're
learning from solving
this is you're
finding E. And that's
really important, the
binding energy of the nucleus
and the electron.
And we saw before
that, if you just
used simple classical
mechanics, you
have a positive and
negative charge that
are close to each other.
Why don't they come and
crash into each other?
We want to know how they are
bonded to each other, what's
the real energy of
that association.
We also saw, with the
photoelectric effect,
that it's not that
easy to get an electron
to eject from a metal surface.
So it's bound in there.
And what is that
actual binding energy?
So that comes out of the
Schrodinger equation.
This E here is the
binding energy.
And also, solving it will tell
you about the wave function
or, as chemists
like to talk about,
orbitals, where the electrons
are, in what orbitals.
So this is the
information you get out.
And importantly, it works.
It matches experiment.
So chemists are
experimentalists.
We love experiments,
and we see this data,
and we want to understand it.
And the Schrodinger equation
helps us understand it.
It correctly predicts binding
energies and wave functions,
and it explains why the hydrogen
atom is, in fact, stable,
where you don't have crashing or
exploding of the hydrogen atom.
So where did this equation
come from that works so well?
How did Schrodinger
come up with this?
And this is always sort of
the puzzle when I teach this.
I feel like I should have
something profound to say
about where this came from.
And so I've done a little
reading and looked,
and I thought the best
explanation for this
that I ever saw came
from Richard Feynman.
And when he was asked
how Schrodinger came up
with this equation,
he said, "it is not
possible to derive it
from anything you know.
It came out of the
mind of Schrodinger."
And I thought that
pretty much summed it up.
So sometimes-- after
class last week,
on Wednesday, someone
came down and said,
you know, the
Thomson experiment,
discovering the electron,
why didn't someone else
do that experiment?
It seemed like it's
not a cathode ray.
And you have to have a
little phosphorous screen.
Why didn't someone else
discover the electron?
And some of these other--
de Broglie rearranged
some equations, did it
in a way that no one else
was thinking, but still.
Or plot solving the
equation of a straight line.
No one else was thinking
about it some way,
using other people's data.
They just sort of saw things in
data that other people didn't.
But you think why didn't
someone else see that, too?
When it comes to the
Schrodinger equation,
the question is why didn't
someone else or lots of people
come up with it?
I think the question really
is, how did Schrodinger
come up with it?
At least, that's
the question to me.
And I have never really-- that's
the best explanation I have.
It just came out of his mind.
OK.
So we're many years later.
We've had the Schrodinger
equation for a while.
So this is an old story, right?
Well, maybe for
the hydrogen atom,
but this is still actually a
very active area of research.
Oh, my startup disk is full.
Thank you.
Let's go back to that.
All right.
So I just thought--
I always like
to give you examples of current
research on these areas.
And so I know a
number of you were
interested in potential of being
chemical engineering majors,
undergrads.
And I'll tell you
about a new professor
who started about a
year ago, Heather Kulik.
And her research group is
really interested in using
a quantum mechanical
approach to study materials
and to study proteins.
But when you get to
things like proteins,
there's thousands and
thousands of atoms around.
Forget multiple electrons, we're
talking about multiple atoms
with multiple electrons,
huge complexes.
How can you give a quantum
mechanical analysis
of things that are so large?
And this is really important.
I mean, I think that one of
the big problems moving forward
is solving the energy
problem and doing it
in a way that doesn't
destroy our environment, so
new batteries, new
electrodes, new materials.
We need to understand the
properties of different metals
to understand what will
make those good electrodes.
And to really
understand them, we
need a quantum
mechanical approach.
But these are big areas.
There's a lot of things
to consider here.
So Heather is
interested in coming up
with improving algorithms,
improving the computation,
to really give a quantum
mechanical analysis to systems
that have a lot
of atoms in them.
So if you're interested in
this area, you're not too late.
You don't have to go
back to the early 1900s.
There's still a lot
to do in this area.
OK.
So very briefly
now, let's just look
at the Schrodinger equation
we saw from the hydrogen atom.
So we'll go back
to understanding
quantum mechanical analysis
of photosynthesis-- amazing,
don't understand how it works.
That would be great if we did.
That would really solve
a lot of energy problems.
But we'll just go to hydrogen
atom, one electron back.
So if you solve the
Schrodinger equation for this--
and I think I did this in
college, not in freshman,
chemistry, but somewhere
along the line--
you'll come up with this term.
So again, this is
the binding energy.
We just want to know
about how the electron is
being held by the nucleus.
And there are some
terms in here.
We have the electrons
mass-- that's known,
the electrons charge.
We have a permittivity
constant and Planck's constant.
And if you look at this,
you go, wait a minute.
That's a constant.
That's a constant.
That's a constant.
That's a constant.
We can simplify that.
And we will.
And that is the Rydberg's
constant, 2.18 times
10 to the minus 18th joules.
So now it doesn't
look quite as scary.
We can just substitute this RH.
That makes us feel a lot better.
It's one number that will be
given on the equation sheet,
so we don't even
have to remember it.
And now we can rewrite this in
terms of the binding energy.
So again, the binding
energy, this is a constant.
So now this turns into
minus RH over n squared.
And n, what is n?
n is a positive integer
1, 2, 3, up to infinity.
And what's its name?
What is n?
You can you yell it out.
Some of you know.
AUDIENCE: [INAUDIBLE]
CATHERINE DRENNAN: Yeah.
The principle quantum
number, that's right.
So the principle
quantum number comes out
of the Schrodinger equation.
And that's how we
can think about it.
And again, here are these ideas.
The binding energies
are quantized.
This is a constant over here.
So the principle
quantum number comes out
of the Schrodinger equation.
All right.
So now, next time, we're
going to think more
about the Rydberg constant.
And we're going to do a
demonstration next Friday
of the hydrogen atom spectrum
to show that the Schrodinger
equation, in fact, can
explain binding energies.
So that's on Friday, and that's
our first clicker competition.
So come.
Be ready with your clickers.
You can sit in recitations.
You can share answers
before clicking in.
It's not cheating.
It's teamwork.
OK.
See you Friday.
