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ROBERT FIELD: That's the outline
of what we're going to cover.
00:00:24.740 --> 00:00:26.880
But before we get
started on that,
00:00:26.880 --> 00:00:29.910
I want to talk about
a couple of things.
00:00:29.910 --> 00:00:33.750
First of all, last time, we
talked about the two slit
00:00:33.750 --> 00:00:36.080
experiment.
00:00:36.080 --> 00:00:37.870
And it's mostly classical.
00:00:37.870 --> 00:00:40.220
There is only a little
bit of quantum in it
00:00:40.220 --> 00:00:45.840
where we talk about
momentum as being
00:00:45.840 --> 00:00:48.161
determined by h over lambda.
00:00:48.161 --> 00:00:48.660
OK.
00:00:48.660 --> 00:00:53.400
But what were the
two surprising things
00:00:53.400 --> 00:00:54.930
about the two slit experiment?
00:00:54.930 --> 00:00:57.860
There are two of
them, two surprises.
00:00:57.860 --> 00:00:58.360
Yes?
00:00:58.360 --> 00:01:01.600
AUDIENCE: [INAUDIBLE]
That a single particle
00:01:01.600 --> 00:01:02.920
can interfere with itself.
00:01:02.920 --> 00:01:03.670
ROBERT FIELD: Yes.
00:01:03.670 --> 00:01:07.360
That's the most
surprising thing.
00:01:07.360 --> 00:01:13.310
And when you go to
really low intensity--
00:01:13.310 --> 00:01:17.200
so there's only one
photon, which is quantum,
00:01:17.200 --> 00:01:20.470
in the apparatus,
somehow, it knows
00:01:20.470 --> 00:01:23.220
enough to interfere with itself.
00:01:23.220 --> 00:01:29.110
And this is the most
mysterious aspect.
00:01:29.110 --> 00:01:32.550
But then there's
one other aspect,
00:01:32.550 --> 00:01:37.275
which is how it communicates
that interference with itself.
00:01:39.799 --> 00:01:40.340
What is that?
00:01:46.510 --> 00:01:47.010
You're hot.
00:01:47.010 --> 00:01:48.375
You want to do another one?
00:01:48.375 --> 00:01:51.080
AUDIENCE: What do you mean
by how it communicates?
00:01:51.080 --> 00:01:56.600
ROBERT FIELD: Well, here
we have the screen on which
00:01:56.600 --> 00:02:00.070
the information is deposited.
00:02:00.070 --> 00:02:07.580
And you have possibly some sort
of probability distribution,
00:02:07.580 --> 00:02:09.820
which is a kind of
a continuous thing.
00:02:09.820 --> 00:02:10.960
But you don't observe that.
00:02:14.780 --> 00:02:16.740
What do you observe?
00:02:16.740 --> 00:02:21.320
So you could say the
state of a particle
00:02:21.320 --> 00:02:26.430
has amplitude everywhere
on this screen.
00:02:26.430 --> 00:02:32.010
But what do you see
in the experiment?
00:02:32.010 --> 00:02:33.410
Yes?
00:02:33.410 --> 00:02:36.290
AUDIENCE: So you just see
one [INAUDIBLE] point.
00:02:36.290 --> 00:02:38.310
[INAUDIBLE] thousands
and thousands.
00:02:38.310 --> 00:02:40.645
And eventually, you see
it mimic that probability
00:02:40.645 --> 00:02:41.270
[? sequence. ?]
00:02:41.270 --> 00:02:43.530
ROBERT FIELD: That's
exactly right.
00:02:43.530 --> 00:02:47.660
So this state of the
system which is distributed
00:02:47.660 --> 00:02:51.490
collapses to a single point.
00:02:51.490 --> 00:02:55.310
We say that what
we are observing--
00:02:55.310 --> 00:02:56.500
and this is mysterious.
00:02:56.500 --> 00:02:59.800
And it should bother you now.
00:02:59.800 --> 00:03:05.950
We're seeing an eigenvalue
of the measurement operator.
00:03:05.950 --> 00:03:08.080
So we have this sort of thing.
00:03:08.080 --> 00:03:10.060
It goes into the
measurement operator.
00:03:10.060 --> 00:03:12.010
And the measurement
operator says,
00:03:12.010 --> 00:03:16.930
this is one of the answers
I'm permitted to give you.
00:03:16.930 --> 00:03:21.970
And another aspect that's
really disturbing or puzzling
00:03:21.970 --> 00:03:27.160
is that you do many
identical experiments.
00:03:27.160 --> 00:03:28.992
And the answer is
always different.
00:03:33.820 --> 00:03:36.090
There is no determinant.
00:03:36.090 --> 00:03:37.920
It's all probabilistic.
00:03:37.920 --> 00:03:41.610
So this really
should bother you.
00:03:41.610 --> 00:03:44.710
And eventually, it won't.
00:03:44.710 --> 00:03:46.970
OK.
00:03:46.970 --> 00:03:49.430
Now, there's another
question I have.
00:03:49.430 --> 00:03:54.940
When I described the two slit
experiment, I intentionally
00:03:54.940 --> 00:03:58.880
put something up on the
diagram that should bother you,
00:03:58.880 --> 00:04:02.420
that should have said,
this is ridiculous.
00:04:02.420 --> 00:04:04.760
I used a candle in the lecture.
00:04:04.760 --> 00:04:08.990
And I used a light
bulb in the notes.
00:04:08.990 --> 00:04:11.490
And why is that ridiculous?
00:04:11.490 --> 00:04:12.240
Yes?
00:04:12.240 --> 00:04:14.250
AUDIENCE: You said
many frequencies.
00:04:14.250 --> 00:04:15.375
ROBERT FIELD: That's right.
00:04:15.375 --> 00:04:22.890
So the sources of light that I
misled you with intentionally
00:04:22.890 --> 00:04:26.890
have a continuous
frequency distribution.
00:04:26.890 --> 00:04:34.640
And the interference depends
on the same frequency.
00:04:34.640 --> 00:04:41.260
So the only way you would
see any kind of diffraction
00:04:41.260 --> 00:04:44.440
pattern, any kind of pattern
on the two slit experiment,
00:04:44.440 --> 00:04:48.180
is if you had
monochromatic light.
00:04:48.180 --> 00:04:50.220
It would all wash out.
00:04:50.220 --> 00:04:52.170
You'd still get dots.
00:04:52.170 --> 00:04:53.640
But the dots would
never give you
00:04:53.640 --> 00:04:58.290
anything except perhaps a
superposition of two or three
00:04:58.290 --> 00:05:02.521
or an infinite
number of patterns.
00:05:02.521 --> 00:05:03.020
OK.
00:05:03.020 --> 00:05:08.300
This is, again, something
that's really bothersome.
00:05:08.300 --> 00:05:13.610
Now, I also use the crude
illustration of an uncertainty
00:05:13.610 --> 00:05:18.320
principle, that the
uncertainty in position z-axis
00:05:18.320 --> 00:05:21.950
and the uncertainty in the
momentum along the z-axis
00:05:21.950 --> 00:05:23.630
was greater than h.
00:05:26.720 --> 00:05:29.240
And I froze.
00:05:29.240 --> 00:05:34.760
And I didn't realize that I
had delta s with the slit.
00:05:34.760 --> 00:05:37.580
But it really is the
same thing as the z.
00:05:37.580 --> 00:05:40.670
Because the slit
is how you define
00:05:40.670 --> 00:05:43.640
the position in the z-axis.
00:05:43.640 --> 00:05:48.710
And so this is the first taste
of the uncertainty principle.
00:05:48.710 --> 00:05:49.910
And I said I didn't like it.
00:05:56.050 --> 00:05:57.770
OK.
00:05:57.770 --> 00:06:03.890
In quantum mechanics,
you're not allowed
00:06:03.890 --> 00:06:05.600
to look inside small stuff.
00:06:05.600 --> 00:06:09.830
You're not allowed to see
the microscopic structure.
00:06:09.830 --> 00:06:12.050
You're only able
to do experiments,
00:06:12.050 --> 00:06:16.070
usually thought experiments,
an infinite number
00:06:16.070 --> 00:06:17.540
of identical experiments.
00:06:17.540 --> 00:06:25.580
And they reveal the structure in
some complicated, encoded way.
00:06:25.580 --> 00:06:29.950
And this is really not what
the textbooks are about.
00:06:29.950 --> 00:06:34.330
Textbooks don't tell
you how you actually
00:06:34.330 --> 00:06:36.640
think about a problem
with quantum mechanics.
00:06:36.640 --> 00:06:39.940
They tell you, here are some
exactly solved problems.
00:06:39.940 --> 00:06:42.214
Memorize them.
00:06:42.214 --> 00:06:43.130
And I don't want you--
00:06:43.130 --> 00:06:44.130
I don't want to do that.
00:06:47.480 --> 00:06:49.870
OK.
00:06:49.870 --> 00:06:53.240
Last part of the
introduction here--
00:06:53.240 --> 00:06:58.810
suppose we have a circular
drum, a square drum,
00:06:58.810 --> 00:07:01.390
and a rectangular drum.
00:07:05.320 --> 00:07:11.510
Have you ever seen a square
drum or a rectangular drum?
00:07:11.510 --> 00:07:13.850
Do you have an idea how a
square drum would sound?
00:07:16.620 --> 00:07:17.120
Yes.
00:07:17.120 --> 00:07:18.191
You do have an idea.
00:07:18.191 --> 00:07:19.190
It would sound terrible.
00:07:22.460 --> 00:07:27.450
Because the frequencies, you
get are not integer multiples.
00:07:27.450 --> 00:07:29.970
It would just sound
amazingly terrible.
00:07:29.970 --> 00:07:35.860
But if you had a square
drum or a rectangular drum,
00:07:35.860 --> 00:07:38.080
you could do an
experiment with some kind
00:07:38.080 --> 00:07:40.660
of acoustic
instrument to find out
00:07:40.660 --> 00:07:45.820
what the frequency distribution
is of the noise you make.
00:07:45.820 --> 00:07:47.470
And you would be able to tell.
00:07:47.470 --> 00:07:49.300
It's not round.
00:07:49.300 --> 00:07:51.310
It might be square.
00:07:51.310 --> 00:07:54.220
Or it might have a certain
ratio of dimensions.
00:07:54.220 --> 00:07:57.880
This is what we're talking about
as far as internal structure is
00:07:57.880 --> 00:07:58.960
concerned.
00:07:58.960 --> 00:08:01.420
And it's very much like what
you would do as a musician.
00:08:04.100 --> 00:08:06.890
I mean, certainly, when
a musical instrument
00:08:06.890 --> 00:08:11.780
is arranged correctly, it's
not like a square drum.
00:08:11.780 --> 00:08:13.980
It sounds good.
00:08:13.980 --> 00:08:16.800
And that's because
you get harmonics
00:08:16.800 --> 00:08:22.320
or you get integer multiples
of some standard frequency.
00:08:24.850 --> 00:08:25.350
OK.
00:08:25.350 --> 00:08:31.250
So now, we're going to talk
about the classical wave
00:08:31.250 --> 00:08:34.960
equation, which is not quantum.
00:08:34.960 --> 00:08:39.880
But it's a very similar sort
of equation to the Schrodinger
00:08:39.880 --> 00:08:41.240
equation.
00:08:41.240 --> 00:08:44.740
And so the methods for solving
this differential equation
00:08:44.740 --> 00:08:46.960
are on display.
00:08:46.960 --> 00:08:52.960
And so the trick is--
well, first of all, where
00:08:52.960 --> 00:08:55.800
does this equation come from?
00:08:55.800 --> 00:08:58.020
And it's always force
is equal to mass times
00:08:58.020 --> 00:09:00.481
acceleration in disguise.
00:09:00.481 --> 00:09:00.980
OK.
00:09:00.980 --> 00:09:05.700
And then you have tricks
for how you solve this.
00:09:05.700 --> 00:09:10.220
And one of the most frequently
used and powerful tricks
00:09:10.220 --> 00:09:12.110
is separation of variables.
00:09:12.110 --> 00:09:15.160
You need to know how that works.
00:09:15.160 --> 00:09:20.760
Then once you solve the problem,
you have the general solution.
00:09:20.760 --> 00:09:25.170
And you then say, well,
OK, for the specific case
00:09:25.170 --> 00:09:29.740
we have, like a string
tied down at both ends,
00:09:29.740 --> 00:09:32.460
we have boundary conditions.
00:09:32.460 --> 00:09:35.480
And we impose those
boundary conditions.
00:09:35.480 --> 00:09:38.950
And then we have
basically what we
00:09:38.950 --> 00:09:42.620
would call the normal
modes of the problem.
00:09:42.620 --> 00:09:46.550
And then we would
ask, OK, well, suppose
00:09:46.550 --> 00:09:51.710
we're doing a
specific experiment
00:09:51.710 --> 00:09:56.660
or doing a specific
preparation of the system.
00:09:56.660 --> 00:09:59.850
And we can call that
the pluck of the system.
00:09:59.850 --> 00:10:04.400
And you might pluck
several normal modes.
00:10:04.400 --> 00:10:06.430
You get a superposition state.
00:10:06.430 --> 00:10:12.170
And that superposition state
behaves in a dynamic way.
00:10:12.170 --> 00:10:15.500
And you want to be able to
understand that dynamics.
00:10:15.500 --> 00:10:19.010
And the most important
thing that I want you to do
00:10:19.010 --> 00:10:24.410
is, instead of trying to draw
the solutions to a differential
00:10:24.410 --> 00:10:27.200
equation, which is a
mathematical equation,
00:10:27.200 --> 00:10:31.010
I want you to draw
cartoons, cartoons
00:10:31.010 --> 00:10:34.077
that embody your
understanding of the problem.
00:10:34.077 --> 00:10:35.910
And I'm going to be
trying to do that today.
00:10:46.624 --> 00:10:47.130
OK.
00:10:47.130 --> 00:10:49.820
In this course, for the
first half of the course,
00:10:49.820 --> 00:10:51.580
most of what we're
going to be doing
00:10:51.580 --> 00:10:55.590
is solving for exactly
soluble problems--
00:10:55.590 --> 00:11:00.060
the particle in a box,
the harmonic oscillator,
00:11:00.060 --> 00:11:07.777
the rigid rotor, and
the hydrogen atom.
00:11:10.980 --> 00:11:14.700
With these four problems,
most of the things
00:11:14.700 --> 00:11:19.170
that we will encounter
in quantum mechanics
00:11:19.170 --> 00:11:20.740
are somehow related to these.
00:11:23.924 --> 00:11:28.140
And in the textbooks, they
treat these things as sacred.
00:11:28.140 --> 00:11:30.480
And they say, OK, well, now
that you've solved them,
00:11:30.480 --> 00:11:32.580
you understand
quantum mechanics.
00:11:32.580 --> 00:11:35.160
But these are really
tools for understanding
00:11:35.160 --> 00:11:37.509
more complicated situations.
00:11:37.509 --> 00:11:39.300
I mean, you might have
a particle in a box.
00:11:39.300 --> 00:11:44.280
Instead of with a square bottom,
it might have a tilted bottom.
00:11:44.280 --> 00:11:47.080
Or it might have
a double minimum.
00:11:47.080 --> 00:11:50.520
But if you understand
that, you then
00:11:50.520 --> 00:11:52.800
can begin to build
an understanding of,
00:11:52.800 --> 00:11:56.190
what are the things in the
experiment that tell you
00:11:56.190 --> 00:12:01.200
about these distortions
of the standard problem?
00:12:01.200 --> 00:12:03.960
And the same thing for
a harmonic oscillator.
00:12:03.960 --> 00:12:05.760
Almost everything
that's vibrating
00:12:05.760 --> 00:12:07.890
is harmonic approximately.
00:12:07.890 --> 00:12:09.600
But there's a little
bit of distortion
00:12:09.600 --> 00:12:11.220
as you stretch it more.
00:12:11.220 --> 00:12:13.560
And again, you
can understand how
00:12:13.560 --> 00:12:17.730
to measure the distortions from
harmonicity by understanding
00:12:17.730 --> 00:12:18.890
the harmonic oscillator.
00:12:18.890 --> 00:12:20.400
We did rotor, H atom.
00:12:20.400 --> 00:12:21.610
It's all the same.
00:12:21.610 --> 00:12:25.650
So I would like to tell you
that these standard problems are
00:12:25.650 --> 00:12:27.540
really important.
00:12:27.540 --> 00:12:30.500
But nothing is like that.
00:12:30.500 --> 00:12:34.650
And what's important is how
it's different from that.
00:12:34.650 --> 00:12:37.160
And this is my
unique perspective.
00:12:37.160 --> 00:12:43.030
And you won't get that from
McQuarrie or any textbook.
00:12:43.030 --> 00:12:45.190
But this is MIT.
00:12:45.190 --> 00:12:47.740
So there are templates for
understanding real quantum
00:12:47.740 --> 00:12:49.400
mechanical system.
00:12:49.400 --> 00:12:59.800
And the big thing, the
most important technique
00:12:59.800 --> 00:13:01.930
for doing that is
perturbation theory.
00:13:06.270 --> 00:13:09.690
And so perturbation
theory is just
00:13:09.690 --> 00:13:15.450
a way of building beyond
the oversimplification.
00:13:15.450 --> 00:13:17.820
And it's mathematically
really ugly.
00:13:17.820 --> 00:13:20.710
But it's tremendously powerful.
00:13:20.710 --> 00:13:23.710
And it's where you get insight.
00:13:23.710 --> 00:13:24.620
OK.
00:13:24.620 --> 00:13:31.250
Now, many people have complained
that they found 5.61 hard.
00:13:31.250 --> 00:13:32.780
Because it's so mathematical.
00:13:36.500 --> 00:13:39.230
And maybe this is going to be
the most mathematical lecture
00:13:39.230 --> 00:13:40.790
in the course.
00:13:40.790 --> 00:13:43.310
But I don't want it to be hard.
00:13:43.310 --> 00:13:49.550
Now, chemists usually derive
insights from pictorial
00:13:49.550 --> 00:13:51.950
rather than mathematical
views of a problem.
00:13:55.190 --> 00:14:00.060
So what are the pictures that
describe these differential
00:14:00.060 --> 00:14:00.930
equations?
00:14:00.930 --> 00:14:04.350
How do we convert
what seems to be
00:14:04.350 --> 00:14:07.800
just straight mathematics to
pictures that mean something
00:14:07.800 --> 00:14:08.790
to us?
00:14:08.790 --> 00:14:12.150
And that's my goal, to
get you to be drawing
00:14:12.150 --> 00:14:15.900
freehand pictures that
embody the important features
00:14:15.900 --> 00:14:19.300
of the solutions
to the problems.
00:14:19.300 --> 00:14:21.060
OK.
00:14:21.060 --> 00:14:25.560
So we're going to be looking
at a differential equation.
00:14:25.560 --> 00:14:28.380
And one of the first
questions you ask, well, where
00:14:28.380 --> 00:14:29.670
did that equation come from?
00:14:32.510 --> 00:14:35.330
And you're not going to derive
a differential equation ever
00:14:35.330 --> 00:14:36.140
in this course.
00:14:36.140 --> 00:14:37.990
But you're going to
want to think, well,
00:14:37.990 --> 00:14:43.850
I pretty much understand what's
in this differential equation.
00:14:43.850 --> 00:14:46.820
And then we'll use
standard methods
00:14:46.820 --> 00:14:47.930
for solving that equation.
00:14:52.630 --> 00:14:56.830
And one such differential
equation is this--
00:14:56.830 --> 00:14:58.720
second derivative
of some function
00:14:58.720 --> 00:15:03.580
with respect to a variable
is equal to a constant times
00:15:03.580 --> 00:15:05.200
that function.
00:15:05.200 --> 00:15:07.250
Now, that you know.
00:15:07.250 --> 00:15:09.980
You know sines and cosines
are solutions to that.
00:15:09.980 --> 00:15:12.190
And you know that exponentials
are solutions to that.
00:15:14.890 --> 00:15:17.730
Now, that pretty much takes
you through a lot of problems
00:15:17.730 --> 00:15:18.740
in quantum mechanics.
00:15:21.340 --> 00:15:22.990
But now, one of the
important things
00:15:22.990 --> 00:15:27.982
is this is a second-order
differential equation.
00:15:27.982 --> 00:15:29.440
And that means that
there are going
00:15:29.440 --> 00:15:35.010
to be two linearly
independent solutions.
00:15:35.010 --> 00:15:38.490
And you need to
know both of them.
00:15:38.490 --> 00:15:41.460
I'll talk about this
some more later.
00:15:41.460 --> 00:15:44.480
Now, sometimes, the
differential equations
00:15:44.480 --> 00:15:47.210
look much more
complicated than this.
00:15:47.210 --> 00:15:52.160
And so the goal is
usually to rewrite it
00:15:52.160 --> 00:15:56.390
in a form which corresponds to
a differential equation that
00:15:56.390 --> 00:16:00.200
is well known and
solved by mathematicians
00:16:00.200 --> 00:16:02.650
whose business is doing that.
00:16:02.650 --> 00:16:06.100
But we won't be doing that.
00:16:06.100 --> 00:16:08.140
OK.
00:16:08.140 --> 00:16:12.590
But usually, when you have
a differential equation,
00:16:12.590 --> 00:16:16.400
the function is of
more than one variable.
00:16:16.400 --> 00:16:18.990
And frequently, it's
position and time.
00:16:18.990 --> 00:16:23.289
And so the first thing you do is
you try to separate variables.
00:16:23.289 --> 00:16:24.830
And so that's what
we're going to do.
00:16:28.247 --> 00:16:29.705
So we have a
differential equation.
00:16:32.250 --> 00:16:35.520
And the first thing
is a general solution.
00:16:41.260 --> 00:16:47.270
And one of the things that this
solution will have is nodes.
00:16:47.270 --> 00:16:50.120
And the distance between nodes--
00:16:53.047 --> 00:16:53.630
here's a node.
00:16:53.630 --> 00:16:54.680
Here's a node.
00:16:54.680 --> 00:16:59.160
That's half the wavelength.
00:16:59.160 --> 00:17:01.650
And we know that in
quantum mechanics,
00:17:01.650 --> 00:17:05.400
if you know the wavelength,
you know the momentum.
00:17:05.400 --> 00:17:06.994
So nodes are really important.
00:17:06.994 --> 00:17:09.160
Because it's telling you
how fast things are moving.
00:17:12.220 --> 00:17:17.109
We can also look
at the envelope.
00:17:17.109 --> 00:17:19.270
And this would be some
kind of classical,
00:17:19.270 --> 00:17:22.210
as opposed to a quantum
mechanical, probability
00:17:22.210 --> 00:17:23.780
distribution.
00:17:23.780 --> 00:17:29.470
And so it might look like this.
00:17:29.470 --> 00:17:31.720
But the important thing
about the envelope
00:17:31.720 --> 00:17:33.250
is that it's always positive.
00:17:33.250 --> 00:17:35.770
Because it's
probability, as opposed
00:17:35.770 --> 00:17:38.200
to a probability
amplitude, which
00:17:38.200 --> 00:17:41.610
can be positive and negative.
00:17:41.610 --> 00:17:44.280
Interference is really
important in quantum mechanics.
00:17:44.280 --> 00:17:49.350
But sometimes, the envelope
tells you all you need to know.
00:17:49.350 --> 00:17:56.475
And the other thing is the
velocity of a stationary phase.
00:18:00.550 --> 00:18:01.470
So you have a wave.
00:18:01.470 --> 00:18:02.340
And it's moving.
00:18:02.340 --> 00:18:04.380
And you sit at a
point on that wave.
00:18:04.380 --> 00:18:07.390
And you ask, how fast
does that point move?
00:18:07.390 --> 00:18:09.700
And I did that last time.
00:18:09.700 --> 00:18:10.510
And OK.
00:18:13.390 --> 00:18:19.390
So I've already talked a little
bit about what we do next.
00:18:19.390 --> 00:18:23.140
But the important thing
is always, at the end,
00:18:23.140 --> 00:18:24.880
you draw a cartoon.
00:18:24.880 --> 00:18:31.060
And you endow that cartoon
with your insights.
00:18:31.060 --> 00:18:35.320
And that enables you to
remember and to understand
00:18:35.320 --> 00:18:37.290
and to organize questions
about the problem.
00:18:40.640 --> 00:18:41.480
OK.
00:18:41.480 --> 00:18:43.160
So let's get to work
on a real problem.
00:18:51.820 --> 00:18:58.020
So we have a string that's
tied down to two points.
00:18:58.020 --> 00:19:06.600
And so let's look at the
distortion of that string.
00:19:06.600 --> 00:19:12.490
And so we chop this region
of space up into regions.
00:19:15.670 --> 00:19:20.460
So this might be the
region at x minus 1.
00:19:20.460 --> 00:19:22.230
And this might be
the region at x0.
00:19:22.230 --> 00:19:26.100
And this might be
the region of x1.
00:19:26.100 --> 00:19:28.270
And we're interested in--
00:19:28.270 --> 00:19:36.150
OK, suppose we have the value
of the displacement of the wave
00:19:36.150 --> 00:19:42.410
here at x minus 1
and here and here.
00:19:42.410 --> 00:19:45.780
OK, so these would be
the amount that the wave
00:19:45.780 --> 00:19:48.560
is displaced from equilibrium.
00:19:48.560 --> 00:19:53.205
And we call those u of x.
00:19:55.820 --> 00:20:03.230
And so the first segment
here, the minus 1 segment,
00:20:03.230 --> 00:20:08.770
this segment is pulling down
on this segment of the string
00:20:08.770 --> 00:20:11.180
by this amount.
00:20:11.180 --> 00:20:17.260
And this one is pulling up on
the segment by that amount.
00:20:17.260 --> 00:20:22.650
So we want to know, what is the
force acting on each segment?
00:20:22.650 --> 00:20:30.240
And so we have the
force constant times
00:20:30.240 --> 00:20:36.930
the displacement at x0 minus
the displacement at x minus 1.
00:20:36.930 --> 00:20:40.360
So this is the difference
between the displacements.
00:20:40.360 --> 00:20:41.730
And this is the force constant.
00:20:41.730 --> 00:20:44.530
We're talking about Hooke's law.
00:20:44.530 --> 00:20:48.830
Hooke's law is the
force is equal to minus
00:20:48.830 --> 00:20:51.198
k times the displacement.
00:20:54.070 --> 00:21:00.270
And so we collect the forces
felt by each particle.
00:21:00.270 --> 00:21:06.690
And the forces felt by each
particle are, again, the force
00:21:06.690 --> 00:21:15.360
constant times the difference
in u at 0 and minus 1
00:21:15.360 --> 00:21:17.730
minus the difference--
00:21:17.730 --> 00:21:23.746
plus 1 minus the difference
in u at 0 and plus 1.
00:21:23.746 --> 00:21:27.400
And this is a second derivative.
00:21:27.400 --> 00:21:32.520
This is the second derivative
of u with respect to x.
00:21:32.520 --> 00:21:36.310
So we've derived
a wave equation.
00:21:36.310 --> 00:21:39.450
And we know it's going to
involve a second derivative.
00:21:39.450 --> 00:21:43.990
So force is equal to
mass times acceleration.
00:21:43.990 --> 00:21:46.950
Well, we already
know the force is
00:21:46.950 --> 00:21:53.610
going to be related to
the second derivative of u
00:21:53.610 --> 00:21:57.030
with respect to x.
00:21:57.030 --> 00:21:58.860
And now, this is something.
00:21:58.860 --> 00:22:01.320
And we know what this is.
00:22:01.320 --> 00:22:03.544
This is going to be
the time derivative.
00:22:08.390 --> 00:22:08.900
OK.
00:22:08.900 --> 00:22:11.320
And this is just something
that gets the units right.
00:22:15.160 --> 00:22:18.610
And it has physical
significance.
00:22:18.610 --> 00:22:22.490
But in the case of this
particle on a string--
00:22:22.490 --> 00:22:25.690
this wave on a
string, it's related
00:22:25.690 --> 00:22:29.010
to the mass of the string and
the tension of the string.
00:22:31.780 --> 00:22:35.820
And it's also related to the
velocity that things move.
00:22:35.820 --> 00:22:36.320
OK.
00:22:36.320 --> 00:22:38.600
So we have a
differential equation
00:22:38.600 --> 00:22:42.200
that is related to forces equal
to mass times acceleration.
00:22:42.200 --> 00:22:47.126
And the differential equation
has the form second derivative
00:22:47.126 --> 00:22:56.030
of u with respect
to x is equal to 1
00:22:56.030 --> 00:23:01.980
over v squared times the
second derivative of u
00:23:01.980 --> 00:23:05.540
with respect to t.
00:23:05.540 --> 00:23:08.880
That's the wave equation.
00:23:08.880 --> 00:23:12.260
So it is really
f is equal to ma.
00:23:12.260 --> 00:23:13.790
But OK.
00:23:13.790 --> 00:23:15.820
And now, the units of this--
00:23:15.820 --> 00:23:17.230
this is x.
00:23:17.230 --> 00:23:18.110
And this is t.
00:23:18.110 --> 00:23:20.480
In order to be
dimensionally consistent,
00:23:20.480 --> 00:23:27.080
this has to be something
that is x over t, OK?
00:23:27.080 --> 00:23:29.700
And so this may be a velocity.
00:23:29.700 --> 00:23:31.150
But it has units of velocity.
00:23:31.150 --> 00:23:33.690
That's the differential
equation we want to solve.
00:23:37.400 --> 00:23:38.170
OK.
00:23:38.170 --> 00:23:41.710
Well, the original differential
equation that I wrote--
00:23:48.020 --> 00:23:49.640
but I'm getting ahead of myself.
00:23:49.640 --> 00:23:50.480
OK.
00:23:50.480 --> 00:23:53.975
So this U of x and t--
00:23:56.590 --> 00:24:03.280
we'd like it to be
X of x times T of t.
00:24:03.280 --> 00:24:07.780
We think we could separate
the variables in this way.
00:24:07.780 --> 00:24:08.950
So we try it.
00:24:08.950 --> 00:24:13.050
If we fail, it says
you can't do that.
00:24:13.050 --> 00:24:16.980
Failure is usually
going to be a result
00:24:16.980 --> 00:24:20.670
that the solution to the
differential equation
00:24:20.670 --> 00:24:22.200
in this form is nothing.
00:24:22.200 --> 00:24:23.440
It's a straight line.
00:24:23.440 --> 00:24:25.770
Nothing's happening.
00:24:25.770 --> 00:24:28.290
So failure is acceptable.
00:24:28.290 --> 00:24:31.730
But if we're
successful, we're going
00:24:31.730 --> 00:24:36.100
to get two separate
differential equations.
00:24:36.100 --> 00:24:36.610
OK.
00:24:36.610 --> 00:24:42.690
So what we do then is take
this differential equation,
00:24:42.690 --> 00:24:45.480
substitute this in,
and divide on the left.
00:24:45.480 --> 00:24:54.650
So we have 1 over X of
x times T of t times
00:24:54.650 --> 00:25:02.150
the second derivative
with respect to x of xt
00:25:02.150 --> 00:25:10.740
is equal to 1 over xt times the
second derivative with respect
00:25:10.740 --> 00:25:13.200
to t of xt.
00:25:16.360 --> 00:25:18.610
OK.
00:25:18.610 --> 00:25:24.190
Well, on this side
of the equation,
00:25:24.190 --> 00:25:27.710
the only thing that
involves time is here.
00:25:27.710 --> 00:25:29.440
This doesn't operate on time.
00:25:29.440 --> 00:25:35.560
And so we can cancel the time
dependence from this side.
00:25:35.560 --> 00:25:40.490
And over on this
side, this derivative
00:25:40.490 --> 00:25:42.620
operates on t but not x.
00:25:42.620 --> 00:25:45.890
And so we can cancel the x part.
00:25:45.890 --> 00:25:50.150
And so what we have
now is an equation
00:25:50.150 --> 00:25:56.310
X of x second derivative
with respect to x squared
00:25:56.310 --> 00:26:07.640
of x is equal to this constant,
1 over v squared, times 1
00:26:07.640 --> 00:26:15.630
over t times the derivative of
T with respect to little t, OK?
00:26:15.630 --> 00:26:17.550
So this is interesting.
00:26:17.550 --> 00:26:20.600
We have a function
of x on this side
00:26:20.600 --> 00:26:23.330
and a function of
t on this side.
00:26:23.330 --> 00:26:25.510
They are independent variables.
00:26:25.510 --> 00:26:28.510
This can only be
true if both sides
00:26:28.510 --> 00:26:29.965
are equal to the same constant.
00:26:34.970 --> 00:26:38.710
So now, we have to
differential equations.
00:26:38.710 --> 00:26:42.940
We have 1 over x second
derivative with respect
00:26:42.940 --> 00:26:47.570
to x of x is equal to a concept.
00:26:47.570 --> 00:26:54.400
And we have 1 over v squared
1 over t second derivative
00:26:54.400 --> 00:27:00.160
with respect to t
of T is equal to K.
00:27:00.160 --> 00:27:03.290
So now, we're on firmer ground.
00:27:03.290 --> 00:27:05.835
We know about solutions
to this kind of equation.
00:27:11.200 --> 00:27:13.320
And so there's two cases.
00:27:13.320 --> 00:27:15.800
One is this K is greater than 0.
00:27:15.800 --> 00:27:19.440
And the other is K less than 0.
00:27:19.440 --> 00:27:21.120
So let's look at this equation.
00:27:21.120 --> 00:27:28.750
If K is greater than 0, then if
we plug in a sine or a cosine,
00:27:28.750 --> 00:27:31.330
we get something
that's less than 0.
00:27:31.330 --> 00:27:35.260
Because the derivative of sine
with respect to its variable
00:27:35.260 --> 00:27:37.240
is negative cosine.
00:27:37.240 --> 00:27:40.430
And then we do it again,
we get back to sine.
00:27:40.430 --> 00:27:44.830
And so sines and cosines are
no good for this equation
00:27:44.830 --> 00:27:48.090
if K is greater than 0.
00:27:48.090 --> 00:27:57.100
But exponentials-- so we can
have e to some constant x
00:27:57.100 --> 00:28:00.730
or e to the minus
some constant x.
00:28:00.730 --> 00:28:05.980
Or here, for the
negative value of K,
00:28:05.980 --> 00:28:09.690
we could have sine
some constant x
00:28:09.690 --> 00:28:12.600
and cosine of some constant x.
00:28:12.600 --> 00:28:15.176
We know that.
00:28:15.176 --> 00:28:16.050
So we have two cases.
00:28:19.480 --> 00:28:24.600
So to make life
simple, we say K is
00:28:24.600 --> 00:28:27.450
going to be equal to
lowercase k squared.
00:28:30.400 --> 00:28:35.230
Because we want to use this
lowercase k in our solutions.
00:28:35.230 --> 00:28:35.980
All right.
00:28:35.980 --> 00:28:39.610
So I may have confused matters.
00:28:39.610 --> 00:28:47.830
But the solution for the time
equation and the position
00:28:47.830 --> 00:28:50.680
equation are clear.
00:28:50.680 --> 00:28:53.980
And so depending on whether
K is positive or negative,
00:28:53.980 --> 00:29:00.511
we're dealing with sines
and cosines or exponentials.
00:29:00.511 --> 00:29:01.010
OK.
00:29:01.010 --> 00:29:04.760
So I don't want to belabor
this, but the next stage
00:29:04.760 --> 00:29:06.444
is boundary conditions.
00:29:13.220 --> 00:29:17.740
We don't know whether K positive
or K negative is possible.
00:29:20.350 --> 00:29:22.360
But we do know
boundary conditions.
00:29:22.360 --> 00:29:26.300
And so if we have a string
which is tied down at the end--
00:29:26.300 --> 00:29:28.560
so this is x equals 0.
00:29:28.560 --> 00:29:33.030
And this is x equals
L. Then we input
00:29:33.030 --> 00:29:36.016
impose the boundary conditions.
00:29:36.016 --> 00:29:43.750
Well, the boundary conditions
are u of 0t is equal to 0.
00:29:43.750 --> 00:29:49.960
And u of Lt is equal to 0.
00:29:49.960 --> 00:30:00.930
So if we take the K greater than
0 case, u of 0t is equal to--
00:30:11.110 --> 00:30:13.390
well, let's just do this again.
00:30:13.390 --> 00:30:16.120
0t.
00:30:16.120 --> 00:30:22.570
We have x of 0 times T of t.
00:30:22.570 --> 00:30:25.180
OK, we don't really
care about this.
00:30:25.180 --> 00:30:31.830
But x of 0 has to be 0--
00:30:31.830 --> 00:30:32.620
I'm sorry.
00:30:32.620 --> 00:30:33.120
OK.
00:30:33.120 --> 00:30:36.310
So we have two solutions.
00:30:36.310 --> 00:30:44.030
If K is greater than 0, we
have the exponential terms.
00:30:44.030 --> 00:30:50.830
So we have Ae to the 0
plus B to the minus 0.
00:30:54.800 --> 00:30:56.180
And this has to be equal to 0.
00:30:59.630 --> 00:31:01.610
That's x of 0.
00:31:04.295 --> 00:31:06.200
Well, e to the 0 is 1.
00:31:06.200 --> 00:31:07.850
E to the minus 0 is 1.
00:31:07.850 --> 00:31:09.590
And so this is good.
00:31:09.590 --> 00:31:15.850
A has to be equal to
minus B. And then we
00:31:15.850 --> 00:31:22.060
have the other boundary
condition, X of L.
00:31:22.060 --> 00:31:35.190
We have A e to the L plus B e
to the minus L. Well, I'm sorry.
00:31:35.190 --> 00:31:38.290
Let's just put what
we already know.
00:31:38.290 --> 00:31:43.320
Minus A. So we can write
this as A e to the L minus
00:31:43.320 --> 00:31:47.400
e to the minus L. And
that has to be equal to 0.
00:31:47.400 --> 00:31:49.530
Can't do it.
00:31:49.530 --> 00:31:50.610
This can never be 0.
00:31:56.580 --> 00:32:00.170
So that means the K greater
than 0 solutions are illegal.
00:32:03.950 --> 00:32:06.170
Well, that's kind of bad news.
00:32:06.170 --> 00:32:08.240
Because it sounds like
separation of variables
00:32:08.240 --> 00:32:09.660
is failing.
00:32:09.660 --> 00:32:10.340
But it doesn't.
00:32:10.340 --> 00:32:13.115
Because the K less
than 0 solution works.
00:32:17.060 --> 00:32:28.980
So for K less than 0, X of 0 is
equal to C sine 0 plus D cosine
00:32:28.980 --> 00:32:31.070
0.
00:32:31.070 --> 00:32:34.790
And so that means
D is equal to 0.
00:32:34.790 --> 00:32:36.710
Things are dying.
00:32:36.710 --> 00:32:56.200
And X of L, boundary condition,
is C sine KL is equal to 0.
00:32:56.200 --> 00:32:59.000
And this we can solve.
00:32:59.000 --> 00:33:12.660
So sine is equal to 0 when KL
is equal to 0, 0 pi, 2 pi, et
00:33:12.660 --> 00:33:13.160
cetera.
00:33:13.160 --> 00:33:17.850
And so we have KL
is equal to n pi.
00:33:22.250 --> 00:33:26.180
So we can write this
as Kn is equal to n
00:33:26.180 --> 00:33:30.320
pi over L. We get quantization.
00:33:30.320 --> 00:33:32.990
This isn't quantum mechanics.
00:33:32.990 --> 00:33:38.670
But there are certain allowed
values of this K constant.
00:33:38.670 --> 00:33:41.130
And we have a
bunch of solutions.
00:33:43.940 --> 00:33:45.450
And so what do they look like?
00:33:45.450 --> 00:33:55.920
So n equals 0, n
equals 1, n equals 2.
00:33:55.920 --> 00:33:57.990
So what does the 0
solution look like?
00:34:04.930 --> 00:34:05.848
Yes?
00:34:05.848 --> 00:34:07.184
AUDIENCE: No node.
00:34:07.184 --> 00:34:08.100
ROBERT FIELD: Nothing.
00:34:11.380 --> 00:34:13.070
So we don't even
think about this.
00:34:13.070 --> 00:34:15.650
We say, n equals 0
is not a solution.
00:34:15.650 --> 00:34:19.909
Because the wave isn't there.
00:34:19.909 --> 00:34:24.760
No nodes, one node.
00:34:24.760 --> 00:34:27.320
And if we look at
this carefully,
00:34:27.320 --> 00:34:30.139
the node is always
at a cemetery point.
00:34:30.139 --> 00:34:31.699
It's in the middle.
00:34:31.699 --> 00:34:35.270
If we have the next one,
we'll have two nodes.
00:34:35.270 --> 00:34:38.360
And they'll be at
the 1/3 2/3 point.
00:34:38.360 --> 00:34:41.960
And so we know
where the nodes are.
00:34:41.960 --> 00:34:45.469
We also know that the amplitude
of each loop of the wave
00:34:45.469 --> 00:34:46.460
function is the same.
00:34:46.460 --> 00:34:50.199
But it alternates in sine.
00:34:50.199 --> 00:34:53.500
So you can draw
cartoons now at will.
00:34:53.500 --> 00:35:00.610
Because this spatial part of the
solution to this wave is clear.
00:35:00.610 --> 00:35:04.060
Any value of the quantum
number, or the n,
00:35:04.060 --> 00:35:08.690
gives you a picture that
you can draw in seconds.
00:35:08.690 --> 00:35:13.540
And there are a lot of quantum
mechanical problems like that.
00:35:13.540 --> 00:35:16.590
But sometimes, you
have to keep in mind
00:35:16.590 --> 00:35:19.860
that the node separation,
in other words--
00:35:23.390 --> 00:35:31.580
well, let's just say node
separation is lambda over 2.
00:35:31.580 --> 00:35:41.820
And lambda over 2 is
equal to 1/2 h over p.
00:35:41.820 --> 00:35:47.310
So if we know what
the momentum is, or we
00:35:47.310 --> 00:35:50.430
know what the kinetic energy is,
we know what the momentum is.
00:35:50.430 --> 00:35:54.750
We know how momentum is
encoded in node separations.
00:35:57.470 --> 00:36:00.590
So everything we want to know
about a one-dimensional problem
00:36:00.590 --> 00:36:05.510
is expressed in the
spacing of nodes
00:36:05.510 --> 00:36:08.060
and the amplitude between nodes.
00:36:08.060 --> 00:36:10.070
And the amplitude between
nodes have something
00:36:10.070 --> 00:36:13.076
to do with the momentum, too.
00:36:13.076 --> 00:36:14.450
Because if you're
going from here
00:36:14.450 --> 00:36:22.150
to here at some high velocity,
there's not much amplitude.
00:36:22.150 --> 00:36:26.690
And at a lower velocity,
you get more amplitude.
00:36:26.690 --> 00:36:31.250
And so the amplitude
in each of these node
00:36:31.250 --> 00:36:33.800
to node separate
sections is related
00:36:33.800 --> 00:36:37.280
to the average momentum
of the classical particle
00:36:37.280 --> 00:36:40.070
in that section.
00:36:40.070 --> 00:36:41.860
So the classical
mechanics is going
00:36:41.860 --> 00:36:46.510
to be extremely important
in drawing cartoons
00:36:46.510 --> 00:36:49.180
for quantum mechanical systems.
00:36:49.180 --> 00:36:50.140
Not in the textbooks.
00:36:53.550 --> 00:36:56.400
We supposedly know classical
mechanics pretty well,
00:36:56.400 --> 00:36:59.160
and especially here at MIT.
00:36:59.160 --> 00:37:01.140
So you might as well
use that in order
00:37:01.140 --> 00:37:06.240
to get an idea of how all of
the quantum mechanical problems
00:37:06.240 --> 00:37:09.790
you're facing will be behaving.
00:37:09.790 --> 00:37:10.870
OK.
00:37:10.870 --> 00:37:18.220
So the next thing we want
to do is finish the job.
00:37:18.220 --> 00:37:23.110
And so I can simply write down
the time-dependent solutions.
00:37:23.110 --> 00:37:40.280
They are E sine vkn t
plus F cosine vkn t.
00:37:44.840 --> 00:37:47.120
And we can say that--
00:37:47.120 --> 00:37:49.760
rather than carrying
around all this stuff,
00:37:49.760 --> 00:37:53.750
we can say omega n is vkn.
00:38:00.540 --> 00:38:03.050
Isn't that nice?
00:38:03.050 --> 00:38:07.570
So we have a frequency for
the time-dependent part, which
00:38:07.570 --> 00:38:11.950
is an integer multiple of
this constant V times this
00:38:11.950 --> 00:38:17.570
[? vector. ?] Or this you
can think of as just k sub n.
00:38:17.570 --> 00:38:22.250
So we can rewrite this in
a frequency and phase form.
00:38:22.250 --> 00:38:26.000
We have now the full solution.
00:38:26.000 --> 00:38:30.760
We have A n sine n pi L over x.
00:38:35.210 --> 00:38:46.830
And then this e N sine n pi--
00:38:56.322 --> 00:38:58.350
I'm so used to the
pictures I don't even
00:38:58.350 --> 00:39:01.393
want to look at the
equations anymore--
00:39:01.393 --> 00:39:11.270
n omega t plus F n
cosine n omega t.
00:39:11.270 --> 00:39:11.770
OK.
00:39:20.590 --> 00:39:23.980
So we can also take
this and rewrite it
00:39:23.980 --> 00:39:36.160
in a simpler form, E n prime
cosine n omega t plus phi n.
00:39:36.160 --> 00:39:39.700
So we can combine these two
terms as a single cosine
00:39:39.700 --> 00:39:40.630
with a phase vector.
00:39:46.500 --> 00:39:47.360
OK.
00:39:47.360 --> 00:39:53.860
So now, we're ready to actually
go to the specific thing
00:39:53.860 --> 00:39:59.930
that you do in a real experiment
or a real musical instrument.
00:40:02.870 --> 00:40:07.400
We say, OK, here is the
actual initial condition,
00:40:07.400 --> 00:40:10.880
the pluck of the system.
00:40:10.880 --> 00:40:15.950
And the pluck usually
occurs at t equals 0.
00:40:15.950 --> 00:40:19.440
But I'll just specify it here.
00:40:19.440 --> 00:40:26.630
And what we have is now a
sum over as many normal modes
00:40:26.630 --> 00:40:27.800
as you want.
00:40:27.800 --> 00:40:46.030
We have A n E n prime times sine
n pi over L x times cosine N
00:40:46.030 --> 00:40:51.600
omega t plus phi N.
00:40:51.600 --> 00:40:56.160
So we have a bunch of terms
like this, a spacial factor,
00:40:56.160 --> 00:40:58.424
and a temporal factor.
00:40:58.424 --> 00:40:59.840
And you can draw
pictures of both.
00:41:02.590 --> 00:41:05.000
Now, but there is
another simplification.
00:41:05.000 --> 00:41:15.020
From trigonometry, sine A cosine
B-- we have sine and cosine--
00:41:15.020 --> 00:41:28.730
can be written as 1/2
times sine n pi L x plus
00:41:28.730 --> 00:41:46.810
n omega t plus phi
n plus sine n pi L x
00:41:46.810 --> 00:41:51.250
minus n omega t minus phi n.
00:41:51.250 --> 00:41:55.450
So these are the two
possible solutions.
00:41:55.450 --> 00:41:58.690
And we can write them now
in terms of position factor
00:41:58.690 --> 00:42:04.760
at a time factor in the same
sine or cosine function.
00:42:07.290 --> 00:42:08.550
So these are the things.
00:42:08.550 --> 00:42:10.070
Now, we're ready
to make a picture.
00:42:12.610 --> 00:42:13.530
OK.
00:42:13.530 --> 00:42:16.050
So these are the
actual things that you
00:42:16.050 --> 00:42:22.010
make by exciting the system
not in an eigenfunction.
00:42:22.010 --> 00:42:24.352
But it's a superposition
of eigenfunctions.
00:42:27.250 --> 00:42:29.760
And again, there are
certain things you learn.
00:42:33.030 --> 00:42:37.570
If you have a pure
eigenfunction,
00:42:37.570 --> 00:42:39.560
you have standing waves.
00:42:39.560 --> 00:42:41.360
There's no left-right motion.
00:42:41.360 --> 00:42:42.650
There's no breathing motion.
00:42:42.650 --> 00:42:48.760
There's only up-down motion of
each loop of the wave function.
00:42:48.760 --> 00:42:54.360
If you have a superposition of
two or more functions, which
00:42:54.360 --> 00:43:02.950
are all of even n, then what
happens is you have no motions,
00:43:02.950 --> 00:43:04.450
you just have breathing.
00:43:04.450 --> 00:43:06.220
In other words,
you have a function
00:43:06.220 --> 00:43:08.910
that might look sort of
like this at one time
00:43:08.910 --> 00:43:12.570
and like that at another time.
00:43:12.570 --> 00:43:14.750
So amplitude is moving.
00:43:14.750 --> 00:43:18.740
So it can be moving.
00:43:18.740 --> 00:43:22.530
And in between, it's
sort of like this.
00:43:22.530 --> 00:43:24.500
Now, if you have
a function which
00:43:24.500 --> 00:43:29.410
involves both even and odd n,
you have left-right motion.
00:43:32.030 --> 00:43:34.400
This is true in
quantum mechanics, too.
00:43:34.400 --> 00:43:40.040
So you only get motion if you're
making superposition of eigen--
00:43:40.040 --> 00:43:40.610
Yes?
00:43:40.610 --> 00:43:42.693
AUDIENCE: What is the
difference between breathing
00:43:42.693 --> 00:43:46.580
and the standing
wave with no nodes?
00:43:46.580 --> 00:43:50.930
ROBERT FIELD: Well, for this
picture, this is over-simple.
00:43:50.930 --> 00:43:53.120
So I mean, you could have--
00:43:56.360 --> 00:43:59.390
basically, what's happening is
amplitude is moving from middle
00:43:59.390 --> 00:44:01.340
to the edges and back.
00:44:01.340 --> 00:44:05.430
And so yes.
00:44:05.430 --> 00:44:08.530
But you want to develop
your own language,
00:44:08.530 --> 00:44:12.970
your own set of drawings so that
you understand these things.
00:44:12.970 --> 00:44:16.740
And the important thing is
the understanding, the ability
00:44:16.740 --> 00:44:19.800
to draw these
pictures which contain
00:44:19.800 --> 00:44:24.330
the critical information about
node spacings, amplitudes,
00:44:24.330 --> 00:44:28.290
shapes, and to
anticipate when you're
00:44:28.290 --> 00:44:31.330
going to have left-right
motion or when
00:44:31.330 --> 00:44:35.200
you're just going to have
complicated up-down motions.
00:44:35.200 --> 00:44:36.810
Because there could be nodes.
00:44:36.810 --> 00:44:43.620
But there is no motion of the
center of this wavepacket.
00:44:43.620 --> 00:44:46.500
Now, this is fantastic.
00:44:46.500 --> 00:44:51.170
Because I just said wavepacket.
00:44:51.170 --> 00:45:01.000
Quantum mechanics--
eigenfunctions don't move.
00:45:01.000 --> 00:45:04.420
Superpositions of
eigenfunctions do move.
00:45:04.420 --> 00:45:07.040
If we make a superposition
of many eigenfunctions,
00:45:07.040 --> 00:45:10.370
it is a particle-like state.
00:45:10.370 --> 00:45:12.060
What that particle-like
state will do
00:45:12.060 --> 00:45:17.600
is exactly what you
expect from 8.01.
00:45:17.600 --> 00:45:20.870
The particle-like
states-- the center
00:45:20.870 --> 00:45:25.190
of the wavepacket for a
particle-like state moves
00:45:25.190 --> 00:45:28.080
according to Newton's equations.
00:45:28.080 --> 00:45:30.710
So I'm saying I'm
taking away your ability
00:45:30.710 --> 00:45:33.169
to look at microscopic stuff.
00:45:33.169 --> 00:45:34.710
And I'm going to
give it back to you.
00:45:34.710 --> 00:45:36.590
By the end of the
course, we're going
00:45:36.590 --> 00:45:39.300
to have the time-dependent
Schrodinger equation.
00:45:39.300 --> 00:45:41.640
We're going to be able
to see things move.
00:45:41.640 --> 00:45:43.560
And we're going to
see why they move
00:45:43.560 --> 00:45:46.840
and how they encode that motion.
00:45:46.840 --> 00:45:49.041
Not in the textbooks,
but I think
00:45:49.041 --> 00:45:51.040
it's something you really
want to be able to do.
00:45:51.040 --> 00:45:53.560
If you're going to
understand physical systems
00:45:53.560 --> 00:45:56.780
and use it to guide
your understanding,
00:45:56.780 --> 00:45:59.140
you have to be able
to draw these pictures
00:45:59.140 --> 00:46:01.130
and build a step at a time.
00:46:01.130 --> 00:46:05.110
And so this way the equation,
the classical wave equation,
00:46:05.110 --> 00:46:09.040
gives you almost all of
the tools for artistry
00:46:09.040 --> 00:46:10.040
as well as insight.
00:46:13.230 --> 00:46:15.200
OK.
00:46:15.200 --> 00:46:19.610
Now, in the notes, there is
a time-lapse movie that shows
00:46:19.610 --> 00:46:23.972
what a two-state wave function--
00:46:23.972 --> 00:46:27.290
what a two-state solution
to the wave equation looks
00:46:27.290 --> 00:46:33.360
like if you have even and odd
or only even and even terms.
00:46:33.360 --> 00:46:33.860
OK.
00:46:33.860 --> 00:46:38.740
Now, I'm going to make
some assertions at the end.
00:46:38.740 --> 00:46:40.470
We're coming back
to the drum problem.
00:46:43.000 --> 00:46:45.690
And suppose we have
a rectangular drum.
00:46:48.970 --> 00:46:51.760
Well, solving the
differential equation
00:46:51.760 --> 00:46:54.190
for this rectangular
drum gives you
00:46:54.190 --> 00:46:56.440
a bunch of normal
mode frequencies
00:46:56.440 --> 00:46:57.940
that depend on two indices.
00:47:15.871 --> 00:47:21.900
And so this is the geometric
structure of the drum.
00:47:21.900 --> 00:47:24.410
And these are the
quantum numbers.
00:47:24.410 --> 00:47:27.557
And these are the frequencies.
00:47:27.557 --> 00:47:29.890
And it's going to make a whole
bunch of frequencies that
00:47:29.890 --> 00:47:31.810
are not integer
multiples of each other
00:47:31.810 --> 00:47:34.734
or of any simpler thing.
00:47:34.734 --> 00:47:36.150
And that's why it
sounds horrible.
00:47:39.980 --> 00:47:44.510
It's perhaps a little bit like
playing a violin with a saw.
00:47:44.510 --> 00:47:47.200
It will sound terrible.
00:47:47.200 --> 00:47:48.700
You would never do it.
00:47:48.700 --> 00:47:52.490
But you would also never build
a square or rectangular drum.
00:47:52.490 --> 00:47:57.910
But the noise that you
make tells immediately not
00:47:57.910 --> 00:48:02.440
just what the shape
of the instrument is
00:48:02.440 --> 00:48:03.940
but how it was played.
00:48:03.940 --> 00:48:09.320
For example, suppose you
had an elliptical drum.
00:48:09.320 --> 00:48:10.940
That'll sound terrible, too.
00:48:10.940 --> 00:48:13.070
But here are the two foci.
00:48:13.070 --> 00:48:14.780
I'm not so sure
it'll sound terrible
00:48:14.780 --> 00:48:16.130
if you hit it here or here.
00:48:18.650 --> 00:48:21.720
And certainly, if you
are a circular drum,
00:48:21.720 --> 00:48:24.470
if you hit it in the middle
as opposed to on the edges,
00:48:24.470 --> 00:48:26.670
it'll sound different.
00:48:26.670 --> 00:48:29.110
The spectral content
will be the same.
00:48:29.110 --> 00:48:32.830
But the amplitudes of each
component will be different.
00:48:32.830 --> 00:48:38.370
And so in quantum mechanics, you
use the same sort of instinct
00:48:38.370 --> 00:48:41.070
as you develop as
a musician in order
00:48:41.070 --> 00:48:45.135
to figure out how this
system is going to respond
00:48:45.135 --> 00:48:46.010
to what you do to it.
00:48:48.411 --> 00:48:49.535
And that's pretty powerful.
00:48:52.320 --> 00:48:54.540
So many of you are musicians.
00:48:54.540 --> 00:48:57.640
And you know
instinctively what's
00:48:57.640 --> 00:49:01.450
wrong when you do something
that's not quite right,
00:49:01.450 --> 00:49:04.600
or your instrument
is out of tune.
00:49:04.600 --> 00:49:07.690
But in quantum mechanics,
all of those insights
00:49:07.690 --> 00:49:09.880
will come to bear.
00:49:09.880 --> 00:49:13.240
Not in the textbooks.
00:49:13.240 --> 00:49:17.140
Because the textbooks tell you
about exactly solved problems.
00:49:17.140 --> 00:49:20.050
And then they tell
you how to do spectra
00:49:20.050 --> 00:49:23.940
that are too perfect for
anybody else to observe.
00:49:23.940 --> 00:49:26.490
And you won't see those spectra.
00:49:26.490 --> 00:49:29.970
And they don't tell you what
the spectra you will observe
00:49:29.970 --> 00:49:32.610
tell you about the
system in question.
00:49:32.610 --> 00:49:33.300
OK.
00:49:33.300 --> 00:49:35.070
So I should stop now.
00:49:35.070 --> 00:49:41.550
And I'm going to be generating
Problem Set 2, which
00:49:41.550 --> 00:49:43.230
will be posted on Friday.
00:49:43.230 --> 00:49:46.130
Problem Set 1 is
due this Friday.
00:49:46.130 --> 00:49:49.080
And-- Good.
00:49:49.080 --> 00:49:50.870
Thank you.