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ROBERT FIELD: Today, I'm going
to go over a lot of material
00:00:26.900 --> 00:00:31.850
that you already know
and work an example
00:00:31.850 --> 00:00:37.390
with non-degenerate
perturbation theory.
00:00:37.390 --> 00:00:41.920
This will give me an excuse
to use interspersed matrix
00:00:41.920 --> 00:00:47.500
and operator notation,
and one of the things
00:00:47.500 --> 00:00:53.850
that you will be doing is
dealing with infinite matrices.
00:00:53.850 --> 00:00:58.980
And we don't diagonalize
infinite matrices,
00:00:58.980 --> 00:01:02.130
but the picture of
an infinite matrix
00:01:02.130 --> 00:01:05.620
is useful in guiding intuition.
00:01:05.620 --> 00:01:10.800
And so we have to know about
what those pictures mean
00:01:10.800 --> 00:01:15.640
and how to construct them from
things that seem to be obvious.
00:01:15.640 --> 00:01:21.090
OK, the problem I want to
spend most of my time on
00:01:21.090 --> 00:01:26.550
is an example of a real
molecular potential which
00:01:26.550 --> 00:01:30.420
is quadratic but with a cubic
and a quartic perturbation
00:01:30.420 --> 00:01:39.020
term, and this is a
simpler problem than what
00:01:39.020 --> 00:01:41.870
I talked about last time.
00:01:41.870 --> 00:01:43.370
The last time I
really wanted to set
00:01:43.370 --> 00:01:47.450
the stage for what you
can do with non-degenerate
00:01:47.450 --> 00:01:49.520
perturbation theory,
and this is going
00:01:49.520 --> 00:01:57.860
to be more of dealing
with a specific problem.
00:01:57.860 --> 00:02:01.850
Now, the algebra
does not get simpler,
00:02:01.850 --> 00:02:04.770
and so it's easy to get
lost in the algebra.
00:02:04.770 --> 00:02:08.210
And so I will give
you some rules
00:02:08.210 --> 00:02:12.320
about what to expect, when you
have a potential like this,
00:02:12.320 --> 00:02:15.770
and how does that come out of
non-degenerate perturbation
00:02:15.770 --> 00:02:17.300
theory?
00:02:17.300 --> 00:02:22.060
Now, everything that
we know about molecules
00:02:22.060 --> 00:02:24.560
comes from spectroscopy.
00:02:24.560 --> 00:02:31.590
That means transitions between
energy levels, and so well,
00:02:31.590 --> 00:02:33.450
how do they happen?
00:02:33.450 --> 00:02:36.870
And we can lead
into that by talking
00:02:36.870 --> 00:02:39.720
about the stark effect
which is the interaction
00:02:39.720 --> 00:02:44.145
of an electric field with
a dipole in the molecule.
00:02:46.910 --> 00:02:52.770
And so most
transitions are caused
00:02:52.770 --> 00:02:57.390
by a time-dependent electric
field which somehow interacts
00:02:57.390 --> 00:03:01.540
with the molecule through
its dipole moment,
00:03:01.540 --> 00:03:05.700
and so you'll begin
to see that today.
00:03:11.310 --> 00:03:14.330
Last time, I
derived the formulas
00:03:14.330 --> 00:03:16.470
for non-degenerate
perturbation theory.
00:03:29.180 --> 00:03:33.990
Now, I'm a little bit
crazy, and I really
00:03:33.990 --> 00:03:36.030
believe in perturbation theory.
00:03:36.030 --> 00:03:38.730
Most of the textbooks
say, well, let's just
00:03:38.730 --> 00:03:41.190
stop here at the
first-order correction
00:03:41.190 --> 00:03:44.980
to the energy levels,
and you get nothing
00:03:44.980 --> 00:03:48.220
from that except
the Zeeman Effect
00:03:48.220 --> 00:03:50.380
which is kind of
important for NMR.
00:03:50.380 --> 00:03:57.850
But in terms of what molecules
do, you need this too,
00:03:57.850 --> 00:04:03.520
and this is equal to
the zero-order energy.
00:04:06.730 --> 00:04:09.490
I'm sorry, the
zero-order energy which
00:04:09.490 --> 00:04:14.530
comes from an exactly
solved problem
00:04:14.530 --> 00:04:23.230
and then the diagonal matrix
element of the everything
00:04:23.230 --> 00:04:27.070
that's bad in the world,
the first-order Hamiltonian.
00:04:27.070 --> 00:04:32.070
And then, we have this
complicated-looking thing,
00:04:32.070 --> 00:04:43.228
m not equal to n of H n m
squared, E n 0 minus E m 0.
00:04:46.480 --> 00:04:48.400
Now, the only thing
that you might
00:04:48.400 --> 00:04:51.670
forget about in this
kind of a formula
00:04:51.670 --> 00:04:57.280
is is it n before m or
the other way around?
00:04:57.280 --> 00:05:01.470
And what we know is, OK,
here is the nth level,
00:05:01.470 --> 00:05:04.290
and here is the mth level,
and the interaction causes
00:05:04.290 --> 00:05:06.790
this level to be pushed down.
00:05:06.790 --> 00:05:13.480
So if the mth level is
above the nth level,
00:05:13.480 --> 00:05:16.100
this denominator
will be negative,
00:05:16.100 --> 00:05:18.440
and that causes
the pushing down.
00:05:18.440 --> 00:05:21.730
And so if you remember the
idea of perturbation theory--
00:05:21.730 --> 00:05:26.350
that things interact, and
one level gets pushed down,
00:05:26.350 --> 00:05:30.040
the other level gets pushed
up, equal and opposite--
00:05:30.040 --> 00:05:36.580
you can correct your flaws of
memory on which comes first.
00:05:36.580 --> 00:05:42.790
And we can write
the wave function
00:05:42.790 --> 00:05:47.680
as an eigenfunction of
the exactly solved problem
00:05:47.680 --> 00:06:05.460
plus a term, m not equal to n of
H n m 1 E n 0 minus E m 0 times
00:06:05.460 --> 00:06:09.082
psi m 0.
00:06:09.082 --> 00:06:14.260
Now, the reason this sum does
not include the nth level
00:06:14.260 --> 00:06:18.370
is, well, if you did, this
denominator would blow up,
00:06:18.370 --> 00:06:21.370
and you don't need the nth
level, because you've already
00:06:21.370 --> 00:06:21.940
got it here.
00:06:25.570 --> 00:06:30.450
So these are the
formulas, and we had said,
00:06:30.450 --> 00:06:38.880
OK, we're going to write the
Hamiltonian as H0 plus H1,
00:06:38.880 --> 00:06:42.360
and this is an exactly
solved problem.
00:06:42.360 --> 00:06:44.940
This is everything else.
00:06:44.940 --> 00:06:49.020
Now, some people will sort this
out into small everything else
00:06:49.020 --> 00:06:51.010
and big everything else.
00:06:51.010 --> 00:06:53.400
But it's foolish,
because all that does
00:06:53.400 --> 00:06:57.180
is multiply the algebra
in a horrible way,
00:06:57.180 --> 00:06:58.740
and you don't care, anyway.
00:06:58.740 --> 00:07:02.550
You just know that this stuff
that's not in this exactly
00:07:02.550 --> 00:07:05.070
solved problem
gets treated here,
00:07:05.070 --> 00:07:09.060
and you're going to know, in
principle, how to deal with it,
00:07:09.060 --> 00:07:10.830
and there'll be
some small problems
00:07:10.830 --> 00:07:14.000
that you know how to deal with.
00:07:14.000 --> 00:07:24.630
OK, and there is a rule that
H n m 1 over E n 0 minus E m
00:07:24.630 --> 00:07:29.677
0 absolute value is
much less than 1.
00:07:29.677 --> 00:07:30.510
That's what we mean.
00:07:30.510 --> 00:07:31.227
Yes?
00:07:31.227 --> 00:07:33.060
AUDIENCE: Are you making
a deliberate choice
00:07:33.060 --> 00:07:36.971
to put all the badness in
the first-order perturbation?
00:07:36.971 --> 00:07:37.720
ROBERT FIELD: Yes.
00:07:37.720 --> 00:07:41.650
AUDIENCE: Then, what
goes into higher order,
00:07:41.650 --> 00:07:43.210
if we make that deliberate?
00:07:43.210 --> 00:07:44.960
ROBERT FIELD: OK, you
can say, well, we've
00:07:44.960 --> 00:07:54.890
got things that are
important, that obviously
00:07:54.890 --> 00:07:56.480
affect the energy
levels, and there's
00:07:56.480 --> 00:07:59.940
things that are smaller,
like hyperfine structure.
00:07:59.940 --> 00:08:03.490
And when you use the word hyper,
you generally mean it's small,
00:08:03.490 --> 00:08:07.610
and so what ends up happening
is that you do the big picture,
00:08:07.610 --> 00:08:10.820
and then you see
some more details.
00:08:10.820 --> 00:08:14.840
But it's only a
very small number
00:08:14.840 --> 00:08:18.140
of people who actually
segregate the badness
00:08:18.140 --> 00:08:20.720
into real bad and not so bad.
00:08:24.470 --> 00:08:31.470
OK, so this works only when
this off-diagonal matrix element
00:08:31.470 --> 00:08:34.530
is smaller than the
energy difference,
00:08:34.530 --> 00:08:36.090
and that's going to be true.
00:08:39.549 --> 00:08:42.850
Now, here is an infinite
matrix, and we're
00:08:42.850 --> 00:08:48.670
interested in a certain
space, a state space, which
00:08:48.670 --> 00:08:51.300
our experiments are
designed to measure,
00:08:51.300 --> 00:08:53.860
and it depends on what
experiment you do.
00:08:53.860 --> 00:08:55.870
This might be here,
it might be somewhere
00:08:55.870 --> 00:09:02.440
in the middle, and the
rest, we're not interested,
00:09:02.440 --> 00:09:03.790
because it's far away.
00:09:03.790 --> 00:09:07.300
Because the energy
denominator is so large
00:09:07.300 --> 00:09:11.140
that the effect on the
energy levels can be ignored,
00:09:11.140 --> 00:09:14.890
or you can say, well, we're
not really ignoring it.
00:09:14.890 --> 00:09:17.560
We're folding the effect
of all these levels
00:09:17.560 --> 00:09:19.960
through the matrix
elements in here
00:09:19.960 --> 00:09:24.150
into here by second-order
perturbation theory.
00:09:24.150 --> 00:09:26.270
We could in principle do that.
00:09:26.270 --> 00:09:28.310
We certainly are
allowed to think
00:09:28.310 --> 00:09:32.630
that we are getting rid of this
state space by, in principle,
00:09:32.630 --> 00:09:35.480
folding it into this block.
00:09:35.480 --> 00:09:40.580
Now, in this block of
levels that we care about--
00:09:40.580 --> 00:09:44.480
now I always like to draw
these infinite matrices,
00:09:44.480 --> 00:09:46.707
or these matrices,
by just saying, OK,
00:09:46.707 --> 00:09:47.540
here's the diagonal.
00:09:50.220 --> 00:09:58.380
And there might be a couple of
levels within this state space
00:09:58.380 --> 00:10:01.770
we're interested in
which, because of accident
00:10:01.770 --> 00:10:06.840
or because of something evil,
their energy denominator
00:10:06.840 --> 00:10:12.900
is small compared to the
zero-order energy differences.
00:10:12.900 --> 00:10:17.980
But that's usually true
for only a few accidentally
00:10:17.980 --> 00:10:21.300
degenerate states,
and we deal with them
00:10:21.300 --> 00:10:24.600
by using the
machinery we obtained
00:10:24.600 --> 00:10:29.060
from the two-level
problem, and so there, we
00:10:29.060 --> 00:10:33.280
are actually diagonalizing
a small dimension matrix.
00:10:33.280 --> 00:10:36.820
We're not diagonalizing it, our
computer is diagonalizing it.
00:10:36.820 --> 00:10:39.550
You really don't
care how it's done,
00:10:39.550 --> 00:10:43.030
because you have a
machine that will solve
00:10:43.030 --> 00:10:46.990
this like difficulty, and
so it's OK for this rule
00:10:46.990 --> 00:10:49.030
to be violated.
00:10:49.030 --> 00:10:58.660
But so what happens, suppose
we have two levels, and--
00:10:58.660 --> 00:11:01.080
I just want to make sure
I use the same notation
00:11:01.080 --> 00:11:03.910
as in the notes.
00:11:03.910 --> 00:11:07.690
So here, we have the
two zero-order levels,
00:11:07.690 --> 00:11:12.940
and they interact and repel
each other equal and opposite
00:11:12.940 --> 00:11:14.470
amounts.
00:11:14.470 --> 00:11:25.430
And these states, which are E2
and E1, are not pure state 2.
00:11:25.430 --> 00:11:28.230
They have a mixture
of state 1 in them.
00:11:28.230 --> 00:11:33.720
Now, in spectroscopy, we
always observe these levels
00:11:33.720 --> 00:11:35.290
by transitions.
00:11:35.290 --> 00:11:39.300
And so suppose we have
state 0 down here,
00:11:39.300 --> 00:11:48.390
and let us say that this
transition between the zero
00:11:48.390 --> 00:11:51.105
level and the zero-order
level 1 is allowed.
00:11:54.510 --> 00:12:02.010
I'll symbolize that by
mu 1,0 is not equal to 0.
00:12:02.010 --> 00:12:08.550
And the transition between this
level and this zero-order level
00:12:08.550 --> 00:12:14.160
is forbidden, mu
2,0 is equal to 0.
00:12:14.160 --> 00:12:18.270
So we call this a dark
state and a bright state,
00:12:18.270 --> 00:12:20.360
and now these two
states interact,
00:12:20.360 --> 00:12:22.920
and we have mixed states.
00:12:22.920 --> 00:12:26.100
So the real eigenstates,
you wouldn't
00:12:26.100 --> 00:12:30.960
have expected to see this
level in the spectrum.
00:12:30.960 --> 00:12:34.950
But you do, because it's
borrowed some bright character
00:12:34.950 --> 00:12:39.360
through the perturbation
interaction,
00:12:39.360 --> 00:12:42.190
and so there's two surprises.
00:12:42.190 --> 00:12:46.710
One is everything is sort of
describable by a simple set
00:12:46.710 --> 00:12:51.760
of equations, and then there's
a couple of deviations,
00:12:51.760 --> 00:12:54.220
and there's some extra
levels that appear.
00:12:57.330 --> 00:13:00.090
That's information very
rich, because if it's dark,
00:13:00.090 --> 00:13:01.410
you can't see it.
00:13:01.410 --> 00:13:02.160
It's not a ghost.
00:13:02.160 --> 00:13:04.080
It's there, but
you can't see it.
00:13:04.080 --> 00:13:06.780
Well, the perturbation
somehow pulls--
00:13:06.780 --> 00:13:09.300
sometimes-- pulls
the curtain back
00:13:09.300 --> 00:13:13.160
and enables you to see stuff
you need to know about.
00:13:13.160 --> 00:13:18.680
Now, you can imagine exciting
this two-level system
00:13:18.680 --> 00:13:20.390
with a short pulse
of light, where
00:13:20.390 --> 00:13:23.660
the uncertainty broadening
of the short pulse of light
00:13:23.660 --> 00:13:27.360
covers these two eigenstates.
00:13:27.360 --> 00:13:29.982
What happens then is
you get quantum beats,
00:13:29.982 --> 00:13:31.440
and that's going
to be on the exam.
00:13:36.310 --> 00:13:38.750
OK.
00:13:38.750 --> 00:13:43.210
So this is a local perturbation,
and it's just something in here
00:13:43.210 --> 00:13:49.630
that spoils the general rule
but it's very information-rich,
00:13:49.630 --> 00:13:54.160
and it also is
pedagogically fun.
00:13:54.160 --> 00:13:55.810
OK.
00:13:55.810 --> 00:13:58.750
So now, I'm going to talk
about stuff that I've
00:13:58.750 --> 00:14:00.610
talked about before,
but I'm going
00:14:00.610 --> 00:14:03.430
to go a little
deeper and slower,
00:14:03.430 --> 00:14:08.680
and so let's do this problem.
00:14:08.680 --> 00:14:12.160
We have a potential,
and for molecules, we
00:14:12.160 --> 00:14:16.810
tend to use Q rather than
R or X as the displacement
00:14:16.810 --> 00:14:17.670
from equilibrium.
00:14:17.670 --> 00:14:20.170
The harmonic
oscillator coordinate,
00:14:20.170 --> 00:14:24.070
and it's the same
thing and so we have--
00:14:34.320 --> 00:14:38.030
OK, so this is an
exactly solved problem,
00:14:38.030 --> 00:14:39.920
and this is something extra.
00:14:39.920 --> 00:14:42.370
And so this is a
cubic anharmonicity,
00:14:42.370 --> 00:14:44.650
and this is a quartic
anharmonicity.
00:14:44.650 --> 00:14:46.480
In the previous
lecture, I talked
00:14:46.480 --> 00:14:50.560
about anharmonic couplings
between different modes
00:14:50.560 --> 00:14:52.850
of the same polyatomic molecule.
00:14:52.850 --> 00:14:55.000
Here, this is really
a simpler problem,
00:14:55.000 --> 00:14:57.430
and I probably should have
talked about it first.
00:14:57.430 --> 00:14:59.290
But I didn't, because
I want to go deeper
00:14:59.290 --> 00:15:05.360
here than the big picture,
which I described last time.
00:15:05.360 --> 00:15:09.380
So we're going to
have two terms that we
00:15:09.380 --> 00:15:13.247
are going to treat by
perturbation theory,
00:15:13.247 --> 00:15:15.080
and there are several
important things here.
00:15:17.980 --> 00:15:23.660
Molecular potential looks
like this, not like that.
00:15:23.660 --> 00:15:24.160
Right?
00:15:24.160 --> 00:15:29.310
This is bond breaking, and it
breaks at large displacement.
00:15:29.310 --> 00:15:31.400
There is no such thing
as a bond breaking as you
00:15:31.400 --> 00:15:33.800
squeeze the molecules together.
00:15:33.800 --> 00:15:37.310
So this is a crazy
idea, and how do you
00:15:37.310 --> 00:15:38.770
get an asymmetric potential?
00:15:38.770 --> 00:15:47.340
Well, it's cubic, and which sign
of b is going to lead to this?
00:15:47.340 --> 00:15:47.840
Yes.
00:15:47.840 --> 00:15:48.715
AUDIENCE: [INAUDIBLE]
00:15:48.715 --> 00:15:49.920
ROBERT FIELD: Right.
00:15:49.920 --> 00:15:51.120
OK.
00:15:51.120 --> 00:15:59.230
Now, if we're really naive, we
say, yeah, negative is good,
00:15:59.230 --> 00:16:06.130
but boom, we ignore that.
00:16:06.130 --> 00:16:11.290
We don't worry that, if
we took this seriously
00:16:11.290 --> 00:16:14.000
at large enough
displacement, the potential
00:16:14.000 --> 00:16:16.280
will go to minus infinity.
00:16:16.280 --> 00:16:20.830
We don't worry about tunneling
through this barrier.
00:16:20.830 --> 00:16:24.280
We just use this to
give us something
00:16:24.280 --> 00:16:26.350
that has the right
shape, and that we can
00:16:26.350 --> 00:16:28.480
apply perturbation theory to.
00:16:28.480 --> 00:16:31.960
And we're not going to worry
about using perturbation theory
00:16:31.960 --> 00:16:37.650
to capture this tunneling,
at least not now and not
00:16:37.650 --> 00:16:40.205
in this course.
00:16:40.205 --> 00:16:42.480
OK.
00:16:42.480 --> 00:16:47.250
So one thing is
we have this term,
00:16:47.250 --> 00:16:53.500
and there is something you know
immediately about odd powers.
00:16:53.500 --> 00:16:57.850
They never have delta v
equals 0 matrix elements,
00:16:57.850 --> 00:16:59.950
they never have
diagonal elements,
00:16:59.950 --> 00:17:04.720
and so they do not contribute
to the energy in first order.
00:17:08.319 --> 00:17:13.420
The only way you know the
sign of this perturbation
00:17:13.420 --> 00:17:18.460
is from a non-zero
first-order contribution,
00:17:18.460 --> 00:17:22.030
because when you do second-order
perturbation theory,
00:17:22.030 --> 00:17:24.650
the matrix element gets squared,
and the sign information
00:17:24.650 --> 00:17:26.950
is lost.
00:17:26.950 --> 00:17:31.540
So there's nothing
so far that tells you
00:17:31.540 --> 00:17:37.910
what the sign of b is, and the
energy level pattern that you
00:17:37.910 --> 00:17:43.610
would obtain from either
sign of b would be the same.
00:17:43.610 --> 00:17:48.170
It's just one of the signs
is completely ridiculous.
00:17:48.170 --> 00:17:56.320
Now Q to the 4th has selection
rules delta v of 4, 2, 0,
00:17:56.320 --> 00:17:59.370
minus 2, and minus 4.
00:17:59.370 --> 00:18:02.940
So that delta v of
0 term does enter
00:18:02.940 --> 00:18:05.430
in first-order
perturbation theory,
00:18:05.430 --> 00:18:13.600
and so you can determine the
sign of any even perturbation
00:18:13.600 --> 00:18:15.695
which is useful.
00:18:15.695 --> 00:18:18.840
Now, what does this do?
00:18:18.840 --> 00:18:30.380
Well, one thing that Q
does is, if c is positive,
00:18:30.380 --> 00:18:33.080
instead of having a
harmonic oscillator,
00:18:33.080 --> 00:18:33.995
it makes it steeper.
00:18:37.800 --> 00:18:42.520
And if Q is negative,
it makes it flatter.
00:18:42.520 --> 00:18:47.060
This is typical of
bending vibrations,
00:18:47.060 --> 00:18:50.967
and so bending vibrations tend
to have flat bottom potentials,
00:18:50.967 --> 00:18:52.550
but there there's
also something else.
00:18:55.950 --> 00:18:58.440
Suppose we have two
electronic states.
00:18:58.440 --> 00:19:01.490
Now, I am cheating,
because I'm assuming
00:19:01.490 --> 00:19:04.080
you'll accept the idea
that there is something
00:19:04.080 --> 00:19:06.320
we haven't talked about yet.
00:19:06.320 --> 00:19:09.040
But there are different
potential curves,
00:19:09.040 --> 00:19:14.660
and it's mostly true
for atomic molecules.
00:19:14.660 --> 00:19:17.230
These two states can't
talk to each other
00:19:17.230 --> 00:19:19.330
at equilibrium,
because a symmetry
00:19:19.330 --> 00:19:20.950
exists that prevents that.
00:19:20.950 --> 00:19:23.400
And as you move away
from equilibrium,
00:19:23.400 --> 00:19:25.870
there's a perturbation that
gets larger and larger.
00:19:25.870 --> 00:19:32.020
And as a result, what happens
is that this potential
00:19:32.020 --> 00:19:33.395
does something like that.
00:19:38.330 --> 00:19:43.040
So you get either a flattening
or an actual extra pair
00:19:43.040 --> 00:19:46.940
of minima, and this
one gets sharper.
00:19:46.940 --> 00:19:48.880
That's called a
vibronic interaction,
00:19:48.880 --> 00:19:53.690
and that's fairly important
in polyatomic molecules.
00:19:53.690 --> 00:19:56.990
But so we can begin to
understand these things
00:19:56.990 --> 00:20:04.510
just by dealing with these
terms in the potential.
00:20:04.510 --> 00:20:06.720
OK.
00:20:06.720 --> 00:20:12.570
So now, we're off to the
races, and my goal here
00:20:12.570 --> 00:20:16.830
is to make you comfortable
with either the operator
00:20:16.830 --> 00:20:20.140
or the matrix
notation, and so I'm
00:20:20.140 --> 00:20:23.440
going to go back and forth
between them in what might
00:20:23.440 --> 00:20:26.190
seem to be a random manner.
00:20:26.190 --> 00:20:26.950
OK.
00:20:26.950 --> 00:20:31.420
Now, you know that a dagger.
00:20:31.420 --> 00:20:33.720
Now, we can call it
a dagger with a hat,
00:20:33.720 --> 00:20:36.370
or we can call it a
dagger double underline--
00:20:36.370 --> 00:20:40.240
bold, hat, matrix, operator.
00:20:40.240 --> 00:20:43.970
And you know that
this operates on v
00:20:43.970 --> 00:20:52.710
to give v plus 1 square
root psi v plus 1,
00:20:52.710 --> 00:20:57.420
or in bracket notation,
we could write this as v
00:20:57.420 --> 00:21:03.400
plus 1 a dagger v.
00:21:03.400 --> 00:21:05.260
Now this is a shorthand.
00:21:05.260 --> 00:21:07.240
It's a wonderful shorthand.
00:21:07.240 --> 00:21:11.130
It's easier to draw this
than wave functions.
00:21:11.130 --> 00:21:12.630
But we have to
know, OK, what does
00:21:12.630 --> 00:21:19.440
this look like in the
matrix for a dagger?
00:21:19.440 --> 00:21:29.380
So here is a dagger, and it's a
matrix, and what goes in here?
00:21:29.380 --> 00:21:33.350
Well, the first thing
you do, this is infinite,
00:21:33.350 --> 00:21:35.900
so you need some sort of
a way of drawing something
00:21:35.900 --> 00:21:38.930
that's infinite so that
you understand what it is.
00:21:38.930 --> 00:21:42.860
And so the first thing you do
is you know that there is no--
00:21:42.860 --> 00:21:47.280
all of the diagonal
elements are 0.
00:21:47.280 --> 00:21:51.120
Now, most of the elements
in this matrix are 0,
00:21:51.120 --> 00:21:54.690
and you don't want to draw them,
because you'll just be spending
00:21:54.690 --> 00:21:57.390
all your time writing 0's.
00:21:57.390 --> 00:22:03.410
So we want a shorthand, and so
now this is a matrix element.
00:22:03.410 --> 00:22:09.860
This is the row, and this
is the column, and so where
00:22:09.860 --> 00:22:15.870
do I put this square
root of E plus 1?
00:22:15.870 --> 00:22:18.310
AUDIENCE: [INAUDIBLE]
00:22:21.740 --> 00:22:25.720
ROBERT FIELD: So
here, we have the row.
00:22:29.300 --> 00:22:31.880
I get confused about
this, so before I
00:22:31.880 --> 00:22:35.780
accept your answer,
which I want to reject,
00:22:35.780 --> 00:22:39.810
I have to think
carefully about it.
00:22:39.810 --> 00:22:46.385
So let us say this is 0, and
this is 1, so we have the 1--
00:22:50.880 --> 00:22:51.580
you're right.
00:22:54.260 --> 00:22:58.735
OK, and so we have
the square root of 1,
00:22:58.735 --> 00:23:06.010
the square root of
2, square root of n.
00:23:06.010 --> 00:23:08.620
And everything else is 0,
so we can write big 0's.
00:23:11.267 --> 00:23:12.850
So that's what this
matrix looks like.
00:23:15.910 --> 00:23:21.280
Now, if we're using computers,
instead of multiplying
00:23:21.280 --> 00:23:24.850
these matrices to have
say Q to the 13th,
00:23:24.850 --> 00:23:30.080
or a dagger to the 13th, you can
just multiply these matrices.
00:23:30.080 --> 00:23:33.660
That's an easy request
for the computer.
00:23:33.660 --> 00:23:36.450
It's not such an
easy request for you,
00:23:36.450 --> 00:23:38.700
but you end up getting matrices.
00:23:38.700 --> 00:23:41.610
When you have integer
powers of these,
00:23:41.610 --> 00:23:47.100
you get a matrix with
a diagonal and then
00:23:47.100 --> 00:23:53.154
another diagonal separated by
a diagonal of 0's, and so you
00:23:53.154 --> 00:23:55.070
know what these things
are going to look like.
00:24:00.240 --> 00:24:06.580
So you might ask, OK,
what does a look like?
00:24:06.580 --> 00:24:08.260
And there are
several ways to say
00:24:08.260 --> 00:24:10.870
what a is going to
look like, because it's
00:24:10.870 --> 00:24:13.975
the conjugate
transpose of a dagger.
00:24:17.010 --> 00:24:19.650
Well, these are
all real numbers,
00:24:19.650 --> 00:24:21.790
and so all you do is flip
this on its diagonal,
00:24:21.790 --> 00:24:22.590
and now you get a.
00:24:27.590 --> 00:24:29.450
Then there's another
actor in this game,
00:24:29.450 --> 00:24:33.160
and that's n, the
number operator,
00:24:33.160 --> 00:24:41.310
and the number operator
is going to be a, a dagger
00:24:41.310 --> 00:24:43.470
or is it going to be a dagger a?
00:24:49.900 --> 00:24:51.992
So which is it?
00:24:51.992 --> 00:24:53.896
AUDIENCE: [INAUDIBLE]
00:24:55.324 --> 00:24:56.276
ROBERT FIELD: This.
00:24:56.276 --> 00:24:57.228
AUDIENCE: [INAUDIBLE]
00:24:57.228 --> 00:25:00.100
ROBERT FIELD: Yes.
00:25:00.100 --> 00:25:05.650
OK, one way to
remember this is when
00:25:05.650 --> 00:25:09.040
you operate with
either a or a dagger,
00:25:09.040 --> 00:25:11.484
it connects two
vibrational levels,
00:25:11.484 --> 00:25:13.150
and the thing you put
in the square root
00:25:13.150 --> 00:25:15.010
is the larger of the
two quantum numbers.
00:25:18.740 --> 00:25:21.300
OK.
00:25:21.300 --> 00:25:25.860
So what would the number
operator matrix look like?
00:25:36.060 --> 00:25:37.790
So what do I put here?
00:25:40.664 --> 00:25:42.101
Yes.
00:25:42.101 --> 00:25:44.496
AUDIENCE: [INAUDIBLE]
00:25:47.219 --> 00:25:49.510
ROBERT FIELD: But what's the
lowest vibrational quantum
00:25:49.510 --> 00:25:49.870
number?
00:25:49.870 --> 00:25:50.730
AUDIENCE: [INAUDIBLE]
00:25:50.730 --> 00:25:51.563
ROBERT FIELD: Right.
00:25:56.920 --> 00:26:01.490
OK, and so once you've
practiced a little bit,
00:26:01.490 --> 00:26:04.610
you can write these things.
00:26:04.610 --> 00:26:06.720
And it's not just
an arbitrary thing,
00:26:06.720 --> 00:26:10.420
because you want to be able to
visualize what you're doing.
00:26:10.420 --> 00:26:13.930
Because you're dealing with
multiple infinities of objects,
00:26:13.930 --> 00:26:16.600
and you want to focus only
on the ones you care about.
00:26:16.600 --> 00:26:20.500
And with a little bit of
guidance from these pictures,
00:26:20.500 --> 00:26:22.480
you can do what you need to do.
00:26:27.824 --> 00:26:28.324
OK.
00:26:36.430 --> 00:26:39.850
Well, we know what
the operator Q is
00:26:39.850 --> 00:26:43.660
in terms of a
dimensionalist version of Q,
00:26:43.660 --> 00:26:49.570
and that's h bar over-- now,
I'm going to be using this.
00:26:56.900 --> 00:26:58.710
OK, this is the
dimensionalist Q.
00:26:58.710 --> 00:27:01.980
That's what the twiddle means,
and these are constants.
00:27:01.980 --> 00:27:11.000
And now I'm using omega twiddle,
because spectroscopists always
00:27:11.000 --> 00:27:16.140
use omega in wave number units,
reciprocal centimeter units.
00:27:16.140 --> 00:27:19.340
Which is a terrible thing,
because first of all, wave
00:27:19.340 --> 00:27:20.730
number doesn't have a unit.
00:27:20.730 --> 00:27:22.130
It's a quantity.
00:27:22.130 --> 00:27:24.680
And centimeters are things
that we're not supposed to use,
00:27:24.680 --> 00:27:26.600
because we use MKS.
00:27:26.600 --> 00:27:29.120
But spectroscopists
are stubborn,
00:27:29.120 --> 00:27:33.050
and when we observe
transitions, we always
00:27:33.050 --> 00:27:37.620
talk about wave
numbers not energy.
00:27:37.620 --> 00:27:41.790
And so the difference between
wave numbers and energies
00:27:41.790 --> 00:27:46.350
is the factor of
hc, not h bar c.
00:27:46.350 --> 00:27:49.630
So anyway, this is the
conversion factor, when
00:27:49.630 --> 00:27:52.030
omega is in wave number units.
00:27:56.020 --> 00:28:07.480
And we can go further and
relate Q twiddle well,
00:28:07.480 --> 00:28:11.070
let's just not do that.
00:28:11.070 --> 00:28:20.280
We can relate this
to h bar over 4 pi
00:28:20.280 --> 00:28:26.680
c u omega square root
times a plus a dagger.
00:28:26.680 --> 00:28:28.540
OK?
00:28:28.540 --> 00:28:32.680
So this is something
we did before.
00:28:32.680 --> 00:28:46.940
Omega twiddle is k over mu
square root 1 over 2 pi c--
00:28:46.940 --> 00:28:49.400
hc.
00:28:49.400 --> 00:28:52.280
So this what you're
used to, and this
00:28:52.280 --> 00:28:55.910
is the extra stuff that we
have to carry along in order
00:28:55.910 --> 00:28:59.675
to work in wave number units.
00:28:59.675 --> 00:29:00.175
OK.
00:29:06.200 --> 00:29:09.170
So we have operators
that can be expressed
00:29:09.170 --> 00:29:20.050
like a, a, a dagger, a or a
dagger, a, a, a or anywhere you
00:29:20.050 --> 00:29:21.790
put the dagger anywhere.
00:29:21.790 --> 00:29:27.140
You look at this, and you say
immediately I know two things.
00:29:27.140 --> 00:29:29.810
I know the selection rule.
00:29:29.810 --> 00:29:35.100
The selection rule is count
up the a's and count up
00:29:35.100 --> 00:29:44.330
the a daggers, and so this
is delta v of minus 2.
00:29:44.330 --> 00:29:46.100
So is this.
00:29:46.100 --> 00:29:50.730
All of the positions of the a
dagger are delta v of minus 2,
00:29:50.730 --> 00:29:53.870
but the numbers, the matrix
elements, are different.
00:29:53.870 --> 00:29:55.490
You know this from
the last exam.
00:29:58.090 --> 00:30:07.300
So we can have v minus
2 v and whatever comes
00:30:07.300 --> 00:30:10.370
in here in one of those forms.
00:30:10.370 --> 00:30:12.460
So those are the only
non-zero elements,
00:30:12.460 --> 00:30:16.210
and you know how to
mechanically figure out
00:30:16.210 --> 00:30:17.930
what is the value of
the matrix element.
00:30:21.130 --> 00:30:25.150
Now, in order to simplify
the algebra, which is not
00:30:25.150 --> 00:30:30.400
essential to the physics, you
want to take all of the terms
00:30:30.400 --> 00:30:37.120
that results say
from Q to the 4th
00:30:37.120 --> 00:30:40.190
and arrange them according
to selection rule.
00:30:40.190 --> 00:30:43.610
And then take all of the terms
that have the same selection
00:30:43.610 --> 00:30:47.350
rule and combining them
to a single number.
00:30:53.060 --> 00:30:56.610
And you use the
computation rule to be
00:30:56.610 --> 00:30:59.745
able to reverse
the order of terms.
00:31:03.371 --> 00:31:04.620
Well, I don't want to do that.
00:31:08.240 --> 00:31:13.780
So suppose we have an a,
a dagger, and we can write
00:31:13.780 --> 00:31:19.072
that as a, a dagger
plus a dagger, a.
00:31:19.072 --> 00:31:22.350
And so if we want
to convert something
00:31:22.350 --> 00:31:25.590
like this to
something like that,
00:31:25.590 --> 00:31:32.930
we know that this has a value
of plus 1, so we can do that.
00:31:32.930 --> 00:31:36.080
And that's tedious,
and you have to do it.
00:31:36.080 --> 00:31:39.290
But one of the things
that is kind of nice
00:31:39.290 --> 00:31:41.750
is when you have
a problem that's
00:31:41.750 --> 00:31:47.360
cubic or quartic or quintic, you
mess around with this operator
00:31:47.360 --> 00:31:51.470
algebra once in your life, and
you put it on a sheet of paper,
00:31:51.470 --> 00:31:52.530
and you refer to it.
00:31:52.530 --> 00:31:54.230
It doesn't matter
what the molecule
00:31:54.230 --> 00:31:58.580
is, what the constant in front
of Q to the 3rd or 4th or 13th
00:31:58.580 --> 00:31:59.540
is.
00:31:59.540 --> 00:32:02.777
If you've done the operator
algebra, you're fine.
00:32:02.777 --> 00:32:04.860
Now, you might say, well,
I don't want to do that.
00:32:04.860 --> 00:32:07.410
I'm going to have
the computer do that.
00:32:07.410 --> 00:32:09.360
Well, fine, you can have
the computer do that,
00:32:09.360 --> 00:32:13.020
and then a computer will tell
you what the matrix looks like,
00:32:13.020 --> 00:32:15.260
and you can do what
you need to do.
00:32:15.260 --> 00:32:15.930
OK.
00:32:15.930 --> 00:32:21.330
So spectroscopists call the
vibrational energy formula
00:32:21.330 --> 00:32:26.360
G of v. I don't know why
the letter G is always use,
00:32:26.360 --> 00:32:30.530
but it is, and so
this is the same thing
00:32:30.530 --> 00:32:32.870
as vibrational energy.
00:32:32.870 --> 00:32:35.750
And the vibrational energy, I'm
going to put the tilde on it.
00:32:35.750 --> 00:32:41.030
You will never find a tilde in
any spectroscopy note paper.
00:32:41.030 --> 00:32:45.260
We assume that you understand
that the only units
00:32:45.260 --> 00:32:50.230
for spectroscopic quantities
are reciprocal centimeters,
00:32:50.230 --> 00:32:54.720
but for this purposes,
I have finally caved,
00:32:54.720 --> 00:32:57.350
and I said, OK, I'm going
to put the tildes on.
00:32:57.350 --> 00:33:10.730
So the energy levels, now
you might ask, why this?
00:33:10.730 --> 00:33:13.140
This is the first
anharmonicity constant.
00:33:13.140 --> 00:33:16.410
It's not a product
of two numbers.
00:33:16.410 --> 00:33:19.580
It's just what people
wrote originally,
00:33:19.580 --> 00:33:23.450
because they sort of thought of
it as a product of two numbers,
00:33:23.450 --> 00:33:26.870
but it's really only one.
00:33:26.870 --> 00:33:33.470
And that's times v
plus 1/2 squared,
00:33:33.470 --> 00:33:45.760
and then the next term is
omega e, ye, e plus 1/2 cubed.
00:33:45.760 --> 00:33:48.370
So this is a dumb power series
in the vibrational quantum
00:33:48.370 --> 00:33:52.860
number, and so in
the spectrum, one
00:33:52.860 --> 00:33:58.050
is able to fit the spectrum
to these sorts of things.
00:33:58.050 --> 00:34:00.270
So that's what you
get experimentally,
00:34:00.270 --> 00:34:07.640
but what you want
to know is we want
00:34:07.640 --> 00:34:11.210
to know the force
constant, the reduced
00:34:11.210 --> 00:34:16.850
mass, the cubic anharmonicity
constant, the quartic
00:34:16.850 --> 00:34:18.870
anharmonicity
constant, and whatever.
00:34:18.870 --> 00:34:21.730
So these are
structural parameters,
00:34:21.730 --> 00:34:24.050
and these are
molecular parameters.
00:34:26.750 --> 00:34:29.540
People like to call them
spectroscopic parameters,
00:34:29.540 --> 00:34:32.115
but that implies something
more fundamental.
00:34:32.115 --> 00:34:35.429
These are just what you
measure in the spectrum.
00:34:35.429 --> 00:34:39.110
And so we want to know the
relationship between the things
00:34:39.110 --> 00:34:42.674
we measure and the
things we want to know,
00:34:42.674 --> 00:34:44.590
and so that's what
perturbation theory is for.
00:34:52.960 --> 00:34:54.070
OK.
00:34:54.070 --> 00:34:56.469
Now, to risk boring
you, I'm just
00:34:56.469 --> 00:35:00.410
going to go over material
that we've done before.
00:35:00.410 --> 00:35:06.720
So if we have Q
to the n, we have
00:35:06.720 --> 00:35:12.440
this constant out in
front, h bar over 4 pi c
00:35:12.440 --> 00:35:23.170
mu omega twiddle to the n over
2, a plus a dagger to the n.
00:35:28.090 --> 00:35:31.000
And so we're going to
be constantly dealing
00:35:31.000 --> 00:35:34.060
with terms like a
plus a dagger squared
00:35:34.060 --> 00:35:37.745
and plus a dagger
cubed and so on.
00:35:37.745 --> 00:35:40.120
These are the things I said
you're going to work out once
00:35:40.120 --> 00:35:42.610
in your life and
either remember or just
00:35:42.610 --> 00:35:45.520
become so practiced with
it, you'll do it faster.
00:35:48.470 --> 00:35:52.430
And so these contain the
values of the matrix elements
00:35:52.430 --> 00:35:53.540
and the selection rules.
00:35:57.200 --> 00:35:58.200
OK.
00:35:58.200 --> 00:36:06.170
So I'm going to do an
example of the cube,
00:36:06.170 --> 00:36:10.850
and that's, of course, this
thing in constant the 3/2
00:36:10.850 --> 00:36:15.870
and then a cubed
plus a dagger cubed
00:36:15.870 --> 00:36:19.430
plus a whole bunch of
terms which have two a's
00:36:19.430 --> 00:36:21.770
and one a dagger.
00:36:21.770 --> 00:36:27.990
So let's just put it a
squared and a dagger.
00:36:27.990 --> 00:36:29.810
This isn't a computation rule.
00:36:29.810 --> 00:36:32.540
This is just three terms
that you have to deal with
00:36:32.540 --> 00:36:36.020
and three terms that have
a dagger squared and a.
00:36:38.570 --> 00:36:41.300
So that's what you
do your work on,
00:36:41.300 --> 00:36:44.420
because you don't want
to be messing around
00:36:44.420 --> 00:36:48.700
once you start doing
the perturbation sums,
00:36:48.700 --> 00:36:50.980
because it's ugly enough.
00:36:50.980 --> 00:36:53.800
So you want to simplify this as
much as possible, and you do.
00:36:57.880 --> 00:37:01.460
And so the purpose is this
is delta v of minus 3.
00:37:01.460 --> 00:37:03.770
This is delta v of plus 3.
00:37:03.770 --> 00:37:05.520
This is delta v of minus 1.
00:37:05.520 --> 00:37:08.300
This is delta v of plus 1.
00:37:08.300 --> 00:37:14.060
You arrange the terms
according to selection rules,
00:37:14.060 --> 00:37:16.450
and so what you
end up getting is
00:37:16.450 --> 00:37:25.230
Q cubed is equal to this
thing, to the 3/2 times
00:37:25.230 --> 00:37:41.840
a cubed plus 3 a n plus 3
a dagger n plus 1 plus a.
00:37:41.840 --> 00:37:46.470
OK, the algebra is
tedious, pretty simple.
00:37:46.470 --> 00:37:49.080
And what you usually
want to do is
00:37:49.080 --> 00:37:52.170
have the thing that changes
the vibrational quantum
00:37:52.170 --> 00:37:55.170
number after the thing
that preserves it,
00:37:55.170 --> 00:37:58.230
because then it's easy just
to write down these matrix
00:37:58.230 --> 00:38:00.920
elements just by inspection.
00:38:00.920 --> 00:38:01.730
OK?
00:38:01.730 --> 00:38:03.330
You could do it the
other way around.
00:38:03.330 --> 00:38:07.550
It's more complicated if you
put the n first and then the a,
00:38:07.550 --> 00:38:11.000
but it's up to you.
00:38:11.000 --> 00:38:14.760
So these then are
what you work on,
00:38:14.760 --> 00:38:17.371
and now we start doing
non-degenerate perturbation
00:38:17.371 --> 00:38:17.870
theory.
00:38:17.870 --> 00:38:21.200
The first thing
you do is you want
00:38:21.200 --> 00:38:24.620
to know, well, what is
the first-order correction
00:38:24.620 --> 00:38:26.210
to the energy?
00:38:26.210 --> 00:38:37.995
And if this is Q to the 3rd
power, then this, that's 0.
00:38:40.710 --> 00:38:42.870
So you like 0's,
but in this case,
00:38:42.870 --> 00:38:45.570
you're kind of
disappointed, because you
00:38:45.570 --> 00:38:49.230
don't know what the sign
of the coefficient of Q
00:38:49.230 --> 00:38:54.210
to the 3rd power is from
any experiment or at least
00:38:54.210 --> 00:38:57.240
any experiment at the
level we've described.
00:38:57.240 --> 00:39:02.760
When we introduce rotation
as well as vibration,
00:39:02.760 --> 00:39:05.130
there will be something
that reports the sign
00:39:05.130 --> 00:39:09.300
of the coefficient of Q 3rd.
00:39:09.300 --> 00:39:13.470
So now, we're stuck, and
we have to start doing
00:39:13.470 --> 00:39:15.930
all of this second-order stuff.
00:39:24.610 --> 00:39:25.270
OK.
00:39:25.270 --> 00:39:31.220
Well, we have this
bQ to the 3rd power,
00:39:31.220 --> 00:39:35.480
and so we get squares
of the matrix element.
00:39:35.480 --> 00:39:37.510
So we get a b squared.
00:39:37.510 --> 00:39:45.615
We get this thing
to the 3rd power,
00:39:45.615 --> 00:39:47.490
because we're squaring
the matrix element, so
00:39:47.490 --> 00:39:51.030
this bunch of constants.
00:39:51.030 --> 00:39:56.370
And then, you get a
matrix element v prime,
00:39:56.370 --> 00:40:07.360
some operator v, then we
get an energy denominator.
00:40:07.360 --> 00:40:10.380
Now, there's a lot
of symbols, but we
00:40:10.380 --> 00:40:17.940
want to simplify things as much
as possible, hc omega times v
00:40:17.940 --> 00:40:20.670
minus v prime.
00:40:20.670 --> 00:40:23.670
So this is matrix element
squared over an energy
00:40:23.670 --> 00:40:27.150
denominator, and what
you really want to know
00:40:27.150 --> 00:40:30.580
is what is the quantum number
dependence of everything?
00:40:30.580 --> 00:40:34.710
OK, so the operator here
is either Q dagger cubed,
00:40:34.710 --> 00:40:37.770
Q dagger--
00:40:37.770 --> 00:40:47.940
I'm sorry, a cubed, a times n,
a dagger n plus 1, or a dagger
00:40:47.940 --> 00:40:49.140
cubed.
00:40:49.140 --> 00:40:52.980
And so we know how to write
all of these matrix elements,
00:40:52.980 --> 00:40:55.590
trivially, no work.
00:40:55.590 --> 00:40:59.070
Once we've simplified
here, it's really trivial
00:40:59.070 --> 00:41:04.910
to write the squared
matrix element.
00:41:07.620 --> 00:41:18.030
OK, so we do it, and
so we arrange things
00:41:18.030 --> 00:41:23.985
according to delta v
and we have delta v
00:41:23.985 --> 00:41:30.600
of plus 3, plus 1,
minus 1, and minus 3.
00:41:30.600 --> 00:41:36.450
And so what we get from the
square of the matrix element
00:41:36.450 --> 00:41:47.470
a dagger cubed, we get v
plus 1, v plus 2, v plus 3,
00:41:47.470 --> 00:41:52.180
and we also have an
energy denominator.
00:41:52.180 --> 00:41:57.820
And we're going to get for
this one 1 over minus 3,
00:41:57.820 --> 00:42:05.380
because the initial
quantum number is v,
00:42:05.380 --> 00:42:10.420
and the second quantum
number, v prime, is v plus 3,
00:42:10.420 --> 00:42:14.520
and so we get a minus 3
in the energy denominator.
00:42:14.520 --> 00:42:17.430
And then the plus 1,
that comes out to be--
00:42:23.106 --> 00:42:25.800
I've got it in a different
order in my notes--
00:42:25.800 --> 00:42:41.550
that comes out to be 9, v plus
1, v plus 2, v plus 1 squared,
00:42:41.550 --> 00:42:45.423
and the energy denominator
for this is 1 over minus 1.
00:42:48.261 --> 00:42:49.090
OK.
00:42:49.090 --> 00:42:50.630
Then, we have the--
00:42:50.630 --> 00:42:54.210
I'm going to skip this one-- we
have the matrix element here,
00:42:54.210 --> 00:42:59.537
and that's going to be
v minus 1, v minus 2.
00:42:59.537 --> 00:43:01.120
Remember, we're
squaring these things.
00:43:01.120 --> 00:43:05.420
So that's why we don't have
those square roots anymore,
00:43:05.420 --> 00:43:07.560
and we have an energy
denominator 1 over 3.
00:43:10.430 --> 00:43:14.320
OK, now advice--
you don't like this,
00:43:14.320 --> 00:43:17.740
and you want to
minimize your effort.
00:43:17.740 --> 00:43:20.470
And so it turns out that
if you take the terms
00:43:20.470 --> 00:43:23.020
with the equal and opposite
energy denominators
00:43:23.020 --> 00:43:26.975
and combine them,
simplifications occur.
00:43:26.975 --> 00:43:28.600
One of the things
you can see is you're
00:43:28.600 --> 00:43:31.060
going to have a v
cubed here, and you're
00:43:31.060 --> 00:43:33.010
going to have a v cubed here.
00:43:33.010 --> 00:43:39.930
They're going to cancel,
because we have a 1 over minus 3
00:43:39.930 --> 00:43:42.750
and a 1 over plus 3.
00:43:42.750 --> 00:43:45.000
So you get an algebraic
simplification
00:43:45.000 --> 00:43:50.280
when you take these terms
pairwise and combine them.
00:43:50.280 --> 00:43:54.700
Now, you're never in your
life going to do this,
00:43:54.700 --> 00:43:58.660
but if you did do it,
this is how you end up
00:43:58.660 --> 00:44:04.390
with formulas which are simple.
00:44:04.390 --> 00:44:06.670
They're horrible getting
there, unless you
00:44:06.670 --> 00:44:07.910
know how to get there.
00:44:07.910 --> 00:44:08.860
So now there's a rule.
00:44:15.920 --> 00:44:19.990
So if the perturbation
is Q to the n,
00:44:19.990 --> 00:44:33.800
then the highest-order
term involves v plus 1/2
00:44:33.800 --> 00:44:35.133
to the n minus 1.
00:44:38.837 --> 00:44:42.240
The reason for that is
in the matrix element
00:44:42.240 --> 00:44:49.910
the highest-order term is
v plus 1/2 to the 3 over 2,
00:44:49.910 --> 00:44:54.399
and then you square it,
you get to the 3rd power,
00:44:54.399 --> 00:44:56.065
and then the highest-order
term cancels.
00:45:01.640 --> 00:45:09.730
So we know that if we're
dealing with Q to the 3rd power,
00:45:09.730 --> 00:45:14.750
we're going to get a
term v plus 1/2 squared.
00:45:14.750 --> 00:45:19.550
If we're dealing with
Q to the 4th power,
00:45:19.550 --> 00:45:22.270
well, then we get
something from second order
00:45:22.270 --> 00:45:25.040
from the first-order
correction to the energy
00:45:25.040 --> 00:45:29.270
a delta v of one matrix element.
00:45:29.270 --> 00:45:34.760
What we've done is square
it, and we get v plus 1/2
00:45:34.760 --> 00:45:35.930
squared also.
00:45:39.260 --> 00:45:45.070
OK and the off-diagonal
matrix element,
00:45:45.070 --> 00:45:49.660
we're going to get
from the highest order
00:45:49.660 --> 00:45:55.140
from the off-diagonal
is v plus 1/2 cubed.
00:45:55.140 --> 00:45:58.500
So one of the things that
is great about the algebra
00:45:58.500 --> 00:46:00.420
is with a little
bit of practice,
00:46:00.420 --> 00:46:07.470
you know how to organize things,
and if you do the algebra,
00:46:07.470 --> 00:46:09.660
you collect the
highest-order terms.
00:46:09.660 --> 00:46:12.060
Then, there's that kind
of cascading result,
00:46:12.060 --> 00:46:16.410
and you get the lowest-order
terms in simplified form.
00:46:16.410 --> 00:46:19.500
So this is irrelevant,
but this is how you do it,
00:46:19.500 --> 00:46:21.822
if you're a professional.
00:46:21.822 --> 00:46:27.110
So with these results,
we can determine
00:46:27.110 --> 00:46:30.920
the relationship between
these molecular constants.
00:46:30.920 --> 00:46:31.910
Where did I put them?
00:46:31.910 --> 00:46:33.110
Oh, probably on this board.
00:46:36.120 --> 00:46:38.580
No.
00:46:38.580 --> 00:46:39.840
Oh yeah, it's right here.
00:46:39.840 --> 00:46:43.130
So we have omega e,
omega e xe, omega e ye,
00:46:43.130 --> 00:46:46.010
and we have then
the relationships
00:46:46.010 --> 00:46:48.770
between these things
which you measure
00:46:48.770 --> 00:46:50.990
and these things that
you want to know.
00:46:50.990 --> 00:46:53.240
So the stuff you want
to know is encoded
00:46:53.240 --> 00:46:56.420
in the spectrum in a not
particularly complicated way--
00:46:56.420 --> 00:46:58.070
it's just not a very
interesting way,
00:46:58.070 --> 00:47:02.990
but you have to do it
in order to get it.
00:47:02.990 --> 00:47:06.960
OK, so I've got just
a few minutes left,
00:47:06.960 --> 00:47:09.230
but I want to get to the
really interesting stuff.
00:47:12.330 --> 00:47:24.720
So if you have some
expression for the potential,
00:47:24.720 --> 00:47:29.910
it's easy to go and get the
expression for the energy
00:47:29.910 --> 00:47:34.700
levels and the wave functions.
00:47:34.700 --> 00:47:37.950
And you can use the vector
picture or the wave function
00:47:37.950 --> 00:47:38.450
picture.
00:47:38.450 --> 00:47:42.860
It doesn't matter,
and so this is
00:47:42.860 --> 00:47:45.320
enough to do the spectroscopy.
00:47:45.320 --> 00:47:49.410
It tells you not only
where are the energies,
00:47:49.410 --> 00:47:52.670
but because some
transitions are supposed
00:47:52.670 --> 00:47:56.600
to be weak because of
the dipole selection rule
00:47:56.600 --> 00:47:58.970
or whatever, it
says, well, there
00:47:58.970 --> 00:48:01.460
are some transitions that
have borrowed intensity.
00:48:05.780 --> 00:48:08.930
It also tells you--
00:48:08.930 --> 00:48:12.970
suppose we make a coherent
superposition state at T
00:48:12.970 --> 00:48:18.410
equals 0 using a short
pulse, and suppose
00:48:18.410 --> 00:48:23.520
in the linear combination
of zero-order states,
00:48:23.520 --> 00:48:27.380
there is only one
that is bright.
00:48:27.380 --> 00:48:30.020
And so then it
tells us, if we know
00:48:30.020 --> 00:48:34.880
how to go from basis
states to eigenstates,
00:48:34.880 --> 00:48:39.800
we can go backwards, and
we can write the expression
00:48:39.800 --> 00:48:44.240
for the T equals 0
superposition in terms
00:48:44.240 --> 00:48:48.520
of some sum over psi v0 cv.
00:48:54.030 --> 00:48:55.770
And if we have
this, as long as we
00:48:55.770 --> 00:49:00.730
are writing a superposition
in terms of eigenstates,
00:49:00.730 --> 00:49:05.156
we know immediately
how to get to this,
00:49:05.156 --> 00:49:06.655
and then we've got
all the dynamics.
00:49:10.060 --> 00:49:13.870
So the perturbation
theory enables
00:49:13.870 --> 00:49:17.590
you to say, if I want to
work in the time domain,
00:49:17.590 --> 00:49:19.420
I know what to do.
00:49:23.920 --> 00:49:25.710
You're going to get--
00:49:25.710 --> 00:49:26.620
in the time domain--
00:49:26.620 --> 00:49:30.670
you're going to get a
signal that oscillates,
00:49:30.670 --> 00:49:35.080
and it oscillates at frequencies
corresponding to energy level
00:49:35.080 --> 00:49:39.590
differences divided by h bar.
00:49:39.590 --> 00:49:44.640
And so what frequencies will
appear in the Fourier transform
00:49:44.640 --> 00:49:51.780
of the spectrum, and what are
the amplitudes of those Fourier
00:49:51.780 --> 00:49:53.190
components?
00:49:53.190 --> 00:49:55.860
You can calculate
all of those stuff.
00:49:55.860 --> 00:49:57.870
It all comes from
perturbation theory.
00:49:57.870 --> 00:50:02.670
These mixing coefficients you
get by perturbation theory.
00:50:02.670 --> 00:50:07.340
And remember, if you have
a transformation that
00:50:07.340 --> 00:50:15.420
diagonalizes the Hamiltonian
that the eigenstates correspond
00:50:15.420 --> 00:50:21.720
to, the columns of T
dagger, and the expression
00:50:21.720 --> 00:50:25.320
of the zero-order states,
in terms of the eigenstates,
00:50:25.320 --> 00:50:28.440
corresponds to either
the columns of T
00:50:28.440 --> 00:50:30.216
or the rows of T dagger.
00:50:33.120 --> 00:50:35.430
So once you do the
perturbation theory,
00:50:35.430 --> 00:50:42.630
you can go to the
frequency domain
00:50:42.630 --> 00:50:46.140
spectrum with
intensities and frequency
00:50:46.140 --> 00:50:51.160
or the time domain spectrum
with amplitude and frequencies.
00:50:51.160 --> 00:50:52.110
It's all there.
00:50:52.110 --> 00:50:53.980
This is a complete tool.
00:50:53.980 --> 00:50:56.220
It's the kind of
tool that you can
00:50:56.220 --> 00:50:59.910
use for an enormous
number of problems,
00:50:59.910 --> 00:51:02.640
and so you'd better get
comfortable with perturbation
00:51:02.640 --> 00:51:06.090
theory, because the people
who aren't comfortable
00:51:06.090 --> 00:51:09.242
can't do anything
except talk about it.
00:51:09.242 --> 00:51:12.860
But if you want to actually
solve problems, especially
00:51:12.860 --> 00:51:15.530
problems on an exam,
you want to know
00:51:15.530 --> 00:51:17.750
how to use perturbation theory.
00:51:17.750 --> 00:51:22.130
And you also want to know
how to read and construct
00:51:22.130 --> 00:51:26.090
the relevant notation
in the vector picture,
00:51:26.090 --> 00:51:29.090
because a vector picture
and the matrix picture
00:51:29.090 --> 00:51:31.790
is the one where you
see the entire structure
00:51:31.790 --> 00:51:33.900
of the problem.
00:51:33.900 --> 00:51:38.610
And you can decide on how you're
going to organize your time
00:51:38.610 --> 00:51:41.110
or what are the important
things that I'm going
00:51:41.110 --> 00:51:43.460
to get from this analysis.
00:51:43.460 --> 00:51:46.780
And so it's much better than
the Schrodinger picture,
00:51:46.780 --> 00:51:49.690
because with the
Schrodinger picture,
00:51:49.690 --> 00:51:52.210
you're just solving a
differential equation and one
00:51:52.210 --> 00:51:55.060
problem at a time, whereas
with the matrix picture,
00:51:55.060 --> 00:51:58.510
you're solving all
problems at once.
00:51:58.510 --> 00:52:02.310
This is really a
wonderful thing,
00:52:02.310 --> 00:52:05.780
and so that's why I'm
taking a very different path
00:52:05.780 --> 00:52:08.730
from what is in the textbooks.
00:52:08.730 --> 00:52:10.980
Your wonderful
textbook McQuarrie
00:52:10.980 --> 00:52:14.650
does not do second-order
perturbation theory.
00:52:14.650 --> 00:52:19.190
So nothing you want can be
calculated, unless you're
00:52:19.190 --> 00:52:24.560
dealing with NMR, and you're
dealing with magnetic dipole
00:52:24.560 --> 00:52:25.730
transitions.
00:52:25.730 --> 00:52:31.400
And then, you can get
a lot of good stuff
00:52:31.400 --> 00:52:33.170
from first-order
perturbation theory,
00:52:33.170 --> 00:52:38.370
and you can avoid second
order until you grow up.
00:52:38.370 --> 00:52:39.700
OK, I'm done.
00:52:39.700 --> 00:52:42.390
I'll be talking about
rigid rotor next time,
00:52:42.390 --> 00:52:46.310
and I will be talking about it
also in an unconventional way.