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PROFESSOR: Then let's talk
about exciting physics, the Lamb

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00:00:23,531 --> 00:00:24,030
shift.

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00:00:27,130 --> 00:00:32,170
So we discussed the
Lamb shift last Friday,

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00:00:32,170 --> 00:00:35,570
and the Lamb shift is
really due to the fact

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that, if you have
a atom consisting

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of an electron, a Coulomb
field, and the proton,

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this is not the
complete description.

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00:00:43,820 --> 00:00:48,040
The atom lives in a
vacuum, and the vacuum

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00:00:48,040 --> 00:00:51,010
is filled with
electromagnetic waves.

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00:00:51,010 --> 00:00:54,070
So what we have to include,
for an accurate description

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00:00:54,070 --> 00:00:58,050
of atomic level
structure, is the coupling

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of the atomic system, of the
electron, to all of the modes

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00:01:02,390 --> 00:01:07,350
the electromagnetic field,
and this is radiation.

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We will talk about the quantized
electromagnetic field later,

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00:01:11,480 --> 00:01:14,230
at this point I
could introduce you

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00:01:14,230 --> 00:01:17,810
to a simple model, fairly
accurate model of the Lamb

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00:01:17,810 --> 00:01:21,190
shift, by simply
assuming, and that's

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00:01:21,190 --> 00:01:26,280
what we did on Friday,
that we have fluctuating

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00:01:26,280 --> 00:01:30,200
electric fields, those
fluctuating electric fields

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00:01:30,200 --> 00:01:34,310
shake, accelerate, the
electron and the electron

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00:01:34,310 --> 00:01:39,590
performs some oscillatory motion
and this oscillatory motion

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00:01:39,590 --> 00:01:44,890
leads to an ever reaching
of the Coulomb potential.

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00:01:44,890 --> 00:01:56,820
And similarly what we
saw for the Darwin term,

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00:01:56,820 --> 00:01:59,580
this ever reaching of
the Coulomb potential

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00:01:59,580 --> 00:02:03,750
takes away this singularity
of the Coulomb potential

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00:02:03,750 --> 00:02:10,850
and therefore lowers the
binding energy of the electron.

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00:02:10,850 --> 00:02:14,190
So today I want to just say a
few more words about the result

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00:02:14,190 --> 00:02:20,960
we derived, and
then we have done

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00:02:20,960 --> 00:02:24,900
what happens to an electron in
a Coulomb field plus radiation.

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00:02:24,900 --> 00:02:29,770
The next thing is, then, to
discuss hyperfine structure.

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00:02:29,770 --> 00:02:32,890
So let me first make
one comment, when

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00:02:32,890 --> 00:02:36,550
we integrated over all modes of
the electromagnetic spectrum,

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00:02:36,550 --> 00:02:38,580
we needed an upper
cutoff, and a lower

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00:02:38,580 --> 00:02:42,040
cutoff due to logarithmic
singularities.

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00:02:42,040 --> 00:02:46,410
Eventually, an upper cutoff
is relativistic rest mass

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00:02:46,410 --> 00:02:48,870
of the electron we
have to cut off things.

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00:02:48,870 --> 00:02:53,150
I just want to say one more
word about the lower cutoff.

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00:02:53,150 --> 00:02:56,220
I suggested, as a cut
off, the orbital frequency

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00:02:56,220 --> 00:02:57,940
of the electron.

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00:02:57,940 --> 00:03:04,690
So the justification for that
is the following, the electron,

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00:03:04,690 --> 00:03:15,610
the free electron-- the free
electron, when it's driven,

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00:03:15,610 --> 00:03:20,420
has an amplitude which is 1
over the frequency squared.

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00:03:20,420 --> 00:03:22,990
So if you drive it
slower and slower,

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00:03:22,990 --> 00:03:24,890
it's amplitude
increases because it

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00:03:24,890 --> 00:03:27,430
has more time to go
in one direction.

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00:03:27,430 --> 00:03:30,640
So this divergence at
low frequency, of course,

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00:03:30,640 --> 00:03:32,900
happens only for
the free system.

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00:03:32,900 --> 00:03:36,000
When you have a bound system,
like a mnemonic oscillate,

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00:03:36,000 --> 00:03:39,140
and you drive it at lower
and lower frequency,

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the response converges
to a constant

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00:03:43,270 --> 00:03:47,550
and not to 1 over high
frequency squared singularity.

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00:03:47,550 --> 00:03:52,160
So therefore, when we reach the
drive and the high frequencies

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00:03:52,160 --> 00:03:58,770
on the order of-- oscillatory
frequency of the bound system,

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the behavior changes.

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00:04:00,800 --> 00:04:04,960
So when we mimic this with
an effect cutoff because what

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you want to get rid of is a
singularity, but in reality,

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00:04:07,970 --> 00:04:11,870
of course, it should approach
a constant at low frequency,

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and that is essentially the
physics of the AC and then

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the DC Stark effect.

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00:04:19,390 --> 00:04:27,853
So that was a rational
for the cutoff

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and let me just
annotate it here.

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00:04:32,610 --> 00:04:37,690
And let's say, for
a free particle

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we had the situation that the
amplitude was proportional to 1

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00:04:41,490 --> 00:04:44,390
over the high frequency.

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00:04:44,390 --> 00:04:48,750
Whereas four bound
particle is, you

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00:04:48,750 --> 00:04:54,679
should approach a constant
for low frequencies.

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00:04:54,679 --> 00:04:56,720
And that's what we have
introduced with a cutoff.

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00:05:02,451 --> 00:05:02,950
OK.

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00:05:05,550 --> 00:05:09,040
So we have discussed
the Lamb shift,

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but you've already discussed one
contribution to the Lamb shift.

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So this was a contribution
that the Coulomb potential

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is effectively smeared
out, and the result of this

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is that there is a
weaker binding energy.

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However, there is a second
contribution to the Lamb shift.

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00:05:54,790 --> 00:05:56,700
I've sometimes
seen sources where

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00:05:56,700 --> 00:06:00,170
this is discussed as the main
contribution to the Lamb shift,

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but this is not correct.

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00:06:02,710 --> 00:06:08,800
This is only 3% off the total
and contributes 27 megahertz.

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00:06:08,800 --> 00:06:11,460
This is what is called
the vacuum polarization.

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00:06:18,452 --> 00:06:28,570
So if you have a-- the
proton positive charge

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00:06:28,570 --> 00:06:33,460
and you have the electron, and
they scatter off each other,

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00:06:33,460 --> 00:06:40,230
and we use this kind of
diagram to indicate that,

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00:06:40,230 --> 00:06:45,060
now that there is an
additional diagram which

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has this bubble which is
this production of e minus

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00:06:49,120 --> 00:06:52,100
and e plus pairs.

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00:06:52,100 --> 00:06:56,420
And you can say that if an
electron and proton attract

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00:06:56,420 --> 00:07:03,180
each other and you create,
by virtual pair production--

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00:07:03,180 --> 00:07:05,940
because the vacuum
is alive, things

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00:07:05,940 --> 00:07:07,850
can happen in the
vacuum-- it will virutally

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00:07:07,850 --> 00:07:10,810
[INAUDIBLE] electron
positron pair that now you

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00:07:10,810 --> 00:07:16,970
create electron positron pairs,
which shield the Coulomb field.

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00:07:16,970 --> 00:07:18,960
And this is a
second contribution

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00:07:18,960 --> 00:07:22,300
in addition to the shaking
motion of the electron.

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00:07:22,300 --> 00:07:24,880
So I want you to just think
about it for 10 seconds

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before I tell you the answer.

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00:07:26,700 --> 00:07:33,870
Does this vacuum polarization
strengthen or weaken

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00:07:33,870 --> 00:07:34,670
the binding energy?

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00:07:43,946 --> 00:07:45,320
Anybody want to
offer an opinion?

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00:07:49,670 --> 00:07:51,465
Whatever you say,
you can't be wrong

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00:07:51,465 --> 00:07:53,423
because there are to
aspects to the answers so.

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00:07:58,060 --> 00:08:00,240
Well then the actual
answer is you would say,

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00:08:00,240 --> 00:08:03,290
if you have charged--
if you create charges

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00:08:03,290 --> 00:08:06,140
between the electron
and the proton,

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00:08:06,140 --> 00:08:09,100
you have a shielding effect
and the shielding effect

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00:08:09,100 --> 00:08:11,270
should weaken the Coulomb field.

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00:08:11,270 --> 00:08:14,110
But the question is,
what do we regard

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00:08:14,110 --> 00:08:18,180
as the elementary charge, e,
in our Schrodinger equation.

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00:08:18,180 --> 00:08:20,645
And what we regard
as the charge, what

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00:08:20,645 --> 00:08:23,530
is measured in a Millikan
Droplet experiment,

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00:08:23,530 --> 00:08:26,250
is already the shielded charge.

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00:08:26,250 --> 00:08:29,330
So therefore, the fact
that vacuum polarization

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00:08:29,330 --> 00:08:33,000
exists means we always
measure the shielded charge,

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00:08:33,000 --> 00:08:36,970
but because vacuum polarization
happens at a finite distance,

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00:08:36,970 --> 00:08:41,419
the electron, in an s state,
can sort of penetrate the shield

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00:08:41,419 --> 00:08:44,980
and feel a somewhat
stronger Coulomb potential.

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00:08:44,980 --> 00:08:47,930
So therefore,
accounting for the fact

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00:08:47,930 --> 00:08:50,720
that we have these
virtual electron

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00:08:50,720 --> 00:08:53,906
positron pairs actually
means that the binding energy

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00:08:53,906 --> 00:08:57,350
is increased and the
vacuum polarization

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00:08:57,350 --> 00:09:00,480
has the opposite sign
as the dominant effect

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00:09:00,480 --> 00:09:01,480
we've mentioned earlier.

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00:09:04,160 --> 00:09:20,910
So the sum of e is, we observe
the shielded charge and vacuum

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00:09:20,910 --> 00:09:29,620
polarization implies that
the s electron can penetrate

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00:09:29,620 --> 00:09:34,375
the shield and sees
a higher charge.

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00:09:40,250 --> 00:09:43,910
So we normally observe
the shielded charge,

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but the electron can
see the higher charge.

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00:09:47,000 --> 00:09:51,740
So therefore, that means, now,
that we have, not an upshift

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00:09:51,740 --> 00:09:58,060
in energy, but a downshift in
energy for-- which for the two

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00:09:58,060 --> 00:10:03,860
is one half state,
is 27 megahertz,

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00:10:03,860 --> 00:10:07,060
as I mentioned earlier.

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00:10:07,060 --> 00:10:09,290
Anyway, I'm not
really deriving it,

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00:10:09,290 --> 00:10:16,500
but I want to sort of uncover
certain myths, so the vacuum

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00:10:16,500 --> 00:10:20,240
polarization, number one, is
not dominant and number two,

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00:10:20,240 --> 00:10:23,420
has the opposite sign as
everybody would naively assume.

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00:10:25,845 --> 00:10:26,345
Questions?

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00:10:29,140 --> 00:10:33,430
I hope not because I don't
know anything more about that.

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00:10:33,430 --> 00:10:34,600
All right.

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00:10:34,600 --> 00:10:40,160
So we have dealt
with Lamb shift.

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00:10:40,160 --> 00:10:43,310
So is next now, in revealing
the atomic structure,

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00:10:43,310 --> 00:10:48,160
is we want to go beyond
the Coulomb field created

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by a point charge.

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00:10:49,970 --> 00:10:54,380
And that means we want to
address the fact that we don't

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00:10:54,380 --> 00:10:57,900
have a point charge,
but we have a nucleolus.

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00:10:57,900 --> 00:11:08,200
And we are discussing, now,
effects of the nucleus, which

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00:11:08,200 --> 00:11:12,410
also go by the name
hyperfine structure.

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00:11:12,410 --> 00:11:16,610
So, just to summarize,
so far we have

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00:11:16,610 --> 00:11:23,580
treated, in pretty much
complete detail, what

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00:11:23,580 --> 00:11:30,040
happens for an atom which
consists of a point charge

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00:11:30,040 --> 00:11:30,680
and electrons.

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00:11:33,900 --> 00:11:36,870
But now we want
to bring in that,

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00:11:36,870 --> 00:11:40,285
what creates a Coulomb field,
the nucleolus, has structure.

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00:11:47,340 --> 00:11:51,090
And there are actually
four different ways

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00:11:51,090 --> 00:11:55,730
how the nucleolus has
structure and contributions

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00:11:55,730 --> 00:11:59,580
to observable effects
on the atomic structure

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00:11:59,580 --> 00:12:01,700
and atomic energy levels.

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00:12:01,700 --> 00:12:05,560
The most important one
is that the nucleus

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has a magnetic moment
associated with the angular

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00:12:09,010 --> 00:12:13,560
momentum of the nucleus I.

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00:12:13,560 --> 00:12:18,180
The second contribution is, in
addition to magnetic moment,

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00:12:18,180 --> 00:12:19,950
there may be a
quadrupole moment.

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00:12:22,700 --> 00:12:27,080
And since those effects can
lead to a splitting-- this

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00:12:27,080 --> 00:12:31,190
is actually, usually,
called hyperfine structure,

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00:12:31,190 --> 00:12:33,030
but then there are
two more effects.

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00:12:33,030 --> 00:12:39,180
One is the nucleus
has finite mass,

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00:12:39,180 --> 00:12:41,295
and the nucleus has
a finite volume.

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00:12:44,200 --> 00:12:51,390
Both the mass and volume
effect lead to energy shifts.

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00:12:51,390 --> 00:12:54,540
But tiny energy shifts
are hard to measure

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00:12:54,540 --> 00:12:56,680
unless you have two
different shifts,

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00:12:56,680 --> 00:13:01,860
and therefore those effects
go as isotope shifts

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00:13:01,860 --> 00:13:03,750
because when you
have an atom which

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00:13:03,750 --> 00:13:05,850
comes in two
different isotopes you

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00:13:05,850 --> 00:13:08,660
find that, due to
those two effects,

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00:13:08,660 --> 00:13:12,580
the energy levels
are not the same.

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00:13:12,580 --> 00:13:14,950
So this goes by the
name of isotope shifts.

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00:13:17,730 --> 00:13:23,780
By far the most important
phenomenon is the first one.

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00:13:23,780 --> 00:13:27,040
The fact that if a nucleus
has angular momentum,

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00:13:27,040 --> 00:13:35,170
we have hyperfine structure,
and for the hydrogen atom

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00:13:35,170 --> 00:13:41,770
that means that the ground
state, the singlet S 1/2 state,

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00:13:41,770 --> 00:13:46,760
actually splits into two states
with total angular momentum

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00:13:46,760 --> 00:13:58,180
quantum number F.

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00:13:58,180 --> 00:14:04,610
So the relevance of
hyperfine splitting

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00:14:04,610 --> 00:14:08,870
is, it's actually
a huge relevance,

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00:14:08,870 --> 00:14:13,390
one is, you don't have a single
ground state of many atoms,

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00:14:13,390 --> 00:14:15,435
you have several ground states.

192
00:14:21,350 --> 00:14:24,470
So the lowest electronic state
has several ground states,

193
00:14:24,470 --> 00:14:29,360
has several hyperfine states,
due to angular momentum

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00:14:29,360 --> 00:14:30,850
selection, where
you can actually

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00:14:30,850 --> 00:14:35,010
talk to them individually, you
can prepare them individually,

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00:14:35,010 --> 00:14:37,840
and many of you who do
magnetic trapping know when

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00:14:37,840 --> 00:14:39,990
need magnetic trapping
you better prepare

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00:14:39,990 --> 00:14:43,110
the atom in a single
hyperfine state,

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00:14:43,110 --> 00:14:46,700
otherwise you are in trouble.

200
00:14:46,700 --> 00:14:50,790
So you can prepare
individual states.

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00:14:50,790 --> 00:14:57,570
In the old days this was
done by optical pumping,

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00:14:57,570 --> 00:15:00,500
and you can use several
hyperfine states

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00:15:00,500 --> 00:15:07,360
to great advantage for
the manipulation of atoms.

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00:15:07,360 --> 00:15:10,200
For instance, if
you want to, you

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00:15:10,200 --> 00:15:13,380
can put atoms into
a hyperfine state

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00:15:13,380 --> 00:15:15,610
where they don't absorb
light, and then you

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00:15:15,610 --> 00:15:20,329
can have resonant light for the
other ones blast those away.

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00:15:20,329 --> 00:15:22,370
So you can play, sort of,
your tricks because you

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00:15:22,370 --> 00:15:28,270
have two states between which
you can juggle at the atoms.

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00:15:28,270 --> 00:15:36,360
Well, what else is relevance
of hyperfine structure?

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00:15:36,360 --> 00:15:42,000
OK if you have two levels,
F equals 1, F equals 0,

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00:15:42,000 --> 00:15:44,140
you can observe a transition.

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00:15:44,140 --> 00:15:48,410
And the famous 21
centimeter line

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00:15:48,410 --> 00:15:51,335
is used for astronomical
observations.

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00:15:58,250 --> 00:16:01,430
Hydrogen is the most abundant
element in the universe,

216
00:16:01,430 --> 00:16:04,360
and how do you see
hydrogen out there.

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00:16:04,360 --> 00:16:08,560
Well it is due to hyperfine
transition, the 21 centimeter

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00:16:08,560 --> 00:16:10,970
line.

219
00:16:10,970 --> 00:16:16,780
And finally, another aspect
why hyperfine structure

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00:16:16,780 --> 00:16:18,930
is relevant, where
people use it,

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00:16:18,930 --> 00:16:21,970
is for the determination
of nuclear properties.

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00:16:21,970 --> 00:16:24,890
How do you know what the
properties of nuclei are?

223
00:16:24,890 --> 00:16:27,540
Well there are techniques
in nuclear physics,

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00:16:27,540 --> 00:16:32,060
but a lot, a lot about
the knowledge of nuclei

225
00:16:32,060 --> 00:16:35,440
comes from atomic spectroscopy.

226
00:16:35,440 --> 00:16:39,480
If you measure atomic energy
levels with high accuracy,

227
00:16:39,480 --> 00:16:43,320
you figure out what the
properties of the nucleus is,

228
00:16:43,320 --> 00:16:46,730
and one of the most outstanding
examples we will discuss later

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00:16:46,730 --> 00:16:49,800
on, and some of which is also
on your homework assignment,

230
00:16:49,800 --> 00:16:53,390
is you can use atomic
spectroscopy of hydrogen

231
00:16:53,390 --> 00:16:56,350
to contain the most
accurate measurement, how

232
00:16:56,350 --> 00:16:58,820
big is the proton.

233
00:16:58,820 --> 00:17:02,260
And the big surprise is that
there is, that there was,

234
00:17:02,260 --> 00:17:04,060
a surprise that
people figured out

235
00:17:04,060 --> 00:17:09,050
that, until now, even
in 2014, we do not fully

236
00:17:09,050 --> 00:17:11,869
understand how big the
proton is but we'll

237
00:17:11,869 --> 00:17:12,930
talk about that later.

238
00:17:15,480 --> 00:17:19,119
So for the level of this-- the
level of this introduction,

239
00:17:19,119 --> 00:17:27,930
we can use it for determination
of nuclear properties

240
00:17:27,930 --> 00:17:31,910
and actually, you cannot only
determine properties of stable

241
00:17:31,910 --> 00:17:36,370
nuclei, you can also determine
properties of unstable nuclei.

242
00:17:36,370 --> 00:17:40,340
At various accelerators,
they have a facility

243
00:17:40,340 --> 00:17:43,330
when, by, you know,
energy collisions

244
00:17:43,330 --> 00:17:47,050
they create unstable nuclei.

245
00:17:47,050 --> 00:17:52,490
Maybe helium six, helium
with four newtons, it exists.

246
00:17:52,490 --> 00:17:57,129
And you can take helium
six, extract it, utilize it

247
00:17:57,129 --> 00:17:58,670
and then you've have
a neutral helium

248
00:17:58,670 --> 00:18:01,960
atom which looks like your
every days helium atom

249
00:18:01,960 --> 00:18:04,270
but it has two more
neutrons in the nucleus.

250
00:18:04,270 --> 00:18:06,860
And by performing
atomic spectroscopy,

251
00:18:06,860 --> 00:18:09,320
you can figure out what
is the deformation, what

252
00:18:09,320 --> 00:18:14,780
is the structure, of this--
I want to say alpha particle,

253
00:18:14,780 --> 00:18:17,420
but it's an alpha particle
plus two neutrons.

254
00:18:17,420 --> 00:18:20,440
So people have really learned
to do those atomic physics

255
00:18:20,440 --> 00:18:22,910
measurements within
a few seconds

256
00:18:22,910 --> 00:18:25,030
after the element
has been produced,

257
00:18:25,030 --> 00:18:28,600
and such determined
nuclear properties even

258
00:18:28,600 --> 00:18:29,530
of unstable nuclei.

259
00:18:44,100 --> 00:18:44,610
OK.

260
00:18:44,610 --> 00:18:45,980
So that's my introduction.

261
00:18:49,550 --> 00:18:54,780
So we are now discussing
the most important effect

262
00:18:54,780 --> 00:18:57,050
due to the hyperfine structure.

263
00:18:57,050 --> 00:19:02,190
And this is the fact
that the nucleolus has

264
00:19:02,190 --> 00:19:07,040
a magnetic moment, and
this magnetic moment

265
00:19:07,040 --> 00:19:12,370
couples to the
magnetic field even

266
00:19:12,370 --> 00:19:15,610
if you don't apply an
external magnetic field,

267
00:19:15,610 --> 00:19:17,660
we talked about that
on Wednesday, there

268
00:19:17,660 --> 00:19:21,090
is an internal
magnetic field created

269
00:19:21,090 --> 00:19:26,250
by the electron with total
angular momentum change.

270
00:19:26,250 --> 00:19:29,450
So this is, so to speak,
the Zeeman Hamiltonian

271
00:19:29,450 --> 00:19:35,280
of the nucleus in the magnetic
field created by the electron.

272
00:19:35,280 --> 00:19:41,280
And I will show you quickly
to, is a simple derivation,

273
00:19:41,280 --> 00:19:43,240
what the result of that is.

274
00:19:43,240 --> 00:19:46,740
But before I do that, I
also want to mention out--

275
00:19:46,740 --> 00:19:49,600
mention that there is--
I want to point out

276
00:19:49,600 --> 00:19:51,830
that there is an alternative.

277
00:19:51,830 --> 00:19:56,290
Right now, I said we say
the nucleus experiences

278
00:19:56,290 --> 00:20:00,620
the magnetic field
created by the electron.

279
00:20:00,620 --> 00:20:02,870
But we can also take
the other approach,

280
00:20:02,870 --> 00:20:05,750
the nucleus creates a
vector potential because

281
00:20:05,750 --> 00:20:08,690
of its magnetic moment,
and the electron,

282
00:20:08,690 --> 00:20:11,280
which goes around the
nucleus, is not only

283
00:20:11,280 --> 00:20:13,760
feeling the Coulomb
potential but also

284
00:20:13,760 --> 00:20:15,920
feeling a vector potential.

285
00:20:15,920 --> 00:20:18,910
And of course, both
different perspectives,

286
00:20:18,910 --> 00:20:21,580
whether the electron moves in
the magnetic field the nucleus,

287
00:20:21,580 --> 00:20:24,520
or the nucleus experiences the
magnetic field of the electron,

288
00:20:24,520 --> 00:20:27,130
both treatments have to agree.

289
00:20:27,130 --> 00:20:29,530
I follow the more
standard treatment,

290
00:20:29,530 --> 00:20:38,230
but the alternative treatment,
where the electron moves

291
00:20:38,230 --> 00:20:40,940
in this electric and magnetic
potential of the nucleus,

292
00:20:40,940 --> 00:20:45,770
is fully elaborated on
the atomic physics wiki.

293
00:20:45,770 --> 00:20:51,870
So alternatively, electron
moves in the potential

294
00:20:51,870 --> 00:21:02,720
of the nucleus, which is
the Coulomb potential,

295
00:21:02,720 --> 00:21:05,750
we've discussed that,
but then there is also

296
00:21:05,750 --> 00:21:10,775
vector potential created by the
magnetic moment of the nucleus.

297
00:21:16,840 --> 00:21:19,090
So you simply assume this
is a potential created

298
00:21:19,090 --> 00:21:21,000
with a nucleus,
and then you just

299
00:21:21,000 --> 00:21:26,120
sort of Schrodinger's
equation and this approach

300
00:21:26,120 --> 00:21:27,960
is carried out on the wiki.

301
00:21:31,320 --> 00:21:34,580
However, since it's a
little bit more standard,

302
00:21:34,580 --> 00:21:37,410
and there's an easy
semi-classical derivation,

303
00:21:37,410 --> 00:21:40,330
let me now discuss this one.

304
00:21:40,330 --> 00:21:42,590
Because what I
like about it is it

305
00:21:42,590 --> 00:21:47,090
addresses one intuitive
quantity, namely the fact

306
00:21:47,090 --> 00:21:49,635
that there is an
internal magnetic field.

307
00:21:49,635 --> 00:21:51,510
We're not just using
the Schrodinger equation

308
00:21:51,510 --> 00:21:52,968
for the whole
system, we are really

309
00:21:52,968 --> 00:21:55,720
estimating what is the
magnetic field, which

310
00:21:55,720 --> 00:21:59,228
the electron creates, at
the position of the nucleus.

311
00:22:04,010 --> 00:22:13,090
So let's now do a
semi-classical derivation

312
00:22:13,090 --> 00:22:17,040
of this internal magnetic field.

313
00:22:19,580 --> 00:22:21,810
And I have to-- I will
immediately tell you,

314
00:22:21,810 --> 00:22:24,370
this derivation
agrees quantitatively

315
00:22:24,370 --> 00:22:28,760
with the fully
thermomechanical treatment.

316
00:22:28,760 --> 00:22:35,710
So, as often as these
semi-classical derivations,

317
00:22:35,710 --> 00:22:38,240
we have to separate two parts.

318
00:22:40,960 --> 00:22:44,600
There are two ways how
the electron creates

319
00:22:44,600 --> 00:22:46,630
a magnetic field at the nucleus.

320
00:22:46,630 --> 00:22:49,810
One is due to it's
orbital motion,

321
00:22:49,810 --> 00:22:53,510
the electron is a ring current
and creates a magnetic field,

322
00:22:53,510 --> 00:22:57,800
but then the electron has
magnetic moment for-- due

323
00:22:57,800 --> 00:22:58,560
to it's spin.

324
00:23:03,170 --> 00:23:18,110
The spin part is simply the
potential of a magnetic dipole,

325
00:23:18,110 --> 00:23:24,440
you will need to vector--
you will need to vector where

326
00:23:24,440 --> 00:23:30,490
the magnetic dipole
moment of the electron

327
00:23:30,490 --> 00:23:33,630
is proportional to it's
spin with a g factor.

328
00:23:36,490 --> 00:23:41,920
However, and you can find
that in all textbooks

329
00:23:41,920 --> 00:23:44,820
on classical
electrodynamics but, there

330
00:23:44,820 --> 00:23:48,910
is one important term
which we have to add here,

331
00:23:48,910 --> 00:23:51,500
which is also part
of classical E and M,

332
00:23:51,500 --> 00:23:57,560
and this is the delta
function contribution.

333
00:24:00,400 --> 00:24:03,830
You've probably seen it,
you find it in Jackson,

334
00:24:03,830 --> 00:24:06,190
if you haven't the
model is that you

335
00:24:06,190 --> 00:24:09,440
can assume that a
magnetic moment is created

336
00:24:09,440 --> 00:24:11,880
by ring current,
and the ring current

337
00:24:11,880 --> 00:24:14,490
has-- creates a
magnetic moment and you

338
00:24:14,490 --> 00:24:17,580
have the dipole potentially
due to the magnetic moment.

339
00:24:17,580 --> 00:24:19,760
However, if you
have a ring current,

340
00:24:19,760 --> 00:24:22,570
there is-- you
can also ask, what

341
00:24:22,570 --> 00:24:27,160
is the magnetic field
inside the current loop

342
00:24:27,160 --> 00:24:29,350
and then eventually do
the transition where

343
00:24:29,350 --> 00:24:32,320
you allow the current
loop to go to 0.

344
00:24:32,320 --> 00:24:36,090
That's how you make a point
model of a magnetic dipole,

345
00:24:36,090 --> 00:24:39,770
but what remains is, sorts
of-- as a delta function,

346
00:24:39,770 --> 00:24:42,480
the location inside the loop.

347
00:24:42,480 --> 00:24:45,330
I'm emphasizing it
because it will be,

348
00:24:45,330 --> 00:24:48,460
eventually, the delta
function contribution,

349
00:24:48,460 --> 00:24:51,090
which is important
for s electrons,

350
00:24:51,090 --> 00:24:54,900
and therefore it is
this contribution

351
00:24:54,900 --> 00:24:57,555
which is the dominant
effect in many situations.

352
00:25:03,071 --> 00:25:03,570
OK.

353
00:25:03,570 --> 00:25:07,339
So this is the magnetic
field created--

354
00:25:07,339 --> 00:25:09,880
it's a classical expression,
but it's buried in [? quantum ?]

355
00:25:09,880 --> 00:25:13,830
mechanics, the expression for
the magnetic field created

356
00:25:13,830 --> 00:25:17,150
by the spin.

357
00:25:17,150 --> 00:25:22,890
The second contribution is
the orbital contribution,

358
00:25:22,890 --> 00:25:28,500
and for that semi-classical,
we just use Biot-Savat.

359
00:25:34,360 --> 00:25:40,060
So Biot-Savat is usually
the 3-Dimensional integral

360
00:25:40,060 --> 00:25:43,010
over the current density.

361
00:25:48,660 --> 00:25:50,590
The volume integral,
or you can rewrite it

362
00:25:50,590 --> 00:25:59,380
as the current I, d, r
cross r, over r cubed.

363
00:25:59,380 --> 00:26:06,470
And eventually, if you now
put in the electron charge

364
00:26:06,470 --> 00:26:13,670
distribution, velocity
course r, well,

365
00:26:13,670 --> 00:26:18,860
velocity cross r means we
get, and that's what we want,

366
00:26:18,860 --> 00:26:21,610
the orbital angular momentum.

367
00:26:21,610 --> 00:26:24,680
The 1 over r cubed
term means we have

368
00:26:24,680 --> 00:26:27,960
to calculate an expectation
value over the wave

369
00:26:27,960 --> 00:26:31,030
function which,
is 1 over r cubed.

370
00:26:31,030 --> 00:26:37,040
And the prefactor
leads us-- it's

371
00:26:37,040 --> 00:26:40,130
nothing else than two
times the Bohr magenton.

372
00:26:46,720 --> 00:26:59,950
So, with those two terms, we can
now obtain our final expression

373
00:26:59,950 --> 00:27:03,450
for the total magnetic
field generated

374
00:27:03,450 --> 00:27:05,400
by the electron at the origin.

375
00:27:07,930 --> 00:27:11,460
And for that I use
the g factor of two

376
00:27:11,460 --> 00:27:15,560
as it comes out of Dirac theory.

377
00:27:15,560 --> 00:27:18,120
So now we have the
total magnetic field.

378
00:27:22,790 --> 00:27:28,240
We had this contribution
L over r cubed.

379
00:27:28,240 --> 00:27:32,340
If you inspect the dipole
potential of the spin

380
00:27:32,340 --> 00:27:39,310
it has a contribution
S over r cubed,

381
00:27:39,310 --> 00:27:47,920
then it is the
second contribution

382
00:27:47,920 --> 00:27:50,220
to the dipole potential.

383
00:27:50,220 --> 00:27:57,070
And finally, most importantly
for the following discussion,

384
00:27:57,070 --> 00:28:01,740
the delta function contribution
which I discussed earlier.

385
00:28:10,790 --> 00:28:20,250
If you have an s state,
these first terms

386
00:28:20,250 --> 00:28:27,410
are 0 for L equals 0 because
these are, sort of, terms which

387
00:28:27,410 --> 00:28:32,420
have dipole potential
where positive and negative

388
00:28:32,420 --> 00:28:36,100
contributions cancel out when
you do a spherical average,

389
00:28:36,100 --> 00:28:38,540
and the s electron performs
a spherical average.

390
00:28:43,400 --> 00:28:49,840
So it is 0 for L equals 0
due to the spherical average.

391
00:28:53,410 --> 00:29:01,940
Whereas the second part, it
would be 0 for L non-equals

392
00:29:01,940 --> 00:29:06,870
to 0 because the probability
for a non-s electron

393
00:29:06,870 --> 00:29:08,802
to be at the nucleus is 0.

394
00:29:11,700 --> 00:29:16,014
So pretty much this
describes that.

395
00:29:16,014 --> 00:29:17,680
So this describes a
hyperfine structure.

396
00:29:21,940 --> 00:29:29,520
Well, it describes the magnetic
field created by the electron,

397
00:29:29,520 --> 00:29:36,360
and now we have to do
the usual projection

398
00:29:36,360 --> 00:29:38,770
in the following way.

399
00:29:38,770 --> 00:29:45,790
That the hyperfine structure
is that Zeeman Hamiltonian

400
00:29:45,790 --> 00:29:48,240
of the internal magnetic
field with a magnetic moment

401
00:29:48,240 --> 00:29:59,050
of the nucleus, and the
magnetic moment of the nucleus

402
00:29:59,050 --> 00:30:04,120
is proportional to the angular
momentum of the nucleus.

403
00:30:04,120 --> 00:30:07,510
Sort of this argument that even
if it were not proportional,

404
00:30:07,510 --> 00:30:10,460
it would rapidly precess
and eventually project it,

405
00:30:10,460 --> 00:30:12,090
and the only direction
which survives

406
00:30:12,090 --> 00:30:14,550
is the direction of
the angular momentum.

407
00:30:14,550 --> 00:30:18,240
And similarly, you can--
the magnetic field,

408
00:30:18,240 --> 00:30:20,640
you have a contribution
of S and L,

409
00:30:20,640 --> 00:30:24,610
but S and L rapidly precess
around the result and angular

410
00:30:24,610 --> 00:30:27,480
momentum, J, and
therefore, as a result,

411
00:30:27,480 --> 00:30:31,720
the internal magnetic
field must be,

412
00:30:31,720 --> 00:30:35,550
can only be, parallel to
the angular momentum chain.

413
00:30:39,260 --> 00:30:41,310
If you do a fully
[? quantum ?] treatment,

414
00:30:41,310 --> 00:30:42,550
it comes out immediately.

415
00:30:42,550 --> 00:30:44,750
But if you do it
semi-classically, you calculate

416
00:30:44,750 --> 00:30:48,560
a magnetic field, you sort of
have to fall in this argument

417
00:30:48,560 --> 00:30:52,730
that you always project on
the axis of angular momentum

418
00:30:52,730 --> 00:30:56,706
and that means that the
hyperfine interaction will

419
00:30:56,706 --> 00:31:00,050
be the Hamiltonian for
it, or the operator,

420
00:31:00,050 --> 00:31:06,450
will be the dot product of
I dot J. For fine structure,

421
00:31:06,450 --> 00:31:11,140
we had L dot S, for hyperfine
structure, we had L dot J,

422
00:31:11,140 --> 00:31:13,630
this is always how we couple
angular momentum with a dot

423
00:31:13,630 --> 00:31:15,080
product.

424
00:31:15,080 --> 00:31:24,320
The hyperfine constant
cause by the letter

425
00:31:24,320 --> 00:31:29,830
a, and since historically a is
measured in frequency units,

426
00:31:29,830 --> 00:31:34,530
in Hertz, I have to put
in h, Planck's quantum.

427
00:31:34,530 --> 00:31:36,150
No it's not h bar.

428
00:31:36,150 --> 00:31:37,760
For historical reasons, it's h.

429
00:31:43,896 --> 00:31:44,860
AUDIENCE: Question.

430
00:31:44,860 --> 00:31:45,725
PROFESSOR: Yes?

431
00:31:45,725 --> 00:31:47,224
AUDIENCE: Are I and
J dimensionless,

432
00:31:47,224 --> 00:31:49,679
or will they carry
units in h bar?

433
00:31:55,080 --> 00:31:57,130
PROFESSOR: Here they
are dimensionless,

434
00:31:57,130 --> 00:32:00,320
thanks for the question, because
each is in frequency units,

435
00:32:00,320 --> 00:32:03,140
it's in Hertz, and if
you multiple with h

436
00:32:03,140 --> 00:32:04,860
you have an energy.

437
00:32:04,860 --> 00:32:08,330
So therefore, I and J
measure the angular momentum

438
00:32:08,330 --> 00:32:10,150
in unit of h bar.

439
00:32:10,150 --> 00:32:13,390
So it's not in that sense, it's
a normalized angular momentum

440
00:32:13,390 --> 00:32:14,470
operator.

441
00:32:14,470 --> 00:32:18,580
The quantum numbers of I and
J are not 1/2, or 1h bar,

442
00:32:18,580 --> 00:32:22,279
it's just 1/2 or 1.

443
00:32:22,279 --> 00:32:22,945
Other questions?

444
00:32:28,431 --> 00:32:28,930
OK.

445
00:32:28,930 --> 00:32:34,820
I can now take this expression
with, you know, L and S

446
00:32:34,820 --> 00:32:37,650
and S.r and evaluate
further, but I

447
00:32:37,650 --> 00:32:40,350
feel I'm not
providing any insight

448
00:32:40,350 --> 00:32:43,600
and you can read
about it on the wiki.

449
00:32:43,600 --> 00:32:51,350
So for a non-s state, how
to simplify this expression

450
00:32:51,350 --> 00:32:55,650
and get the final textbook
result, I defer to the wiki.

451
00:32:55,650 --> 00:33:00,730
I want to discuss the most
important part, namely for s v

452
00:33:00,730 --> 00:33:04,210
electrons because hydrogen,
all the alkaloids,

453
00:33:04,210 --> 00:33:05,660
have an s ground state.

454
00:33:08,690 --> 00:33:15,990
So, in that case, all we have to
consider is the delta function

455
00:33:15,990 --> 00:33:21,750
part and if we project
the magnetic field

456
00:33:21,750 --> 00:33:31,570
onto the angular
momentum axis, we

457
00:33:31,570 --> 00:33:37,925
get the probability of the s
electron to be at the origin.

458
00:33:43,120 --> 00:33:54,060
And therefore, for s states,
the hyperfine constant

459
00:33:54,060 --> 00:34:06,130
is-- oh, I forgot one thing.

460
00:34:06,130 --> 00:34:11,889
We have to parametrize the
magnetic moment of the nucleus,

461
00:34:11,889 --> 00:34:17,420
and that is done by using
a nuclear magnetron.

462
00:34:17,420 --> 00:34:19,620
It's the same as a
Bhor magneton, where

463
00:34:19,620 --> 00:34:26,520
you have replaced the electron
mass by the proton mass,

464
00:34:26,520 --> 00:34:29,250
and you have the
nuclear g factor.

465
00:34:29,250 --> 00:34:33,620
Just as a reminder, the g
factor of the proton is 5.6,

466
00:34:33,620 --> 00:34:36,600
the g factor of the
neutron is minus 3.8.

467
00:34:36,600 --> 00:34:40,290
So the g factor has nothing
to do, not even close,

468
00:34:40,290 --> 00:34:43,920
to the factor of 2, which we
obtained in the Dirac equation

469
00:34:43,920 --> 00:34:45,030
for the electron.

470
00:34:45,030 --> 00:34:49,389
That just shows that the
nucleons, protons and neutrons,

471
00:34:49,389 --> 00:34:50,889
are more complicated.

472
00:34:50,889 --> 00:34:53,080
Well, they have
quarks inside, they

473
00:34:53,080 --> 00:34:54,780
have a complicated
internal structure.

474
00:34:58,090 --> 00:34:58,590
OK.

475
00:34:58,590 --> 00:35:03,930
So therefore the hyperfine
constant involves,

476
00:35:03,930 --> 00:35:08,890
now, the g factor
of the nucleus,

477
00:35:08,890 --> 00:35:12,580
the product of the
nuclear magnetron,

478
00:35:12,580 --> 00:35:22,530
with a Bohr magneton,
and for hydrogen.

479
00:35:29,630 --> 00:35:43,680
This gives the famous
result of 1420 megahertz.

480
00:35:43,680 --> 00:35:47,670
So this is hydrogen, and this
h is now the Hamiltonian.

481
00:35:47,670 --> 00:35:50,900
So the hyperfine coupling
Hamiltonian, which has I

482
00:35:50,900 --> 00:36:02,730
dot J, by using the expression
for the total angular momentum

483
00:36:02,730 --> 00:36:08,990
I plus J and then we square it.

484
00:36:08,990 --> 00:36:11,850
When you evaluate
this square you get,

485
00:36:11,850 --> 00:36:19,010
on the right hand side,
an expression for I dot J.

486
00:36:19,010 --> 00:36:23,920
So I dot J is nothing
else than 1/2.

487
00:36:23,920 --> 00:36:32,050
F squared, minus I
squared, minus J squared.

488
00:36:35,090 --> 00:36:43,950
And therefore, for hydrogen,
where I, J and S are all 1/2,

489
00:36:43,950 --> 00:36:45,970
the proton has been
1/2, the electron

490
00:36:45,970 --> 00:36:48,730
has been 1/2, that's it.

491
00:36:48,730 --> 00:36:53,790
You have only two values of
the result and total angular

492
00:36:53,790 --> 00:36:59,400
momentum, 1/2 and 1/2
can add up to 1 or 0.

493
00:36:59,400 --> 00:37:13,130
And now the hyperfine
splitting is into an F equals 1

494
00:37:13,130 --> 00:37:17,070
and F equals 0 state.

495
00:37:17,070 --> 00:37:21,630
And one thing to remember is,
if you inspect the above formula

496
00:37:21,630 --> 00:37:25,110
with the [? quantum ?]
numbers you find immediately

497
00:37:25,110 --> 00:37:34,530
that, compared to
the degenerate line,

498
00:37:34,530 --> 00:37:37,220
without hyperfine
splitting, this--

499
00:37:37,220 --> 00:37:44,030
so what comes out of
the Dirac equation,

500
00:37:44,030 --> 00:37:53,890
the splitting is the
fight-- is 1/4 and 3/4

501
00:37:53,890 --> 00:37:56,770
of the hyperfine constant.

502
00:37:56,770 --> 00:38:01,040
Since F equals 1 has a
multiplicity of 3, 2 and F

503
00:38:01,040 --> 00:38:03,860
quantum numbers,
plus minus 1 and 0,

504
00:38:03,860 --> 00:38:09,640
the rule is that the center
of mass of this level

505
00:38:09,640 --> 00:38:11,595
does not change due to
hyperfine splitting.

506
00:38:16,300 --> 00:38:24,160
So the center of mass
of hyperfine states

507
00:38:24,160 --> 00:38:25,990
is not changing.

508
00:38:29,800 --> 00:38:32,500
So we introduce a
level splitting,

509
00:38:32,500 --> 00:38:33,860
but no overall shift.

510
00:38:40,450 --> 00:38:44,950
Any questions about magnetic
hyperfine structure?

511
00:38:44,950 --> 00:38:45,450
Yes?

512
00:38:45,450 --> 00:38:48,200
AUDIENCE: Would you explain
again the center of mass

513
00:38:48,200 --> 00:38:48,940
is not changing.

514
00:38:48,940 --> 00:38:50,740
Is this just for hydrogen?

515
00:38:50,740 --> 00:38:52,090
Or is this a general rule?

516
00:38:52,090 --> 00:38:53,920
PROFESSOR: No, this
is a general property,

517
00:38:53,920 --> 00:38:57,330
and you could actually
show that when you evaluate

518
00:38:57,330 --> 00:39:00,130
the product of I dot
J. So, the product of I

519
00:39:00,130 --> 00:39:05,270
dot J means that for arbitrary
I and arbitrary J, if you

520
00:39:05,270 --> 00:39:07,980
calculate the hyperfine
structure, the center of mass

521
00:39:07,980 --> 00:39:08,560
is the same.

522
00:39:14,480 --> 00:39:14,980
OK.

523
00:39:22,110 --> 00:39:26,400
I've not done any
experiment in my life

524
00:39:26,400 --> 00:39:33,260
where higher order
moments became important,

525
00:39:33,260 --> 00:39:37,240
but I want to teach you about
it because the discussion

526
00:39:37,240 --> 00:39:38,970
about whether those
higher moments exist

527
00:39:38,970 --> 00:39:42,360
or not is really an interesting
discussion about what

528
00:39:42,360 --> 00:39:44,580
is allowed by symmetry
and what's not.

529
00:39:44,580 --> 00:39:48,490
So I'm bringing in
a higher moments,

530
00:39:48,490 --> 00:39:51,990
not so much because you need
it to understand the level

531
00:39:51,990 --> 00:39:53,910
structure of your
favourite atom,

532
00:39:53,910 --> 00:39:58,510
but because it teaches us a
really nice piece of physics.

533
00:39:58,510 --> 00:40:03,310
So let me now discuss higher
order moments and the leading

534
00:40:03,310 --> 00:40:06,960
one is the electric
quadripole moment.

535
00:40:06,960 --> 00:40:09,810
So I want to raise, in
general, the question,

536
00:40:09,810 --> 00:40:14,190
what further moments
can a nucleus have.

537
00:40:14,190 --> 00:40:19,120
What we have discussed so far
is the magnetic moment, mu.

538
00:40:19,120 --> 00:40:39,290
So beyond mu, what further
electric or magnetic moments

539
00:40:39,290 --> 00:40:42,250
can a nucleus have?

540
00:40:42,250 --> 00:40:46,780
Well, it's a question
about symmetry

541
00:40:46,780 --> 00:40:58,410
and if you look at the
parity of multiples,

542
00:40:58,410 --> 00:41:05,790
if you have an electric
multiple-- well,

543
00:41:05,790 --> 00:41:09,300
you know if you have a dipole,
plus minus, you invert it,

544
00:41:09,300 --> 00:41:12,460
the dipole becomes
minus the dipole.

545
00:41:12,460 --> 00:41:17,070
So for L equals 1, it
is minus 1, L is 1,

546
00:41:17,070 --> 00:41:19,200
but in general
the it is minus L.

547
00:41:19,200 --> 00:41:22,030
If you have a quadripole, you
invert your coordinate system.

548
00:41:22,030 --> 00:41:25,780
It's L equals 2, plus
plus, minus minus,

549
00:41:25,780 --> 00:41:27,735
you invert it nothing changes.

550
00:41:27,735 --> 00:41:29,860
So these I've shown you,
for dipole and quadripole,

551
00:41:29,860 --> 00:41:32,040
that this formula is correct.

552
00:41:32,040 --> 00:41:36,630
So an electric multiple
has this parity,

553
00:41:36,630 --> 00:41:41,990
and for, magnetic
keys-- now, you

554
00:41:41,990 --> 00:41:46,310
know, magnetic we have axial
vectors versus polar vectors,

555
00:41:46,310 --> 00:41:49,150
there's always an extra
factor of minus 1.

556
00:41:49,150 --> 00:41:57,390
So therefore if you go magnetic,
magnetic multiples with L

557
00:41:57,390 --> 00:41:59,410
have a parity of minus L plus 1.

558
00:42:02,710 --> 00:42:08,120
So this really restricts
what multiples are possible.

559
00:42:08,120 --> 00:42:11,020
Instead of giving you
a general discussion,

560
00:42:11,020 --> 00:42:16,160
let me just look at the
very important keys.

561
00:42:16,160 --> 00:42:20,210
Whether a magnet with a nucleus
can have a electric dipole

562
00:42:20,210 --> 00:42:23,430
moment, and you will immediately
see what it needs to-- and then

563
00:42:23,430 --> 00:42:26,610
I give you a general result.

564
00:42:26,610 --> 00:42:39,710
So let's assume a nucleus
has angular momentum I,

565
00:42:39,710 --> 00:42:48,080
and there is a magnetic
moment, mu associated with it.

566
00:42:51,540 --> 00:43:02,100
So the general result is
that odd electric and even

567
00:43:02,100 --> 00:43:18,010
magnetic multiple moments
would violate not just one,

568
00:43:18,010 --> 00:43:28,410
but two symmetries, would
violate parity and time

569
00:43:28,410 --> 00:43:32,055
reversal symmetry.

570
00:43:34,720 --> 00:43:37,920
So the argument goes as follows.

571
00:43:37,920 --> 00:43:41,450
Let us assume this is
the magnetic moment, mu

572
00:43:41,450 --> 00:43:43,930
or the angular momentum
I, it's a vector.

573
00:43:46,810 --> 00:43:52,590
And now we are asking,
is it possible to have

574
00:43:52,590 --> 00:43:56,560
a vector of the dipole moment.

575
00:43:56,560 --> 00:43:59,820
And dipole moment, you
should just think about it

576
00:43:59,820 --> 00:44:02,795
as a plus minus
charge, separated.

577
00:44:05,680 --> 00:44:09,280
We can now do the parity
operation and the time reversal

578
00:44:09,280 --> 00:44:10,590
operation.

579
00:44:10,590 --> 00:44:12,900
If you do the time
reversal operation,

580
00:44:12,900 --> 00:44:16,120
the current, which
generates a magnetic moment

581
00:44:16,120 --> 00:44:18,420
if you want to think
about it in this picture,

582
00:44:18,420 --> 00:44:19,680
goes the other way.

583
00:44:19,680 --> 00:44:22,920
So mu flips but nothing
moves, of course,

584
00:44:22,920 --> 00:44:25,880
for an electric dipole
moment, so reversing time

585
00:44:25,880 --> 00:44:29,120
is not changing anything.

586
00:44:29,120 --> 00:44:31,410
So in other words,
time reversal symmetry

587
00:44:31,410 --> 00:44:36,670
transforms parallel mu and d
into anti-parallel mu and d.

588
00:44:36,670 --> 00:44:41,830
And similar, parity
is not changing mu

589
00:44:41,830 --> 00:44:44,250
but it is changing d.

590
00:44:44,250 --> 00:44:50,420
So in both cases, would parity,
or time reversal symmetry,

591
00:44:50,420 --> 00:44:55,640
if you had mu-d, a
scalar product of mu

592
00:44:55,640 --> 00:45:10,970
and d which would be
known 0, then both P or P

593
00:45:10,970 --> 00:45:14,330
would change the sign.

594
00:45:14,330 --> 00:45:18,150
But that would mean that two
kinds of particles would exist,

595
00:45:18,150 --> 00:45:21,330
one where the sign is positive,
one where the sign is negative.

596
00:45:21,330 --> 00:45:24,290
But we have assumed that we have
1 nucleus and only 1 nucleus

597
00:45:24,290 --> 00:45:27,800
of this kind, so we cannot
have one nucleus which has

598
00:45:27,800 --> 00:45:29,960
the properties of
having, simultaneously,

599
00:45:29,960 --> 00:45:33,180
a magnetic and
electric dipole moment.

600
00:45:33,180 --> 00:45:42,010
So therefore, we conclude
that mu times d has to be 0.

601
00:45:42,010 --> 00:45:44,440
And if you generalize
this argument,

602
00:45:44,440 --> 00:45:48,820
we have ruled out that
there is an electric dipole

603
00:45:48,820 --> 00:45:51,820
moment, an odd electric moment.

604
00:45:51,820 --> 00:45:55,080
The first, the lowest,
possible electric moment

605
00:45:55,080 --> 00:46:04,570
is the quadripole moment, so the
leading electric moment is not

606
00:46:04,570 --> 00:46:08,180
L equals 2, L equals
2, it's not L equals 1,

607
00:46:08,180 --> 00:46:12,560
the dipole, it's L
equals 2 the quadripole.

608
00:46:12,560 --> 00:46:15,080
And of course if you
generalize the argument,

609
00:46:15,080 --> 00:46:17,340
L equals 4, L equals
6, would be possible

610
00:46:17,340 --> 00:46:19,940
but those effects
would be very small.

611
00:46:28,530 --> 00:46:29,450
Questions so far?

612
00:46:31,891 --> 00:46:32,390
OK.

613
00:46:32,390 --> 00:46:36,470
So we've talked about parity and
time reversal symmetry, which

614
00:46:36,470 --> 00:46:41,190
restricts what kind of magnetic
and electric dipole moments

615
00:46:41,190 --> 00:46:42,940
particles may have.

616
00:46:42,940 --> 00:46:44,650
And maybe in this
context, I should just

617
00:46:44,650 --> 00:46:47,820
mention that John Doyle, Jerry
Gabrielse, and Dave Demille

618
00:46:47,820 --> 00:46:51,420
at Harvard and Yale, they just
published the most accurate

619
00:46:51,420 --> 00:46:56,800
result for the electric
dipole moment of the electron.

620
00:46:56,800 --> 00:46:59,920
They found a bound, which was
more than an order of magnitude

621
00:46:59,920 --> 00:47:04,550
lower than the best
upper bound before

622
00:47:04,550 --> 00:47:06,380
and this has really
made headlines.

623
00:47:06,380 --> 00:47:11,210
So to measure, accurately,
that the electric dipole

624
00:47:11,210 --> 00:47:14,230
moment vanishes, in this
case of the electron,

625
00:47:14,230 --> 00:47:17,020
but other people do
it also for neutrons,

626
00:47:17,020 --> 00:47:19,530
is testing fundamental
symmetries.

627
00:47:19,530 --> 00:47:22,810
In particular, it
tests whether nature

628
00:47:22,810 --> 00:47:24,740
is time reversal invariant.

629
00:47:28,240 --> 00:47:32,070
And the reason why,
until now, everybody

630
00:47:32,070 --> 00:47:35,380
has found that the results
are compatible with 0

631
00:47:35,380 --> 00:47:40,440
is pretty much based on the
argument I just gave you.

632
00:47:40,440 --> 00:47:45,140
OK so we have discussed
those fundamental symmetries,

633
00:47:45,140 --> 00:47:52,420
but now I want to discuss
something else related to it.

634
00:47:52,420 --> 00:47:57,350
So let's assume
you have a nucleus,

635
00:47:57,350 --> 00:48:01,150
and I want to discuss with
you, is a minimum requirement

636
00:48:01,150 --> 00:48:04,830
for the nucleus for the
angular momentum of the nucleus

637
00:48:04,830 --> 00:48:13,410
in order to have
a magnetic dipole,

638
00:48:13,410 --> 00:48:26,220
or to have an
electric quadripole.

639
00:48:26,220 --> 00:48:29,700
So let's formulate it
as a quicker question.

640
00:48:29,700 --> 00:48:36,160
So here let's assume a is--
it's possible for any nucleus

641
00:48:36,160 --> 00:48:40,230
no matter what the
angular momentum is.

642
00:48:40,230 --> 00:48:46,700
Here, we put in 1/2,
or larger than 1,

643
00:48:46,700 --> 00:48:51,800
and for the electric quadripole
we have the same choices.

644
00:48:51,800 --> 00:48:54,820
So you should decide
if I tell you,

645
00:48:54,820 --> 00:48:57,010
a nucleolus has a
magnetic dipole.

646
00:48:57,010 --> 00:48:59,750
Does that imply that
there is a minimum amount

647
00:48:59,750 --> 00:49:03,590
of angular momentum
of the nucleus there?

648
00:49:03,590 --> 00:49:05,120
And then repeat
the same question

649
00:49:05,120 --> 00:49:06,286
for the electric quadripole.

650
00:49:09,860 --> 00:49:12,240
So please tell me your opinion.

651
00:49:21,610 --> 00:49:24,600
So right now, what is
a minimum requirement

652
00:49:24,600 --> 00:49:28,677
for I, for magnetic dipole,
and if yes, what is it?

653
00:49:35,140 --> 00:49:37,630
Three, two, one.

654
00:49:37,630 --> 00:49:38,130
Stop.

655
00:49:42,440 --> 00:49:53,950
OK, so-- and let's
immediately consider

656
00:49:53,950 --> 00:49:56,650
the electric quadripole moment.

657
00:49:56,650 --> 00:49:58,170
So this was a majority opinion.

658
00:49:58,170 --> 00:50:01,246
I gave this answer for
both of them together.

659
00:50:07,240 --> 00:50:10,870
So the question is, what
is the minimum requirement

660
00:50:10,870 --> 00:50:14,881
for a nucleus to have an
electric quadripole moment.

661
00:50:29,980 --> 00:50:31,680
OK, stop.

662
00:50:31,680 --> 00:50:32,180
Display.

663
00:50:35,360 --> 00:50:37,410
No requirement.

664
00:50:37,410 --> 00:50:39,570
OK.

665
00:50:39,570 --> 00:50:43,650
So the majority
onset was a here.

666
00:50:43,650 --> 00:50:47,360
So let me to discuss it,
and I know some of you

667
00:50:47,360 --> 00:50:50,900
will have-- I always get into
heated discussions with it.

668
00:50:50,900 --> 00:50:54,410
I actually just had a few
minute discussion with one

669
00:50:54,410 --> 00:50:59,310
of my colleagues about
it, who at least wanted

670
00:50:59,310 --> 00:51:01,720
to look at it from a different
perspective than I did.

671
00:51:01,720 --> 00:51:03,700
So let me give you
the short answer

672
00:51:03,700 --> 00:51:05,650
and we can go from there.

673
00:51:05,650 --> 00:51:10,310
In order to the magnetic dipole
you have to take an object,

674
00:51:10,310 --> 00:51:13,350
turn it around, and
figure out that there's

675
00:51:13,350 --> 00:51:15,380
a different energy.

676
00:51:15,380 --> 00:51:18,270
So therefore, unless
you have an object,

677
00:51:18,270 --> 00:51:21,950
which is has two orientations,
you cannot figure out if it has

678
00:51:21,950 --> 00:51:23,800
a magnetic dipole or not.

679
00:51:23,800 --> 00:51:28,560
If you're in a state without
angular momentum, I equals 0,

680
00:51:28,560 --> 00:51:30,650
there is no
distinguishable states.

681
00:51:30,650 --> 00:51:35,000
You cannot orient a 0 angular
momentum object in space.

682
00:51:35,000 --> 00:51:36,990
So therefore you
can never figure out

683
00:51:36,990 --> 00:51:39,490
that there's a magnetic
moment, and now I'll

684
00:51:39,490 --> 00:51:41,350
make a bold statement,
and this means

685
00:51:41,350 --> 00:51:44,300
there is no magnetic moment.

686
00:51:44,300 --> 00:51:53,120
So therefore, you
need this because you

687
00:51:53,120 --> 00:51:59,766
need a minimum number
of true orientations.

688
00:52:02,310 --> 00:52:03,120
OK.

689
00:52:03,120 --> 00:52:08,490
Now an electric quadripole
means that the charge density

690
00:52:08,490 --> 00:52:10,050
is like ellipse.

691
00:52:10,050 --> 00:52:11,178
It has--

692
00:52:11,178 --> 00:52:12,053
AUDIENCE: [INAUDIBLE]

693
00:52:14,940 --> 00:52:17,250
PROFESSOR: Oh, 1/2, sorry.

694
00:52:17,250 --> 00:52:17,750
Thank you.

695
00:52:27,160 --> 00:52:30,440
So I should get larger
than 1/2 because we

696
00:52:30,440 --> 00:52:33,670
need a minimum of two
possible orientations.

697
00:52:33,670 --> 00:52:38,160
Now my question for you is, you
think something is elliptical

698
00:52:38,160 --> 00:52:41,550
but how many different
orientations do

699
00:52:41,550 --> 00:52:45,783
you need to figure out that it
is elliptical and not round.

700
00:52:54,960 --> 00:52:58,830
If I equal 0, you can only look
at it, you can't rotate it,

701
00:52:58,830 --> 00:53:01,520
so will you never find any
energies [? breathing. ?]

702
00:53:01,520 --> 00:53:04,960
It's just there and you cannot
say where it's round or whether

703
00:53:04,960 --> 00:53:06,950
it has a quadripole
or deformation.

704
00:53:06,950 --> 00:53:11,680
If I is equal to 1/2, you can
take it and flip it around

705
00:53:11,680 --> 00:53:15,700
but can you tell from
that that it's an ellipse.

706
00:53:15,700 --> 00:53:19,130
No because if you turn an
ellipse around nothing changes.

707
00:53:19,130 --> 00:53:21,660
So it could be,
as well, a sphere.

708
00:53:21,660 --> 00:53:23,330
You can only figure
out that it's

709
00:53:23,330 --> 00:53:26,610
an ellipse if you have
an intermediate rotation,

710
00:53:26,610 --> 00:53:28,840
let's say, by 90 degrees.

711
00:53:28,840 --> 00:53:32,090
So in order to assess that
something is elliptical,

712
00:53:32,090 --> 00:53:36,390
you have to at least resolve
three positions, three angles,

713
00:53:36,390 --> 00:53:40,290
and three angles require that
you have to three sublevels

714
00:53:40,290 --> 00:53:43,700
and this requires that I is
larger, or equal, than one.

715
00:53:47,330 --> 00:53:48,710
Sorry.

716
00:53:48,710 --> 00:53:55,260
Then one goes-- OK.

717
00:54:00,350 --> 00:54:03,280
I'm want to give you a
formal argument using

718
00:54:03,280 --> 00:54:08,560
this spherical
tensor, but I guess

719
00:54:08,560 --> 00:54:11,490
someone you are waiting
for something simpler.

720
00:54:11,490 --> 00:54:12,420
So.

721
00:54:12,420 --> 00:54:14,770
I mean who wants to know
the answer of the question,

722
00:54:14,770 --> 00:54:17,670
but what happens if
it has a deformation.

723
00:54:17,670 --> 00:54:19,290
Does it just mean
we can't measure it,

724
00:54:19,290 --> 00:54:20,331
but it has a deformation?

725
00:54:23,220 --> 00:54:24,620
Let me explain that.

726
00:54:24,620 --> 00:54:29,380
So if I have a pencil and
the pencil has zero angular

727
00:54:29,380 --> 00:54:31,900
momentum, I can't
really figure out

728
00:54:31,900 --> 00:54:34,440
that it's an elongated object,
because all I measure is

729
00:54:34,440 --> 00:54:35,960
a symmetric wave function.

730
00:54:35,960 --> 00:54:38,150
It's completely
spherically symmetric.

731
00:54:38,150 --> 00:54:40,840
The only way to figure
out that it's a pencil

732
00:54:40,840 --> 00:54:43,540
is I have to localize
it, that I can

733
00:54:43,540 --> 00:54:45,550
see that it's
pointing somewhere.

734
00:54:45,550 --> 00:54:48,680
But to localize
an object in space

735
00:54:48,680 --> 00:54:51,070
is actually an
angular wave packet.

736
00:54:51,070 --> 00:54:52,280
It's not isotropic.

737
00:54:52,280 --> 00:54:53,700
It points somewhere.

738
00:54:53,700 --> 00:54:57,030
And an angular wave
packet is a superposition

739
00:54:57,030 --> 00:55:00,870
of states of different
angular momenta.

740
00:55:00,870 --> 00:55:04,270
So therefore, without assuming
that there is a state with

741
00:55:04,270 --> 00:55:06,730
angular momentum, I
cannot orient this pencil.

742
00:55:09,280 --> 00:55:10,020
OK.

743
00:55:10,020 --> 00:55:14,940
I know you would all agree that
even if this pencil is cooled

744
00:55:14,940 --> 00:55:16,920
to the ground state with
zero angular momentum,

745
00:55:16,920 --> 00:55:19,040
it is a pencil.

746
00:55:19,040 --> 00:55:21,600
But what you're using
here is now your knowledge

747
00:55:21,600 --> 00:55:25,540
that this object has higher
angular momentum states.

748
00:55:25,540 --> 00:55:27,220
And those higher
angular momentum

749
00:55:27,220 --> 00:55:30,730
states have nothing to do with
the structure or the appearance

750
00:55:30,730 --> 00:55:33,020
of this object of
being a pencil.

751
00:55:33,020 --> 00:55:35,670
So you sort of know that
in addition to the i

752
00:55:35,670 --> 00:55:39,560
equals 0 state, there are i
equals 1, 2, 3, 4, 5 states,

753
00:55:39,560 --> 00:55:42,040
and the pencil looks the same.

754
00:55:42,040 --> 00:55:44,230
But if you have a
nucleus, an i equals

755
00:55:44,230 --> 00:55:48,270
0 state requires a certain
configuration of quarks.

756
00:55:48,270 --> 00:55:51,390
And you cannot create an i
equals 2, and i equals 4,

757
00:55:51,390 --> 00:55:55,000
higher states without messing
around with the internal

758
00:55:55,000 --> 00:55:55,740
structure.

759
00:55:55,740 --> 00:55:59,540
So with a nucleus, all you
have is an i equals 0 state.

760
00:55:59,540 --> 00:56:03,560
And to say that this i equals
0 state has a quadrupolar

761
00:56:03,560 --> 00:56:06,180
deformation doesn't
make any sense.

762
00:56:06,180 --> 00:56:11,310
If you would know that this
nuclear state could be rotated

763
00:56:11,310 --> 00:56:14,640
without changing its internal
structure, then you would say,

764
00:56:14,640 --> 00:56:18,550
yes, it has a quadrupole
moment, I just can't see it.

765
00:56:18,550 --> 00:56:20,745
But usually, you cannot
make this assumption.

766
00:56:20,745 --> 00:56:23,870
If all you have is an i
equals 0 ground state,

767
00:56:23,870 --> 00:56:27,030
and the i equals 2 state
is very, very different,

768
00:56:27,030 --> 00:56:29,310
because the quarks are
spinning around each other

769
00:56:29,310 --> 00:56:31,820
in a different way,
you have to see,

770
00:56:31,820 --> 00:56:34,950
i equals 0 is
completely spherical.

771
00:56:34,950 --> 00:56:37,690
It doesn't couple to
anything externally.

772
00:56:37,690 --> 00:56:40,500
And therefore, it has
no moments whatsoever.

773
00:56:44,350 --> 00:56:46,160
So that's the story.

774
00:56:46,160 --> 00:56:48,130
A lot of people get
confused, because they

775
00:56:48,130 --> 00:56:52,610
think an object can have a
deformation without rotating.

776
00:56:52,610 --> 00:56:55,080
But you need the
rotation to resolve it.

777
00:56:55,080 --> 00:56:58,550
If you cannot create
an angular wave packet,

778
00:56:58,550 --> 00:57:01,070
which is a superposition
state of angular momenta,

779
00:57:01,070 --> 00:57:05,080
you can never figure out
that there is a deformation.

780
00:57:05,080 --> 00:57:08,980
And quantum mechanically, if you
cannot figure out that there is

781
00:57:08,980 --> 00:57:12,140
a deformation, there
is no deformation,

782
00:57:12,140 --> 00:57:16,680
because you can only use, in the
language of quantum mechanics,

783
00:57:16,680 --> 00:57:21,605
where you have at least the
possibility to measure it.

784
00:57:21,605 --> 00:57:23,060
Questions about that?

785
00:57:25,881 --> 00:57:26,380
OK.

786
00:57:26,380 --> 00:57:31,916
I think that makes it now--
let me now kind of just

787
00:57:31,916 --> 00:57:32,665
give you a formal.

788
00:57:40,070 --> 00:57:41,150
A formal derivation.

789
00:57:41,150 --> 00:57:41,650
But I agree.

790
00:57:41,650 --> 00:57:43,066
I mean, the formal
derivation, I'm

791
00:57:43,066 --> 00:57:45,460
just throwing a few equations
at you, and say, that's it

792
00:57:45,460 --> 00:57:46,870
and everything
follows from that.

793
00:57:46,870 --> 00:57:49,160
But I provided the
insight for you.

794
00:57:49,160 --> 00:57:55,390
Formally, you can define
the quadrupole moment

795
00:57:55,390 --> 00:58:01,110
by the expectation operator.

796
00:58:01,110 --> 00:58:05,290
You take the nucleus
with maximum MI.

797
00:58:05,290 --> 00:58:12,875
And now you calculate
the expectation value

798
00:58:12,875 --> 00:58:14,125
of this operator.

799
00:58:18,170 --> 00:58:24,200
This is, of course, motivated
by just electrostatics.

800
00:58:24,200 --> 00:58:34,550
If you take an expansion of the
classical electrostatic energy

801
00:58:34,550 --> 00:58:39,580
into multi-poles, you find
the quadrupole configuration

802
00:58:39,580 --> 00:58:44,600
to be related to a
quadripole moment.

803
00:58:44,600 --> 00:58:50,200
Quadrupole moments couple to the
derivative of electric fields.

804
00:58:50,200 --> 00:58:53,910
And then in this purely
classical description,

805
00:58:53,910 --> 00:59:02,290
you have this
term, where beta is

806
00:59:02,290 --> 00:59:08,400
the angle between two
symmetry axis, namely,

807
00:59:08,400 --> 00:59:23,440
between the symmetry axis of
the electric field gradient

808
00:59:23,440 --> 00:59:30,010
and the quadrupole tensor-- just
the classic quadrupole tensor

809
00:59:30,010 --> 00:59:31,230
as it comes out of Jackson.

810
00:59:41,100 --> 00:59:42,130
Yes.

811
00:59:42,130 --> 00:59:44,390
You can see the quantum
mechanical definition

812
00:59:44,390 --> 00:59:48,240
of the quadrupole moment,
or more generally.

813
00:59:48,240 --> 00:59:49,940
So this is quadrupole moment.

814
00:59:49,940 --> 00:59:53,800
If you have a moment
with l, the operator,

815
00:59:53,800 --> 00:59:56,760
which tells you whether you
have a non-vanishing moment,

816
00:59:56,760 --> 01:00:00,460
a non-vanishing
deformation, is actually

817
01:00:00,460 --> 01:00:02,726
a spherical tensor, Tlm.

818
01:00:08,360 --> 01:00:10,610
And what you see above
is a spherical tensor,

819
01:00:10,610 --> 01:00:15,830
T20-- l equals 2, m equals 0.

820
01:00:15,830 --> 01:00:18,970
And those spherical tensors
are defined by the fact

821
01:00:18,970 --> 01:00:22,190
that they transform as
spherical harmonics.

822
01:00:22,190 --> 01:00:25,300
And now you sort of
realize what it means.

823
01:00:25,300 --> 01:00:32,030
If you want a magnetic or
electric moment with l,

824
01:00:32,030 --> 01:00:35,920
the operator for the
moment transforms

825
01:00:35,920 --> 01:00:38,530
like angular momentum l.

826
01:00:38,530 --> 01:00:41,370
And now you realize that
you have the triangle rule.

827
01:00:41,370 --> 01:00:47,460
If you want a matrix element
where I and l overlap with I,

828
01:00:47,460 --> 01:00:50,600
you want to make sure
that I, l, and I couple.

829
01:00:50,600 --> 01:00:52,870
And you have a triangle rule.

830
01:00:52,870 --> 01:00:57,530
So therefore, if you
want a magnetic moment,

831
01:00:57,530 --> 01:01:02,800
or electric moment, of l, and
you evaluate this expectation

832
01:01:02,800 --> 01:01:08,050
value, well, at least
the triangle rules

833
01:01:08,050 --> 01:01:10,600
can only be justified like this.

834
01:01:10,600 --> 01:01:14,740
Or in other words, you can
only get a non-vanishing moment

835
01:01:14,740 --> 01:01:18,020
if l is smaller than 2I.

836
01:01:18,020 --> 01:01:20,940
And this is what we discussed
in the clicker question

837
01:01:20,940 --> 01:01:25,520
for the two cases of l
equals 1 and l equals 2.

838
01:01:25,520 --> 01:01:28,490
So ultimately, it's
a selection rule

839
01:01:28,490 --> 01:01:31,460
which is related to
the triangle rule

840
01:01:31,460 --> 01:01:35,470
for the addition
of angular momenta.

841
01:01:35,470 --> 01:01:38,190
But I like much
better the argument,

842
01:01:38,190 --> 01:01:39,870
how many orientations
do you need

843
01:01:39,870 --> 01:01:43,970
to find out that
something is elliptical?

844
01:01:43,970 --> 01:01:46,960
It's formalized here.

845
01:01:46,960 --> 01:01:47,760
All right.

846
01:01:51,780 --> 01:01:57,930
Let's just spend one more minute
on the quadrupolar structure.

847
01:01:57,930 --> 01:02:04,330
So based on the expansion of
the electrostatic energy, what

848
01:02:04,330 --> 01:02:07,410
determines the
quadrupolar structure

849
01:02:07,410 --> 01:02:13,660
is this cosine angle,
which is the angle

850
01:02:13,660 --> 01:02:20,100
between the axis of the
nucleus and the axis

851
01:02:20,100 --> 01:02:21,966
of an electric field gradient.

852
01:02:24,560 --> 01:02:26,920
And that means it is the angle.

853
01:02:30,490 --> 01:02:35,100
It's a cosine of
the angle between J,

854
01:02:35,100 --> 01:02:41,120
the outer environment, and
I, the axis of the nucleus.

855
01:02:41,120 --> 01:02:46,630
So therefore, when we would
derive-- I'm not deriving it,

856
01:02:46,630 --> 01:02:49,480
but if we would derive an
expression for quadrupolar

857
01:02:49,480 --> 01:02:52,480
structure, the
quadrupolar structure

858
01:02:52,480 --> 01:02:56,290
would be proportional
to a quantity C, which

859
01:02:56,290 --> 01:03:03,020
is nothing else than the
dot product of I and J.

860
01:03:03,020 --> 01:03:06,460
And at least in my notes now,
I and J have units of h bar.

861
01:03:06,460 --> 01:03:09,150
So I'm dividing it out here.

862
01:03:09,150 --> 01:03:14,720
And as you know, I,
dot, J can be expressed

863
01:03:14,720 --> 01:03:19,730
by quantum numbers F, F
plus 1, minus I, I plus 1,

864
01:03:19,730 --> 01:03:24,670
minus J, J plus 1.

865
01:03:24,670 --> 01:03:32,810
So therefore, the
quadrupolar energies-- E2,

866
01:03:32,810 --> 01:03:39,650
El equals 2-- involve
the classical expression

867
01:03:39,650 --> 01:03:42,060
at cosine square.

868
01:03:42,060 --> 01:03:49,260
So therefore, you would expect
there is a quadrupole constant.

869
01:03:49,260 --> 01:03:52,640
And then it is cosine square.

870
01:03:52,640 --> 01:03:55,040
But well, usually,
quantum mechanic

871
01:03:55,040 --> 01:03:57,460
even, we have the
square of a quantity,

872
01:03:57,460 --> 01:04:03,480
we have to write it as
quantity times quantity plus 1.

873
01:04:03,480 --> 01:04:06,780
So this is the
quadrupolar structure.

874
01:04:06,780 --> 01:04:09,010
And to remind you,
we just discussed

875
01:04:09,010 --> 01:04:13,670
that for the hydrogen atom, the
magnetic hyperfine structure

876
01:04:13,670 --> 01:04:21,480
had invoiced the same product of
I, dot, J, but in a linear way.

877
01:04:25,100 --> 01:04:28,570
So the reason why I'm not
discussing quadrupolar

878
01:04:28,570 --> 01:04:33,390
structure in more
detail is usually

879
01:04:33,390 --> 01:04:36,340
the hyperfine
structure associated

880
01:04:36,340 --> 01:04:41,040
with quadrupole moments
is much, much smaller

881
01:04:41,040 --> 01:04:43,450
than the hyperfine
structure associated

882
01:04:43,450 --> 01:04:53,050
with magnetic moment,
typically by a factor of 100.

883
01:04:53,050 --> 01:04:55,840
The only exceptions
are molecules,

884
01:04:55,840 --> 01:04:58,960
because molecules
can have-- because

885
01:04:58,960 --> 01:05:01,590
of molecular binding
mechanisms-- a much, much

886
01:05:01,590 --> 01:05:04,830
larger electric field gradient.

887
01:05:04,830 --> 01:05:07,800
So therefore, in molecules,
quadrupolar structure

888
01:05:07,800 --> 01:05:10,830
is more important than in atoms.

889
01:05:16,910 --> 01:05:19,848
Questions about that?

890
01:05:26,930 --> 01:05:32,620
With that, we have
discussed the two effects

891
01:05:32,620 --> 01:05:37,050
of hyperfine structure due to
magnetic moment of the nucleus.

892
01:05:37,050 --> 01:05:42,120
And we also discussed further
deformations of the nucleus,

893
01:05:42,120 --> 01:05:45,050
in particular, the
quadrupolar deformation.

894
01:05:45,050 --> 01:05:48,330
Let me now use the
last 10 minutes

895
01:05:48,330 --> 01:05:50,525
to quickly discuss with
you isotope effects.

896
01:06:09,600 --> 01:06:12,900
And I know that many people
here know about isotope effects,

897
01:06:12,900 --> 01:06:15,450
because if you lock your
laser to a lithium cell

898
01:06:15,450 --> 01:06:20,000
or to a rubidium cell, you find
lithium-6 and lithium-7 peaks.

899
01:06:20,000 --> 01:06:24,590
And in rubidium,
rubidium-85 and rubidium-87.

900
01:06:24,590 --> 01:06:28,440
So there are two peaks which
are spectrally very, very

901
01:06:28,440 --> 01:06:30,150
well resolved.

902
01:06:30,150 --> 01:06:34,870
And now I tell you what causes
the splitting between the lines

903
01:06:34,870 --> 01:06:36,590
of rubidium-85 and rubidium-87.

904
01:06:39,520 --> 01:06:42,650
Well, the first effect
is really trivial.

905
01:06:42,650 --> 01:06:45,200
It's the mass effect.

906
01:06:45,200 --> 01:06:48,270
And I have to remind
you that the Rydberg

907
01:06:48,270 --> 01:06:56,640
formula for a single
electron energy level

908
01:06:56,640 --> 01:06:59,300
contains the reduced mass.

909
01:06:59,300 --> 01:07:01,900
In other words,
the energy levels

910
01:07:01,900 --> 01:07:04,600
are the energy
levels-- if you assume

911
01:07:04,600 --> 01:07:08,670
that the mass of the nucleus is
infinite, that means you just

912
01:07:08,670 --> 01:07:12,460
take for the electron mass
in the Rydberg constant

913
01:07:12,460 --> 01:07:14,050
the electron mass.

914
01:07:14,050 --> 01:07:23,450
But in general, you have
to take the reduced mass--

915
01:07:23,450 --> 01:07:28,030
the big M is the nucleus.

916
01:07:28,030 --> 01:07:29,825
And small m is the electron.

917
01:07:32,340 --> 01:07:38,020
The simplest case is if you set
the nuclear mass to infinity.

918
01:07:38,020 --> 01:07:41,015
Then you simply have the Rydberg
constant with the electron

919
01:07:41,015 --> 01:07:43,530
mass.

920
01:07:43,530 --> 01:07:47,080
So in the limit that
the nuclear mass is

921
01:07:47,080 --> 01:07:49,020
much larger than
the electron mass,

922
01:07:49,020 --> 01:07:54,220
this correction factor is 1
minus little m over big M.

923
01:07:54,220 --> 01:07:57,840
So therefore, the correction
factor is on the order of 10

924
01:07:57,840 --> 01:08:02,290
to the 4 or 10 to the 5.

925
01:08:02,290 --> 01:08:06,750
Visible frequencies are on
the order of 10 to the 14.

926
01:08:06,750 --> 01:08:09,910
So 10 to the minus 4 or
10 to the minus 5 of it

927
01:08:09,910 --> 01:08:12,850
is between 1 and 10 gigahertz.

928
01:08:12,850 --> 01:08:16,446
So this is the scale
for mass corrections

929
01:08:16,446 --> 01:08:17,529
due to the isotope effect.

930
01:08:24,140 --> 01:08:27,950
What is the sine of
the isotope effect?

931
01:08:27,950 --> 01:08:33,890
Does the fact that the
nucleus has a finite mass--

932
01:08:33,890 --> 01:08:36,970
does that mean that the
binding energy of the electron

933
01:08:36,970 --> 01:08:38,290
is smaller or larger?

934
01:08:52,540 --> 01:08:56,824
AUDIENCE: [INAUDIBLE]
it would be larger.

935
01:08:56,824 --> 01:09:02,620
PROFESSOR: The absolute
value of the binding energy--

936
01:09:02,620 --> 01:09:03,990
look at your signs.

937
01:09:03,990 --> 01:09:06,090
I would say it in the following.

938
01:09:06,090 --> 01:09:08,010
Let's start out with
an infinite nucleus,

939
01:09:08,010 --> 01:09:10,290
and only the electron is moving.

940
01:09:10,290 --> 01:09:12,580
But now we make the
nucleus lighter.

941
01:09:12,580 --> 01:09:15,290
And that means the
nucleus also has to move,

942
01:09:15,290 --> 01:09:16,990
because it's a two-body problem.

943
01:09:16,990 --> 01:09:19,260
And there is additional
kinetic energy,

944
01:09:19,260 --> 01:09:21,550
additional kinetic
energy of the nucleus.

945
01:09:21,550 --> 01:09:25,500
And kinetic energy is positive
and weakens the binding energy

946
01:09:25,500 --> 01:09:27,260
of the total system.

947
01:09:27,260 --> 01:09:31,770
So therefore, the fact that the
effective mass correction means

948
01:09:31,770 --> 01:09:36,040
the lighter the nucleus
is, the more kinetic energy

949
01:09:36,040 --> 01:09:38,810
has to be added to the system
for the nuclear motion,

950
01:09:38,810 --> 01:09:41,284
and the more the binding
energy is weakened.

951
01:09:43,790 --> 01:09:50,920
The most dramatic example
is not the hydrogen atom.

952
01:09:50,920 --> 01:09:59,510
It is positronium, where your
nucleus is not a nucleus,

953
01:09:59,510 --> 01:10:02,260
it's a positively charged
electron to the positronium.

954
01:10:02,260 --> 01:10:06,610
And in this situation,
you have an effective mass

955
01:10:06,610 --> 01:10:11,350
which is only 1/2 of
the electron mass.

956
01:10:11,350 --> 01:10:30,090
And the 1s-2s transition, which
is Lyman alpha for hydrogen,

957
01:10:30,090 --> 01:10:32,260
is now not even
in the vacuum UV,

958
01:10:32,260 --> 01:10:35,710
it just happens at
ordinary UV transitions.

959
01:10:35,710 --> 01:10:39,610
And that means now
the 1s-2s energy

960
01:10:39,610 --> 01:10:41,570
is smaller by a factor of 2.

961
01:10:41,570 --> 01:10:44,770
That really means the binding
energy between an electron

962
01:10:44,770 --> 01:10:49,050
and a positron is only
50% of the binding energy

963
01:10:49,050 --> 01:10:50,710
of an electron in
the hydrogen atom.

964
01:10:54,040 --> 01:10:54,540
OK.

965
01:10:54,540 --> 01:10:56,630
That's all I want to say
about the mass effect.

966
01:11:02,470 --> 01:11:04,235
Let's now talk about
the volume effect.

967
01:11:11,690 --> 01:11:16,420
So if you would look at
the charge distribution

968
01:11:16,420 --> 01:11:27,410
in a nucleus as a function of
r, if we go from one isotope

969
01:11:27,410 --> 01:11:30,810
to a heavier isotope
with more neutrons,

970
01:11:30,810 --> 01:11:34,450
the nuclear radius
becomes larger

971
01:11:34,450 --> 01:11:36,521
and the charge becomes
more spread out.

972
01:11:40,850 --> 01:11:50,040
So if I plot now the
electrostatic potential,

973
01:11:50,040 --> 01:11:54,780
the electrostatic potential
is, of course, the Coulomb

974
01:11:54,780 --> 01:12:01,770
potential, 1 over r, until we
enter the charge distribution.

975
01:12:01,770 --> 01:12:10,130
And then, as you know from
electrostatics, it continues.

976
01:12:14,240 --> 01:12:15,770
This is 1 over r.

977
01:12:15,770 --> 01:12:16,995
And then it's flattened off.

978
01:12:16,995 --> 01:12:21,230
It continues quadratically
for the heavier-- oops,

979
01:12:21,230 --> 01:12:22,940
I wanted to change color.

980
01:12:22,940 --> 01:12:25,800
For the heavier nucleus,
it is like this,

981
01:12:25,800 --> 01:12:29,630
and for the lighter
nucleus, it is like this.

982
01:12:29,630 --> 01:12:32,160
So in other words, the
finite size of the nucleus

983
01:12:32,160 --> 01:12:34,960
is cutting off the
Coulomb potential

984
01:12:34,960 --> 01:12:36,250
where it is strongest.

985
01:12:36,250 --> 01:12:38,870
And this happens the
earlier, the larger,

986
01:12:38,870 --> 01:12:40,900
or the heavier the nucleus is.

987
01:12:43,530 --> 01:12:47,195
So therefore, what you
obtain is you obtain,

988
01:12:47,195 --> 01:12:51,250
in perturbation
theory, a level shift.

989
01:12:51,250 --> 01:12:53,880
Since it only affects
the electron when

990
01:12:53,880 --> 01:12:56,990
it's very close to the
origin, this level shift

991
01:12:56,990 --> 01:13:00,750
is, as other effects we
have discussed today,

992
01:13:00,750 --> 01:13:03,130
proportional to the
probability of the electron

993
01:13:03,130 --> 01:13:04,900
to be at the center.

994
01:13:04,900 --> 01:13:09,320
This is only s
electrons are effected.

995
01:13:20,180 --> 01:13:23,040
What is the effect
in terms of energy?

996
01:13:23,040 --> 01:13:26,690
Well, it's clear the Coulomb
potential is weakened.

997
01:13:26,690 --> 01:13:32,740
Therefore, this effect,
the volume effect weakens,

998
01:13:32,740 --> 01:13:42,380
decreases the binding
energy of the electron.

999
01:13:42,380 --> 01:13:48,090
So we have two effects now-- we
have the volume effect, which

1000
01:13:48,090 --> 01:13:52,180
is the stronger the bigger and
the heavier the nucleus is.

1001
01:13:59,710 --> 01:14:06,760
So it's largest
for heavy nuclei.

1002
01:14:06,760 --> 01:14:12,020
And the mass effect, or
the effective mass effect,

1003
01:14:12,020 --> 01:14:14,715
is of course largest
for the lightest nuclei,

1004
01:14:14,715 --> 01:14:17,210
with the extreme
example of positronium.

1005
01:14:33,560 --> 01:14:34,485
Any questions?

1006
01:14:42,410 --> 01:14:43,880
Well, we have five minutes left.

1007
01:14:43,880 --> 01:14:44,496
But Cody.

1008
01:14:44,496 --> 01:14:45,870
AUDIENCE: What
sort of scales are

1009
01:14:45,870 --> 01:14:46,690
associated with
the volume effect?

1010
01:14:46,690 --> 01:14:47,582
PROFESSOR: Pardon?

1011
01:14:47,582 --> 01:14:48,920
AUDIENCE: What sort
of energy scales

1012
01:14:48,920 --> 01:14:51,503
are associated with the volume
effect? [INAUDIBLE] comparable?

1013
01:14:51,503 --> 01:14:55,130
PROFESSOR: What energy is--
no, it's actually-- wait.

1014
01:14:55,130 --> 01:14:56,902
Let's estimate it.

1015
01:14:56,902 --> 01:14:58,110
You're working with rubidium.

1016
01:14:58,110 --> 01:15:01,270
What is your isotope shift in
rubidium between 85 and 87?

1017
01:15:01,270 --> 01:15:04,336
AUDIENCE: I actually don't know.

1018
01:15:04,336 --> 01:15:08,171
PROFESSOR: Isn't
it 170, 90-- no,

1019
01:15:08,171 --> 01:15:09,920
I'm getting confused
now with [INAUDIBLE].

1020
01:15:13,694 --> 01:15:15,110
So many people
work with rubidium.

1021
01:15:15,110 --> 01:15:17,810
What is the difference
in transition frequency

1022
01:15:17,810 --> 01:15:20,114
between rubidium-85 and -87?

1023
01:15:20,114 --> 01:15:21,922
AUDIENCE: [INAUDIBLE].

1024
01:15:21,922 --> 01:15:24,558
PROFESSOR: It's much
more than gigahertz.

1025
01:15:24,558 --> 01:15:26,480
You really have to
tune your laser.

1026
01:15:26,480 --> 01:15:28,370
I think if you look
at the wave per cell,

1027
01:15:28,370 --> 01:15:32,400
you will never accidentally
see a rubidium-85 line.

1028
01:15:32,400 --> 01:15:35,206
I just don't recall the number.

1029
01:15:35,206 --> 01:15:36,190
AUDIENCE: [INAUDIBLE].

1030
01:15:42,094 --> 01:15:46,030
AUDIENCE: [INAUDIBLE] together,
you can see both of them.

1031
01:15:46,030 --> 01:15:49,490
PROFESSOR: OK.

1032
01:15:49,490 --> 01:15:51,840
And for the mass
effect, we said it's

1033
01:15:51,840 --> 01:15:55,980
sort of-- then they
would be comparable.

1034
01:15:55,980 --> 01:15:57,980
The mass effect is
easy to estimate,

1035
01:15:57,980 --> 01:16:01,730
because the mass correction
is one part in a few thousand.

1036
01:16:01,730 --> 01:16:06,530
So that would mean on the order
of 10 to the 10 gigahertz.

1037
01:16:06,530 --> 01:16:08,880
So the volume effect--
it really depends.

1038
01:16:08,880 --> 01:16:11,520
It's tiny for light elements.

1039
01:16:11,520 --> 01:16:14,510
Rubidium is already heavy.

1040
01:16:14,510 --> 01:16:17,170
So right now I would
say they are comparable.

1041
01:16:17,170 --> 01:16:20,170
But since it's of
interest, I will give you

1042
01:16:20,170 --> 01:16:23,070
more accurate
numbers on Wednesday.

1043
01:16:23,070 --> 01:16:24,745
Any other questions?

1044
01:16:24,745 --> 01:16:25,245
Yes?

1045
01:16:25,245 --> 01:16:25,911
AUDIENCE: Sorry.

1046
01:16:25,911 --> 01:16:28,040
Could you explain the
graph of the previous page,

1047
01:16:28,040 --> 01:16:29,992
how the lines are joined off?

1048
01:16:32,920 --> 01:16:34,090
PROFESSOR: The potential?

1049
01:16:34,090 --> 01:16:35,374
AUDIENCE: Yes.

1050
01:16:35,374 --> 01:16:36,040
PROFESSOR: Yeah.

1051
01:16:36,040 --> 01:16:41,900
So what happens is-- so
if we're being cryptic,

1052
01:16:41,900 --> 01:16:44,540
this is the charge distribution.

1053
01:16:44,540 --> 01:16:47,710
And what I'm doing now is
I'm solving Laplace equation.

1054
01:16:47,710 --> 01:16:50,000
I'm solving Laplace
equation and integrating

1055
01:16:50,000 --> 01:16:51,720
from r equals infinity.

1056
01:16:51,720 --> 01:16:54,980
And as long as I'm
outside the charge radius,

1057
01:16:54,980 --> 01:16:57,540
I get the 1 over r
Coulomb potential.

1058
01:16:57,540 --> 01:17:01,370
But the moment I heat the
surface of the nucleus,

1059
01:17:01,370 --> 01:17:04,600
I continue to indicate
Laplace equation.

1060
01:17:04,600 --> 01:17:12,040
But what is now inside is a
smaller and smaller charge.

1061
01:17:12,040 --> 01:17:14,280
I can also say I use
a form of Gauss's law.

1062
01:17:14,280 --> 01:17:18,670
So I'm therefore not
continuing on the 1 over r.

1063
01:17:18,670 --> 01:17:20,750
And if you look at
Jackson, or if you

1064
01:17:20,750 --> 01:17:26,460
look at maybe a PSET you have
solved, the 1 over r potential

1065
01:17:26,460 --> 01:17:29,260
becomes now a parabola.

1066
01:17:29,260 --> 01:17:31,270
And so what I wanted to
sort of indicate here

1067
01:17:31,270 --> 01:17:34,360
is once you heat the
surface of the nucleus,

1068
01:17:34,360 --> 01:17:38,030
you're not continuing on
the 1 over r trajectory.

1069
01:17:38,030 --> 01:17:41,640
You have in a wave which
is continuous and has

1070
01:17:41,640 --> 01:17:45,900
a continuous deriviative--
you have to fit in a parabola.

1071
01:17:45,900 --> 01:17:47,640
And for the heavier
nucleus, this

1072
01:17:47,640 --> 01:17:51,460
leads us to this potential.

1073
01:17:51,460 --> 01:17:54,490
As for the lighter nucleus,
the potential is deeper.

1074
01:17:54,490 --> 01:17:58,040
And that explains why the
binding energy for heavier

1075
01:17:58,040 --> 01:18:00,140
nuclei is smaller than
for lighter nuclei.

1076
01:18:02,930 --> 01:18:03,430
Yes?

1077
01:18:03,430 --> 01:18:05,406
AUDIENCE: So for the
other orbitals, is it

1078
01:18:05,406 --> 01:18:07,382
exactly zero or
[? near to ?] zero?

1079
01:18:10,346 --> 01:18:11,420
PROFESSOR: OK.

1080
01:18:11,420 --> 01:18:14,970
For the other orbitals,
what we have to do

1081
01:18:14,970 --> 01:18:18,370
is-- and actually, you
have a homework assignment

1082
01:18:18,370 --> 01:18:20,720
to do it for hydrogen
and the proton.

1083
01:18:20,720 --> 01:18:23,530
You take the difference
between the actual potential

1084
01:18:23,530 --> 01:18:26,630
and the Coulomb potential.

1085
01:18:26,630 --> 01:18:28,710
So you take the difference.

1086
01:18:28,710 --> 01:18:31,950
And the difference is
your perturbation operator

1087
01:18:31,950 --> 01:18:34,230
for the finite size
of the nucleus.

1088
01:18:34,230 --> 01:18:36,730
And now you take this
perturbation operator

1089
01:18:36,730 --> 01:18:39,385
between your wave function
and calculate the lowest order

1090
01:18:39,385 --> 01:18:42,230
of the expectation value.

1091
01:18:42,230 --> 01:18:44,900
For the s electron,
you can immediately

1092
01:18:44,900 --> 01:18:49,280
factor out the probability
for the electronic s equals 0.

1093
01:18:49,280 --> 01:18:53,430
But s-- we discussed, if you
have orbital angular momentum,

1094
01:18:53,430 --> 01:18:56,510
let me just scribble
it down here,

1095
01:18:56,510 --> 01:19:00,961
the wave function is
proportional to r to the l.

1096
01:19:00,961 --> 01:19:04,730
So therefore, you
have actually an r

1097
01:19:04,730 --> 01:19:13,320
to the l-- the wave function is
exactly 0 only at r equals 0.

1098
01:19:13,320 --> 01:19:15,830
And then it slowly grows.

1099
01:19:15,830 --> 01:19:19,370
So therefore, given the
finite size of the nucleus,

1100
01:19:19,370 --> 01:19:23,590
you will get a tiny, an
absolutely tiny effect

1101
01:19:23,590 --> 01:19:26,180
if you integrate a wave
function, r to the l,

1102
01:19:26,180 --> 01:19:29,290
over your perturbation operator.

1103
01:19:29,290 --> 01:19:31,520
So it's not mathematically zero.

1104
01:19:31,520 --> 01:19:35,300
But for all practical
purposes, it vanishes.

1105
01:19:35,300 --> 01:19:37,328
Nancy.

1106
01:19:37,328 --> 01:19:40,590
AUDIENCE: For the different
nuclei-- for example,

1107
01:19:40,590 --> 01:19:43,789
in rubidium-- is mass and
volume affecting anything?

1108
01:19:43,789 --> 01:19:48,719
Or is [INAUDIBLE] structure also
important in these isotopes?

1109
01:19:48,719 --> 01:19:52,170
Because the nuclear structure
would be different [INAUDIBLE].

1110
01:19:55,128 --> 01:19:56,390
PROFESSOR: Oh, yeah.

1111
01:19:56,390 --> 01:19:58,850
When we talk isotope
effects, I was

1112
01:19:58,850 --> 01:20:01,360
talking about the isotope
effects of mass and volume.

1113
01:20:01,360 --> 01:20:04,530
But different isotopes
will, in general,

1114
01:20:04,530 --> 01:20:10,470
have different magnetic moments
or different quadrupolar

1115
01:20:10,470 --> 01:20:11,370
deformations.

1116
01:20:11,370 --> 01:20:14,630
So what I discussed about
hyperfine structure also

1117
01:20:14,630 --> 01:20:16,420
applies to isotopes.

1118
01:20:16,420 --> 01:20:19,140
I only separated it,
because usually when

1119
01:20:19,140 --> 01:20:21,790
you don't have isotopes, you
don't talk about, like, sodium.

1120
01:20:21,790 --> 01:20:24,300
Who has ever talked about
the mass shift or volume

1121
01:20:24,300 --> 01:20:25,820
effect in sodium?

1122
01:20:25,820 --> 01:20:27,780
Usually you don't,
because sodium

1123
01:20:27,780 --> 01:20:31,590
has 100% natural
abundance in sodium-23.

1124
01:20:31,590 --> 01:20:33,890
So therefore, the
hyperfine effects,

1125
01:20:33,890 --> 01:20:36,200
they lead to
observable splittings

1126
01:20:36,200 --> 01:20:39,710
even if you have
only one isotope.

1127
01:20:39,710 --> 01:20:42,490
But in general, yes,
different isotopes

1128
01:20:42,490 --> 01:20:45,516
differ in all four effects-- the
mass effect, the volume effect,

1129
01:20:45,516 --> 01:20:47,807
the deformation effect, and
the magnetic moment effect.

1130
01:20:47,807 --> 01:20:52,101
AUDIENCE: [INAUDIBLE] mass and
volume effects are used more

1131
01:20:52,101 --> 01:20:54,586
than [INAUDIBLE] splitting?

1132
01:20:54,586 --> 01:20:57,071
Because I find there
is a different rate

1133
01:20:57,071 --> 01:21:00,053
for different isotopes.

1134
01:21:00,053 --> 01:21:02,041
Like, in the starting
of the lecture,

1135
01:21:02,041 --> 01:21:04,526
you were talking about
adding two positrons

1136
01:21:04,526 --> 01:21:06,017
to the alpha particles.

1137
01:21:06,017 --> 01:21:09,496
That would change [INAUDIBLE].

1138
01:21:09,496 --> 01:21:11,510
PROFESSOR: Oh, yeah,
this would change.

1139
01:21:11,510 --> 01:21:14,080
But you can separate
those effects,

1140
01:21:14,080 --> 01:21:17,070
because I mentioned that
in hyperfine structure,

1141
01:21:17,070 --> 01:21:20,560
the center of mass of the
energy levels is the same.

1142
01:21:20,560 --> 01:21:24,180
So if you see a splitting
but take the center of mass,

1143
01:21:24,180 --> 01:21:26,310
then the center
of mass will only

1144
01:21:26,310 --> 01:21:28,800
depend on volume
and mass effects.

1145
01:21:31,790 --> 01:21:32,290
OK.

1146
01:21:32,290 --> 01:21:35,560
Let's continue on Wednesday.