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ROBERT FIELD: OK.
00:00:23.250 --> 00:00:26.580
So this is going to be a fun
lecture because it's mostly
00:00:26.580 --> 00:00:29.130
pictures and language.
00:00:29.130 --> 00:00:33.780
And it's also an area
of physical chemistry,
00:00:33.780 --> 00:00:35.880
which is pretty much--
00:00:35.880 --> 00:00:39.780
I mean, there's three
areas of physical chemistry
00:00:39.780 --> 00:00:42.150
where people have trouble
talking to each other.
00:00:42.150 --> 00:00:44.007
There's statistical mechanics.
00:00:44.007 --> 00:00:45.090
There's quantum mechanics.
00:00:45.090 --> 00:00:48.000
And there's the time dependent,
time independent forms
00:00:48.000 --> 00:00:49.470
of quantum mechanics.
00:00:49.470 --> 00:00:51.990
And these three
communities have to learn
00:00:51.990 --> 00:00:53.520
how to talk to each other.
00:00:53.520 --> 00:00:57.370
And I'm hoping that I will
help to bridge that gap.
00:00:57.370 --> 00:00:57.870
OK.
00:00:57.870 --> 00:01:01.721
So last time, we talked about
the time dependent Schrodinger
00:01:01.721 --> 00:01:02.220
equation.
00:01:02.220 --> 00:01:05.060
It's a simple
looking little thing.
00:01:05.060 --> 00:01:07.610
Remember though, it's an
extra level of complexity
00:01:07.610 --> 00:01:10.670
on what you already understand.
00:01:10.670 --> 00:01:15.020
Now, in the special case
where the Hamiltonian is time
00:01:15.020 --> 00:01:21.650
independent, then, if you
know all of the energy levels
00:01:21.650 --> 00:01:26.360
and wave functions for the
time independent Hamiltonian,
00:01:26.360 --> 00:01:29.180
you can immediately
write down the solution
00:01:29.180 --> 00:01:33.470
to the time dependent
Hamiltonian.
00:01:33.470 --> 00:01:37.970
And so you have a complete
set of wave functions.
00:01:37.970 --> 00:01:40.910
And you have a complete
set of energy levels.
00:01:40.910 --> 00:01:44.900
And with that, you can
basically describe something
00:01:44.900 --> 00:01:47.480
that satisfies this thing.
00:01:47.480 --> 00:01:49.730
Another set of coefficients
is often something
00:01:49.730 --> 00:01:56.090
that you arrange for
simplicity or for insight.
00:01:56.090 --> 00:01:57.260
And I want to understand--
00:01:57.260 --> 00:01:59.240
I want to talk about that.
00:01:59.240 --> 00:02:00.020
OK.
00:02:00.020 --> 00:02:02.180
And so in the
previous lecture, we
00:02:02.180 --> 00:02:10.250
talked about the probability
density, which is psi star psi.
00:02:10.250 --> 00:02:12.300
And that can move.
00:02:12.300 --> 00:02:16.230
And that can move in two ways.
00:02:19.800 --> 00:02:24.270
In the wave function, if
there are two or more states
00:02:24.270 --> 00:02:29.140
belonging to two or
more different energies,
00:02:29.140 --> 00:02:30.780
then you can have motion.
00:02:30.780 --> 00:02:36.700
And you can have motion like
breathing, where probability
00:02:36.700 --> 00:02:41.890
moves towards the
extremes, or motion
00:02:41.890 --> 00:02:44.920
where there is actually a wave
packet going from one side
00:02:44.920 --> 00:02:47.580
to another.
00:02:47.580 --> 00:02:50.600
So for anything
interesting to happen,
00:02:50.600 --> 00:02:53.870
this probability
density has to include
00:02:53.870 --> 00:02:56.945
at least two different
eigenstates belonging
00:02:56.945 --> 00:02:59.300
to two different energy levels.
00:02:59.300 --> 00:03:02.540
Now, the total probability
is just the integral
00:03:02.540 --> 00:03:08.540
over all coordinates of
the probability density.
00:03:08.540 --> 00:03:12.180
And probability is conserved.
00:03:12.180 --> 00:03:14.000
So one of the things
that happens--
00:03:14.000 --> 00:03:16.550
when you do an integral,
the wave functions go away.
00:03:16.550 --> 00:03:20.660
And in this particular
case, integral psi star psi,
00:03:20.660 --> 00:03:22.550
all you get is one.
00:03:22.550 --> 00:03:25.800
Or you get a constant, depending
on how you normalize things.
00:03:25.800 --> 00:03:28.930
But it's not time dependent.
00:03:28.930 --> 00:03:34.640
Now, in understanding motion
from classical mechanics,
00:03:34.640 --> 00:03:36.310
we know Newton's equations.
00:03:36.310 --> 00:03:42.880
We know how the coordinate and
the momentum depend on time.
00:03:42.880 --> 00:03:47.290
And so if we calculate
the expectation
00:03:47.290 --> 00:03:49.900
value of the coordinate
or the expectation
00:03:49.900 --> 00:03:53.650
value of the momentum,
we get some results.
00:03:53.650 --> 00:03:58.180
And it turns out that these
things describe the motion
00:03:58.180 --> 00:04:00.490
of the center of a wave packet.
00:04:00.490 --> 00:04:04.240
A wave packet is anything that's
not just a single eigenstate,
00:04:04.240 --> 00:04:05.740
so it moves.
00:04:05.740 --> 00:04:10.560
And so the motion of
a single eigenstate--
00:04:10.560 --> 00:04:15.130
the motion of a wave packet
is described by Newton's laws.
00:04:15.130 --> 00:04:17.050
And this is Ehrenfest theorem.
00:04:17.050 --> 00:04:18.940
And it can easily be proven.
00:04:18.940 --> 00:04:20.709
But we're not going to prove it.
00:04:20.709 --> 00:04:22.240
We're just going to use it.
00:04:22.240 --> 00:04:27.010
So we are very interested
in expectation values
00:04:27.010 --> 00:04:29.410
of the coordinate
and the momentum.
00:04:29.410 --> 00:04:33.700
And for a harmonic
oscillator, this is duck soup.
00:04:33.700 --> 00:04:38.500
Then we have nothing but
trivial integrals involving
00:04:38.500 --> 00:04:40.870
the a's and a-daggers.
00:04:40.870 --> 00:04:45.580
And so even though in some
ways the harmonic oscillator
00:04:45.580 --> 00:04:49.880
is a more complicated problem
than the particle in a box,
00:04:49.880 --> 00:04:51.610
the harmonic oscillator
is the problem
00:04:51.610 --> 00:05:00.190
of choice for dealing with
motion and developing insight.
00:05:00.190 --> 00:05:02.110
There's another useful--
00:05:02.110 --> 00:05:02.990
OK.
00:05:02.990 --> 00:05:05.780
We have this wave function here.
00:05:05.780 --> 00:05:09.290
There is a huge amount of
information embedded in it,
00:05:09.290 --> 00:05:12.050
too much to just look at.
00:05:12.050 --> 00:05:16.230
We need to have ways of
reducing that information.
00:05:16.230 --> 00:05:19.880
And one of the nice ways is
the survival probability.
00:05:19.880 --> 00:05:22.010
The survival
probability tells you
00:05:22.010 --> 00:05:27.600
how fast does the initial
state move away from itself.
00:05:27.600 --> 00:05:30.240
And this is very revealing.
00:05:30.240 --> 00:05:34.766
Another very revealing thing is
if this survival probability,
00:05:34.766 --> 00:05:36.390
which is a function
of time and nothing
00:05:36.390 --> 00:05:40.350
else, because we've integrated
over the wave functions--
00:05:40.350 --> 00:05:42.840
if this survival
probability goes up and down
00:05:42.840 --> 00:05:47.160
and up and down,
there are recurrences.
00:05:47.160 --> 00:05:49.860
So at a maximum,
which is usually
00:05:49.860 --> 00:05:53.520
a maximum at t equals 0
because the wave function is it
00:05:53.520 --> 00:05:57.060
at its birthplace, the
survival probability
00:05:57.060 --> 00:06:00.630
will start high and go down,
and come up, and go down.
00:06:00.630 --> 00:06:03.780
And there's all sorts of
information about recurrences.
00:06:03.780 --> 00:06:07.800
As the wave function
returns to its birthplace,
00:06:07.800 --> 00:06:11.040
it might not return completely.
00:06:11.040 --> 00:06:13.420
And so we get
partial recurrences.
00:06:13.420 --> 00:06:15.910
And these tell us something.
00:06:15.910 --> 00:06:20.830
And the times at which
the maxima in the survival
00:06:20.830 --> 00:06:27.120
probability occur tell us
a lot about the potential.
00:06:27.120 --> 00:06:29.280
And there are some
grand recurrences
00:06:29.280 --> 00:06:33.390
where, because all of the
energy level differences
00:06:33.390 --> 00:06:36.130
are an integer multiple
of a common factor,
00:06:36.130 --> 00:06:39.120
then you get a
perfect recurrence.
00:06:39.120 --> 00:06:43.150
And these are wonderful
for drawing pictures.
00:06:43.150 --> 00:06:50.590
Now, you really want to
understand the concepts of time
00:06:50.590 --> 00:06:53.900
dependent quantum
mechanics pictorially
00:06:53.900 --> 00:06:56.720
and to develop a language
that describes it.
00:06:56.720 --> 00:06:58.220
And there are a lot
of little pieces
00:06:58.220 --> 00:07:02.904
here that you need to convince
yourself you understand
00:07:02.904 --> 00:07:04.070
so you can use the language.
00:07:21.230 --> 00:07:22.080
OK.
00:07:22.080 --> 00:07:24.840
One of the things I said at
the beginning of the course
00:07:24.840 --> 00:07:32.310
is the central object in quantum
mechanics is the wave function.
00:07:32.310 --> 00:07:36.900
Or we could generalize to this.
00:07:36.900 --> 00:07:39.760
This is actually the truth.
00:07:39.760 --> 00:07:42.360
This is a partial truth.
00:07:42.360 --> 00:07:46.050
This is everything that
we can possibly know.
00:07:46.050 --> 00:07:47.910
But we can't know this.
00:07:47.910 --> 00:07:50.560
We can never observe
the wave function.
00:07:50.560 --> 00:07:53.490
However, we do observations.
00:07:53.490 --> 00:07:58.980
And we construct
a picture, which
00:07:58.980 --> 00:08:03.040
is what we call the
effective Hamiltonian.
00:08:03.040 --> 00:08:06.660
This effective Hamiltonian
describes everything
00:08:06.660 --> 00:08:09.930
we know, mostly energy levels.
00:08:09.930 --> 00:08:12.640
We have formulas for
the energy levels.
00:08:12.640 --> 00:08:14.490
And we determine the constants.
00:08:14.490 --> 00:08:18.130
There's all sorts of stuff
that we collect by experiment.
00:08:18.130 --> 00:08:20.820
And so we create an
object which we think
00:08:20.820 --> 00:08:23.730
is like the true Hamiltonian.
00:08:23.730 --> 00:08:27.660
And by having the
effective Hamiltonian,
00:08:27.660 --> 00:08:32.309
we can get these things.
00:08:32.309 --> 00:08:36.330
So we observe we make
some observations.
00:08:36.330 --> 00:08:40.320
And then this is a reduced
version of the truth.
00:08:40.320 --> 00:08:43.770
And we can get a pretty
good representation
00:08:43.770 --> 00:08:46.440
of the thing that is
supposedly hidden from us.
00:08:50.201 --> 00:08:50.700
OK.
00:08:53.352 --> 00:08:58.085
So let's start
now at t equals 0.
00:08:58.085 --> 00:09:01.430
At t equals 0, which I
like to call the pluck--
00:09:01.430 --> 00:09:04.430
and it really makes it
contact with those of you who
00:09:04.430 --> 00:09:08.330
are musicians, that
you know how to create
00:09:08.330 --> 00:09:12.700
a particular kind of sound,
which will evolve in time.
00:09:12.700 --> 00:09:18.200
And it depends on the details of
how you handle the instrument.
00:09:18.200 --> 00:09:24.080
So at t equals 0,
we have this thing
00:09:24.080 --> 00:09:29.030
which, if we have a complete
set of eigenfunctions
00:09:29.030 --> 00:09:34.160
of the Hamiltonian, of the
time independent Hamiltonian,
00:09:34.160 --> 00:09:39.146
we know we can write
this as cj psi j of x.
00:09:45.930 --> 00:09:48.960
Now, often we don't need an
infinite number of terms.
00:09:48.960 --> 00:09:53.190
But we know that there are lots
of different kinds of plucks.
00:09:53.190 --> 00:10:00.390
And you can find out what
these coefficients are
00:10:00.390 --> 00:10:05.190
by just calculating overlap
integral of these functions
00:10:05.190 --> 00:10:06.570
with this initial state.
00:10:10.961 --> 00:10:11.460
OK.
00:10:11.460 --> 00:10:14.730
So this is telling you
what happens at t equals 0.
00:10:14.730 --> 00:10:18.630
And if you can write
the initial state
00:10:18.630 --> 00:10:20.970
as a superposition
of eigenstates,
00:10:20.970 --> 00:10:24.030
then you can immediately
write the time
00:10:24.030 --> 00:10:46.620
evolving object this way.
00:10:46.620 --> 00:10:52.190
Now, we have a minus i.
00:10:52.190 --> 00:10:55.190
Well, why do we need a minus i?
00:10:55.190 --> 00:10:56.700
Well, because
we're going to take
00:10:56.700 --> 00:10:59.760
the time derivative
of the Hamiltonian
00:10:59.760 --> 00:11:02.430
and multiply it by iH bar.
00:11:02.430 --> 00:11:06.780
We're going to want to get the
energy, not minus the energy.
00:11:09.380 --> 00:11:12.440
And so we need a minus i here.
00:11:12.440 --> 00:11:15.970
And we need a
divided by H bar here
00:11:15.970 --> 00:11:20.240
in order to satisfy the
time independence or time
00:11:20.240 --> 00:11:22.130
dependence.
00:11:22.130 --> 00:11:24.681
So this always bothers people.
00:11:24.681 --> 00:11:25.930
Why should there be an i here?
00:11:25.930 --> 00:11:28.210
And why should it be minus i?
00:11:28.210 --> 00:11:31.330
And I recommend not
trying to memorize it,
00:11:31.330 --> 00:11:32.890
but to just convince yourself.
00:11:32.890 --> 00:11:37.060
I need the minus i because I'm
going to bring down a minus i
00:11:37.060 --> 00:11:39.880
over H bar times e.
00:11:39.880 --> 00:11:42.310
And we have this iH bar.
00:11:42.310 --> 00:11:48.170
The minus i times
i gives plus 1.
00:11:48.170 --> 00:11:49.600
OK.
00:11:49.600 --> 00:11:59.570
So now, in understanding how the
time dependent Hamiltonian can
00:11:59.570 --> 00:12:05.660
be sampled, we build
the superposition states
00:12:05.660 --> 00:12:10.550
with a minimum number of
terms, like two or three.
00:12:10.550 --> 00:12:15.530
Even though, in order to create
a very, very sharply localized
00:12:15.530 --> 00:12:20.420
state, it needs a huge
number of terms here.
00:12:20.420 --> 00:12:23.300
But you always
build your insight
00:12:23.300 --> 00:12:25.890
with something you
can do in your head.
00:12:25.890 --> 00:12:32.010
And so for some problems,
you need only two states.
00:12:32.010 --> 00:12:34.440
And for others, you need three.
00:12:34.440 --> 00:12:37.440
And those are the
ones you want to kill.
00:12:37.440 --> 00:12:39.330
You want to
understand everything
00:12:39.330 --> 00:12:43.166
you can build with two and
three state superpositions.
00:12:46.290 --> 00:12:47.130
OK.
00:12:47.130 --> 00:12:53.505
So there's two
classes of problems.
00:13:07.500 --> 00:13:12.740
OK so for a half
harmonic oscillator--
00:13:24.710 --> 00:13:28.400
so for a half
harmonic oscillator,
00:13:28.400 --> 00:13:30.440
we start with
harmonic oscillator
00:13:30.440 --> 00:13:35.670
and divide it in half and
say this goes to infinity.
00:13:35.670 --> 00:13:39.840
And so this side
is not accessible.
00:13:39.840 --> 00:13:43.640
And so if the potential is
half of a harmonic oscillator
00:13:43.640 --> 00:13:46.490
potential, then we
know immediately what
00:13:46.490 --> 00:13:50.540
the eigenvalues and
eigenfunctions are.
00:13:50.540 --> 00:13:52.519
They are the harmonic
oscillator functions
00:13:52.519 --> 00:13:53.810
that have a node in the middle.
00:13:56.760 --> 00:14:07.230
So we know that we have
v equals 1, v equals 3.
00:14:07.230 --> 00:14:10.600
No v equals 0 or 2.
00:14:10.600 --> 00:14:13.220
I mean, this corresponds
to the lowest level, v
00:14:13.220 --> 00:14:14.740
equals 0 for the
half oscillator.
00:14:14.740 --> 00:14:16.630
But this is
basically-- now, what
00:14:16.630 --> 00:14:20.740
we're going to do is we
create some state of a half
00:14:20.740 --> 00:14:23.250
oscillator.
00:14:23.250 --> 00:14:26.050
And then we take
away the barrier.
00:14:26.050 --> 00:14:29.250
So that's a cheap way of
creating localization.
00:14:29.250 --> 00:14:31.760
Of course, we can't do
this experimentally.
00:14:31.760 --> 00:14:34.840
But you could, in
principle, do it.
00:14:34.840 --> 00:14:46.650
And so basically, you
want an initial state,
00:14:46.650 --> 00:14:50.885
which is a superposition
of vibrational levels.
00:14:56.290 --> 00:14:56.970
OK.
00:14:56.970 --> 00:15:02.660
And this initial state
needs to have the wave
00:15:02.660 --> 00:15:04.525
function be 0 at the barrier.
00:15:07.860 --> 00:15:13.740
So since we're going to be
looking at the full potential,
00:15:13.740 --> 00:15:15.480
we're taking away the barrier.
00:15:15.480 --> 00:15:20.460
We're allowed to use
the even and odd states.
00:15:20.460 --> 00:15:24.200
And so let's take the
three lowest states.
00:15:24.200 --> 00:15:32.390
And so we have c0 psi 0
plus c1 psi 1 plus c2 psi 2.
00:15:34.811 --> 00:15:35.310
OK.
00:15:35.310 --> 00:15:37.140
This guy is OK.
00:15:37.140 --> 00:15:38.310
Yes?
00:15:38.310 --> 00:15:42.260
AUDIENCE: When you create the
states of the half harmonic
00:15:42.260 --> 00:15:45.360
oscillator, do you
have to re-scale them
00:15:45.360 --> 00:15:46.600
so that they normalize?
00:15:46.600 --> 00:15:47.350
ROBERT FIELD: Yes.
00:15:50.400 --> 00:15:52.980
We play fast and loose with
normalization constants.
00:15:52.980 --> 00:15:55.530
We already always
know that, whenever
00:15:55.530 --> 00:15:57.480
you want to calculate
anything, you're
00:15:57.480 --> 00:16:02.130
going to divide by the
normalization integral.
00:16:02.130 --> 00:16:04.890
However, it's OK.
00:16:04.890 --> 00:16:08.010
Because we're going to be using
the full harmonic oscillator
00:16:08.010 --> 00:16:12.390
functions, which
do exist over here.
00:16:12.390 --> 00:16:15.960
And it's fine once we
remove the barrier.
00:16:15.960 --> 00:16:21.810
But we want to choose a problem
which is as simple as possible.
00:16:21.810 --> 00:16:26.460
OK, now, this guy is
not 0 at x equals 0.
00:16:26.460 --> 00:16:28.530
And this guy is not
0 at x equals 0.
00:16:28.530 --> 00:16:40.710
But you can say 0 is equal to c0
psi 0 of 0 plus c2 psi 2 of 0.
00:16:40.710 --> 00:16:46.590
We choose the coefficients so
that these two guys together
00:16:46.590 --> 00:16:49.890
make 0 at the boundary.
00:16:49.890 --> 00:16:50.850
And why do we do this?
00:16:50.850 --> 00:16:54.720
We're going to be calculating
expectation values of x and p.
00:16:54.720 --> 00:16:59.970
And if we only have half
of the energy levels,
00:16:59.970 --> 00:17:03.940
all of the intricacies
are going to be 0.
00:17:03.940 --> 00:17:05.950
So we have to have
three consecutive levels
00:17:05.950 --> 00:17:07.511
to have anything interesting.
00:17:10.220 --> 00:17:14.640
OK, so the artifice
of making these two--
00:17:14.640 --> 00:17:20.190
arranging the coefficients so
that you get a temporary node
00:17:20.190 --> 00:17:24.170
at x equals 0--
00:17:24.170 --> 00:17:25.210
it won't stay a node.
00:17:25.210 --> 00:17:27.760
Because when we let things
oscillate with time,
00:17:27.760 --> 00:17:30.670
these two will
oscillate differently.
00:17:30.670 --> 00:17:33.409
And the node will go away.
00:17:33.409 --> 00:17:34.450
But that's how you do it.
00:17:34.450 --> 00:17:37.571
That's how you build
a superposition.
00:17:37.571 --> 00:17:38.070
OK.
00:17:46.111 --> 00:17:46.610
OK.
00:17:46.610 --> 00:17:51.710
Now, what we want to do is
to be able to draw pictures
00:17:51.710 --> 00:17:57.240
of what's happening and also
to calculate what's going on.
00:17:57.240 --> 00:18:00.230
And one of the things
we use for our pictures
00:18:00.230 --> 00:18:05.990
is the expectation value of x
and the expectation value of p.
00:18:09.100 --> 00:18:12.460
So you know how to calculate
the expectation value of x.
00:18:12.460 --> 00:18:18.650
We have this capital psi star
x capital psi star capital
00:18:18.650 --> 00:18:21.610
psi integrate over x.
00:18:21.610 --> 00:18:35.535
And so we get to c0 c1
x01 cosine omega t plus 2
00:18:35.535 --> 00:18:43.710
c1 c2 x12 cosine omega t.
00:18:43.710 --> 00:18:45.000
Now, how did I do that?
00:18:48.784 --> 00:18:49.290
Oh, come on.
00:18:49.290 --> 00:18:51.330
You can do it, too.
00:18:51.330 --> 00:18:52.550
So what is this x01?
00:18:52.550 --> 00:19:03.020
Well, x01 is the integral
psi 0 star x psi 1 dx.
00:19:03.020 --> 00:19:06.050
And we know we can replace
x by a plus a-dagger
00:19:06.050 --> 00:19:08.370
times the constant.
00:19:08.370 --> 00:19:10.050
We always like to
forget that concept.
00:19:10.050 --> 00:19:13.080
We only bring it in
at the end anyway.
00:19:13.080 --> 00:19:16.700
And so, well, this has a value--
00:19:16.700 --> 00:19:22.370
it's a constant, the constant
that we're forgetting--
00:19:22.370 --> 00:19:26.990
times 1 square root, right?
00:19:30.150 --> 00:19:34.080
And x12 is the
same sort of thing.
00:19:34.080 --> 00:19:41.520
It's going to be the same
constant times 2 square root.
00:19:41.520 --> 00:19:43.380
This is easy.
00:19:43.380 --> 00:19:48.450
So getting from
this symbol to this,
00:19:48.450 --> 00:19:52.640
that just requires
a little practice,
00:19:52.640 --> 00:19:54.730
and then simplifying
further to know
00:19:54.730 --> 00:20:00.110
what these x's are, and then
the constraints on c2 and c0.
00:20:00.110 --> 00:20:03.030
We have all that stuff.
00:20:03.030 --> 00:20:03.530
OK.
00:20:03.530 --> 00:20:10.230
And now, we can draw a picture
of what's going to happen.
00:20:10.230 --> 00:20:12.380
So here we have the
full oscillator.
00:20:12.380 --> 00:20:15.950
And let's just draw
some energy which is--
00:20:15.950 --> 00:20:17.070
now that's complicated.
00:20:17.070 --> 00:20:20.790
We have a time
dependent wave function,
00:20:20.790 --> 00:20:25.910
which is composed of several
different energy eigenstates.
00:20:25.910 --> 00:20:27.470
So what is its energy?
00:20:27.470 --> 00:20:29.210
Well, you can evaluate
what the energy
00:20:29.210 --> 00:20:34.380
is by taking the expectation
value of the Hamiltonian.
00:20:34.380 --> 00:20:38.300
And so you can do that.
00:20:38.300 --> 00:20:40.149
So what we've made
at t equals 0 is
00:20:40.149 --> 00:20:41.440
something that looks like this.
00:20:47.060 --> 00:20:48.904
It's localized on the left side.
00:20:48.904 --> 00:20:50.570
Or it's more localized
on the left side.
00:20:50.570 --> 00:20:54.655
Now, sometimes, you're
going to worry about phase.
00:20:57.360 --> 00:21:02.280
And so many times when
you're working symbolically
00:21:02.280 --> 00:21:05.160
rather than actually
evaluating integrals,
00:21:05.160 --> 00:21:12.860
there are symbolic phase
choices that what you're using
00:21:12.860 --> 00:21:13.990
has made.
00:21:13.990 --> 00:21:15.980
And for example, for
the harmonic oscillator,
00:21:15.980 --> 00:21:18.740
if you look in the
book, you'll see
00:21:18.740 --> 00:21:24.410
that, for all of the harmonic
oscillator functions,
00:21:24.410 --> 00:21:26.780
the outer lobe is
always positive.
00:21:26.780 --> 00:21:30.080
And the inner lobe
is alternating
00:21:30.080 --> 00:21:33.330
with the quantum number.
00:21:33.330 --> 00:21:38.090
So that's a phase convention
that's implicit in everything
00:21:38.090 --> 00:21:41.150
that people have derived.
00:21:41.150 --> 00:21:44.180
And as long as the
different things
00:21:44.180 --> 00:21:48.560
you combine in doing
a calculation involve
00:21:48.560 --> 00:21:52.190
the same phase convention,
which is implicit-- we don't
00:21:52.190 --> 00:21:54.330
want to look at what functions.
00:21:54.330 --> 00:21:57.731
We want to look at these xij's.
00:21:57.731 --> 00:21:58.230
OK.
00:21:58.230 --> 00:22:03.570
And so those things
that we're manipulating
00:22:03.570 --> 00:22:07.020
have a phase implicit
in how you define them.
00:22:07.020 --> 00:22:09.350
But that's gone in
your manipulation.
00:22:09.350 --> 00:22:11.340
So you want to be
a little careful.
00:22:11.340 --> 00:22:13.350
OK.
00:22:13.350 --> 00:22:21.090
So we start out and
the expectation value
00:22:21.090 --> 00:22:23.460
is on this side.
00:22:23.460 --> 00:22:30.370
And the psi star psi is
localized on this side mostly.
00:22:30.370 --> 00:22:33.330
And at a later time--
00:22:33.330 --> 00:22:35.950
so this is a later time.
00:22:35.950 --> 00:22:39.320
I shouldn't make it bigger.
00:22:39.320 --> 00:22:42.380
At a later time, this thing
has moved to the other turning
00:22:42.380 --> 00:22:44.260
point.
00:22:44.260 --> 00:22:46.360
Back and forth, back and forth.
00:22:49.690 --> 00:22:55.610
Now, since we have
three energy levels--
00:22:55.610 --> 00:22:59.780
and well, actually, we
have a coherence term
00:22:59.780 --> 00:23:04.970
which involves the
product of psi 0 and psi 1
00:23:04.970 --> 00:23:07.610
and psi 1 and psi 2--
00:23:07.610 --> 00:23:11.420
they differ in
energy both by omega.
00:23:11.420 --> 00:23:15.680
So these two things have the
same oscillation frequency.
00:23:15.680 --> 00:23:18.950
And so what's going to happen
is the wave function, the wave
00:23:18.950 --> 00:23:21.620
packet, is going to move
from this side to that side,
00:23:21.620 --> 00:23:26.690
back and forth always forever
at the same frequency,
00:23:26.690 --> 00:23:28.660
no dephasing.
00:23:28.660 --> 00:23:32.470
In the middle, I can't tell you
what it's going to look like.
00:23:32.470 --> 00:23:33.670
I don't want to tell you.
00:23:33.670 --> 00:23:34.650
Because you don't care.
00:23:37.320 --> 00:23:41.580
You care mostly about what's it
going to look like at a turning
00:23:41.580 --> 00:23:42.870
point.
00:23:42.870 --> 00:23:47.920
Or what is it going to look like
when it returns to home base?
00:23:47.920 --> 00:23:50.050
And all sorts of
insights come from that.
00:23:52.600 --> 00:23:54.340
OK.
00:23:54.340 --> 00:23:59.940
Now, I said, well, if we want
to know the energy of this wave
00:23:59.940 --> 00:24:03.300
packet, well, we take
the expectation value
00:24:03.300 --> 00:24:04.890
of the Hamiltonian.
00:24:04.890 --> 00:24:07.020
And the expectation
of the Hamiltonian
00:24:07.020 --> 00:24:29.450
is c0 squared e0 plus c1
squared e1 plus c2 squared e2.
00:24:35.240 --> 00:24:37.010
These things are
easy once you've gone
00:24:37.010 --> 00:24:38.930
through it a couple of times.
00:24:38.930 --> 00:24:42.760
Because what's happening
is you have these factors,
00:24:42.760 --> 00:24:50.310
e to the minus i
ej t over H bar.
00:24:50.310 --> 00:24:54.960
And they cancel when
you do a psi star psi.
00:24:54.960 --> 00:24:58.830
Or they generate a
difference, omega H bar omega,
00:24:58.830 --> 00:25:01.750
when you have different
values of the energy.
00:25:01.750 --> 00:25:05.700
So this is something that you
can derive really quickly.
00:25:05.700 --> 00:25:09.090
And the fact that
your eigenfunction--
00:25:09.090 --> 00:25:13.780
the psi's in your
linear combination
00:25:13.780 --> 00:25:16.530
up there are eigenfunctions
of the Hamiltonian.
00:25:16.530 --> 00:25:21.460
So every time the Hamiltonian
operates in a wave function,
00:25:21.460 --> 00:25:24.510
it gives the energy
times that wave function.
00:25:24.510 --> 00:25:27.930
And then you're taking
the expectation value.
00:25:27.930 --> 00:25:32.010
And so we have the wave
function time itself integrated.
00:25:32.010 --> 00:25:33.490
And that goes away.
00:25:33.490 --> 00:25:36.870
So after doing this
a few times, you
00:25:36.870 --> 00:25:40.000
don't need to write
the intermediate steps.
00:25:40.000 --> 00:25:41.760
And you shouldn't.
00:25:41.760 --> 00:25:45.397
Because you'll get lost
in the forest of notation.
00:25:45.397 --> 00:25:46.980
Because one of the
things you probably
00:25:46.980 --> 00:25:50.790
noticed in the last lecture is
the equations got really big.
00:25:50.790 --> 00:25:52.500
And then we calculate
something else.
00:25:52.500 --> 00:25:54.660
And it gets twice as big.
00:25:54.660 --> 00:25:56.750
And then all of a
sudden, it all goes away.
00:25:56.750 --> 00:25:59.588
And that's what you want
to be able to anticipate.
00:26:09.160 --> 00:26:09.660
OK.
00:26:09.660 --> 00:26:14.260
So you can really kill this
half oscillator problem.
00:26:14.260 --> 00:26:16.210
There's nothing much
happening except you
00:26:16.210 --> 00:26:18.430
create a localization.
00:26:18.430 --> 00:26:20.080
And when you take
away the other--
00:26:20.080 --> 00:26:23.410
when you restore
the full oscillator,
00:26:23.410 --> 00:26:28.320
everything oscillates at omega.
00:26:28.320 --> 00:26:30.350
And so you have a
whole bunch of terms
00:26:30.350 --> 00:26:35.240
contributing to the
motion of the wave packet.
00:26:35.240 --> 00:26:36.410
And they're all very simple.
00:26:36.410 --> 00:26:38.659
Because they're all oscillating
at the same frequency.
00:26:42.780 --> 00:26:47.420
Now, as an aside, I want to say
there's a huge number of stuff
00:26:47.420 --> 00:26:49.590
that's up in this lecture
that's going to be
00:26:49.590 --> 00:26:53.850
on the exam, a huge amount.
00:26:53.850 --> 00:26:58.260
So if for example you wanted
to calculate something
00:26:58.260 --> 00:27:02.770
like x squared, well, fine.
00:27:02.770 --> 00:27:06.750
You know what the selection
rule for x squared is.
00:27:06.750 --> 00:27:10.560
It's delta v of 0
plus or minus 2.
00:27:10.560 --> 00:27:14.820
And so this thing is going to
generate some constant terms
00:27:14.820 --> 00:27:16.320
and some terms at 2 omega.
00:27:21.940 --> 00:27:22.620
OK.
00:27:22.620 --> 00:27:26.750
So there are a lot of things
about the harmonic oscillator
00:27:26.750 --> 00:27:31.370
that make it really wonderful
to consider a problem, even
00:27:31.370 --> 00:27:34.100
a complicated problem,
which is not explicitly
00:27:34.100 --> 00:27:35.690
a harmonic oscillator problem.
00:27:35.690 --> 00:27:39.740
Because you can get everything
so quickly without any
00:27:39.740 --> 00:27:42.215
thought after a little
bit of investment.
00:27:44.830 --> 00:27:46.640
OK.
00:27:46.640 --> 00:27:50.120
Now, we haven't talked
about electronic transitions
00:27:50.120 --> 00:27:51.590
and potential energy curves.
00:27:51.590 --> 00:27:52.910
But we will.
00:27:52.910 --> 00:27:55.130
And I think you know about them.
00:27:55.130 --> 00:27:57.850
And so we have some
electronic ground state.
00:27:57.850 --> 00:28:01.020
And we have some
electronically excited state.
00:28:01.020 --> 00:28:07.790
And so each of these states
has a vibrational coordinate.
00:28:07.790 --> 00:28:13.420
And we can pretend that it's
harmonic even if it's not.
00:28:13.420 --> 00:28:15.520
Because we build a
framework treating
00:28:15.520 --> 00:28:17.260
them as harmonic,
and then discover
00:28:17.260 --> 00:28:23.320
that there's discrepancies
which we can fit to a model.
00:28:23.320 --> 00:28:25.780
And we can determine
from the time
00:28:25.780 --> 00:28:27.340
dependence what that model is.
00:28:27.340 --> 00:28:28.090
OK.
00:28:28.090 --> 00:28:34.220
So we start out with a
molecule in v equals 0.
00:28:37.890 --> 00:28:45.420
And there is a much
repeated truism.
00:28:45.420 --> 00:28:48.870
Electrons move
fast, nuclei slow.
00:28:48.870 --> 00:28:52.450
Transition is instantaneous,
or nearly instantaneous,
00:28:52.450 --> 00:28:54.400
because it involves
the electrons.
00:28:54.400 --> 00:28:59.280
So what ends up happening
is you draw a vertical line.
00:28:59.280 --> 00:29:05.750
And now, you transfer this wave
function to the upper state.
00:29:05.750 --> 00:29:08.930
You just move-- this is
the probability amplitude
00:29:08.930 --> 00:29:15.170
distribution of the vibration.
00:29:15.170 --> 00:29:17.360
We transfer that to
the excited state.
00:29:17.360 --> 00:29:19.880
And that's not an eigenstate,
the excited state.
00:29:19.880 --> 00:29:23.770
It's a localized state,
localized at a turning point.
00:29:23.770 --> 00:29:26.560
Now, you want to know--
00:29:26.560 --> 00:29:34.330
so there's the
Franck-Condon principle,
00:29:34.330 --> 00:29:36.930
that is just another
way of saying electrons
00:29:36.930 --> 00:29:38.950
move fast, nuclei slow.
00:29:38.950 --> 00:29:44.460
And there is delta x
equals 0 delta p equals 0.
00:29:47.980 --> 00:29:51.160
If the nuclear state can't
change while the electron is
00:29:51.160 --> 00:29:58.810
jumping, then the coordinate and
the momentum are both constant.
00:29:58.810 --> 00:30:01.480
So you're creating
a wave packet.
00:30:01.480 --> 00:30:04.600
And the best place to create
it is near a turning point.
00:30:04.600 --> 00:30:08.080
Because then, you can match
the momentum of this guy
00:30:08.080 --> 00:30:11.890
to that zero point momentum,
which you can calculate.
00:30:11.890 --> 00:30:13.660
You know how to calculate this.
00:30:13.660 --> 00:30:25.540
Because the potential energy
curve is 1/2 k x squared.
00:30:25.540 --> 00:30:34.100
And the momentum is given by
this energy difference here.
00:30:34.100 --> 00:30:35.530
And so you can work it out.
00:30:35.530 --> 00:30:36.280
It's in the notes.
00:30:36.280 --> 00:30:38.290
I don't want to write it down.
00:30:38.290 --> 00:30:40.330
So you're going to
create something
00:30:40.330 --> 00:30:43.960
which is vertical and
is not quite exactly
00:30:43.960 --> 00:30:45.550
at the turning point.
00:30:45.550 --> 00:30:48.130
Because you have to have
a little bit of momentum
00:30:48.130 --> 00:30:51.270
to match the zero
point momentum here.
00:30:51.270 --> 00:30:55.220
So you know everything
about the initial state.
00:30:55.220 --> 00:30:59.035
And so you can calculate what
it is by taking the overlap of v
00:30:59.035 --> 00:31:03.550
equals 0 of the
ground state with all
00:31:03.550 --> 00:31:06.620
of the vibrational levels
of the excited state.
00:31:06.620 --> 00:31:09.730
And so we have the
coefficients of each
00:31:09.730 --> 00:31:12.760
of those vibrational levels
in the excited state that
00:31:12.760 --> 00:31:15.530
makes this wave packet.
00:31:15.530 --> 00:31:17.690
Now, of course,
this wave packet is
00:31:17.690 --> 00:31:20.660
going to move back and
forth, back and forth.
00:31:23.450 --> 00:31:25.570
And if this were a
harmonic oscillator,
00:31:25.570 --> 00:31:28.070
it would move harmonically.
00:31:28.070 --> 00:31:30.400
And so the only thing
that would appear
00:31:30.400 --> 00:31:33.700
in the expectation
value of x is going
00:31:33.700 --> 00:31:38.450
to be this motion at omega.
00:31:38.450 --> 00:31:41.000
Now, this is usually
a relatively high
00:31:41.000 --> 00:31:42.110
vibrational level.
00:31:42.110 --> 00:31:45.860
And the molecule is not
being harmonic here.
00:31:45.860 --> 00:31:49.460
So there are correction terms
called anharmonicity terms.
00:31:52.550 --> 00:32:05.320
So we have the energy level
expression plus H bar omega e
00:32:05.320 --> 00:32:06.620
xe--
00:32:06.620 --> 00:32:13.600
that's one number--
plus 1/2 squared.
00:32:13.600 --> 00:32:15.960
So we have a linear term
and a quadratic term.
00:32:15.960 --> 00:32:24.180
And this omega e xe is on
the order of 0.02 times omega
00:32:24.180 --> 00:32:26.550
e, 2%.
00:32:26.550 --> 00:32:28.870
So it's a small thing.
00:32:28.870 --> 00:32:31.800
But if you're going
to be allowing a wave
00:32:31.800 --> 00:32:36.520
packet to be built out of
many vibrational levels,
00:32:36.520 --> 00:32:38.500
this guy is going to
de-phase a little bit.
00:32:38.500 --> 00:32:40.300
So you go around and come back.
00:32:40.300 --> 00:32:42.460
And you can't quite
have everybody
00:32:42.460 --> 00:32:45.230
back where they started.
00:32:45.230 --> 00:32:53.900
And so you'll see a
decreasing amplitude here.
00:32:53.900 --> 00:33:00.480
And that's best looked at
by the survival probability.
00:33:00.480 --> 00:33:03.151
And you'll see characteristic
behavior in the survival
00:33:03.151 --> 00:33:03.650
probably.
00:33:10.570 --> 00:33:12.010
OK.
00:33:12.010 --> 00:33:14.620
But let's go a little
bit deeper before I
00:33:14.620 --> 00:33:17.240
move on to-- what time is it?
00:33:17.240 --> 00:33:20.860
Oh, I'm doing OK.
00:33:20.860 --> 00:33:23.380
So let's just say
our superposition
00:33:23.380 --> 00:33:30.790
state, the electronically
excited state,
00:33:30.790 --> 00:33:35.410
is a combination of v
equals 10 and v equals 11.
00:33:51.710 --> 00:34:02.660
So we know immediately
that psi star t psi t--
00:34:02.660 --> 00:34:07.370
this probability amplitude--
probability, yes,
00:34:07.370 --> 00:34:16.389
this probability is
c10 squared psi 10
00:34:16.389 --> 00:34:31.380
squared plus c11 squared psi
11 squared plus 2 c10 c11
00:34:31.380 --> 00:34:38.510
psi 10 psi 11 cosine omega t.
00:34:38.510 --> 00:34:40.100
That's not very legible.
00:34:45.540 --> 00:34:48.670
Now, I'm playing
fast and loose here.
00:34:48.670 --> 00:34:50.050
Because I've done this before.
00:34:50.050 --> 00:34:52.000
I know what disappears.
00:34:52.000 --> 00:34:55.030
And I know, if it's
harmonic, you just get this.
00:34:55.030 --> 00:34:57.250
Well, if there's two
states, this thing
00:34:57.250 --> 00:35:02.380
is really just the
frequency associated
00:35:02.380 --> 00:35:05.220
with these two levels.
00:35:05.220 --> 00:35:07.440
OK.
00:35:07.440 --> 00:35:13.560
Now, once we have this, we
can also generate the survival
00:35:13.560 --> 00:35:15.240
probability.
00:35:15.240 --> 00:35:17.570
Well, the survival
probability, capital P
00:35:17.570 --> 00:35:30.390
of t, that's defined as the
square modulus of psi star xt
00:35:30.390 --> 00:35:34.768
psi x0 dx.
00:35:38.920 --> 00:35:39.420
OK.
00:35:39.420 --> 00:35:42.100
And you can immediately write
what that is going to be.
00:35:42.100 --> 00:35:47.680
It's going to be c10 squared.
00:35:47.680 --> 00:35:53.010
And we've integrated so
the wave functions go away.
00:35:53.010 --> 00:36:08.220
And so we have c10 e to
the i H bar 10.5 omega
00:36:08.220 --> 00:36:29.970
t divided by H bar plus c11
squared e to the minus i plus H
00:36:29.970 --> 00:36:37.001
bar 11.5 omega e over H bar.
00:36:37.001 --> 00:36:37.500
OK.
00:36:37.500 --> 00:36:38.980
Why did I made the mistake here?
00:36:42.010 --> 00:36:43.350
Well, we have a psi star.
00:36:45.960 --> 00:36:53.490
And so we're going to get
the complex conjugate of e
00:36:53.490 --> 00:36:56.660
to the minus i omega.
00:36:56.660 --> 00:36:59.730
So you end up getting this.
00:36:59.730 --> 00:37:00.755
Practice that.
00:37:00.755 --> 00:37:02.380
AUDIENCE: That whole
thing is [? the ?]
00:37:02.380 --> 00:37:02.700
modulus squared, right?
00:37:02.700 --> 00:37:03.450
ROBERT FIELD: Yes.
00:37:03.450 --> 00:37:04.721
I have that right.
00:37:04.721 --> 00:37:05.220
Exactly.
00:37:08.100 --> 00:37:16.320
So and now, lo and behold, we
know what to do with this too.
00:37:16.320 --> 00:37:23.790
And so we're going to get
c1 0 to the 4 plus c--
00:37:23.790 --> 00:37:33.700
not 1-0, 10, c11
to the 4 plus 2 c10
00:37:33.700 --> 00:37:39.430
squared c11 squared
cosine omega t.
00:37:43.690 --> 00:37:45.200
Isn't that neat?
00:37:45.200 --> 00:37:48.872
So we've got a whole
bunch of constant terms.
00:37:48.872 --> 00:37:49.580
This is constant.
00:37:49.580 --> 00:37:53.660
Because it's square modulus
and it's to the fourth power.
00:37:53.660 --> 00:37:59.480
And so all of these
coefficients are positive.
00:37:59.480 --> 00:38:02.640
At t equals 0, this is 1.
00:38:02.640 --> 00:38:09.110
And so at t equals 0,
the survival probability
00:38:09.110 --> 00:38:09.800
is at a maximum.
00:38:13.680 --> 00:38:16.775
And at some later time,
that survival probability
00:38:16.775 --> 00:38:17.650
will be at a minimum.
00:38:24.530 --> 00:38:25.030
OK.
00:38:25.030 --> 00:38:31.540
And so you can say the maximum
will occur at integer when
00:38:31.540 --> 00:38:43.500
omega t is equal to 2 n pi.
00:38:48.290 --> 00:38:52.100
So then the exponential
factor is always 1.
00:38:55.690 --> 00:39:04.210
And we have a minimum when omega
t is an odd multiple of pi.
00:39:04.210 --> 00:39:06.900
And all the exponential
factors are minus 1.
00:39:14.660 --> 00:39:19.120
So we can also now
look at the expectation
00:39:19.120 --> 00:39:24.960
value for x of t and p of t.
00:39:24.960 --> 00:39:26.860
And I'm going to
just draw sketches.
00:39:33.950 --> 00:39:41.690
So for x of t, we have
something that looks like this.
00:39:41.690 --> 00:39:45.387
This is at the
left turning point.
00:39:45.387 --> 00:39:47.470
This is the right turning
point, or near the right
00:39:47.470 --> 00:39:48.970
turning point.
00:39:48.970 --> 00:39:56.250
And this is at pi over 2 omega.
00:39:56.250 --> 00:39:59.920
This is at pi over omega.
00:39:59.920 --> 00:40:01.950
So that's the half
oscillator point.
00:40:01.950 --> 00:40:07.110
Now, it starts out and
the expectation value
00:40:07.110 --> 00:40:08.730
is at the left turning point.
00:40:08.730 --> 00:40:11.210
And it's not changing.
00:40:11.210 --> 00:40:13.970
The derivative of
the expectation value
00:40:13.970 --> 00:40:17.138
with respect to t is 0.
00:40:17.138 --> 00:40:24.970
The momentum, which
has a different phase--
00:40:24.970 --> 00:40:28.940
the momentum starts
out at 0 also.
00:40:28.940 --> 00:40:35.270
And at pi over 2 omega,
it reaches a maximum.
00:40:35.270 --> 00:40:40.460
And at pi over omega,
it reaches 9 again.
00:40:40.460 --> 00:40:44.270
But the important
thing here is at t
00:40:44.270 --> 00:40:47.420
equals 0 the derivative
of the momentum
00:40:47.420 --> 00:40:48.770
is as big as it can get.
00:40:52.510 --> 00:40:56.600
So what's that telling you?
00:40:56.600 --> 00:40:58.660
It's telling you
this wave packet,
00:40:58.660 --> 00:41:02.320
as far as coordinate space is
concerned, it's not moving at t
00:41:02.320 --> 00:41:04.120
equals 0.
00:41:04.120 --> 00:41:09.470
As far as momentum is concerned,
it's moving like crazy.
00:41:09.470 --> 00:41:16.990
And so this survival probability
is changing entirely dominated
00:41:16.990 --> 00:41:19.420
by the change in momentum,
which is encoded in the wave
00:41:19.420 --> 00:41:23.780
function, which is neat.
00:41:23.780 --> 00:41:31.340
Because Newton's equation
says that the time derivative
00:41:31.340 --> 00:41:39.200
of the momentum is equal to--
00:41:39.200 --> 00:41:44.330
that's the mass
times acceleration--
00:41:44.330 --> 00:41:47.660
is equal to the
force, which is minus
00:41:47.660 --> 00:41:55.912
the gradient of the potential--
00:41:55.912 --> 00:41:57.860
the function of time.
00:41:57.860 --> 00:42:04.530
And for a harmonic oscillator,
the gradient potential is x kx.
00:42:04.530 --> 00:42:15.760
And so it's telling you that
we have this relationship
00:42:15.760 --> 00:42:19.060
between x and t and p.
00:42:19.060 --> 00:42:23.350
And we know what
the momentum is.
00:42:23.350 --> 00:42:24.760
And I'm sorry.
00:42:24.760 --> 00:42:27.320
And so the change
in the momentum,
00:42:27.320 --> 00:42:30.760
which is responsible for
the change in the survival
00:42:30.760 --> 00:42:35.890
probability at t equals
0, is due to the gradient
00:42:35.890 --> 00:42:37.550
of the potential.
00:42:37.550 --> 00:42:40.390
So we're actually
sampling what is
00:42:40.390 --> 00:42:44.320
the gradient of the potential
at the left turning point.
00:42:44.320 --> 00:42:46.150
And often, for any
kind of a problem,
00:42:46.150 --> 00:42:49.060
we want to know
what kind of force
00:42:49.060 --> 00:42:53.470
is acting on the wave packet.
00:42:53.470 --> 00:42:57.540
There it is, classic mechanics
embedded in quantum mechanics.
00:42:57.540 --> 00:43:00.640
It's really amazing.
00:43:00.640 --> 00:43:04.110
So if you have some
way of measuring
00:43:04.110 --> 00:43:07.560
the survival probability
near t equals 0,
00:43:07.560 --> 00:43:11.460
it's telling you what the
slope of the potential
00:43:11.460 --> 00:43:12.660
is at the turning point.
00:43:22.575 --> 00:43:23.075
OK.
00:43:39.990 --> 00:43:44.490
Well, if we have an
initial state which
00:43:44.490 --> 00:43:49.830
involves many eigenstates,
the Hamiltonian,
00:43:49.830 --> 00:43:54.840
we still know that there's going
to be some complicated behavior
00:43:54.840 --> 00:43:58.540
which is modulated by omega.
00:43:58.540 --> 00:44:03.620
So we have a cosine
omega t always.
00:44:03.620 --> 00:44:05.610
And so no matter
what kind of a wave
00:44:05.610 --> 00:44:09.750
function we make, if we start
out at one turning point,
00:44:09.750 --> 00:44:13.230
it's going to go to the
other turning point.
00:44:13.230 --> 00:44:16.230
And it'll keep coming
back, and back, and back.
00:44:16.230 --> 00:44:20.610
And so one could imagine
doing an experiment where--
00:44:20.610 --> 00:44:24.881
let's just draw the excited
state and some other repulsive
00:44:24.881 --> 00:44:25.380
state.
00:44:29.980 --> 00:44:33.520
So at this turning point,
vertical transition
00:44:33.520 --> 00:44:37.690
to that repulsive state is
within the range of your laser.
00:44:37.690 --> 00:44:41.230
And at this turning
point, it's way high.
00:44:41.230 --> 00:44:43.810
And so what you
can imagine doing
00:44:43.810 --> 00:44:52.830
is probing where this wave
packet is as a function of time
00:44:52.830 --> 00:44:59.570
by having a probe pulse which
creates dissociating fragments.
00:44:59.570 --> 00:45:05.040
And now, that is what
Ahmed Zewail did.
00:45:05.040 --> 00:45:09.500
He talked about real
dynamics in real time.
00:45:09.500 --> 00:45:12.620
So when the wave packet is
here, it can't be dissociated.
00:45:12.620 --> 00:45:14.380
When it's here, it can.
00:45:14.380 --> 00:45:16.650
And you look at the fragments.
00:45:16.650 --> 00:45:17.890
Very simple.
00:45:17.890 --> 00:45:21.810
So there are lots of ways of
taking these simple pictures
00:45:21.810 --> 00:45:26.080
of wave packets
moving and saying,
00:45:26.080 --> 00:45:29.120
OK, I can use them to set
up an experiment where
00:45:29.120 --> 00:45:32.490
I ask a very specific question.
00:45:32.490 --> 00:45:33.270
And I get answers.
00:45:33.270 --> 00:45:34.894
And I know what to
do with the answers.
00:45:37.530 --> 00:45:47.560
Now, if it's not
harmonic, you can still
00:45:47.560 --> 00:45:49.930
do Zewail's experiment
or some other experiment.
00:45:49.930 --> 00:45:54.970
And you can ask, OK, the
wave packet moved over here.
00:45:54.970 --> 00:45:56.960
And there's some
dynamics that occurs.
00:45:56.960 --> 00:45:59.230
It dissociates or
you do something.
00:45:59.230 --> 00:46:04.240
And when it comes back, it's
not at the same amplitude.
00:46:04.240 --> 00:46:13.600
And so you can use the time
history of these recurrences
00:46:13.600 --> 00:46:14.500
if it's harmonic.
00:46:14.500 --> 00:46:18.700
If it's not harmonic, then the
recurrences-- even if nothing
00:46:18.700 --> 00:46:21.250
happens over here,
the recurrences
00:46:21.250 --> 00:46:24.930
will be increasingly
less perfect.
00:46:24.930 --> 00:46:27.202
And so you measure
the anharmonicity.
00:46:35.240 --> 00:46:36.290
OK.
00:46:36.290 --> 00:46:36.990
Five minutes.
00:46:50.980 --> 00:46:54.460
Tunneling is a quantum
mechanical phenomenon.
00:46:54.460 --> 00:46:58.450
And again, you want to use
the simplest possible picture
00:46:58.450 --> 00:47:01.450
to understand the
signature of tunneling.
00:47:01.450 --> 00:47:02.450
And so what do you do?
00:47:02.450 --> 00:47:04.300
Well, the simplest
thing you can do
00:47:04.300 --> 00:47:07.460
is start with a
harmonic oscillator.
00:47:07.460 --> 00:47:11.740
The next thing you do is you
put a barrier in the middle.
00:47:11.740 --> 00:47:14.400
And you make it really thin.
00:47:14.400 --> 00:47:17.100
Thin is because you don't want
to calculate the integral.
00:47:17.100 --> 00:47:22.410
You want to just say, oh yeah,
I know that the wave function--
00:47:22.410 --> 00:47:23.640
forget about the barrier.
00:47:23.640 --> 00:47:31.230
We know the magnitude of the
wave function at the barrier,
00:47:31.230 --> 00:47:31.770
OK?
00:47:31.770 --> 00:47:41.040
And so if you have a state
which has a maximum here,
00:47:41.040 --> 00:47:44.720
well, that means that
it's feeling the barrier.
00:47:44.720 --> 00:47:47.480
And so you know what to do.
00:47:47.480 --> 00:47:51.760
If there is a node here, it
doesn't know about the barrier.
00:47:51.760 --> 00:47:53.961
So you can do all sorts
of things really fast.
00:47:53.961 --> 00:47:54.460
OK.
00:47:54.460 --> 00:47:58.520
So and it's harmonic because
you're going to want to use
00:47:58.520 --> 00:47:59.835
a's and a-dagger's.
00:48:04.950 --> 00:48:09.930
So v equals 1, 3, 5.
00:48:09.930 --> 00:48:12.790
They have nodes here.
00:48:12.790 --> 00:48:14.620
They don't know
about the barrier.
00:48:14.620 --> 00:48:19.460
Or if they do, it's
a very modest effect.
00:48:19.460 --> 00:48:21.920
If you make this thin enough,
you won't know about it all.
00:48:25.110 --> 00:48:36.650
And then v equals 0,
2, 2 4, et cetera--
00:48:36.650 --> 00:48:37.850
well, they're maxima.
00:48:37.850 --> 00:48:40.857
Well, they have a
local maximum here.
00:48:40.857 --> 00:48:42.690
And remember, this is
a harmonic oscillator.
00:48:42.690 --> 00:48:45.770
So the big lobes
are on the extremes.
00:48:45.770 --> 00:48:49.900
The smallest lobe is in
the middle, but it's not 0.
00:48:49.900 --> 00:48:50.400
OK.
00:48:50.400 --> 00:48:52.600
There are also
symmetry restrictions.
00:48:52.600 --> 00:48:56.230
This is a problem
that has symmetry.
00:48:56.230 --> 00:48:59.280
We have even functions
and odd functions.
00:48:59.280 --> 00:49:11.970
And for the even functions,
d psi dx is equal to 0.
00:49:11.970 --> 00:49:18.970
And for the odd functions,
psi of 0 is equal to 0.
00:49:24.080 --> 00:49:24.730
OK.
00:49:24.730 --> 00:49:28.121
So this is something that-- you
choose the simplest problem.
00:49:28.121 --> 00:49:28.995
And you use symmetry.
00:49:31.670 --> 00:49:34.040
And bang, all of a sudden,
you get fantastic insights.
00:49:45.500 --> 00:49:48.390
So let's look at what happens
to the even functions.
00:49:48.390 --> 00:49:53.100
So let's just draw a picture of
the harmonic oscillator energy
00:49:53.100 --> 00:49:53.600
levels.
00:49:59.940 --> 00:50:04.120
And we have a barrier, which
we say maybe goes that high.
00:50:07.170 --> 00:50:10.180
And so we have 1.
00:50:10.180 --> 00:50:12.380
And we have 3.
00:50:12.380 --> 00:50:14.180
And we have 5.
00:50:14.180 --> 00:50:16.804
They're basically not
affected by the barrier.
00:50:16.804 --> 00:50:18.470
Because they have a
node at the barrier.
00:50:21.300 --> 00:50:25.060
And then we have v equals 0.
00:50:25.060 --> 00:50:26.250
What happens to v equals 0?
00:50:26.250 --> 00:50:29.000
Well, it hits this barrier.
00:50:29.000 --> 00:50:33.340
And it's got a large amplitude.
00:50:33.340 --> 00:50:38.890
And it can't accrue more phase
as you go through the barrier.
00:50:38.890 --> 00:50:41.050
So when it hits the
other turning point,
00:50:41.050 --> 00:50:43.130
it doesn't satisfy the
boundary conditions.
00:50:43.130 --> 00:50:47.420
It'll go to infinity at x
equals positive infinity.
00:50:47.420 --> 00:50:54.010
So it has to be shifted so
that the boundary conditions
00:50:54.010 --> 00:50:56.230
at the turning points are met.
00:50:56.230 --> 00:50:58.450
And the only way
that can happen is
00:50:58.450 --> 00:51:01.420
it gets shifted up
in energy a lot.
00:51:01.420 --> 00:51:05.520
So v equals 0 is here.
00:51:05.520 --> 00:51:08.025
Now, it can't be shifted
above v equals 1.
00:51:08.025 --> 00:51:10.710
Because that would
violate the node rule.
00:51:10.710 --> 00:51:13.650
And if you draw a picture
of the wave function for v
00:51:13.650 --> 00:51:17.400
equals 0 in the boundary
region, the wave function
00:51:17.400 --> 00:51:19.710
has to look something--
well, it has
00:51:19.710 --> 00:51:23.110
to look something like that.
00:51:23.110 --> 00:51:25.830
So we have two lobes.
00:51:25.830 --> 00:51:29.550
And it's trying to make
a node, but it can't.
00:51:29.550 --> 00:51:32.040
It's a decreasing exponential.
00:51:32.040 --> 00:51:36.180
I mean, if you have a wave
going into this region
00:51:36.180 --> 00:51:39.300
of the barrier, you can have
an increasing exponential
00:51:39.300 --> 00:51:41.770
or a decreasing exponential.
00:51:41.770 --> 00:51:44.400
You can never satisfy
continuity of the wave function
00:51:44.400 --> 00:51:46.560
with the increasing exponential.
00:51:46.560 --> 00:51:49.440
But you can with the
decreasing exponential.
00:51:49.440 --> 00:51:53.550
And so the height of the barrier
determines how close this guy
00:51:53.550 --> 00:51:56.520
comes to having a node.
00:51:56.520 --> 00:51:57.360
It never will.
00:52:00.080 --> 00:52:03.890
And how does this compare to the
wave function for v equals 1?
00:52:03.890 --> 00:52:07.670
Well, the wave function for
v equals 1 looks like this.
00:52:07.670 --> 00:52:11.300
The amplitude in this lobe
and the amplitude in that lobe
00:52:11.300 --> 00:52:12.170
are the same.
00:52:12.170 --> 00:52:14.360
But the sign is reversed.
00:52:14.360 --> 00:52:17.750
And so basically,
this picture tells you
00:52:17.750 --> 00:52:20.900
that these two levels have
almost exactly the same energy.
00:52:23.470 --> 00:52:27.010
So v equals 0 is shifted up.
00:52:27.010 --> 00:52:30.940
v equals 2 is shifted up,
but not quite so much.
00:52:30.940 --> 00:52:36.410
And v 4 is shifted
up hardly at all.
00:52:36.410 --> 00:52:40.070
And so you get what's
called level staggering.
00:52:40.070 --> 00:52:45.360
And this level staggering is
the signature of tunneling.
00:52:45.360 --> 00:52:50.070
This is how we know about
tunneling, the only way we
00:52:50.070 --> 00:52:52.510
know about tunneling.
00:52:52.510 --> 00:52:54.210
So we measure the energy levels.
00:52:54.210 --> 00:52:59.040
And it tells us how are
the wave functions sampling
00:52:59.040 --> 00:53:00.750
this barrier.
00:53:00.750 --> 00:53:03.710
Now, this is a childish barrier.
00:53:03.710 --> 00:53:05.820
And there is a real barrier--
00:53:05.820 --> 00:53:08.610
molecules isomerize.
00:53:08.610 --> 00:53:10.770
And they isomerize
over a barrier.
00:53:10.770 --> 00:53:13.410
And the barrier isn't
necessarily at x equals 0.
00:53:13.410 --> 00:53:15.030
But if you understand
this problem,
00:53:15.030 --> 00:53:17.650
you can deal with isomerization.
00:53:17.650 --> 00:53:20.400
Now, I'm an author
of a paper that's
00:53:20.400 --> 00:53:23.570
just appearing in Science
in the next few weeks
00:53:23.570 --> 00:53:28.510
on the isomerization from
vinylidene to acetylene.
00:53:28.510 --> 00:53:29.635
There's a barrier involved.
00:53:32.470 --> 00:53:37.510
So these sorts of pictures are
important for understanding
00:53:37.510 --> 00:53:39.650
those sorts of phenomena.
00:53:39.650 --> 00:53:43.210
Now, so I'm done really.
00:53:43.210 --> 00:53:48.340
So what we are now
encountering with the time
00:53:48.340 --> 00:53:52.300
dependent Schrodinger
equation is discovering
00:53:52.300 --> 00:53:58.680
how dynamics, like
tunneling, is encoded
00:53:58.680 --> 00:54:01.060
in an eigenstate spectrum.
00:54:01.060 --> 00:54:06.850
Because the encoding is
level staggering eigenstates.
00:54:06.850 --> 00:54:11.080
So people normally
have this naive idea
00:54:11.080 --> 00:54:19.420
that eigenstate time independent
Hamiltonian spectroscopy
00:54:19.420 --> 00:54:21.820
does not sample dynamics
because it's only
00:54:21.820 --> 00:54:23.620
measuring energy levels.
00:54:23.620 --> 00:54:26.890
Real dynamical processes
are time dependent.
00:54:26.890 --> 00:54:31.000
But the dynamics is encoded
in energy level patterns.
00:54:31.000 --> 00:54:36.650
And that is actually
my signature
00:54:36.650 --> 00:54:39.440
in experimental spectroscopy.
00:54:39.440 --> 00:54:42.050
I've looked for ways
in which dynamics is
00:54:42.050 --> 00:54:45.140
encoded in eigenstate spectra.
00:54:45.140 --> 00:54:49.340
And chemists are interested
in dynamics much more than
00:54:49.340 --> 00:54:51.520
in structure.
00:54:51.520 --> 00:54:53.540
So that's it for today.
00:54:53.540 --> 00:54:57.660
I will see you on Wednesday.