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ROBERT FIELD: Now the
main topic of this lecture
00:00:26.570 --> 00:00:31.460
is so important and
so beautiful that I
00:00:31.460 --> 00:00:35.810
don't want to spend any time
reviewing what I did last time.
00:00:35.810 --> 00:00:40.280
At the beginning when we
talked about the rigid rotor,
00:00:40.280 --> 00:00:47.120
I said that this is not just a
simple, exactly solved problem
00:00:47.120 --> 00:00:49.790
but it tells you
about the angular part
00:00:49.790 --> 00:00:52.980
of every central force problem.
00:00:52.980 --> 00:00:55.640
And it's even more than that.
00:00:55.640 --> 00:00:59.370
It enables you to do a
certain kind of algebra
00:00:59.370 --> 00:01:05.610
with operators which
enables you to minimize
00:01:05.610 --> 00:01:10.710
the effort of calculating
matrix elements
00:01:10.710 --> 00:01:14.010
and predicting
selection rules simply
00:01:14.010 --> 00:01:18.120
by the basis of commutation
rules of the operators
00:01:18.120 --> 00:01:21.510
without ever looking
at wave functions,
00:01:21.510 --> 00:01:25.530
without ever looking at
differential operators.
00:01:25.530 --> 00:01:33.030
This is a really beautiful
thing about angular momentum
00:01:33.030 --> 00:01:38.160
that if we define the angular
momentum in this abstract way--
00:01:38.160 --> 00:01:42.000
and I'll describe what I
mean by this epsilon ijk.
00:01:42.000 --> 00:01:48.300
If we say we have an operator
which obeys this commutation
00:01:48.300 --> 00:01:53.240
rule, we will call it
an angular momentum.
00:01:53.240 --> 00:01:57.110
And we go through
some arguments and we
00:01:57.110 --> 00:02:02.940
discover the properties of all
"angular momentum," in quotes.
00:02:02.940 --> 00:02:05.340
Now we define an angular
momentum classically
00:02:05.340 --> 00:02:06.440
as r cross p.
00:02:09.267 --> 00:02:09.850
It's a vector.
00:02:16.620 --> 00:02:23.210
Now there are things, operators,
where there is no r or p
00:02:23.210 --> 00:02:26.860
but we would like to describe
them as angular momentum.
00:02:26.860 --> 00:02:30.040
One of them is electron spin,
something that we sort of take
00:02:30.040 --> 00:02:33.730
for granted, and nuclear spin.
00:02:33.730 --> 00:02:38.260
NMR is based on these things
we call angular momentum
00:02:38.260 --> 00:02:39.850
because they obey some rules.
00:02:42.620 --> 00:02:45.400
And so what I'm going
to do is show the rules
00:02:45.400 --> 00:02:48.060
and show where all
of this comes from
00:02:48.060 --> 00:02:55.530
and that this is an abstract
and so kind of dry derivation,
00:02:55.530 --> 00:03:00.486
but it has astonishing
consequences.
00:03:00.486 --> 00:03:03.910
And basically it
means that if you've
00:03:03.910 --> 00:03:09.065
got angular momenta, if
you know these rules,
00:03:09.065 --> 00:03:10.940
you're never going to
evaluate another matrix
00:03:10.940 --> 00:03:13.760
element in your life.
00:03:13.760 --> 00:03:16.820
Now, it has another
level of complexity.
00:03:16.820 --> 00:03:18.830
Sometimes you have
operators that
00:03:18.830 --> 00:03:23.440
are made out of combinations
of angular momenta,
00:03:23.440 --> 00:03:25.840
and you can use these
sorts of arguments
00:03:25.840 --> 00:03:28.720
to derive the matrix
elements of them.
00:03:28.720 --> 00:03:31.270
That's called the
Wigner-Eckart theorem,
00:03:31.270 --> 00:03:35.470
and it means that the angular
part of every operator
00:03:35.470 --> 00:03:39.550
is in your hands without ever
looking at a wave function
00:03:39.550 --> 00:03:42.270
or a differential operator.
00:03:42.270 --> 00:03:44.290
Now we're not going
to go there, but this
00:03:44.290 --> 00:03:47.470
is a very important area
of quantum mechanics.
00:03:47.470 --> 00:03:51.780
And you've heard of three j
symbols and Racah coefficients.
00:03:51.780 --> 00:03:52.580
Maybe you haven't.
00:03:52.580 --> 00:03:54.970
But there is just
a rich literature
00:03:54.970 --> 00:03:57.550
of this sort of stuff.
00:03:57.550 --> 00:04:02.100
So today I'm going
to talk briefly
00:04:02.100 --> 00:04:06.360
about rotational spectra
because I'm a spectroscopist.
00:04:06.360 --> 00:04:13.040
And from rotational spectra we
learn about molecular geometry.
00:04:13.040 --> 00:04:16.089
Now it's really strange
because why don't we just look
00:04:16.089 --> 00:04:18.025
at a molecule and measure it?
00:04:18.025 --> 00:04:20.140
Well, we can't
because it's smaller
00:04:20.140 --> 00:04:22.570
than the wavelength
of light that we
00:04:22.570 --> 00:04:25.330
would use to
illuminate our ruler,
00:04:25.330 --> 00:04:28.150
and so we get the
structure of a molecule,
00:04:28.150 --> 00:04:31.390
the geometric structure,
at least part of it,
00:04:31.390 --> 00:04:34.050
from the rotational spectrum.
00:04:34.050 --> 00:04:36.520
Now I'm only going to talk
about the rotational spectrum
00:04:36.520 --> 00:04:38.680
of a diatomic molecule.
00:04:38.680 --> 00:04:43.810
You don't want me to go further
because polyatomic molecules
00:04:43.810 --> 00:04:47.130
have extra complexity
which you could understand,
00:04:47.130 --> 00:04:49.180
but I don't want to go
there because we have
00:04:49.180 --> 00:04:52.720
a lot of other stuff to do.
00:04:52.720 --> 00:04:56.350
I also had promised to
talk about visualization
00:04:56.350 --> 00:04:59.200
of wave functions,
and I'll leave that
00:04:59.200 --> 00:05:02.270
to your previous experience.
00:05:02.270 --> 00:05:04.780
But I do want to comment.
00:05:04.780 --> 00:05:09.940
We often take a sum
of wave functions
00:05:09.940 --> 00:05:14.380
for positive and negative
projection quantum numbers
00:05:14.380 --> 00:05:18.820
to make them real
or pure imaginary.
00:05:18.820 --> 00:05:22.630
When we do that, they are still
eigenfunctions of the angular
00:05:22.630 --> 00:05:25.300
momentum of squared,
but they're not
00:05:25.300 --> 00:05:30.090
eigenfunctions of a projection.
00:05:30.090 --> 00:05:33.620
And you could-- in fact,
maybe you will on Thursday--
00:05:33.620 --> 00:05:39.740
actually evaluate Lz times
a symmetrized function.
00:05:43.060 --> 00:05:47.510
So by symmetrizing them, you
get to see the nodal structure,
00:05:47.510 --> 00:05:51.510
which is nice, but
you lose the fact
00:05:51.510 --> 00:05:56.070
that you have eigenfunctions
of a projection of angular
00:05:56.070 --> 00:06:00.170
momentum, and
that's kind of sad.
00:06:00.170 --> 00:06:15.720
OK, so spectra-- so we have
a diatomic molecule, mA,
00:06:15.720 --> 00:06:19.880
mB, and we have a center mass.
00:06:19.880 --> 00:06:23.760
And so we're going to be
interested in the energy
00:06:23.760 --> 00:06:30.060
levels for a rotation of the
molecule around that axis which
00:06:30.060 --> 00:06:31.785
is perpendicular
to the bond axis.
00:06:38.590 --> 00:06:46.680
When we do that, we discover
that the energy levels
00:06:46.680 --> 00:06:49.260
are given by the
rotational Hamiltonian.
00:06:49.260 --> 00:06:51.390
And for a rotation--
00:06:51.390 --> 00:06:56.850
it's free rotation, so
there's no potential.
00:06:56.850 --> 00:07:02.100
And the operator is L
squared or J squared.
00:07:02.100 --> 00:07:04.020
That's another thing.
00:07:04.020 --> 00:07:10.680
You already have experienced
my use of L and J and maybe
00:07:10.680 --> 00:07:12.242
some other things.
00:07:12.242 --> 00:07:13.450
They're all angular momentum.
00:07:13.450 --> 00:07:16.470
They're all the
same sort of thing,
00:07:16.470 --> 00:07:21.060
although L is usually referring
to electronic coordinates,
00:07:21.060 --> 00:07:24.690
and J is usually referring
to nuclear coordinates.
00:07:24.690 --> 00:07:26.670
Big deal.
00:07:26.670 --> 00:07:28.900
But you get a sense
that we're talking
00:07:28.900 --> 00:07:34.030
about a very rich idea
where it doesn't matter
00:07:34.030 --> 00:07:35.710
what you name the things.
00:07:35.710 --> 00:07:36.835
They follow the same rules.
00:07:40.210 --> 00:07:50.020
So we have a single term in
the Hamiltonian, mu r0 squared,
00:07:50.020 --> 00:07:54.010
or this might be the
equilibrium instead of
00:07:54.010 --> 00:07:56.177
just the fixed
internuclear distance.
00:07:56.177 --> 00:07:57.760
But we're talking
about a rigid rotor,
00:07:57.760 --> 00:08:02.850
so r0 is the
internuclear distance.
00:08:02.850 --> 00:08:11.595
And now I want to
be able to write--
00:08:18.520 --> 00:08:22.525
OK, so I want to
have this quantity.
00:08:25.120 --> 00:08:28.720
I want to have this quantity
in reciprocal centimeter units
00:08:28.720 --> 00:08:31.900
because that's what
all spectroscopists do,
00:08:31.900 --> 00:08:35.340
or sometimes they use megahertz.
00:08:35.340 --> 00:08:38.760
In that case, the
speed of light is gone.
00:08:38.760 --> 00:08:44.790
And when I evaluate the effect
of this operator on a wave
00:08:44.790 --> 00:08:48.280
function, we get an h bar
squared, which cancels that.
00:08:48.280 --> 00:08:54.870
We would like to have an
energy level expression
00:08:54.870 --> 00:09:09.570
EJM is equal to hcBL L plus 1.
00:09:09.570 --> 00:09:13.790
So the units of B just
accommodate the fact
00:09:13.790 --> 00:09:16.090
that we want it in wave numbers.
00:09:16.090 --> 00:09:20.270
But this is energy,
So we need the hc.
00:09:20.270 --> 00:09:25.520
And when the operator operates,
we get an h bar squared,
00:09:25.520 --> 00:09:30.980
and that's canceled
by this factor here.
00:09:30.980 --> 00:09:35.290
And so the handy
dandy expression
00:09:35.290 --> 00:09:46.300
for the rotational
constant is 16.85673 times
00:09:46.300 --> 00:09:53.180
the reduced mass in AMU
units times the internuclear
00:09:53.180 --> 00:10:02.540
distance in angstrom
units squared reciprocal.
00:10:02.540 --> 00:10:08.110
So if you want to
know the energy--
00:10:08.110 --> 00:10:11.950
if you want to know
the rotational constant
00:10:11.950 --> 00:10:15.490
in wave number units,
this is the conversion.
00:10:15.490 --> 00:10:16.240
Big deal.
00:10:28.690 --> 00:10:32.995
So the energy
levels are simply--
00:10:35.830 --> 00:10:39.830
I'm going to stick with LM even
though I'm hardwired to call it
00:10:39.830 --> 00:10:42.920
J. Now if I go back and
forth between J and L,
00:10:42.920 --> 00:10:46.970
you'll have to forgive
me because I just can't--
00:10:46.970 --> 00:10:47.960
yes.
00:10:47.960 --> 00:10:56.210
All right, so we have the energy
levels, hcB times L L plus 1.
00:10:56.210 --> 00:10:59.804
Now L is an integer,
and for the simple
00:10:59.804 --> 00:11:02.220
diatomics that you're going
to deal with, it's an integer.
00:11:02.220 --> 00:11:04.460
You can start at zero.
00:11:04.460 --> 00:11:15.650
And so the energy levels, the L
L plus 1, the L L plus 1 is 2.
00:11:15.650 --> 00:11:17.210
I want to make this look right.
00:11:20.550 --> 00:11:24.950
B-- this is 1 times 2.
00:11:24.950 --> 00:11:26.650
This is 2 times 3.
00:11:26.650 --> 00:11:30.210
And the 3 times 4 is 12.
00:11:30.210 --> 00:11:34.360
And the important thing is that
this energy differences is 2B.
00:11:37.580 --> 00:11:41.160
This energy difference is 4B.
00:11:41.160 --> 00:11:42.700
This energy difference is 6B.
00:11:45.550 --> 00:11:47.450
And so what happens
in the spectrum--
00:11:50.320 --> 00:11:52.390
here's energy.
00:11:52.390 --> 00:11:54.370
Here's zero.
00:11:54.370 --> 00:12:04.910
We have a line here at 2B,
a line here at 4B, 6B, 8B.
00:12:04.910 --> 00:12:08.960
So if you were able to look
at the rotational spectrum,
00:12:08.960 --> 00:12:13.660
the lines in the spectrum
would be evenly spaced.
00:12:13.660 --> 00:12:16.380
The levels are not.
00:12:16.380 --> 00:12:18.510
That's very
important, especially
00:12:18.510 --> 00:12:20.340
when you start doing
perturbation theory
00:12:20.340 --> 00:12:23.010
because you're going to have
energy denominators which
00:12:23.010 --> 00:12:25.480
are multiples of
a common factor,
00:12:25.480 --> 00:12:29.100
but they're not
equal to each other.
00:12:29.100 --> 00:12:34.600
But we have a spectrum, and it
looks really, really trivial.
00:12:34.600 --> 00:12:37.100
And textbooks don't
talk about this,
00:12:37.100 --> 00:12:40.310
but if you have a relatively
light diatomic molecule
00:12:40.310 --> 00:12:44.930
and you have a laboratory which
is equipped with a microwave
00:12:44.930 --> 00:12:49.220
spectrometer which is able
to generate data that got you
00:12:49.220 --> 00:12:52.520
tenure and whatever,
it's probably
00:12:52.520 --> 00:12:56.900
a spectrometer where the
tuning range of the microwave
00:12:56.900 --> 00:12:58.100
oscillator is about 30%.
00:13:00.989 --> 00:13:01.530
That's a lot.
00:13:01.530 --> 00:13:06.470
If you think about NMR,
the tuning range is--
00:13:06.470 --> 00:13:09.740
30% is huge.
00:13:09.740 --> 00:13:12.560
So, you see this and
you say, oh yeah.
00:13:12.560 --> 00:13:16.710
I could assign that spectrum
because an obvious pattern.
00:13:16.710 --> 00:13:19.580
But what happens in the
spectrum is you get one line.
00:13:23.310 --> 00:13:27.030
And so you say, well, I need
to know the internuclear
00:13:27.030 --> 00:13:33.230
distance of this molecule
to 6 or 8 or 10 digits,
00:13:33.230 --> 00:13:34.810
but I get one line.
00:13:34.810 --> 00:13:35.870
There's no pattern.
00:13:38.790 --> 00:13:40.810
The textbooks are
so full of formulas
00:13:40.810 --> 00:13:44.880
that they don't indicate
that, in reality, you've
00:13:44.880 --> 00:13:47.150
got a problem.
00:13:47.150 --> 00:13:48.730
And, in fact, in
reality you've got
00:13:48.730 --> 00:13:50.110
something that's also a gift.
00:13:53.250 --> 00:14:07.850
So there's two things that
happen, isotopes and vibration.
00:14:11.020 --> 00:14:12.470
So we have this one line.
00:14:12.470 --> 00:14:16.570
We have a very, very
strong, very narrow--
00:14:16.570 --> 00:14:19.180
you can measure the daylights
out of it if you wanted to.
00:14:22.170 --> 00:14:26.280
And then down here,
there's going to be--
00:14:26.280 --> 00:14:33.150
well, actually, sometimes
like in chlorine and bromine,
00:14:33.150 --> 00:14:36.720
there's a heavy isotope
and a light isotope,
00:14:36.720 --> 00:14:38.790
and they have
similar abundances.
00:14:38.790 --> 00:14:40.470
And so you get
isotope splittings,
00:14:40.470 --> 00:14:46.380
and that's expressed
in the reduced mass mA
00:14:46.380 --> 00:14:51.420
mB over mA plus mB.
00:14:51.420 --> 00:14:55.980
Now the isotope splittings
can be really, really small,
00:14:55.980 --> 00:15:02.000
but these lines have a width
of a part in a million,
00:15:02.000 --> 00:15:05.210
maybe even narrower.
00:15:05.210 --> 00:15:07.565
And so you can
see isotope stuff.
00:15:10.640 --> 00:15:12.710
That doesn't tell you
anything at all that
00:15:12.710 --> 00:15:17.270
you didn't know except
maybe that you were confused
00:15:17.270 --> 00:15:20.390
about what molecule it
was because if you have
00:15:20.390 --> 00:15:23.510
a particular atom,
it's always born
00:15:23.510 --> 00:15:27.950
with the normal isotope ratios.
00:15:27.950 --> 00:15:30.230
Except here we have
a little problem
00:15:30.230 --> 00:15:34.520
where, in sulfur, if
you look in minerals,
00:15:34.520 --> 00:15:38.990
the isotope ratios are
not the naturally abundant
00:15:38.990 --> 00:15:41.000
of sulfur isotope.
00:15:41.000 --> 00:15:43.580
And this has to do with
something really important that
00:15:43.580 --> 00:15:46.550
happened 2 and 1/2
billion years ago.
00:15:46.550 --> 00:15:48.350
Oxygen happened.
00:15:48.350 --> 00:15:56.450
And so isotope ratios are
of some geological chemical
00:15:56.450 --> 00:16:01.100
significance, but here, if
you know what the molecule is,
00:16:01.100 --> 00:16:03.980
there will be isotope lines.
00:16:03.980 --> 00:16:07.310
And they can be pretty
strong depending
00:16:07.310 --> 00:16:10.130
on the relative abundance
of the different isotopes,
00:16:10.130 --> 00:16:12.440
or they can be extremely weak.
00:16:12.440 --> 00:16:18.200
So there's stuff, so some grass
to be mowed on the baseline.
00:16:18.200 --> 00:16:20.270
In addition-- and
this is something
00:16:20.270 --> 00:16:22.970
that really surprises people.
00:16:22.970 --> 00:16:29.300
So here is v equals 0, and
way up high is v equals 1.
00:16:29.300 --> 00:16:32.780
Typically, the
vibrational intervals
00:16:32.780 --> 00:16:35.060
are on the order of a
thousand times bigger
00:16:35.060 --> 00:16:38.580
than the rotational intervals.
00:16:38.580 --> 00:16:43.050
And typically, the
rotational constant
00:16:43.050 --> 00:16:45.180
decreases in steps
of about a tenth
00:16:45.180 --> 00:16:49.040
of a percent per vibration.
00:16:49.040 --> 00:16:52.940
Now we do care about how much
it decreases because that allows
00:16:52.940 --> 00:16:54.800
us to know a whole
bunch of stuff
00:16:54.800 --> 00:17:00.490
about how rotation and
vibration interact.
00:17:00.490 --> 00:17:04.824
And I'm not probably going to
do the lecture on the rotation
00:17:04.824 --> 00:17:07.030
and vibration
interaction unless I
00:17:07.030 --> 00:17:09.280
have to give a lecture on
something that I can't do
00:17:09.280 --> 00:17:12.310
and I'll slip in that one.
00:17:12.310 --> 00:17:17.829
So what happens is there
are vibrational satellites.
00:17:17.829 --> 00:17:19.150
So here's v equals 0.
00:17:19.150 --> 00:17:21.290
It has rotational structure.
00:17:21.290 --> 00:17:22.420
And here is v equals 1.
00:17:22.420 --> 00:17:25.390
It has rotational structure.
00:17:25.390 --> 00:17:29.020
The v equals 1 stuff
is typically a hundred
00:17:29.020 --> 00:17:34.590
to a thousand times weaker
than the v equals 0 stuff.
00:17:34.590 --> 00:17:36.090
And that's basically
telling you,
00:17:36.090 --> 00:17:40.800
how does a molecule
changes its average 1
00:17:40.800 --> 00:17:44.880
over r squared as it vibrates,
and that's a useful thing.
00:17:44.880 --> 00:17:46.710
It may even be
useful on Thursday.
00:17:50.120 --> 00:17:54.140
So in addition to hyperfine,
there's other small stuff
00:17:54.140 --> 00:17:56.000
having to do with vibrations.
00:17:56.000 --> 00:17:59.120
And in some
experiments that I do,
00:17:59.120 --> 00:18:03.140
we use UV light to
break a molecule.
00:18:03.140 --> 00:18:07.910
And the fragments that we make
are born vibrationally excited.
00:18:07.910 --> 00:18:14.060
And so by looking at the stuff
near the v equals 0 frequency,
00:18:14.060 --> 00:18:15.890
you see a whole
bunch of stuff which
00:18:15.890 --> 00:18:20.090
tells you the populations of the
different vibrational levels.
00:18:20.090 --> 00:18:22.560
And that's strange because
vibration is not part
00:18:22.560 --> 00:18:24.180
of the rotational spectrum.
00:18:24.180 --> 00:18:27.330
Vibration is big, but we
get vibrational information
00:18:27.330 --> 00:18:29.220
from the rotational spectrum.
00:18:29.220 --> 00:18:32.220
And because the
rotational spectrum
00:18:32.220 --> 00:18:36.960
is at such high resolution,
it's trivial to resolve
00:18:36.960 --> 00:18:40.170
and to detect these
weak other features.
00:18:40.170 --> 00:18:43.410
So as much as I'm going to
talk about spectroscopy,
00:18:43.410 --> 00:18:47.520
it's a little bit more than
I had originally planned.
00:18:47.520 --> 00:18:51.180
And now we're going to move
to this topic which is dear
00:18:51.180 --> 00:18:56.160
to my heart, and it's an
example of an abstract algebra
00:18:56.160 --> 00:18:58.980
that you use in
quantum mechanics.
00:18:58.980 --> 00:19:02.350
And there are people who
only do this kind of thing
00:19:02.350 --> 00:19:05.430
as opposed to solving
Schrodinger equation
00:19:05.430 --> 00:19:11.220
or even just doing perturbation
theory on matrices.
00:19:11.220 --> 00:19:14.010
So the rest of
today's lecture is
00:19:14.010 --> 00:19:19.330
going to be an excursion through
here as much as I can do.
00:19:19.330 --> 00:19:22.320
It's all clear in the
notes, but I think
00:19:22.320 --> 00:19:24.450
it's a little bit strange.
00:19:24.450 --> 00:19:25.950
Oh, I want to say
one more thing.
00:19:25.950 --> 00:19:27.960
How do we make assignments?
00:19:27.960 --> 00:19:33.252
You all took 5.111
or 5.112 or 3.091,
00:19:33.252 --> 00:19:39.080
and there are things that you
learn about how big atoms are.
00:19:39.080 --> 00:19:41.650
And so you can sort
of estimate what
00:19:41.650 --> 00:19:44.500
the internuclear distance is--
00:19:44.500 --> 00:19:46.750
maybe to 10% or 20%.
00:19:46.750 --> 00:19:49.660
That's not of any
chemical use, but it's
00:19:49.660 --> 00:19:52.370
enough to assign the spectrum.
00:19:52.370 --> 00:19:57.000
So what you do is you say OK, I
guess the internuclear distance
00:19:57.000 --> 00:19:57.730
is this.
00:19:57.730 --> 00:20:01.900
That determines what rotational
transition you were observing.
00:20:01.900 --> 00:20:04.110
And that has
consequences of suppose
00:20:04.110 --> 00:20:06.970
you're observing L to L plus 1.
00:20:06.970 --> 00:20:11.060
Well, what about L plus to
L plus 2 or L minus 1 to L?
00:20:11.060 --> 00:20:13.190
So if you make an
assignment, you
00:20:13.190 --> 00:20:16.370
can predict where
the other guys are.
00:20:16.370 --> 00:20:19.460
And that would require
going to one of your friends
00:20:19.460 --> 00:20:21.680
who has a different
spectrometer and getting
00:20:21.680 --> 00:20:23.420
him to record a
spectrum for you,
00:20:23.420 --> 00:20:26.000
and that's good for
human relations.
00:20:26.000 --> 00:20:30.050
And that then enables you
to make assignments and know
00:20:30.050 --> 00:20:32.570
the rotational constants
to as many digits
00:20:32.570 --> 00:20:37.480
as you possibly could want,
includ-- all the way up to 10.
00:20:37.480 --> 00:20:38.500
It's just crazy.
00:20:38.500 --> 00:20:42.160
You really don't care about
internuclear distances
00:20:42.160 --> 00:20:44.350
beyond about a thousandth
of an angstrom,
00:20:44.350 --> 00:20:45.400
but you can have them.
00:20:49.810 --> 00:20:54.100
So first of all, you know
we can define an angular
00:20:54.100 --> 00:21:00.985
momentum as r cross p, and we
can write that as a matrix.
00:21:12.150 --> 00:21:14.340
Now I suspect you've
all seen this.
00:21:14.340 --> 00:21:17.980
These are unit vectors along
the x, y, and z directions.
00:21:17.980 --> 00:21:23.040
And this is a vector, so
there are three components,
00:21:23.040 --> 00:21:25.540
and we get three
components here.
00:21:25.540 --> 00:21:28.142
Now you do want to make
sure you know this notation
00:21:28.142 --> 00:21:29.100
and know how to use it.
00:21:34.360 --> 00:21:39.020
So here is the magic equation.
00:21:39.020 --> 00:21:51.606
Li, Lj is equal to ih bar
sum over k epsilon ijk Lk.
00:21:51.606 --> 00:21:57.540
Well, what is epsilon ijk?
00:21:57.540 --> 00:21:59.580
Well, it's got many
names, but it's
00:21:59.580 --> 00:22:03.900
a really neat tool which is
very wonderful in enabling
00:22:03.900 --> 00:22:07.900
you to derive new equations.
00:22:07.900 --> 00:22:13.660
So if i, j, and k correspond
to xyz in cyclic order--
00:22:13.660 --> 00:22:17.790
in other words, xyz,
yzx, et cetera--
00:22:17.790 --> 00:22:20.790
then this is plus 1.
00:22:20.790 --> 00:22:24.340
If it's in anticyclic
order, it's minus 1.
00:22:24.340 --> 00:22:27.720
And if any index is
repeated, it's 0.
00:22:27.720 --> 00:22:31.050
So it packs a real
punch, but it enables
00:22:31.050 --> 00:22:33.300
you to do fantastic things.
00:22:33.300 --> 00:22:40.170
So if we have Lx, Ly, it's
equal to ih bar plus 1 times Lz.
00:22:48.400 --> 00:22:52.510
And the point of this
lecture is with this,
00:22:52.510 --> 00:22:56.800
you can derive all of the
matrix elements of an angular
00:22:56.800 --> 00:22:57.400
momentum--
00:22:57.400 --> 00:23:04.610
L squared, Lz, L plus
minus, and anything else.
00:23:04.610 --> 00:23:06.214
But these are the
important ones,
00:23:06.214 --> 00:23:07.630
and this is what
we want to derive
00:23:07.630 --> 00:23:12.120
from our excursion in
matrix element land.
00:23:15.160 --> 00:23:18.220
So the first thing
we do is we extract
00:23:18.220 --> 00:23:22.900
some fundamental equations
from this commutator.
00:23:31.020 --> 00:23:37.990
So the first equation
is that L squared
00:23:37.990 --> 00:23:55.140
Lz is equal to Lx squared Lz
plus Ly squared Lz plus Lz
00:23:55.140 --> 00:23:56.730
squared Lz.
00:24:01.240 --> 00:24:04.910
And we know this
one is 0, right?
00:24:04.910 --> 00:24:06.890
This one, you have to
do a little practice,
00:24:06.890 --> 00:24:13.320
but you can write this
commutation rule as Lx times Lx
00:24:13.320 --> 00:24:22.590
comma Lz plus Lx comma Lz Lx.
00:24:22.590 --> 00:24:26.610
So if you have a square, you
take it out the front side
00:24:26.610 --> 00:24:27.810
then the back side.
00:24:27.810 --> 00:24:29.565
And now we know what this.
00:24:29.565 --> 00:24:31.630
This is minus ih bar Ly.
00:24:34.390 --> 00:24:40.810
And this is minus ih bar Ly.
00:24:45.130 --> 00:24:48.310
And we do this one, and we
discover we have the same thing
00:24:48.310 --> 00:24:52.030
except with the opposite sign.
00:24:52.030 --> 00:24:56.820
And so what we end up
getting is that this is 0.
00:24:56.820 --> 00:24:58.480
Now I skipped some steps.
00:24:58.480 --> 00:25:04.030
I said them, but I want you to
just go through that and see.
00:25:04.030 --> 00:25:06.770
So you know what this is.
00:25:06.770 --> 00:25:09.130
It's going to be
Ly, and it's going
00:25:09.130 --> 00:25:12.490
to be minus Ly times ih bar.
00:25:12.490 --> 00:25:14.140
And you get the same thing here.
00:25:14.140 --> 00:25:17.340
But then you have an LxLy,
and you have an LxLy.
00:25:24.080 --> 00:25:26.990
And when you do the
same trick with this,
00:25:26.990 --> 00:25:30.110
you're going to get
an Ly and an Lx again,
00:25:30.110 --> 00:25:32.360
and they'll be
the opposite sign.
00:25:32.360 --> 00:25:39.450
So this one is really
important because what it says
00:25:39.450 --> 00:25:47.790
is that you can take any
projection quantum number
00:25:47.790 --> 00:25:50.770
and it will commute with
the magnitude squared.
00:25:50.770 --> 00:25:57.740
The same argument works
for Ly and Lz and Lx.
00:25:57.740 --> 00:26:01.130
So we have one really
powerful commutator
00:26:01.130 --> 00:26:13.940
which is that L squared Li
equals 0 for x, y, and z,
00:26:13.940 --> 00:26:19.070
which means since we
like L squared and Lz--
00:26:19.070 --> 00:26:21.890
we could add like
Lx instead of Lz,
00:26:21.890 --> 00:26:24.530
but we tend to favor these--
00:26:24.530 --> 00:26:31.070
that L squared and
Lz are operators
00:26:31.070 --> 00:26:35.430
that can have a common
set of eigenfunctions.
00:26:35.430 --> 00:26:38.090
If we have two
operators that commute,
00:26:38.090 --> 00:26:40.790
the eigenfunctions of one
can be the eigenfunctions
00:26:40.790 --> 00:26:41.360
of the other.
00:26:41.360 --> 00:26:42.440
Very convenient.
00:26:46.760 --> 00:26:49.580
Then there's another
operator that we can derive,
00:26:49.580 --> 00:26:50.270
and that is--
00:26:55.430 --> 00:26:59.680
let's define this
thing, a step up
00:26:59.680 --> 00:27:04.450
or step down or our raising
or lowering operator--
00:27:04.450 --> 00:27:06.340
we don't know that yet--
00:27:06.340 --> 00:27:07.960
Lx plus or minus iLy.
00:27:17.620 --> 00:27:22.360
So we might want to know
the commutation rule of Lz
00:27:22.360 --> 00:27:23.350
with L plus minus.
00:27:27.430 --> 00:27:34.010
We know how to write this out
because we have Lz with Lx,
00:27:34.010 --> 00:27:37.270
and we know that's going
to be a minus ih bar Ly.
00:27:40.340 --> 00:27:49.390
And we have Lz with iLy, and
that's going to be a minus Lx.
00:27:49.390 --> 00:27:51.885
Anyway, I'm going to just
write down the final result,
00:27:51.885 --> 00:28:00.290
that this is equal to plus or
minus h bar times L plus minus.
00:28:04.100 --> 00:28:09.050
The algebra of this
operator enables
00:28:09.050 --> 00:28:11.870
you to slice through
any derivation
00:28:11.870 --> 00:28:14.960
as fast as you can write
once you've loaded this
00:28:14.960 --> 00:28:16.190
into your head.
00:28:16.190 --> 00:28:16.925
Yes?
00:28:16.925 --> 00:28:19.730
AUDIENCE: So for the epsilon,
how do you [INAUDIBLE]??
00:28:19.730 --> 00:28:21.740
Is it like xy becomes--
00:28:21.740 --> 00:28:23.380
if it's cyclical it's positive?
00:28:23.380 --> 00:28:24.380
ROBERT FIELD: I'm sorry?
00:28:24.380 --> 00:28:28.590
AUDIENCE: When you say the
epsilon thing, epsilon ijk,
00:28:28.590 --> 00:28:32.164
so you're saying that if
it's in order, it's 1?
00:28:32.164 --> 00:28:34.080
ROBERT FIELD: Let's just
do this a little bit.
00:28:34.080 --> 00:28:38.720
Let's say we have Lx and Ly.
00:28:38.720 --> 00:28:42.680
Well, we know that
that's going to give Lz.
00:28:42.680 --> 00:28:49.520
And xyz, ijk,
that's cyclic order.
00:28:49.520 --> 00:28:52.340
We say that's the home base.
00:28:52.340 --> 00:28:59.004
And if we have yxz, that
would be anticyclic,
00:28:59.004 --> 00:29:00.420
and so that would
be a minus sign.
00:29:00.420 --> 00:29:02.070
You know that just
by looking at this,
00:29:02.070 --> 00:29:04.860
and you say if we switch this,
the sign of the commutator
00:29:04.860 --> 00:29:07.140
has to switch.
00:29:07.140 --> 00:29:09.850
There's a lot of
stuff loaded in there.
00:29:09.850 --> 00:29:12.820
And once you've sort
of processed it,
00:29:12.820 --> 00:29:14.550
it becomes automatic.
00:29:14.550 --> 00:29:15.850
You forget the beauty of it.
00:29:18.560 --> 00:29:21.800
So are you satisfied?
00:29:21.800 --> 00:29:23.580
Everybody else?
00:29:23.580 --> 00:29:26.120
All right, so now
let's do another one.
00:29:30.120 --> 00:29:33.160
Let's look at L
squared L plus minus.
00:29:36.760 --> 00:29:40.210
Well, this one is super
easy because we already
00:29:40.210 --> 00:29:44.440
know that L squared commutes
with Lx, Ly, and Lz.
00:29:44.440 --> 00:29:46.810
So I just need to
just write 0 here
00:29:46.810 --> 00:29:50.710
because this is Lx
plus or minus iLy,
00:29:50.710 --> 00:29:53.270
and we know L squared
commutes with both of them.
00:30:02.930 --> 00:30:05.915
Now comes the abstract
and weird stuff.
00:30:10.970 --> 00:30:13.550
We're starting to use
the commutators to derive
00:30:13.550 --> 00:30:15.470
the matrix elements
and selection rules.
00:30:18.180 --> 00:30:23.760
So let us say that we
have some function which
00:30:23.760 --> 00:30:28.440
is an eigenfunction
of L squared and Lz.
00:30:28.440 --> 00:30:33.330
And so we're entitled to
say that L squared operating
00:30:33.330 --> 00:30:40.520
on this function gives an
eigenvalue we call lambda.
00:30:40.520 --> 00:30:44.920
And we can also say that Lz
operating on the function
00:30:44.920 --> 00:30:49.020
gives a different concept mu.
00:30:49.020 --> 00:30:52.190
Now this lambda and mu
have no significance.
00:30:52.190 --> 00:30:53.940
They're just numbers.
00:30:53.940 --> 00:30:56.440
There's not something that's
going to pop up here that says,
00:30:56.440 --> 00:30:59.114
oh yeah, this means something.
00:31:02.290 --> 00:31:06.580
So now we're going to use the
fact that this function, which
00:31:06.580 --> 00:31:10.000
we're allowed to have as a
simultaneous eigenfunction of L
00:31:10.000 --> 00:31:15.780
squared and Lz with its own set
of eigenvalues, this function,
00:31:15.780 --> 00:31:20.260
we are going to operate on it
and derive some useful results
00:31:20.260 --> 00:31:23.590
that all are based on
the commutation rules.
00:31:29.060 --> 00:31:35.390
So let us take L squared
operating on L plus minus
00:31:35.390 --> 00:31:36.330
times f.
00:31:39.520 --> 00:31:50.510
And we know that L plus minus
commutes with L squared.
00:31:50.510 --> 00:31:55.420
So we can write L plus
minus times L squared f.
00:31:58.200 --> 00:32:02.720
But L squared operating
on f gives lambda.
00:32:02.720 --> 00:32:05.620
We have L plus minus lambda f.
00:32:13.990 --> 00:32:16.110
Oh, isn't that interesting?
00:32:16.110 --> 00:32:24.256
We have-- I'll just write it--
lambda times L plus minus f--
00:32:24.256 --> 00:32:27.220
L plus minus f.
00:32:27.220 --> 00:32:34.150
So it's saying that this
thing is an eigenfunction of L
00:32:34.150 --> 00:32:38.219
squared with eigenvalue lambda.
00:32:38.219 --> 00:32:39.010
Well, we knew that.
00:32:48.040 --> 00:32:58.340
So L plus minus
operating on f does not
00:32:58.340 --> 00:33:05.040
change lambda, the
eigenvalue of L squared.
00:33:21.640 --> 00:33:23.850
Now let's use another one.
00:33:23.850 --> 00:33:28.732
Let's use Lz L plus minus.
00:33:28.732 --> 00:33:29.990
Well, I derived it.
00:33:29.990 --> 00:33:36.626
It's plus or minus h
bar times L plus minus.
00:33:36.626 --> 00:33:38.470
And if I didn't derive
it, I should have,
00:33:38.470 --> 00:33:40.190
but I'm pretty sure I did.
00:33:40.190 --> 00:33:42.850
And so now what
we can do is write
00:33:42.850 --> 00:33:48.880
Lz L plus minus
minus L plus minus Lz
00:33:48.880 --> 00:33:53.550
is equal to h plus or
minus h bar L plus minus.
00:34:10.400 --> 00:34:16.514
Let's stick in a function
on the right, f, f, f.
00:34:19.820 --> 00:34:24.260
So now we have these operators
operating on the same function.
00:34:27.900 --> 00:34:32.100
Well, we don't yet know
what L plus minus does to f,
00:34:32.100 --> 00:34:35.260
but we know what Lz does to it.
00:34:35.260 --> 00:34:38.130
And so what we can
write immediately
00:34:38.130 --> 00:34:43.620
is that Lz operating
on L plus minus
00:34:43.620 --> 00:34:51.679
f is equal to plus
or minus h bar
00:34:51.679 --> 00:35:01.850
L plus minus f plus
mu L plus minus f.
00:35:01.850 --> 00:35:04.560
Well, that's interesting.
00:35:04.560 --> 00:35:09.000
So we see that we
can rearrange this,
00:35:09.000 --> 00:35:15.060
and we could write plus
minus h bar L plus minus f is
00:35:15.060 --> 00:35:25.520
equal to mu L plus minus
f plus Lz L plus minus f--
00:35:35.747 --> 00:35:37.610
that's h bar, OK.
00:35:40.190 --> 00:35:42.910
Oh, I'm sorry, L
plus minus f there.
00:35:46.476 --> 00:35:47.970
So what's this telling us?
00:35:56.540 --> 00:35:58.430
So we can simply
combine these terms.
00:35:58.430 --> 00:36:00.020
We have the L plus minus f here.
00:36:05.030 --> 00:36:12.650
And so we can write mu plus
h bar times L plus minus f.
00:36:12.650 --> 00:36:15.680
That's the point.
00:36:15.680 --> 00:36:18.624
So we have this
operator operating--
00:36:22.206 --> 00:36:24.330
AUDIENCE: I don't think
you want the whole second--
00:36:24.330 --> 00:36:24.840
ROBERT FIELD: I'm sorry?
00:36:24.840 --> 00:36:26.839
AUDIENCE: The first line
goes straight to there.
00:36:26.839 --> 00:36:28.722
I think your second
line's [INAUDIBLE]..
00:36:34.720 --> 00:36:37.780
ROBERT FIELD: I took
this thing over to here.
00:36:37.780 --> 00:36:39.730
So let's just
rewrite that again.
00:36:39.730 --> 00:36:47.030
We have Lz L plus minus
f is equal to this.
00:36:47.030 --> 00:36:52.300
And so here we have
mu, the eigenvalue,
00:36:52.300 --> 00:36:55.550
and it's been
increased by h bar.
00:36:55.550 --> 00:36:59.390
And so what that
tells us is that we
00:36:59.390 --> 00:37:03.120
have a manifold of levels--
00:37:03.120 --> 00:37:10.770
mu, et cetera.
00:37:10.770 --> 00:37:13.710
So we get a manifold of levels
that are equally spaced,
00:37:13.710 --> 00:37:16.620
spaced by h bar.
00:37:19.590 --> 00:37:23.190
AUDIENCE: I think it also should
be plus or minus h bar, right?
00:37:26.190 --> 00:37:28.730
ROBERT FIELD: Plus minus h bar--
00:37:28.730 --> 00:37:29.230
yeah.
00:37:32.670 --> 00:37:36.160
So we have this
manifold of levels,
00:37:36.160 --> 00:37:40.110
and so what we can say is, well,
this isn't going to go forever.
00:37:43.290 --> 00:37:47.240
This is a ladder of
equally spaced levels,
00:37:47.240 --> 00:37:52.020
and it will have a highest
and a lowest member.
00:37:52.020 --> 00:38:03.380
And so we can say, all right,
well, suppose we have f max mu,
00:38:03.380 --> 00:38:06.400
and we have L plus
operating on it.
00:38:06.400 --> 00:38:07.410
That's going to give 0.
00:38:10.280 --> 00:38:15.186
And at the same time we can
say we have L minus min mu
00:38:15.186 --> 00:38:16.310
and that's going to give 0.
00:38:16.310 --> 00:38:17.730
We're going to
use both of these.
00:38:23.030 --> 00:38:26.606
Now I'm just going
to leave that there.
00:38:26.606 --> 00:38:27.670
Oh, I'm not.
00:38:27.670 --> 00:38:34.130
I'm going to say, all
right, so since we
00:38:34.130 --> 00:38:35.420
have this arrangement--
00:38:41.570 --> 00:38:43.050
all right, I am
skipping something,
00:38:43.050 --> 00:38:44.216
and I don't want to skip it.
00:38:47.650 --> 00:38:57.790
So if we have L plus operating
on the maximum value of mu,
00:38:57.790 --> 00:38:59.080
we get 0.
00:38:59.080 --> 00:39:04.690
And the next one down is down
by an integer number of L,
00:39:04.690 --> 00:39:15.630
and so we can say that
Lz operating on f max mu
00:39:15.630 --> 00:39:19.260
is equal to h bar
L, some integer.
00:39:19.260 --> 00:39:24.010
Now this L is chosen
with some prejudice.
00:39:24.010 --> 00:39:24.550
Yes?
00:39:24.550 --> 00:39:26.480
AUDIENCE: Why is
there an f of x?
00:39:31.610 --> 00:39:33.990
ROBERT FIELD: Now
I have to cheat.
00:39:33.990 --> 00:39:39.740
I'm going to apply an
argument which is not
00:39:39.740 --> 00:39:42.564
based on just abstract vectors.
00:39:42.564 --> 00:39:43.730
We have an angular momentum.
00:39:43.730 --> 00:39:46.930
It has a certain length.
00:39:46.930 --> 00:39:50.920
We know the projection of that
angular momentum on some axis
00:39:50.920 --> 00:39:52.540
cannot be longer
than its length.
00:39:57.144 --> 00:39:59.060
I mean, I'm uncomfortable
making that argument
00:39:59.060 --> 00:40:04.100
because I should be able to
say it in a more abstract way,
00:40:04.100 --> 00:40:05.240
but this is, in fact--
00:40:08.810 --> 00:40:12.080
we know there cannot be an
infinite number of projection
00:40:12.080 --> 00:40:16.640
quantum numbers, values of the
projection quantum number that
00:40:16.640 --> 00:40:21.660
aren't reached by applying
L plus and L minus.
00:40:21.660 --> 00:40:23.450
It must be limited.
00:40:23.450 --> 00:40:33.260
And so we're going to call the
maximum value of mu h bar L
00:40:33.260 --> 00:40:36.470
or L.
00:40:36.470 --> 00:40:39.260
Now I have to derive
a new commutation
00:40:39.260 --> 00:40:42.090
rule based on the original one.
00:40:42.090 --> 00:40:43.820
No, let's not erase this.
00:40:43.820 --> 00:40:45.220
We might want to see it again.
00:40:49.200 --> 00:40:54.400
So let's ask, well, what does
this combination of operators
00:40:54.400 --> 00:40:54.900
do?
00:41:07.980 --> 00:41:11.640
Well, this is surely
equal to Lx squared,
00:41:11.640 --> 00:41:16.140
and we get a plus
i and a minus i,
00:41:16.140 --> 00:41:20.620
and so it's going to
be plus Ly squared.
00:41:20.620 --> 00:41:29.180
And then we get i times
LyLx, and we get a minus i
00:41:29.180 --> 00:41:32.100
times LxLy.
00:41:41.710 --> 00:41:45.770
This is L squared
minus Lx squared.
00:41:45.770 --> 00:41:49.750
We have the square root of
2 components of L squared,
00:41:49.750 --> 00:41:53.160
and so this is equal
to the difference.
00:41:56.100 --> 00:42:04.680
And now we express
this as i times LyLx.
00:42:09.180 --> 00:42:10.926
And what is this?
00:42:10.926 --> 00:42:15.778
This is plus ih bar Lx.
00:42:21.030 --> 00:42:23.830
AUDIENCE: I think you
wrote an x [INAUDIBLE]..
00:42:23.830 --> 00:42:33.460
ROBERT FIELD: OK, this
is, yes, x, and that's Lz.
00:42:33.460 --> 00:42:35.860
I didn't like what
I wrote because I
00:42:35.860 --> 00:42:41.980
want to have everything
but the z and the L squared
00:42:41.980 --> 00:42:45.850
disappearing, and so
we get that we have
00:42:45.850 --> 00:42:49.600
L squared minus Lz squared.
00:42:49.600 --> 00:42:55.020
And then we have plus ih bar Lz.
00:42:58.606 --> 00:43:00.550
I lost the plus and minus.
00:43:00.550 --> 00:43:01.540
No I didn't.
00:43:01.540 --> 00:43:02.590
OK, that's it.
00:43:06.310 --> 00:43:10.540
And so we can rearrange
this and say L squared
00:43:10.540 --> 00:43:22.930
is equal to Lz squared minus
or plus h bar Lz plus L
00:43:22.930 --> 00:43:25.230
plus minus L minus plus.
00:43:28.390 --> 00:43:32.350
So we can use this equation--
00:43:32.350 --> 00:43:34.770
OK, I'd better not--
00:43:34.770 --> 00:43:36.260
to derive some good stuff.
00:43:36.260 --> 00:43:39.560
I better erase some
stuff or access a board.
00:43:42.835 --> 00:43:44.460
We're actually pretty
close to the end,
00:43:44.460 --> 00:43:47.800
so I might actually finish this.
00:43:47.800 --> 00:43:50.610
So we're going to use
this equation to find--
00:43:50.610 --> 00:43:57.300
so we want lambda, the value
of lambda for the top rung
00:43:57.300 --> 00:43:59.890
of the manifold over here.
00:43:59.890 --> 00:44:07.740
So we apply L
squared to f max mu.
00:44:07.740 --> 00:44:12.780
And we know we have
an equation here
00:44:12.780 --> 00:44:17.100
which enables us to evaluate
what the consequences of that
00:44:17.100 --> 00:44:19.900
will be, and it
will be Lz squared
00:44:19.900 --> 00:44:35.460
f max mu minus and plus
h bar Lz f max mu plus L
00:44:35.460 --> 00:44:41.864
plus L minus f max mu.
00:44:45.650 --> 00:44:51.170
So if we take the bottom
sign, that L plus on f max
00:44:51.170 --> 00:44:53.130
is going to give 0.
00:44:53.130 --> 00:44:57.060
So we're looking at the bottom
sign, and we have a 0 here,
00:44:57.060 --> 00:45:07.960
and so we have L squared
f max mu is equal to Lz
00:45:07.960 --> 00:45:21.110
squared f max mu minus or
plus h bar Lz f max mu plus 0.
00:45:21.110 --> 00:45:22.660
Isn't that interesting?
00:45:26.570 --> 00:45:31.670
So we know that Lz
is going to give--
00:45:31.670 --> 00:45:43.050
so we're going to get an h bar
squared and mu max squared.
00:45:43.050 --> 00:45:49.140
We're going to get a
minus h bar h bar mu max.
00:45:56.640 --> 00:46:01.020
So what this is telling us is
that L squared operating on f
00:46:01.020 --> 00:46:13.630
max mu is given by l
because we said that we're
00:46:13.630 --> 00:46:20.300
going to take the maximum
value of mu to be h bar L.
00:46:20.300 --> 00:46:24.590
So I shouldn't have had
an extra h bar here.
00:46:24.590 --> 00:46:31.280
So we get this
result. So lambda--
00:46:34.470 --> 00:46:37.930
so L squared operating
on this gives
00:46:37.930 --> 00:46:41.890
the-- oh yeah, maximum mu.
00:46:41.890 --> 00:46:49.260
So it's telling
us that L squared
00:46:49.260 --> 00:46:57.020
f max mu is equal to h
bar squared l l plus 1.
00:46:57.020 --> 00:47:00.290
Now that is why we chose
that constant to be l.
00:47:06.630 --> 00:47:09.900
And we do a similar argument for
the lowest rung of the ladder.
00:47:15.190 --> 00:47:19.070
And for the lowest
rung of the ladder,
00:47:19.070 --> 00:47:21.860
we know there must
be a lowest rung,
00:47:21.860 --> 00:47:27.090
and so we will simply
say, OK, for the lowest
00:47:27.090 --> 00:47:37.550
rung of the ladder we're going
to get that mu is equal to h
00:47:37.550 --> 00:47:43.730
bar lambda bar mu min.
00:47:43.730 --> 00:47:47.570
And we do some stuff,
and we discover
00:47:47.570 --> 00:47:54.260
that lambda has to be equal
to h bar squared l l plus 1.
00:47:54.260 --> 00:47:58.940
And using this other
relationship and the top sign,
00:47:58.940 --> 00:48:03.747
we get h bar squared
l bar l bar minus 1.
00:48:08.220 --> 00:48:10.180
And there's two
ways to solve this.
00:48:10.180 --> 00:48:13.570
One is that l is
equal to minus l bar,
00:48:13.570 --> 00:48:18.790
and the other is that l
bar is equal to l plus 1.
00:48:25.290 --> 00:48:28.700
Well, this is the lowest
rung of the ladder.
00:48:34.072 --> 00:48:36.590
Wait a minute, let
me just make sure I'm
00:48:36.590 --> 00:48:39.420
doing the logic correctly.
00:48:39.420 --> 00:48:42.580
It's this one, OK, here.
00:48:42.580 --> 00:48:44.410
So this is the lowest
rung of the ladder,
00:48:44.410 --> 00:48:51.400
and l bar is supposedly
larger than l.
00:48:51.400 --> 00:48:53.960
Can't be, so this is impossible.
00:48:53.960 --> 00:48:55.180
This is correct.
00:48:55.180 --> 00:49:00.590
And what we end up getting
is this relationship,
00:49:00.590 --> 00:49:03.410
and so mu can be equal to--
00:49:03.410 --> 00:49:09.700
and this is l l minus
1 minus l stepped to 1.
00:49:14.670 --> 00:49:17.370
This seems very weird
and not very interesting
00:49:17.370 --> 00:49:22.200
until you say, well,
how do I satisfy this?
00:49:22.200 --> 00:49:24.435
Well, if l is an
integer, it's obvious.
00:49:27.030 --> 00:49:30.200
If l is a half integer, it
shouldn't be quite so obvious,
00:49:30.200 --> 00:49:31.540
but it's true.
00:49:31.540 --> 00:49:37.150
So we can have integer
l and half-integer l.
00:49:42.780 --> 00:49:44.690
That's weird.
00:49:44.690 --> 00:49:47.270
We can show no connection
between the integer
00:49:47.270 --> 00:49:48.685
l's and the half-integer l's.
00:49:51.220 --> 00:49:53.770
They belong to completely
different problems,
00:49:53.770 --> 00:49:56.980
but this abstract
argument says, yeah,
00:49:56.980 --> 00:50:00.430
we can have integer l's
and half-integer l's.
00:50:00.430 --> 00:50:05.160
And if we have electron--
00:50:05.160 --> 00:50:09.280
well, we call it electron
spin because we want
00:50:09.280 --> 00:50:11.460
it to be an angular momentum.
00:50:11.460 --> 00:50:15.580
Spin is sort of an angular
momentum or nuclear spin.
00:50:15.580 --> 00:50:19.120
And we discover that there
are patterns of energy levels
00:50:19.120 --> 00:50:22.660
which enable us to count
the number of projection
00:50:22.660 --> 00:50:23.410
components.
00:50:26.210 --> 00:50:29.530
And if you have an
integer l, you'll
00:50:29.530 --> 00:50:37.580
get 2l plus 1 components,
which is an odd number.
00:50:37.580 --> 00:50:39.170
And if it's a half
integer you get
00:50:39.170 --> 00:50:42.290
2l plus 1 components,
which is an even number.
00:50:45.500 --> 00:50:48.290
And so it turns out that
our definition of an angular
00:50:48.290 --> 00:50:51.110
momentum is more
general than we thought.
00:50:51.110 --> 00:50:53.270
It allows there
to be both integer
00:50:53.270 --> 00:50:56.870
and half-integer
angular momentum.
00:50:56.870 --> 00:51:00.740
And this means we can
have angular momenta where
00:51:00.740 --> 00:51:04.550
we can't define it in
terms of r cross b.
00:51:04.550 --> 00:51:06.500
It's defined by the
commutation rule.
00:51:06.500 --> 00:51:08.380
It's more general.
00:51:08.380 --> 00:51:09.280
It's more abstract.
00:51:09.280 --> 00:51:11.670
It's beautiful.
00:51:11.670 --> 00:51:18.720
And I don't have time to finish
the job, but in the notes
00:51:18.720 --> 00:51:20.880
you can see that we
can derive the matrix
00:51:20.880 --> 00:51:23.370
elements for the raising
and lowering operators too.
00:51:23.370 --> 00:51:26.430
And the angular
momentum matrix elements
00:51:26.430 --> 00:51:30.300
are that L plus minus
operating on a function
00:51:30.300 --> 00:51:33.660
gives this combination
square root,
00:51:33.660 --> 00:51:36.790
and it raises or lowers m.
00:51:36.790 --> 00:51:39.940
So it's sort of like what we
have for the a's and a daggers
00:51:39.940 --> 00:51:46.630
for the harmonic oscillator,
but it's not as good because you
00:51:46.630 --> 00:51:48.970
can't generate all the L's.
00:51:48.970 --> 00:51:52.450
You can generate the m sub L's.
00:51:52.450 --> 00:51:56.870
And that's great, but there's
still something that remains
00:51:56.870 --> 00:52:00.050
to be done to generate
the different L's.
00:52:00.050 --> 00:52:02.100
That's not a problem.
00:52:02.100 --> 00:52:05.480
It's just there's not
a simple way to do it,
00:52:05.480 --> 00:52:07.170
at least not simple to me.
00:52:07.170 --> 00:52:12.890
And so now anytime we're
faced with a problem involving
00:52:12.890 --> 00:52:15.080
angular momenta,
we have a prayer
00:52:15.080 --> 00:52:19.120
of writing down the matrix
elements without ever looking
00:52:19.120 --> 00:52:23.400
at the wave function, without
ever looking at a differential
00:52:23.400 --> 00:52:24.422
operator.
00:52:27.134 --> 00:52:29.870
And we can also say,
well, let's suppose
00:52:29.870 --> 00:52:41.100
we had some operator
that involves L and S,
00:52:41.100 --> 00:52:43.380
now that we know that
we have these things,
00:52:43.380 --> 00:52:46.470
and L plus S can be
called J. So now we
00:52:46.470 --> 00:52:49.140
have two different operators,
two different angular momenta.
00:52:49.140 --> 00:52:54.090
We have S and we have
the total of J. Well,
00:52:54.090 --> 00:52:55.270
they're all angular momenta.
00:52:55.270 --> 00:52:58.620
They're going to satisfy their
selection rules and the matrix
00:52:58.620 --> 00:53:02.670
elements, and we
can calculate all
00:53:02.670 --> 00:53:06.870
of these matrix elements,
including things like L dot S
00:53:06.870 --> 00:53:09.180
and whatever.
00:53:09.180 --> 00:53:12.780
So it just opens up
a huge area where
00:53:12.780 --> 00:53:14.820
before you would
say, well, I got
00:53:14.820 --> 00:53:16.224
to look at the wave function.
00:53:16.224 --> 00:53:17.640
I've got to look
at this integral.
00:53:17.640 --> 00:53:19.230
No more.
00:53:19.230 --> 00:53:24.010
But there is one thing,
and that is these arguments
00:53:24.010 --> 00:53:25.600
do not determine--
00:53:25.600 --> 00:53:27.790
I mean, when you take the
square root of something
00:53:27.790 --> 00:53:31.340
you can have a positive
value and a negative value.
00:53:31.340 --> 00:53:33.820
That corresponds to
a phase ambiguity,
00:53:33.820 --> 00:53:37.430
and these arguments
don't resolve that.
00:53:37.430 --> 00:53:41.660
At some point you have to decide
on the phase and be consistent.
00:53:41.660 --> 00:53:43.730
And since you're never
looking at wave functions,
00:53:43.730 --> 00:53:48.980
that actually is a
frequent source of error.
00:53:48.980 --> 00:53:52.340
But that's the only defect
in this whole thing.
00:53:52.340 --> 00:53:55.610
So that's it for the exam.
00:53:55.610 --> 00:53:58.700
I will talk about something
that will make you a little bit
00:53:58.700 --> 00:54:01.220
more comfortable
about some of the exam
00:54:01.220 --> 00:54:05.140
questions on Wednesday, but
it's not going to be tested.