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PROFESSOR: So we're
now starting to get
00:00:25.700 --> 00:00:28.310
into a really familiar
territory, which
00:00:28.310 --> 00:00:29.451
is the hydrogen atom.
00:00:32.159 --> 00:00:40.770
And it's a short step from a
hydrogen atom to molecules.
00:00:40.770 --> 00:00:41.790
And we're chemists.
00:00:41.790 --> 00:00:45.280
We make molecules.
00:00:45.280 --> 00:00:49.530
But one question that would be
a legitimate question to ask
00:00:49.530 --> 00:00:52.810
is, what does a hydrogen atom
have to do with molecules,
00:00:52.810 --> 00:00:55.070
because it's an atom and
it's the simplest atom.
00:00:55.070 --> 00:01:00.390
And what I hope to
show is that what
00:01:00.390 --> 00:01:03.090
we learn from looking
at the hydrogen atom
00:01:03.090 --> 00:01:06.870
has all sorts of good
non textbook stuff
00:01:06.870 --> 00:01:09.660
that prepares you for
understanding stuff
00:01:09.660 --> 00:01:11.580
in molecules.
00:01:11.580 --> 00:01:15.960
And I am going to stress
that because you're not
00:01:15.960 --> 00:01:19.350
going to see it in McQuarrie
and you've probably never seen
00:01:19.350 --> 00:01:24.120
it elsewhere except in freshman
chemistry when they expect
00:01:24.120 --> 00:01:26.790
that you understand the
periodic table based
00:01:26.790 --> 00:01:28.830
on some simple concepts.
00:01:28.830 --> 00:01:31.480
And they come from
the hydrogen atom.
00:01:31.480 --> 00:01:39.600
And so I'm going to attempt
to make those connections.
00:01:39.600 --> 00:01:42.990
My last lecture was I told
you something you're not
00:01:42.990 --> 00:01:44.110
going to be tested on.
00:01:44.110 --> 00:01:47.520
But it's about spectroscopy.
00:01:47.520 --> 00:01:49.770
That's how we learned
almost everything
00:01:49.770 --> 00:01:53.730
we know about small
molecules and a lot of stuff
00:01:53.730 --> 00:01:55.860
about big molecules too.
00:01:55.860 --> 00:01:59.540
The crucial
approximations in that
00:01:59.540 --> 00:02:05.820
were the dipole approximation
where the radiation field
00:02:05.820 --> 00:02:08.250
is such that the
molecule field's
00:02:08.250 --> 00:02:11.580
a uniform but oscillating
electromagnetic field.
00:02:14.620 --> 00:02:18.110
In developing the
theory that I presented,
00:02:18.110 --> 00:02:24.140
we assume that only one
of the initial eigenstates
00:02:24.140 --> 00:02:26.780
is populated at t equal zero.
00:02:26.780 --> 00:02:32.210
And only one final
eigenstate gets tickled so
00:02:32.210 --> 00:02:34.100
that it gets populated.
00:02:34.100 --> 00:02:38.060
And so even though there
is an infinity of states,
00:02:38.060 --> 00:02:43.280
the theory specifies we start
with one definite one and only
00:02:43.280 --> 00:02:49.960
one final state gets selected
because of resonance.
00:02:49.960 --> 00:02:52.030
And this is what
we do all the time.
00:02:52.030 --> 00:02:55.120
We have an infinite
dimension problem,
00:02:55.120 --> 00:02:58.780
and we discover that
we only really need
00:02:58.780 --> 00:03:02.490
to worry about a very
small number of states.
00:03:02.490 --> 00:03:06.420
And then we deal with that.
00:03:06.420 --> 00:03:07.920
There are a couple
other assumptions
00:03:07.920 --> 00:03:11.280
that were essential
for this first step
00:03:11.280 --> 00:03:14.360
into a time-dependent
Hamiltonian.
00:03:14.360 --> 00:03:18.080
And that is the field is weak
and so you get linear response.
00:03:18.080 --> 00:03:22.550
That means that the increase
in the mixing coefficient
00:03:22.550 --> 00:03:26.600
for the final state is
proportional to the coupling
00:03:26.600 --> 00:03:30.450
matrix element times time.
00:03:30.450 --> 00:03:34.930
Now mixing coefficients
can't get larger than 1,
00:03:34.930 --> 00:03:40.900
and so clearly we can't use a
simple theory without saying,
00:03:40.900 --> 00:03:44.200
OK, we don't care about
the mixing coefficients.
00:03:44.200 --> 00:03:47.170
We care about the
rate of increase.
00:03:47.170 --> 00:03:53.260
And so by going from
amplitude to rates,
00:03:53.260 --> 00:03:57.940
we are able to have a theory
that's generally applicable.
00:03:57.940 --> 00:04:02.830
Now what I did was to talk
about a CW radiation field,
00:04:02.830 --> 00:04:05.400
continuous.
00:04:05.400 --> 00:04:10.350
Many experiments
use pulsed fields.
00:04:10.350 --> 00:04:15.040
Many experiments use an
extremely strong pulsed field.
00:04:15.040 --> 00:04:18.320
That I hope to revisit
later in the course.
00:04:18.320 --> 00:04:23.770
But the CW probably is
the best way to begin.
00:04:23.770 --> 00:04:25.540
So now let's talk about
the hydrogen atom.
00:04:30.180 --> 00:04:41.780
We did the rigid rotor, and that
led to some angular momenta.
00:04:41.780 --> 00:04:44.160
Well, let's just picture.
00:04:44.160 --> 00:04:48.940
So we have this rigid
rotor, and it's rotating
00:04:48.940 --> 00:04:50.980
about the center of mass.
00:04:50.980 --> 00:04:53.980
And the Schrodinger
equation tells us
00:04:53.980 --> 00:04:58.030
the probability
amplitudes of the rotor
00:04:58.030 --> 00:05:03.790
axis relative to the
laboratory frame.
00:05:03.790 --> 00:05:05.980
And we got angular
momenta, which
00:05:05.980 --> 00:05:12.680
we can denote by a little l,
big L, J, and many other things.
00:05:12.680 --> 00:05:16.990
And this is important because,
if it's an angular momentum,
00:05:16.990 --> 00:05:18.340
you know it.
00:05:18.340 --> 00:05:21.680
You know everything about it.
00:05:21.680 --> 00:05:25.760
And you don't know yet that, if
you have two angular momenta,
00:05:25.760 --> 00:05:29.194
you could know everything about
the interactions between them.
00:05:29.194 --> 00:05:30.860
That's called the
Wigner-Eckart theorem.
00:05:30.860 --> 00:05:32.820
That's not in this course.
00:05:32.820 --> 00:05:35.450
But the important thing is,
if you've got angular momenta,
00:05:35.450 --> 00:05:41.280
you're on solid ground and you
can do stuff that doesn't ever
00:05:41.280 --> 00:05:44.140
need to be repeated.
00:05:44.140 --> 00:05:47.650
And so one of the things that
we learned about the angular
00:05:47.650 --> 00:05:53.350
momentum was that we have--
00:05:53.350 --> 00:05:58.180
even though it's going to be L
as the angular momentum mostly
00:05:58.180 --> 00:06:02.050
in the hydrogen atom,
I'm going to switch to J,
00:06:02.050 --> 00:06:03.850
which is my favorite notation.
00:06:03.850 --> 00:06:08.740
So J squared
operating on a state--
00:06:08.740 --> 00:06:13.600
now I can denote it like this
or I can denote it like this.
00:06:13.600 --> 00:06:16.180
It's the same general idea.
00:06:16.180 --> 00:06:18.080
These are not quite
equivalent things.
00:06:22.620 --> 00:06:30.450
And you get H bar
squared J A plus one JM.
00:06:30.450 --> 00:06:36.440
And we have JZ operating on JM.
00:06:36.440 --> 00:06:41.100
And you get H bar M JM.
00:06:41.100 --> 00:06:43.930
And we can have J plus
minus operating on JM.
00:06:46.600 --> 00:06:49.620
And that's H bar times
this more complicated
00:06:49.620 --> 00:06:57.327
looking thing, J plus 1
minus M, M plus or minus 1.
00:06:57.327 --> 00:06:59.160
So that's the only bit
you have to remember.
00:07:02.440 --> 00:07:04.900
And you might say, well,
is it plus or minus 1
00:07:04.900 --> 00:07:07.480
or minus or plus 1?
00:07:07.480 --> 00:07:09.360
And so if you choose--
00:07:09.360 --> 00:07:17.160
M is equal to J. Then
you know that if you
00:07:17.160 --> 00:07:20.010
have M is equal to
J, this is going
00:07:20.010 --> 00:07:25.030
to be 0 if we're doing
a raising operator.
00:07:25.030 --> 00:07:29.260
And so there's
nothing to remember
00:07:29.260 --> 00:07:32.530
if you are willing to go
to an extreme situation
00:07:32.530 --> 00:07:35.560
and decide whether it's
plus minus or minus or plus.
00:07:43.990 --> 00:07:49.160
Now the rigid rotor is
a universal problem,
00:07:49.160 --> 00:07:50.020
which is solved.
00:07:50.020 --> 00:07:53.510
It deals with all
central force problems.
00:07:53.510 --> 00:07:56.800
Everything that's
round, the rigid rotor
00:07:56.800 --> 00:07:59.360
is a fantastic starting point.
00:07:59.360 --> 00:08:03.250
So one-electron atoms for sure--
00:08:03.250 --> 00:08:05.620
many-electron atoms, maybe.
00:08:05.620 --> 00:08:07.860
We'll see.
00:08:07.860 --> 00:08:11.220
But anything that's
round, the rigid rotor
00:08:11.220 --> 00:08:14.910
is a good zero point for
dealing with the angular
00:08:14.910 --> 00:08:17.840
part of the problem.
00:08:17.840 --> 00:08:21.580
So for the hydrogen atom,
we have a potential.
00:08:24.390 --> 00:08:25.990
It looks like this.
00:08:28.850 --> 00:08:32.679
And so the radial
potential is new.
00:08:32.679 --> 00:08:36.440
For that rigid rotor, the
radial potential was--
00:08:36.440 --> 00:08:42.970
it's just 0 for the
R is equal to R zero.
00:08:42.970 --> 00:08:47.270
And not zero-- then the
potential is infinite.
00:08:47.270 --> 00:08:49.840
Now here we have
potential, which
00:08:49.840 --> 00:08:53.200
doesn't go to infinity here and
it does something strange here
00:08:53.200 --> 00:08:56.700
because you can't
get to negative R
00:08:56.700 --> 00:08:59.270
in spherical polar coordinates.
00:08:59.270 --> 00:09:02.200
So is it a boundary
condition or is it
00:09:02.200 --> 00:09:09.430
just an accident of the
way we use coordinates?
00:09:09.430 --> 00:09:13.510
So we're going to apply what
we know about angular momenta
00:09:13.510 --> 00:09:15.280
to the hydrogen atom.
00:09:15.280 --> 00:09:18.310
So our potential for
the hydrogen atom
00:09:18.310 --> 00:09:23.860
is going to be
expressed in terms
00:09:23.860 --> 00:09:27.230
of the distance of the
electron from the nucleus
00:09:27.230 --> 00:09:30.220
and then the theta phi
coordinates, which you already
00:09:30.220 --> 00:09:30.970
know.
00:09:30.970 --> 00:09:33.480
And the theta phi
part is universal,
00:09:33.480 --> 00:09:37.900
and the R part is special
to each problem that
00:09:37.900 --> 00:09:39.970
has physical symmetry.
00:09:39.970 --> 00:09:43.960
And there are different kinds
of approximations you use
00:09:43.960 --> 00:09:45.790
to be able to deal with them.
00:09:45.790 --> 00:09:49.540
The nice thing about it
is it's one dimensional,
00:09:49.540 --> 00:09:53.830
and we're very good at thinking
about one dimensional problems.
00:09:53.830 --> 00:09:56.950
And even if the problem
isn't exactly one dimensional
00:09:56.950 --> 00:09:59.470
or it has some
hidden stuff to it,
00:09:59.470 --> 00:10:04.960
we can extend what we know
from one dimensional problems
00:10:04.960 --> 00:10:06.700
and get a great deal of insight.
00:10:06.700 --> 00:10:09.710
And if you have a one
dimensional problem,
00:10:09.710 --> 00:10:14.380
it's very easy to describe
the potential in one dimension
00:10:14.380 --> 00:10:18.430
and the eigen functions
in one dimension.
00:10:18.430 --> 00:10:22.698
And so we can begin to
really understand everything.
00:10:26.180 --> 00:10:31.410
So we're going to have a
wave function, which is
00:10:31.410 --> 00:10:32.660
going to have quantum numbers.
00:10:38.690 --> 00:10:40.770
And we're going to
be able to write it
00:10:40.770 --> 00:10:43.185
as a product of two parts.
00:10:51.270 --> 00:10:55.950
Well, this is the same thing we
had before for the rigid rotor.
00:10:55.950 --> 00:10:59.580
So we're going to be able
to take this rotary equation
00:10:59.580 --> 00:11:04.610
and separate the wave function
into two parts, a radial part
00:11:04.610 --> 00:11:06.030
and an angular part.
00:11:06.030 --> 00:11:07.620
And this is old and this is new.
00:11:13.800 --> 00:11:19.350
Because we've got this, we
can use all of that stuff
00:11:19.350 --> 00:11:21.150
without taking a
breath, except maybe
00:11:21.150 --> 00:11:25.575
getting the right letter
L, J, S, whatever.
00:11:29.330 --> 00:11:33.910
So really the hydrogen
atom is just one thing
00:11:33.910 --> 00:11:37.585
with some curve balls thrown
at you in the latter stages.
00:11:45.350 --> 00:11:48.620
One of the things you also
want to be able to do--
00:11:48.620 --> 00:11:52.310
because nodes are so
important in determining
00:11:52.310 --> 00:11:57.290
both the names of the
states and how they behave
00:11:57.290 --> 00:12:05.240
in various situations, including
external fields and excitation
00:12:05.240 --> 00:12:07.830
by electromagnetic radiation--
00:12:07.830 --> 00:12:10.730
you want to be able to
understand the nodal surfaces.
00:12:10.730 --> 00:12:14.230
And you already know this one.
00:12:14.230 --> 00:12:16.770
So how many nodal
surfaces are there
00:12:16.770 --> 00:12:18.630
if you have a
particular value of L?
00:12:21.440 --> 00:12:21.950
Yes.
00:12:21.950 --> 00:12:22.640
AUDIENCE: L.
00:12:22.640 --> 00:12:27.505
PROFESSOR: Right, and if you
have a particular value of M,
00:12:27.505 --> 00:12:29.770
how many nodes are
there in the xy plane?
00:12:37.790 --> 00:12:39.223
You're hot--
00:12:39.223 --> 00:12:42.980
AUDIENCE: I mean,
I know if M is 0,
00:12:42.980 --> 00:12:47.150
the entire angular momentum
is tipped into the xy plane.
00:12:47.150 --> 00:12:52.760
So that means that
the axis of the rotor
00:12:52.760 --> 00:12:56.655
has to be orthogonal to
the angular momentum.
00:12:56.655 --> 00:12:58.670
So that means that the
probability density is
00:12:58.670 --> 00:13:02.252
oriented along Z, I think.
00:13:02.252 --> 00:13:03.710
PROFESSOR: Now
you're saying things
00:13:03.710 --> 00:13:05.168
that I have to stop
and think about
00:13:05.168 --> 00:13:07.760
because you're not telling
me what I expected to know.
00:13:07.760 --> 00:13:13.670
So if M is equal to
0, L is perpendicular
00:13:13.670 --> 00:13:19.685
to the quantization axis,
and there are no nodes.
00:13:19.685 --> 00:13:21.310
AUDIENCE: Well, it
depends on if you're
00:13:21.310 --> 00:13:25.000
talking about the nodes
of the probability
00:13:25.000 --> 00:13:28.750
density of the rotor,
where the axis is.
00:13:28.750 --> 00:13:32.410
PROFESSOR: That's true,
but where the axis
00:13:32.410 --> 00:13:36.670
is determined by theta.
00:13:36.670 --> 00:13:42.100
And when theta is equal to pi
over 2, you're in the xy plane.
00:13:42.100 --> 00:13:48.430
And the number of
nodes in the xy plane
00:13:48.430 --> 00:13:58.370
is M, or absolute value of M.
The phi part of the rigid rotor
00:13:58.370 --> 00:13:59.420
is simple.
00:13:59.420 --> 00:14:03.300
It's a differential equation
that everybody can solve.
00:14:03.300 --> 00:14:05.570
We already know that
one, and we know what
00:14:05.570 --> 00:14:06.810
the wave functions look like.
00:14:15.240 --> 00:14:26.550
We can write the hydrogen
atom Schrodinger equation,
00:14:26.550 --> 00:14:32.500
and it's very quick to show
separation of variables.
00:14:32.500 --> 00:14:34.480
And I'll do that in a minute.
00:14:34.480 --> 00:14:43.230
And then we have the pictures
of the separated parts--
00:14:43.230 --> 00:14:48.140
RNL of R and YLM of theta phi.
00:14:50.900 --> 00:14:55.670
And essential in these pictures
is the number of nodes.
00:14:55.670 --> 00:14:57.920
In the spacing between
nodes, remember
00:14:57.920 --> 00:15:01.210
the semiclassical approximation.
00:15:01.210 --> 00:15:04.620
We know that Mr. DeBroglie
really hit it out of the park
00:15:04.620 --> 00:15:12.070
by saying that the
wavelength is H over P.
00:15:12.070 --> 00:15:13.600
For every one
dimensional problem,
00:15:13.600 --> 00:15:16.250
we know what to do with that.
00:15:16.250 --> 00:15:19.970
And we're going to discover
that, for the radial problem,
00:15:19.970 --> 00:15:22.640
we have a very simple
way of determining
00:15:22.640 --> 00:15:25.770
what the classical momentum is.
00:15:25.770 --> 00:15:29.790
And so we know everything about
the nodes and the node spacing
00:15:29.790 --> 00:15:32.250
and the amplitudes
between nodes and how
00:15:32.250 --> 00:15:39.220
to evaluate every integral
of some power of R or Z.
00:15:39.220 --> 00:15:41.580
And so there's just
an enormous amount
00:15:41.580 --> 00:15:45.270
gotten from the
semiclassical picture
00:15:45.270 --> 00:15:48.450
once you are familiar with this.
00:15:48.450 --> 00:15:55.560
And so we can say that we
have the classical wavelength,
00:15:55.560 --> 00:15:57.480
or the semiclassical wavelength.
00:15:57.480 --> 00:16:12.460
It has an index R. So we have
a momentum, a linear momentum
00:16:12.460 --> 00:16:14.596
with a quantum number
on it for L. See,
00:16:14.596 --> 00:16:15.595
that's a little strange.
00:16:18.500 --> 00:16:20.960
But that tells us what the
potential is going to be
00:16:20.960 --> 00:16:22.860
and that tells us
what to use in order
00:16:22.860 --> 00:16:25.500
to determine the wavelength.
00:16:25.500 --> 00:16:27.890
And this is really
the core of how
00:16:27.890 --> 00:16:30.230
we can go way beyond textbooks.
00:16:30.230 --> 00:16:34.105
We can do-- in our heads or
on a simple piece of paper,
00:16:34.105 --> 00:16:34.730
we can do this.
00:16:34.730 --> 00:16:41.645
And we can draw pictures, and
we can evaluate matrix elements.
00:16:45.040 --> 00:16:49.210
Without any complicated
integral tables,
00:16:49.210 --> 00:16:54.310
you can make estimates that
are incredibly important.
00:16:54.310 --> 00:17:03.960
And from that you can get
expectation values and also
00:17:03.960 --> 00:17:05.910
off-diagonal matrix
elements of integer
00:17:05.910 --> 00:17:07.871
powers of the coordinate.
00:17:10.460 --> 00:17:13.079
That's an enormous amount
of stuff that you can do.
00:17:13.079 --> 00:17:15.700
Now you can't do it yet.
00:17:15.700 --> 00:17:19.240
But after Wednesday's
lecture, you will.
00:17:19.240 --> 00:17:23.869
And so this will be Wednesday.
00:17:23.869 --> 00:17:31.655
And then we'll have
evidence of electron spin.
00:17:34.940 --> 00:17:39.130
And I will get to that today.
00:17:39.130 --> 00:17:40.430
So this is the menu.
00:17:40.430 --> 00:17:43.130
Now let's start delivering
some of this stuff.
00:17:48.360 --> 00:17:51.510
I have to write
some big equations.
00:17:51.510 --> 00:17:57.920
So the Hamiltonian is kinetic
energy plus potential energy,
00:17:57.920 --> 00:18:00.510
and kinetic energy was--
00:18:00.510 --> 00:18:01.790
I'm sorry.
00:18:01.790 --> 00:18:06.710
For the rigid rotor, V was zero.
00:18:06.710 --> 00:18:08.650
And everything was in
the kinetic energy.
00:18:08.650 --> 00:18:10.660
Well, it's not quite true.
00:18:10.660 --> 00:18:15.420
And so, for
hydrogen-- so we know
00:18:15.420 --> 00:18:21.720
this is P squared over 2 mu,
and mu for the hydrogen atom,
00:18:21.720 --> 00:18:23.130
reduced mass.
00:18:23.130 --> 00:18:32.750
And we know this is just
Coulomb's Law, minus e
00:18:32.750 --> 00:18:42.240
squared over 4 pi epsilon
zero R, H bar squared.
00:18:45.750 --> 00:18:51.570
I'm sorry-- it was
supposed to be e squared.
00:18:51.570 --> 00:18:52.800
So this is the classical--
00:18:55.610 --> 00:18:57.820
So these are the parts.
00:18:57.820 --> 00:19:00.470
But the since it's
spherical, we're
00:19:00.470 --> 00:19:04.840
not going to work in
Cartesian coordinates.
00:19:04.840 --> 00:19:09.560
And in fact, that's a
very strong statement.
00:19:09.560 --> 00:19:12.270
If you have a spherical
problem, don't
00:19:12.270 --> 00:19:14.190
start in Cartesian
coordinates because it's
00:19:14.190 --> 00:19:18.720
a horrible mess transforming
to spherical polar coordinates.
00:19:18.720 --> 00:19:22.820
Just remember the spherical
polar coordinates.
00:19:22.820 --> 00:19:24.990
So you know how spherical
polar coordinates work.
00:19:24.990 --> 00:19:26.031
I'm not going to draw it.
00:19:28.260 --> 00:19:36.850
So the kinetic energy term is--
00:19:44.690 --> 00:19:51.590
and this is the Laplacian,
and that's a terrible thing.
00:19:51.590 --> 00:19:58.070
And Del squared--
it looks like it's
00:19:58.070 --> 00:20:02.428
going to be a real nightmare
partial with respect to R,
00:20:02.428 --> 00:20:08.550
R squared partial with respect
to R. And we have 1 over R
00:20:08.550 --> 00:20:13.460
squared sine squared
theta partial with respect
00:20:13.460 --> 00:20:21.130
to theta, sine theta partial
with respect to theta.
00:20:21.130 --> 00:20:23.560
And then a third term--
00:20:23.560 --> 00:20:28.640
1 over R squared sine squared.
00:20:28.640 --> 00:20:30.100
I got a square here.
00:20:30.100 --> 00:20:32.170
That's wrong.
00:20:32.170 --> 00:20:34.570
And this is sine squared theta.
00:20:34.570 --> 00:20:36.340
I heard a mumble over there.
00:20:43.420 --> 00:20:48.005
Yeah, but I'm just
doing Del squared.
00:20:48.005 --> 00:20:50.130
AUDIENCE: With your first
Del, it's H plus squared.
00:20:50.130 --> 00:20:50.921
PROFESSOR: My what?
00:20:50.921 --> 00:20:53.410
AUDIENCE: Your first time you
used the Del operator, it's H
00:20:53.410 --> 00:20:55.280
plus squared.
00:20:55.280 --> 00:20:56.662
PROFESSOR: Oh, yes, yes.
00:20:56.662 --> 00:20:58.078
AUDIENCE: And then
the second one,
00:20:58.078 --> 00:21:00.165
if you could clarify
that that's a Del--
00:21:00.165 --> 00:21:01.040
PROFESSOR: I'm sorry.
00:21:01.040 --> 00:21:01.920
AUDIENCE: The
second Del squared,
00:21:01.920 --> 00:21:03.764
if you could clarify
that's a Del squared.
00:21:03.764 --> 00:21:05.120
It looks like a--
00:21:05.120 --> 00:21:06.870
PROFESSOR: It looks
like a terrible thing.
00:21:11.640 --> 00:21:18.690
And the last part is a second
derivative with respect to phi.
00:21:18.690 --> 00:21:22.710
This looks like a terrible
thing to build on.
00:21:22.710 --> 00:21:27.670
But with a little
bit of trickery--
00:21:27.670 --> 00:21:32.900
and that is, suppose we multiply
this equation by R squared--
00:21:32.900 --> 00:21:41.340
then we have killed the R
squared terms here and here.
00:21:41.340 --> 00:21:44.020
And we're going to be
able to separate it.
00:21:44.020 --> 00:21:50.980
And so we are able to
write an equation which
00:21:50.980 --> 00:21:53.230
has the separability built in--
00:21:53.230 --> 00:21:56.784
and so partial
with respect to R,
00:21:56.784 --> 00:22:05.300
R squared partial with
respect to R plus L
00:22:05.300 --> 00:22:23.980
squared plus 2 mu H R
squared V of R minus E psi.
00:22:23.980 --> 00:22:26.150
That's a Schrodinger equation.
00:22:26.150 --> 00:22:32.020
So we have an R dependent term
and another R dependent term
00:22:32.020 --> 00:22:36.510
and a theta phi dependent term
all in this one, nice operator
00:22:36.510 --> 00:22:37.570
that we've understood.
00:22:41.380 --> 00:22:44.780
So now there's one more trick.
00:22:44.780 --> 00:22:47.330
In order for the
shorter equation
00:22:47.330 --> 00:22:51.860
to be separated into a
theta phi part and an R part
00:22:51.860 --> 00:22:54.230
is we need a commutator.
00:22:54.230 --> 00:23:00.170
And so that commutator is this.
00:23:00.170 --> 00:23:05.000
What is a commutator between L
squared and any function of R?
00:23:08.330 --> 00:23:08.910
Yes.
00:23:08.910 --> 00:23:11.243
AUDIENCE: Their operators
depend on different variables?
00:23:11.243 --> 00:23:13.250
PROFESSOR: Absolutely,
that's really important.
00:23:13.250 --> 00:23:16.310
We often encounter
operators that
00:23:16.310 --> 00:23:18.780
depend on different variables.
00:23:18.780 --> 00:23:21.420
And when they do, they
commute with each other,
00:23:21.420 --> 00:23:24.200
which is an incredibly
convenient thing because that
00:23:24.200 --> 00:23:26.690
means we can set up the
problem as a product
00:23:26.690 --> 00:23:31.238
of the eigenfunctions of
the different operators.
00:23:31.238 --> 00:23:38.880
So this means that we can
write a theta phi term, which
00:23:38.880 --> 00:23:44.190
we completely know, and an
R term, which we don't know
00:23:44.190 --> 00:23:46.200
and contains all of
the interesting stuff.
00:23:50.450 --> 00:23:52.890
So let's forget
about theta and phi.
00:23:52.890 --> 00:23:54.760
Let's just look at the R part.
00:24:02.710 --> 00:24:04.750
Well, the way we
separated variables,
00:24:04.750 --> 00:24:06.670
we had the Schrodinger
equation and we
00:24:06.670 --> 00:24:10.360
divided by the wave
function, which would be
00:24:10.360 --> 00:24:17.210
of R of R, Y L M of theta phi.
00:24:17.210 --> 00:24:22.790
And on one side of the equality,
we have only the angle part.
00:24:22.790 --> 00:24:25.400
And so when we do that,
when we divide by R,
00:24:25.400 --> 00:24:27.680
we kill the R part on this side.
00:24:27.680 --> 00:24:30.900
And on the other side,
it's the opposite.
00:24:30.900 --> 00:24:33.710
And so we get two pieces--
00:24:33.710 --> 00:24:36.230
one is only dependent
on theta and phi
00:24:36.230 --> 00:24:38.330
and one is only dependent
on R. So they both
00:24:38.330 --> 00:24:40.040
have to be a constant.
00:24:40.040 --> 00:24:42.680
So we get two separate
differential equations.
00:24:42.680 --> 00:24:45.570
And we've already
dealt with one of them.
00:24:45.570 --> 00:24:57.430
So what you end up getting
is 1 over R of R times stuff
00:24:57.430 --> 00:25:09.131
times R of R is equal to
1 over YLM L squared YLM--
00:25:09.131 --> 00:25:09.630
sorry.
00:25:13.700 --> 00:25:15.710
So this is the separation.
00:25:15.710 --> 00:25:17.900
This is all R stuff.
00:25:17.900 --> 00:25:21.960
This is all theta stuff
and separation constant.
00:25:21.960 --> 00:25:24.350
Well, this one is inviting
a separation concept
00:25:24.350 --> 00:25:27.830
because L squared
operating on YLM
00:25:27.830 --> 00:25:30.679
is H bar squared, LL plus 1.
00:25:30.679 --> 00:25:31.970
That's the separation constant.
00:25:38.320 --> 00:25:44.140
So now let's look
for the only time
00:25:44.140 --> 00:25:48.350
at the R part of the
differential equation.
00:25:48.350 --> 00:25:53.080
And so this writing out
all the pieces honestly--
00:26:31.600 --> 00:26:33.070
we have-- there's one more.
00:26:40.090 --> 00:26:43.269
So that's the Schrodinger
equation for the radial part.
00:26:43.269 --> 00:26:44.560
It looks a little bit annoying.
00:26:47.680 --> 00:26:52.600
The important
trick is that we've
00:26:52.600 --> 00:26:58.410
taken the separation constant,
and it has an R dependence
00:26:58.410 --> 00:27:04.020
but when we divide through
by 2 mu HR squared.
00:27:04.020 --> 00:27:07.650
And we say, oh, well, let's
call these two things together--
00:27:07.650 --> 00:27:15.510
VL of R. This is the
effect of potential,
00:27:15.510 --> 00:27:18.790
and it depends on
the value of L.
00:27:18.790 --> 00:27:22.390
So this is just like an
ordinary one dimensional problem
00:27:22.390 --> 00:27:24.250
except now, for
every value of L,
00:27:24.250 --> 00:27:27.140
we have a different potential.
00:27:27.140 --> 00:27:30.520
And this potential is--
00:27:30.520 --> 00:27:36.470
OK, when L is not equal
to zero, this potential
00:27:36.470 --> 00:27:40.350
goes to infinity at
R equals 0, which
00:27:40.350 --> 00:27:44.340
is bad, except it's good because
it keeps the particle ever
00:27:44.340 --> 00:27:46.425
from getting close
to the nucleus.
00:27:49.710 --> 00:27:51.870
And that's what
Mr. Schrodinger--
00:27:51.870 --> 00:27:53.970
I'm sorry what Mr. Bohr
was thinking about,
00:27:53.970 --> 00:27:56.150
that we have only
circular orbits
00:27:56.150 --> 00:28:03.570
or we have only orbits that
are away from the place
00:28:03.570 --> 00:28:08.740
where the Coulomb interaction
would be infinite.
00:28:08.740 --> 00:28:11.130
And I mean, there are
several things that's
00:28:11.130 --> 00:28:12.390
wrong with Bohr's picture.
00:28:12.390 --> 00:28:15.580
One is that we have orbits.
00:28:15.580 --> 00:28:20.280
And the other is that we
don't include L equals 0,
00:28:20.280 --> 00:28:28.020
but this is telling you a
lot of really important stuff
00:28:28.020 --> 00:28:32.490
because the value
of L determines
00:28:32.490 --> 00:28:36.930
the importance of this thing
that goes to infinity at R
00:28:36.930 --> 00:28:39.490
equals 0.
00:28:39.490 --> 00:28:44.450
And that has very significant
consequences as far as
00:28:44.450 --> 00:28:47.270
which orbital angular momentum
states we're dealing with.
00:28:54.420 --> 00:29:01.440
So we're going to get the
usual LML quantum numbers,
00:29:01.440 --> 00:29:04.200
and we're going to get another
one from the radial part.
00:29:04.200 --> 00:29:08.100
And since this is a
1D equation, there's
00:29:08.100 --> 00:29:10.420
only one quantum number.
00:29:10.420 --> 00:29:12.790
And we're going to
call it L. And now
00:29:12.790 --> 00:29:17.180
there's this word principal,
and there's really two words--
00:29:17.180 --> 00:29:19.835
principle with an LE and
principal with an AL.
00:29:22.880 --> 00:29:27.200
Which one do you
think is appropriate?
00:29:27.200 --> 00:29:28.680
I'm sorry.
00:29:28.680 --> 00:29:30.229
I can't hear.
00:29:30.229 --> 00:29:30.770
AUDIENCE: AL.
00:29:30.770 --> 00:29:32.710
PROFESSOR: AL is
the appropriate one.
00:29:32.710 --> 00:29:36.570
Principle has to do with
something fundamental.
00:29:36.570 --> 00:29:40.980
Principal has to do with
something that's important.
00:29:40.980 --> 00:29:44.720
And I can't tell
you how many times
00:29:44.720 --> 00:29:48.498
people who should know better
use the wrong principal.
00:29:52.470 --> 00:29:55.620
You'll never do that now
because it was like, what's nu?
00:29:55.620 --> 00:29:56.460
C over lambda.
00:29:59.730 --> 00:30:01.590
So this is the right principal.
00:30:10.464 --> 00:30:18.110
If we do something clever
and we have the radial part
00:30:18.110 --> 00:30:20.240
and we say, let
us replace it by 1
00:30:20.240 --> 00:30:29.900
over R times this new
function chi L of R, well,
00:30:29.900 --> 00:30:35.380
when we do that, this equation
becomes really simple.
00:30:35.380 --> 00:30:40.530
So what we get when we make that
substitution is H bar squared
00:30:40.530 --> 00:30:52.040
over 2 mu H second derivative
with respect to R plus VL of R
00:30:52.040 --> 00:30:57.950
minus E chi L of
R is equal to 0.
00:30:57.950 --> 00:31:02.250
That looks like a differential
equation we've seen before.
00:31:02.250 --> 00:31:04.820
It's a simple one dimensional
differential equation--
00:31:04.820 --> 00:31:06.710
kinetic energy,
potential energy.
00:31:06.710 --> 00:31:07.690
But it's not.
00:31:07.690 --> 00:31:13.140
There's some kinetic energy
hidden in the potential energy.
00:31:13.140 --> 00:31:16.580
But this is simple, and we
can deal with this a lot.
00:31:16.580 --> 00:31:18.080
I'm not going to.
00:31:18.080 --> 00:31:20.709
But if you're going
to actually do stuff,
00:31:20.709 --> 00:31:23.000
you're going to be wanting
to look at this differential
00:31:23.000 --> 00:31:24.680
equation.
00:31:24.680 --> 00:31:26.975
But I'm going to
forego that pleasure.
00:31:31.100 --> 00:31:34.810
One of the problems
with the radial equation
00:31:34.810 --> 00:31:39.130
is the fact that R is a
special kind of coordinate.
00:31:39.130 --> 00:31:41.860
It can't go negative.
00:31:41.860 --> 00:31:48.730
And so treating the boundary
condition for R is equal to 0
00:31:48.730 --> 00:31:49.840
is a little subtle.
00:31:53.280 --> 00:31:58.920
And it turns out that when
L is equal to 0, then R of 0
00:31:58.920 --> 00:31:59.880
is not 0.
00:32:02.570 --> 00:32:09.340
But for all other values
of L, R of 0 is 0.
00:32:09.340 --> 00:32:18.340
And all of NMR
depends on L being
00:32:18.340 --> 00:32:25.130
0 because the electron
feels the nucleus.
00:32:25.130 --> 00:32:29.450
It doesn't experience
an infinite singularity.
00:32:29.450 --> 00:32:30.950
It feels the nucleus.
00:32:30.950 --> 00:32:34.910
And when L is not
equal to 0, then it's
00:32:34.910 --> 00:32:38.030
sensing the nucleus
at a distance.
00:32:38.030 --> 00:32:41.060
And that leads to some
very small splittings
00:32:41.060 --> 00:32:43.580
called hyperfine.
00:32:43.580 --> 00:32:48.180
And so there's different
kinds of hyperfine structure.
00:32:48.180 --> 00:32:52.200
But anyway, this is a really
subtle and important point.
00:32:57.757 --> 00:33:01.230
How much time left?
00:33:01.230 --> 00:33:02.350
Yes?
00:33:02.350 --> 00:33:04.340
No.
00:33:04.340 --> 00:33:07.151
So now pictures of orbitals--
00:33:09.740 --> 00:33:13.970
So the problem with
pictures of orbitals
00:33:13.970 --> 00:33:18.710
is now we have a function
of three variables
00:33:18.710 --> 00:33:23.460
and it's equal to
some complex number.
00:33:23.460 --> 00:33:25.880
So we need two
degrees of freedom
00:33:25.880 --> 00:33:29.910
to present a complex number
and we have three variables.
00:33:29.910 --> 00:33:34.940
And so representing that on a
two dimensional sheet of paper
00:33:34.940 --> 00:33:35.990
is horrible.
00:33:35.990 --> 00:33:38.150
But we have this
wonderful factorization
00:33:38.150 --> 00:33:42.380
where we have RNL of R. And
we could draw that easily.
00:33:42.380 --> 00:33:44.460
We don't need any special skill.
00:33:44.460 --> 00:33:48.380
And we have YLM of theta phi.
00:33:48.380 --> 00:33:50.120
And we've got lots of
practice with that,
00:33:50.120 --> 00:33:51.830
although what we've
practiced with
00:33:51.830 --> 00:33:54.140
may not be completely
understood yet.
00:33:58.630 --> 00:34:00.760
So we have ways of
representing these.
00:34:00.760 --> 00:34:05.770
And so we go through
the understanding
00:34:05.770 --> 00:34:09.030
of the hydrogen atom by looking
at these two things separately.
00:34:12.330 --> 00:34:16.365
Now for the radial part,
the energy levels--
00:34:27.409 --> 00:34:29.429
where this is the
Rydberg constant.
00:34:29.429 --> 00:34:32.540
It's a combination or
fundamental constants.
00:34:32.540 --> 00:34:37.070
And for hydrogen,
the Rydberg constant
00:34:37.070 --> 00:34:46.440
is equal to 109,737.319--
00:34:46.440 --> 00:34:49.770
there's actually more
digits, wave numbers--
00:34:49.770 --> 00:34:56.590
times mu H over mu infinite.
00:34:56.590 --> 00:34:59.260
Well, this is actually
the Rydberg constant
00:34:59.260 --> 00:35:00.355
for an infinite mass.
00:35:09.900 --> 00:35:20.610
And so mu H is equal to
the mass of the electron
00:35:20.610 --> 00:35:25.969
times the mass of the proton
over the mass of the electron
00:35:25.969 --> 00:35:27.135
plus the mass of the proton.
00:35:30.180 --> 00:35:33.230
Now the mass of the
proton is much bigger
00:35:33.230 --> 00:35:34.790
than the mass of the electron.
00:35:34.790 --> 00:35:37.610
And so you can use
this as a trick.
00:35:37.610 --> 00:35:39.310
You can say, well, what is it?
00:35:39.310 --> 00:35:41.430
Well, we know what this is.
00:35:41.430 --> 00:35:42.509
It's easy to calculate.
00:35:42.509 --> 00:35:43.550
But there are two limits.
00:35:43.550 --> 00:35:47.810
What is the smallest
possible reduced mass?
00:35:47.810 --> 00:35:49.265
And that would be for--
00:35:55.790 --> 00:36:00.240
if the mass of the
proton is infinite,
00:36:00.240 --> 00:36:02.560
then we just get the
mass of the electron.
00:36:02.560 --> 00:36:05.790
That's the biggest reduced mass.
00:36:05.790 --> 00:36:08.310
And the smallest is,
if we have positronium,
00:36:08.310 --> 00:36:13.380
where we have an electron
bound to a proton--
00:36:13.380 --> 00:36:17.680
and then when we do
that, we get half.
00:36:17.680 --> 00:36:18.180
I'm sorry.
00:36:18.180 --> 00:36:19.520
It's not a proton.
00:36:19.520 --> 00:36:24.590
It's a positively
charged particle,
00:36:24.590 --> 00:36:27.020
which we call a positron.
00:36:27.020 --> 00:36:34.010
And so each of the terms here
is the mass of the electron.
00:36:34.010 --> 00:36:43.820
And so the range is
from 1/2 me to me
00:36:43.820 --> 00:36:47.490
depending on what particles
you're dealing with.
00:36:47.490 --> 00:36:50.690
And so that's a useful thing.
00:36:50.690 --> 00:36:55.040
And so the Rydberg
constant for hydrogen
00:36:55.040 --> 00:36:59.491
is smaller than the reduced
mass of the infinite.
00:36:59.491 --> 00:36:59.990
I'm sorry.
00:36:59.990 --> 00:37:05.270
It's smaller than that for
the infinitely mass nucleus.
00:37:05.270 --> 00:37:11.810
And it is the value that
made Mr. Bohr very happy--
00:37:11.810 --> 00:37:15.890
679.
00:37:15.890 --> 00:37:19.430
So the important thing
is this mass scaled,
00:37:19.430 --> 00:37:25.296
or reduced mass scaled, Rydberg
constant explains to a part
00:37:25.296 --> 00:37:28.970
in 10 to the 10th all the
energy levels of one-electron
00:37:28.970 --> 00:37:29.870
systems--
00:37:29.870 --> 00:37:34.010
hydrogen, helium
plus, lithium 2 plus.
00:37:34.010 --> 00:37:34.540
That's it.
00:37:37.725 --> 00:37:39.200
And that's fantastic.
00:37:45.560 --> 00:37:48.740
Now we have to talk to
become really familiar
00:37:48.740 --> 00:37:53.500
with this Rnl of our function.
00:38:06.670 --> 00:38:09.200
And one of the questions
is, how many radial nodes.
00:38:15.710 --> 00:38:21.570
And you know that for 1S
there aren't any nodes.
00:38:24.990 --> 00:38:28.510
And you know for 2P there
aren't any radial nodes.
00:38:31.110 --> 00:38:34.230
And so what we need
is something that
00:38:34.230 --> 00:38:38.870
goes like n minus L minus 1.
00:38:38.870 --> 00:38:41.740
The number of
radial nodes, which
00:38:41.740 --> 00:38:45.100
is all you need to know
about the radial A function,
00:38:45.100 --> 00:38:50.070
is how many nodes are and
how far are they apart
00:38:50.070 --> 00:38:53.710
and what's the amplitude
of each loop between nodes.
00:38:53.710 --> 00:38:57.770
Semi classical theory
gives you all of that.
00:38:57.770 --> 00:39:01.970
And often when you're
calculating an integral,
00:39:01.970 --> 00:39:04.550
all you care about is the
amplitude in the first loop.
00:39:07.190 --> 00:39:11.880
And so instead of having
to evaluate an integral,
00:39:11.880 --> 00:39:16.770
you just figure out what is
the envelope function based
00:39:16.770 --> 00:39:18.810
on the classical
momentum function.
00:39:24.950 --> 00:39:27.410
Now the thing that
everybody's been waiting for.
00:39:33.280 --> 00:39:33.780
Spin.
00:39:39.660 --> 00:39:49.950
So remember, we know an
angular momentum is R cross P,
00:39:49.950 --> 00:39:53.177
and we know that there isn't
any internal structure.
00:39:53.177 --> 00:39:55.260
Or at least we don't know
about internal structure
00:39:55.260 --> 00:39:58.930
of the electron or a proton.
00:39:58.930 --> 00:40:04.850
And so we can't somehow
say, well, it's R cross P.
00:40:04.850 --> 00:40:06.260
So we have the Zeeman effect.
00:40:14.530 --> 00:40:17.400
So we can look at
the Zeeman effect
00:40:17.400 --> 00:40:20.000
for an atom in a magnetic field.
00:40:20.000 --> 00:40:23.860
And we have several
things that we know.
00:40:23.860 --> 00:40:28.950
So we have the magnetic
moment of the electron
00:40:28.950 --> 00:40:37.470
is equal to minus the charge
on the electron times 2ME times
00:40:37.470 --> 00:40:40.850
L. Well, that's an
angular momentum.
00:40:48.980 --> 00:40:57.070
So if we had circulating charge,
well, that circulating charge
00:40:57.070 --> 00:41:00.680
will produce a magnetic moment,
which has a magnitude which
00:41:00.680 --> 00:41:02.330
is related to the velocity.
00:41:04.890 --> 00:41:09.120
And what is L divided by mass?
00:41:09.120 --> 00:41:11.660
It's a velocity.
00:41:11.660 --> 00:41:13.530
And we have the charge here.
00:41:13.530 --> 00:41:17.440
So this is a perfectly
reasonable thing.
00:41:17.440 --> 00:41:22.940
And that's good because we
know that, if we have electrons
00:41:22.940 --> 00:41:25.520
with some kind of
internal structure,
00:41:25.520 --> 00:41:26.900
we know what this
is going to do.
00:41:29.630 --> 00:41:36.390
Now the magnetic potential
is equal to minus
00:41:36.390 --> 00:41:40.080
the magnetic moment times
the external magnetic field.
00:41:42.840 --> 00:41:51.920
And so what we have is
E, BC, LZ over 2 NE.
00:41:55.110 --> 00:41:56.970
Well, this is another
very good thing
00:41:56.970 --> 00:42:02.440
because, not only do
we know what this is,
00:42:02.440 --> 00:42:06.370
we know that it has
only diagonal elements.
00:42:06.370 --> 00:42:12.000
And so we can do first
order perturbation theory.
00:42:15.330 --> 00:42:18.960
If this is so easy, if
this is the Hamiltonian,
00:42:18.960 --> 00:42:22.410
we can just tack that on and
it adds an extra splitting
00:42:22.410 --> 00:42:23.820
to our energy levels.
00:42:27.230 --> 00:42:32.350
The only tricky thing is,
we don't know that it's LZ.
00:42:32.350 --> 00:42:35.360
We just know that it
has a magnetic moment
00:42:35.360 --> 00:42:39.640
or it could have
a magnetic moment.
00:42:39.640 --> 00:42:41.450
And so this wonderful
experiment--
00:42:41.450 --> 00:42:45.920
suppose we start
with a 1S state,
00:42:45.920 --> 00:42:48.880
and you go to a 2P state.
00:42:48.880 --> 00:42:53.800
Now this implies that
we know something
00:42:53.800 --> 00:42:58.422
about the selection rules for
electromagnetic transitions.
00:42:58.422 --> 00:43:00.130
And so for an electric
dipole transition,
00:43:00.130 --> 00:43:02.500
we go from 1S to 2P.
00:43:02.500 --> 00:43:05.470
Now this is not how
it was done initially
00:43:05.470 --> 00:43:09.820
because the frequency
of this transition
00:43:09.820 --> 00:43:12.370
is well into the
vacuum ultraviolet.
00:43:12.370 --> 00:43:16.090
And in the days when quantum
mechanics was being developed,
00:43:16.090 --> 00:43:18.510
that was a hard experiment.
00:43:18.510 --> 00:43:24.750
So one used an S to P
transition on some other atom,
00:43:24.750 --> 00:43:31.580
like mercury, but let's
pretend it was hydrogen.
00:43:31.580 --> 00:43:35.380
So this is an angular
momentum of 0.
00:43:35.380 --> 00:43:40.490
So we would expect it
would be ML equals 0.
00:43:40.490 --> 00:43:46.970
And here we have it could
split into three components.
00:43:46.970 --> 00:43:56.530
And this is an L equals 1, 0
minus 1, because that's minus--
00:43:56.530 --> 00:43:59.350
there with a minus
sign somewhere.
00:43:59.350 --> 00:44:01.700
That's what we expect.
00:44:01.700 --> 00:44:05.920
And so being naive, you might
expect transitions like this.
00:44:09.070 --> 00:44:12.070
Well, the transitions
where you do not
00:44:12.070 --> 00:44:14.260
change the projection
quantum number
00:44:14.260 --> 00:44:18.040
are done with Z
polarized radiation.
00:44:18.040 --> 00:44:23.360
And so if it's Z polarized,
you get only delta ML equals 0.
00:44:23.360 --> 00:44:25.890
And if you have x
and y, you have delta
00:44:25.890 --> 00:44:29.030
ML equals plus and minus 1.
00:44:29.030 --> 00:44:32.380
So depending on how the
experiment was done,
00:44:32.380 --> 00:44:35.680
you would expect
to see one, two,
00:44:35.680 --> 00:44:38.694
or three Zeeman components.
00:44:41.480 --> 00:44:43.610
And they did the
experiment, and they
00:44:43.610 --> 00:44:45.980
saw more than three components.
00:44:45.980 --> 00:44:48.440
They saw five components.
00:44:48.440 --> 00:44:51.780
And that's possibly for
a number of reasons.
00:44:51.780 --> 00:44:53.770
But they saw five components.
00:44:53.770 --> 00:44:57.040
So they knew that there was
something else going on.
00:44:57.040 --> 00:45:09.110
And so you have to try to
collect enough information
00:45:09.110 --> 00:45:13.490
to have a simple minded picture
that will explain it all.
00:45:16.300 --> 00:45:18.090
So one thing you
might do is say, OK,
00:45:18.090 --> 00:45:24.180
suppose there is another
quantum number, another thing.
00:45:24.180 --> 00:45:26.920
And we're going to call it spin.
00:45:26.920 --> 00:45:29.890
And we don't know whether spin
is integer or half integer.
00:45:29.890 --> 00:45:33.910
We know from our exercise
with computation rules
00:45:33.910 --> 00:45:37.390
that both half integer
and integer angular
00:45:37.390 --> 00:45:39.350
momentum are possible.
00:45:39.350 --> 00:45:43.820
And so we can say
we have a spin.
00:45:43.820 --> 00:45:45.430
And it could be 1/2.
00:45:45.430 --> 00:45:46.150
It could be 1.
00:45:46.150 --> 00:45:47.600
It could be anything.
00:45:47.600 --> 00:45:50.290
And then we start looking at
the details of what we observe.
00:45:53.130 --> 00:45:58.500
And what we find is,
all we need is spin 1/2
00:45:58.500 --> 00:46:00.660
to account for
almost everything.
00:46:03.540 --> 00:46:08.590
However, if it were spin
1/2, then you get two here.
00:46:08.590 --> 00:46:14.830
And you'd get two here,
two, here and two here.
00:46:14.830 --> 00:46:15.790
And that's six.
00:46:15.790 --> 00:46:18.510
That's bigger than five.
00:46:18.510 --> 00:46:21.350
So you need to know
something else.
00:46:21.350 --> 00:46:23.810
And one thing is quite
reasonable-- you can say,
00:46:23.810 --> 00:46:28.790
well, the spin
thing is mysterious,
00:46:28.790 --> 00:46:32.810
and electromagnetic
radiation acts on the spatial
00:46:32.810 --> 00:46:34.584
coordinates.
00:46:34.584 --> 00:46:36.250
And there aren't any
spatial coordinates
00:46:36.250 --> 00:46:39.100
of the internal structure
of the electron.
00:46:39.100 --> 00:46:43.390
And so we have a selection
rule delta MS equals 0.
00:46:47.809 --> 00:46:48.850
That still doesn't do it.
00:46:51.500 --> 00:46:53.630
You still need
something else, and that
00:46:53.630 --> 00:46:58.940
is that the proportionality
concept between the angular
00:46:58.940 --> 00:47:06.320
momentum and the energy,
which is called the G factor.
00:47:06.320 --> 00:47:10.940
For orbital angular
matter, the G factor is 1,
00:47:10.940 --> 00:47:12.620
or the proportionality
constant is 1.
00:47:12.620 --> 00:47:18.070
So the G factor is L. And
the G factor for the electron
00:47:18.070 --> 00:47:20.570
turns out to be 2--
00:47:20.570 --> 00:47:23.330
not exactly 2, just a
little more than 2--
00:47:23.330 --> 00:47:26.510
Nobel Prize for that.
00:47:26.510 --> 00:47:30.360
And so with those
extra little things,
00:47:30.360 --> 00:47:35.390
then every detail of the
Zeeman effect is understood.
00:47:35.390 --> 00:47:38.540
And we say, oh, well, the
electron has a spin of 1/2,
00:47:38.540 --> 00:47:40.480
and it like it acts like
an angular momentum.
00:47:40.480 --> 00:47:44.580
It obeys the angular
momentum computation rules.
00:47:44.580 --> 00:47:48.480
It also obeys the
computation rules.
00:47:48.480 --> 00:47:50.600
Well, I won't say that.
00:47:50.600 --> 00:47:52.940
So I should stop--
00:47:52.940 --> 00:47:56.660
there are other things
that provide us information
00:47:56.660 --> 00:47:58.700
that there is something else.
00:47:58.700 --> 00:48:03.980
And one, I'm called by some
people the spin orbit kid
00:48:03.980 --> 00:48:07.550
because I've made a whole lot
of mileage on using spin orbits
00:48:07.550 --> 00:48:08.315
splittings.
00:48:08.315 --> 00:48:13.690
And the spin orbit
Hamiltonian has the form L.S
00:48:13.690 --> 00:48:19.700
And this gives rise to
splittings of a doublet P
00:48:19.700 --> 00:48:21.290
state, for example,
as what you would
00:48:21.290 --> 00:48:24.620
see in the excited
state of hydrogen
00:48:24.620 --> 00:48:27.110
into two components
at zero field.
00:48:29.620 --> 00:48:33.310
And so there's all sorts
of really wonderful stuff.
00:48:33.310 --> 00:48:36.070
And I'm going to cheat you
out of almost all of it
00:48:36.070 --> 00:48:39.580
because we're going to go over
to the semiclassical picture
00:48:39.580 --> 00:48:41.660
next time.
00:48:41.660 --> 00:48:45.410
And we'll understand
everything about Rydberg states
00:48:45.410 --> 00:48:48.560
and about how do we
estimate everything
00:48:48.560 --> 00:48:52.110
having to do with the radial
part of the wave function.
00:48:52.110 --> 00:48:54.640
And there are some
astonishing things.