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ROBERT FIELD: This
lecture is not
00:00:24.550 --> 00:00:28.060
relevant to this
exam or any exam.
00:00:28.060 --> 00:00:33.670
It's time-dependent quantum
mechanics, which you probably
00:00:33.670 --> 00:00:37.060
want to know about,
but it's a lot
00:00:37.060 --> 00:00:39.700
to digest at the
level of this course.
00:00:39.700 --> 00:00:47.530
So I'm going to introduce a lot
of the tricks and terminology,
00:00:47.530 --> 00:00:50.410
and I hope that some of
you will care about that
00:00:50.410 --> 00:00:52.870
and will go on to use this.
00:00:52.870 --> 00:00:57.020
But mostly, this is
a first exposure,
00:00:57.020 --> 00:00:59.520
and there's a lot of derivation.
00:00:59.520 --> 00:01:02.510
And it's hard to see the
forest for the trees.
00:01:02.510 --> 00:01:05.379
OK, so these are
the important things
00:01:05.379 --> 00:01:07.390
that I'm going to
cover in the lecture.
00:01:07.390 --> 00:01:10.540
First, the dipole
approximation--
00:01:10.540 --> 00:01:14.950
how can we simplify the
interaction between molecules
00:01:14.950 --> 00:01:17.090
and electromagnetic radiation?
00:01:17.090 --> 00:01:20.050
This is the main
simplification, and I'll
00:01:20.050 --> 00:01:21.970
explain where it comes from.
00:01:21.970 --> 00:01:24.490
Then we have
transitions that occur.
00:01:24.490 --> 00:01:28.000
And they're caused by a
time-dependent perturbation
00:01:28.000 --> 00:01:36.190
where the zero-order
Hamiltonian is time-independent,
00:01:36.190 --> 00:01:39.169
but the perturbation
term is time-dependent.
00:01:39.169 --> 00:01:40.210
And what does that cause?
00:01:40.210 --> 00:01:43.120
It causes transitions.
00:01:43.120 --> 00:01:45.700
We're going to express
the problem in terms
00:01:45.700 --> 00:01:49.510
of the eigenstates of the
time-independent Hamiltonian,
00:01:49.510 --> 00:01:52.750
the zero-order Hamiltonian,
and we know that these always
00:01:52.750 --> 00:01:55.600
have this time-dependent
factor if we're
00:01:55.600 --> 00:01:59.100
doing time-dependent
quantum mechanics.
00:01:59.100 --> 00:02:00.810
The two crucial
approximations are
00:02:00.810 --> 00:02:06.030
going to be the electromagnetic
field is weak and continuous.
00:02:06.030 --> 00:02:09.389
Now many experiments involve
short pulses and very intense
00:02:09.389 --> 00:02:13.260
pulses, and the time-dependent
quantum mechanics
00:02:13.260 --> 00:02:16.170
for those problems is
completely different,
00:02:16.170 --> 00:02:18.480
but you need to
understand this in order
00:02:18.480 --> 00:02:22.440
to understand what's
different about it.
00:02:22.440 --> 00:02:24.270
We also assume
that we're starting
00:02:24.270 --> 00:02:26.250
the system in a
single eigenstate,
00:02:26.250 --> 00:02:28.680
and that's pretty normal.
00:02:28.680 --> 00:02:33.030
But often, you're starting
the system in many eigenstates
00:02:33.030 --> 00:02:34.080
that are uncorrelated.
00:02:34.080 --> 00:02:35.350
We don't talk about that.
00:02:35.350 --> 00:02:38.100
That's something that has to do
with the density matrix, which
00:02:38.100 --> 00:02:41.560
is beyond the level
of this course.
00:02:41.560 --> 00:02:43.350
And one of the
things that happens
00:02:43.350 --> 00:02:47.700
is we get this thing
called linear response.
00:02:47.700 --> 00:02:51.540
Now I went for years
hearing the reverence
00:02:51.540 --> 00:02:55.150
that people apply
to linear response,
00:02:55.150 --> 00:02:58.020
but I hadn't a clue what it was.
00:02:58.020 --> 00:03:00.270
So you can start out
knowing something
00:03:00.270 --> 00:03:02.220
about linear response.
00:03:02.220 --> 00:03:05.770
Now this all leads up
to Fermi's golden rule,
00:03:05.770 --> 00:03:10.050
which explains the rate
at which transitions
00:03:10.050 --> 00:03:13.690
occur between some initial
state and some final state.
00:03:13.690 --> 00:03:15.420
And there is a lot
more complexity
00:03:15.420 --> 00:03:18.130
in Fermi's golden rule than
what I'm going to present,
00:03:18.130 --> 00:03:23.670
but this is the first
step in understanding it.
00:03:23.670 --> 00:03:26.920
Then I'm going to
talk about where
00:03:26.920 --> 00:03:30.460
do pure rotation transitions
come from and vibrational
00:03:30.460 --> 00:03:31.190
transitions.
00:03:31.190 --> 00:03:33.460
Then at the end,
I'll show a movie
00:03:33.460 --> 00:03:37.420
which gives you a
sense of what goes on
00:03:37.420 --> 00:03:42.220
in making a transition
be strong and sharp.
00:03:45.020 --> 00:03:49.520
OK, I'm a spectroscopist,
and I use spectroscopy
00:03:49.520 --> 00:03:53.840
to learn all sorts of
secrets that molecules keep.
00:03:53.840 --> 00:03:56.690
And in order to do
that, I need to record
00:03:56.690 --> 00:04:01.190
a spectrum, which basically is
you have some radiation source.
00:04:01.190 --> 00:04:05.450
And you tune its frequency,
and things happen.
00:04:05.450 --> 00:04:08.610
And why do the things happen?
00:04:08.610 --> 00:04:12.860
How do we understand
the interaction
00:04:12.860 --> 00:04:16.990
of electromagnetic
radiation and a molecule?
00:04:16.990 --> 00:04:19.010
And there's really two
ways to understand it.
00:04:26.150 --> 00:04:42.020
We have one-way molecules as
targets, photons as bullets,
00:04:42.020 --> 00:04:44.580
and it's a simple
geometric picture.
00:04:44.580 --> 00:04:47.600
And the size of the target
is related to the transition
00:04:47.600 --> 00:04:50.820
moments, and it works.
00:04:50.820 --> 00:04:52.170
It's very, very simple.
00:04:52.170 --> 00:04:54.410
There's no time-dependent
quantum mechanics.
00:04:54.410 --> 00:04:56.270
It's probabilistic.
00:04:56.270 --> 00:05:01.200
And for the first 45
years of my career,
00:05:01.200 --> 00:05:07.130
this is the way I handled an
understanding of transitions
00:05:07.130 --> 00:05:09.597
caused by electromagnetic
radiation.
00:05:09.597 --> 00:05:10.180
This is wrong.
00:05:12.910 --> 00:05:15.170
It has a wide applicability.
00:05:15.170 --> 00:05:18.170
But if you try to
take it too seriously,
00:05:18.170 --> 00:05:20.320
you will miss a
lot of good stuff.
00:05:24.030 --> 00:05:28.970
The other way is to
use the time-dependent
00:05:28.970 --> 00:05:34.530
in your equation, and
it looks complicated
00:05:34.530 --> 00:05:37.196
because we're going
to be combining
00:05:37.196 --> 00:05:38.820
the time-dependent
Schrodinger equation
00:05:38.820 --> 00:05:41.217
and the time-independent
Schrodinger equation.
00:05:41.217 --> 00:05:43.800
We're going to be thinking about
the electromagnetic radiation
00:05:43.800 --> 00:05:47.730
as waves rather than
photons, and that
00:05:47.730 --> 00:05:50.250
means there is constructive
and destructive interference.
00:05:50.250 --> 00:05:52.680
There's phase
information, which is not
00:05:52.680 --> 00:05:55.140
present in the
molecules-as-targets,
00:05:55.140 --> 00:05:57.610
photons-as-bullets picture.
00:05:57.610 --> 00:05:59.960
Now I don't want you
to say, well, I'm
00:05:59.960 --> 00:06:01.710
never going to think
this way because it's
00:06:01.710 --> 00:06:04.430
so easy to think about trends.
00:06:04.430 --> 00:06:07.080
And, you know, the Beer-Lambert
law, all these things that you
00:06:07.080 --> 00:06:12.930
use to describe the probability
of an absorption or emission
00:06:12.930 --> 00:06:18.190
transition, this sort of
thing is really useful.
00:06:18.190 --> 00:06:31.540
OK, so this is the right
way, and the crucial step
00:06:31.540 --> 00:06:33.430
is the dipole approximation.
00:06:36.370 --> 00:06:47.140
So we have
electromagnetic radiation
00:06:47.140 --> 00:06:51.920
being a combination of electric
field and magnetic field,
00:06:51.920 --> 00:06:57.250
and we can describe the
electric field in terms of--
00:07:11.571 --> 00:07:12.070
OK.
00:07:14.710 --> 00:07:21.230
So this is a vector, and it's
a function of a vector in time.
00:07:21.230 --> 00:07:24.760
And there is some magnitude,
which is a vector.
00:07:24.760 --> 00:07:27.430
And its cosine of
this thing, this
00:07:27.430 --> 00:07:34.900
is the wave vector, which
is 2 pi over the wavelength,
00:07:34.900 --> 00:07:37.960
but it also has a direction.
00:07:37.960 --> 00:07:40.030
And it points in the
direction that the radiation
00:07:40.030 --> 00:07:42.360
is propagating.
00:07:42.360 --> 00:07:46.390
And this is the
position coordinate,
00:07:46.390 --> 00:07:47.720
and this is the frequency.
00:07:47.720 --> 00:07:54.150
So there's a similar expression
for the magnetic part--
00:07:59.130 --> 00:08:00.126
same thing.
00:08:07.120 --> 00:08:11.090
I should leave this exposed.
00:08:11.090 --> 00:08:13.910
So we know several things.
00:08:13.910 --> 00:08:17.060
We know for
electromagnetic radiation
00:08:17.060 --> 00:08:21.500
that the electric
field is always
00:08:21.500 --> 00:08:25.010
perpendicular to
the magnetic field.
00:08:25.010 --> 00:08:33.580
We know a relationship
between the constant term,
00:08:33.580 --> 00:08:38.090
and we have this k vector,
which points in the propagation
00:08:38.090 --> 00:08:40.659
direction.
00:08:40.659 --> 00:08:41.799
Now the question is--
00:08:50.710 --> 00:08:54.250
because we have--
we have a molecule
00:08:54.250 --> 00:08:56.530
and we have the
electromagnetic radiation.
00:08:59.910 --> 00:09:04.700
And so the question is, what's
a typical size for a molecule
00:09:04.700 --> 00:09:05.510
in the gas field?
00:09:05.510 --> 00:09:06.820
Well, anywhere?
00:09:06.820 --> 00:09:08.040
Pick a number.
00:09:08.040 --> 00:09:11.250
How big is a molecule?
00:09:11.250 --> 00:09:13.664
STUDENT: A couple angstroms?
00:09:13.664 --> 00:09:15.330
ROBERT FIELD: I like
a couple angstroms.
00:09:15.330 --> 00:09:17.010
That's a diatomic molecule.
00:09:17.010 --> 00:09:19.080
They're going to be
people who like proteins,
00:09:19.080 --> 00:09:22.560
and they're going to talk
about 10 or 100 nanometers.
00:09:22.560 --> 00:09:28.140
But typically, you can say
1 nanometer or 2 angstroms
00:09:28.140 --> 00:09:30.960
or something like that.
00:09:30.960 --> 00:09:33.940
Now we're going to shine
light at a molecule.
00:09:33.940 --> 00:09:37.550
What's the typical
wavelength of light that we
00:09:37.550 --> 00:09:41.740
use to record a spectrum?
00:09:41.740 --> 00:09:45.020
Visible wavelength.
00:09:45.020 --> 00:09:46.950
What is that?
00:09:46.950 --> 00:09:48.950
STUDENT: 400 to 700 nanometers?
00:09:48.950 --> 00:09:52.340
ROBERT FIELD: Yeah, so
the wavelength of light
00:09:52.340 --> 00:09:55.850
is on the order of,
say, 500 nanometers.
00:09:55.850 --> 00:10:04.130
And if it's in the infrared,
it might be 10,000 nanometers.
00:10:04.130 --> 00:10:07.100
If it were in the
visible, it might be 100--
00:10:07.100 --> 00:10:10.550
in the ultraviolet, it might
be as short as 100 nanometers.
00:10:10.550 --> 00:10:15.170
But the point is that this
wavelength is much, much larger
00:10:15.170 --> 00:10:18.470
than the size of a molecule,
so this picture here
00:10:18.470 --> 00:10:19.400
is complete garbage.
00:10:22.290 --> 00:10:27.550
The picture for the ratio--
00:10:27.550 --> 00:10:30.330
so even this is garbage.
00:10:30.330 --> 00:10:32.250
The electric field
or the magnetic field
00:10:32.250 --> 00:10:36.090
that the molecule sees is
constant over the length
00:10:36.090 --> 00:10:43.750
of the molecule to a
very good approximation.
00:10:43.750 --> 00:10:52.580
So now we have this
expression for the field,
00:10:52.580 --> 00:10:57.640
and this is a number which
is a very, very small number
00:10:57.640 --> 00:11:01.480
times a still
pretty small number.
00:11:01.480 --> 00:11:05.500
This k dot r is very small.
00:11:05.500 --> 00:11:11.750
It says we can expand
this in a power series
00:11:11.750 --> 00:11:16.520
and throw away everything
except the omega t.
00:11:16.520 --> 00:11:19.840
That's the dipole approximation.
00:11:19.840 --> 00:11:28.000
So all of a sudden, we
have as our electric field
00:11:28.000 --> 00:11:32.290
just E0 cosine omega t.
00:11:34.970 --> 00:11:37.430
That's fantastic.
00:11:37.430 --> 00:11:41.630
So we've gotten rid of the
spatial degree of freedom,
00:11:41.630 --> 00:11:44.430
and that enables us to
do all sorts of things
00:11:44.430 --> 00:11:47.700
that would have required
a lot more justification.
00:11:47.700 --> 00:11:54.320
Now sometimes we need to
keep higher order terms
00:11:54.320 --> 00:11:55.070
in this expansion.
00:11:55.070 --> 00:11:57.920
We've kept none of them,
just the zero order term.
00:11:57.920 --> 00:12:03.350
And so if we do, that's called
quadrupole or octopole or
00:12:03.350 --> 00:12:06.380
hexadecapole, and
there are transitions
00:12:06.380 --> 00:12:10.520
that are not dipole allowed
but are quadrupole allowed.
00:12:10.520 --> 00:12:14.120
And they're incredibly
weak because k
00:12:14.120 --> 00:12:16.100
dot r is really, really small.
00:12:19.520 --> 00:12:25.140
Now the intensity of
quadrupole-allowed transitions
00:12:25.140 --> 00:12:30.230
is on the order of a million
times smaller than dipole.
00:12:30.230 --> 00:12:31.570
So why go there?
00:12:31.570 --> 00:12:35.890
Well, sometimes the dipole
transitions are forbidden.
00:12:35.890 --> 00:12:38.550
And so if you're going to get
the molecule to talk to you,
00:12:38.550 --> 00:12:41.120
you're going to have
to somehow make use
00:12:41.120 --> 00:12:43.010
of the quadrupole transitions.
00:12:43.010 --> 00:12:45.770
But it's a completely
different kind of experiment
00:12:45.770 --> 00:12:48.290
because you have to have
an incredibly long path
00:12:48.290 --> 00:12:51.720
length and a relatively
high number density.
00:12:51.720 --> 00:12:54.170
And so you don't
want to go there,
00:12:54.170 --> 00:12:56.270
and that's something
that's beside--
00:12:56.270 --> 00:12:58.800
aside from what we care about.
00:12:58.800 --> 00:13:05.420
So now many of you are going to
be doing experiments involving
00:13:05.420 --> 00:13:13.280
light, and that will
involve the electric field.
00:13:13.280 --> 00:13:16.830
Some of you will be
doing magnetic resonance,
00:13:16.830 --> 00:13:19.680
and they will be
thinking entirely
00:13:19.680 --> 00:13:21.450
about the magnetic field.
00:13:24.690 --> 00:13:26.560
The theory is the same.
00:13:26.560 --> 00:13:32.330
It's just the main actor
is a little bit different.
00:13:32.330 --> 00:13:36.740
Now if we're dealing
with an electric field,
00:13:36.740 --> 00:13:40.670
we are interested
in the symmetry
00:13:40.670 --> 00:13:45.410
of this operator, which is
the electric field dotted
00:13:45.410 --> 00:13:51.390
into the molecular
dipole moment,
00:13:51.390 --> 00:13:56.610
and that operator
has odd parity.
00:13:56.610 --> 00:14:00.000
And so now I'm not going
to tell you what parity is.
00:14:00.000 --> 00:14:03.150
But because this has
odd parity, there
00:14:03.150 --> 00:14:06.840
are only transitions between
states of opposite parity,
00:14:06.840 --> 00:14:12.180
whereas this, the magnetic
operator, has even parity.
00:14:12.180 --> 00:14:15.000
And so they only have
transitions between states
00:14:15.000 --> 00:14:16.410
of the same parity.
00:14:16.410 --> 00:14:18.710
Now you want to be curious
about what parity is,
00:14:18.710 --> 00:14:20.250
and I'm not going to tell you.
00:14:20.250 --> 00:14:26.160
OK, so the problem
is tremendously
00:14:26.160 --> 00:14:29.280
simplified by the
fact that now we just
00:14:29.280 --> 00:14:33.900
have a time-dependent
field, which
00:14:33.900 --> 00:14:36.490
is constant over the molecule.
00:14:36.490 --> 00:14:40.110
So the molecule is seeing
an oscillatory field,
00:14:40.110 --> 00:14:43.140
but the whole molecule is
feeling that same field.
00:14:46.870 --> 00:14:50.010
OK, now we're ready to start
doing quantum mechanics.
00:14:57.540 --> 00:15:02.040
So the interaction term,
the thing that causes
00:15:02.040 --> 00:15:03.330
transitions to occur--
00:15:05.970 --> 00:15:11.570
the electric interaction term,
which we're going to call
00:15:11.570 --> 00:15:15.720
H1 because it's a perturbation.
00:15:15.720 --> 00:15:18.480
We're going to be doing
something in perturbation
00:15:18.480 --> 00:15:21.550
theory, but it's time-dependent
perturbation theory,
00:15:21.550 --> 00:15:24.300
which is a whole lot
more complicated and rich
00:15:24.300 --> 00:15:26.040
than ordinary time-independent.
00:15:26.040 --> 00:15:31.500
Now many of you have found
time-independent perturbation
00:15:31.500 --> 00:15:36.270
theory tedious and
algebraically complicated.
00:15:36.270 --> 00:15:37.770
Time-dependent
perturbation theory
00:15:37.770 --> 00:15:41.460
for these kinds of
operators is not tedious.
00:15:41.460 --> 00:15:43.050
It's really beautiful.
00:15:43.050 --> 00:15:45.510
And there are many, many cases.
00:15:45.510 --> 00:15:47.580
It's not just having
another variable.
00:15:47.580 --> 00:15:50.550
There's a lot of
really neat stuff.
00:15:50.550 --> 00:15:55.290
And what I'm going to present
today or I am presenting today
00:15:55.290 --> 00:16:00.180
is the theory for CW radiation--
that's continuous radiation--
00:16:00.180 --> 00:16:05.430
really weak, interacting
with a molecule or a system
00:16:05.430 --> 00:16:09.190
in a single quantum
state initially.
00:16:09.190 --> 00:16:11.560
And it's important.
00:16:11.560 --> 00:16:16.850
The really weak and the CW are
two really important features.
00:16:16.850 --> 00:16:20.030
And the single quantum
state is just a convenience.
00:16:20.030 --> 00:16:20.990
We can deal with that.
00:16:20.990 --> 00:16:22.930
That's not a big
deal, but it does
00:16:22.930 --> 00:16:29.050
involve using a different, more
physical, or a more correct
00:16:29.050 --> 00:16:32.050
definition of what we mean
by an average measurement
00:16:32.050 --> 00:16:35.240
on a system of many particles.
00:16:35.240 --> 00:16:38.170
And you'll hear the word
"density matrix" if you go on
00:16:38.170 --> 00:16:39.421
in physical chemistry.
00:16:39.421 --> 00:16:41.170
But I'm not going to
do anything about it,
00:16:41.170 --> 00:16:43.400
but that's how we deal with it.
00:16:43.400 --> 00:16:53.230
OK, so this is going
to be minus mu--
00:16:53.230 --> 00:16:59.720
it's a vector-- dot E of
t, which is also a vector.
00:16:59.720 --> 00:17:02.720
Now a dot product,
that looks really neat.
00:17:02.720 --> 00:17:06.500
However, this is a vector
in the molecular frame,
00:17:06.500 --> 00:17:09.150
and this is a vector in
the laboratory frame.
00:17:09.150 --> 00:17:12.530
So this dot product is a
whole bunch more complicated
00:17:12.530 --> 00:17:15.720
than you would think.
00:17:15.720 --> 00:17:17.910
Now I do want to
mention that when
00:17:17.910 --> 00:17:22.680
we talk about the rigid rotor,
the rigid rotor is telling
00:17:22.680 --> 00:17:26.550
us what is the probability
amplitude of the orientation
00:17:26.550 --> 00:17:30.420
of the molecular frame relative
to the laboratory frame.
00:17:30.420 --> 00:17:34.710
So that is where all this
information about these two
00:17:34.710 --> 00:17:38.980
different coordinate
systems reside,
00:17:38.980 --> 00:17:42.330
and we'll see a
little bit of that.
00:17:42.330 --> 00:17:49.012
OK, there's a similar expression
for the magnetic term.
00:17:49.012 --> 00:17:50.470
I'm just not going
to write it down
00:17:50.470 --> 00:17:53.290
because it's just too
much stuff to write down.
00:17:53.290 --> 00:17:57.580
So the Hamiltonian, the
time-independent Hamiltonian,
00:17:57.580 --> 00:18:04.200
can be expressed
as H0 plus H1 of t.
00:18:07.190 --> 00:18:10.460
This looks exactly like
time-independent perturbation
00:18:10.460 --> 00:18:14.420
theory, except this guy, which
makes all the complications is
00:18:14.420 --> 00:18:17.990
time dependent.
00:18:17.990 --> 00:18:21.690
But this says, OK, we
can find a whole set,
00:18:21.690 --> 00:18:26.280
a complete set of eignenenergies
and eigenfunctions.
00:18:26.280 --> 00:18:29.892
And we know how to write the
time-dependent Schrodinger--
00:18:29.892 --> 00:18:31.850
the solutions of the
time-dependent Schrodinger
00:18:31.850 --> 00:18:34.790
equation if this
is the whole game.
00:18:34.790 --> 00:18:37.250
So we're going to use
these as basis functions
00:18:37.250 --> 00:18:40.340
just as we did in ordinary
perturbation theory.
00:18:45.697 --> 00:18:55.220
So H0 times some
eigenfunction, which now I'm
00:18:55.220 --> 00:19:02.493
writing as explicitly
time-dependent is En phi n t
00:19:02.493 --> 00:19:08.180
equals 0 e to the minus
i En t over h-bar.
00:19:08.180 --> 00:19:09.890
So this is a solution.
00:19:09.890 --> 00:19:13.201
This thing is a solution to
the time-dependent Schrodinger
00:19:13.201 --> 00:19:13.700
equation.
00:19:18.870 --> 00:19:23.930
And so when the
external field is off,
00:19:23.930 --> 00:19:28.450
then the only states
that we consider
00:19:28.450 --> 00:19:31.840
are eigenstates of the
zero-order Hamiltonian,
00:19:31.840 --> 00:19:35.260
and they can be time dependent.
00:19:35.260 --> 00:19:49.570
But if we write psi n star
of t times psi n of t,
00:19:49.570 --> 00:19:54.380
well, that's not time dependent
if this is an eigenstate.
00:19:54.380 --> 00:19:58.220
So the only way we
get time dependence
00:19:58.220 --> 00:20:01.760
is by having this time-dependent
perturbation term.
00:20:06.050 --> 00:20:13.820
OK, so let's take
some initial state.
00:20:13.820 --> 00:20:20.870
And let us call that initial
state some arbitrary state.
00:20:20.870 --> 00:20:24.830
And we can always write
this as a superposition
00:20:24.830 --> 00:20:30.455
of zero-order states.
00:20:37.270 --> 00:20:45.260
OK, and now, unfortunately,
both the coefficients
00:20:45.260 --> 00:20:48.020
in this linear combination
and the functions
00:20:48.020 --> 00:20:49.261
are time dependent.
00:20:51.910 --> 00:20:54.480
So this means when we're
going to be applying
00:20:54.480 --> 00:20:56.400
the time-dependent
Schrodinger equation,
00:20:56.400 --> 00:20:59.500
we take a partial derivative
with respect to t,
00:20:59.500 --> 00:21:03.310
we get derivatives
with this and this.
00:21:03.310 --> 00:21:05.260
So it's an extra
level of complexity,
00:21:05.260 --> 00:21:08.130
but we can deal
with it, because one
00:21:08.130 --> 00:21:10.560
of the things that we
keep coming back to
00:21:10.560 --> 00:21:14.920
is that everything we
talk about is expressed
00:21:14.920 --> 00:21:19.290
as a linear combination of
t equals zero eigenstates
00:21:19.290 --> 00:21:21.489
of the zero-order Hamiltonian.
00:21:26.540 --> 00:21:28.590
OK, so the time-dependent
Schrodinger equation--
00:21:28.590 --> 00:21:34.120
i h-bar partial with
respect to t of the--
00:21:38.800 --> 00:21:42.120
yeah, of the wave
function is equal to--
00:21:56.600 --> 00:22:00.570
OK, that's our friend
or our new friend
00:22:00.570 --> 00:22:04.600
because the old
friend was too simple.
00:22:04.600 --> 00:22:10.130
And so, well, we can represent
this partial derivative
00:22:10.130 --> 00:22:15.210
just using dots because
the equations I'm
00:22:15.210 --> 00:22:18.000
going to be putting on
the board are hideous,
00:22:18.000 --> 00:22:23.940
and so we want to use
every abbreviation we can.
00:22:23.940 --> 00:22:27.960
This is written as a product
of time-dependent coefficients
00:22:27.960 --> 00:22:29.670
and time-dependent functions.
00:22:29.670 --> 00:22:32.580
When we apply the
derivative to it,
00:22:32.580 --> 00:22:36.580
we're going to get
derivatives of each.
00:22:36.580 --> 00:23:03.205
And so that's the
left-hand side.
00:23:07.630 --> 00:23:10.600
OK, and let's look at this
left-hand side for a minute.
00:23:15.760 --> 00:23:18.230
OK, so we've got something
that we don't really
00:23:18.230 --> 00:23:22.671
know what to do with, but this
guy, we know that this is--
00:23:22.671 --> 00:23:26.750
this time-dependent
wave function
00:23:26.750 --> 00:23:30.860
is something that we can use
the time-dependent Schrodinger
00:23:30.860 --> 00:23:33.860
equation on and get
a simplification.
00:23:33.860 --> 00:23:34.819
So the left-hand side--
00:23:34.819 --> 00:23:36.401
I haven't written
the right-hand side.
00:23:36.401 --> 00:23:38.090
I'm just working on
the left-hand side
00:23:38.090 --> 00:23:40.890
of what we get when we start
to write this equation.
00:23:40.890 --> 00:23:53.230
And what we get is we know that
the time dependence of this
00:23:53.230 --> 00:24:01.570
is equal to 1 over i h-bar
times the Hamiltonian operating
00:24:01.570 --> 00:24:04.790
on phi n.
00:24:07.736 --> 00:24:10.620
Is that what I want?
00:24:10.620 --> 00:24:12.310
I can't read my
notes so I have to--
00:24:12.310 --> 00:24:14.350
I have to be--
00:24:14.350 --> 00:24:16.930
yeah, so we've just taken
that 1 over i h-bar.
00:24:20.740 --> 00:24:25.150
This is going to be the
time-independent Hamiltonian,
00:24:25.150 --> 00:24:26.370
the zero-order Hamiltonian.
00:24:26.370 --> 00:24:27.970
And we know what we get here.
00:24:33.000 --> 00:24:33.500
Yes?
00:24:33.500 --> 00:24:35.060
STUDENT: So all
your phi n's, those
00:24:35.060 --> 00:24:36.642
are the zero-order solutions?
00:24:36.642 --> 00:24:37.850
ROBERT FIELD: That's correct.
00:24:37.850 --> 00:24:39.475
STUDENT: So they're
unperturbed states?
00:24:39.475 --> 00:24:43.370
ROBERT FIELD: They're
unperturbed eigenstates of H0.
00:24:43.370 --> 00:24:50.210
And if it's psi n of t, it
has the e to the i En of t--
00:24:52.940 --> 00:24:56.510
En t over h-bar factor
implicit, and we're
00:24:56.510 --> 00:24:58.700
going to be using that.
00:24:58.700 --> 00:25:06.910
All right, so what
we get when we
00:25:06.910 --> 00:25:09.400
take that partial derivative,
we get a simplification.
00:25:19.510 --> 00:25:22.120
OK, let me just write
the right-hand side
00:25:22.120 --> 00:25:23.140
of this equation too.
00:25:23.140 --> 00:25:30.140
So we have the simplified
left-hand side,
00:25:30.140 --> 00:25:34.210
which is psi n c--
00:25:34.210 --> 00:25:37.270
I've never lectured on
time-dependent perturbation
00:25:37.270 --> 00:25:38.650
theory before.
00:25:38.650 --> 00:25:41.740
And so although I
think I understand it,
00:25:41.740 --> 00:25:47.140
it's not as available in
core as it ought to be.
00:25:47.140 --> 00:25:52.030
OK, so we have this minus--
00:25:56.829 --> 00:25:58.120
where did the wave function go?
00:26:04.630 --> 00:26:07.930
Well, there's got
to be a phi in here
00:26:07.930 --> 00:26:21.380
and then minus i over h-bar En
cn t over the times phi n of t.
00:26:21.380 --> 00:26:30.010
That's the left-hand
side in the bracket here.
00:26:30.010 --> 00:26:33.050
OK, and the right-hand side
of the original equation,
00:26:33.050 --> 00:26:56.940
that is just some n cn t
En plus H1 of t phi n t.
00:26:56.940 --> 00:27:01.420
OK, it takes a
little imagination,
00:27:01.420 --> 00:27:05.260
but this and the
terms associated
00:27:05.260 --> 00:27:07.360
with that are the same.
00:27:07.360 --> 00:27:10.330
This happened when we did
non-degenerate perturbation
00:27:10.330 --> 00:27:11.110
theory.
00:27:11.110 --> 00:27:14.320
We looked at the
lambdas of one equation.
00:27:14.320 --> 00:27:18.910
There was a cancellation
of two ugly terms.
00:27:18.910 --> 00:27:21.820
And so what ends
up happening is we
00:27:21.820 --> 00:27:26.720
get a tremendous
simplification of the problem.
00:27:26.720 --> 00:27:49.530
And so the left-hand side of
the equation has the form,
00:27:49.530 --> 00:27:53.640
and the right-hand side has
the form over here without
00:27:53.640 --> 00:27:54.680
the extra term--
00:27:54.680 --> 00:28:09.820
sum over n, cn of t
H1 of t psi n of t.
00:28:18.640 --> 00:28:23.337
OK, and now we
have this equation.
00:28:23.337 --> 00:28:24.920
We have this simple
thing here, and we
00:28:24.920 --> 00:28:27.110
have this ugly thing here.
00:28:27.110 --> 00:28:40.160
And we want to simplify this
by multiplying on the left
00:28:40.160 --> 00:28:43.230
by psi F of t so--
00:28:46.520 --> 00:28:48.206
and integrating
with respect to tau.
00:28:51.860 --> 00:28:54.750
F is for final.
00:28:54.750 --> 00:28:56.820
So we're interested
in the transition
00:28:56.820 --> 00:28:59.560
from some initial state
to some final state.
00:28:59.560 --> 00:29:02.270
So we're going to massage this.
00:29:02.270 --> 00:29:05.340
And when we do that, we get--
00:29:09.690 --> 00:29:12.510
I've clearly skipped a step,
but it doesn't matter--
00:29:12.510 --> 00:29:26.790
i h-bar cf dot of t is equal
to this integral sum c n of t
00:29:26.790 --> 00:29:30.060
integral cf of--
00:29:30.060 --> 00:29:42.820
phi f of t H1 f of
t phi n of t, e tau.
00:29:46.170 --> 00:29:50.940
This is a very
important equation
00:29:50.940 --> 00:29:54.240
because we have a simple
derivative of the coefficient
00:29:54.240 --> 00:29:57.220
that we want, and it's
expressed as an integral.
00:29:57.220 --> 00:30:00.940
And we have an integral
between an eigenstate
00:30:00.940 --> 00:30:04.440
of the zero-order Hamiltonian
and another eigenstate.
00:30:04.440 --> 00:30:17.430
And this is just H1 f n of t.
00:30:17.430 --> 00:30:21.715
OK, so we have these guys.
00:30:27.810 --> 00:30:29.830
So what we want to
know is, all right,
00:30:29.830 --> 00:30:33.590
this is the thing that's
making stuff happen.
00:30:33.590 --> 00:30:38.010
This is a matrix
element of this term.
00:30:38.010 --> 00:30:48.600
Well, H1 of t, which is
equal to v cosine omega t
00:30:48.600 --> 00:30:57.810
can be written as v times
1/2 e to the i omega t plus e
00:30:57.810 --> 00:31:00.930
to the minus i omega t.
00:31:00.930 --> 00:31:04.530
This is really neat
because you notice
00:31:04.530 --> 00:31:08.520
we have these complex
oscillating field terms,
00:31:08.520 --> 00:31:11.850
and we have on each of
these wave functions
00:31:11.850 --> 00:31:14.970
a complex oscillating term.
00:31:14.970 --> 00:31:20.700
And what ends up happening
is that we get this equation.
00:31:20.700 --> 00:31:27.340
i h-bar cf dot of
t is equal to--
00:31:27.340 --> 00:31:29.390
and this is-- you
know, it's ugly.
00:31:29.390 --> 00:31:30.060
It gets big.
00:31:30.060 --> 00:31:32.040
A lot of stuff
has to be written,
00:31:32.040 --> 00:31:34.750
and I have to transfer
from my notes to here.
00:31:34.750 --> 00:31:36.720
And then you have to
transfer to your paper.
00:31:36.720 --> 00:31:38.670
And there is going to be--
00:31:38.670 --> 00:31:40.290
there will be printed
lecture notes.
00:31:40.290 --> 00:31:42.840
And in fact, there may actually
be printed lecture notes
00:31:42.840 --> 00:31:44.470
for this lecture.
00:31:44.470 --> 00:31:47.430
But if they're not,
they will be soon.
00:31:47.430 --> 00:31:51.600
OK, and so we get this
differential equation, which
00:31:51.600 --> 00:32:03.540
is the sum over n c n of t
integral psi f star of t times
00:32:03.540 --> 00:32:10.710
1/2 v, v to the i omega t
plus e to the minus i omega
00:32:10.710 --> 00:32:18.530
t times psi n of t, e tau.
00:32:21.510 --> 00:32:24.510
Well, these guys
have time dependence,
00:32:24.510 --> 00:32:26.780
and so we can put that in.
00:32:26.780 --> 00:32:39.210
And now this integral has
the form psi f star 0 1/2 v,
00:32:39.210 --> 00:32:54.520
and we have e to the minus
I omega and f minus omega t.
00:32:54.520 --> 00:32:58.070
Omega nf, the difference in--
00:32:58.070 --> 00:33:05.210
so omega nf is En
minus Ef over h-bar.
00:33:08.970 --> 00:33:14.670
And so we have minus this
oscillating term, minus omega
00:33:14.670 --> 00:33:24.930
t, and then we have e to the
minus i nft plus omega t.
00:33:30.080 --> 00:33:34.620
So here this isn't well, it's
so ugly because of my stupidity
00:33:34.620 --> 00:33:35.120
here.
00:33:35.120 --> 00:33:38.360
But what we have here
is a resonance integral.
00:33:38.360 --> 00:33:42.740
We have something
that's oscillating fast
00:33:42.740 --> 00:33:45.880
minus something that's
oscillating fast.
00:33:45.880 --> 00:33:48.140
And we have the same
thing plus something
00:33:48.140 --> 00:33:50.930
that's oscillating fast.
00:33:50.930 --> 00:33:55.550
So those terms are zero because
we have an integral that
00:33:55.550 --> 00:33:57.740
is oscillating.
00:33:57.740 --> 00:33:58.392
I'm sorry.
00:33:58.392 --> 00:34:00.350
It's oscillating between
positive and negative,
00:34:00.350 --> 00:34:01.670
positive, negative.
00:34:04.880 --> 00:34:10.489
And as long as omega is
different from omega nf,
00:34:10.489 --> 00:34:16.040
those integrals are zero because
this integrand, as we integrate
00:34:16.040 --> 00:34:20.420
to t equals infinity
or to any time,
00:34:20.420 --> 00:34:24.300
is oscillating about
zero, and it's small.
00:34:24.300 --> 00:34:30.650
However, if omega is the
same as minus omega nf
00:34:30.650 --> 00:34:35.270
or plus omega nf, well, then
this thing is 1 times t.
00:34:38.120 --> 00:34:40.400
It gets really big.
00:34:40.400 --> 00:34:42.920
Now we're talking about
coefficients, which
00:34:42.920 --> 00:34:45.139
are related to probabilities.
00:34:45.139 --> 00:34:47.060
And so these coefficients
had better not
00:34:47.060 --> 00:34:51.989
go get really big because
probability is always
00:34:51.989 --> 00:34:54.980
going to be less than 1.
00:34:54.980 --> 00:34:57.200
OK, so what we're
going to do now
00:34:57.200 --> 00:35:00.410
is collect the rubble in
a form that it turns out
00:35:00.410 --> 00:35:01.380
to be really useful.
00:35:13.510 --> 00:35:17.310
So we have an
equation for the time
00:35:17.310 --> 00:35:23.580
dependence of a final state, and
it's expressed as a sum over n.
00:35:23.580 --> 00:35:27.570
But if we say, oh, let's
make our initial state
00:35:27.570 --> 00:35:30.850
just one of those.
00:35:30.850 --> 00:35:33.690
So our initial state is--
00:35:33.690 --> 00:35:37.990
let's call it ci.
00:35:37.990 --> 00:35:41.380
And we say, well,
the system is not
00:35:41.380 --> 00:35:47.540
in any other state other than
the i state, and this is weak.
00:35:47.540 --> 00:35:51.040
So we can neglect all
of the other states
00:35:51.040 --> 00:35:54.370
where n is not equal to i.
00:35:54.370 --> 00:36:01.870
And if they're not
there, cn has to be 1,
00:36:01.870 --> 00:36:04.240
so we can forget about it.
00:36:04.240 --> 00:36:07.150
So we end up with this
incredibly wonderful
00:36:07.150 --> 00:36:09.170
simple equation.
00:36:09.170 --> 00:36:10.630
So we make the two
approximations.
00:36:10.630 --> 00:36:13.690
Single state, the
perturbation is really weak,
00:36:13.690 --> 00:36:20.720
and we get cf of t
is equal to the vfi--
00:36:20.720 --> 00:36:22.510
the off-diagonal
matrix element--
00:36:22.510 --> 00:36:30.578
over 2i h-bar times the
integral from 0 to t e
00:36:30.578 --> 00:36:44.720
to the minus i omega i
f t minus omega times
00:36:44.720 --> 00:36:52.930
e to the i omega
i f plus omega dt.
00:36:57.350 --> 00:37:00.260
Well, all complexity is gone.
00:37:00.260 --> 00:37:03.690
We have the amount
of the final state,
00:37:03.690 --> 00:37:07.650
and it's expressed by a
matrix element and some time
00:37:07.650 --> 00:37:09.500
dependence.
00:37:09.500 --> 00:37:18.580
And this is a resonant situation
where if omega t, omega t--
00:37:18.580 --> 00:37:24.570
if omega is equal
to omega i f, fine.
00:37:24.570 --> 00:37:29.490
Then this is zero,
the exponent is zero,
00:37:29.490 --> 00:37:32.770
we get t here from that.
00:37:32.770 --> 00:37:35.150
And we get zero from that one
because that's oscillating
00:37:35.150 --> 00:37:36.441
so fast it doesn't do anything.
00:37:39.330 --> 00:37:45.430
But that's a problem because
this c is a probability.
00:37:45.430 --> 00:37:50.580
And so the square of c had
better not be larger than one,
00:37:50.580 --> 00:37:54.240
and this is cruising
to be larger than 1.
00:37:54.240 --> 00:37:56.940
But we don't care about cw.
00:37:56.940 --> 00:37:58.530
What we really
care about-- well,
00:37:58.530 --> 00:38:03.000
what is the rate as
opposed to the probability?
00:38:03.000 --> 00:38:11.130
OK, because the rate
of increase of state f
00:38:11.130 --> 00:38:16.080
is something that we can
calculate from this integral
00:38:16.080 --> 00:38:19.880
simply by taking the--
00:38:19.880 --> 00:38:23.695
we multiply the
integral by 1 over T
00:38:23.695 --> 00:38:29.850
if the limit T goes to infinity.
00:38:32.820 --> 00:38:34.550
And now we get a
new equation, which
00:38:34.550 --> 00:38:40.210
is called Fermi's golden rule.
00:38:40.210 --> 00:38:43.100
OK, so I'm skipping
some steps, and I'm
00:38:43.100 --> 00:38:45.260
doing things in the wrong order.
00:38:45.260 --> 00:38:48.920
But so first of
all, the probability
00:38:48.920 --> 00:38:52.250
of the transition from
the i state to the f state
00:38:52.250 --> 00:38:53.820
as a function of time.
00:38:53.820 --> 00:38:56.330
So the probability is
going to keep growing.
00:38:56.330 --> 00:39:00.230
That's why we want to do this
trick with dividing by t.
00:39:00.230 --> 00:39:01.801
What time is it?
00:39:01.801 --> 00:39:02.300
OK.
00:39:09.740 --> 00:39:17.150
That's just cf of t
squared, and that's just
00:39:17.150 --> 00:39:28.760
the fi over 4 h-bar squared
times this integral 0 to t e
00:39:28.760 --> 00:39:39.200
to the plus and e to the
minus term dt squared.
00:39:39.200 --> 00:39:45.340
OK, the integrals survive only
if omega is equal to omega i f.
00:39:49.630 --> 00:39:52.580
And if we convert to a
rate so that the rate is
00:39:52.580 --> 00:40:04.890
going to be Wfi, which is
going to be Vfi over 4 h-bar
00:40:04.890 --> 00:40:14.331
squared times the sum of
two delta functions Vi
00:40:14.331 --> 00:40:27.440
minus Ef minus omega plus sum
of Ei minus Ef plus omega.
00:40:30.420 --> 00:40:34.530
So the rate is just
this simple thing--
00:40:34.530 --> 00:40:36.990
the square matrix
element and a delta
00:40:36.990 --> 00:40:39.570
function-- saying either it's
an absorption or emission
00:40:39.570 --> 00:40:42.250
transition on resonance,
and we're cooked.
00:40:42.250 --> 00:40:46.700
OK, so now I want to
show some pictures
00:40:46.700 --> 00:40:54.540
of a movie, which will make this
whole thing make more sense.
00:40:54.540 --> 00:40:57.240
This is for a
vibrational transition.
00:40:57.240 --> 00:41:00.815
So we have the electric field--
00:41:03.770 --> 00:41:06.530
the dipole interacting
with the electric field.
00:41:06.530 --> 00:41:10.410
And now let's just turn
on the time dependence.
00:41:10.410 --> 00:41:14.100
OK, so this is the
interaction term.
00:41:14.100 --> 00:41:18.050
We add that interaction term
to the zero-order Hamiltonian,
00:41:18.050 --> 00:41:24.650
and so we end up getting a
big effect of the potential.
00:41:24.650 --> 00:41:27.450
The potential's going
like this, like that.
00:41:27.450 --> 00:41:30.660
And so the eigenfunctions
of that potential
00:41:30.660 --> 00:41:35.260
are going to be profoundly
affected, and so let's do that.
00:41:35.260 --> 00:41:38.760
Let's go to the next.
00:41:38.760 --> 00:41:42.980
All right, so here now we
have a realistic small field,
00:41:42.980 --> 00:41:44.870
and now this is small.
00:41:44.870 --> 00:41:46.640
And you can hardly
see this thing moving.
00:41:51.520 --> 00:42:00.830
OK, now what we have is
the wave function of this.
00:42:00.830 --> 00:42:12.610
And what we see is if omega
is 1/4 the energy, if omega
00:42:12.610 --> 00:42:16.120
is much smaller than the
vibrational frequency
00:42:16.120 --> 00:42:21.730
or much larger, we get very
little effect of the time
00:42:21.730 --> 00:42:24.750
dependent.
00:42:24.750 --> 00:42:27.330
You can see that the
wave function is just
00:42:27.330 --> 00:42:28.560
moving a little bit.
00:42:28.560 --> 00:42:30.630
The potential is
jiggling around,
00:42:30.630 --> 00:42:33.990
whether the perturbation
is strong or weak.
00:42:33.990 --> 00:42:35.590
It's not on resonance.
00:42:35.590 --> 00:42:41.810
And now let's go
to the resonance.
00:42:41.810 --> 00:42:43.940
Now what's happening
is the potential
00:42:43.940 --> 00:42:46.640
is moving not too much,
but the wave function
00:42:46.640 --> 00:42:50.050
is diving all over the place.
00:42:50.050 --> 00:42:51.810
And if we ask, well,
what does that really
00:42:51.810 --> 00:42:59.230
look like as a sum
of terms, the thing
00:42:59.230 --> 00:43:01.810
that's different from the
zero-order wave function
00:43:01.810 --> 00:43:03.730
is this.
00:43:03.730 --> 00:43:08.070
So zero-order wave function
is one nodeless thing.
00:43:08.070 --> 00:43:14.850
This is the time-dependent term,
and it looks like V equals 1 So
00:43:14.850 --> 00:43:19.060
what this shows is,
yes, there are--
00:43:19.060 --> 00:43:23.890
if we have a time-dependent
field, and it's resonant,
00:43:23.890 --> 00:43:25.720
then we get a very
strong interaction
00:43:25.720 --> 00:43:27.700
even though the field is weak.
00:43:27.700 --> 00:43:31.570
And it causes the appearance
of the other level
00:43:31.570 --> 00:43:32.841
but oscillating.
00:43:35.610 --> 00:43:39.030
And so resonance is
really important,
00:43:39.030 --> 00:43:41.450
and selection rule
is really important.
00:43:41.450 --> 00:43:46.650
The selection rule for the
vibrational transitions
00:43:46.650 --> 00:43:49.620
has to do with the form.
00:43:49.620 --> 00:43:54.790
Oh, I shouldn't
be rushing at all.
00:43:54.790 --> 00:43:59.915
OK, so let's draw a picture.
00:44:02.840 --> 00:44:06.210
And this is the part that has
puzzled me for a long time,
00:44:06.210 --> 00:44:09.620
but I've got it now.
00:44:09.620 --> 00:44:20.170
So here we have a
picture of the molecule.
00:44:20.170 --> 00:44:22.800
And this end is positive,
and this end is negative.
00:44:22.800 --> 00:44:25.280
And we have a
positive electrode,
00:44:25.280 --> 00:44:27.570
and we have a
negative electrode.
00:44:27.570 --> 00:44:30.050
So that's an electric field.
00:44:30.050 --> 00:44:35.800
And so now the positive
electrode is saying,
00:44:35.800 --> 00:44:38.690
you better go away, and
you better come here.
00:44:38.690 --> 00:44:42.370
So it's trying to
use compressed bond.
00:44:42.370 --> 00:44:44.470
And now this field
oscillates, and so it's
00:44:44.470 --> 00:44:48.120
compressing and expanding.
00:44:48.120 --> 00:44:50.960
Now that's what's going on.
00:44:50.960 --> 00:44:53.260
But how does quantum
mechanics account for it?
00:44:53.260 --> 00:44:57.100
Well, quantum mechanics says
in order for the bond length
00:44:57.100 --> 00:45:00.710
to change, we have to
mix in some other state.
00:45:00.710 --> 00:45:06.460
So we have the ground state,
and we have an excited state
00:45:06.460 --> 00:45:08.630
that looks like that.
00:45:08.630 --> 00:45:11.560
And so the field is
mixing these two.
00:45:11.560 --> 00:45:19.600
Now that means that the operator
is mu 0 plus derivative of mu
00:45:19.600 --> 00:45:25.600
with respect to the
electric field times q.
00:45:25.600 --> 00:45:31.750
So this is the thing that
allows some mixing of an excited
00:45:31.750 --> 00:45:33.680
state into the ground state.
00:45:33.680 --> 00:45:37.210
This is our friend the
harmonic oscillator--
00:45:37.210 --> 00:45:39.310
operator, displacement operator.
00:45:39.310 --> 00:45:42.440
It has selection rules delta
v equals plus or minus 1,
00:45:42.440 --> 00:45:45.740
and that's all.
00:45:45.740 --> 00:45:50.630
So a vibrational
transition is caused
00:45:50.630 --> 00:45:55.640
by the derivative of the--
00:45:58.839 --> 00:46:05.150
yeah, that's-- no, derivative of
the dipole moment with respect
00:46:05.150 --> 00:46:07.650
to Q.
00:46:07.650 --> 00:46:12.750
So did I have it right?
00:46:12.750 --> 00:46:14.100
Yes.
00:46:14.100 --> 00:46:16.170
So this is something--
00:46:16.170 --> 00:46:20.040
we can calculate how the
dipole moment depends
00:46:20.040 --> 00:46:21.810
on the displacement
from equilibrium,
00:46:21.810 --> 00:46:26.060
but this is the operator that
causes the mixing of states.
00:46:26.060 --> 00:46:30.690
So one of the things I've
loved to do over the years
00:46:30.690 --> 00:46:34.670
is to write a cumulative
exam in which I ask, well,
00:46:34.670 --> 00:46:38.110
what is it that causes a
vibrational transition?
00:46:38.110 --> 00:46:40.290
What does a molecule
have to have in order
00:46:40.290 --> 00:46:42.480
to have a vibrational
transition?
00:46:42.480 --> 00:46:44.250
And also what does
a molecule have
00:46:44.250 --> 00:46:47.610
to have to have a
rotational transition?
00:46:47.610 --> 00:46:50.100
Well, this is what causes
the rotational transition
00:46:50.100 --> 00:46:55.210
because we can think of the
dipole moment interacting
00:46:55.210 --> 00:46:59.950
with a field, which is going
like that or like that.
00:46:59.950 --> 00:47:04.720
And so what that does is it
causes a torque on the system.
00:47:04.720 --> 00:47:07.330
It doesn't change
the dipole moment,
00:47:07.330 --> 00:47:09.640
doesn't stretch the molecule.
00:47:09.640 --> 00:47:17.650
It causes a
transition, and this is
00:47:17.650 --> 00:47:23.920
expressed in terms
of the interaction
00:47:23.920 --> 00:47:32.230
mu dot E. This dot
product, this cosine theta,
00:47:32.230 --> 00:47:34.060
is the operator
that causes this.
00:47:34.060 --> 00:47:44.830
We call the relationship between
the laboratory and the body
00:47:44.830 --> 00:47:46.690
fixed coordinate
system is determined
00:47:46.690 --> 00:47:50.720
by the cosine of some angle,
and the cosine of the angle
00:47:50.720 --> 00:47:55.970
is what's responsible for a
pure rotational transition.
00:47:55.970 --> 00:47:57.720
And we have
vibrational transitions
00:47:57.720 --> 00:48:00.240
where they are derivative
of the dipole with respect
00:48:00.240 --> 00:48:02.160
to the coordinate.
00:48:02.160 --> 00:48:06.980
Now let's say we have nitrogen--
00:48:06.980 --> 00:48:11.830
no dipole moment, no derivative
of the dipole moment.
00:48:11.830 --> 00:48:13.380
Suppose we have CO.
00:48:13.380 --> 00:48:15.850
CO has a very
small dipole moment
00:48:15.850 --> 00:48:19.570
and a huge derivative of the
dipole moment with respect
00:48:19.570 --> 00:48:21.100
to displacement.
00:48:21.100 --> 00:48:24.670
And so CO has really strong
vibrational transitions
00:48:24.670 --> 00:48:27.430
and rather weak
rotational transitions.
00:48:27.430 --> 00:48:34.180
So if it happened that CO had
zero permanent dipole moment,
00:48:34.180 --> 00:48:36.490
it would have no
rotational transition.
00:48:36.490 --> 00:48:39.100
But as you go up to higher
V's, then it would not be zero.
00:48:39.100 --> 00:48:41.840
And you would see
rotational transitions.
00:48:41.840 --> 00:48:44.800
And so there's all sorts of
insights that come from this.
00:48:44.800 --> 00:48:48.460
And so now we know what
causes transitions.
00:48:48.460 --> 00:48:51.140
There is some
operator, which causes
00:48:51.140 --> 00:48:53.980
mixing of some wave functions.
00:48:53.980 --> 00:48:55.810
And the time-dependent
perturbation theory
00:48:55.810 --> 00:49:00.910
when it's resonant
mixes only one state.
00:49:00.910 --> 00:49:03.130
We have selection rules
which we understand just
00:49:03.130 --> 00:49:04.510
by looking at the wave fun--
00:49:04.510 --> 00:49:08.290
looking at the matrix
elements, and now we
00:49:08.290 --> 00:49:11.770
have a big
understanding of what is
00:49:11.770 --> 00:49:13.330
going to appear in a spectrum.
00:49:13.330 --> 00:49:15.170
What are the intensities
in the spectrum?
00:49:15.170 --> 00:49:17.140
What are the transitions?
00:49:17.140 --> 00:49:18.980
Which transitions are
going to be allowed?
00:49:18.980 --> 00:49:22.600
Which are going to be forbidden?
00:49:22.600 --> 00:49:24.170
And that's kind of useful.
00:49:24.170 --> 00:49:28.790
So there is this
tremendously tedious algebra,
00:49:28.790 --> 00:49:31.540
which I didn't do a very
good job displaying,
00:49:31.540 --> 00:49:33.760
but you don't need it
because, at the end,
00:49:33.760 --> 00:49:36.640
you get Fermi's golden
rule, which says
00:49:36.640 --> 00:49:40.310
transitions occur on resonance.
00:49:40.310 --> 00:49:42.400
Now if you're a little
bit off resonance,
00:49:42.400 --> 00:49:46.780
well, then the stationary phase
in the oscillating exponential
00:49:46.780 --> 00:49:50.590
persists for a while,
and then it goes away.
00:49:50.590 --> 00:49:54.100
And so you get a little bit
of slightly off-resonance
00:49:54.100 --> 00:49:58.180
transition probability, and
you get other things too.
00:49:58.180 --> 00:50:01.780
But you already now have
enough to understand basically
00:50:01.780 --> 00:50:06.940
everything you need
to begin to make sense
00:50:06.940 --> 00:50:13.190
of the interaction of radiation
with molecules correctly,
00:50:13.190 --> 00:50:15.020
and this isn't
bullets and targets.
00:50:15.020 --> 00:50:19.370
This is waves with
phases, and so there
00:50:19.370 --> 00:50:21.980
are all sorts of things you have
to do to be honest about it.
00:50:21.980 --> 00:50:24.530
But you know what
the actors are,
00:50:24.530 --> 00:50:26.810
and that's really
a useful thing.
00:50:26.810 --> 00:50:31.150
And you're never going to
be tested on this from me.
00:50:31.150 --> 00:50:34.390
OK, good luck on the
exam tomorrow night.