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ROBERT FIELD: This is one of
my favorite lectures for 5.61.
00:00:26.310 --> 00:00:29.190
And I hope you like it too.
00:00:29.190 --> 00:00:34.540
OK, last time, we talked
about the hydrogen atom.
00:00:34.540 --> 00:00:37.830
And it's the last
exactly solved problem.
00:00:37.830 --> 00:00:40.680
And you've probably
noticed that like the way
00:00:40.680 --> 00:00:44.400
I treated each of the
exactly solved problems,
00:00:44.400 --> 00:00:51.150
I avoided looking at
the exact solutions.
00:00:51.150 --> 00:00:58.860
I said we should be able to
build our own picture based
00:00:58.860 --> 00:01:00.990
on simplifications.
00:01:00.990 --> 00:01:03.090
And this is something
that I really
00:01:03.090 --> 00:01:08.670
want to stress that, yes,
the exact solutions are
00:01:08.670 --> 00:01:09.870
in textbooks.
00:01:09.870 --> 00:01:13.380
You can program them
into your computers
00:01:13.380 --> 00:01:15.960
and ask your computer
to calculate anything.
00:01:15.960 --> 00:01:18.150
But sometimes you
want to know what
00:01:18.150 --> 00:01:20.730
it is you want to
calculate and what
00:01:20.730 --> 00:01:25.620
is the right answer or
approximately the right answer
00:01:25.620 --> 00:01:26.550
to build insights.
00:01:26.550 --> 00:01:34.410
Now, in 5.112 and possibly
5.111 and certainly not 3.091,
00:01:34.410 --> 00:01:38.790
you learn something
about the periodic table.
00:01:38.790 --> 00:01:40.830
You learn how to
make predictions
00:01:40.830 --> 00:01:45.769
about all properties of atoms
based on some simple ideas.
00:01:45.769 --> 00:01:46.935
What are those simple ideas?
00:01:50.927 --> 00:01:52.385
I'm looking for
three simple ideas.
00:01:58.430 --> 00:01:59.327
Yes.
00:01:59.327 --> 00:02:01.452
STUDENT: Things with similar
valence configurations
00:02:01.452 --> 00:02:03.140
behave similarly.
00:02:03.140 --> 00:02:09.350
ROBERT FIELD: Yes, that's sort
of a second order simple idea,
00:02:09.350 --> 00:02:13.640
that because--
00:02:13.640 --> 00:02:16.070
I mean, it's using
the idea that if you
00:02:16.070 --> 00:02:19.850
know that quantum numbers of
the electrons and what orbitals
00:02:19.850 --> 00:02:27.620
are occupied, you can make
connections between atomic
00:02:27.620 --> 00:02:29.180
and some molecular properties.
00:02:33.690 --> 00:02:36.330
Yes.
00:02:36.330 --> 00:02:38.830
STUDENT: If you
understand the interplay
00:02:38.830 --> 00:02:41.070
between nuclear charge
and electric charge,
00:02:41.070 --> 00:02:44.085
you can derive large
physical magnitudes.
00:02:44.085 --> 00:02:44.835
ROBERT FIELD: Yes.
00:02:47.430 --> 00:02:53.440
I mean, there are things like
electronegativity, shielding.
00:02:53.440 --> 00:02:58.510
And these are things where
if we have some intuitive
00:02:58.510 --> 00:03:04.010
sense for how big
orbitals are and what
00:03:04.010 --> 00:03:09.620
is the relative attractiveness
to different nuclei
00:03:09.620 --> 00:03:13.370
of various orbitals,
you build up everything.
00:03:13.370 --> 00:03:16.310
And most of it comes
from this lecture.
00:03:19.150 --> 00:03:24.210
And so I'm going to talk
about the hydrogen atom
00:03:24.210 --> 00:03:27.600
as a model for
electronic structure.
00:03:27.600 --> 00:03:34.165
And in particular, I'm to talk
about quantum number scaling
00:03:34.165 --> 00:03:34.665
effects.
00:03:38.140 --> 00:03:41.780
And I'm going to be using this
semi-classical method a lot.
00:03:45.410 --> 00:03:51.170
So we know that for
the hydrogen atom,
00:03:51.170 --> 00:03:54.290
you can factor the wave function
into an angular part, which
00:03:54.290 --> 00:03:56.260
is universal.
00:03:56.260 --> 00:03:59.830
So if you understand it
once for any problem,
00:03:59.830 --> 00:04:01.490
you understand it all the time.
00:04:01.490 --> 00:04:02.960
And so we can just
put that aside,
00:04:02.960 --> 00:04:05.650
because that's a done deal.
00:04:05.650 --> 00:04:08.530
And then there's
the radial part.
00:04:08.530 --> 00:04:11.740
And the radial part is different
for every central force
00:04:11.740 --> 00:04:13.120
problem.
00:04:13.120 --> 00:04:14.680
And so we want to
be able to know
00:04:14.680 --> 00:04:17.620
what goes into the
radial part and how
00:04:17.620 --> 00:04:22.840
to use those ideas to be
able to understand almost
00:04:22.840 --> 00:04:25.732
any problem, even
problems where there
00:04:25.732 --> 00:04:26.815
is more than one electron.
00:04:30.850 --> 00:04:36.440
So I'm going to be making
use of the classical,
00:04:36.440 --> 00:04:39.270
the semiclassical
approach, where
00:04:39.270 --> 00:04:46.630
the classical momentum that's
conjugate to the radius,
00:04:46.630 --> 00:04:49.480
in other words, the
radial momentum,
00:04:49.480 --> 00:04:53.130
that we know what that is.
00:04:53.130 --> 00:04:56.790
It's related to the
difference between the energy
00:04:56.790 --> 00:04:59.240
and the potential energy.
00:04:59.240 --> 00:05:04.530
And we get the momentum from
that, the classical momentum
00:05:04.530 --> 00:05:05.430
function.
00:05:05.430 --> 00:05:08.310
And that gives us
everything, because we
00:05:08.310 --> 00:05:12.210
use the DeBroglie relationship
between the wavelength
00:05:12.210 --> 00:05:15.020
and the momentum.
00:05:15.020 --> 00:05:19.730
And I hope you
believe what I'm going
00:05:19.730 --> 00:05:24.260
to try to do in the
time we have for this.
00:05:24.260 --> 00:05:27.890
So there are several
important things
00:05:27.890 --> 00:05:33.020
that I didn't get to have
time to do in lecture.
00:05:33.020 --> 00:05:36.410
And I want to mention
them, because I probably
00:05:36.410 --> 00:05:40.190
will write a question on
the final connected to spin
00:05:40.190 --> 00:05:43.667
orbit and Stern-Gerlach.
00:05:43.667 --> 00:05:45.000
So you want to understand those.
00:05:48.240 --> 00:05:53.010
So the schedule for
today is suppose
00:05:53.010 --> 00:05:56.760
you record a spectrum
of the hydrogen atom.
00:05:59.520 --> 00:06:03.420
Now, that spectrum is
going to reveal something
00:06:03.420 --> 00:06:09.880
that I regard as electronic
structure, structure
00:06:09.880 --> 00:06:17.020
in this sense, not
of bricks and mortar
00:06:17.020 --> 00:06:19.640
as you would talk about
the structure of building,
00:06:19.640 --> 00:06:22.150
but what the
architect had in mind
00:06:22.150 --> 00:06:24.580
and what the function
of the building was.
00:06:24.580 --> 00:06:28.900
And so the structure is
kind of a mystical thing,
00:06:28.900 --> 00:06:31.540
in which if you look
at a little bit of it,
00:06:31.540 --> 00:06:34.550
you sort of get the
idea of all of the rest.
00:06:34.550 --> 00:06:37.270
And so here, I'm
going to show you
00:06:37.270 --> 00:06:41.880
how you can approach
a spectrum and to know
00:06:41.880 --> 00:06:44.340
where you are, because
the spectrum can
00:06:44.340 --> 00:06:45.960
be really complicated.
00:06:45.960 --> 00:06:48.480
And there are certain
patterns that knowing
00:06:48.480 --> 00:06:52.200
about electronic structure
enables you to say, yeah,
00:06:52.200 --> 00:06:53.190
I've got a Google map.
00:06:53.190 --> 00:06:55.790
I know exactly where I am.
00:06:55.790 --> 00:06:59.080
And it's based on
some simple ideas.
00:06:59.080 --> 00:07:03.330
So this is first
illustration of structure.
00:07:03.330 --> 00:07:07.320
Then I'm going to go and deal
with the semiclassical methods
00:07:07.320 --> 00:07:10.830
for calculating all electronic
properties of the hydrogen
00:07:10.830 --> 00:07:12.540
atom.
00:07:12.540 --> 00:07:15.500
And we get this
business of scaling
00:07:15.500 --> 00:07:17.940
of properties with the
principal quantum number.
00:07:21.130 --> 00:07:24.620
And that extends to Rydberg
states of everything--
00:07:24.620 --> 00:07:28.750
atoms and molecules-- not just
the one electron spectrum.
00:07:28.750 --> 00:07:33.040
And the Rydberg states are
special because one electron
00:07:33.040 --> 00:07:33.740
is special.
00:07:33.740 --> 00:07:36.040
It's outside of
all of the others.
00:07:36.040 --> 00:07:39.370
And what we're going to
learn about next time when
00:07:39.370 --> 00:07:42.370
we have many electrons
or more than one,
00:07:42.370 --> 00:07:45.910
we don't have to think
about antisymmetrization.
00:07:45.910 --> 00:07:47.980
We don't have--
we can still take
00:07:47.980 --> 00:07:52.090
our simple minded picture
of an electron interacting
00:07:52.090 --> 00:07:53.770
with something.
00:07:53.770 --> 00:07:58.960
And so we can then take
what we knew from hydrogen
00:07:58.960 --> 00:08:05.140
and describe everything about
Rydberg states of molecules.
00:08:05.140 --> 00:08:08.180
And that's kind of exciting.
00:08:08.180 --> 00:08:12.300
And this will be
followed by the bad news
00:08:12.300 --> 00:08:15.650
that when you have
more than one electron,
00:08:15.650 --> 00:08:18.560
you have to do something else.
00:08:18.560 --> 00:08:24.180
You have to write
antisymmetric wave functions,
00:08:24.180 --> 00:08:28.167
antisymmetric with respect to
the exchange of all electrons.
00:08:28.167 --> 00:08:29.750
And that looks like
it's going to lead
00:08:29.750 --> 00:08:32.760
to a tremendous headache.
00:08:32.760 --> 00:08:35.130
But it doesn't.
00:08:35.130 --> 00:08:38.380
But you do have to
learn a new algebra.
00:08:38.380 --> 00:08:41.126
OK, so let's go on to this.
00:08:48.070 --> 00:08:54.390
OK, so for the
hydrogen atom, it's
00:08:54.390 --> 00:09:00.290
easy to solve the time
independent Schrodinger
00:09:00.290 --> 00:09:01.330
equation.
00:09:01.330 --> 00:09:06.720
And we get a complete set of
these Rnl radial functions,
00:09:06.720 --> 00:09:08.520
complete set, an
infinite number.
00:09:11.720 --> 00:09:13.870
We don't care.
00:09:13.870 --> 00:09:21.960
You can take this-- it's
a simple one-dimensional
00:09:21.960 --> 00:09:23.065
differential equation.
00:09:23.065 --> 00:09:23.940
And you can solve it.
00:09:23.940 --> 00:09:25.564
You can tell your
computer to solve it.
00:09:25.564 --> 00:09:27.540
You don't have to have
any tricks at all.
00:09:27.540 --> 00:09:30.630
You can just use
whatever numerical method
00:09:30.630 --> 00:09:34.260
to find the wave functions and
calculate whatever you want.
00:09:34.260 --> 00:09:35.880
There's no insight there.
00:09:35.880 --> 00:09:39.270
It's the same thing as
recording a spectrum, where
00:09:39.270 --> 00:09:41.520
you have tables
of observed lines
00:09:41.520 --> 00:09:43.320
and maybe observed intensities.
00:09:43.320 --> 00:09:45.950
You don't know anything.
00:09:45.950 --> 00:09:47.570
It's a description.
00:09:47.570 --> 00:09:49.940
And you can have a
mathematical description,
00:09:49.940 --> 00:09:53.030
or you can have an
experimental description.
00:09:53.030 --> 00:09:55.010
You don't know anything.
00:09:55.010 --> 00:10:00.110
You want to have your insight
telling you what's important
00:10:00.110 --> 00:10:03.620
and how to use what you know
about part of the spectrum
00:10:03.620 --> 00:10:08.850
to determine other things.
00:10:08.850 --> 00:10:13.800
For example, we have
two energy levels.
00:10:13.800 --> 00:10:18.640
Let's say n equals
1 and n equals 2.
00:10:18.640 --> 00:10:24.200
And there are several
things that you
00:10:24.200 --> 00:10:28.100
would need to do in order
to know what the selection
00:10:28.100 --> 00:10:33.530
rule for l is from this, because
the l, the orbital angular
00:10:33.530 --> 00:10:36.991
momentum states of n
equals 2 are degenerate.
00:10:39.880 --> 00:10:46.370
And so you could use
the Zeeman effect.
00:10:46.370 --> 00:10:49.920
You can use radiative lifetimes.
00:10:49.920 --> 00:10:54.710
And what you would find is we
have a p state and an s state.
00:10:54.710 --> 00:10:59.890
And the p state
fluoresces to the s state
00:10:59.890 --> 00:11:03.450
and the s state doesn't, unless
you apply an electric field
00:11:03.450 --> 00:11:07.330
and it mixes with a p state
and it then fluoresces.
00:11:07.330 --> 00:11:10.570
So there are all sorts of things
that will alert you to the fact
00:11:10.570 --> 00:11:13.040
that there's more going on here
than just this simple level
00:11:13.040 --> 00:11:13.539
diagram.
00:11:18.580 --> 00:11:22.190
OK, so let's do some
elementary stuff
00:11:22.190 --> 00:11:29.410
and then get to the Google
map for the spectrum.
00:11:32.990 --> 00:11:37.940
We have this thing that makes
the hydrogen atom interesting.
00:11:37.940 --> 00:11:41.090
The potential, the
radial potential
00:11:41.090 --> 00:11:44.750
is a function of the
orbital angular momentum.
00:11:44.750 --> 00:11:50.930
And the effective
potential is given
00:11:50.930 --> 00:11:58.910
by h bar squared l, l plus
1, over 2 mu r squared--
00:11:58.910 --> 00:12:01.550
and that's with a plus sign.
00:12:01.550 --> 00:12:08.910
And we have z e squared
over 4 pi epsilon 0--
00:12:08.910 --> 00:12:17.160
that's just stuff from the
MKS units for the Coulomb
00:12:17.160 --> 00:12:20.480
equation-- and 1 over r.
00:12:20.480 --> 00:12:23.330
So we have an
attractive thing that
00:12:23.330 --> 00:12:25.760
pulls the electron
towards the nucleus.
00:12:25.760 --> 00:12:30.140
And we have a repulsion part,
which keeps the electron away
00:12:30.140 --> 00:12:33.790
as long as l is not equal to 0.
00:12:33.790 --> 00:12:39.390
So this is the actual
effective potential.
00:12:39.390 --> 00:12:43.010
This is the thing we would use
for solving the Schrodinger
00:12:43.010 --> 00:12:48.320
equation for the
Rnl of R functions
00:12:48.320 --> 00:12:52.520
exactly if you wanted to.
00:12:52.520 --> 00:12:57.950
And first surprise is that the
energy levels come out to be--
00:13:05.650 --> 00:13:10.700
for n equals 1, 2, 3, etc.
00:13:10.700 --> 00:13:14.600
So why are the energy
levels not dependent on l?
00:13:14.600 --> 00:13:19.360
Because it takes more ink
to write this than that.
00:13:19.360 --> 00:13:22.190
And this can be
really important.
00:13:22.190 --> 00:13:24.270
But there is no l dependence.
00:13:24.270 --> 00:13:26.040
That's a surprise.
00:13:26.040 --> 00:13:30.860
It's not true for anything
other than one-electron spectra.
00:13:30.860 --> 00:13:35.570
But if it's true
and it has to be,
00:13:35.570 --> 00:13:38.930
you have to have something that
says, OK, for something that's
00:13:38.930 --> 00:13:42.430
not hydrogen. Maybe there
will be some l splitting.
00:13:42.430 --> 00:13:43.040
Why?
00:13:43.040 --> 00:13:43.800
Why is that?
00:13:43.800 --> 00:13:48.070
How does this
effective potential
00:13:48.070 --> 00:13:50.600
have an effect for something
other than hydrogen?
00:13:53.230 --> 00:13:57.960
And we'll understand
that in a second.
00:13:57.960 --> 00:14:02.280
OK, now there's the
question of degeneracy.
00:14:07.830 --> 00:14:15.750
So if we pick a value of n,
we can have l equals 0, 1,
00:14:15.750 --> 00:14:16.920
up to n minus 1.
00:14:20.240 --> 00:14:21.860
That comes out of
the mathematics.
00:14:21.860 --> 00:14:24.950
And I'm just assuming that
you can accept that as a fact.
00:14:27.580 --> 00:14:34.180
We know from our study of
the rigid rotor for the Ylm
00:14:34.180 --> 00:14:40.930
functions that the degeneracy of
the l part of the wave function
00:14:40.930 --> 00:14:45.840
goes as 2l plus 1.
00:14:45.840 --> 00:14:47.790
If we have an orbital
angular momentum of 0,
00:14:47.790 --> 00:14:50.260
there's a degeneracy of 1.
00:14:50.260 --> 00:14:52.930
Angular momentum of 1,
it has a degeneracy of 3.
00:14:52.930 --> 00:14:53.590
We know that.
00:14:57.120 --> 00:15:02.230
So if we do a little game.
00:15:02.230 --> 00:15:06.790
And we look at the degeneracies
for l equals 0, 1, 2,
00:15:06.790 --> 00:15:10.290
3, the degeneracies--
00:15:10.290 --> 00:15:16.970
so g sub l, is 1, 3, 5, 7.
00:15:16.970 --> 00:15:25.620
OK, now, the degeneracy for n is
going to be all of the possible
00:15:25.620 --> 00:15:26.520
l's.
00:15:26.520 --> 00:15:30.090
And so if we have n equals
1, we only have one l.
00:15:30.090 --> 00:15:31.770
And so the degeneracy is 1.
00:15:34.760 --> 00:15:40.100
If we have n equals
2, we have p and s.
00:15:40.100 --> 00:15:42.320
And so we have 4.
00:15:42.320 --> 00:15:47.600
And for n equals 2, we have--
00:15:52.900 --> 00:16:05.960
I'm sorry, for n equals
1, we have only 1.
00:16:05.960 --> 00:16:08.230
And then we have
for n equals 2, 4.
00:16:08.230 --> 00:16:12.040
And for n equals 3, 9.
00:16:12.040 --> 00:16:14.380
So you see 1 plus 3 is 4.
00:16:14.380 --> 00:16:17.620
1 plus 3 plus 5 is 9.
00:16:17.620 --> 00:16:22.150
And so the degeneracy
for n is n squared.
00:16:25.870 --> 00:16:28.770
So as you go really high
end, you get lots of states.
00:16:28.770 --> 00:16:33.820
And for hydrogen,
they're all degenerate.
00:16:33.820 --> 00:16:37.930
But if they're not split,
maybe you don't care.
00:16:37.930 --> 00:16:38.680
Of course, you do.
00:16:41.500 --> 00:16:43.050
I better leave this down.
00:16:47.180 --> 00:16:50.210
Now, here we are for--
00:16:50.210 --> 00:16:51.440
suppose you take a spectrum.
00:16:54.230 --> 00:16:56.630
And suppose it's an
emission spectrum.
00:16:56.630 --> 00:17:00.050
You run a discharge
through gas of something
00:17:00.050 --> 00:17:04.839
that contains hydrogen. And
you'll see a bunch of lines.
00:17:04.839 --> 00:17:08.740
And so one of the
things you know
00:17:08.740 --> 00:17:15.069
is that the energy
levels are separated--
00:17:15.069 --> 00:17:17.349
they go as 1 over n squared.
00:17:17.349 --> 00:17:21.354
And so the energy levels--
00:17:23.945 --> 00:17:26.430
well, actually,
they're converging.
00:17:26.430 --> 00:17:29.470
And so let's just
indicate that like this.
00:17:29.470 --> 00:17:35.677
So suppose you ask, well,
suppose I'm at this level,
00:17:35.677 --> 00:17:38.010
there are going to be a series
of transitions converging
00:17:38.010 --> 00:17:42.220
to a common limit from it.
00:17:42.220 --> 00:17:46.520
And that's a marker, a
pattern in the spectrum.
00:17:46.520 --> 00:17:49.090
If you have essentially
an infinite number
00:17:49.090 --> 00:17:53.560
of levels converging to a fixed
point, that's easy to see.
00:17:53.560 --> 00:17:55.800
And so for each
level, there will
00:17:55.800 --> 00:17:59.440
be convergence to
the same fixed point.
00:17:59.440 --> 00:18:03.990
And so that tells you there's
a kind of pattern recognition
00:18:03.990 --> 00:18:05.670
you would do.
00:18:05.670 --> 00:18:08.940
And you would begin to build
up the energy level diagram
00:18:08.940 --> 00:18:13.410
and to know which levels you're
looking at from these existence
00:18:13.410 --> 00:18:15.210
of convergent series.
00:18:18.410 --> 00:18:25.240
So if we're interested
in the spectrum,
00:18:25.240 --> 00:18:30.040
we have the energy
level differences, z
00:18:30.040 --> 00:18:38.279
squared Rydberg 1 over n squared
minus 1 over n prime squared.
00:18:38.279 --> 00:18:39.445
That's the Rydberg equation.
00:18:42.120 --> 00:18:45.520
And so this also
tells you something.
00:18:45.520 --> 00:18:49.670
So suppose you're in the nth
level, well, you know this.
00:18:49.670 --> 00:18:54.010
And now this then converges
to the ionization limit.
00:18:54.010 --> 00:18:55.710
But there's something else.
00:18:55.710 --> 00:19:01.990
And that is suppose we observe
two consecutive members,
00:19:01.990 --> 00:19:03.610
n an plus 1.
00:19:03.610 --> 00:19:05.800
You don't know what
n is, but because you
00:19:05.800 --> 00:19:08.230
see the pattern of
this convergent series,
00:19:08.230 --> 00:19:12.010
you can see two levels that are
obviously consecutive members
00:19:12.010 --> 00:19:13.900
of a series.
00:19:13.900 --> 00:19:17.750
And the question is what is n?
00:19:17.750 --> 00:19:26.040
And, again, you can solve that
by knowing that the energy
00:19:26.040 --> 00:19:29.080
levels go as 1 over n squared.
00:19:29.080 --> 00:19:32.010
So the difference
between energy levels
00:19:32.010 --> 00:19:43.030
goes as 2 z squared
r over n cubed.
00:19:43.030 --> 00:19:46.590
This is just taking the
derivative of 1 over n squared.
00:19:46.590 --> 00:19:49.290
We get 2 times 1 over n cubed.
00:19:52.290 --> 00:19:57.280
And so that kind of a pattern
enables you to say, oh, yeah,
00:19:57.280 --> 00:19:58.210
I know what n is.
00:20:02.380 --> 00:20:05.440
So this is a kind
of a hand waving,
00:20:05.440 --> 00:20:08.350
but this is what we
do as spectroscopists.
00:20:08.350 --> 00:20:12.520
We're looking for a
clue as to how to begin
00:20:12.520 --> 00:20:14.770
to put assignments on levels.
00:20:14.770 --> 00:20:17.710
And because there is this
simple structure, which
00:20:17.710 --> 00:20:21.800
is represented by
this Rydberg equation,
00:20:21.800 --> 00:20:23.675
we can say where we are.
00:20:23.675 --> 00:20:24.550
We know where we are.
00:20:24.550 --> 00:20:26.270
We have two choices.
00:20:26.270 --> 00:20:29.110
One is this 1 over n cubed
scaling of the energy
00:20:29.110 --> 00:20:30.160
differences.
00:20:30.160 --> 00:20:33.130
And the other is the
existence of a convergence
00:20:33.130 --> 00:20:37.480
of every initial level
to a common final level.
00:20:37.480 --> 00:20:39.320
And that's very important.
00:20:39.320 --> 00:20:41.170
You can't assign a
spectrum unless you
00:20:41.170 --> 00:20:42.820
know what the patterns are.
00:20:42.820 --> 00:20:45.130
And these are two things
that come out of the fact
00:20:45.130 --> 00:20:48.070
that there is a structure
associated with the hydrogen
00:20:48.070 --> 00:20:50.670
atom.
00:20:50.670 --> 00:20:55.830
OK, now, I'm going to have to
put some numbers on the board.
00:20:55.830 --> 00:20:58.328
And it's another
illustration of structure.
00:21:02.540 --> 00:21:08.320
Now, I'm just taking things
from a table in McQuarrie.
00:21:08.320 --> 00:21:14.050
And this table is a table
of expectation values
00:21:14.050 --> 00:21:17.900
of integer powers of r.
00:21:17.900 --> 00:21:18.670
And what is this?
00:21:18.670 --> 00:21:22.060
Every electronic property,
anything you would measure
00:21:22.060 --> 00:21:26.370
is going to be a
function of the--
00:21:26.370 --> 00:21:29.680
is going to involve an integral
involving a function of r.
00:21:32.270 --> 00:21:34.820
That's what we mean by
electronic properties.
00:21:34.820 --> 00:21:40.010
And these electronic properties
have a wonderful behavior.
00:21:40.010 --> 00:21:41.630
So let's make a little table.
00:21:41.630 --> 00:21:43.640
Here is the integer power.
00:21:43.640 --> 00:21:47.060
Here is the expectation value.
00:21:47.060 --> 00:21:48.140
And that's l.
00:21:51.330 --> 00:21:54.150
So we start with n equals 2.
00:21:58.580 --> 00:22:04.050
The result you get by
actually accurately evaluating
00:22:04.050 --> 00:22:05.250
an integral--
00:22:05.250 --> 00:22:06.655
and these are doable integrals.
00:22:06.655 --> 00:22:08.910
They are not
numerically evaluated.
00:22:08.910 --> 00:22:11.260
And they have a simple formula--
00:22:11.260 --> 00:22:20.390
a0 squared n to the
fourth z squared times 1
00:22:20.390 --> 00:22:31.380
plus 3/2 times 1 minus
l, l plus 1 minus 1/3.
00:22:31.380 --> 00:22:32.460
These are done integrals.
00:22:32.460 --> 00:22:33.390
I can't do them.
00:22:33.390 --> 00:22:36.300
I wouldn't care to do them.
00:22:36.300 --> 00:22:42.270
So that's how any
electronic property that
00:22:42.270 --> 00:22:46.320
goes as the square of
r behaves, we get this.
00:22:46.320 --> 00:22:48.120
This is the Bohr radius.
00:22:48.120 --> 00:22:51.120
This is approximately
half an angstrom.
00:22:51.120 --> 00:22:55.830
And it's the radius of
the n equals 1 Bohr orbit,
00:22:55.830 --> 00:22:58.020
which doesn't even exist.
00:22:58.020 --> 00:22:58.520
Right?
00:22:58.520 --> 00:23:01.040
There are no Bohr orbits.
00:23:01.040 --> 00:23:03.260
But we have this
Bohr model, which
00:23:03.260 --> 00:23:05.040
explains these energy levels.
00:23:05.040 --> 00:23:07.130
And so we use that.
00:23:07.130 --> 00:23:09.560
And there's nothing else here.
00:23:09.560 --> 00:23:11.210
That's the charge
on the nucleus.
00:23:11.210 --> 00:23:17.240
Well, it's 1 for hydrogen. But
there's nothing empirical here.
00:23:17.240 --> 00:23:20.200
And this is a sort of a
fundamental constant now,
00:23:20.200 --> 00:23:21.810
but it's sort of empirical.
00:23:24.390 --> 00:23:31.470
And then the next one, it has a
very interesting initial term,
00:23:31.470 --> 00:23:37.560
a0 not squared n squared over z.
00:23:37.560 --> 00:23:40.570
We've lost a power here.
00:23:40.570 --> 00:23:46.250
And it again is one of these
complicated looking functions.
00:23:58.450 --> 00:24:04.560
OK, this r to the 1,
that's how big an atom
00:24:04.560 --> 00:24:06.510
is, the radius of the atom.
00:24:06.510 --> 00:24:09.430
And again, we have
a closed form.
00:24:09.430 --> 00:24:12.030
Notice that the l
is present here,
00:24:12.030 --> 00:24:16.070
even though it's not
present in the energies.
00:24:16.070 --> 00:24:20.020
0, that just goes
n to the 0 power.
00:24:20.020 --> 00:24:26.902
It's just a constant
minus 1, minus 2, minus 3.
00:24:26.902 --> 00:24:30.790
Well, minus 1,
that's what you have
00:24:30.790 --> 00:24:35.530
for the Coulomb attraction.
00:24:35.530 --> 00:24:42.180
And that goes as z
over a0 n squared.
00:24:42.180 --> 00:24:50.020
And for minus 2, that goes
as z squared a0 n cubed.
00:24:53.580 --> 00:24:56.400
And all that plus 1/2.
00:24:56.400 --> 00:25:07.503
And for minus 3, we get z
cubed a0 cubed n cubed l,
00:25:07.503 --> 00:25:14.270
l plus 1/2, l plus 1.
00:25:14.270 --> 00:25:17.890
So there are all these formulas.
00:25:17.890 --> 00:25:20.170
Well, the important thing
is the leading term.
00:25:20.170 --> 00:25:26.510
And what you discover is that
right here something happens.
00:25:26.510 --> 00:25:29.472
The leading term goes
as 1 over n cubed.
00:25:29.472 --> 00:25:34.540
It doesn't matter what the
negative power of r is.
00:25:34.540 --> 00:25:37.870
The leading term goes
as 1 over n cubed.
00:25:41.530 --> 00:25:42.146
Why's that?
00:25:45.226 --> 00:25:47.100
Well, that's really
important, because this 1
00:25:47.100 --> 00:25:51.080
over n cubed behavior is telling
you something very important.
00:25:55.310 --> 00:25:58.230
So here's the nucleus.
00:25:58.230 --> 00:26:00.600
And here's the electron.
00:26:00.600 --> 00:26:02.120
We can think of it
as a Bohr orbit.
00:26:05.220 --> 00:26:08.710
If we have a
negative power of r,
00:26:08.710 --> 00:26:13.390
then as you get farther
and farther away from r,
00:26:13.390 --> 00:26:16.740
the property gets small.
00:26:16.740 --> 00:26:22.740
And so what happens is for large
enough negative powers of r,
00:26:22.740 --> 00:26:25.200
the only thing that
matters is the first lobe,
00:26:25.200 --> 00:26:28.189
the innermost lobe
of the wave function.
00:26:28.189 --> 00:26:29.980
And it turns out that's
the only thing that
00:26:29.980 --> 00:26:31.524
matters for almost everything.
00:26:34.130 --> 00:26:37.500
And so one of the things
I'm going to show you
00:26:37.500 --> 00:26:41.160
is why does the
innermost lobe matter
00:26:41.160 --> 00:26:45.380
and how do we use that
to understand everything,
00:26:45.380 --> 00:26:47.330
not just about hydrogen,
but about Rydberg
00:26:47.330 --> 00:26:48.564
states of everything.
00:26:51.290 --> 00:26:55.480
So the scaling of
electronic properties
00:26:55.480 --> 00:27:00.590
depending on the power of r is
another example of structure.
00:27:00.590 --> 00:27:06.320
And if you know one thing
about the hydrogen atom,
00:27:06.320 --> 00:27:09.590
if you make one measurement,
if you know how to use it,
00:27:09.590 --> 00:27:12.010
you know everything.
00:27:12.010 --> 00:27:14.750
You don't have to do
all this other stuff.
00:27:14.750 --> 00:27:19.320
That's a really beautiful
example of structure.
00:27:19.320 --> 00:27:23.630
And this is insight too,
because sometimes the structure
00:27:23.630 --> 00:27:27.020
that you have for hydrogen
is only approximately
00:27:27.020 --> 00:27:28.700
valid for other things.
00:27:28.700 --> 00:27:32.030
And you're going to want to
know what you can use and know
00:27:32.030 --> 00:27:35.070
how to modify it.
00:27:35.070 --> 00:27:37.550
And I guarantee you that
nobody teaching a course
00:27:37.550 --> 00:27:40.550
like this would ever talk
about that kind of stuff,
00:27:40.550 --> 00:27:44.150
because, you know,
it's approximate.
00:27:44.150 --> 00:27:44.990
It's intuitive.
00:27:44.990 --> 00:27:48.680
And all the good stuff is
tabulated in the textbooks
00:27:48.680 --> 00:27:50.030
and you have to memorize it.
00:27:50.030 --> 00:27:52.450
But you don't know
how to use it.
00:27:52.450 --> 00:27:53.600
And this is how you use it.
00:27:53.600 --> 00:27:55.500
This is really important.
00:27:58.040 --> 00:28:01.550
So now let me draw
some pictures.
00:28:06.320 --> 00:28:13.165
So this is hydrogen.
This is sodium.
00:28:16.060 --> 00:28:17.390
This is carbon monoxide.
00:28:21.720 --> 00:28:29.020
So the electron sees a
point charge in hydrogen.
00:28:29.020 --> 00:28:31.510
That's easy.
00:28:31.510 --> 00:28:35.380
Now in sodium-- we
don't know yet why,
00:28:35.380 --> 00:28:38.660
but once I've talked about
helium and many atoms,
00:28:38.660 --> 00:28:41.110
you will know why--
00:28:41.110 --> 00:28:46.270
there are a whole bunch of
electrons in the nucleus
00:28:46.270 --> 00:28:49.930
or in the core of sodium.
00:28:49.930 --> 00:28:52.540
And outside the core,
you have something
00:28:52.540 --> 00:28:54.520
that looks hydrogenic.
00:28:54.520 --> 00:28:55.900
But there is this core.
00:28:55.900 --> 00:28:57.490
It's not a point.
00:28:57.490 --> 00:29:03.580
And that leads to the appearance
of l dependence in the energy
00:29:03.580 --> 00:29:07.330
levels, because what
you're going to find
00:29:07.330 --> 00:29:11.860
is that l equals 0
penetrates into the core.
00:29:11.860 --> 00:29:13.720
l equals 1 can't.
00:29:13.720 --> 00:29:16.360
And so as you have
higher and higher l,
00:29:16.360 --> 00:29:18.860
you're seeing less
and less of this.
00:29:18.860 --> 00:29:22.000
And so as a result,
there is an l dependence.
00:29:22.000 --> 00:29:24.130
But it's an s emphatically
decreasing one.
00:29:24.130 --> 00:29:27.520
As l increases, you get
less and less sensitivity
00:29:27.520 --> 00:29:29.260
to this inner part.
00:29:29.260 --> 00:29:30.985
Now, CO isn't even round.
00:29:35.200 --> 00:29:38.250
But there are nuclei.
00:29:38.250 --> 00:29:43.000
But if you have an electron
at a high enough end,
00:29:43.000 --> 00:29:44.940
it's outside of this core.
00:29:48.010 --> 00:29:51.460
But again, if you pick l--
00:29:51.460 --> 00:29:54.540
now, you can't have l for
something that's not round.
00:29:54.540 --> 00:29:56.460
But if you're far
enough away from it,
00:29:56.460 --> 00:29:59.700
you can pretend it's round
and use the not roundness
00:29:59.700 --> 00:30:01.830
as a perturbation.
00:30:01.830 --> 00:30:04.830
And so you can have
Rydberg states of CO.
00:30:04.830 --> 00:30:07.470
And you can use the same
ideas that we developed
00:30:07.470 --> 00:30:12.030
for the hydrogen atom to explain
the Rydberg states of CO,
00:30:12.030 --> 00:30:12.870
anthracene--
00:30:15.810 --> 00:30:19.220
I don't even know how to say the
names of biological molecules.
00:30:19.220 --> 00:30:21.060
So just consider it.
00:30:21.060 --> 00:30:22.710
If you had it in the
gas phase and you
00:30:22.710 --> 00:30:28.110
could make a high enough
Rydberg, high enough end state,
00:30:28.110 --> 00:30:34.380
it would look following
this kind of extrapolation
00:30:34.380 --> 00:30:39.280
with features
common to hydrogen.
00:30:39.280 --> 00:30:42.780
OK, so now, semiclassical--
00:30:58.190 --> 00:31:03.240
OK, remember, you
can calculate this.
00:31:03.240 --> 00:31:06.570
If you want to use
rigorous mathematics,
00:31:06.570 --> 00:31:08.640
you can find an
analytic solution
00:31:08.640 --> 00:31:12.530
for this, for hydrogen.
And that's good,
00:31:12.530 --> 00:31:14.720
but you haven't extracted
any insight from it.
00:31:14.720 --> 00:31:15.440
You just have it.
00:31:18.910 --> 00:31:21.460
But the semiclassical
theory extracts insight.
00:31:27.020 --> 00:31:29.600
So semiclassical
theory starts out
00:31:29.600 --> 00:31:34.310
with writing the classical
equation for the momentum.
00:31:34.310 --> 00:31:36.460
And the momentum as a
function of r, that's
00:31:36.460 --> 00:31:38.720
a no-no from quantum mechanics.
00:31:38.720 --> 00:31:44.200
You can't ever say we know the
momentum as a function of r.
00:31:44.200 --> 00:31:46.100
And it's also true
in quantum mechanics,
00:31:46.100 --> 00:31:48.770
whenever you
calculate an integral
00:31:48.770 --> 00:31:53.490
that you need to
evaluate some property,
00:31:53.490 --> 00:31:56.560
you're evaluating
it over all space.
00:31:56.560 --> 00:31:58.620
But what I'm going to
show you is that many
00:31:58.620 --> 00:32:02.670
of these integrals
accumulated a specific point
00:32:02.670 --> 00:32:09.370
in space, which is a reminder
of the classical mechanics.
00:32:09.370 --> 00:32:12.760
So there are things where
quantum mechanics says
00:32:12.760 --> 00:32:15.100
you have to do something,
but there's insight
00:32:15.100 --> 00:32:20.590
waiting to be harvested because
many of the things you have
00:32:20.590 --> 00:32:25.360
to calculate are trivially
interpretable in terms
00:32:25.360 --> 00:32:27.280
of semiclassical ideas.
00:32:27.280 --> 00:32:30.880
And the semiclassical
means classical.
00:32:30.880 --> 00:32:32.650
You can do classical mechanics.
00:32:32.650 --> 00:32:34.720
You can calculate
the probability
00:32:34.720 --> 00:32:37.307
of finding a particle
at a particular place.
00:32:37.307 --> 00:32:38.890
And you can use that
information here.
00:32:41.520 --> 00:32:48.780
And so always the relationship
between the classical momentum
00:32:48.780 --> 00:32:54.790
function and the potential
is this equation.
00:33:04.510 --> 00:33:07.300
p squared over 2
mu is the momentum.
00:33:07.300 --> 00:33:12.360
And energy minus
potential is the momentum,
00:33:12.360 --> 00:33:14.590
is the kinetic energy.
00:33:14.590 --> 00:33:15.959
And this is how you get there.
00:33:15.959 --> 00:33:17.125
OK, so this is the function.
00:33:19.720 --> 00:33:22.150
And so that's one of
the important pieces.
00:33:22.150 --> 00:33:25.810
And the other important
piece is really unexpected,
00:33:25.810 --> 00:33:27.730
but wonderful.
00:33:27.730 --> 00:33:34.630
And so we have this Vl of r.
00:33:34.630 --> 00:33:37.780
If l is 0, the inner wall of
the potential is vertical.
00:33:41.460 --> 00:33:44.880
If l is not 0, the inner
wall of the potential
00:33:44.880 --> 00:33:48.000
isn't quite vertical,
but it might as well
00:33:48.000 --> 00:34:01.650
be because when r gets small,
even the l l plus 1 over r
00:34:01.650 --> 00:34:04.770
squared term gets small.
00:34:04.770 --> 00:34:08.969
And so we get this
vertical potential.
00:34:08.969 --> 00:34:14.070
And so what happens is we can
use the idea from DeBroglie
00:34:14.070 --> 00:34:19.400
and say, OK, where
is the first lobe?
00:34:19.400 --> 00:34:20.400
Where is the first node?
00:34:20.400 --> 00:34:24.040
Well, it's half a wavelength
from the turning point.
00:34:24.040 --> 00:34:26.159
So if we know where
the turning point is we
00:34:26.159 --> 00:34:30.320
can draw something
that looks like that.
00:34:30.320 --> 00:34:34.670
Now, it's a wave function, so
it's going to be oscillating.
00:34:34.670 --> 00:34:39.860
So the important thing
is that the first lobe,
00:34:39.860 --> 00:34:49.630
the innermost lobe, of all n
and l lines up because this
00:34:49.630 --> 00:34:54.070
is nearly vertical and the
energy distance from here
00:34:54.070 --> 00:34:58.060
to the bottom of the
potential at r not equal to 0
00:34:58.060 --> 00:35:01.120
is really large.
00:35:01.120 --> 00:35:06.400
And so if you change n or l by
1 or 2, nothing much happens.
00:35:06.400 --> 00:35:09.670
And so if you just say,
OK, for n greater than
00:35:09.670 --> 00:35:17.590
or equal to 6, 6, 36, the
Rydberg constant divided by 36
00:35:17.590 --> 00:35:26.800
is not a big number,
3,000, 3,000 wave numbers.
00:35:26.800 --> 00:35:33.630
The vertical distance is more
than 100,000 wave numbers.
00:35:33.630 --> 00:35:41.340
So the correction
from n equals 6--
00:35:41.340 --> 00:35:43.800
so if we have n
equals 6, n equals 7,
00:35:43.800 --> 00:35:45.090
there isn't much change.
00:35:45.090 --> 00:35:51.070
And so the first lobe is going
to always be at the same place.
00:35:51.070 --> 00:35:54.000
And so one thing that
happens is they line up.
00:35:54.000 --> 00:36:00.310
The first node for all n
and l is at the same place,
00:36:00.310 --> 00:36:02.520
to a good approximation.
00:36:02.520 --> 00:36:09.110
And the amplitude in this
first lobe scales as n--
00:36:09.110 --> 00:36:14.190
so the amplitude scales
n to the minus 3/2.
00:36:14.190 --> 00:36:16.530
I'm going to prove that.
00:36:16.530 --> 00:36:19.630
This is where all the
insight comes from.
00:36:19.630 --> 00:36:25.000
If you have a property, which
is essentially determined
00:36:25.000 --> 00:36:29.750
close to the
nucleus, because you
00:36:29.750 --> 00:36:38.150
have a negative power of the
r in the electronic property,
00:36:38.150 --> 00:36:41.540
then the amplitude
of the wave function
00:36:41.540 --> 00:36:44.420
will scale as n
to the minus 3/2.
00:36:44.420 --> 00:36:51.790
And we can use this in
evaluating all integrals,
00:36:51.790 --> 00:36:57.290
because everybody has a lobe
here, at the same point.
00:36:57.290 --> 00:36:59.890
And the only thing that's
different from one state,
00:36:59.890 --> 00:37:05.440
at one n l and another is
the amplitude of this lobe.
00:37:05.440 --> 00:37:09.220
And that goes as 1
over n to 3/2 power.
00:37:13.960 --> 00:37:21.031
OK, that's what I was afraid of.
00:37:21.031 --> 00:37:21.530
OK.
00:37:31.420 --> 00:37:33.625
So we're interested
in turning points.
00:37:44.040 --> 00:37:46.680
And what's the definition
of a turning point?
00:37:46.680 --> 00:37:49.560
Or what is the mathematical
equation that tells you
00:37:49.560 --> 00:37:54.240
how to get the inner
nuclear dis-- the r
00:37:54.240 --> 00:37:59.510
value for the turning point.
00:37:59.510 --> 00:38:00.160
Yes.
00:38:00.160 --> 00:38:01.370
STUDENT: Derivative equals 0.
00:38:01.370 --> 00:38:02.370
ROBERT FIELD: I'm sorry.
00:38:02.370 --> 00:38:05.210
STUDENT: When the derivative
of V l d r equals 0.
00:38:07.546 --> 00:38:09.170
ROBERT FIELD: I'm
not looking for that.
00:38:09.170 --> 00:38:13.310
I'm looking for--
so when the energy
00:38:13.310 --> 00:38:20.910
is equal to V l of
r, plus or minus.
00:38:20.910 --> 00:38:22.430
That's the equation
that tells you
00:38:22.430 --> 00:38:24.020
where the turning points are.
00:38:24.020 --> 00:38:26.060
And these are simple equations.
00:38:26.060 --> 00:38:32.130
So it's child's play,
adult child's, to calculate
00:38:32.130 --> 00:38:34.380
what the inner and
outer turning point is
00:38:34.380 --> 00:38:38.320
as a function of n and l.
00:38:38.320 --> 00:38:43.370
And so we know the
effective potential.
00:38:43.370 --> 00:38:44.300
We pick an energy.
00:38:44.300 --> 00:38:48.660
We know what it is because it's
the Rydberg over n squared.
00:38:48.660 --> 00:38:55.950
So we have enough to
calculate this function, which
00:38:55.950 --> 00:39:07.370
is a0 n squared 1 plus or
minus 1 minus l l plus 1
00:39:07.370 --> 00:39:12.100
over n squared square root.
00:39:12.100 --> 00:39:13.300
Now I could derive this.
00:39:13.300 --> 00:39:15.850
You could derive this.
00:39:15.850 --> 00:39:18.080
It's a simple closed
form equation.
00:39:18.080 --> 00:39:21.950
And that's the foundation
of all of this.
00:39:21.950 --> 00:39:25.960
Now, the thing that's
common for everything
00:39:25.960 --> 00:39:32.570
is the inner turning
point, r minus,
00:39:32.570 --> 00:39:38.100
because what we're
interested in is this.
00:39:38.100 --> 00:39:44.910
Where is the maximum or
where is the first node
00:39:44.910 --> 00:39:46.817
and what is the amplitude?
00:39:46.817 --> 00:39:49.150
And so we're going to use
this equation to get all that.
00:39:56.610 --> 00:40:03.030
All right, so the semiclassical
theory lambda n l--
00:40:03.030 --> 00:40:05.110
that's the wavelength--
00:40:05.110 --> 00:40:07.906
is equal to h over Pr of r.
00:40:10.570 --> 00:40:12.400
That's DeBroglie.
00:40:12.400 --> 00:40:16.840
But it's generalized
to a potential, which
00:40:16.840 --> 00:40:20.290
is dependent on r, or
a momentum function,
00:40:20.290 --> 00:40:24.520
which is dependent on r.
00:40:24.520 --> 00:40:28.540
So now, we have to calculate
several important things
00:40:28.540 --> 00:40:29.860
in order to build our model.
00:40:36.620 --> 00:40:42.420
One is the classical
oscillation period.
00:40:48.170 --> 00:40:53.480
So if I told you we have
a harmonic oscillator,
00:40:53.480 --> 00:41:00.780
we know the energy level
spacing things are h bar omega.
00:41:00.780 --> 00:41:03.600
And they're constant.
00:41:03.600 --> 00:41:08.340
If we make a superposition
state of a harmonic oscillator,
00:41:08.340 --> 00:41:13.490
there's going to be beat nodes
at integer multiples of omega.
00:41:13.490 --> 00:41:20.610
So we can simply say, well, for
Rydberg states or for anything,
00:41:20.610 --> 00:41:26.210
we can define the period
of oscillation as h over,
00:41:26.210 --> 00:41:33.410
in this case, n plus 1/2
minus n Vn minus 1/2.
00:41:33.410 --> 00:41:38.660
This is the energetic
separation between levels--
00:41:38.660 --> 00:41:40.130
or even not levels.
00:41:40.130 --> 00:41:44.310
We have a formula that's giving
the independence of the energy.
00:41:44.310 --> 00:41:47.810
So this is related to
the period that we get
00:41:47.810 --> 00:41:50.580
from the harmonic oscillator.
00:41:50.580 --> 00:41:53.000
So this is a perfectly
legitimate way
00:41:53.000 --> 00:41:54.840
of knowing the period.
00:41:58.750 --> 00:42:03.450
Well, when we do that, this is
something I wrote down before,
00:42:03.450 --> 00:42:07.710
the energy separation of
two levels differing in n
00:42:07.710 --> 00:42:17.890
by 1 centered at n is going to
be the Rydberg divided by 2n--
00:42:17.890 --> 00:42:19.400
well, I'll just write it down.
00:42:19.400 --> 00:42:20.720
So we have h.
00:42:20.720 --> 00:42:27.180
And now this hcR 2 over n cubed.
00:42:30.440 --> 00:42:34.840
And n cubed is really important.
00:42:34.840 --> 00:42:38.360
It came just from taking
the derivative of the 1
00:42:38.360 --> 00:42:42.230
over n squared energy dependent.
00:42:42.230 --> 00:42:45.260
So this is a formula that's
perfectly legitimate.
00:42:45.260 --> 00:42:48.990
What n tells us is the period
is proportional to n cubed.
00:42:51.970 --> 00:42:53.680
The higher you go,
the slower you go.
00:43:00.260 --> 00:43:10.972
Second, node to
node probability.
00:43:13.630 --> 00:43:18.570
So how long does it
take-- you have this well.
00:43:18.570 --> 00:43:22.980
And I don't mean to draw it
like a harmonic oscillator,
00:43:22.980 --> 00:43:25.270
but it's a well.
00:43:25.270 --> 00:43:28.000
And so you have a node here,
and you have a node here.
00:43:28.000 --> 00:43:31.086
And you have a particle
that's moving, classically.
00:43:31.086 --> 00:43:33.210
How long did it take for
the center of the particle
00:43:33.210 --> 00:43:35.610
to go from here to here?
00:43:35.610 --> 00:43:37.770
That's an easy thing
to calculate too,
00:43:37.770 --> 00:43:40.110
because we know classically
what the momentum is
00:43:40.110 --> 00:43:41.940
at any point in space.
00:43:41.940 --> 00:43:45.000
The momentum is related
to the energy difference
00:43:45.000 --> 00:43:46.170
between here and here.
00:43:52.610 --> 00:43:56.920
So what we're going
for is the ratio
00:43:56.920 --> 00:44:06.840
of the time node to
node, or to next node,
00:44:06.840 --> 00:44:13.100
to delta t turning
to turning point.
00:44:15.780 --> 00:44:20.420
OK, well, this is 1/2
the period, right?
00:44:20.420 --> 00:44:22.640
So we know what
1/2 the period is.
00:44:22.640 --> 00:44:28.700
And the period goes as n cubed.
00:44:28.700 --> 00:44:34.450
So this is the probability of
finding the particle within one
00:44:34.450 --> 00:44:38.000
lobe of the wave function.
00:44:38.000 --> 00:44:44.530
So we can calculate the
probability of finding
00:44:44.530 --> 00:44:47.880
the particle within one lobe.
00:44:47.880 --> 00:44:50.940
And it's clearly going
as 1 over n cubed.
00:44:53.550 --> 00:44:55.544
Now, we want an amplitude.
00:44:55.544 --> 00:44:56.460
What is the amplitude?
00:44:56.460 --> 00:44:59.620
It's square root
of the probability.
00:44:59.620 --> 00:45:04.240
All of a sudden, we start to see
that the amplitude in whatever
00:45:04.240 --> 00:45:09.670
lobe we want goes as
1 over n to the 3/2.
00:45:09.670 --> 00:45:14.620
Now, because we have this lining
up of nodes, the first node
00:45:14.620 --> 00:45:16.620
and the first lobe
being identical,
00:45:16.620 --> 00:45:20.500
what we have now is the
scaling of the probability
00:45:20.500 --> 00:45:23.440
or the amplitude in the first
lobe of the wave function.
00:45:27.000 --> 00:45:34.110
And the next thing
is again something
00:45:34.110 --> 00:45:36.790
that's almost never
talked about in textbooks,
00:45:36.790 --> 00:45:39.930
but it's a really fantastic
tool for understanding
00:45:39.930 --> 00:45:47.640
stuff is suppose we want to
know the value of an integral.
00:45:51.430 --> 00:45:54.340
So we have an integral.
00:45:54.340 --> 00:46:04.430
And that will be, say, the
electronic wave function
00:46:04.430 --> 00:46:06.490
in this chi
representation, as opposed
00:46:06.490 --> 00:46:08.650
to the Rnl representation.
00:46:18.030 --> 00:46:21.060
OK, so this is the
integral we want.
00:46:21.060 --> 00:46:24.450
But what we really want
is a way of saying,
00:46:24.450 --> 00:46:26.790
I can convert this
integral to a number
00:46:26.790 --> 00:46:30.800
that I can figure out on
the back of a postage stamp.
00:46:30.800 --> 00:46:32.210
No methods for integration.
00:46:32.210 --> 00:46:33.650
No numerical methods.
00:46:33.650 --> 00:46:35.420
Just insight.
00:46:35.420 --> 00:46:45.450
So we have some
electronic property,
00:46:45.450 --> 00:46:47.680
which comes from this integral.
00:46:47.680 --> 00:46:52.010
But now, this is a rapidly
oscillating function.
00:46:52.010 --> 00:46:55.650
This is another rapidly
isolating function.
00:46:55.650 --> 00:46:57.960
And so if we were
to ask, well, how
00:46:57.960 --> 00:46:59.880
does the integral accumulate?
00:46:59.880 --> 00:47:01.770
In other words, let's
say we're integrating
00:47:01.770 --> 00:47:12.810
from r minus to r prime chi
n l of r r of k chi n prime l
00:47:12.810 --> 00:47:17.640
prime of r Vr.
00:47:17.640 --> 00:47:25.890
So if we plotted the integral,
the value of the integral,
00:47:25.890 --> 00:47:28.550
as it accumulates--
00:47:28.550 --> 00:47:31.555
so we're not evaluating
the whole integral.
00:47:31.555 --> 00:47:33.530
We're evaluating the
integral up to the r
00:47:33.530 --> 00:47:35.210
point, the r prime point.
00:47:39.380 --> 00:47:50.700
And what we see that there
is a stationary phase point,
00:47:50.700 --> 00:47:55.650
where at some point in
space, the two functions
00:47:55.650 --> 00:47:59.760
are oscillating spatially
at the same frequency.
00:47:59.760 --> 00:48:02.670
And we already saw this for
Fermi's golden rule, right.
00:48:02.670 --> 00:48:04.680
You have a time integral
where you have rapidly
00:48:04.680 --> 00:48:06.060
oscillating functions.
00:48:06.060 --> 00:48:09.960
And then when the
applied frequency
00:48:09.960 --> 00:48:14.310
is resonant with the
difference in frequency,
00:48:14.310 --> 00:48:16.690
we get the integral
accumulating.
00:48:16.690 --> 00:48:17.940
So same thing.
00:48:17.940 --> 00:48:19.830
It's easier.
00:48:19.830 --> 00:48:25.380
And so we have this idea that
there is a stationary phase
00:48:25.380 --> 00:48:26.520
point.
00:48:26.520 --> 00:48:29.580
And the integral accumulates
from 0 to its final value
00:48:29.580 --> 00:48:31.120
there.
00:48:31.120 --> 00:48:33.870
And then there's
just little dithering
00:48:33.870 --> 00:48:36.140
as you go out the
rest of the way.
00:48:36.140 --> 00:48:38.610
You get in complications when
they're two stationary phase
00:48:38.610 --> 00:48:41.490
points, because then there's
destructive or constructive
00:48:41.490 --> 00:48:42.820
interference between them.
00:48:42.820 --> 00:48:45.060
But it's very rare
that you have two.
00:48:45.060 --> 00:48:46.330
You only have one.
00:48:46.330 --> 00:48:50.850
And so if you know the
amplitude of the wave functions
00:48:50.850 --> 00:48:55.140
at a stationary phase point,
you can calculate the integral
00:48:55.140 --> 00:48:56.790
as a product of three things.
00:48:59.460 --> 00:49:02.180
No integration, just one number.
00:49:02.180 --> 00:49:04.730
And the thing that
we care about is
00:49:04.730 --> 00:49:10.070
that almost all
electronic properties
00:49:10.070 --> 00:49:16.530
accumulate in the innermost
lobe, because they're the same.
00:49:16.530 --> 00:49:18.650
The wave functions are the same.
00:49:18.650 --> 00:49:20.150
And they get more
and more different
00:49:20.150 --> 00:49:23.310
as you go farther out.
00:49:23.310 --> 00:49:25.100
And as a result,
since the amplitude
00:49:25.100 --> 00:49:31.090
in this innermost lobe
goes as 1 over n to the 3/2
00:49:31.090 --> 00:49:32.700
and you've got two functions--
00:49:32.700 --> 00:49:34.630
you've got the nl
function and the n prime l
00:49:34.630 --> 00:49:37.310
prime function-- you
have that the integral
00:49:37.310 --> 00:49:45.053
is proportional to 1 over
n to the 3/2 1 over n
00:49:45.053 --> 00:49:48.310
prime to the 3/2.
00:49:48.310 --> 00:49:50.710
So not only do you get
expectation values,
00:49:50.710 --> 00:49:54.030
you get off diagonal
matrix elements.
00:49:54.030 --> 00:49:56.230
Now this is not exact.
00:49:56.230 --> 00:50:00.370
But it tells you this is the
structure of the problem.
00:50:00.370 --> 00:50:03.400
And if I look at
enough stuff, this
00:50:03.400 --> 00:50:09.156
is really important to say,
how does everything scale?
00:50:09.156 --> 00:50:10.530
And if you know
how things scale,
00:50:10.530 --> 00:50:13.020
you know that if
you're combining
00:50:13.020 --> 00:50:15.300
things that aren't part
of that scaling rule,
00:50:15.300 --> 00:50:19.030
that they're just not
going to be relevant.
00:50:19.030 --> 00:50:22.670
This is a fantastic
simplification,
00:50:22.670 --> 00:50:26.000
because, yes, you
can do it exactly
00:50:26.000 --> 00:50:28.010
by programming your
computer to calculate
00:50:28.010 --> 00:50:29.550
the integral numerically.
00:50:29.550 --> 00:50:31.010
And that's not a big deal.
00:50:31.010 --> 00:50:33.080
But you don't know anything.
00:50:33.080 --> 00:50:35.930
And here, you know
what to expect.
00:50:35.930 --> 00:50:39.140
And this is a professional
spectroscopist
00:50:39.140 --> 00:50:43.800
who survives by recognizing
patterns and using patterns.
00:50:43.800 --> 00:50:46.760
This is beyond
numerical integrals.
00:50:46.760 --> 00:50:50.360
This is actually understanding
the structure of the problem.
00:50:50.360 --> 00:50:55.050
And the fact that the molecule
gives you this beautiful--
00:50:55.050 --> 00:50:57.770
or the atom gives you
this beautiful lobe
00:50:57.770 --> 00:51:01.520
always at the same place,
you know stationary phase.
00:51:01.520 --> 00:51:03.020
Now, there are
other problems where
00:51:03.020 --> 00:51:05.900
you use the stationary
phase approximation.
00:51:05.900 --> 00:51:09.080
You always want to be trying
to use stationary phase.
00:51:09.080 --> 00:51:13.100
For example-- and I'm
going to stop soon--
00:51:13.100 --> 00:51:16.349
suppose you have two
potential energy curves.
00:51:16.349 --> 00:51:18.140
Now, you don't know
what a potential energy
00:51:18.140 --> 00:51:20.080
curve for a molecule is yet.
00:51:20.080 --> 00:51:22.100
But you can imagine
what there is.
00:51:22.100 --> 00:51:24.200
And so they cross.
00:51:24.200 --> 00:51:32.120
And this is the place
where at this energy,
00:51:32.120 --> 00:51:36.290
this curve is exactly the same
as the momentum on that curve.
00:51:36.290 --> 00:51:40.310
And so the integral accumulates
at this curve crossing point.
00:51:40.310 --> 00:51:42.246
Isn't that neat?
00:51:42.246 --> 00:51:44.120
If you know what the
curve crossing point is,
00:51:44.120 --> 00:51:46.310
you know the amplitude
of the wave function.
00:51:46.310 --> 00:51:49.930
And you can estimate
any integral.
00:51:49.930 --> 00:51:54.250
So this is something you
would use if you're actually
00:51:54.250 --> 00:51:59.080
creating new knowledge again
and again, because it's deeper
00:51:59.080 --> 00:52:01.480
than the actual
experimental observation.
00:52:01.480 --> 00:52:03.220
It's the explanation for it.
00:52:03.220 --> 00:52:05.680
And it's what you're
looking for and how
00:52:05.680 --> 00:52:10.310
you use your knowledge of
what you're looking for.
00:52:10.310 --> 00:52:13.640
That's why I like this
lecture, because it really
00:52:13.640 --> 00:52:20.520
gives the approach that I take
to all scientific problems.
00:52:20.520 --> 00:52:23.210
I look for an
approximate way where
00:52:23.210 --> 00:52:26.690
I don't have to do any
complicated mathematics
00:52:26.690 --> 00:52:28.040
or numerical integration.
00:52:28.040 --> 00:52:32.880
I can do that just
sitting at my desk.
00:52:32.880 --> 00:52:37.420
OK, so on Friday, we'll hear
about helium and why helium
00:52:37.420 --> 00:52:39.840
looks horrible, but isn't.