WEBVTT

00:00:00.750 --> 00:00:03.780
PROFESSOR: Today we have to
continue with our discussion

00:00:03.780 --> 00:00:06.300
of the hydrogen atom.

00:00:06.300 --> 00:00:11.100
We had derived or
explained how you

00:00:11.100 --> 00:00:16.410
would derive the corrections
to the original Hamiltonian,

00:00:16.410 --> 00:00:21.210
the Hamiltonian you've studied
already in a couple of courses,

00:00:21.210 --> 00:00:25.200
this h0 Hamiltonian
for the hydrogen atom

00:00:25.200 --> 00:00:31.290
that has a kinetic term
and a potential term.

00:00:34.310 --> 00:00:37.320
This Hamilton we've
studied for a while.

00:00:37.320 --> 00:00:44.360
And what we've said was that the
Dirac equation provided a way

00:00:44.360 --> 00:00:49.520
to get systematically all the
corrections, the first order

00:00:49.520 --> 00:00:53.480
of that Hamiltonian, and
that's what we have here.

00:00:53.480 --> 00:00:58.880
That was the end product of that
discussion in which the better

00:00:58.880 --> 00:01:02.420
Hamiltonian or the perturbed
Hamiltonian including more

00:01:02.420 --> 00:01:06.860
effects was the original one
plus a relativistic correction

00:01:06.860 --> 00:01:11.330
because this is not the
relativistic kinetic energy.

00:01:11.330 --> 00:01:13.930
Then there was a
spin orbit coupling

00:01:13.930 --> 00:01:16.820
in which you couple the
spin of the electron

00:01:16.820 --> 00:01:19.370
to the orbital angular
momentum of the vector.

00:01:21.900 --> 00:01:25.050
And finally, there
is a Darwin term.

00:01:25.050 --> 00:01:26.640
It's a surprising term.

00:01:26.640 --> 00:01:30.660
This is Charles Gaston
Darwin, a British physicist,

00:01:30.660 --> 00:01:34.660
that discovered this term.

00:01:34.660 --> 00:01:43.120
And all these terms were
suppressed with respect to h0,

00:01:43.120 --> 00:01:46.650
in the sense that
h0 had energies.

00:01:46.650 --> 00:01:49.450
The energies
associated to h0 went

00:01:49.450 --> 00:01:54.550
like alpha squared mc squared.

00:01:54.550 --> 00:01:59.830
The fine structure constant,
that's for h0 veera.

00:01:59.830 --> 00:02:06.210
But for h1 or for
delta h, the energies

00:02:06.210 --> 00:02:09.449
go like alpha to the
fourth mc squared,

00:02:09.449 --> 00:02:11.670
where m is the mass
of the electron,

00:02:11.670 --> 00:02:17.970
and that's about 20,000,
19,000 times smaller.

00:02:17.970 --> 00:02:20.420
So this is our
perturbation, and that's

00:02:20.420 --> 00:02:21.740
what you have to understand.

00:02:21.740 --> 00:02:26.810
So we'll begin with
the Darwin term.

00:02:26.810 --> 00:02:29.600
Then we'll turn to
the relativistic term,

00:02:29.600 --> 00:02:32.960
then to the spin orbit
term, and, by the end

00:02:32.960 --> 00:02:36.180
of today's lecture, we'll
put it all together.

00:02:36.180 --> 00:02:39.630
So that's our plan.

00:02:39.630 --> 00:02:42.620
So let's start with
the Darwin term.

00:02:42.620 --> 00:02:45.490
What is this term?

00:02:45.490 --> 00:02:55.690
So Darwin term, I will
try to evaluate it.

00:02:55.690 --> 00:02:58.240
So it depends on the
potential energy.

00:02:58.240 --> 00:03:02.110
It has the Laplacian of
the potential energy.

00:03:02.110 --> 00:03:07.820
So the potential energy
is minus e squared over r.

00:03:07.820 --> 00:03:12.260
So how much is the Laplacian
of the potential energy?

00:03:12.260 --> 00:03:16.820
Well, the Laplacian of v
would be minus e squared

00:03:16.820 --> 00:03:21.610
times the Leplacian of 1 over r.

00:03:21.610 --> 00:03:30.190
And that's minus e squared times
minus 4 pi delta function of r.

00:03:30.190 --> 00:03:35.390
That's something you study in
in E and M. The Laplacian of 1

00:03:35.390 --> 00:03:39.910
over r is related
to a delta function.

00:03:39.910 --> 00:03:44.080
It's the charge density
of a point particle

00:03:44.080 --> 00:03:46.420
that produces that potential.

00:03:46.420 --> 00:03:51.190
So our result here is that
this Laplacian is e square 4

00:03:51.190 --> 00:03:57.470
pi e squared delta of r.

00:03:57.470 --> 00:04:03.080
It's a delta function
contribution.

00:04:03.080 --> 00:04:05.910
So let's write it here.

00:04:12.290 --> 00:04:25.510
The delta h Darwin
is pi over 2 e

00:04:25.510 --> 00:04:32.110
squared h squared over m
squared c squared delta of r.

00:04:35.720 --> 00:04:40.180
All right, so that's
our correction.

00:04:40.180 --> 00:04:42.910
And we want to do
perturbation theory with it,

00:04:42.910 --> 00:04:48.370
how it affects the energy
levels of the various states

00:04:48.370 --> 00:04:51.640
of hydrogen. So how
are they changed?

00:04:51.640 --> 00:04:54.380
What does it do to them?

00:04:54.380 --> 00:04:59.310
Now, there is a simplifying
fact here, the delta function.

00:04:59.310 --> 00:05:03.280
So you remember, first
order corrections

00:05:03.280 --> 00:05:07.510
are always obtained by taking
the interactive Hamiltonian

00:05:07.510 --> 00:05:09.130
and putting states in it.

00:05:11.960 --> 00:05:15.310
And the first thing we notice
is that unless the wave

00:05:15.310 --> 00:05:19.900
function of the states
does not vanish at 0,

00:05:19.900 --> 00:05:21.470
the correction vanishes.

00:05:21.470 --> 00:05:29.480
So this will pick up the
value of the wave functions.

00:05:29.480 --> 00:05:32.210
When you have two
states, arbitrary

00:05:32.210 --> 00:05:36.980
state 1 and state 2 and
delta h Darwin here--

00:05:39.850 --> 00:05:41.480
there's two wave
functions that you're

00:05:41.480 --> 00:05:44.500
going to put here if you're
trying to compute matrix

00:05:44.500 --> 00:05:46.960
elements of the perturbation.

00:05:46.960 --> 00:05:54.310
And if either of
these wave functions

00:05:54.310 --> 00:05:57.310
vanishes at the origin,
that's not possible.

00:05:57.310 --> 00:06:00.910
But all wave functions
vanish at the origin,

00:06:00.910 --> 00:06:05.980
unless the orbital angular
momentum l is equal to 0.

00:06:05.980 --> 00:06:08.920
Remember, all wave
functions go like r

00:06:08.920 --> 00:06:11.860
to the l near the origin.

00:06:11.860 --> 00:06:16.720
So for l equals 0, that
goes like 1, a constant,

00:06:16.720 --> 00:06:19.930
and you get a possibility.

00:06:19.930 --> 00:06:30.540
So this only affects
l equals 0 states.

00:06:30.540 --> 00:06:34.140
That's a great simplicity.

00:06:34.140 --> 00:06:39.600
Not only does that, but
that's another simplification.

00:06:39.600 --> 00:06:47.860
Because at any energy
level, l equals

00:06:47.860 --> 00:06:50.290
0 states is just one of them.

00:06:50.290 --> 00:06:53.290
If you consider spin,
there are two of them.

00:06:53.290 --> 00:06:56.370
Remember in our table
of hydrogen atoms

00:06:56.370 --> 00:07:00.770
states go like that.

00:07:00.770 --> 00:07:04.800
So here are the l
equals 0 states.

00:07:04.800 --> 00:07:08.640
And those are the
only ones that matter.

00:07:08.640 --> 00:07:12.209
So the problem is
really very simple.

00:07:12.209 --> 00:07:14.250
We don't have to consider
the fact that there are

00:07:14.250 --> 00:07:17.310
other degenerate states here.

00:07:17.310 --> 00:07:19.350
We just need to
focus on l equals

00:07:19.350 --> 00:07:23.350
0 states because both
states have to be l equals 0

00:07:23.350 --> 00:07:26.910
and l equals 0.

00:07:26.910 --> 00:07:39.452
So our correction that we can
call e1 Darwin for n 0 0--

00:07:39.452 --> 00:07:44.570
because l is equal to 0, and
therefore m is equal to 0--

00:07:44.570 --> 00:07:59.020
would be equal to psi n zero
zero delta h Darwin psi n 0 0.

00:08:06.700 --> 00:08:09.970
So what is this?

00:08:09.970 --> 00:08:13.550
We have this expression
for delta h Darwin.

00:08:13.550 --> 00:08:19.840
So let's put it here, pi over
2 e squared h squared over m

00:08:19.840 --> 00:08:23.000
squared c squared.

00:08:23.000 --> 00:08:29.270
And this will pick up the
values of the wave functions.

00:08:29.270 --> 00:08:34.270
This overlap means integral
over all of space of this wave

00:08:34.270 --> 00:08:38.330
function complex conjugated,
this wave function and delta

00:08:38.330 --> 00:08:39.289
h Darwin.

00:08:39.289 --> 00:08:43.380
So at the end of the day,
due the delta function,

00:08:43.380 --> 00:08:49.935
it gives us just psi n 0
0 at the origin squared.

00:08:53.080 --> 00:08:55.710
That's all it is.

00:08:55.710 --> 00:08:56.620
Simple enough.

00:09:00.750 --> 00:09:09.290
Now, finding this
number is not that easy.

00:09:09.290 --> 00:09:16.020
Because while the wave
function here for l equals 0

00:09:16.020 --> 00:09:18.720
is simple for the
ground state, it already

00:09:18.720 --> 00:09:21.030
involves more and
more complicated

00:09:21.030 --> 00:09:22.830
polynomials over here.

00:09:22.830 --> 00:09:26.370
And the value of the wave
function at the origin

00:09:26.370 --> 00:09:30.419
requires that you normalize
the wave function correctly.

00:09:30.419 --> 00:09:32.460
So if you have the wave
function and you have not

00:09:32.460 --> 00:09:36.330
normalized it
properly, how are you

00:09:36.330 --> 00:09:40.390
going to get the value of the
wave function at the origin?

00:09:40.390 --> 00:09:42.570
This whole perturbation
theory, we always

00:09:42.570 --> 00:09:45.150
assume we have an
orthonormal basis.

00:09:45.150 --> 00:09:48.780
And indeed, if you change
the normalization--

00:09:48.780 --> 00:09:53.250
if the normalization
was irrelevant here,

00:09:53.250 --> 00:09:55.770
this number would change
with the normalization,

00:09:55.770 --> 00:09:57.300
and the correction would change.

00:09:57.300 --> 00:10:00.810
So you really have to
be normalized for this

00:10:00.810 --> 00:10:02.520
to make sense.

00:10:02.520 --> 00:10:09.000
And finding this
normalization is complicated.

00:10:09.000 --> 00:10:12.720
You could for a few problems
maybe look look up tables

00:10:12.720 --> 00:10:16.150
and see the normalized
wave function, what it is.

00:10:16.150 --> 00:10:20.250
But in general here,
there is a method

00:10:20.250 --> 00:10:25.090
that can be used to find this
normalization analytically.

00:10:25.090 --> 00:10:27.720
And it's something you will
explore in the homework.

00:10:27.720 --> 00:10:31.050
So in the homework
there is a way

00:10:31.050 --> 00:10:38.490
to do this, a very clever way,
in which it actually turns out.

00:10:38.490 --> 00:10:44.920
So in the homework, you will
see that the wave function

00:10:44.920 --> 00:10:53.460
at the origin for l
equals 0 problems is

00:10:53.460 --> 00:10:56.490
proportional to the
expectation value

00:10:56.490 --> 00:11:00.180
of the derivative of the
potential with respect to r.

00:11:02.710 --> 00:11:06.700
So it's something you will do.

00:11:06.700 --> 00:11:10.540
This potential, of
course, is the 1

00:11:10.540 --> 00:11:12.796
over r potential in our case.

00:11:12.796 --> 00:11:14.170
And the derivative
means that you

00:11:14.170 --> 00:11:16.810
need to evaluate the
expectation value of 1

00:11:16.810 --> 00:11:21.940
over r squared in this state.

00:11:21.940 --> 00:11:25.990
But the expectation values of 1
over r squared in the hydrogen

00:11:25.990 --> 00:11:30.774
atom is something you've already
done in the previous homework.

00:11:30.774 --> 00:11:32.430
It's not that difficult.

00:11:32.430 --> 00:11:39.110
So the end result is that
this term is calculable.

00:11:39.110 --> 00:11:48.680
And psi of n 0 0 at
the origin squared

00:11:48.680 --> 00:11:53.800
is actually 1 over
pi n cube a0 cube.

00:11:58.730 --> 00:12:05.190
With that in, one can
substitute this value

00:12:05.190 --> 00:12:07.250
and get the Darwin correction.

00:12:10.070 --> 00:12:13.520
You can see there is no
big calculation to be done.

00:12:13.520 --> 00:12:16.760
But you can rewrite it in
terms of the fine structure

00:12:16.760 --> 00:12:21.430
constant Darwin.

00:12:21.430 --> 00:12:26.200
And it's equal to alpha
to the fourth mc squared 1

00:12:26.200 --> 00:12:28.840
over 2n cubed.

00:12:28.840 --> 00:12:31.735
And it's valid for
l equals 0 states.

00:12:39.780 --> 00:12:43.860
So this number goes in
here, and then write

00:12:43.860 --> 00:12:46.920
things in terms of the
fine structure constant,

00:12:46.920 --> 00:12:50.660
and then out comes
this resolved.