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PROFESSOR: The outline for today
is a little bit more review
00:00:26.332 --> 00:00:28.540
of feeling the power of the
creation and annihilation
00:00:28.540 --> 00:00:31.420
operators.
00:00:31.420 --> 00:00:35.320
They enable you to do anything
with harmonic oscillators,
00:00:35.320 --> 00:00:37.310
really fast.
00:00:37.310 --> 00:00:42.760
And so you don't
spend time thinking
00:00:42.760 --> 00:00:44.860
about how to do the math.
00:00:44.860 --> 00:00:48.760
You think about the meaning of
the problem and understanding
00:00:48.760 --> 00:00:51.940
how to manipulate,
or how to understand
00:00:51.940 --> 00:00:54.780
harmonic oscillators.
00:00:54.780 --> 00:00:56.740
And then the exciting thing.
00:00:56.740 --> 00:00:59.790
The real Schrodinger
equation is not the one
00:00:59.790 --> 00:01:01.160
we've been playing with.
00:01:01.160 --> 00:01:04.060
It's the time dependent
Schrodinger equation.
00:01:04.060 --> 00:01:09.330
And I like to introduce
the time independent one
00:01:09.330 --> 00:01:15.840
first, because it enables you
to sort of develop some insight
00:01:15.840 --> 00:01:18.030
and assemble some of
the tools before we
00:01:18.030 --> 00:01:21.450
hit the really serious stuff.
00:01:21.450 --> 00:01:25.970
And so I will talk about this
and use the time dependent
00:01:25.970 --> 00:01:28.740
Schrodinger equation
to show what
00:01:28.740 --> 00:01:37.600
it takes to get a motion of this
product of the psi star psi.
00:01:37.600 --> 00:01:38.350
What does it take?
00:01:38.350 --> 00:01:42.750
It takes a superposition
state consisting of at least
00:01:42.750 --> 00:01:45.660
two different energies.
00:01:45.660 --> 00:01:48.830
Then this normalization
integral--
00:01:48.830 --> 00:01:52.940
well, you'd expect
normalization, or something
00:01:52.940 --> 00:01:56.150
like this to be preserved.
00:01:56.150 --> 00:02:00.910
And in fact, the normalization
is time independent.
00:02:00.910 --> 00:02:04.910
Then, if we calculate--
00:02:04.910 --> 00:02:08.570
now, this is a vector rather
than just a simple coordinate.
00:02:08.570 --> 00:02:13.810
If we calculate the expectation
value of the position,
00:02:13.810 --> 00:02:19.610
and the expectation
value of the momentum,
00:02:19.610 --> 00:02:23.030
Ehrenfest's theorem
tells you that this
00:02:23.030 --> 00:02:25.820
is going to describe the motion
of the center of the wave
00:02:25.820 --> 00:02:27.170
packet.
00:02:27.170 --> 00:02:30.380
And the motion of the
center of the wave packet
00:02:30.380 --> 00:02:32.720
is following Newton's laws.
00:02:32.720 --> 00:02:34.050
No surprise.
00:02:34.050 --> 00:02:37.490
And so we're seeing
how classical mechanics
00:02:37.490 --> 00:02:40.680
is given back to
you once we start
00:02:40.680 --> 00:02:43.470
making particle-like states.
00:02:43.470 --> 00:02:46.380
We can see how these
particle-like states evolve.
00:02:46.380 --> 00:02:51.510
But there is a lot more than
just particle-like behavior.
00:02:51.510 --> 00:02:56.070
You would, perhaps, challenge
any outfielder for the Red Sox
00:02:56.070 --> 00:03:00.810
to go beyond calculating the
center of the wave packet.
00:03:00.810 --> 00:03:02.149
That's what they know to do.
00:03:02.149 --> 00:03:03.690
But they don't know
some other things
00:03:03.690 --> 00:03:06.280
like survival
probability, how fast
00:03:06.280 --> 00:03:09.870
does the wave packet move
away from its birthplace,
00:03:09.870 --> 00:03:15.360
or the wave packet does stuff
and wanders around and comes
00:03:15.360 --> 00:03:18.450
back and rephases sometimes.
00:03:18.450 --> 00:03:22.080
And sometimes it just dephases.
00:03:22.080 --> 00:03:25.260
And these are words that
you're going to want
00:03:25.260 --> 00:03:28.030
to put into your vocabulary.
00:03:28.030 --> 00:03:30.900
So there's a lot of
really beautiful stuff
00:03:30.900 --> 00:03:33.480
when we start looking at the
time dependent Schrodinger
00:03:33.480 --> 00:03:34.860
equation.
00:03:34.860 --> 00:03:40.770
And we are going to mostly
consider the time dependent
00:03:40.770 --> 00:03:43.950
Schrodinger equation
when the Hamiltonian is
00:03:43.950 --> 00:03:47.460
independent of time, because
we can get our arms around that
00:03:47.460 --> 00:03:48.800
really easily.
00:03:48.800 --> 00:03:51.390
But when the Hamiltonian
is dependent on time,
00:03:51.390 --> 00:03:54.450
then it opens up a
world of complexity
00:03:54.450 --> 00:03:59.400
that is really best left
to a more advanced course.
00:04:02.860 --> 00:04:09.210
But how do molecules get excited
from one state to another?
00:04:09.210 --> 00:04:11.300
A time dependent Hamiltonian.
00:04:11.300 --> 00:04:15.440
So we are going to have
to at least briefly talk
00:04:15.440 --> 00:04:18.800
about a time dependent
Hamiltonian and what it does.
00:04:18.800 --> 00:04:20.959
And you'll see that eventually.
00:04:26.480 --> 00:04:29.800
I'm first going to do
a little bit of review
00:04:29.800 --> 00:04:33.280
of the a's and a daggers.
00:04:33.280 --> 00:04:36.310
So the most important
thing is, we're
00:04:36.310 --> 00:04:41.860
going to have some problem
where we have this x operator
00:04:41.860 --> 00:04:44.810
to some integer power.
00:04:44.810 --> 00:04:48.220
And so we need to be
able to relate that
00:04:48.220 --> 00:04:49.840
to the a's and a daggers.
00:05:00.750 --> 00:05:04.800
I'll drop the hats, because
we know they're there.
00:05:04.800 --> 00:05:09.390
And one of the nice things
is, if we have this operator
00:05:09.390 --> 00:05:16.010
to the nth power, we have this
co-factor to the nth power.
00:05:16.010 --> 00:05:17.762
So we don't ever
have to deal with it
00:05:17.762 --> 00:05:19.970
until the end of the problem
when we say, oh, wow, we
00:05:19.970 --> 00:05:21.230
had x to the n.
00:05:21.230 --> 00:05:25.621
Well, we have then n h bar.
00:05:25.621 --> 00:05:26.120
I'm sorry.
00:05:26.120 --> 00:05:30.042
We have h bar over 2
mu n n over 2 power.
00:05:32.900 --> 00:05:34.940
So it really saves
a lot of writing.
00:05:34.940 --> 00:05:38.150
And it means you
solve one problem
00:05:38.150 --> 00:05:40.160
for a harmonic oscillator.
00:05:40.160 --> 00:05:43.110
And you've solved it for
all harmonic oscillators.
00:05:43.110 --> 00:05:45.990
And this is just details
about what is the mass
00:05:45.990 --> 00:05:50.830
and what is the force constant,
which you need to know.
00:05:50.830 --> 00:05:52.430
And there is a
similar thing for p.
00:05:54.990 --> 00:05:57.360
So that opens the door.
00:05:57.360 --> 00:06:00.720
It says, OK, we have
some problem involving
00:06:00.720 --> 00:06:03.765
the coordinate and the momentum,
some function of the coordinate
00:06:03.765 --> 00:06:08.130
and momentum, and we
want to know things
00:06:08.130 --> 00:06:10.290
about the coordinate
and momentum.
00:06:10.290 --> 00:06:15.070
And so we use this
operator algebra.
00:06:15.070 --> 00:06:27.760
And so a operating on psi v
gives v square root v minus 1.
00:06:27.760 --> 00:06:29.490
And a dagger, we know that.
00:06:29.490 --> 00:06:31.080
So that's hardwired.
00:06:31.080 --> 00:06:35.400
But suppose we
wanted a to the 5th,
00:06:35.400 --> 00:06:40.810
operating on psi v.
Well, what do we do?
00:06:40.810 --> 00:06:46.720
Well, we start, and we
start operating on...
00:06:46.720 --> 00:06:48.900
You know, if this were
a complicated operator,
00:06:48.900 --> 00:06:52.440
we would take the rightmost
piece of that operator
00:06:52.440 --> 00:06:54.430
and operate on
the wave function.
00:06:54.430 --> 00:07:02.260
And that will give us v and then
v minus 1 and then v minus 2,
00:07:02.260 --> 00:07:04.154
v minus 3.
00:07:04.154 --> 00:07:05.070
I've got five of them.
00:07:05.070 --> 00:07:06.495
1, 2, 3, 4, 5.
00:07:06.495 --> 00:07:08.640
v minus 4.
00:07:08.640 --> 00:07:12.080
Square root psi v minus 5.
00:07:15.080 --> 00:07:15.860
OK.
00:07:15.860 --> 00:07:17.450
It's mechanical.
00:07:17.450 --> 00:07:19.370
You don't remember this.
00:07:19.370 --> 00:07:23.720
You generate this
one step at a time.
00:07:23.720 --> 00:07:25.570
And it's automatic.
00:07:25.570 --> 00:07:27.110
And so it doesn't
stress your brain.
00:07:27.110 --> 00:07:28.430
You can be thinking
about the next thing
00:07:28.430 --> 00:07:29.846
while you're writing
that garbage.
00:07:37.290 --> 00:07:40.320
We have this number
operator, which
00:07:40.320 --> 00:07:43.920
is a friend, because
it enables you to just
00:07:43.920 --> 00:07:45.840
get rid of a bunch of terms.
00:07:45.840 --> 00:07:50.340
The number operator is a dagger
a, and the number operator
00:07:50.340 --> 00:07:58.660
operating on psi
v gives v psi v.
00:07:58.660 --> 00:08:03.790
If we want any harmonic
oscillator function,
00:08:03.790 --> 00:08:15.190
we can operate on psi 0 with
a dagger to the v power.
00:08:15.190 --> 00:08:18.580
And then we have to
correct, because you
00:08:18.580 --> 00:08:24.420
know that this operating v
times on this will give psi v.
00:08:24.420 --> 00:08:27.730
But it will also give a
bunch of garbage, right?
00:08:27.730 --> 00:08:29.800
And you want to
cancel that garbage.
00:08:29.800 --> 00:08:32.919
And so you write v
factorial minus 1/2.
00:08:36.980 --> 00:08:39.409
And so that gives you
the normalized function.
00:08:39.409 --> 00:08:43.100
So these are really,
really simple things.
00:08:43.100 --> 00:08:45.954
And most of them, once you've
thought about it a little bit,
00:08:45.954 --> 00:08:47.120
you can figure out yourself.
00:08:51.790 --> 00:08:53.580
OK.
00:08:53.580 --> 00:09:01.590
Now, many problems involve
x, x cubed, x to the 4th.
00:09:01.590 --> 00:09:05.340
And we know that x is this.
00:09:05.340 --> 00:09:08.046
And x squared is this squared.
00:09:08.046 --> 00:09:10.020
And x cubed is this cubed.
00:09:10.020 --> 00:09:12.580
And so we have to do a
little algebra to simplify.
00:09:12.580 --> 00:09:24.060
And what we want to do is
simplify to sum of terms
00:09:24.060 --> 00:09:31.000
according to delta
v selection rule.
00:09:35.790 --> 00:09:38.460
So there is some
algebra that we do,
00:09:38.460 --> 00:09:44.780
when we have, say, a dagger,
a dagger, a, a, a dagger.
00:09:44.780 --> 00:09:49.010
But we know that three a
daggers and two a's means
00:09:49.010 --> 00:09:50.330
delta v of plus 1.
00:09:54.780 --> 00:09:59.110
And that came from
x to the 5th power.
00:09:59.110 --> 00:10:02.290
But you get a lot of terms
from x to the 5th power.
00:10:02.290 --> 00:10:04.130
And you have to simplify them.
00:10:04.130 --> 00:10:08.260
And in order to do that, you
use this commutation rule--
00:10:08.260 --> 00:10:13.857
a, a dagger is equal to 1.
00:10:13.857 --> 00:10:14.940
And that rearranges terms.
00:10:14.940 --> 00:10:16.830
So all of the work
you do when you're
00:10:16.830 --> 00:10:20.820
faced with a problem
involving integrals
00:10:20.820 --> 00:10:24.990
of integer powers of x and
p for a harmonic oscillator
00:10:24.990 --> 00:10:29.700
is playing around with
moving the a's and a daggers
00:10:29.700 --> 00:10:32.220
around, so that you
have all the terms that
00:10:32.220 --> 00:10:37.680
have the same selection rule
compressed into one term.
00:10:37.680 --> 00:10:38.650
That's the work.
00:10:38.650 --> 00:10:39.990
It's not much.
00:10:39.990 --> 00:10:44.430
And once you've done it for
x squared, x cubed and x 4th,
00:10:44.430 --> 00:10:47.040
you've done it as much as
you'll ever need to do.
00:10:47.040 --> 00:10:48.220
And that's it.
00:10:48.220 --> 00:10:49.370
That's the end.
00:10:49.370 --> 00:10:49.870
OK.
00:10:54.390 --> 00:11:00.070
Now here's an
example of a problem
00:11:00.070 --> 00:11:01.850
that's a little bit tricky.
00:11:01.850 --> 00:11:06.350
And it's sort of right
at the borderline of what
00:11:06.350 --> 00:11:08.920
I might use on the exam.
00:11:08.920 --> 00:11:15.840
So we have this, we have
a dagger to the m power.
00:11:15.840 --> 00:11:18.430
A to the n power.
00:11:18.430 --> 00:11:20.330
Psi.
00:11:20.330 --> 00:11:20.910
OK.
00:11:20.910 --> 00:11:22.870
Now, here.
00:11:22.870 --> 00:11:26.290
What v is going to give
a non-zero integral?
00:11:26.290 --> 00:11:28.690
We have a dagger to the m.
00:11:28.690 --> 00:11:31.120
And we have a to the n.
00:11:31.120 --> 00:11:41.570
And so that's going to be
v minus n plus m, right?
00:11:41.570 --> 00:11:47.430
Because we lose n
quanta, because of this.
00:11:47.430 --> 00:11:49.440
And we gain m quanta
because of that.
00:11:52.180 --> 00:11:54.220
And now, well, that's good.
00:11:54.220 --> 00:11:55.810
You've used the selection rule.
00:11:55.810 --> 00:11:58.890
Now, how do we write
out this interval?
00:11:58.890 --> 00:12:01.960
And there is a little
bit of art there too.
00:12:01.960 --> 00:12:07.240
Because we have a
whole bunch of terms.
00:12:07.240 --> 00:12:11.360
We have n plus m terms
in the square root.
00:12:11.360 --> 00:12:15.250
And how do you generate
them without getting lost?
00:12:18.721 --> 00:12:19.220
Yes?
00:12:19.220 --> 00:12:20.886
AUDIENCE: I think you
have it backwards.
00:12:20.886 --> 00:12:24.512
Shouldn't it be plus n minus n?
00:12:24.512 --> 00:12:26.920
It means you're going
to lower it n times.
00:12:26.920 --> 00:12:27.950
PROFESSOR: Yes.
00:12:27.950 --> 00:12:30.410
Yes.
00:12:30.410 --> 00:12:32.190
I wonder what I had in my notes.
00:12:32.190 --> 00:12:33.290
This is wrong.
00:12:39.980 --> 00:12:41.690
OK.
00:12:41.690 --> 00:12:49.900
So this has to
withstand this and this.
00:12:49.900 --> 00:12:51.730
And so, yes.
00:12:51.730 --> 00:12:52.230
OK.
00:12:52.230 --> 00:12:54.810
However you remember it,
you've got to do it right.
00:12:54.810 --> 00:13:00.430
And now we have the
actual matrix element.
00:13:00.430 --> 00:13:07.950
And so the first term is going
to be what does a n do to that?
00:13:07.950 --> 00:13:10.080
So you start on the right.
00:13:10.080 --> 00:13:13.460
And you start
building up this way.
00:13:13.460 --> 00:13:19.830
And so the first term is going
to be, what does a do to this?
00:13:19.830 --> 00:13:24.950
Well, it's going
to leave it alone.
00:13:24.950 --> 00:13:27.440
But it's going to lower v.
00:13:27.440 --> 00:13:32.060
So we have v plus n minus m.
00:13:32.060 --> 00:13:37.790
And then we have v
plus n minus m minus 1,
00:13:37.790 --> 00:13:39.995
et cetera, until
we have n terms.
00:13:45.740 --> 00:13:49.940
And then we start going
back up, because we're now
00:13:49.940 --> 00:13:53.160
dealing with this.
00:13:53.160 --> 00:13:56.410
And so I'm not going to
write the rest of this.
00:13:56.410 --> 00:14:00.760
Maybe this is going to be a
problem I start the exam with.
00:14:04.180 --> 00:14:05.500
So you don't want to get lost.
00:14:05.500 --> 00:14:08.080
And the main thing is,
you have these operators.
00:14:08.080 --> 00:14:12.910
And you start operating on the
right, and one step at a time.
00:14:12.910 --> 00:14:16.060
And this is a little
tricky because you're
00:14:16.060 --> 00:14:18.130
changing the quantum
number, and you're
00:14:18.130 --> 00:14:20.100
changing the wave function.
00:14:20.100 --> 00:14:22.420
And you have to
keep both in mind,
00:14:22.420 --> 00:14:25.610
but you're only
writing down this.
00:14:25.610 --> 00:14:26.110
OK?
00:14:35.330 --> 00:14:38.330
I want to save enough time
so that we can actually
00:14:38.330 --> 00:14:41.510
do the time dependent
Schrodinger equation.
00:14:41.510 --> 00:14:43.470
The time dependent
Schrodinger equation.
00:14:43.470 --> 00:14:45.340
H psi.
00:14:55.250 --> 00:14:57.530
It still looks pretty simple.
00:14:57.530 --> 00:15:02.130
Instead of e psi here,
we have this thing.
00:15:02.130 --> 00:15:04.500
This is the time dependent
Schrodinger equation.
00:15:04.500 --> 00:15:05.460
This is it.
00:15:05.460 --> 00:15:07.170
This is quantum mechanics.
00:15:07.170 --> 00:15:09.900
Everything that comes
from quantum mechanics
00:15:09.900 --> 00:15:11.850
starts with this.
00:15:11.850 --> 00:15:15.270
When we don't care
about time, we
00:15:15.270 --> 00:15:17.760
can use the time independent
Schrodinger equation.
00:15:17.760 --> 00:15:19.950
But when we do
care about time, we
00:15:19.950 --> 00:15:22.480
have to be a little bit careful.
00:15:22.480 --> 00:15:25.030
So this is the real
Schrodinger equation.
00:15:25.030 --> 00:15:27.250
And notice that
I'm using a capital
00:15:27.250 --> 00:15:33.900
psi, rather than a lower
case, or less decorated psi.
00:15:33.900 --> 00:15:37.890
And so this is usually used
to indicate the time dependent
00:15:37.890 --> 00:15:38.910
Schrodinger equation.
00:15:38.910 --> 00:15:40.680
It's time dependent
wave function.
00:15:40.680 --> 00:15:46.200
This is used to indicate the
time independent equation.
00:15:46.200 --> 00:15:49.520
Now, if the Hamiltonian--
00:15:49.520 --> 00:15:51.180
and this is wonderful--
00:15:51.180 --> 00:15:54.940
if the Hamiltonian is
independent of time,
00:15:54.940 --> 00:16:03.010
then if we know the
solutions psi n en.
00:16:03.010 --> 00:16:05.950
If we know all of
these solutions,
00:16:05.950 --> 00:16:07.940
then there's nothing new.
00:16:07.940 --> 00:16:10.900
We just are
repackaging the stuff
00:16:10.900 --> 00:16:13.900
that we know from the time
independent Schrodinger
00:16:13.900 --> 00:16:15.430
equation.
00:16:15.430 --> 00:16:17.840
So the first thing I
want to do is to show you
00:16:17.840 --> 00:16:24.180
that if the Hamiltonian
is independent,
00:16:24.180 --> 00:16:30.090
we can always write a
solution to the time
00:16:30.090 --> 00:16:36.180
dependent Schrodinger
equation-- e to the minus i e n
00:16:36.180 --> 00:16:42.690
t over h bar psi n x.
00:16:45.330 --> 00:16:52.110
So this, for a time
independent Hamiltonian,
00:16:52.110 --> 00:16:55.661
is always the solution of the
time dependent Schrodinger
00:16:55.661 --> 00:16:56.160
equation.
00:17:02.010 --> 00:17:04.319
So we're just using
this stuff that we know,
00:17:04.319 --> 00:17:06.210
or at least we barely
know, because we just
00:17:06.210 --> 00:17:08.450
started playing the game.
00:17:08.450 --> 00:17:12.089
But we can manipulate to see
all sorts of useful stuff.
00:17:12.089 --> 00:17:12.589
OK.
00:17:12.589 --> 00:17:16.319
So let's show if this form
satisfies the Schrodinger
00:17:16.319 --> 00:17:17.359
equation.
00:17:17.359 --> 00:17:25.339
So we have ih bar partial
PSI with respect to t.
00:17:25.339 --> 00:17:28.339
So we get an ih bar.
00:17:28.339 --> 00:17:32.660
And then we take the
partial with respect to t.
00:17:32.660 --> 00:17:34.370
This is independent of time.
00:17:34.370 --> 00:17:36.290
This has time
[INAUDIBLE] And so we
00:17:36.290 --> 00:17:42.830
get a minus i e n over h bar.
00:17:42.830 --> 00:17:50.060
And then we get e to the minus
i e n t over h bar psi n of x.
00:17:56.090 --> 00:17:56.900
Well.
00:17:56.900 --> 00:17:59.040
So, let's put this together.
00:17:59.040 --> 00:18:01.520
We have an i times
i times minus 1.
00:18:01.520 --> 00:18:02.840
So that's plus 1.
00:18:02.840 --> 00:18:04.540
We have an h bar in
the numerator and h
00:18:04.540 --> 00:18:06.180
bar in the denominator.
00:18:06.180 --> 00:18:07.160
That's 1.
00:18:07.160 --> 00:18:14.690
And so what we end up getting
is e n e to the minus i e n
00:18:14.690 --> 00:18:17.090
t over h bar psi n.
00:18:20.054 --> 00:18:24.980
Well, this is psi.
00:18:28.320 --> 00:18:37.020
This function here is
the solution to the time
00:18:37.020 --> 00:18:38.850
dependent Schrodinger equation.
00:18:38.850 --> 00:18:40.080
How do we know that?
00:18:40.080 --> 00:18:44.700
Well, we have this
factor, h psi.
00:18:44.700 --> 00:18:50.200
Well, h doesn't operate
on either the i e n t.
00:18:50.200 --> 00:18:57.280
So what we have is that we
get E n e to the i omega t.
00:19:01.118 --> 00:19:01.618
Sorry.
00:19:01.618 --> 00:19:03.546
I'm jumping ahead.
00:19:03.546 --> 00:19:07.386
i e n t over h bar psi.
00:19:16.810 --> 00:19:18.190
All right.
00:19:18.190 --> 00:19:23.770
So, if we apply the
Hamiltonian to this function,
00:19:23.770 --> 00:19:26.500
we get just e n
times the function.
00:19:31.530 --> 00:19:35.240
And that's what we got when
we did i h bar times a partial
00:19:35.240 --> 00:19:37.040
with respect to t.
00:19:37.040 --> 00:19:41.900
So what that shows is
that this form always
00:19:41.900 --> 00:19:46.220
satisfies the time dependent
Schrodinger equation, provided
00:19:46.220 --> 00:19:49.630
that the Hamiltonian
is independent of time.
00:19:49.630 --> 00:19:53.400
Now, that's a large
range of problems, things
00:19:53.400 --> 00:19:55.630
that we need to understand.
00:19:55.630 --> 00:19:58.920
But it's not the whole potato,
because the Hamiltonian
00:19:58.920 --> 00:20:01.560
is often dependent on time.
00:20:01.560 --> 00:20:06.570
But it enables us to
build up insight, and then
00:20:06.570 --> 00:20:10.890
treat the time dependence
as a perturbation.
00:20:10.890 --> 00:20:13.860
And we're going to do
perturbation theory in the time
00:20:13.860 --> 00:20:15.994
independent world.
00:20:15.994 --> 00:20:17.910
And then we're going to
do perturbation theory
00:20:17.910 --> 00:20:21.470
a little bit in the
time dependent world.
00:20:21.470 --> 00:20:21.970
OK.
00:20:21.970 --> 00:20:26.440
So what we always do here is
we solve a familiar problem,
00:20:26.440 --> 00:20:29.050
and then we say, OK, well,
there's something more
00:20:29.050 --> 00:20:31.400
to this familiar problem.
00:20:31.400 --> 00:20:34.410
And so we treat that
as something extra.
00:20:34.410 --> 00:20:36.250
And we work out the
formalism for dealing
00:20:36.250 --> 00:20:38.210
with that extra thing.
00:20:38.210 --> 00:20:39.700
But before we do
the extra thing,
00:20:39.700 --> 00:20:42.460
we have to really
kill the problem that
00:20:42.460 --> 00:20:45.950
is within our grasp.
00:20:45.950 --> 00:20:46.450
OK.
00:20:49.470 --> 00:20:54.630
So now, our job
is to just explore
00:20:54.630 --> 00:20:55.920
what we've really got here.
00:20:58.840 --> 00:21:04.750
So the first problem is motion.
00:21:08.140 --> 00:21:16.975
So we have psi star
x and t psi x and t.
00:21:20.640 --> 00:21:22.350
So we have this
thing, which we're not
00:21:22.350 --> 00:21:23.640
going to integrate yet.
00:21:29.790 --> 00:21:32.220
Well, when is this
thing going to move?
00:21:35.000 --> 00:21:39.010
Well, the only way this
probability density
00:21:39.010 --> 00:21:42.190
is going to evolve
in time is going
00:21:42.190 --> 00:21:46.650
to be if we have a
wave function, psi,
00:21:46.650 --> 00:21:54.400
which is of x and t is equal to
c1 psi 1, e to the minus i e 1,
00:21:54.400 --> 00:22:00.720
e over h bar, plus c2,
psi 2, e to the minus i e
00:22:00.720 --> 00:22:10.290
2, t over h bar, where
e1 is not equal to e2.
00:22:10.290 --> 00:22:12.780
So this is the first,
most elementary step.
00:22:12.780 --> 00:22:18.040
And remember, we have this
notation e to the i something.
00:22:18.040 --> 00:22:20.140
And e to the minus i something.
00:22:20.140 --> 00:22:24.200
And when we take a plus 1 and
a minus and put them together,
00:22:24.200 --> 00:22:26.040
we get 1.
00:22:26.040 --> 00:22:28.820
So this notation, this
exponential notation,
00:22:28.820 --> 00:22:30.450
is really valuable.
00:22:30.450 --> 00:22:30.950
OK.
00:22:30.950 --> 00:22:34.910
So now let's just look at
this quantity, psi star psi.
00:22:39.050 --> 00:22:39.550
OK.
00:22:39.550 --> 00:22:40.870
Psi star psi.
00:22:44.980 --> 00:22:54.040
Well, it's going to be c1
squared, psi 1 squared--
00:22:54.040 --> 00:23:01.430
square modulus-- and we
get c2 square modulus,
00:23:01.430 --> 00:23:05.360
psi 2 square modulus.
00:23:05.360 --> 00:23:06.120
Are we done?
00:23:06.120 --> 00:23:07.280
No.
00:23:07.280 --> 00:23:14.360
So, what we did is, I put
in c1, psi 1 plus c2, psi 2.
00:23:14.360 --> 00:23:17.150
And I looked at the
easy terms, the terms
00:23:17.150 --> 00:23:20.980
where the exponential
factor goes away.
00:23:20.980 --> 00:23:22.690
And then there's
two cross terms.
00:23:22.690 --> 00:23:25.730
Those two cross
terms are c1 star--
00:23:25.730 --> 00:23:49.080
whoops-- c2 e to the minus
i e 2 minus e1 t over h bar
00:23:49.080 --> 00:23:54.370
psi 1 star, psi 2.
00:23:54.370 --> 00:24:04.780
And we have c1, c2 star e
to the plus i e2 minus e1
00:24:04.780 --> 00:24:10.930
t over h bar, psi 1 star.
00:24:10.930 --> 00:24:14.370
psi 1, psi 2 star.
00:24:14.370 --> 00:24:17.371
Now, this is just the
automatic writing.
00:24:17.371 --> 00:24:17.870
OK.
00:24:17.870 --> 00:24:23.760
So we have two terms that
are time independent.
00:24:23.760 --> 00:24:26.570
So, this is no big surprise.
00:24:26.570 --> 00:24:28.520
But then we have
this stuff here.
00:24:28.520 --> 00:24:38.970
And if we say e2 minus e1
over h bar is omega 2, 1,
00:24:38.970 --> 00:24:42.830
everything becomes
very transparent,
00:24:42.830 --> 00:24:44.630
because now we
have something that
00:24:44.630 --> 00:24:52.070
looks like it looks like it's
trying to be a cosine omega t.
00:24:52.070 --> 00:24:55.050
We have to be a
little bit careful.
00:24:55.050 --> 00:24:57.440
These are the two
time dependent terms.
00:24:57.440 --> 00:25:01.180
And they are the complex
conjugate of each other.
00:25:01.180 --> 00:25:12.690
And so we know that if we
have two complex numbers,
00:25:12.690 --> 00:25:21.840
c plus c star, we're going to
get twice the real part of c.
00:25:27.140 --> 00:25:31.460
So this enables us to take these
two terms and combine them.
00:25:31.460 --> 00:25:35.900
And we know it had
better be real,
00:25:35.900 --> 00:25:38.360
because we're talking
about a probability here.
00:25:38.360 --> 00:25:42.010
This probability
has got to be real.
00:25:42.010 --> 00:25:44.410
And it's got to be positive.
00:25:44.410 --> 00:25:48.825
It's got to be real and
positive everywhere and forever.
00:25:51.580 --> 00:25:55.410
Because there's no such thing
as a negative probability.
00:25:55.410 --> 00:25:57.650
There's no such thing as
a negative probability
00:25:57.650 --> 00:26:00.170
in a little region of
space, and we say, well,
00:26:00.170 --> 00:26:04.170
we integrate over all space,
and so that goes away.
00:26:04.170 --> 00:26:12.330
This psi star psi is going to
be real everywhere for all time.
00:26:12.330 --> 00:26:14.060
And so that's a good
thing, because we
00:26:14.060 --> 00:26:21.450
have a sum of a plus
its complex conjugate.
00:26:21.450 --> 00:26:23.270
And so this is real.
00:26:23.270 --> 00:26:32.900
And we can simplify everything,
and we can write it simply.
00:26:32.900 --> 00:26:35.400
But suppose we choose
a particular case--
00:26:35.400 --> 00:26:42.980
c1 star is equal to
c1, which is equal to 1
00:26:42.980 --> 00:26:45.560
over square root of 2.
00:26:45.560 --> 00:26:52.910
And we can say psi 1
and psi 2 are real.
00:26:52.910 --> 00:26:57.330
When we do that, then this
complicated-looking thing, psi
00:26:57.330 --> 00:27:03.760
star psi, becomes
1/2 psi 1 squared,
00:27:03.760 --> 00:27:15.430
plus 1/2 psi 2 squared, plus
cosine omega 1 2 t psi 1 psi 2.
00:27:20.950 --> 00:27:22.760
OK.
00:27:22.760 --> 00:27:25.490
Well, these two
guys aren't moving.
00:27:25.490 --> 00:27:27.540
And they're real.
00:27:27.540 --> 00:27:28.590
And positive.
00:27:28.590 --> 00:27:29.240
Yeah.
00:27:29.240 --> 00:27:32.160
AUDIENCE: Wait, is c2 also
1 over square root of 2?
00:27:35.932 --> 00:27:37.640
PROFESSOR: I had 1
over square root of 2,
00:27:37.640 --> 00:27:38.490
one over square root of 2.
00:27:38.490 --> 00:27:39.060
That's 1/2.
00:27:41.880 --> 00:27:46.560
But then when you
add the two terms,
00:27:46.560 --> 00:27:48.600
you get twice the real part.
00:27:48.600 --> 00:27:50.640
So we get a 1/2 times 2.
00:27:50.640 --> 00:27:54.000
And so this is not a mistake.
00:27:54.000 --> 00:27:56.730
It comes out this way, OK?
00:27:56.730 --> 00:27:58.800
AUDIENCE: I think you meant c2.
00:27:58.800 --> 00:28:05.340
You're only finding c1 star and
c1 to be 2 over square root--
00:28:05.340 --> 00:28:07.580
1 over square root of 2.
00:28:07.580 --> 00:28:09.467
You didn't say
anything about c2.
00:28:09.467 --> 00:28:10.050
PROFESSOR: Oh.
00:28:10.050 --> 00:28:10.860
OK.
00:28:10.860 --> 00:28:12.010
I want c1 and c2.
00:28:16.710 --> 00:28:18.640
That's what I wanted.
00:28:18.640 --> 00:28:19.140
OK.
00:28:24.450 --> 00:28:25.560
Thank you.
00:28:25.560 --> 00:28:27.360
That shows you're listening.
00:28:27.360 --> 00:28:30.570
And it shows that
I'm sufficiently here
00:28:30.570 --> 00:28:32.460
to understand your
questions, which
00:28:32.460 --> 00:28:34.350
is another wonderful thing.
00:28:34.350 --> 00:28:36.960
So this is now,
we have something
00:28:36.960 --> 00:28:40.460
that's positive everywhere.
00:28:40.460 --> 00:28:41.860
It's time independent.
00:28:41.860 --> 00:28:45.800
And we have something
that's oscillating.
00:28:45.800 --> 00:28:49.080
And so this term
can be negative.
00:28:49.080 --> 00:28:51.150
But you can show--
00:28:51.150 --> 00:28:52.750
I don't choose to do that.
00:28:52.750 --> 00:28:55.500
You can show that this
term is never larger
00:28:55.500 --> 00:28:57.480
than psi 1 squared
plus psi 2 squared.
00:29:00.770 --> 00:29:05.720
And so even though this term
can be negative at some points,
00:29:05.720 --> 00:29:10.970
it never is a negative enough
to make the evolving probability
00:29:10.970 --> 00:29:12.211
go negative.
00:29:15.880 --> 00:29:18.780
Now, you may want to play with
that just to convince yourself.
00:29:18.780 --> 00:29:19.780
And it's an easy proof.
00:29:19.780 --> 00:29:22.600
And I'm just not going to do it.
00:29:22.600 --> 00:29:25.080
OK.
00:29:25.080 --> 00:29:29.400
So we have something
that says, just
00:29:29.400 --> 00:29:35.270
like for the wave equation,
what does it take to get motion?
00:29:35.270 --> 00:29:37.040
And to get motion
you had to have
00:29:37.040 --> 00:29:41.810
a superposition of two
waves of different energy
00:29:41.810 --> 00:29:43.070
or different wave vector.
00:29:45.630 --> 00:29:51.180
And so here we have, if
e1 is not equal to e2,
00:29:51.180 --> 00:29:53.695
we have a non-zero omega 1, 2.
00:29:53.695 --> 00:29:56.340
It doesn't matter whether it's
omega 1, 2, or omega 2, 1,
00:29:56.340 --> 00:29:59.100
because it's cosine.
00:29:59.100 --> 00:30:02.700
And so that's motion.
00:30:02.700 --> 00:30:07.890
So we get a standing
wave sort of situation.
00:30:07.890 --> 00:30:10.950
And then we get this motion.
00:30:10.950 --> 00:30:14.430
OK, now, the fun begins.
00:30:14.430 --> 00:30:23.614
Suppose we want to calculate the
expectation value of x and p.
00:30:30.290 --> 00:30:30.790
OK.
00:30:30.790 --> 00:30:35.160
And let's just take
it as a 1D problem.
00:30:35.160 --> 00:30:37.300
And so x of t.
00:30:40.090 --> 00:30:41.640
No, there was no question.
00:30:41.640 --> 00:30:42.870
All right.
00:30:42.870 --> 00:30:43.410
We have--
00:30:59.122 --> 00:30:59.940
OK.
00:30:59.940 --> 00:31:02.100
And again, we take
this thing apart.
00:31:02.100 --> 00:31:05.460
And we say, all right, suppose
we have the same superposition
00:31:05.460 --> 00:31:09.160
of psi is equal to c1.
00:31:09.160 --> 00:31:10.330
psi 1.
00:31:21.640 --> 00:31:23.050
OK.
00:31:23.050 --> 00:31:30.650
Actually, these should
be the time independent.
00:31:30.650 --> 00:31:35.120
So what we get is, when we
do this integral, we get c1
00:31:35.120 --> 00:31:47.940
squared integral psi
star x, psi 1 dx.
00:31:47.940 --> 00:31:59.890
And we get c2 squared integral
psi 2 star x, psi 2 dx.
00:31:59.890 --> 00:32:03.260
And then we get cross
terms. c1 star, c2.
00:32:05.980 --> 00:32:18.970
e to the minus i omega 2, 1 t
times the integral psi 1 star
00:32:18.970 --> 00:32:23.170
x psi 2 dx.
00:32:23.170 --> 00:32:31.330
And we have c1, c2 star,
e to the plus i omega 2 1
00:32:31.330 --> 00:32:38.090
t integral psi 2
star x psi 1 dx.
00:32:40.741 --> 00:32:43.170
A lot of stuff here.
00:32:43.170 --> 00:32:47.810
Now, we're talking about
the harmonic oscillator.
00:32:47.810 --> 00:32:49.385
This integral is 0.
00:32:52.360 --> 00:32:55.340
Because x is a plus a dagger.
00:32:55.340 --> 00:32:58.750
The selection rule is delta
v of plus and minus 1.
00:32:58.750 --> 00:33:01.190
This one is 0.
00:33:01.190 --> 00:33:05.790
For the particle in
a box, one can also
00:33:05.790 --> 00:33:08.610
ask, what about this integral?
00:33:08.610 --> 00:33:11.880
And for the particle in a
box, it's a little bit more
00:33:11.880 --> 00:33:12.510
complicated.
00:33:12.510 --> 00:33:15.510
Because we've chosen a
mathematically simple way
00:33:15.510 --> 00:33:19.110
to solve the particle in a
box, with the box having a zero
00:33:19.110 --> 00:33:21.500
left edge.
00:33:21.500 --> 00:33:25.310
If we make the box symmetric,
then we can make judgments
00:33:25.310 --> 00:33:26.930
and say, oh, yeah.
00:33:26.930 --> 00:33:30.930
This integral is also 0
for the particle in a box.
00:33:36.010 --> 00:33:38.490
And that's a little bit more
complicated, the argument,
00:33:38.490 --> 00:33:40.200
than what I just did.
00:33:40.200 --> 00:33:45.070
So these two terms are 0.
00:33:45.070 --> 00:33:51.100
And now we have motion of the
expectation value, which is
00:33:51.100 --> 00:33:52.720
described by these two terms.
00:33:52.720 --> 00:33:56.180
And again, they're the complex
conjugate of each other.
00:33:56.180 --> 00:34:13.949
And so what we have is
x of t is equal to twice
00:34:13.949 --> 00:34:22.900
the real part of c1 star, c2
e to the minus i omega 2, 1 t.
00:34:28.500 --> 00:34:32.131
Times x 1, 2.
00:34:32.131 --> 00:34:32.630
OK.
00:34:32.630 --> 00:34:34.310
This x 1, 2 is an integral.
00:34:37.210 --> 00:34:42.969
So x 1, 2 if these are the
vibrational quantum numbers,
00:34:42.969 --> 00:34:46.310
well, then this is non-zero.
00:34:46.310 --> 00:34:52.370
This is just the square root
of 2, or square root of 1.
00:35:00.051 --> 00:35:00.550
OK.
00:35:00.550 --> 00:35:05.005
So we have motion
described by this.
00:35:07.990 --> 00:35:14.440
And so the only
time we get motion
00:35:14.440 --> 00:35:20.650
is if v1 is equal to
v2 plus or minus 1,
00:35:20.650 --> 00:35:22.065
for the harmonic oscillator.
00:35:25.320 --> 00:35:29.720
For the particle in a box,
there's different rules.
00:35:29.720 --> 00:35:32.010
And often, for these
simple problems,
00:35:32.010 --> 00:35:35.090
you want to go through in your
head all of the simple cases.
00:35:48.810 --> 00:35:52.630
Now, we already can see that--
00:35:59.950 --> 00:36:02.270
I don't want to talk about
the particle in a box.
00:36:02.270 --> 00:36:05.320
So now let's just
take another step.
00:36:05.320 --> 00:36:17.780
And we're going to have
Ehrenfest's theorem, which
00:36:17.780 --> 00:36:26.800
you can prove, says that m times
the derivative of vector p--
00:36:31.790 --> 00:36:35.210
so this is a time dependent
expectation value--
00:36:35.210 --> 00:36:37.681
is equal to--
00:36:37.681 --> 00:36:38.180
I'm sorry.
00:36:38.180 --> 00:36:39.200
This is not vector p.
00:36:39.200 --> 00:36:41.420
This is vector r--
00:36:41.420 --> 00:36:42.080
is equal to--
00:36:48.550 --> 00:36:49.720
So this is the vector p.
00:36:49.720 --> 00:36:53.500
So this is the derivative
of the coordinate,
00:36:53.500 --> 00:36:54.380
with respect to time.
00:36:54.380 --> 00:36:55.600
That's velocity.
00:36:55.600 --> 00:36:59.080
Velocity times mass is momentum.
00:36:59.080 --> 00:37:02.380
And so we have a relationship
between the expectation
00:37:02.380 --> 00:37:06.070
value of the position
and the expectation
00:37:06.070 --> 00:37:08.480
value of the momentum.
00:37:08.480 --> 00:37:10.520
And that's for
Newton's first law.
00:37:10.520 --> 00:37:14.170
And then there is another
Newton's equation translated
00:37:14.170 --> 00:37:16.580
into quantum mechanics.
00:37:16.580 --> 00:37:25.270
So we have the expectation
value of the momentum, dt,
00:37:25.270 --> 00:37:30.520
is equal to minus
the expectation
00:37:30.520 --> 00:37:33.250
value of the potential.
00:37:37.310 --> 00:37:43.540
While this is acceleration
times mass, and this is force--
00:37:43.540 --> 00:37:46.020
minus the gradient of a
potential is the force--
00:37:46.020 --> 00:37:47.860
these are Newton's
two equations.
00:37:47.860 --> 00:37:58.280
And what they're saying is,
if we know these things,
00:37:58.280 --> 00:38:01.520
we know something about the
center of the wave packet,
00:38:01.520 --> 00:38:03.570
and how the center of
the wave packet moves.
00:38:03.570 --> 00:38:06.540
Now, the wave packet might
be localized at one time.
00:38:06.540 --> 00:38:09.210
It might be mushed
out at another time.
00:38:09.210 --> 00:38:11.710
But you can always calculate
the center of the wave packet.
00:38:11.710 --> 00:38:13.501
It's just that there
are only certain times
00:38:13.501 --> 00:38:16.650
that it looks like a particle.
00:38:16.650 --> 00:38:19.830
But this thing,
these quantities,
00:38:19.830 --> 00:38:24.380
which you define by an integral,
they evolve classically.
00:38:24.380 --> 00:38:26.000
So I told you at
the beginning, you
00:38:26.000 --> 00:38:28.790
had to give up
classical mechanics.
00:38:28.790 --> 00:38:31.130
It's all coming back.
00:38:31.130 --> 00:38:34.320
But it's coming back in a
quantum mechanical framework,
00:38:34.320 --> 00:38:38.360
because we're talking now
about wave functions, which
00:38:38.360 --> 00:38:43.350
have amplitudes and phases,
and can do terrible things.
00:38:43.350 --> 00:38:49.110
But at certain limits, they're
going to act like particles.
00:38:49.110 --> 00:38:57.650
But if you were to
ask a question, well--
00:38:57.650 --> 00:39:00.230
suppose we do this experiment.
00:39:00.230 --> 00:39:06.770
And so here we have an
electronic ground state
00:39:06.770 --> 00:39:07.970
potential surface.
00:39:07.970 --> 00:39:10.610
Now, I'm jumping way ahead.
00:39:10.610 --> 00:39:16.730
But this is the wave function
for that vibrational level.
00:39:16.730 --> 00:39:21.780
And you excite the molecule with
a time dependent Hamiltonian,
00:39:21.780 --> 00:39:25.200
a time dependent
radiation field light.
00:39:25.200 --> 00:39:29.880
And you vertically
transport this wave function
00:39:29.880 --> 00:39:32.520
to the upper state,
until you get something
00:39:32.520 --> 00:39:35.690
which is not an eigenstate.
00:39:35.690 --> 00:39:37.470
It's a pluck.
00:39:37.470 --> 00:39:41.560
This pluck is a
superposition of eigenstates.
00:39:41.560 --> 00:39:43.360
And we can ask, how
does this evolve?
00:39:43.360 --> 00:39:45.130
And what it's going
to do is, it's
00:39:45.130 --> 00:39:47.910
going to start out localized.
00:39:47.910 --> 00:39:51.280
And at some point, it'll
do terrible things.
00:39:51.280 --> 00:39:53.140
And then at some
other time it'll
00:39:53.140 --> 00:39:55.380
be localized again at
the other turning point.
00:39:55.380 --> 00:39:57.780
And it will come back and forth.
00:39:57.780 --> 00:40:00.810
And now, if it's not
a harmonic oscillator,
00:40:00.810 --> 00:40:03.780
it won't quite relocalize.
00:40:03.780 --> 00:40:05.810
It'll mush out a little
bit, and it'll come back
00:40:05.810 --> 00:40:07.660
and it'll mush out more.
00:40:07.660 --> 00:40:11.900
And so again, we can use the
evolution of the wave packet
00:40:11.900 --> 00:40:15.530
to sample the shape
of the potential.
00:40:15.530 --> 00:40:17.940
We can measure
the anharmonicity.
00:40:17.940 --> 00:40:20.780
And so let's now
talk about that.
00:40:24.970 --> 00:40:27.400
What are other
quantities that we
00:40:27.400 --> 00:40:34.039
can calculate from the
wave function of the time
00:40:34.039 --> 00:40:35.330
dependent Schrodinger equation?
00:40:35.330 --> 00:40:38.960
So let's talk about the
survival probability.
00:40:38.960 --> 00:40:41.500
So this is a capital P.
Survival probability.
00:40:41.500 --> 00:40:45.790
It's going to be
the wave function.
00:40:45.790 --> 00:40:50.440
The time dependent wave function
is created at t equals 0.
00:40:50.440 --> 00:40:51.970
It has some shape.
00:40:51.970 --> 00:40:55.990
Now, we always like to have
a simple shape at t equals 0.
00:40:55.990 --> 00:41:00.670
And we'd like to know how
fast that thing moves away
00:41:00.670 --> 00:41:04.210
from its birthplace.
00:41:04.210 --> 00:41:10.900
So we have this
survival probability.
00:41:10.900 --> 00:41:12.280
It's a probability.
00:41:12.280 --> 00:41:25.640
So we're going to have
integral psi star xt psi xt dx.
00:41:25.640 --> 00:41:28.730
And this integral is
going to look like that.
00:41:32.490 --> 00:41:40.610
Now, whoops.
00:41:40.610 --> 00:41:42.800
I knew I was going to screw up.
00:41:42.800 --> 00:41:46.370
If I put a t here,
then it would just
00:41:46.370 --> 00:41:48.320
be the normalization interval.
00:41:48.320 --> 00:41:51.250
And we already
know what that is.
00:41:51.250 --> 00:41:52.690
But this is the birthplace.
00:41:52.690 --> 00:41:55.460
This is what was
created at t equals 0.
00:41:55.460 --> 00:41:58.300
And this is time evolving thing.
00:41:58.300 --> 00:42:01.150
And so we can calculate
how that behaves.
00:42:01.150 --> 00:42:04.790
And I'm just going to
write the solution.
00:42:09.770 --> 00:42:19.150
So the result is, if we have the
same kind of two state c1 psi
00:42:19.150 --> 00:42:27.530
1, e to the minus
i e1 t for h bar
00:42:27.530 --> 00:42:35.140
plus c2 psi 2 e to the
minus i e2 e over h bar,
00:42:35.140 --> 00:42:41.330
well then, what we get
is c1 to the 4th power
00:42:41.330 --> 00:42:55.380
plus c2 to the 4th power plus
c1 squared, c2 squared, times
00:42:55.380 --> 00:43:06.210
e to the i omega 2, 1 t plus
e to the minus i omega 2, 1 t.
00:43:10.390 --> 00:43:14.090
Now we're integrating.
00:43:14.090 --> 00:43:18.520
When we integrate, the
wave functions go away.
00:43:18.520 --> 00:43:23.010
The wave functions
become either 1 or 0.
00:43:23.010 --> 00:43:24.750
And so we're integrating.
00:43:24.750 --> 00:43:27.030
We're making the wave
functions go away.
00:43:27.030 --> 00:43:32.490
And we have some amplitudes
of the wave functions.
00:43:32.490 --> 00:43:36.210
And we have a time independent
part and a time dependent part.
00:43:36.210 --> 00:43:41.472
And this is 2
cosine omega 2, 1 t.
00:43:44.070 --> 00:43:49.530
So what we see is this survival
probability, the wave function,
00:43:49.530 --> 00:43:51.870
starts out at some place.
00:43:51.870 --> 00:43:53.560
And it goes away.
00:43:53.560 --> 00:43:54.700
And it comes back.
00:43:54.700 --> 00:43:55.800
And it goes away.
00:43:55.800 --> 00:43:59.060
And it comes back.
00:43:59.060 --> 00:43:59.930
And it does that.
00:43:59.930 --> 00:44:04.670
It completely rephases,
because there's only two terms.
00:44:04.670 --> 00:44:07.070
But if there were three
terms that are not
00:44:07.070 --> 00:44:09.400
satisfying a
certain requirement,
00:44:09.400 --> 00:44:11.870
then when it comes back,
it can't completely
00:44:11.870 --> 00:44:12.960
reconstruct itself.
00:44:17.080 --> 00:44:20.560
And this is the basis
for doing experiments.
00:44:20.560 --> 00:44:26.770
One can observe the
periodic rephasings
00:44:26.770 --> 00:44:29.500
of some initial pluck.
00:44:29.500 --> 00:44:31.840
And you can look at
the decay of them,
00:44:31.840 --> 00:44:33.850
and that gives you
something about the shape
00:44:33.850 --> 00:44:36.650
of the potential,
the anharmonicity.
00:44:36.650 --> 00:44:40.490
And you can do all sorts
of fantastic things.
00:44:40.490 --> 00:44:43.760
You can create a
wave packet, and you
00:44:43.760 --> 00:44:47.250
can wait until it reaches
the other turning point.
00:44:47.250 --> 00:44:48.920
And at the other
turning point, it's
00:44:48.920 --> 00:44:50.495
possible that you
could excite it
00:44:50.495 --> 00:44:53.900
to a different
higher excited state.
00:44:53.900 --> 00:44:55.770
Ahmed Zewail got the
Nobel Prize for that.
00:44:59.550 --> 00:45:06.690
So we're right at the
frontier of what you can do,
00:45:06.690 --> 00:45:08.430
and what you can
understand using
00:45:08.430 --> 00:45:11.810
this very simple problem of a
time independent Hamiltonian
00:45:11.810 --> 00:45:17.400
and a 2 or a 3
term superposition.
00:45:17.400 --> 00:45:17.900
OK.
00:45:27.850 --> 00:45:28.510
Recurrence.
00:45:35.910 --> 00:45:38.750
This is a special property.
00:45:38.750 --> 00:45:50.960
When all of the wave functions,
or all of the difference--
00:45:50.960 --> 00:45:54.410
all of the energy levels, or
differences in energy levels
00:45:54.410 --> 00:46:06.140
are integer multiples of
a common factor, well then
00:46:06.140 --> 00:46:09.830
any coherent superposition
state you make
00:46:09.830 --> 00:46:13.940
will rephase at a special time.
00:46:13.940 --> 00:46:19.440
And so what we can do is say,
OK, for a particle in a box,
00:46:19.440 --> 00:46:26.360
the energy levels are
e1 times n squared.
00:46:26.360 --> 00:46:29.450
For our harmonic oscillator,
the energy level difference
00:46:29.450 --> 00:46:39.360
is ev plus n minus
ev our n h bar omega.
00:46:39.360 --> 00:46:41.970
For a rigid rotor, which
we haven't seen yet,
00:46:41.970 --> 00:46:49.520
the energy levels are a function
of this quantum number j hcb.
00:46:49.520 --> 00:46:54.410
This is the rotational
constant-- times j plus 1.
00:46:56.940 --> 00:47:01.970
So all of the energy
levels are related.
00:47:01.970 --> 00:47:03.590
I'm jumping ahead.
00:47:03.590 --> 00:47:04.550
Sorry.
00:47:04.550 --> 00:47:12.380
So the energy levels
are j times j plus 1.
00:47:12.380 --> 00:47:18.980
And the differences,
ej plus 1 minus ej,
00:47:18.980 --> 00:47:27.620
are given by 2 hcb
times j plus 1.
00:47:27.620 --> 00:47:31.880
So for these three problems,
we have this perfect situation
00:47:31.880 --> 00:47:34.820
where one can have
these oscillating
00:47:34.820 --> 00:47:37.330
terms all have a common factor.
00:47:37.330 --> 00:47:42.130
And at certain times, the
oscillating terms are all 1.
00:47:42.130 --> 00:47:44.510
Or some are 1 and
some are minus 1.
00:47:44.510 --> 00:47:47.660
And we get a really
wonderful simplification.
00:47:47.660 --> 00:47:52.190
So at a time which
we call, or I call,
00:47:52.190 --> 00:47:59.860
the grand recurrence
time, where it's
00:47:59.860 --> 00:48:09.010
equal to h over e1 for this
case of the particle in a box,
00:48:09.010 --> 00:48:24.060
or h over h bar omega,
or h over 2 hcb,
00:48:24.060 --> 00:48:27.540
we get all of the phase
factors becoming 1.
00:48:33.040 --> 00:48:38.740
And sometimes, something special
happens at the grand recurrence
00:48:38.740 --> 00:48:41.170
time divided by 2.
00:48:41.170 --> 00:48:44.410
And that's a little
bit like this.
00:48:44.410 --> 00:48:47.900
We have a wave function here.
00:48:47.900 --> 00:48:50.010
And at half the recurrence
time, it's over here.
00:48:53.720 --> 00:48:56.050
And so you want to work
your way through the algebra
00:48:56.050 --> 00:49:00.590
to convince yourself that
that is in fact true.
00:49:00.590 --> 00:49:04.280
And in between, if
you have enough terms
00:49:04.280 --> 00:49:08.380
in your superposition,
which is usually the case,
00:49:08.380 --> 00:49:09.610
in between you get this.
00:49:09.610 --> 00:49:13.080
You get garbage looking.
00:49:13.080 --> 00:49:15.330
It's still moving.
00:49:15.330 --> 00:49:18.390
It's still satisfying
the Ehrenfest theorem,
00:49:18.390 --> 00:49:20.010
but it doesn't look
like a particle.
00:49:20.010 --> 00:49:23.470
It just looks like garbage.
00:49:23.470 --> 00:49:26.140
But you get these
wonderful things happening.
00:49:26.140 --> 00:49:28.480
And now, if they're
not perfectly
00:49:28.480 --> 00:49:32.350
satisfying this integer
rule, then each time you
00:49:32.350 --> 00:49:36.130
get to a turning
point, the amplitude
00:49:36.130 --> 00:49:40.440
has decreased and
decreased and decreased.
00:49:40.440 --> 00:49:46.830
And at infinite
time, it might recur.
00:49:46.830 --> 00:49:48.580
But nobody waits around
for infinite time,
00:49:48.580 --> 00:49:51.970
because other things happen and
destroy the coherence that you
00:49:51.970 --> 00:49:55.005
built. So this is a glimpse.
00:49:59.310 --> 00:50:01.020
Time independent
quantum mechanics
00:50:01.020 --> 00:50:03.980
is complicated enough.
00:50:03.980 --> 00:50:07.610
Well, we can embed
what we understand
00:50:07.610 --> 00:50:10.730
in the time dependent mechanics.
00:50:10.730 --> 00:50:14.560
And there are a lot of beautiful
things that we can anticipate.
00:50:14.560 --> 00:50:19.210
Now, we can use these
beautiful things
00:50:19.210 --> 00:50:22.420
to do experiments
which measure stuff
00:50:22.420 --> 00:50:27.020
that is related to, how does
energy move in the molecule?
00:50:27.020 --> 00:50:28.550
What is going on
in the molecule?
00:50:28.550 --> 00:50:31.964
What are the mechanisms
for stuff happening?
00:50:31.964 --> 00:50:33.880
And in magnetic resonance,
there are all sorts
00:50:33.880 --> 00:50:37.150
of pulse sequences that
interrogate distances
00:50:37.150 --> 00:50:39.830
and correlated motions.
00:50:39.830 --> 00:50:43.570
And it is really a laboratory
for time dependent quantum
00:50:43.570 --> 00:50:48.130
mechanics with a time
dependent Hamiltonian.
00:50:48.130 --> 00:50:54.700
But a lot of stuff that we do
with ordinary laser experiments
00:50:54.700 --> 00:51:01.880
are usually understandable
in a time independent way.
00:51:01.880 --> 00:51:07.590
Now, we want to get rid of
things that are complicated.
00:51:07.590 --> 00:51:11.220
And so one of the
things that we do
00:51:11.220 --> 00:51:14.970
is, we map a problem
onto something
00:51:14.970 --> 00:51:17.460
which is time independent.
00:51:17.460 --> 00:51:20.970
And so one of the
tricks that you will see
00:51:20.970 --> 00:51:25.890
is that if we go into what's
called a rotating coordinate
00:51:25.890 --> 00:51:28.890
system, this rotating
coordinate system
00:51:28.890 --> 00:51:36.330
is rotating at an energy level
different divided by h bar.
00:51:36.330 --> 00:51:39.984
And that converts the problem
into a time independent problem
00:51:39.984 --> 00:51:41.400
in the rotating
coordinate system.
00:51:41.400 --> 00:51:43.660
So that you're in a
rotating coordinate system
00:51:43.660 --> 00:51:47.610
and, basically, you use the
ordinary perturbation theory.
00:51:47.610 --> 00:51:50.670
And then you go back to the
non-rotating coordinate system.
00:51:50.670 --> 00:51:53.880
So there's all sorts of tricks
where we build on the stuff
00:51:53.880 --> 00:51:56.480
that we understood.
00:51:56.480 --> 00:52:00.560
And we can have a picture
which is intuitive,
00:52:00.560 --> 00:52:02.870
because we want to strip
away a lot of the mathematics
00:52:02.870 --> 00:52:05.540
and see the universal stuff.
00:52:05.540 --> 00:52:08.660
And so I'm going to try
to present as much of that
00:52:08.660 --> 00:52:10.950
as I can during this course.
00:52:10.950 --> 00:52:13.882
But a lot of the time
independent stuff
00:52:13.882 --> 00:52:16.340
is heavy lifting, and I'm going
to have to do a lot of that
00:52:16.340 --> 00:52:17.250
too.
00:52:17.250 --> 00:52:18.080
OK.
00:52:18.080 --> 00:52:20.600
Have a nice long weekend.
00:52:20.600 --> 00:52:22.250
I'll see you on Monday.