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Let's get going,
here.
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Remember where we were?
We were trying to figure out
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the structure of the atom.
At the beginning of the course,
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we saw classical physics,
classical mechanics fail to
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describe how that electron in
the nucleus hung together.
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Then we started talking about
this wave-particle duality of
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light and matter.
We saw that radiation and
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matter both can exhibit both
wave-like properties and
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particle-like properties.
And it was really important,
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this observation of Davisson
and Germer, and George Thompson,
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this observation that electrons
exhibited inference phenomena.
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That is when you took electrons
and scattered them from a nickel
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single crystal.
The electrons scattered back as
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if they were behaving as waves.
There were diffraction
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phenomena or interference
phenomena, bright,
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dark, bright,
dark patterns of electrons.
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Actually, that Davisson and
Germer paper is on our website.
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You are welcome to take a look
at that.
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It was just that observation,
coupled with de Broglie's
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insight into Schrdinger's
relativistic equations of motion
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that led Schrdinger to say,
well, maybe what I need to do
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is I need to treat the wave-like
properties of that electron in a
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hydrogen atom.
Maybe that is the key.
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In particular,
maybe that is the key because
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the electron has a de Broglie
wavelength that is on the order
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of the size of its environment.
Maybe, in those cases,
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I need to treat the particle as
a wave and not as a particle
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with classical mechanics.
He wrote down this wave
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equation, an equation of motion
for waves, this H Psi equals E
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Psi, where we said last time we
are going to represent the
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electron, our particle,
by this Psi,
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the wave.
We are going to call it a wavef
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unction because we are going to
put a functional form to it very
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soon.
And there was some kind of
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operator, here,
called the Hamiltonian
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operator, that operated on this
wave function.
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And, when it did,
you got back the same wave
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function times a constant E.
And this constant,
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as we are going to see,
is going to be the binding
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energy of the electron to the
nucleus.
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But then, we took a little
detour and I said,
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well, let's see if we can
derive, in a sense,
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the Schrdinger equation.
And that is what we started to
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do.
And I am really doing this for
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fun, you are not responsible for
it, but I am doing it because I
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want you to see just how easy
this is.
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To illustrate this,
I am just going to take a
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one-dimensional problem.
I am going to let my electron
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be represented by this wave,
one-dimension,
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Psi of x.
2 a cosine 2 pi x over lambda.
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And then I said,
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suppose I want an equation of
motion, I want to know how that
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Psi changes with x.
Well, you already know that if
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I take the derivative of Psi
with x, that is going to tell me
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how Psi changes with x.
And we did that last time.
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And then I said,
well, I want to know the rate
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of change of Psi with x.
I am going to take the
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derivative again.
I have the second derivative of
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Psi of x.
And that is what we got last
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time.
And then, I noticed that in the
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second derivative,
and you noticed,
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too, somebody said this was
recursive, that we have our
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original wave function back in
this expression.
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I can rewrite that whole second
derivative here just as minus 2
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pi squared, quantity squared,
over lambda,
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psi of x.
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So far,
this is just any old wave
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equation.
Nothing special about this.
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This anybody could,
and had, written down before.
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What is special is that
Schrdinger realized,
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here, that if this is going to
be a wave equation for a
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particle, then maybe this lambda
here, maybe I ought to put in
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for lambda what de Broglie told
me.
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And that is h over p.
Maybe this lambda here is the
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wavelength of a matter wave,
so let me write this expression
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in terms of the momentum of the
particle, where the momentum has
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this mass m in it.
And so when he did this,
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this became minus p squared
over h bar squared,
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we said hbar is h over 2pi,
times psi of x.
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Hey, this is getting good
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because now we have a Psi of x
over here.
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But then what he said was,
well, I want to write this
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momentum in terms of the total
energy.
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Total energy is always kinetic
plus potential.
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The kinetic energy,
we said the other day,
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can be written in terms of the
momentum.
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The kinetic energy is p squared
over 2m-- plus the potential
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energy. And I
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am going to make this as a
function of x,
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the potential energy.
Now, I am just going to solve
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this for p squared.
p squared is equal to 2m times
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the total energy minus this
potential energy.
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Now, I am going to
plug this into
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here right in there.
And, when I do that,
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I am going to get the second
derivative of Psi of x with
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respect to x equals minus 2m
over h bar squared times E minus
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U of x times Psi of x.
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Just simple
substitution for p
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squared there.
Nothing else.
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Now, I am going to do some
rearranging.
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And the rearranging,
on the right-hand side,
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is I am going to have only E
times Psi of x.
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E times Psi of x looks like the
right-hand side of the
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Schrdinger as I wrote it down.
That is good.
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When I rearrange this,
I get minus hbar squared over
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2m times the second derivative
of Psi of x with respect to x
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plus U of x times Psi of x
equals E times Psi of x.
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And now, I am going to pull out
a Psi of x here,
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so that is minus hbar squared
over 2m second derivative with
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respect to x plus U of x,
the quantity times Psi of x
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equals E times Psi of x.
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And guess what?
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We've got it.
We got it because all of this
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is what we define as the
Hamiltonian.
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All of this is h hat.
H hat, operating on Psi of x,
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gives us E times Psi of x.
This Hamiltonian,
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as you will learn
later on, is a kinetic energy
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operator.
This is the potential energy
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operator operating on psi.
That is the Schrdinger
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equation.
It is hardly a derivation.
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It is taking derivatives.
It is a wave equation.
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The insight came right here,
this substitution of the de
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Broglie wavelength in an
ordinary wave equation.
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This is the insight,
getting that momentum in there
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with the mass,
making this,
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then, an equation for a matter
wave.
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That is it.
You just "derived the
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Schrdinger equation." Easy.
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Bottom line here is that the
Schrdinger equation is to
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quantum mechanics like Newton's
equations are to classical
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mechanics.
When the wavelength of a
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particle is on the order of the
size of its environment,
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00:10:17 --> 00:10:22
the equation of motion that you
have to use to describe that
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particle moving within some
potential field U of x
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or U, you have to us this
equation of motion and not
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00:10:30 --> 00:10:35
Newton's equations.
Newton's equations don't work
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to describe the motion of any
particle whose wavelength is on
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the order of the size of the
environment.
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It just does not work.
Now, just as an aside,
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classical mechanics really is
embedded in quantum mechanics.
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That is, if you took a problem
and solved it quantum
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mechanically,
and you solved the problem,
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00:11:04 --> 00:11:08
a problem for which the
wavelength of a particle was
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00:11:08 --> 00:11:13
much, much greater than the size
of the environment,
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00:11:13 --> 00:11:18
which is the classical limit,
quantum mechanics would give
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you the right answer.
In other words,
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say you took some problem where
the wavelength of the particle
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00:11:26 --> 00:11:32
is larger than the size of the
environment.
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00:11:32 --> 00:11:35
That is, a problem where you
would normally use classical
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mechanics.
But if you use quantum
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mechanics, you would get the
right answer,
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if you could solve the problem
because the equations are very
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difficult.
But, in principle,
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you would get the right answer.
However, if you took a quantum
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mechanical problem,
that is, a problem where the
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wavelength of the particle is on
the order of the size of the
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environment and you used
classical mechanics,
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00:11:59 --> 00:12:03
well, you won't get the right
answer.
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00:12:03 --> 00:12:08
Because classical mechanics is,
in a sense, a subset.
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It is contained within quantum
mechanics.
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It is a limit of quantum
mechanics.
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We have to learn a new kind of
mechanics, here,
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this mechanics for the motion
of waves.
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Now, for a hydrogen atom,
we have to think of the wave
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function in three dimensions
instead of just one dimension,
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here.
And so we are going to have to
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describe the particle in terms
of three position coordinates.
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Usually you use Cartesian
coordinates, x,
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y, and z.
But this problem is solvable
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exactly if we use spherical
coordinates.
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How many of you had spherical
coordinates before and know what
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we are talking about?
Not everybody.
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If I gave you an x,
y and z for this electron in
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this atom, where the nucleus was
pinned at the origin.
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If I gave you an x,
y and z coordinate,
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you would know where that
electron was.
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But, alternatively,
I could tell you the position
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of this electron using spherical
coordinates.
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That is, I could tell you what
the distance is of the electron
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from the nucleus.
I am going to call that r.
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That will be one of the
variables.
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I could then also tell you this
angle theta.
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Theta is the angle that r makes
from the z-axis.
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That is the second coordinate.
And then, finally,
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the third coordinate is phi.
Phi is the angle made by the
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following.
If I take the electron and drop
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it perpendicular to the
x,y-plane and then I draw a line
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here, well, the angle between
that line and the x-axis is the
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00:14:04 --> 00:14:09
angle phi.
Instead of giving you x,
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y, and z, I am just going to
give you r, theta,
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and phi in spherical
coordinates.
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And now, this wave function is
also a function of time,
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and I will talk about that a
little bit probably next time.
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So, that is the wave function
that is in some way going to
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represent our electron.
I haven't told you exactly yet
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how Psi represents the electron,
and I won't tell you that for
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another few days.
Question?
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No.
I actually thought this was the
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way the book set it up.
I may have it backwards.
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00:15:04 --> 00:15:09
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00:15:09 --> 00:15:12
I think this is the way our
book sets it up.
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00:15:12 --> 00:15:16
It won't make a difference in
any of the problems we solve
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00:15:16 --> 00:15:18
here.
But, in general,
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00:15:18 --> 00:15:23
if you are given a problem to
solve, you are going to have to
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00:15:23 --> 00:15:28
look and see how they define
their coordinate system.
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00:15:28 --> 00:15:33
Actually, somebody else asked
me that the other day.
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Now, what we have to do is
actually set up the Hamiltonian
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00:15:39 --> 00:15:45
for the hydrogen atom.
That is, we have to set up the
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00:15:45 --> 00:15:50
Hamiltonian specifically for the
hydrogen atom.
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00:15:50 --> 00:15:56
And we have to set it up in
terms of spherical coordinates,
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00:15:56 --> 00:16:01
r, theta, and phi.
And when we do that,
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00:16:01 --> 00:16:06
and we are not actually going
to do that, that is what the
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00:16:06 --> 00:16:10
Hamiltonian looks like.
Actually, this is in three
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00:16:10 --> 00:16:14
dimensions, and so if I were
doing it in the x,
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00:16:14 --> 00:16:20
y, z, what I would have here is
a second derivative with respect
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00:16:20 --> 00:16:24
to y, plus a second derivative
with respect to z.
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00:16:24 --> 00:16:29
That is what it would look like
in three dimensions in Cartesian
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00:16:29 --> 00:16:33
coordinates.
But when I transform from
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Cartesian coordinates to
spherical coordinates,
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00:16:37 --> 00:16:41
an exercise that takes five
pages, and everybody should have
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00:16:41 --> 00:16:45
that experience once in their
life, but maybe now is not the
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right time for that experience,
you do get this.
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00:16:48 --> 00:16:51
And essentially,
the Hamiltonian is a sum of
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00:16:51 --> 00:16:56
second derivatives with respect
to each one of the coordinates,
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00:16:56 --> 00:17:00
because essentially this term
is a second derivative with
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00:17:00 --> 00:17:03
respect to r.
This term is a second
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00:17:03 --> 00:17:06
derivative with respect to
theta.
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00:17:06 --> 00:17:10
This term is a second
derivative with respect to phi.
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00:17:10 --> 00:17:12
That is the specific
Hamiltonian.
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00:17:12 --> 00:17:17
These are all kinetic energy
operators, as you will learn
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00:17:17 --> 00:17:20
later on.
And then there is this term
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00:17:20 --> 00:17:22
here, U of r.
This U of r,
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00:17:22 --> 00:17:24
what is that?
What is U(r)?
241
00:17:24 --> 00:17:27
Potential energy.
It is the Coulomb potential
242
00:17:27 --> 00:17:32
energy of interaction.
It is isotropic,
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00:17:32 --> 00:17:35
meaning it is the same at all
angles.
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00:17:35 --> 00:17:40
The only thing it depends on is
the distance of the electron
245
00:17:40 --> 00:17:43
from the nucleus.
It only depends on r.
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00:17:43 --> 00:17:46
It doesn't depend on theta and
phi.
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00:17:46 --> 00:17:50
So, that is the Schrdinger
equation for the hydrogen atom.
248
00:17:50 --> 00:17:55
It is a differential equation,
second-order ordinary
249
00:17:55 --> 00:18:00
differential equation.
In 18.03, you are going to
250
00:18:00 --> 00:18:04
learn how to solve that.
In 5.61, or 8.04 I think it is,
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00:18:04 --> 00:18:08
in physics, in quantum
mechanics, you are going to
252
00:18:08 --> 00:18:10
solve that.
You can do that.
253
00:18:10 --> 00:18:14
It is not hard.
But what do I mean when I say
254
00:18:14 --> 00:18:16
solve?
What I mean is that we are
255
00:18:16 --> 00:18:21
going to calculate these Es,
these binding energies,
256
00:18:21 --> 00:18:23
this constant in front of the
psi.
257
00:18:23 --> 00:18:28
That is called an eigenvalue,
for those of you who might know
258
00:18:28 --> 00:18:35
something already about these
kinds of differential equations.
259
00:18:35 --> 00:18:40
This E is the binding energy of
the electron to the nucleus.
260
00:18:40 --> 00:18:45
We are going to look at those
results in just a moment.
261
00:18:45 --> 00:18:50
And the other quantity we are
going to solve for,
262
00:18:50 --> 00:18:53
here, is psi.
What we are going to want to
263
00:18:53 --> 00:19:00
find is the actual functional
form for the wave functions.
264
00:19:00 --> 00:19:04
That, we can get out of solving
this differential equation.
265
00:19:04 --> 00:19:07
The actual functional form for
psi.
266
00:19:07 --> 00:19:13
The functional form is going to
be more complicated than what I
267
00:19:13 --> 00:19:18
wrote here, but we can get that.
We will look at that in a few
268
00:19:18 --> 00:19:22
days from now.
And do you know what those wave
269
00:19:22 --> 00:19:25
functions are?
They are what you studied in
270
00:19:25 --> 00:19:31
high school as orbitals.
You talked about an s-orbital,
271
00:19:31 --> 00:19:34
p-orbital, d-orbital.
Orbitals are wave functions.
272
00:19:34 --> 00:19:38
That where they come from,
orbitals, from solving the
273
00:19:38 --> 00:19:41
Schrdinger equation.
Now, specifically,
274
00:19:41 --> 00:19:45
orbitals are the spatial part
of a wave function.
275
00:19:45 --> 00:19:48
There is also a spin part to
the wave function,
276
00:19:48 --> 00:19:52
but for all intents and
purposes, we are going to use
277
00:19:52 --> 00:19:56
the term orbital and wave
function interchangeably because
278
00:19:56 --> 00:20:00
they are the same thing.
Question?
279
00:20:00 --> 00:20:03
We are going to get to that in
just a moment,
280
00:20:03 --> 00:20:09
and if you think I don't answer
it then we will go back to it.
281
00:20:09 --> 00:20:13
We are going to solve this.
And this equation is going to
282
00:20:13 --> 00:20:18
predict binding energies,
and it is going to predict
283
00:20:18 --> 00:20:22
these wave functions in
agreement with our observations,
284
00:20:22 --> 00:20:29
and it is going to predict that
the hydrogen atom is stable.
285
00:20:29 --> 00:20:33
Remember when we tried to
predict the hydrogen atom using
286
00:20:33 --> 00:20:36
classical ideas?
We found it lived a whole
287
00:20:36 --> 00:20:40
whopping 10^-10 seconds.
Not so when we treat the
288
00:20:40 --> 00:20:45
hydrogen atom with the quantum
mechanical equations of motion.
289
00:20:45 --> 00:20:50
Let's write down the results
for solving the Schrdinger
290
00:20:50 --> 00:20:53
equation.
And the part I am going to
291
00:20:53 --> 00:20:58
concentrate on today is these
binding energies.
292
00:20:58 --> 00:21:00
And, on Friday,
we will talk about the wave
293
00:21:00 --> 00:21:03
function, solving the
Schrdinger equation for those
294
00:21:03 --> 00:21:05
wave functions.
295
00:21:05 --> 00:21:16
296
00:21:16 --> 00:21:19
What does the binding energy
look like?
297
00:21:19 --> 00:21:23
Well, the binding energies,
here, coming out of the
298
00:21:23 --> 00:21:28
Schrdinger equation look like
this, minus 1 over n squared
299
00:21:28 --> 00:21:32
times m e to the 4 over 8
epsilon nought squared time h
300
00:21:32 --> 00:21:35
squared.
301
00:21:35 --> 00:21:40
m is the mass of the electron.
302
00:21:40 --> 00:21:43
e is the charge on the
electron.
303
00:21:43 --> 00:21:48
Epsilon nought is this
permittivity of vacuum we talked
304
00:21:48 --> 00:21:52
about before.
This is a conversion between
305
00:21:52 --> 00:21:56
ESU units and SI units.
h is Planck's constant.
306
00:21:56 --> 00:21:59
Planck's constant is
ubiquitous.
307
00:21:59 --> 00:22:05
It is everywhere.
What we typically do is we lump
308
00:22:05 --> 00:22:11
these constants together.
And we call those constants a
309
00:22:11 --> 00:22:15
new constant,
the Rydberg constant,
310
00:22:15 --> 00:22:21
R sub H.
And so our expression is equal
311
00:22:21 --> 00:22:28
to minus R sub H over n squared.
312
00:22:28 --> 00:22:35
The value of R sub H is equal
to 2.17987x10^-18 joules.
313
00:22:35 --> 00:22:43
That is a number that you are
going to use a lot in the next
314
00:22:43 --> 00:22:48
few days.
But what you also see,
315
00:22:48 --> 00:22:51
here, is an n.
What is n?
316
00:22:51 --> 00:23:00
Well, n is what we call the
principle quantum number.
317
00:23:00 --> 00:23:07
And its allowed values are
integers, where the integers
318
00:23:07 --> 00:23:13
start with 1,
2, 3, all the way up to n is
319
00:23:13 --> 00:23:19
equal to infinity.
Well, let's try to understand
320
00:23:19 --> 00:23:26
this a little bit more by
looking at an energy level
321
00:23:26 --> 00:23:32
diagram again.
I am just plotting here energy.
322
00:23:32 --> 00:23:38
Here is the zero of energy.
Here is the expression for the
323
00:23:38 --> 00:23:43
energy levels that come out of
the Schrdinger equation.
324
00:23:43 --> 00:23:48
When n is equal to one,
that is the lowest allowed
325
00:23:48 --> 00:23:54
value for n, binding energy is
essentially equal to minus the
326
00:23:54 --> 00:23:58
Rydberg constant.
**E = -(R)H**
327
00:23:58 --> 00:24:02
But our equation says the
binding energy of the electron
328
00:24:02 --> 00:24:07
can also be this value because
when n is equal to 2,
329
00:24:07 --> 00:24:11
well, the binding energy now is
only a quarter of the Rydberg
330
00:24:11 --> 00:24:15
constant. It is higher
331
00:24:15 --> 00:24:17
in energy.
When n is equal to 3,
332
00:24:17 --> 00:24:22
well, the binding energy of
that electron to the nucleus is
333
00:24:22 --> 00:24:26
a ninth to the Rydberg constant.
When it is equal to 4,
334
00:24:26 --> 00:24:30
it is a sixteenth.
When it is equal to 5,
335
00:24:30 --> 00:24:34
it is a twenty-fifth.
When it is equal to six,
336
00:24:34 --> 00:24:37
it is a thirty-sixth.
So on, so on,
337
00:24:37 --> 00:24:40
and so on until it gets n equal
to infinity.
338
00:24:40 --> 00:24:45
When n is equal to infinity,
then the binding energy is
339
00:24:45 --> 00:24:47
zero.
When n is equal to infinity,
340
00:24:47 --> 00:24:51
the electron and the nucleus
are no longer bound.
341
00:24:51 --> 00:24:54
They are separated from each
other.
342
00:24:54 --> 00:25:00
They don't hang together when n
is equal to infinity.
343
00:25:00 --> 00:25:05
Now, this is rather peculiar.
This is saying that the binding
344
00:25:05 --> 00:25:10
energy of the electron to the
nucleus can have essentially an
345
00:25:10 --> 00:25:16
infinite number of values,
except it is a discrete number
346
00:25:16 --> 00:25:21
of infinite numbers of values.
In the sense that the binding
347
00:25:21 --> 00:25:27
energy can be this or this or
this, but it cannot be something
348
00:25:27 --> 00:25:31
in between, here or here or
here.
349
00:25:31 --> 00:25:34
This is the quantum nature of
the hydrogen atom.
350
00:25:34 --> 00:25:37
There are allowed energy
levels.
351
00:25:37 --> 00:25:40
Where did this quantization
come from?
352
00:25:40 --> 00:25:44
It came from solving the
Schrdinger equation.
353
00:25:44 --> 00:25:49
When you solve a differential
equation, as you will learn,
354
00:25:49 --> 00:25:54
and that differential equation
applies to some physical
355
00:25:54 --> 00:25:59
problem, in order to make that
differential equation specific
356
00:25:59 --> 00:26:03
to your physical problem,
you often have to apply
357
00:26:03 --> 00:26:09
something called boundary
conditions to the problem.
358
00:26:09 --> 00:26:13
When you do that,
that is when this quantization
359
00:26:13 --> 00:26:16
comes out.
It drops out of solving the
360
00:26:16 --> 00:26:21
differential equation.
What are boundary conditions?
361
00:26:21 --> 00:26:27
Well, remember I told you in
the spherical coordinate system,
362
00:26:27 --> 00:26:30
r, theta, phi?
Well, phi sweeps from zero to
363
00:26:30 --> 00:26:35
360 degrees.
Well, if you just went 90
364
00:26:35 --> 00:26:38
degrees further,
so if you went to 450 degrees,
365
00:26:38 --> 00:26:43
you really have the same
situation as you had when phi
366
00:26:43 --> 00:26:47
was equal to 90 degrees.
What you have to do in a
367
00:26:47 --> 00:26:51
differential equation,
which usually has a series as a
368
00:26:51 --> 00:26:54
solution, is that you have to
cut it off.
369
00:26:54 --> 00:26:57
You have to apply boundary
conditions.
370
00:26:57 --> 00:27:02
You have to cut it off at
degrees.
371
00:27:02 --> 00:27:06
Because otherwise you just have
the same problem that you had
372
00:27:06 --> 00:27:09
before.
This is a cyclical boundary
373
00:27:09 --> 00:27:12
condition.
And it is that cutting it off
374
00:27:12 --> 00:27:16
to make the equation be really
pertinent or apply to your
375
00:27:16 --> 00:27:20
physical problem,
that is what leads to these
376
00:27:20 --> 00:27:23
boundary conditions,
mathematically.
377
00:27:23 --> 00:27:27
That is where it comes from.
That is where all the
378
00:27:27 --> 00:27:32
quantization comes from.
Now, let's talk a little bit
379
00:27:32 --> 00:27:38
about the significance here of
these binding energies because
380
00:27:38 --> 00:27:41
somebody asked me about it
already.
381
00:27:41 --> 00:27:47
When the electron is bound to
the nucleus with this much
382
00:27:47 --> 00:27:52
energy, we say that the electron
is in the n equals 1 state,
383
00:27:52 --> 00:27:57
or equivalently,
we say that the hydrogen atom
384
00:27:57 --> 00:28:02
is in the n equals 1 state.
We use both kinds of
385
00:28:02 --> 00:28:07
expressions equivalently.
When the hydrogen atom or the
386
00:28:07 --> 00:28:13
electron is in the n equals 1
state, we call that the ground
387
00:28:13 --> 00:28:16
state.
The ground state is the lowest
388
00:28:16 --> 00:28:20
energy state.
The electron is most strongly
389
00:28:20 --> 00:28:23
bound there.
The binding energy is most
390
00:28:23 --> 00:28:26
negative.
And the physical significance
391
00:28:26 --> 00:28:32
of that binding energy is that
it is minus the ionization
392
00:28:32 --> 00:28:35
energy.
Or, alternatively,
393
00:28:35 --> 00:28:40
the ionization energy is minus
the binding energy.
394
00:28:40 --> 00:28:46
It is going to require this
much energy, from here to here,
395
00:28:46 --> 00:28:50
to rip the electron off of the
nucleus.
396
00:28:50 --> 00:28:56
The binding energy here is
minus the ionization energy.
397
00:28:56 --> 00:29:01
That is the physical
significance of it.
398
00:29:01 --> 00:29:04
Now, did you ask me about the
work function?
399
00:29:04 --> 00:29:07
Okay.
The work function is the
400
00:29:07 --> 00:29:11
ionization energy when we talk
about a solid.
401
00:29:11 --> 00:29:15
That is just a terminology,
that is historical.
402
00:29:15 --> 00:29:20
When we talk about ripping and
electron off of a solid,
403
00:29:20 --> 00:29:25
we call it the work function.
When we talk about ripping it
404
00:29:25 --> 00:29:30
off of an atom or a molecule,
we call it the ionization
405
00:29:30 --> 00:29:34
energy.
And the other important thing
406
00:29:34 --> 00:29:39
to know here is that when we
talk about ionization energy,
407
00:29:39 --> 00:29:44
we are usually talking about
the energy required to pull the
408
00:29:44 --> 00:29:50
electron off when the molecule
or the atom is in the lowest
409
00:29:50 --> 00:29:52
energy state,
the ground state.
410
00:29:52 --> 00:29:57
That is also important.
But our equations tell us we
411
00:29:57 --> 00:30:03
also can have a hydrogen atom in
the n equals 2 state.
412
00:30:03 --> 00:30:06
If the hydrogen atom is in the
n equals 2 state,
413
00:30:06 --> 00:30:11
it is in an excited state.
Actually, it is the first
414
00:30:11 --> 00:30:14
excited state.
The electron is bound less
415
00:30:14 --> 00:30:17
strongly.
It is bound less strongly,
416
00:30:17 --> 00:30:21
and consequently,
the ionization energy from an
417
00:30:21 --> 00:30:26
excited state hydrogen atom is
less because the electron is not
418
00:30:26 --> 00:30:30
bound so strongly.
And, of course,
419
00:30:30 --> 00:30:34
we could have a hydrogen at n
equals 3, n equals 4 and n
420
00:30:34 --> 00:30:38
equals 5, any of these allowed
energy levels.
421
00:30:38 --> 00:30:42
Not all at the same time,
but, at any given time,
422
00:30:42 --> 00:30:47
if you had a lot of hydrogen
atoms, you could have hydrogen
423
00:30:47 --> 00:30:50
atoms in all of these different
states.
424
00:30:50 --> 00:30:54
Now, the other point that I
want to make is that this
425
00:30:54 --> 00:30:58
Schrdinger result,
here, for the energy levels
426
00:30:58 --> 00:31:04
predicts the energy levels of
all one electron atoms.
427
00:31:04 --> 00:31:07
What is a one electron atom?
Well, helium plus,
428
00:31:07 --> 00:31:11
that is a one electron atom or
a one electron ion.
429
00:31:11 --> 00:31:15
Helium usually has two
electrons, but if we pull one
430
00:31:15 --> 00:31:19
off it only has one left,
so it is a one electron atom.
431
00:31:19 --> 00:31:23
Lithium double plus
is a one electron atom because
432
00:31:23 --> 00:31:26
we pulled two of its three
electrons off.
433
00:31:26 --> 00:31:30
One electron is left,
and that is a one electron atom
434
00:31:30 --> 00:31:34
or an ion.
Uranium plus 91 is
435
00:31:34 --> 00:31:38
a one electron atom.
We pulled 91 electrons off,
436
00:31:38 --> 00:31:42
we have one left,
and that is a one electron
437
00:31:42 --> 00:31:45
atom.
The Schrdinger equation will
438
00:31:45 --> 00:31:51
predict what those energy levels
are, as long as you remember the
439
00:31:51 --> 00:31:53
Z squared up here.
For hydrogen,
440
00:31:53 --> 00:31:56
Z is 1, and so it doesn't
appear.
441
00:31:56 --> 00:32:03
But Z is not one for all of
these other one electron atoms.
442
00:32:03 --> 00:32:07
Where does the Z come from?
It comes from the Coulomb
443
00:32:07 --> 00:32:10
interaction.
That potential energy of
444
00:32:10 --> 00:32:15
interaction is the charge on the
electron times the charge on the
445
00:32:15 --> 00:32:19
nucleus, which is Z times e.
That is where the Z squared
446
00:32:19 --> 00:32:22
comes from.
We have to remember that.
447
00:32:22 --> 00:32:27
Now, how do we know that the
Schrdinger's predictions for
448
00:32:27 --> 00:32:32
these energy levels are correct,
are accurate?
449
00:32:32 --> 00:32:34
Well, we have to do an
experiment.
450
00:32:34 --> 00:32:39
The experiment we are going to
do is we are going to take a
451
00:32:39 --> 00:32:43
bulb here, pump it out,
and then we are going to fill
452
00:32:43 --> 00:32:47
it up with molecular hydrogen.
And then in this bulb,
453
00:32:47 --> 00:32:51
there is a negative and a
positive electrode.
454
00:32:51 --> 00:32:55
What we are going to do is
crank up the potential energy
455
00:32:55 --> 00:33:00
difference between those two so
high until finally that gas is
456
00:33:00 --> 00:33:04
going to ignite,
just like that.
457
00:33:04 --> 00:33:08
And we are going to look at the
light coming out.
458
00:33:08 --> 00:33:13
And what we are going to do is
we are going to disperse that
459
00:33:13 --> 00:33:16
light.
We are going to send it through
460
00:33:16 --> 00:33:19
a diffraction grating,
essentially.
461
00:33:19 --> 00:33:22
And that is like an array of
slits.
462
00:33:22 --> 00:33:26
What you are going to see are
points of constructive and
463
00:33:26 --> 00:33:31
destructive interference.
But in the constructive
464
00:33:31 --> 00:33:34
interference,
you are going to see the color
465
00:33:34 --> 00:33:37
separated.
There is going to be purple,
466
00:33:37 --> 00:33:38
blue, green,
etc.
467
00:33:38 --> 00:33:41
The reason is that the
different radiation,
468
00:33:41 --> 00:33:45
the different colors have
different wavelengths.
469
00:33:45 --> 00:33:48
They have different
wavelengths, then they have
470
00:33:48 --> 00:33:52
slightly different points in
space at which constructive
471
00:33:52 --> 00:33:56
interference occurs,
and so the light is dispersed
472
00:33:56 --> 00:34:00
in space.
We are going to analyze what
473
00:34:00 --> 00:34:04
the wavelengths of the light
that are being emitted from
474
00:34:04 --> 00:34:08
these hydrogen atoms are.
See, the discharge pulls apart
475
00:34:08 --> 00:34:12
the H two and makes
hydrogen atoms.
476
00:34:12 --> 00:34:16
We have to do this by taking a
diffraction grating.
477
00:34:16 --> 00:34:20
The TAs are going to give you a
pair of diffraction grating
478
00:34:20 --> 00:34:23
glasses.
You are supposed to come and
479
00:34:23 --> 00:34:28
look at the discharge here.
You can see it.
480
00:34:28 --> 00:34:32
I will turn it a little bit for
those of you on the sides here.
481
00:34:32 --> 00:34:35
And we will see what we see.
482
00:34:35 --> 00:35:00
483
00:35:00 --> 00:35:03
If you cannot see the lamp,
you are welcome to get up and
484
00:35:03 --> 00:35:06
move around so that you can see
it.
485
00:35:06 --> 00:35:20
486
00:35:20 --> 00:35:21
Can you do this light,
too?
487
00:35:21 --> 00:35:23
This one over here,
sir.
488
00:35:23 --> 00:35:26
Can you move this one away?
489
00:35:26 --> 00:35:35
490
00:35:35 --> 00:35:38
If you look up at the lights in
the room, you can see the whole
491
00:35:38 --> 00:35:40
spectrum, because that is white
light.
492
00:35:40 --> 00:36:08
493
00:36:08 --> 00:36:13
I am going to turn the lamp
over here so that you can see
494
00:36:13 --> 00:36:18
this a little better.
The spectrum that you should
495
00:36:18 --> 00:36:21
see is what is shown on the
board.
496
00:36:21 --> 00:36:27
If you look to the right of the
lamp, here, you should see a
497
00:36:27 --> 00:36:32
purple line.
The purple line is actually
498
00:36:32 --> 00:36:36
very light, so only if you are
up close are you going to see
499
00:36:36 --> 00:36:39
the purple line.
You can see the blue line very
500
00:36:39 --> 00:36:41
well.
That is very intense.
501
00:36:41 --> 00:36:44
The green line is also very
diffuse.
502
00:36:44 --> 00:36:48
Again, only if you are close
are you going to be able to see
503
00:36:48 --> 00:36:52
the green line.
And then, the red line is very
504
00:36:52 --> 00:36:55
bright.
And now I am going to move this
505
00:36:55 --> 00:37:00
over here so you have the
opportunity to see it.
506
00:37:00 --> 00:37:09
Again, you are welcome to get
out of your seats and move
507
00:37:09 --> 00:37:15
around so that you can,
in fact, see it.
508
00:37:15 --> 00:37:24
What you should be seeing,
hey, interference phenomena
509
00:37:24 --> 00:37:29
works.
Useful pointer.
510
00:37:29 --> 00:37:40
511
00:37:40 --> 00:37:43
What you should see,
depending on which one of the
512
00:37:43 --> 00:37:48
constructive interference
patterns you are looking at,
513
00:37:48 --> 00:37:53
there should be a purple line,
there should be a blue line,
514
00:37:53 --> 00:37:55
very intense,
purple is weak,
515
00:37:55 --> 00:38:00
green is rather week,
and the red is very intense.
516
00:38:00 --> 00:38:02
What is happening,
here?
517
00:38:02 --> 00:38:06
Well, what is happening is that
in this discharge,
518
00:38:06 --> 00:38:12
there is enough energy to put
some of the hydrogen atoms in a
519
00:38:12 --> 00:38:17
high energy state.
And we will call that state E
520
00:38:17 --> 00:38:21
sub i,
the energy of the initial
521
00:38:21 --> 00:38:24
state.
Actually, that is an unstable
522
00:38:24 --> 00:38:28
situation.
That hydrogen atom wants to
523
00:38:28 --> 00:38:32
relax.
It wants to be in the lower
524
00:38:32 --> 00:38:36
energy state.
And what happens is that it
525
00:38:36 --> 00:38:38
does relax.
It relaxes.
526
00:38:38 --> 00:38:42
The electron falls into the
lower energy state,
527
00:38:42 --> 00:38:47
but, because it is lower
energy, it has to give up a
528
00:38:47 --> 00:38:50
photon.
And so the photon that is
529
00:38:50 --> 00:38:56
emitted, the energy of that
photon has to be exactly the
530
00:38:56 --> 00:39:02
difference in energy between the
energy of the initial state and
531
00:39:02 --> 00:39:07
the final state.
That is the quantum nature of
532
00:39:07 --> 00:39:12
the hydrogen atom.
The photon that comes out has
533
00:39:12 --> 00:39:15
to have that energy difference
exactly.
534
00:39:15 --> 00:39:18
And, therefore,
the frequency,
535
00:39:18 --> 00:39:24
here, of the radiation coming
out is going to correspond to
536
00:39:24 --> 00:39:29
that exact energy difference.
And, for the different
537
00:39:29 --> 00:39:33
energies, for the different
transitions, you are going to
538
00:39:33 --> 00:39:38
have very specific values of the
frequency of the radiation or
539
00:39:38 --> 00:39:41
the wavelength of the radiation.
For example,
540
00:39:41 --> 00:39:46
some of those hydrogen atoms in
this discharge have been excited
541
00:39:46 --> 00:39:49
to this excited state,
which we will call B.
542
00:39:49 --> 00:39:54
The energy difference between B
and this ground state here is
543
00:39:54 --> 00:39:58
small.
Therefore, we are going to have
544
00:39:58 --> 00:40:01
some radiation that is low
frequency because delta E,
545
00:40:01 --> 00:40:05
the difference in the energy
between the two states is small.
546
00:40:05 --> 00:40:09
And then there are going to be
some hydrogen atoms that are
547
00:40:09 --> 00:40:12
going to be excited to this
state, the A state.
548
00:40:12 --> 00:40:15
And, relatively speaking,
that energy difference is
549
00:40:15 --> 00:40:17
large.
There are going to be some
550
00:40:17 --> 00:40:21
hydrogen atoms that are going to
be relaxing to this ground
551
00:40:21 --> 00:40:23
state.
And when they do,
552
00:40:23 --> 00:40:27
since that energy difference is
large, the frequency of the
553
00:40:27 --> 00:40:31
radiation coming off is going to
be high.
554
00:40:31 --> 00:40:35
Or, correspondingly,
the wavelength of the radiation
555
00:40:35 --> 00:40:39
coming off is going to be low
because the wavelength is
556
00:40:39 --> 00:40:42
inversely proportional to the
frequency.
557
00:40:42 --> 00:40:45
And, likewise,
for this transition we are
558
00:40:45 --> 00:40:50
going to have some long
wavelength radiation emitted.
559
00:40:50 --> 00:40:54
Well, let's see if we can
understand specifically the
560
00:40:54 --> 00:40:57
spectrum, here,
in the visible range for the
561
00:40:57 --> 00:41:03
hydrogen atom.
What I have done is to draw an
562
00:41:03 --> 00:41:09
energy level diagram again.
Here is the energy of n equals
563
00:41:09 --> 00:41:12
1, n equals 2,
n equals 3, etc.
564
00:41:12 --> 00:41:16
Here is the n equals infinity,
up here.
565
00:41:16 --> 00:41:22
It turns out that this purple
line is a transition from a
566
00:41:22 --> 00:41:30
hydrogen atom in the n equals 6
state to the n equals 2 state.
567
00:41:30 --> 00:41:36
That blue line is a transition
from n equals 5 to n equals 2,
568
00:41:36 --> 00:41:43
the green line is a transition
from n equals 4 to n equals 2
569
00:41:43 --> 00:41:48
and the red line from n equals 3
to n equals 2.
570
00:41:48 --> 00:41:54
Notice that since this energy
difference here is small,
571
00:41:54 --> 00:41:59
this line has a long
wavelength.
572
00:41:59 --> 00:42:04
Since this energy difference is
larger, this line has a shorter
573
00:42:04 --> 00:42:08
wavelength.
All of these transitions that
574
00:42:08 --> 00:42:13
you are seeing in the visible
range have the final state of n
575
00:42:13 --> 00:42:16
equals 2.
Now, how do we know that the
576
00:42:16 --> 00:42:22
Schrdinger equation is making
predictions that are consistent
577
00:42:22 --> 00:42:27
with the frequencies or the
wavelengths of the radiation
578
00:42:27 --> 00:42:32
that we observe here?
Well, we have got to do a plug
579
00:42:32 --> 00:42:34
here.
This is the frequency that we
580
00:42:34 --> 00:42:37
would expect.
It is the energy difference
581
00:42:37 --> 00:42:40
between two states over H.
Here is the initial energy.
582
00:42:40 --> 00:42:44
Here is the final energy.
We said that the Schrdinger
583
00:42:44 --> 00:42:48
equation tells us that the
energies of the state are minus
584
00:42:48 --> 00:42:50
R sub H over n.
585
00:42:50 --> 00:42:54
For the initial energy level,
it is minus R sub H over n sub
586
00:42:54 --> 00:42:57
i squared. We plug
587
00:42:57 --> 00:43:01
that in.
For the final energy level,
588
00:43:01 --> 00:43:05
well, it is minus R sub H over
n sub f,
589
00:43:05 --> 00:43:08
the final quantum number
squared.
590
00:43:08 --> 00:43:11
We plug that in.
We rearrange some things.
591
00:43:11 --> 00:43:15
And then this is the prediction
for the frequency.
592
00:43:15 --> 00:43:20
I said that this level right
here is the n equals 2 level,
593
00:43:20 --> 00:43:24
so we will plug in the two for
the final quantum state.
594
00:43:24 --> 00:43:28
And so this is then the
prediction for the frequencies
595
00:43:28 --> 00:43:33
of the radiation to the n equals
2 level.
596
00:43:33 --> 00:43:37
You just plug in n equals 6,
n equals 5, n equals 4,
597
00:43:37 --> 00:43:39
n equals 3.
And you know what?
598
00:43:39 --> 00:43:43
The frequencies that are
predicted match what we observe
599
00:43:43 --> 00:43:46
to one part in 10^8.
There really is precise
600
00:43:46 --> 00:43:51
agreement between the results of
the Schrdinger equation and
601
00:43:51 --> 00:43:55
what we actually observe in
nature, therefore,
602
00:43:55 --> 00:44:01
we think it is correct.
And there are no experiments
603
00:44:01 --> 00:44:06
that cast any doubt,
so far, on the Schrdinger
604
00:44:06 --> 00:44:09
equation.
Now, the set of transitions
605
00:44:09 --> 00:44:14
that we just looked at are these
transitions.
606
00:44:14 --> 00:44:20
Plotted here is another
energy-level diagram for the
607
00:44:20 --> 00:44:24
hydrogen atom.
Here is the n equals 1 state,
608
00:44:24 --> 00:44:30
n equals 2, n equals 3,
n equals 4.
609
00:44:30 --> 00:44:34
And we looked at all the
transitions that end up in the n
610
00:44:34 --> 00:44:36
equals 2 state.
This set of transitions.
611
00:44:36 --> 00:44:41
It is called the Balmer series.
Now, the n equals 2 state is
612
00:44:41 --> 00:44:44
not the ground state.
There is a transition from n
613
00:44:44 --> 00:44:47
equals 6 to n equals 1,
the ground state.
614
00:44:47 --> 00:44:50
It is this one right here.
But, you see,
615
00:44:50 --> 00:44:53
that is a very high energy
transition.
616
00:44:53 --> 00:44:57
Actually, this transition,
from n equals 6 to n equals 1,
617
00:44:57 --> 00:45:03
is in the ultraviolet range of
the electromagnetic spectrum.
618
00:45:03 --> 00:45:06
And so you cannot see it with
the experiment that we did,
619
00:45:06 --> 00:45:09
but it is there.
And then, of course,
620
00:45:09 --> 00:45:12
the hydrogen atoms that relaxed
to n equals 2,
621
00:45:12 --> 00:45:16
well, they actually eventually
relax to n equals 1.
622
00:45:16 --> 00:45:19
And there is a transition
there, it is over here,
623
00:45:19 --> 00:45:23
n equals 2 to n equals 1.
But, again, that is a high
624
00:45:23 --> 00:45:26
energy transition.
It is also in the ultraviolet
625
00:45:26 --> 00:45:30
range of the electromagnetic
spectrum.
626
00:45:30 --> 00:45:34
And so all of these transitions
that end up in n equals 1 are in
627
00:45:34 --> 00:45:36
the UV range.
They are called the Lyman
628
00:45:36 --> 00:45:38
series.
All the transitions that end up
629
00:45:38 --> 00:45:40
in n equals 2 are in the
visible.
630
00:45:40 --> 00:45:44
They are called the Balmer.
n equals 3, the Paschen,
631
00:45:44 --> 00:45:47
is in the infrared.
Brackett is in the infrared.
632
00:45:47 --> 00:45:50
Pfund is in the infrared.
And so we cannot see these
633
00:45:50 --> 00:45:52
easily.
Now, these are named for the
634
00:45:52 --> 00:45:55
different discoverers.
The reason there are so many
635
00:45:55 --> 00:46:01
discovers is because these are
all different frequency ranges.
636
00:46:01 --> 00:46:04
Different frequencies of
radiation require different
637
00:46:04 --> 00:46:08
kinds of detectors.
And so, depending on exactly
638
00:46:08 --> 00:46:11
what kind of detector the
experimentalist had,
639
00:46:11 --> 00:46:16
that will dictate then which
one of these sets of transitions
640
00:46:16 --> 00:46:20
he was able to discover.
But not only does this work for
641
00:46:20 --> 00:46:24
emission, but this also works
for absorption.
642
00:46:24 --> 00:46:28
That is, we can have a hydrogen
atom in the ground state,
643
00:46:28 --> 00:46:33
a low energy state.
And if there is a photon around
644
00:46:33 --> 00:46:37
that is exactly the energy
difference between these two
645
00:46:37 --> 00:46:40
states, well,
that photon can be absorbed.
646
00:46:40 --> 00:46:44
If that photon is a little bit
higher in energy,
647
00:46:44 --> 00:46:47
it won't be absorbed.
If it is a little lower,
648
00:46:47 --> 00:46:50
it won't be absorbed.
It has to be exactly the
649
00:46:50 --> 00:46:54
difference in the energies
between these two states.
650
00:46:54 --> 00:47:00
Again, that is the quantum
nature of the hydrogen atom.
651
00:47:00 --> 00:47:04
And now, if you are calculating
the frequency for absorption,
652
00:47:04 --> 00:47:09
what I have done here is I have
reversed n sub i and n
653
00:47:09 --> 00:47:12
sub f.
This is 1 over n sub i squared
654
00:47:12 --> 00:47:17
instead of 1 over
n sub f squared,
655
00:47:17 --> 00:47:21
which was the case in emission.
And I have reversed this so
656
00:47:21 --> 00:47:25
that you get a positive value
for the frequency of the
657
00:47:25 --> 00:47:28
radiation absorbed.
We will have two different
658
00:47:28 --> 00:47:33
equations for absorption and for
emission.
659
00:47:33 --> 00:47:36
So, those are the Schrdinger
equation results.
660
00:47:36 --> 00:47:40
I forgot to announce that there
is a forum this evening from
661
00:47:40 --> 00:47:44
5:00 to 6:00.
You should have signed up in
662
00:47:44 --> 00:47:45
recitation.
If you didn't,
663
00:47:45 --> 00:47:49
and still want to come,
send us an email and come.
664
00:47:49 --> 00:47:54
And we need the diffraction
glasses back as you are exiting.
665
00:47:54 --> 47:57
See you Friday.