WEBVTT

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Prof: Okay,
so we just glimpsed this at the

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end last time.

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This is a crystal structure of
a complicated molecule that was

00:10.320 --> 00:14.390
performed by these same Swiss
folks that we've talked about,

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and notice how very precise it
is.

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The bond distances between
atoms are reported to

00:24.450 --> 00:27.070
The bonds are like one and a
half angstroms.

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So it's like a part in a
thousand, is the precision to

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which the positions of the atoms
is known.

00:33.780 --> 00:34.810
Okay?

00:34.810 --> 00:39.200
But those positions are average
positions, because the atoms are

00:39.200 --> 00:41.570
constantly in motion,
vibrating.

00:41.570 --> 00:44.970
In fact, the typical vibration
amplitude,

00:44.970 --> 00:48.030
which depends on -- an atom
that's off on the end of

00:48.026 --> 00:51.556
something is more floppy than
one that's held in by a lot of

00:51.562 --> 00:55.592
bonds in various directions,
in a cage sort of thing.

00:55.590 --> 01:00.160
But typically they're vibrating
by about 0.05 angstroms,

01:00.158 --> 01:04.378
which is twenty-five times as
big as the precision to which

01:04.375 --> 01:07.205
the position of the average is
known.

01:07.209 --> 01:08.659
Okay?

01:08.659 --> 01:11.189
So no molecule looks like that
at an instant,

01:11.192 --> 01:13.672
the atoms are all displaced a
little bit.

01:13.670 --> 01:15.070
Now how big is that?

01:15.069 --> 01:17.309
Here, if you look at that
yellow thing,

01:17.313 --> 01:19.793
when it shrinks down,
that's how big it is,

01:19.793 --> 01:21.923
that's how big the vibration
is.

01:21.920 --> 01:23.320
It's very small.

01:23.319 --> 01:25.469
But these are very precise
measurements.

01:25.468 --> 01:25.908
Right?

01:25.909 --> 01:28.569
Now why did they do so precise
measurements?

01:28.569 --> 01:32.499
Did they really care to know
bond distances to that accuracy?

01:32.500 --> 01:35.530
Maybe for some purposes they
did, but that wasn't the main

01:35.534 --> 01:37.774
reason they did the work very
carefully.

01:37.769 --> 01:41.659
They did it carefully in order
to get really precise positions

01:41.659 --> 01:44.629
for the average atom,
so they could subtract

01:44.632 --> 01:47.982
spherical atoms and see the
difference accurately.

01:47.980 --> 01:48.970
Okay?

01:48.970 --> 01:52.490
Because if you have the wrong
position for the atom that

01:52.492 --> 01:54.992
you're subtracting,
you get nonsense.

01:54.989 --> 01:59.629
Okay, and what this is going to
reveal is some pathologies of

01:59.625 --> 02:04.025
bonding, from the point of view
of Lewis concept of shared

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electrons.

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Okay, so here's a picture of
this molecule.

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And remember,
we had -- rubofusarin,

02:10.984 --> 02:14.304
which we looked at last time,
had the great virtue that it

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was planar.

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So you could cut a slice that
went through all the atoms.

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This molecule's definitely not
planar, so you have to cut

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various slices to see different
things.

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So first we'll cut a slice that
goes through those ten atoms.

02:27.759 --> 02:28.729
Okay?

02:28.729 --> 02:32.179
And here is the difference
electron density.

02:32.180 --> 02:39.010
What does the difference
density show?

02:39.009 --> 02:40.429
Somebody?

02:40.430 --> 02:42.590
Yes Alex?

02:42.590 --> 02:43.690
Student:  It's the
electron density minus the

02:43.693 --> 02:45.163
spherical --
Prof: It's the total

02:45.161 --> 02:46.601
electron density minus the
atoms;

02:46.598 --> 02:49.818
that is, how the electron
density shifted when the

02:49.818 --> 02:52.248
molecule was formed from the
atoms.

02:52.250 --> 02:56.190
Okay, and here we see just
exactly what we expected to see,

02:56.190 --> 03:00.000
that the electrons shifted in
between the carbon atoms,

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the benzene ring,
and other pairs of carbon atoms

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as well.

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It also shows the C-H bonds,
because in this case the

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hydrogen atoms were subtracted.

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We showed one last time where
the hydrogen atoms weren't

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subtracted.

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Okay, so this is -- there's
nothing special here,

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everything looks the way you
expect it to be;

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although it's really beautiful,
as you would expect from these

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guys who do such great work.

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Now we'll do a different slice.

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This is sort of the plane of
the screen which divides the

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molecule symmetrically down the
middle,

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cuts through some bonds,
cuts through some atoms and so

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on.

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So here's the difference
density map that appears on that

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slice.

03:41.550 --> 03:44.380
Now the colored atoms,
on the right,

03:44.381 --> 03:49.321
are the positions of the atoms
through which the plane slices,

03:49.316 --> 03:52.306
but the atoms are subtracted
out;

03:52.310 --> 03:55.910
so what you see is the bonding
in that plane.

03:55.910 --> 03:59.070
So you see those bonds,
because both ends of the bond

03:59.073 --> 04:01.933
are in the plane,
so the bonds are in the plane,

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and you see just what you
expect to see.

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But there are other things you
see as well.

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You see the C-H bonds,
although they don't have nearly

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as much electron density as the
C-C bonds did.

04:14.161 --> 04:14.941
Right?

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You also see that lump,
which is the unshared pair on

04:20.132 --> 04:21.032
nitrogen.

04:21.031 --> 04:21.931
Right?

04:21.930 --> 04:25.160
But you see these two things,
which are bonds,

04:25.160 --> 04:28.530
but they're cross-sections of
bonds,

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because this particular plane
cuts through the middle of those

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bonds.

04:32.209 --> 04:36.029
Everybody see that?

04:36.029 --> 04:38.309
Okay, so again that's nothing
surprising.

04:38.310 --> 04:40.340
But here is something
surprising.

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There's another bond through
which that plane cuts,

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which is the one on the right,
through those three-membered

04:47.579 --> 04:48.719
rings, right?

04:48.720 --> 04:54.420
And what do you notice about
that bond?

04:54.420 --> 04:54.520
Student:  It isn't there.

04:54.519 --> 04:56.609
Prof: It isn't there.

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There isn't any electron
density for that bond.

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So it's a missing bond.

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This is what we'll refer to as
pathological bonding,

05:06.459 --> 05:07.059
right?

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It's not what Lewis would have
expected, maybe;

05:11.129 --> 05:14.339
we don't have Lewis to talk to,
so we don't know what he

05:14.339 --> 05:17.259
would've thought about this
particular molecule.

05:17.259 --> 05:21.069
So here's a third plane to
slice, which goes through those

05:21.074 --> 05:24.024
three atoms, and here's the
picture of it.

05:24.019 --> 05:29.579
And again, that bond is
missing, that we saw before.

05:29.579 --> 05:31.309
Previously we looked at a
cross-section.

05:31.310 --> 05:34.720
Here we're looking at a plane
that contains the bond,

05:34.716 --> 05:37.006
and again there's no there
there.

05:37.009 --> 05:37.599
Okay?

05:37.600 --> 05:40.550
But there's something else
that's funny about this slice.

05:40.550 --> 05:44.190
Do you see what?

05:44.190 --> 05:51.200
What's funny about the bonds
that you do see?

05:51.199 --> 05:55.169
Corey?

05:55.170 --> 05:56.450
Speak up so I can hear you.

05:56.449 --> 05:59.049
Student:  They're
connected; they're not totally

05:59.050 --> 05:59.580
separate.

05:59.579 --> 06:02.449
Prof: What do you mean
they're connected separately?

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Student: Usually you see
separate electron densities,

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but they're connected.

06:10.800 --> 06:12.820
Prof: Somebody say it in
different words.

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I think you got the idea but
I'm not sure everybody

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understood it.

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John, do you have an idea?

06:19.079 --> 06:21.699
Student:  The top one
seems to be more dense than the

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bottom one.

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Prof: One,
two, three, four;

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one, two, three four five.

06:25.519 --> 06:27.559
That's true,
it is a little more dense.

06:27.560 --> 06:30.540
That kind of thing could be
experimental error,

06:30.540 --> 06:33.160
because even though this was
done so precisely,

06:33.160 --> 06:37.440
you're subtracting two numbers
that are very large,

06:37.440 --> 06:40.450
so that any error you make in
the experimental one,

06:40.449 --> 06:43.489
or in positioning things for
the theoretical position of the

06:43.485 --> 06:45.785
atoms,
any error you make will really

06:45.788 --> 06:47.808
be amplified in a map like this.

06:47.810 --> 06:49.350
But it's true, you noticed.

06:49.350 --> 06:52.080
But there's something I think
more interesting about those --

06:52.079 --> 06:52.489
that...

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Yes John?

06:53.629 --> 06:56.529
Student:  The contour
lines, they're connected,

06:56.526 --> 06:59.636
the contour lines between the
top and the bottom bonds are

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connected.

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So maybe the electrons,
maybe -- I don't know if

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they're connected.

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Prof: Yes,
they sort of overlap one

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another.

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But of course if they're sort
of close to one another,

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that doesn't surprise you too
much because as you go out and

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out and out,
ultimately you'll get rings

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that do meet,
if you go far enough down.

07:19.230 --> 07:21.120
Yes Chris?

07:21.120 --> 07:24.700
Student:  The center of
density on the bonds doesn't

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intercept the lines connecting
the atoms.

07:27.350 --> 07:28.150
Prof: Ah.

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The bonds are not centered on
the line that connects the

07:32.935 --> 07:33.715
nuclei.

07:33.720 --> 07:40.300
These bonds are bent.

07:40.300 --> 07:42.120
Okay?

07:42.120 --> 07:46.270
So again, pathological bonding;
and in three weeks you'll

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understand this,
from first principles,

07:49.533 --> 07:52.283
but you've got to be patient.

07:52.279 --> 07:56.369
Okay, so Lewis pairs and octets
provide a pretty good

07:56.369 --> 08:00.379
bookkeeping device for keeping
track of valence,

08:00.379 --> 08:03.379
but they're hopelessly crude
when it comes to describing the

08:03.380 --> 08:06.480
actual electron distribution,
which you can see

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experimentally here.

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There is electron
sharing.

08:10.610 --> 08:14.750
There's a distortion of the
spheres of electron density that

08:14.747 --> 08:17.887
are the atoms,
but it's only about 5% as big

08:17.886 --> 08:21.306
as Lewis would've predicted,
had he predicted that two

08:21.310 --> 08:23.230
electrons would be right
between.

08:23.230 --> 08:24.330
Okay?

08:24.329 --> 08:27.969
And there are unshared pairs,
as Lewis predicted.

08:27.970 --> 08:31.830
And again they're less -- but
in this case they're even less

08:31.827 --> 08:34.637
than 5% of what Lewis would've
predicted.

08:34.639 --> 08:36.009
But you can see them.

08:36.009 --> 08:39.779
Now this raises the question,
is there a better bond theory

08:39.783 --> 08:42.333
than Lewis theory,
maybe even one that's

08:42.327 --> 08:44.537
quantitative,
that would give you numbers for

08:44.543 --> 08:46.933
these things,
rather than just say there are

08:46.928 --> 08:48.128
pairs here and there.

08:48.130 --> 08:48.590
Right?

08:48.590 --> 08:50.430
And the answer,
thank goodness,

08:50.428 --> 08:53.858
is yes, there is a great theory
for this, and what it is,

08:53.864 --> 08:55.954
is chemical quantum mechanics.

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Now you can study quantum
mechanics in this department,

08:58.327 --> 09:00.927
you can study quantum mechanics
in physics, you can probably

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study it other places,
right?

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And different people use the
same quantum mechanics but apply

09:06.636 --> 09:08.406
it to different problems.

09:08.410 --> 09:09.050
Right?

09:09.048 --> 09:14.408
So what we're going to discuss
in this course is quantum

09:14.407 --> 09:17.717
mechanics as applied to bonding.

09:17.720 --> 09:20.170
So it'll be somewhat different
in its flavor --

09:20.168 --> 09:22.428
in fact, a lot different in its
flavor --

09:22.428 --> 09:26.468
from what you do in physics,
or even what you do in physical

09:26.469 --> 09:29.219
chemistry,
because we're more interested

09:29.217 --> 09:32.737
-- we're not so interested in
getting numbers or solving

09:32.743 --> 09:36.063
mathematical problems,
we're interested in getting

09:36.059 --> 09:39.239
insight to what's really
involved in forming bonds.

09:39.240 --> 09:42.700
We want it to be rigorous but
we don't need it to be

09:42.703 --> 09:43.523
numerical.

09:43.519 --> 09:44.479
Okay?

09:44.480 --> 09:47.390
So it'll be much more pictorial
than numerical.

09:50.953 --> 09:54.133
that was discovered in,
or invented perhaps we should

09:54.133 --> 09:56.243
say, in --
I don't know whether -- it's

09:56.235 --> 09:59.085
hard to know whether to say
discovered or invented;

09:59.090 --> 10:03.460
I think invented is probably
better -- in 1926.

10:06.904 --> 10:10.804
the sort of ninety-seven-pound
weakling on the beach.

10:10.799 --> 10:11.549
Right?

10:11.548 --> 10:14.848
He's this guy back here with
the glasses on.

10:14.850 --> 10:15.390
Okay?

10:15.389 --> 10:19.139
He was actually a well-known
physicist but he hadn't done

10:19.143 --> 10:21.693
anything really earthshaking at
all.

10:21.690 --> 10:25.480
He was at the University of
Zurich.

10:25.480 --> 10:28.120
And Felix Bloch,
who was a student then --

10:28.120 --> 10:32.380
two years before he had come as
an undergraduate to the

10:32.384 --> 10:35.944
University of Zurich to study
engineering,

10:35.940 --> 10:38.330
and after a year and a half he
decided he would do physics,

10:38.330 --> 10:41.470
which was completely
impractical and not to the taste

10:41.465 --> 10:42.485
of his parents.

10:42.490 --> 10:46.490
But anyhow, as an undergraduate
he went to these colloquia that

10:46.493 --> 10:50.603
the Physics Department had,
and he wrote fifty years later

10:50.596 --> 10:54.406
-- see this was 1976,
so it's the 50th anniversary of

10:54.410 --> 10:56.690
the discovery of,
or invention,

10:56.692 --> 10:58.322
of quantum mechanics.

10:58.320 --> 11:00.240
So he said:
"At the end of a

11:00.240 --> 11:02.600
colloquium I heard
Debye"(there's a picture of

11:05.515 --> 11:10.015
you're not working right now on
very important problems anyway.

11:10.019 --> 11:14.119
Why don't you tell us something
about that thesis of de

11:14.115 --> 11:14.945
Broglie?'

11:17.842 --> 11:21.232
a beautifully clear account of
how de Broglie associated a wave

11:21.227 --> 11:22.317
with a particle.

11:22.320 --> 11:25.690
When he had finished,
Debye casually remarked that he

11:25.693 --> 11:28.943
thought this way of talking was
rather childish.

11:28.940 --> 11:32.230
He had learned that,
to deal properly with waves,

11:32.229 --> 11:34.559
one had to have a wave
equation.

11:34.558 --> 11:37.118
It sounded rather trivial and
did not seem to make a great

11:39.292 --> 11:41.712
thought a bit more about the
idea afterwards,

11:41.710 --> 11:44.240
and just a few weeks later he
gave another talk in the

11:44.240 --> 11:47.010
colloquium,
which he started by saying,

11:47.006 --> 11:51.356
'My colleague Debye suggested
that one should have a wave

11:51.356 --> 11:52.286
equation.

11:52.289 --> 11:57.169
Well I have found one.'"
And we write it now:

11:57.167 --> 11:58.867
HΨ=EΨ.

11:58.870 --> 12:01.290
He actually wrote it in
different terms,

12:04.528 --> 12:08.338
The reason the one we write is
a little different from his is

12:08.342 --> 12:10.952
he included time as a variable
in his,

12:10.950 --> 12:13.340
whereas we're not interested
in, for this purpose,

12:13.340 --> 12:15.000
in changes in time.

12:15.000 --> 12:18.330
We want to see how molecules
are when they're just sitting

12:18.331 --> 12:18.801
there.

12:18.799 --> 12:20.889
We'll talk about time later.

12:20.889 --> 12:23.829
So within, what?

12:27.682 --> 12:29.472
a good deal sharper.

12:29.472 --> 12:30.282
Right?

12:30.279 --> 12:32.509
And where is he standing?

12:32.509 --> 12:35.659
He's standing at the tram stop
in Stockholm,

12:35.663 --> 12:39.483
where he's going to pick up his
Nobel Prize for this.

12:39.476 --> 12:40.206
Right?

12:40.210 --> 12:45.140
And he's standing with Dirac,
with whom he shared the Nobel

12:45.144 --> 12:47.114
Prize,
and with Heisenberg,

12:47.106 --> 12:50.156
who got the Nobel Prize the
previous year but hadn't

12:50.162 --> 12:53.382
collected it yet,
so he came at the same time.

12:53.379 --> 12:54.939
Okay?

12:58.167 --> 13:00.347
HΨ=EΨ.

13:00.350 --> 13:03.230
H and E you've seen,
but Ψ may be new to

13:03.225 --> 13:03.565
you.

13:03.570 --> 13:04.470
It's a Greek letter.

13:04.470 --> 13:09.320
We can call it Sigh or P-sighi,
some people call it Psee.

13:09.317 --> 13:10.007
Right?

13:10.009 --> 13:12.119
I'll probably call it Psi.

13:12.120 --> 13:13.010
Okay.

13:13.009 --> 13:16.859
And it's a wave function.

13:16.860 --> 13:20.930
Well what in the world is a
wave function?

13:20.929 --> 13:21.599
Okay?

13:21.600 --> 13:26.250
So this is a stumbling block
for people that come into the

13:26.254 --> 13:28.594
field,
and it's not just a stumbling

13:28.594 --> 13:31.264
block for you,
it was a stumbling block for

13:31.258 --> 13:33.988
the greatest minds there were at
the time.

13:33.990 --> 13:36.750
So, for example,
this is five years later in

13:36.753 --> 13:38.953
Leipzig,
and it's the research group of

13:38.952 --> 13:40.792
Werner Heisenberg,
who's sitting there in the

13:40.793 --> 13:43.273
front, the guy that --
this was about the time he was

13:43.269 --> 13:45.979
being nominated or selected for
the Nobel Prize.

13:45.980 --> 13:46.500
Right?

13:46.500 --> 13:51.630
So he's there with his research
group, and right behind him is

13:51.634 --> 13:55.854
seated Felix Bloch,
who himself got the Nobel Prize

13:55.845 --> 13:58.535
for discovering NMR in 1952.

13:58.538 --> 14:01.908
So he's quite a young guy here,
and he's with these other --

14:01.908 --> 14:05.488
there's a guy who became famous
at Oxford and another one who

14:05.493 --> 14:08.543
became the head of the Physics
Department at MIT.

14:08.539 --> 14:10.519
Bloch was at Stanford.

14:10.519 --> 14:14.169
So these guys know they're
pretty hot stuff,

14:14.168 --> 14:16.138
so they're looking right into
the camera,

14:16.139 --> 14:18.939
to record themselves for
posterity,

14:18.940 --> 14:20.570
as part of this distinguished
group;

14:20.570 --> 14:22.130
except for Bloch.

14:22.129 --> 14:23.849
What's he thinking about?

14:23.850 --> 14:28.310
>

14:28.308 --> 14:30.798
What in the world is
Ψ?

14:30.797 --> 14:31.397
Right?

14:31.399 --> 14:33.279
Now, in fact,
in that same year,

14:36.438 --> 14:38.318
wave equation,
and Ψ.

14:38.321 --> 14:39.051
Right?

14:39.048 --> 14:43.868
And that summer these smart
guys, who were hanging around

14:43.870 --> 14:47.290
Zurich at that time,
theoretical physicists,

14:47.289 --> 14:49.919
the young guys went out on an
excursion,

14:49.918 --> 14:53.618
on the lake of Zurich,
and they made up doggerel

14:53.621 --> 14:58.351
rhymes for fun about different
things that were going on,

15:05.490 --> 15:07.400
whom we'll talk about next
semester,

15:07.399 --> 15:08.939
was about Ψ.

15:08.940 --> 15:12.010
"Gar Manches rechnet
Erwin schon, Mit seiner

15:12.006 --> 15:12.966
Wellenfuction.

15:16.779 --> 15:18.719
vorstell'n soll."

15:18.720 --> 15:22.960
Which means:
"Erwin with his Psi can do

15:22.958 --> 15:25.718
calculations,
quite a few.

15:25.720 --> 15:30.040
We only wish that we could
glean an inkling of what Psi

15:30.038 --> 15:31.398
could mean."

15:31.399 --> 15:32.119
Right?

15:32.120 --> 15:35.560
You can do calculations with
it, but what is it?

15:35.559 --> 15:36.929
-- was the question.

15:36.929 --> 15:37.959
Okay?

15:37.960 --> 15:42.010
And it wasn't just these young
guys who were confused.

15:46.345 --> 15:48.675
Ψ really means.

15:48.678 --> 15:51.138
Now if we're lucky,
this'll play this time.

15:54.000 --> 15:56.860
"What is Matter",
from 1952.

15:56.860 --> 15:59.130
<<Short film clip is
played>>

16:02.256 --> 16:05.376
einmal den Sancho Panza,
sein liebes Eselchen auf dem er

16:11.130 --> 16:13.640
vergessen und das gute Tier ist
wieder da.

16:13.639 --> 16:18.169
Nun werden sie mich vielleicht
zuletzt fragen,

16:18.174 --> 16:23.324
ja was sind denn nun aber
wirklich diese Korpuskeln,

16:31.630 --> 16:35.250
wo Sancho Panzas zweites
Eselchen hergekommen ist."

16:41.042 --> 16:43.712
really know what Ψ
was.

16:43.710 --> 16:44.470
Okay?

16:44.470 --> 16:49.040
So don't be depressed when it
seems a little curious what

16:49.041 --> 16:51.001
Ψ might be.

16:51.000 --> 16:51.730
Okay?

16:53.922 --> 16:56.932
other guys --
first we'll learn how to find

16:56.931 --> 17:01.211
Ψ and use it,
and then later we'll learn what

17:01.210 --> 17:01.910
it means.

17:01.909 --> 17:04.529
Okay?

17:04.528 --> 17:07.778
So Ψ is a function,
a wave function.

17:07.778 --> 17:12.368
What do you want to know,
from what I've shown here?

17:12.369 --> 17:14.089
What is a function?

17:14.089 --> 17:16.029
Student:  A relationship.

17:16.028 --> 17:17.588
Prof: Like sine is a
function;

17:17.589 --> 17:19.909
what does that mean?

17:19.910 --> 17:21.430
Yes?

17:21.430 --> 17:24.020
I can't hear very well.

17:24.019 --> 17:26.399
Student:  You put an
input into a function and you

17:26.400 --> 17:27.080
get an output.

17:27.078 --> 17:29.308
Prof: Yes,
it's like a little machine.

17:29.308 --> 17:32.448
You put a number in,
or maybe several numbers,

17:32.453 --> 17:34.063
and a number comes out.

17:34.060 --> 17:34.690
Right?

17:34.690 --> 17:40.600
That's what the function does;
okay, you put in ninety degrees

17:40.599 --> 17:42.679
and sin says one.

17:42.680 --> 17:43.830
Okay?

17:43.828 --> 17:47.258
So what do you want to know
about Ψ

17:47.263 --> 17:48.273
first?

17:48.269 --> 17:50.909
Student:  What does it
do?

17:50.910 --> 17:53.270
Prof: What's it a
function of?

17:53.269 --> 17:57.219
What are the things you have to
put in, in order to get a number

17:57.222 --> 17:57.602
out?

17:57.599 --> 17:59.829
Okay?

17:59.829 --> 18:03.079
So it's different from the name.

18:03.079 --> 18:06.549
The wave functions have names.

18:06.548 --> 18:09.398
That's not what they're a
function of.

18:09.403 --> 18:10.023
Right?

18:10.019 --> 18:13.689
You can have sine,
sine^2, cosine.

18:13.690 --> 18:17.550
Those are different functions,
but they can be functions of

18:17.548 --> 18:19.138
the same thing,
an angle.

18:19.144 --> 18:19.814
Right?

18:19.808 --> 18:22.958
So we're interested in what's
it a function of;

18:22.960 --> 18:26.210
not what the function is but
what's it a function of?

18:26.210 --> 18:28.640
So you can have different
Ψs.

18:28.640 --> 18:34.880
They have names and quantum
numbers give them their names.

18:34.880 --> 18:36.360
For example,
you can have n,

18:36.355 --> 18:37.435
l, and m.

18:37.440 --> 18:39.480
You've seen those before,
n, l,

18:39.478 --> 18:41.098
m, to name wave
functions.

18:41.099 --> 18:42.329
Those are just their names.

18:42.328 --> 18:44.058
It's not what they're a
function of.

18:44.058 --> 18:46.908
Or you can have 1s or
3d_xy,

18:46.906 --> 18:49.126
or σ, or π,
or π*.

18:49.130 --> 18:51.460
Those are all names of
functions.

18:51.458 --> 18:52.038
Right?

18:52.038 --> 18:55.568
But they're not what it's a
function of.

18:55.568 --> 19:01.538
What it's a function of is the
position of a particle,

19:01.544 --> 19:04.254
or a set of particles.

19:04.250 --> 19:09.120
It's a function of position,
and it's also a function of

19:09.122 --> 19:13.522
time, and sometimes of spin;
some particles have spin and it

19:13.522 --> 19:15.382
could be a function of that too.

19:15.380 --> 19:18.890
But you'll be happy to know
that for purposes of this course

19:18.891 --> 19:21.451
we're not so interested in time
and spin.

19:21.450 --> 19:26.940
So for our purposes it's just a
function of position.

19:26.940 --> 19:29.980
So if you have N
particles, how many positions do

19:29.980 --> 19:32.690
you have to specify to know
where they all are?

19:32.690 --> 19:34.970
How many numbers do you need?

19:34.970 --> 19:37.370
You need x,
y, z for every

19:37.369 --> 19:37.839
particle.

19:37.839 --> 19:38.309
Right?

19:38.308 --> 19:42.808
So you need 3N arguments
for Ψ.

19:42.808 --> 19:47.338
So Ψ is a function
that when you tell it where all

19:47.340 --> 19:50.740
these positions are,
it gives you a number.

19:50.740 --> 19:53.830
Now curiously enough,
the number can be positive,

19:53.828 --> 19:56.188
it can be zero,
it can be negative,

19:56.190 --> 20:00.170
it can even be complex,
right, although we won't talk

20:00.165 --> 20:02.685
about cases where it's complex.

20:02.690 --> 20:06.680
The physicists will tell you
about those, or physical

20:06.683 --> 20:07.533
chemists.

20:07.529 --> 20:08.129
Okay?

20:08.130 --> 20:11.880
And sometimes it can be as many
as 4N+1 arguments.

20:11.880 --> 20:14.060
How could it be 4N+1?

20:14.058 --> 20:16.028
Student:
>

20:16.028 --> 20:18.258
Prof: Because if each
particle also had a spin,

20:18.259 --> 20:20.659
then it would be x,
y, z and spin;

20:20.660 --> 20:22.030
that'd be four.

20:22.028 --> 20:25.888
And if time is also included,
it's plus one.

20:25.890 --> 20:29.050
Okay, so how are we going to go
through this?

20:29.048 --> 20:32.228
First we'll try -- this is
unfamiliar territory,

20:32.233 --> 20:34.203
I bet, to every one of you.

20:34.200 --> 20:35.090
Okay?

20:35.088 --> 20:39.048
So first we're going to talk
about just one particle and one

20:39.045 --> 20:42.125
dimension, so the function is
fairly simple.

20:42.130 --> 20:42.990
Okay?

20:42.990 --> 20:46.810
And then we'll go on to three
dimensions, but still one

20:46.806 --> 20:49.346
particle, the electron in an
atom;

20:49.348 --> 20:52.538
so a one-electron atom,
but now three dimensions,

20:52.542 --> 20:54.342
so it's more complicated.

20:54.338 --> 20:58.488
Then we'll go on to atoms that
have several electrons.

20:58.490 --> 21:01.890
So you have now more than three
variables, because you have at

21:01.892 --> 21:04.702
least two electrons;
that would be six variables

21:04.704 --> 21:08.234
that you have to put into the
function to get a number out.

21:08.230 --> 21:12.650
Then we'll go into molecules --
that is, more than one atom --

21:12.645 --> 21:14.235
and what bonding is.

21:14.240 --> 21:17.980
And then finally we get to the
payoff for organic chemistry,

21:17.980 --> 21:21.130
which is talking about what
makes a group a functional group

21:21.134 --> 21:23.384
and what does it mean to be
functional,

21:23.380 --> 21:24.550
what makes it reactive?

21:24.549 --> 21:26.019
That's where we're heading.

21:26.019 --> 21:29.739
But first we have to understand
what quantum mechanics is.

21:31.380 --> 21:34.590
ΗΨ=EΨ,
and we're talking about the

21:36.710 --> 21:41.500
so time is not a variable,
and that means what we're

21:41.501 --> 21:44.981
talking about is stationary
states.

21:44.980 --> 21:47.750
We don't mean that the atoms
aren't moving,

21:47.748 --> 21:51.368
but just that they're in a
cloud and we're going to find

21:51.374 --> 21:53.554
how is the cloud distributed.

21:53.548 --> 21:57.488
If a molecule reacts,
the electrons shift their

21:57.491 --> 22:00.151
clouds and so on;
it changes.

22:00.150 --> 22:02.340
We're not interested in
reaction now,

22:02.339 --> 22:05.629
we're just interested in
understanding the cloud that's

22:05.625 --> 22:08.055
sitting there,
not changing in time.

22:08.058 --> 22:12.358
Okay, now the right part of the
equation is E times

22:12.359 --> 22:14.019
Ψ,
right?

22:14.019 --> 22:16.969
And E will turn out to
be the energy of the system;

22:16.970 --> 22:19.050
maybe you won't be surprised at
that.

22:19.049 --> 22:20.679
So that's quite simple.

22:20.680 --> 22:22.710
What's the left?

22:22.710 --> 22:25.510
It looks like H times
Ψ.

22:25.509 --> 22:28.299
If that were true,
what could you do to simplify

22:28.304 --> 22:28.844
things?

22:28.839 --> 22:31.149
Knock out Ψ.

22:31.150 --> 22:36.680
But HΨ is not
H times Ψ.

22:36.680 --> 22:39.430
H is sort of recipe for
doing something with

22:39.425 --> 22:41.795
Ψ;
we'll see that right away.

22:41.798 --> 22:45.858
So you can't just cancel out
the Ψ,

22:45.864 --> 22:47.314
unfortunately.

22:47.308 --> 22:50.008
Okay, so
ΗΨ=EΨ.

22:50.009 --> 22:52.919
Oops sorry, what did I do,
there we go.

22:52.920 --> 22:56.070
Now we can divide,
you can divide the right by

22:56.068 --> 22:58.178
Ψ,
and since it was E times

22:58.180 --> 22:59.970
Ψ,
the Ψ goes away.

22:59.970 --> 23:04.980
But when you divide on the
left, you don't cancel the

23:04.976 --> 23:09.016
Ψs,
because the top doesn't mean

23:09.019 --> 23:10.849
multiplication.

23:10.848 --> 23:14.758
Now I already told you the
right side of this equation is

23:14.761 --> 23:16.091
the total energy.

23:16.088 --> 23:19.568
So when you see a system,
what does the total energy

23:19.571 --> 23:20.461
consist of?

23:20.460 --> 23:25.450
Potential energy and kinetic
energy.

23:25.450 --> 23:29.230
So somehow this part on the
left, ΗΨ/Ψ,

23:29.226 --> 23:32.336
must be kinetic energy plus
potential energy.

23:32.338 --> 23:35.588
That recipe,
H, must somehow tell you

23:35.592 --> 23:40.512
how to work with Ψ in
order to get something which,

23:40.509 --> 23:43.669
divided by Ψ,
gives kinetic energy plus

23:43.673 --> 23:44.903
potential energy.

23:44.900 --> 23:46.190
So there are two parts of it.

23:46.190 --> 23:48.640
There's the part that's
potential energy,

23:48.636 --> 23:51.146
of the recipe,
and there's the part that's

23:51.146 --> 23:52.306
kinetic energy.

23:52.308 --> 23:58.548
Now, the potential energy part
is in fact easy because it's

23:58.551 --> 23:59.951
given to you.

23:59.951 --> 24:00.921
Right?

24:00.920 --> 24:04.270
What's Ψ a function
of?

24:04.269 --> 24:05.289
Students:  Position.

24:05.288 --> 24:06.748
Prof: Position of the
particles.

24:06.750 --> 24:08.930
Now if you know the charges of
the particles,

24:08.930 --> 24:12.280
and their positions,
and know Couloumb's Law,

24:12.278 --> 24:14.378
then you know the potential
energy,

24:14.380 --> 24:16.360
if Couloumb's Law is right.

24:16.359 --> 24:17.559
Is everybody with me on that?

24:17.558 --> 24:19.688
If you know there's a unit
positive charge here,

24:19.690 --> 24:22.390
a unit negative charge here,
another unit positive charge

24:22.390 --> 24:24.320
here and a unit negative charge
here,

24:24.318 --> 24:27.678
or something like that,
you -- it might be complicated,

24:27.680 --> 24:30.110
you might have to write an
Excel program or something to do

24:30.106 --> 24:32.156
it --
but you could calculate the

24:32.156 --> 24:34.626
distances and the charges and so
on,

24:34.630 --> 24:36.870
and what the energy is,
due to that.

24:36.868 --> 24:40.558
So that part is really given to
you, once you know what system

24:40.557 --> 24:43.337
you're dealing with,
the recipe for finding the

24:43.339 --> 24:44.609
potential energy.

24:44.608 --> 24:48.058
So that part of
HΨ/Ψ is no

24:48.061 --> 24:49.531
problem at all.

24:49.529 --> 24:52.439
But hold your breath on kinetic
energy.

24:52.440 --> 24:52.970
Sam?

24:52.970 --> 24:56.330
Student:  Didn't we just
throw away an equation?

24:56.328 --> 25:00.278
There was an adjusted
Couloumb's Law equation.

25:00.278 --> 25:01.758
Prof: Yes,
that was wrong.

25:01.759 --> 25:04.109
That was three years earlier,
remember?

25:04.109 --> 25:07.289
1923 Thomson proposed that.

25:07.289 --> 25:08.869
But it was wrong.

25:08.869 --> 25:11.079
This is what was right.

25:11.078 --> 25:12.598
Student:  How did they
prove it wrong?

25:12.599 --> 25:13.579
Prof: How did what?

25:13.578 --> 25:14.768
Student: Did they prove
it wrong or just --

25:14.769 --> 25:17.199
Prof: Yes,
they proved this right,

25:17.204 --> 25:21.004
that Couloumb's Law held,
because it agreed with a whole

25:20.997 --> 25:24.157
lot of spectroscopic evidence
that had been collected about

25:24.162 --> 25:26.762
atomic spectra,
and then everything else that's

25:26.760 --> 25:28.020
tried with it works too.

25:28.019 --> 25:30.839
So we believe it now.

25:30.838 --> 25:32.898
So how do you handle kinetic
energy?

25:32.900 --> 25:35.050
Well that's an old one,
you did that already in high

25:35.046 --> 25:35.716
school, right?

25:35.720 --> 25:37.580
Forget kinetic energy,
here it is.

25:37.578 --> 25:42.158
It's some constant,
which will get the units right,

25:42.160 --> 25:44.500
depending on what energy units
you want,

25:44.500 --> 25:48.510
times the sum over all the
particles of the kinetic energy

25:48.505 --> 25:49.835
of each particle.

25:49.838 --> 25:51.678
So if you know the kinetic
energy of this particle,

25:51.682 --> 25:53.382
kinetic energy of this
particle, this particle,

25:53.375 --> 25:55.275
this particle;
you add them all up and you get

25:55.279 --> 25:56.319
the total kinetic energy.

25:56.319 --> 25:58.129
No problem there, right?

25:58.130 --> 26:01.260
Now what is the kinetic energy
that you're summing up over each

26:01.255 --> 26:01.805
particle?

26:03.880 --> 26:07.100
Has everybody seen that before?

26:07.098 --> 26:11.458
Okay, so that's the sum of
classical kinetic energy over

26:11.457 --> 26:15.257
all the particles of interest in
the problem,

26:15.259 --> 26:18.139
and the constant is just some
number you put in to get the

26:18.142 --> 26:21.402
right units for your energy,
depending on whether you use

26:21.396 --> 26:24.346
feet per second or meters per
year or whatever,

26:24.349 --> 26:26.139
for the velocity.

26:26.140 --> 26:31.160
Okay, but it turned out that
although this was fine for our

26:31.159 --> 26:35.309
great-grandparents,
it's not right when you start

26:35.314 --> 26:37.484
dealing with tiny things.

26:37.477 --> 26:38.427
Right?

26:38.430 --> 26:43.330
Here's what kinetic energy
really is.

26:43.329 --> 26:44.579
It's a constant.

26:44.578 --> 26:48.418
This is the thing that gets it
in the right units:

26:48.415 --> 26:53.265
(h^2)/8(π^2) times a
sum over all the particles --

26:53.269 --> 26:57.979
it's looking promising, right?

26:57.980 --> 27:02.450
-- of one over the mass -- not
the mass mv^2,

27:02.450 --> 27:06.950
but one over the mass of each
particle --

27:06.950 --> 27:12.040
and here's where we get it --
>

27:12.038 --> 27:21.168
-- times second derivatives of
a wave function.

27:21.170 --> 27:24.340
That's weird.

27:24.338 --> 27:27.348
I mean, at least it has twos in
it, like v^2,

27:27.347 --> 27:27.817
right?

27:27.819 --> 27:29.099
>

27:29.099 --> 27:30.769
That's something.

27:30.769 --> 27:33.999
And in fact it's not completely
coincidental that it has twos in

27:34.002 --> 27:34.262
it.

27:34.259 --> 27:38.279
There was an analogy that was
being followed that allowed them

27:38.280 --> 27:39.600
to formulate this.

27:39.598 --> 27:44.698
And you divide it by the number
Ψ.

27:44.700 --> 27:47.910
So that's a pretty complicated
thing.

27:47.910 --> 27:51.490
So if we want to get our heads
around it, we'd better simplify

27:51.486 --> 27:51.776
it.

27:51.779 --> 27:55.309
And oh also there's a minus
sign;

27:55.308 --> 27:59.448
it's minus, the constant is
negative that you use.

27:59.450 --> 28:03.360
Okay, now let's simplify it by
using just one particle,

28:03.358 --> 28:05.818
so we don't have to sum over a
bunch of particles,

28:05.818 --> 28:08.368
and we'll use just one
dimension, x;

28:08.369 --> 28:10.099
forget y and z.

28:10.099 --> 28:11.539
Okay?

28:11.539 --> 28:13.429
So now we see something simpler.

28:13.430 --> 28:18.390
So it's a negative constant
times one over the mass of the

28:18.385 --> 28:21.315
particle,
times the second derivative of

28:21.316 --> 28:23.906
the function,
the wave function,

28:23.909 --> 28:26.199
divided by Ψ.

28:31.539 --> 28:32.429
Okay?

28:32.430 --> 28:34.080
Or here it is,
written just a little

28:34.078 --> 28:34.738
differently.

28:34.740 --> 28:37.720
So there's a constant,
C, over the mass,

28:37.721 --> 28:38.241
right?

28:38.240 --> 28:42.120
And then we have the important
part, is the second derivative.

28:42.118 --> 28:45.268
Does everybody know that the
second derivative is a curvature

28:45.270 --> 28:46.010
of a function.

28:46.007 --> 28:46.477
Right?

28:46.480 --> 28:47.920
What's the first derivative?

28:47.920 --> 28:48.780
Students:  Slope.

28:48.779 --> 28:50.249
Prof: Slope,
and the second derivative is

28:50.246 --> 28:50.836
how curved it is.

28:50.838 --> 28:54.368
It can be curving down,
that's negative curvature;

28:54.368 --> 28:57.128
or curving up,
that's positive curvature.

28:57.130 --> 28:58.380
So it can be positive or
negative;

28:58.380 --> 29:01.070
it can be zero if the line is
straight.

29:01.069 --> 29:02.809
Okay.

29:02.808 --> 29:07.158
So note that the kinetic energy
involves the shape of

29:07.159 --> 29:09.669
Ψ,
how curved it is,

29:09.667 --> 29:13.597
not just what the value of
Ψ is;

29:13.599 --> 29:18.439
although it involves that too.

29:18.440 --> 29:22.100
Maybe it's not too early to
point out something interesting

29:22.098 --> 29:22.918
about this.

29:22.920 --> 29:26.300
So suppose that's the kinetic
energy.

29:26.298 --> 29:32.518
What would happen if you
multiplied Ψ by two?

29:32.519 --> 29:35.859
Obviously the denominator would
get twice as large,

29:35.862 --> 29:38.672
if you made Ψ twice
as large.

29:38.670 --> 29:40.410
What would happen to the
curvature?

29:40.410 --> 29:42.210
What happens to the slope?

29:42.210 --> 29:44.930
Suppose you have a function and
you make it twice as big and

29:44.932 --> 29:46.872
look at the slope at a
particular point?

29:46.868 --> 29:50.248
How does the slope change if
you've stretched the paper on

29:50.251 --> 29:51.441
which you drew it?

29:51.440 --> 29:52.780
Student: It's sharper.

29:52.779 --> 29:55.069
Prof: The slope will
double, right,

29:55.074 --> 29:56.534
if you double the scale.

29:56.529 --> 30:01.719
How about the curvature,
the second derivative?

30:01.720 --> 30:06.470
Does it go up by four times?

30:06.470 --> 30:11.420
No it doesn't go up by four
times, it goes up by twice.

30:11.420 --> 30:16.000
So what would happen to the
kinetic energy there if we

30:15.999 --> 30:20.059
doubled the size of Ψ
every place?

30:20.059 --> 30:20.769
Student:  Stay the same.

30:20.769 --> 30:22.179
Prof: It would stay the
same.

30:22.180 --> 30:25.830
The kinetic energy doesn't
depend on how you scale

30:25.827 --> 30:28.957
Ψ,
it only depends on its shape,

30:28.955 --> 30:30.515
how curved it is.

30:30.519 --> 30:32.739
Everybody see the idea?

30:32.740 --> 30:36.340
Curvature divided by the value.

30:36.338 --> 30:39.358
Okay, now solving a quantum
problem.

30:39.358 --> 30:42.748
So if you're in a course and
you're studying quantum

30:42.753 --> 30:45.353
mechanics, you get problems to
solve.

30:45.348 --> 30:50.678
A problem means you're given
something, you have to find

30:50.676 --> 30:51.836
something.

30:51.838 --> 30:56.168
You're given a set of
particles, like a certain nuclei

30:56.172 --> 31:00.102
of given mass and charge,
and a certain number of

31:00.095 --> 31:04.225
electrons;
that's what you're given.

31:04.230 --> 31:07.580
Okay, the masses of the
particles and the potential law.

31:07.578 --> 31:11.038
When you're given the charge,
and you know Couloumb's Law,

31:11.040 --> 31:14.320
then you know how to calculate
the potential energy;

31:14.319 --> 31:15.949
remember that's part of it.

31:15.950 --> 31:16.930
Okay?

31:16.930 --> 31:19.490
So that part's easy, okay?

31:19.490 --> 31:23.680
Now what do you need to find if
you have a problem to solve?

31:23.680 --> 31:26.330
Oh, for example,
you can have one particle in

31:26.334 --> 31:29.064
one dimension;
so it could be one atomic mass

31:29.061 --> 31:32.351
unit is the weight of the
particle, and Hooke's Law could

31:32.353 --> 31:33.943
be the potential,
right?

31:33.940 --> 31:38.010
It doesn't have to be
realistic, it could be Hooke's

31:38.010 --> 31:42.320
Law, it could be a particle held
by a spring, to find a

31:42.321 --> 31:43.681
Ψ.

31:43.680 --> 31:47.000
You want to find the shape of
this function,

31:46.996 --> 31:49.306
which is a function of what?

31:49.308 --> 31:50.668
Student:  Positions of
the particles.

31:50.670 --> 31:54.660
Prof: Positions of the
particles, and if you're higher,

31:54.659 --> 31:57.209
further on than we are,
time as well;

31:57.210 --> 31:58.760
maybe spin even.

31:58.759 --> 32:04.319
But that function has to be
such that HΨ/Ψ is

32:04.323 --> 32:08.293
the total energy,
and the total energy is the

32:08.292 --> 32:11.382
same, no matter where the
particle is,

32:11.380 --> 32:14.390
right, because the potential
energy and the kinetic energy

32:14.392 --> 32:15.082
cancel out.

32:15.078 --> 32:16.998
It's like a ball rolling back
and forth.

32:17.000 --> 32:20.090
The total energy is constant
but it goes back and forth

32:20.094 --> 32:22.734
between potential and kinetic
energy, right?

32:22.730 --> 32:23.840
Same thing here.

32:23.838 --> 32:26.628
No matter where the particles
are, you have to get the same

32:26.626 --> 32:27.056
energy.

32:27.058 --> 32:31.368
So Ψ has to be such
that when you calculate the

32:31.368 --> 32:34.868
kinetic energy for it,
changes in that kinetic energy,

32:34.869 --> 32:38.539
in different positions,
exactly compensate for the

32:38.538 --> 32:41.178
changes in potential energy.

32:41.180 --> 32:45.950
When you've got that,
then you've got a correct

32:45.954 --> 32:48.244
Ψ;
maybe.

32:48.240 --> 32:51.900
It's also important that
Ψ remain finite,

32:51.904 --> 32:53.994
that it not go to infinity.

32:53.990 --> 32:55.890
And if you're a real
mathematician,

32:55.888 --> 32:59.048
it has to be single valued;
you can't have two values for

32:59.048 --> 33:00.108
the same positions.

33:00.108 --> 33:03.248
It has to be continuous,
you can't get a sudden break in

33:03.251 --> 33:04.111
Ψ.

33:04.108 --> 33:08.838
And Ψ^2^( )has to be
integrable;

33:08.838 --> 33:12.628
you have to be able to tell how
much area is under

33:12.634 --> 33:16.124
Ψ^2,
and you'll see why shortly.

33:16.118 --> 33:20.388
But basically what you need is
to find a Ψ such that

33:20.394 --> 33:24.534
the changes in kinetic energy
compensate changes in potential

33:24.530 --> 33:25.290
energy.

33:25.289 --> 33:27.139
Now what's coming?

33:27.140 --> 33:29.740
Let's just rehearse what we did
before.

33:29.740 --> 33:33.120
So first there'll be one
particle in one dimension;

33:33.118 --> 33:37.328
then it'll be one-electron
atoms, so one particle in three

33:37.328 --> 33:40.648
dimensions;
then it will be many electrons

33:40.647 --> 33:43.397
and the idea of what orbitals
are;

33:43.400 --> 33:45.440
and then it'll be molecules and
bonds;

33:45.440 --> 33:49.050
and finally functional groups
and reactivity.

33:49.048 --> 33:53.268
Okay, but you'll be happy to
hear that by a week from Friday

33:53.270 --> 33:56.420
we'll only get through
one-electron atoms.

33:56.420 --> 33:58.990
So don't worry about the rest
of the stuff now.

33:58.990 --> 34:03.610
But do read the parts on the
webpage that have to do with

34:03.607 --> 34:06.327
what's going to be on the exam.

34:06.328 --> 34:09.228
Okay, so normally you're given
a problem,

34:09.230 --> 34:11.400
the mass and the charges --
that is,

34:11.400 --> 34:14.290
that potential energy as a
function of position --

34:14.289 --> 34:18.009
and you need to find
Ψ.

34:18.010 --> 34:21.250
But at first we're going to try
it a different way.

34:21.250 --> 34:25.810
We're going to play Jeopardy
and we're going to start with

34:25.809 --> 34:29.649
the answer and find out what the
question was.

34:29.650 --> 34:31.090
Okay?

34:31.090 --> 34:36.540
So suppose that Ψ is
the sine of x;

34:36.539 --> 34:41.449
this is one particle in one
dimension, the position of the

34:41.447 --> 34:46.007
particle, and the function of
Ψ is sine.

34:46.010 --> 34:48.720
If you know Ψ,
what can you figure out?

34:48.719 --> 34:50.639
We've just been talking about
it.

34:50.639 --> 34:53.029
What can you use Ψ
to find?

34:53.030 --> 34:54.180
Student:  Kinetic energy.

34:54.179 --> 34:54.779
Prof: Kinetic energy.

34:54.780 --> 34:58.620
How do you find it?

34:58.619 --> 35:01.519
So we can get the kinetic
energy, which is minus a

35:01.516 --> 35:04.886
constant over the mass times the
curvature of Ψ

35:04.887 --> 35:07.547
divided by Ψ at any
given position.

35:07.550 --> 35:08.260
Right?

35:08.260 --> 35:12.320
And once we know how the
kinetic energy varies with

35:12.315 --> 35:14.845
position,
then we know how the potential

35:14.849 --> 35:18.319
energy varies with position,
because it's just the opposite,

35:18.320 --> 35:20.360
in order that the sum be
constant.

35:20.355 --> 35:20.905
Right?

35:20.909 --> 35:23.729
So once we know the kinetic
energy, then we know what the

35:23.726 --> 35:26.286
potential energy was,
which was what the problem was

35:26.293 --> 35:27.353
at the beginning.

35:27.349 --> 35:29.059
Okay?

35:29.059 --> 35:33.789
So suppose our answer is
sin(x).

35:33.789 --> 35:36.169
What is the curvature of
sin(x),

35:36.173 --> 35:37.683
the second derivative?

35:37.679 --> 35:39.149
<<Students speak over
one another>>

35:40.713 --> 35:41.993
-sin(x).

35:41.989 --> 35:44.809
Okay, so what is the kinetic
energy?

35:44.809 --> 35:46.239
Student:  C/m.

35:46.239 --> 35:47.609
Prof: C/m.

35:47.610 --> 35:52.140
Does it depend on where you
are, on the value of x?

35:52.139 --> 35:54.459
No, it's always C/m.

35:54.460 --> 35:57.200
So what was the potential
energy?

35:57.199 --> 36:00.239
How did the potential energy
vary with position?

36:00.239 --> 36:02.109
Student:  It doesn't.

36:02.110 --> 36:04.820
Prof: The potential
energy doesn't vary with

36:04.822 --> 36:05.422
position.

36:05.420 --> 36:10.490
So sin(x) is a solution
for what?

36:10.489 --> 36:15.939
A particle that's not being
influenced by anything else;

36:15.940 --> 36:19.170
so its potential energy doesn't
change with the position,

36:19.172 --> 36:21.022
it's a particle in free space.

36:21.019 --> 36:25.539
Okay?

36:25.539 --> 36:28.329
So the potential energy is
independent of x.

36:28.329 --> 36:31.779
Constant potential energy,
it's a particle in free space.

36:31.780 --> 36:34.960
Now, suppose we take a
different one,

36:34.958 --> 36:36.458
sin(ax).

36:36.460 --> 36:41.260
How does sin(ax) look
different from sin(x)?

36:41.260 --> 36:44.220
Suppose it's sin(2x).

36:44.219 --> 36:47.419
Here's sin(x).

36:47.420 --> 36:52.450
How does sin(2x) look?

36:52.445 --> 36:53.655
Right?

36:53.659 --> 36:56.779
It's shorter wavelength.

36:56.780 --> 36:58.430
Okay?

36:58.429 --> 37:04.599
Now so we need to figure out --
so it's a shortened wave,

37:04.597 --> 37:06.577
if a*>*1.

37:06.579 --> 37:11.159
Okay, now what's the curvature?

37:11.159 --> 37:13.629
Russell?

37:13.630 --> 37:15.080
Student:  It's -a^2
times sin(ax).

37:15.079 --> 37:18.359
Prof: It's -a^2
times sin(ax).

37:18.355 --> 37:18.875
Right?

37:18.880 --> 37:21.110
The a comes out,
that constant,

37:21.105 --> 37:23.145
each time you take a
derivative.

37:23.150 --> 37:30.250
So now what does the kinetic
energy look like?

37:30.250 --> 37:34.140
It's a^2^( )times the
same thing.

37:34.139 --> 37:34.959
Okay?

37:34.960 --> 37:39.120
So again, the potential energy
is constant.

37:39.117 --> 37:39.907
Right?

37:39.909 --> 37:42.619
It doesn't change with position.

37:42.619 --> 37:46.809
But what is different?

37:46.809 --> 37:50.269
It has higher kinetic energy if
it's a shorter wavelength.

37:50.268 --> 37:55.158
And notice that the kinetic
energy is proportional to one

37:55.155 --> 37:58.465
over the wavelength squared,
right?;

37:58.469 --> 38:01.989
a^2;
a shortens the wave,

38:01.985 --> 38:05.035
it's proportional to
a^2, one over the

38:05.041 --> 38:06.571
wavelength squared.

38:06.570 --> 38:07.830
Okay.

38:07.829 --> 38:10.949
Now let's take another
function, exponential,

38:10.954 --> 38:12.094
so e^x.

38:12.090 --> 38:19.430
What's the second derivative of
e^x?

38:19.429 --> 38:20.239
Pardon me?

38:20.239 --> 38:21.159
Student: e^x.

38:21.159 --> 38:21.889
Prof: e^x.

38:21.889 --> 38:24.249
What's the 18th derivative of
e^x?

38:24.250 --> 38:25.080
Students:  e^x.

38:25.079 --> 38:26.209
Prof: Okay, good.

38:26.210 --> 38:27.150
So it's e^x.

38:27.150 --> 38:31.210
So what's this situation,
what's the kinetic energy?

38:31.210 --> 38:38.040
Student:  -C/m.
Prof: -C/m.

38:38.039 --> 38:40.369
Negative kinetic energy.

38:40.369 --> 38:44.569
Your great-grandparents didn't
get that.

38:44.570 --> 38:49.740
You can have kinetic energy
that's less than zero.

38:49.739 --> 38:51.219
What does that mean?

38:51.219 --> 39:00.489
It means the total energy is
lower than the potential energy.

39:00.489 --> 39:04.279
Pause a minute just to let that
sink in.

39:04.280 --> 39:10.910
The total energy is lower than
the potential energy.

39:10.909 --> 39:13.689
The difference is negative.

39:13.690 --> 39:14.910
Okay?

39:14.909 --> 39:17.629
So the kinetic energy,
if that's the difference,

39:17.625 --> 39:20.105
between potential and total,
is negative.

39:22.949 --> 39:24.209
Yes?

39:24.210 --> 39:26.700
Student:  Does this
violate that Ψ has to

39:26.702 --> 39:27.392
remain finite?

39:27.389 --> 39:29.039
Prof: Does it violate
what?

39:29.039 --> 39:30.909
Student:  That
Ψ has to remain

39:30.911 --> 39:31.271
finite?

39:31.269 --> 39:33.179
Prof: No.

39:33.179 --> 39:34.839
You'll see in a second.

39:34.840 --> 39:38.850
Okay, so anyhow the constant
potential energy is greater than

39:38.849 --> 39:40.719
the total energy for that.

39:40.719 --> 39:46.569
Now, how about if it were minus
exponential, e^-x?

39:46.570 --> 39:48.250
Now what would it be?

39:48.250 --> 39:51.970
It would be the same deal
again, it would still be

39:51.972 --> 39:56.612
-C/m, and again it would
be a constant potential energy

39:56.606 --> 39:59.186
greater than the total energy.

39:59.190 --> 40:05.750
This is not just a mathematical
curiosity, it actually happens

40:05.751 --> 40:09.411
for every atom in you,
or in me.

40:09.409 --> 40:13.909
Every atom has the electrons
spend some of their time in

40:13.914 --> 40:18.014
regions where they have negative
kinetic energy.

40:18.010 --> 40:23.110
It's not just something weird
that never happens.

40:23.110 --> 40:27.650
And it happens at large
distance from the nuclei where

40:27.648 --> 40:32.868
1/r -- that's Couloumb's
Law -- where it stops changing

40:32.871 --> 40:34.071
very much.

40:34.070 --> 40:37.860
When you get far enough,
1/r gets really tiny and

40:37.860 --> 40:41.170
it's essentially zero,
it doesn't change anymore.

40:41.170 --> 40:41.860
Right?

40:41.860 --> 40:46.500
Then you have this situation in
any real atom.

40:46.500 --> 40:53.500
So let's look at getting the
potential energy from the shape

40:53.503 --> 40:58.373
of Ψ via the kinetic
energy.

40:58.369 --> 41:00.759
Okay, so here's a map of
Ψ,

41:00.760 --> 41:02.730
or a plot of Ψ,it
could be positive,

41:02.730 --> 41:05.860
negative, zero -- as a function
of the one-dimension x,

41:05.860 --> 41:07.860
wherever the particle is.

41:07.860 --> 41:08.720
Okay?

41:08.719 --> 41:12.669
Now let's suppose that that is
our wave function,

41:12.668 --> 41:15.468
sometimes positive,
sometimes zero,

41:15.467 --> 41:17.357
sometimes negative.

41:17.360 --> 41:17.900
Okay?

41:17.900 --> 41:22.510
And let's look at different
positions and see what the

41:22.512 --> 41:25.572
kinetic energy is,
and then we'll be able to

41:25.570 --> 41:28.050
figure out,
since the total will be

41:28.050 --> 41:31.250
constant, what the potential
energy is.

41:31.250 --> 41:31.770
Okay?

41:31.768 --> 41:34.988
So we'll try to find out what
was the potential energy that

41:34.987 --> 41:36.427
gave this as a solution?

41:36.429 --> 41:38.399
This is again the Jeopardy
approach.

41:38.400 --> 41:40.840
Okay?

41:40.840 --> 41:43.930
Okay, so the curvature minus --
remember it's a negative

41:43.929 --> 41:46.019
constant --
minus the curvature over the

41:46.021 --> 41:49.141
amplitude could be positive --
that's going to be the kinetic

41:49.139 --> 41:50.639
energy;
it could be positive,

41:50.639 --> 41:52.569
it could be zero,
it could be negative,

41:52.574 --> 41:55.684
or it could be that we can't
tell by looking at the graph.

41:55.679 --> 41:58.869
So let's look at different
positions on the graph and see

41:58.865 --> 41:59.715
what it says.

41:59.719 --> 42:01.719
First look at that position.

42:01.719 --> 42:03.759
What is the kinetic energy
there?

42:03.760 --> 42:16.760
Positive, negative, zero?

42:16.760 --> 42:18.940
Ryan, why don't you help me out?

42:18.940 --> 42:20.070
Student:  No
Prof: Well no,

42:20.070 --> 42:20.580
you can help me out.

42:20.579 --> 42:21.909
>

42:21.909 --> 42:22.949
Look, so what do you need to
know?

42:22.949 --> 42:26.549
You need to know -- here's the
complicated thing you have to

42:26.547 --> 42:27.337
figure out.

42:27.340 --> 42:33.260
What is minus the curvature
divided by the amplitude at this

42:33.257 --> 42:34.057
point?

42:34.059 --> 42:36.839
Is it positive,
negative or zero?

42:36.840 --> 42:40.050
So what's the curvature at that
point?

42:40.050 --> 42:48.180
Is it curving up or down at
that point?

42:48.179 --> 42:49.329
No idea.

42:49.329 --> 42:51.409
Anybody got an idea?

42:51.409 --> 42:52.229
Keith?

42:52.230 --> 42:53.790
Kevin?

42:53.789 --> 42:56.179
Student:  It looks like
a saddle point so it's probably

42:56.175 --> 42:56.445
zero.

42:56.449 --> 42:57.339
Prof: It's not a saddle
point.

42:57.340 --> 42:58.170
What do you call it?

42:58.170 --> 42:59.140
Students:
Inflection points.

42:59.139 --> 43:01.569
Prof: A saddle point's
for three dimensions.

43:01.570 --> 43:03.440
In this it's what?

43:03.440 --> 43:04.590
Inflection point.

43:04.590 --> 43:05.740
It's flat there.

43:05.739 --> 43:08.139
It's curving one way on one
side, the other way on the other

43:08.143 --> 43:08.433
side.

43:08.429 --> 43:10.869
So it's got zero curvature
there;

43:10.869 --> 43:12.909
okay, zero curvature.

43:12.909 --> 43:15.569
Now Ryan, can you tell me
anything about that,

43:15.565 --> 43:17.155
if the curvature is zero?

43:17.159 --> 43:18.019
Student:  Zero.

43:18.019 --> 43:18.839
Prof: Ah ha.

43:18.840 --> 43:20.130
>

43:20.130 --> 43:20.820
Prof: Not bad.

43:20.820 --> 43:23.470
So that one we'll color grey
for zero.

43:23.469 --> 43:25.649
The kinetic energy at that
point is zero,

43:25.646 --> 43:27.276
if that's the wave function.

43:27.280 --> 43:28.670
Now let's take another point.

43:28.670 --> 43:31.250
Who's going to help me with
this one?

43:31.250 --> 43:34.640
How about the curvature at this
point right here?

43:34.639 --> 43:35.629
Student:  Negative.

43:35.630 --> 43:36.990
<<Students speak over one
another>>

43:36.989 --> 43:39.849
Prof: It's actually -- I
choose a point that's not

43:39.849 --> 43:40.309
curved.

43:40.309 --> 43:40.949
Student:  Ah.

43:40.949 --> 43:42.559
Prof: It's straight
right there.

43:42.559 --> 43:44.699
I assure you that's true.

43:44.699 --> 43:47.329
So I bet Ryan can help me again
on that one.

43:47.329 --> 43:49.469
How about it?

43:49.469 --> 43:50.179
Student:  Zero.

43:50.179 --> 43:51.479
Prof: Ah ha.

43:51.480 --> 43:53.190
So we'll make that one grey too.

43:53.190 --> 43:54.290
Now I'll go to someone else.

43:54.289 --> 43:55.859
How about there?

43:55.860 --> 43:58.240
What's the curvature at that
point do you think?

43:58.239 --> 43:59.709
Shai?

43:59.710 --> 44:02.050
Student:  It looks
straight, zero curvature.

44:02.050 --> 44:03.410
Prof: It looks straight,
zero curvature.

44:03.409 --> 44:05.659
So does that mean that this
value is zero?

44:05.659 --> 44:06.639
Student:  Not
necessarily, because the

44:06.635 --> 44:08.115
amplitude --
Prof: Ah,

44:08.115 --> 44:10.475
the amplitude is zero there
too.

44:10.480 --> 44:14.180
So really you can't be sure.

44:14.175 --> 44:15.095
Right?

44:15.099 --> 44:17.569
So that one we're going to have
to leave questionable,

44:17.572 --> 44:18.742
that's a question mark.

44:18.739 --> 44:21.009
How about out here?

44:21.010 --> 44:23.260
Not curved.

44:23.260 --> 44:26.460
So what's the kinetic energy?

44:26.460 --> 44:28.930
Josh?

44:28.929 --> 44:29.849
Student:  Questionable.

44:29.849 --> 44:31.089
Prof: Questionable,
right?

44:31.090 --> 44:32.690
Because the amplitude is zero
again;

44:32.690 --> 44:36.110
zero in the numerator,
also zero in the denominator;

44:36.110 --> 44:37.970
we really don't know.

44:37.969 --> 44:44.119
Okay, how about here?

44:44.119 --> 44:46.049
Tyler, what do you say?

44:46.050 --> 44:47.660
Is it curved there?

44:47.659 --> 44:48.519
Student:  Yes.

44:48.519 --> 44:49.369
Prof: Curving up or down?

44:49.369 --> 44:50.109
Student:  Down.

44:50.110 --> 44:53.480
Prof: So negative,
the curvature is negative.

44:53.480 --> 44:55.010
The value of Ψ?

44:55.010 --> 44:55.570
Student:  Positive.

44:55.570 --> 44:56.240
Prof: Positive.

44:56.239 --> 44:57.859
The energy, kinetic energy?

44:57.860 --> 44:58.380
Student:  Positive.

44:58.380 --> 44:59.270
Prof: Positive.

44:59.268 --> 45:02.578
Okay, so we can make that one
green.

45:02.579 --> 45:05.079
Okay, here's another one.

45:05.079 --> 45:08.669
Who's going to help me here?

45:08.670 --> 45:09.950
Kate?

45:09.949 --> 45:11.379
Student:  Yes.

45:11.380 --> 45:13.150
Prof: Okay,
so how about the curvature;

45:13.150 --> 45:14.360
curving up, curving down?

45:14.360 --> 45:16.170
Student:  It's curving
down, that's negative.

45:16.170 --> 45:16.810
Prof: Yes.

45:16.809 --> 45:17.819
Amplitude?

45:17.820 --> 45:19.600
Student:  Zero.

45:19.599 --> 45:20.049
So it should be green.

45:20.050 --> 45:20.840
Prof: Ah, green again.

45:20.840 --> 45:22.110
Okay.

45:22.110 --> 45:24.850
How about here?

45:24.849 --> 45:27.719
Ah, now how about the curvature?

45:27.719 --> 45:30.729
Seth?

45:30.730 --> 45:32.770
Student:  I don't know.

45:32.768 --> 45:34.888
Prof: Which way is it
curving at this point here?

45:34.889 --> 45:35.679
Student:  Curving up.

45:35.679 --> 45:36.389
Prof: Curving up.

45:36.389 --> 45:38.729
So the curvature is --
Student:  Positive.

45:38.730 --> 45:39.940
Prof: Positive.

45:39.940 --> 45:43.220
Student:  The amplitude
is negative, so it's positive.

45:43.219 --> 45:44.749
Prof: Yeah.

45:44.750 --> 45:46.360
So what color would we make it?

45:46.360 --> 45:48.480
Green again.

45:48.480 --> 45:52.720
Okay, so if you're -- you can
have -- be curving down or

45:52.721 --> 45:55.421
curving up and still be
positive;

45:55.420 --> 45:57.880
curving down if you're above
the baseline,

45:57.884 --> 46:00.234
curving up if you're below the
baseline.

46:00.228 --> 46:00.828
Right?

46:00.829 --> 46:04.759
So as long as you're curving
toward the baseline,

46:04.760 --> 46:09.430
towards Ψ=0,
the kinetic energy is positive.

46:09.429 --> 46:13.359
How about here?

46:13.360 --> 46:15.430
Zack?

46:15.429 --> 46:18.709
Which way is it curving?

46:18.710 --> 46:20.080
Curving up or curving down?

46:20.079 --> 46:20.989
Student:  It should be
curving up.

46:20.989 --> 46:22.959
Prof: Curving up,
curvature is positive.

46:22.960 --> 46:24.840
The value?

46:24.840 --> 46:25.620
Student:  Positive.

46:25.619 --> 46:26.219
Prof: Positive.

46:26.219 --> 46:27.309
Student:  I guess it'll
be negative.

46:27.309 --> 46:29.369
Prof: So it's negative
kinetic energy there.

46:29.369 --> 46:33.779
Make that one whatever that
pinkish color is.

46:33.780 --> 46:34.660
Okay?

46:34.659 --> 46:39.209
Here's another one,
how about there?

46:39.210 --> 46:41.150
Alex?

46:41.150 --> 46:43.840
Which way is it curving at the
new place?

46:43.840 --> 46:45.250
Here?

46:45.250 --> 46:46.150
Student:  Curving down.

46:46.150 --> 46:48.970
Prof: Curving down,
negative curvature.

46:48.969 --> 46:50.919
Student:  Negative
amplitude.

46:50.920 --> 46:51.430
Prof: Negative amplitude.

46:51.429 --> 46:52.679
Student:  Negative
kinetic energy.

46:52.679 --> 46:54.289
Prof: Negative kinetic
energy;

46:54.289 --> 46:55.569
pink again.

46:55.570 --> 46:56.570
Is that enough?

46:56.570 --> 47:02.400
Oh, there's one more,
here, the one right here.

47:02.400 --> 47:09.410
Okay?

47:09.409 --> 47:10.289
Student:  Negative.

47:10.289 --> 47:10.919
Prof: Pardon me?

47:10.920 --> 47:11.720
Student:  Negative.

47:11.719 --> 47:13.319
Prof: Negative,
because it's -- how did you do

47:13.318 --> 47:13.748
it so quick?

47:13.750 --> 47:14.850
We didn't have to go through
curvature.

47:14.849 --> 47:16.269
Student:  It goes away
from the line.

47:16.268 --> 47:19.448
Prof: Because it's
curving away from the baseline,

47:19.454 --> 47:20.084
negative.

47:20.079 --> 47:22.369
Okay, pink.

47:22.369 --> 47:24.899
Okay, curving away from
Ψ=0 means that the

47:24.898 --> 47:26.308
kinetic energy is negative.

47:26.309 --> 47:29.619
So now we know at all these
positions whether the kinetic

47:29.615 --> 47:31.735
energy is positive,
negative or zero,

47:31.742 --> 47:34.872
although there are a few that
we aren't certain about.

47:34.871 --> 47:35.581
Right?

47:35.579 --> 47:38.559
So here's the potential energy
that will do that.

47:38.559 --> 47:41.719
If you have this line for the
total energy -- right?

47:41.719 --> 47:48.609
-- then here and here you have
zero.

47:48.614 --> 47:50.194
Right?

47:50.190 --> 47:53.210
Also, incidentally,
here and here,

47:53.206 --> 47:56.036
you have zero kinetic energy.

47:56.039 --> 47:56.989
With me?

47:56.989 --> 47:59.129
Okay.

47:59.130 --> 48:01.740
So no curvature, right?

48:01.739 --> 48:07.179
At these green places,
the total energy is higher than

48:07.179 --> 48:09.539
the potential energy.

48:09.539 --> 48:13.209
So the kinetic energy is
positive.

48:13.210 --> 48:15.550
Okay?

48:15.550 --> 48:19.370
At these places,
the potential energy is higher

48:19.367 --> 48:21.357
than the total energy.

48:21.360 --> 48:24.900
So the kinetic energy is
negative and the thing is

48:24.900 --> 48:27.140
curving away from the baseline.

48:27.139 --> 48:27.789
Right?

48:27.789 --> 48:31.329
And now we know something about
this point.

48:31.329 --> 48:35.829
If the potential energy is a
continuous kind of thing,

48:35.829 --> 48:39.149
then, although we couldn't tell
by looking at the wave function,

48:39.150 --> 48:40.960
it's curving away from the
baseline,

48:40.960 --> 48:43.970
but very slightly, right?

48:43.969 --> 48:47.989
It's negative kinetic energy
there, and also on the right

48:47.989 --> 48:50.429
here is negative kinetic energy.

48:50.429 --> 48:55.909
And here we know,
just by continuity,

48:55.909 --> 48:59.379
that at this point it must've
been positive kinetic energy,

48:59.380 --> 49:01.770
even though we couldn't tell it
by looking at the curve.

49:01.768 --> 49:06.818
There must be an inflection
point when you go through zero,

49:06.820 --> 49:11.260
otherwise you'd get a
discontinuity in the potential

49:11.262 --> 49:12.222
energy.

49:12.219 --> 49:14.209
Okay, so that one was green.

49:14.210 --> 49:17.500
Okay, now I have to stop.

49:17.500 --> 49:23.000