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All right. Let's get going.
Where were we? We were at the point where we
started out the course wondering about the structure of the atom,
how the electron and the nucleus hung together.
And we saw that we could not explain how that nucleus and
electron hung together using classical ideas,
classical physics, classical mechanics and
classical electromagnetism. And so we put that discussion
aside and started to talk about the wave-particle duality of
light and matter. And we saw that both light and
matter can behave as a wave or it can behave as a particle.
And we needed that discussion in order to come back to talk
about the structure of the atom. And, in particular,
what was so important last time that we met is that we saw the
results of an experiment, that Davidson,
Germer, and George Thompson experiment that demonstrated
that massive particles could exhibit wave-like behavior.
We saw the interference pattern of electrons reflected from a
nickel single crystal. An actual original paper that
reports that result is on our website.
You're welcome to take a look at it.
But that really was the impetus, this observation of the
wave-like behavior of matter. That was the impetus for this
gentleman here, Erwin Schrödinger in 1926-27 to
write down an equation of motion for waves.
That is, he thought maybe the answer here is that if particles
can behave like waves, then maybe we have to treat the
wave-like nature of particles, the wave-like nature of
electrons, in particular in the case when the wavelength,
the de Broglie wavelength of the particle or the electron,
is comparable to the size of its environment.
Maybe in those cases we have to use a different kind of equation
of motion, a wave equation of motion.
And that is what he did. So, he wrote down this
equation. We briefly looked at it last
time. This equation has some kind of
operator called the Hamiltonian operator.
It has a hat on it, a carrot on the top of it.
That tells us it is an operator that operates on this thing,
psi. That psi is what's going to
represent our particle. That psi is a wave.
Since we're going to give it a functional form in another day
or so, we're going to call it a wave function.
Somehow that psi represents our particle.
Exactly how it represents our particle is something we're
going to talk about again in a few days.
But right now the important thing is to realize that this
psi represents the presence of a particle, in this case the
electron. And when H operates on psi we
get back psi. We do this operation and out
comes psi again, the original wave function,
but it's multiplied by something.
That something is the energy. It's the binding energy.
It's the energy with which the electron is bound to the
nucleus. This equation is an equation of
motion. This wave equation,
Schršdinger's equation, is to this new kind of
mechanics, called quantum mechanics, like Newton's
equation of motion, and I show you just the second
law here, are to classical mechanics.
This equation here tells us how psi changes with position and
also with time. It tells us something about
where the electron is and also tells us where the electron is
as a function of time. It's an equation of motion.
And this is what Schrödinger realized, is that maybe when the
wavelength of a particle is on the order of the size of its
environment, you have to treat it with a different equation of
motion. You can no longer use F equals ma,
this classical equation of
motion. You have to use a different
equation of motion. And that's Schršdinger's
equation, this wave equation. Can we dim the front lights a
little bit? Because the screen is just a
little bit hard to see, I think.
We are going to let our electron be represented by this
wave psi. And so psi, since it's going to
be representing the electron, this psi is going to be a
function of some position coordinates and also,
in the broadest sense, a function of time.
Now we of course can label psi in Cartesian coordinates,
giving it an x, y and z.
If I gave you x, y and z for this electron in
this kind of coordinate system where the nucleus is at the
origin of the coordinate system, if I give you the x,
y and z coordinates, you'd know where the electron
was. But it turns out that this
problem of the hydrogen atom is really impossible to solve if I
use a Cartesian coordinate system.
So, I am going to use a spherical coordinate system.
How many of you are familiar and have used a spherical
coordinate system before? A few of you.
Not all of you. Well, it's not hard to
understand. And it is important that you
understand it. So, instead of giving you an x,
y and z to locate this electron, this particle in
space, we're going to give you an r, a theta and a phi.
And the definitions of r, theta and phi are the
following. If here is my nucleus at the
origin and here is the electron, r is the distance of that
electron from the nucleus. It's just the length of this
line right here. That's one coordinate.
A second coordinate is theta. Theta is the angle that this r
makes from the Z-axis. And then the third coordinate
is phi. Phi is the following.
If I take that electron and I just drop a perpendicular to the
x,y-plane, and I then draw a line in the x,y-plane from that
point of intersection to the origin, the angle that that line
makes with the x-axis here is phi.
So, I am going to give you an r, a theta, and a phi.
r is just the distance of the electron from the nucleus.
Theta and phi tell us something about the angular position.
And then, as I said, in the largest sense,
there is also time. But we'll talk about time a
little bit later. Psi represents our electron.
Now, what does the Schrödinger equation specifically,
for a hydrogen atom, actually look like?
This H psi times E psi is the kind of generic Schrödinger
equation. And now we've got to write a
specific one, one specific for the hydrogen
atom. We need to know what this is,
H. That's our Hamiltonian,
our operator. And so the operator here for
that hydrogen atom is this. What is it is essentially three
second derivatives, one here is with respect to r,
a second is a second derivative effectively with respect to
theta, and another one, the final one is a second
derivative with respect to phi. In other words,
if this whole Hamiltonian is operating on psi,
what you're going to do is essentially take the first
derivative of psi with respect to r, multiply it by r squared,
-- -- and then take a first
derivative with respect to r again and multiple it by 1 over
r squared.
And add to that the first derivative of psi with
respect to theta multiplied by psi and theta, etc.
You don't have to know this. I'm just showing this to you so
you recognize it later on. This is a differential
equation. In 18.03, you learned how to
solve these differential equations.
And then there is another very important term here,
so it's all of this plus this, u of r.
What is u(r)? Potential energy of
interaction. And the potential energy of
interaction, of course, is the Coulomb potential energy
right here, 1 over r dependence.
We've talked about the Coulomb force.
This is the potential energy of interaction that corresponds to
the Coulomb force. So, that's the specific
Schrödinger equation for the hydrogen atom.
Now, what we have got to do is we have to solve this equation
for the hydrogen atom. And when I say solve this
equation, what I mean is we're going to have to find E,
these binding energies of the electron to the nucleus.
That is part of our goal when we say solve this differential
equation is knowing what E is, is figuring out what E is.
And, actually, this is what we're going to do
today, finding those energies. But then the second goal is to
find psi. We want to find what is the
functional form of psi that represents the electron and the
hydrogen atom? Therefore, we're going to want
to find the wave functions for psi.
And, you know what, those wave functions are
nothing other than what you already sort of know,
and that is orbitals. You talked in high school about
s orbitals and p orbitals and d orbitals.
Those orbitals are nothing other than wave functions.
They come from solving Schršdinger's equation for the
hydrogen atom. That's where they come from.
Now, specifically, the orbital is something called
the spatial part of the wave function as opposed to the spin
part. But for all intents and
purposes they are the same. We're actually going to use
these terms interchangeably, orbital wave function,
wave function orbital. The bottom line is that when
you solve Schršdinger's equation for the energy and the wave
function, it makes predictions for the energies and the wave
functions that agree with our observations,
as we're going to see today, in particular for the case of
the binding energies of the electron to the nucleus.
This equation predicts having a stable hydrogen atom,
a hydrogen atom that seemingly lives forever in contrast to
when we use classical equations of motion.
When we used classical equations of motion we got a
hydrogen atom that lived for all of 10^-10 seconds.
But here we finally have some way to understand the stability
of the hydrogen atom. It makes the Schrödinger
equation, it makes predictions that agree with our observations
of the world we live in. And, therefore,
we believe it to be correct. That is it.
It agrees with the observations that we make.
Let's start. And we're actually not going to
solve the equation, as I said, but you will do so
if you take 5.61, which is the quantum course in
chemistry, or 8.04, I think it is,
which is the quantum course in physics after you take
differential equations, so that you know how to solve
the differential equations. But we're going to write down
the solution in particular here for E, the binding energies of
the electrons to the nucleus. Now we're going to need this.
We've got H psi equals E psi. And when we
solve that equation, we get the following expression
for E, these binding energies. E is equal to 1 over n squared
times m e^4 over 8 epsilon nought squared times h squared.
That is what we get out of it. And there is a minus sign out
in front.
Now, what is m?
m is the mass of the electron. What is e?
e is the charge on the electron.
Epsilon nought is this permittivity of vacuum that we
talked about before. It's really just a unit
conversation factor here. h is Planck's constant.
Here comes Planck's constant again.
It is ubiquitous. It's everywhere.
And what we do is that we typically take all of these
constants and group them together into another constant
that we call the Rydberg constant.
And we denote it as a capital R sub capital H.
All of that is equal to RH, so this is over n squared,
minus 1. And the value of RH,
that Rydberg constant, and this is something you're
going to need to use a lot in the next few weeks,
is 2.17987x10^-8 joules. But you also see,
in this expression for the binding energies of the electron
to the nucleus, that there is this n here.
What's n? N is an integer.
When you solve that differential equation,
you find that n has only certain allowed values.
N can be as low as 1, 2, 3, and n can go all the way
up to infinity. N is what we call the principal
quantum number.
I am going to explain that a little bit more by looking right
now at an energy level diagram. That is the expression.
But now let's plot it out so that we can understand what is
going on here a little more. We are going to be plotting
this expression as n goes from 1 to infinity.
I have the energy access here. Energy is going to be going up
in that direction. When n is equal to 1,
the binding energy of that electron to the nucleus is
effectively the Rydberg constant.
Here I rounded it off, -2.180x10^-18 joules.
But our expression here says that there can be another
binding energy of the electron to the nucleus.
It says that n can be equal to 2.
And if n is equal to 2, well, then the binding energy
of the electron is one-quarter of the Rydberg constant,
because it is the Rydberg constant over 2 squared.
If n is equal to 3, well, our expression says that
the binding energy is minus a ninth of the Rydberg constant.
If n is equal to 4,
it is minus a sixteenth of the Rydberg constant,
n is equal to 5, minus a twenty-fifth,
n equal to 6, minus a thirty-sixth,
n equal to 7, minus a forty-ninth,
all the way up to n equals infinity.
And you know what the value of the binding energy is when n is
equal to infinity? Zero.
Our equation says that the electron can be bound to the
nucleus with this much energy or this much energy or this much
energy and so on, but it cannot be bound to the
nucleus with this much energy, somewhere in between,
or this much energy or that much energy.
It has to be exactly this, exactly this,
exactly this, so on and so forth.
That is important. What we see here is that the
binding energies of the electron to the nucleus are quantized,
that that binding energy can only have specific allowed
values. It doesn't have a continuum of
values for the binding energy. Yes?
Those are identically the same size.
Because this is an operator, right?
I left the hat off here. Remember that we took a second
derivative of psi? So, you cannot cancel this.
This is an operator taking the derivative of psi.
You cannot just cancel that. This isn't a multiply by on
this side. This side is.
This is E times psi, but not over here.
That's really important. We have these quantization of
the allowed binding energies of the electron to the nucleus.
Where did that quantization come from?
That quantization came from solving the Schrödinger
equation. It drops right out of solving
the Schrödinger equation. How did that happen?
Well, in differential equations, as you will see,
when you solve a differential equation, what you have to do to
solve it so that it adequately describes your physical
situation is you have to often impose boundary conditions onto
the problem. And it's that imposition of
boundary conditions that gives you that quantization.
That is where it comes from mathematically.
In other words, remember one of those angles
that I showed you, the phi angle?
You can see it would run from zero to 360 degrees.
But you also know, if you go 90 degrees beyond 360
degrees, suppose you to go to 450 degrees, well,
that should give you the same result as if you had phi equals
90. What you have to do is you have
to cut off your solution at 360 degrees.
When you cut off that solution, well, then that gives you,
in these differential equations, these quantization.
That is physically where it comes from.
Again, this is not something you're responsible for,
but when you do differential equations later on in 18.03,
you will see how that happens. Let's talk some more about
these loud energy levels. When the electron,
or when n is equal to 1, the language we use is that we
say that the hydrogen atom, or we say that the electron of
the hydrogen atom, is in the ground state.
We call this the ground state because this is the lowest
energy state. It has got the most negative
energy. It's the lowest energy state.
We call n equals 1 the ground state of the hydrogen
atom or the ground state for the electron.
We use those terms of the electron or the hydrogen atom
interchangeably. Now, what's the significance of
this binding energy? And this is important.
The significance is that the binding energy is minus the
ionization energy for the hydrogen atom,
because if I put this energy in from here to there into the
system, then I will be ripping off the electron and will have a
free electron. So, the ionization energy is
minus the value of this binding energy.
The ionization energy is always positive.
The binding energy, the way we're going to treat
this, is going to be negative because the electron is bound.
And then the separated limit, the electron far away from the
nucleus, well, that energy is zero.
So, the binding energy is minus the ionization energy,
or conversely, the ionization energy is minus
the binding energy. That is the physical
significance of these binding energies.
And when we talk about an ionization energy for an atom,
we are typically talking about the ionization energy when the
atom is in the ground state. This is the ionization energy
we're talking about. But we also said that the
binding energy of the electron can be this much,
meaning it's in the n equals 2 state.
That can be possible also. Not at the same time as it's in
the n equals 1 state, but you can have a hydrogen
atom in a state, which is the n equals 2 state.
What that means is that the electron is bound by less
energy. When that is the case,
we talk about the hydrogen atom being in the first excited
state. This is the ground,
this is the first excited state, but n is equal to 2.
In that case, the electron is not as strongly
bound because it is going to require less energy to rip that
electron off. The binding energy in n equals
2 is minus the ionization energy if you have a hydrogen atom in
the first excited state. Make sense to you?
Yeah. Okay, so we can have atoms in
this state, too. Then the ionization energy is
less. It takes less energy to pull
the electron off. Yes?
In everything that we are going to deal with,
we are going to have binding energies that are negative.
Let's do that. You can, of course,
have a binding energy that is positive, but the problem is
that isn't a stable situation. Okay.
Good. Other questions?
Yes. When we're dealing with a
solid, we talk about a work function as opposed to an
ionization energy. When we're dealing with an atom
or a molecule, we talk about an ionization
energy as opposed to the work function.
It's really the same thing. Historically,
there is a reason for calling the ionization energy off of a
solid the work function. Oh, one other thing.
I just wanted to point out again right here,
is that when n is equal to infinity, the binding energy is
zero. That is the ionization limit.
That is when the electron is no longer bound to the nucleus.
Now, one other point here is that this solution to the
Schrödinger equation for the hydrogen atom works.
It predicts the allowed energy levels for any one electron
atom. What do I mean by one electron
atom? Well, helium plus is
one electron atom. Because helium usually has two
electrons, but if you take one away you have only one electron
left. And so this helium plus ion,
that's a one electron atom, or if you want to say it more
precisely, one electron ion. Or, lithium double plus,
that's a one electron atom or a one electron ion.
Because lithium usually has three electrons,
but if you take two away and you only have one left,
that's a one electron atom. Uranium plus 91 is a
one electron atom. Because you took 92 of them
away, one is left, that's a one electron atom or
an ion. And the bottom line is that
this expression for the energy levels predicts all of the
binding energies for one electron atoms as long as you
remember to put in the Z squared up here.
For a hydrogen atom that is, of course, Z equals 1,
so we just have minus R sub H over n squared.
But for these other one electron atoms,
you have to have the Z in there, the charge on the
nucleus. Why?
Well, because that Z comes from the potential energy of
interaction. The Coulomb potential energy of
interaction is the charge on the electron times Z times e,
the charge on the nucleus. That is where the Z comes from.
That is important. How do we know that the
Schrödinger equation is making predictions that agree with our
observations? Well, we've got to do an
experiment. And the experiment we're going
to do is we're going to take a glass tube like this.
We're going to pump it out and we're going to fill it with
hydrogen, H two. And then in this glass tube
there are two electrodes, a positive electrode and a
negative electrode. And what I am going to do here
is that I am going to crank up the potential difference between
these positive and negative electrodes, higher,
higher, higher until at a point we're going to have the gas
break down, a discharge is going to be ignited,
just like I am going to do over here.
Did I ignite a discharge? Yes.
There it is. And the gas is going to glow.
We are going to have a plasma formed here.
Oh, and what happens in this plasma is that the H two
is broken down into hydrogen atoms.
And these hydrogen atoms are going to emit radiation.
That is some of the radiation that you're seeing here in this
particular discharge lamp. We are going to take that
radiation and we're going to disperse it.
That is, we're going to send the light to a diffraction
grating. This is kind of like the
two-slit experiment. And when you look at it you're
going to see constructive and destructive interference.
But when you look at the bright spots of constructive
interference you're going to find that those bright spots now
are broken down into different colors, purple,
blue, green, etc.
And that is because the different colors of light have
different wavelengths. And if they have different
wavelengths, well, then the points in space of
maximum constructive interference are going to be a
little different. And so we're going to literally
separate the light out in space depending on their colors.
And we're going to see what colors come out of this.
And so now, in order to help you do that, we've got some
diffraction grating glasses for you.
You should put them on and look into this light.
And you will be able to see off to your left and to the right
some very distinct lines.
And if you look into the lights above, you can see all different
colors from the white light.
All right. Do you see the hydrogen lamp?
I know that the white lights above the room are more
interesting because there is a whole rainbow there.
I am going to turn the lamp a little bit since not all of you,
if you are way on the side, can see it.
I am going to start over here and I am going to turn the lamp
a bit. Can you see that now?
You should see a bunch of lines here to your left and some to
your right. And then, of course,
you will see some up here. But they will probably
dispersed best to your left and to your right.
Pardon? Can we dim the bay lights?
Can we dim those big lights over there?
Probably not. I am going to turn it over
here. Can you see it?
The spectrum that you should see is what I am showing on the
center board there. You see it?
Pardon? You have to look at the light.
Oh, thank you. Thank you very much.
Can you see that better? I will turn it back there.
Do you see the emission spectrum now?
It's a little better.
Let's see if we can try to understand this emission
spectrum that you're seeing. What you should see the
brightest is a purple line. No?
Well, let's see. The purple line is actually
rather weak, I have to say. If you come really close you
can see it. And you're invited to come up a
little bit closer. The purple line is kind of
weak. What did I do?
[LAUGHTER] Oh, I see.
Yes. Interference phenomena work.
Hey, look at that. [LAUGHTER] Fantastic.
All right. The purple line is kind of
weak, but the blue line is really strong.
And then there is a green line, which is also a little bit
weak. And I can see because I'm
really close, well, I'm not going to tell you
that. There is a green line there.
And then there is this red line.
Let's see if we can understand where these lines are coming
from. What is happening is that this
discharge, not only does it pull the H two apart,
break bonds, make hydrogen atoms,
but it puts some of those hydrogen atoms into these
excited states. And so a hydrogen atom might be
in this excited state. This initial excited state,
high energy state. And, of course,
that's a high energy state. It is unstable.
The system wants to relax. It wants to relax to a lower
energy state. And when it does so,
because it's going to lower energy state,
it has to emit radiation. And that radiation is going to
come out as a photon whose energy is exactly the energy
difference between these two states.
That's the quantum nature, here, of the hydrogen atom.
The photon that comes out has to have an energy,
delta E, which is exactly the energy of the initial state
minus that of the final state. And, therefore,
the frequency of that radiation is going to have one value given
by this energy difference divided by Planck's constant h.
That is what's happening in the discharge.
What we've got is some hydrogen atoms excited to say,
for example, this B state,
which is a lower energy state, and so when it relaxes there is
a small energy difference between here and this bottom
state. Therefore, you are going to
have a low frequency of radiation.
If you have some other hydrogen atoms in the discharge that are
excited to this state up here, well, this is a big energy
difference. And so delta E is going to be
large. And, therefore,
you're going to have some radiation emitted that's at a
high frequency because delta E is large.
If it's at a high frequency, it's going to have a short
wavelength. These hydrogen atoms are going
to have a low frequency emission.
It's going to be a long wavelength.
So, we've got a mixture of atoms in this state or in this
state or in any other states in this discharge.
Now, let's try to understand this spectrum.
And to do that I have drawn an energy level diagram for the
hydrogen atom. Here is n equals 1 state,
n equals 2, n equals 3, n equals 4, all the way up to n
equals 0 here on the top. They get closer and closer
together as we go up. This purple line,
it turns out, or the purple color comes from
a transition made from a hydrogen atom in the n equals 6
state to the n equals 2 state. The final n here is 2.
The blue line comes from a hydrogen atom that has made a
transition from n equals 5 also to n equals 2.
The green line is from a hydrogen atom that makes a
transition from n equals 4 to n equals 2 and then the red line
from n equals 3 to n equals 2. Of course, the transition from
n equals 3 to n equals 2 is the smallest energy.
Therefore, it is going to be the longest wavelength.
n equals 6 to n equals 2 largest energy.
Therefore, it is going to have the smallest wavelength.
Now, how do we know that these frequencies agree with what
Schrödinger predicted they should be?
Well, to know that, what we're going to do is we're
going to write down this equation here,
which is just telling us what the frequency of the radiation
should be, it is delta E over H. But we're going to use the
predictions from the Schrödinger equation and plug them into here
to calculate what the frequency should be.
We were told here that the energy, say, of the initial
state given by the Schrödinger equation is minus R sub H over
the initial quantum number squared.
We're going to plug that into
there. The final state,
well, that's also the expression for the energy,
we're going to plug that into there.
We're then going to rearrange that equation,
so we get the frequency is the Rydberg constant over H times
this quantity (1 over n sub f squared minus 1 over n sub i
squared).
And since I told you here that all of these lines --
The final quantum number is 2. We can plug that in.
And then we can just go in and put in 3, 4, 5,
6 and get predictions for what nu is for the frequency.
And what you would find is that predictions that this makes,
the Schrödinger equation makes agrees with the observations of
the frequencies of these lines to one part in 10^8.
There is really just remarkable agreement between the energies
or the frequencies predicted by Schršdinger's equation and what
we actual observe for the hydrogen atom.
Here is another diagram of the energies of the hydrogen atom n
equals 1, n equals 2, n equals 3.
And the four lines that we were looking at where shown right
here. These are the four lines.
Here is n equals 6 to n equals 2, n equals 5 to n equals 2.
These lines are actually called the Balmer series.
I want you to know that there is also a transition from n
equals 6 to n equals 1. It is over here.
But you can see that that transition is a very high energy
transition. That transition occurs in the
ultraviolet range of the electromagnetic spectrum.
And, therefore, you cannot see it,
but it is there. Actually, what you can see is
that there are transitions from these higher energy states to
the ground state, transitions from all of them to
the ground state, but they're all in the
ultraviolet range of the electromagnetic spectrum.
That is why you cannot see that right now.
But those lines are called the Lyman series.
And then there are transitions here to the n equals 3 state.
These transitions from the larger quantum number to n
equals 3 are called the Paschen series.
They occur in the near infrared.
Brackett series in the infrared.
Pfund series in the far infrared.
I got that backwards. And these different series are
all labeled by the final state. And they're labeled by the
names of the discoverers. And the reason there are so
many different discoverers is because in order to see the
different kinds of radiation, you have to have a different
kind of detector. And, depending on what kind of
a detector an experimentalist had, well, that will dictate
then what he actually can see, what kind of radiation,
which ones of these transitions he can view.
Now, we looked at emission. But it is also possible for
there to be absorption between these allowed states of a
hydrogen atom. That is, we can have a hydrogen
atom here in a low energy state, the initial state E sub i.
And if there is a photon around
whose energy matches the energy difference between these two
states, well, then, that photon can be
absorbed by the hydrogen atom. Again, the energy of that
photon has to be exactly the difference in energy between
those two states. It cannot be a little larger.
If it is a little larger, that photon is not going to be
absorbed. That's important.
That's the quantum nature, again, of the hydrogen atom.
There are specific energies that are allowed and nothing in
between. And then, from knowing the
energy of the photon, you can get that frequency.
And then in the case of absorption, the frequencies of
the radiation that can be absorbed by a hydrogen atom are
given by this expression. This expression differs from
the frequencies for emission only in that I've reversed these
two terms. This is 1 over n sub i squared.
This is 1 over n sub f squared.
I have reversed them so that
you come out with a frequency that is a positive number.
Frequencies do have to be positive.
So, we've got two different expressions here for the
frequency depending on whether we're absorbing a photon or
we're emitting a photon. Questions?
I cannot see anybody. There.
Epsilon nought is a conversion factor for electrostatic units.
That is all you need at the moment.
In 8.02 maybe you will go through the unit conversation
there to get you to SI units.