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00:00:26,240 --> 00:00:30,470
PROFESSOR: Let's get started.

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00:00:30,470 --> 00:00:37,670
We want to wrap up today
our discussion of atoms

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00:00:37,670 --> 00:00:40,970
without external
field, and then discuss

11
00:00:40,970 --> 00:00:46,850
what happens when we put atoms
into external magnetic fields.

12
00:00:46,850 --> 00:00:51,400
Last class, we talked
about isotope shifts,

13
00:00:51,400 --> 00:00:55,472
and there was a question, how
big are those isotopes for some

14
00:00:55,472 --> 00:00:56,430
of your favorite atoms?

15
00:00:59,640 --> 00:01:02,150
I just looked up
some information

16
00:01:02,150 --> 00:01:06,330
on lithium, which
is a light atom,

17
00:01:06,330 --> 00:01:10,590
and this paper here
shows calculations

18
00:01:10,590 --> 00:01:13,175
compared with experiments.

19
00:01:13,175 --> 00:01:16,050
The isotope shift between
lithium six and lithium seven

20
00:01:16,050 --> 00:01:18,760
due to the mass, lithium
six to lithium seven,

21
00:01:18,760 --> 00:01:20,940
is about 10 gigahertz.

22
00:01:20,940 --> 00:01:25,360
The volume effect is only two
megahertz, 1,000 times smaller.

23
00:01:25,360 --> 00:01:27,170
However, the precision
of experiments

24
00:01:27,170 --> 00:01:30,170
is such that if you
find an isotope shift,

25
00:01:30,170 --> 00:01:32,972
the mass effect can
be exactly calculated

26
00:01:32,972 --> 00:01:36,350
from the atomic masses, you
can still get information

27
00:01:36,350 --> 00:01:39,610
about the size of the
atomic nucleus out of it.

28
00:01:39,610 --> 00:01:44,360
So this is the example of a
light atom, 10 gigahertz mass

29
00:01:44,360 --> 00:01:46,360
effect, two megahertz
volume effect.

30
00:01:52,240 --> 00:01:55,340
And here is a rubidium atom.

31
00:01:55,340 --> 00:02:00,080
The isotope shift of the
D1 line between 85 and 87

32
00:02:00,080 --> 00:02:03,920
is 77 megahertz only.

33
00:02:03,920 --> 00:02:08,609
The mass shift is 56
megahertz, and the remainder

34
00:02:08,609 --> 00:02:11,700
is mainly the volume effect.

35
00:02:11,700 --> 00:02:14,770
There's a specific mass effect
due to electronic correlations

36
00:02:14,770 --> 00:02:17,800
if you calculate electrons,
which I think is small here

37
00:02:17,800 --> 00:02:19,960
but I don't want to discuss it.

38
00:02:19,960 --> 00:02:25,170
We can now compare
the mass effect

39
00:02:25,170 --> 00:02:28,300
in rubidium versus lithium.

40
00:02:28,300 --> 00:02:33,265
First of all, the shift
compared to infinite mass.

41
00:02:33,265 --> 00:02:37,020
the reduced mass effect,
is much bigger in lithium.

42
00:02:37,020 --> 00:02:41,180
It's 20 times bigger in
lithium than in rubidium.

43
00:02:41,180 --> 00:02:44,400
However, when you go from
lithium six to lithium seven,

44
00:02:44,400 --> 00:02:48,850
the mass changes by 15%,
so the delta m over m

45
00:02:48,850 --> 00:02:51,620
is also much larger for
lithium than rubidium.

46
00:02:51,620 --> 00:02:53,725
And if you add
those two factors,

47
00:02:53,725 --> 00:02:57,285
you find that the mass
effect is 200 times larger

48
00:02:57,285 --> 00:03:03,440
in lithium, 10 gigahertz
versus 50 megahertz.

49
00:03:03,440 --> 00:03:05,960
The nuclear volume
effect is two megahertz

50
00:03:05,960 --> 00:03:08,830
for lithium, 20
megahertz for rubidium,

51
00:03:08,830 --> 00:03:12,660
so that sets the scale.

52
00:03:12,660 --> 00:03:14,680
Any questions about that?

53
00:03:20,010 --> 00:03:27,970
Let me come back to one
other question which

54
00:03:27,970 --> 00:03:37,130
we discussed last
class, and this

55
00:03:37,130 --> 00:03:41,375
was the question
about if you have

56
00:03:41,375 --> 00:03:45,300
a deformation of a
nucleus, or if you have

57
00:03:45,300 --> 00:03:50,360
any kind of isotropic
shape of an object,

58
00:03:50,360 --> 00:03:52,906
what is the minimum
angular momentum in order

59
00:03:52,906 --> 00:03:54,440
to observe it?

60
00:03:54,440 --> 00:03:57,220
I know a lot of you
got confused about it,

61
00:03:57,220 --> 00:04:01,890
so I want to discuss
the same thing again,

62
00:04:01,890 --> 00:04:06,772
but now focusing on two
different frames, the lab frame

63
00:04:06,772 --> 00:04:09,310
and the body-fixed frame.

64
00:04:09,310 --> 00:04:11,410
I hope you find this
discussion insightful.

65
00:04:37,510 --> 00:04:42,828
So let's assume we have an
object, it can be a molecule.

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00:04:42,828 --> 00:04:45,018
Actually, you have a
homework assignment

67
00:04:45,018 --> 00:04:47,770
whether you can observe
a permanent dipole

68
00:04:47,770 --> 00:04:50,100
moment of a molecule,
and this will lead you

69
00:04:50,100 --> 00:04:55,440
into a discussion of lab
frame versus body-fixed frame.

70
00:04:55,440 --> 00:04:57,850
But let's assume
we have an object,

71
00:04:57,850 --> 00:05:01,970
assume it's a nucleus, which
has a really odd shape.

72
00:05:01,970 --> 00:05:07,600
However, if it has
angular momentum of zero,

73
00:05:07,600 --> 00:05:12,950
all you have is one level.

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00:05:12,950 --> 00:05:20,760
If you have an angular momentum
of 1/2, you have two levels,

75
00:05:20,760 --> 00:05:24,530
and you can now define
in the laboratory

76
00:05:24,530 --> 00:05:27,830
that the energy difference
is due to the magnetic dipole

77
00:05:27,830 --> 00:05:30,601
moment, for instance, or
the electric dipole moment

78
00:05:30,601 --> 00:05:34,497
if you put in electric field.

79
00:05:34,497 --> 00:05:43,877
If you have I equals
1, you can have

80
00:05:43,877 --> 00:05:52,215
three levels, E1, E0, E minus 1.

81
00:05:54,850 --> 00:05:58,240
Let's say you have put the
atoms into an electric field

82
00:05:58,240 --> 00:05:59,520
gradient.

83
00:05:59,520 --> 00:06:09,430
You can then ask if E0 is in
the middle between up and down,

84
00:06:09,430 --> 00:06:13,270
plus 1 or minus 1, or whether
it's displaced or not.

85
00:06:13,270 --> 00:06:18,510
And depending whether this is
larger or smaller than zero,

86
00:06:18,510 --> 00:06:21,790
you would say there is a
quadrupole moment which

87
00:06:21,790 --> 00:06:24,400
is larger or smaller than zero.

88
00:06:24,400 --> 00:06:28,220
In other words, what I'm telling
you is if you have only one

89
00:06:28,220 --> 00:06:32,940
level, I equals 0,
you can't say anything

90
00:06:32,940 --> 00:06:34,485
about the shape of the object.

91
00:06:34,485 --> 00:06:38,090
If you have two levels, you
can determine a dipole moment.

92
00:06:38,090 --> 00:06:40,690
If you have three
levels, by the deviation

93
00:06:40,690 --> 00:06:45,340
from the equidistance, you can
find a higher order moment.

94
00:06:50,230 --> 00:06:54,035
But now comes the point.

95
00:06:54,035 --> 00:06:55,670
You can now take two positions.

96
00:06:58,300 --> 00:07:09,680
You can say that for low I,
the deformation of a magnetic

97
00:07:09,680 --> 00:07:13,920
moment for I equals
0 cannot be measured.

98
00:07:27,810 --> 00:07:30,230
Therefore, it is zero.

99
00:07:30,230 --> 00:07:46,880
Or you can say deformation
exists, but only not in the lab

100
00:07:46,880 --> 00:07:51,104
frame but in the
body-fixed frame.

101
00:07:54,010 --> 00:07:58,410
So in other words, you would
say that the deformation exists

102
00:07:58,410 --> 00:07:59,090
always.

103
00:07:59,090 --> 00:08:01,640
It's just a measurement problem.

104
00:08:01,640 --> 00:08:04,370
At low angular momentum,
I cannot measure.

105
00:08:04,370 --> 00:08:08,970
Or the other statement would be,
well, if you can't measure it,

106
00:08:08,970 --> 00:08:12,210
[INAUDIBLE], it means
it doesn't exist.

107
00:08:12,210 --> 00:08:16,847
So which statement or which
conclusion is correct?

108
00:08:34,070 --> 00:08:36,901
This is the body-fixed
frame argument,

109
00:08:36,901 --> 00:08:39,391
and this is more based
on the lab frame.

110
00:08:43,380 --> 00:08:49,090
Well, I would say the lab frame
argument is always correct

111
00:08:49,090 --> 00:08:50,996
because you can't
measure it, you

112
00:08:50,996 --> 00:08:54,730
can't determine it in the
lab, but let me go from there.

113
00:08:54,730 --> 00:09:00,430
Let's assume we have
an object, and we

114
00:09:00,430 --> 00:09:03,614
think it has a deformation but
it has a low angular momentum

115
00:09:03,614 --> 00:09:05,920
phase.

116
00:09:05,920 --> 00:09:10,395
Can we still say that
in its body-fixed frame,

117
00:09:10,395 --> 00:09:15,630
we have an object which
has a deformation or not?

118
00:09:15,630 --> 00:09:21,300
Now, my personal opinion
is the following.

119
00:09:21,300 --> 00:09:22,794
It really depends.

120
00:09:22,794 --> 00:09:26,430
If you have a system
where you can have

121
00:09:26,430 --> 00:09:30,000
angular momentum
without changing

122
00:09:30,000 --> 00:09:31,530
the internal structure.

123
00:09:31,530 --> 00:09:34,865
For instance, I have this stick,
and I can just spin it up,

124
00:09:34,865 --> 00:09:37,740
and if I can spin it up, I can
get an angular momentum wave

125
00:09:37,740 --> 00:09:41,840
packet, I can orient it and
measure its deformation.

126
00:09:41,840 --> 00:09:47,060
Then I think I would say, even
if this state has zero angular

127
00:09:47,060 --> 00:09:50,230
momentum, it has a
deformation, and the way

128
00:09:50,230 --> 00:09:53,110
I know it because if I
add angular momentum,

129
00:09:53,110 --> 00:09:54,940
I can measure the deformation.

130
00:09:54,940 --> 00:09:57,570
I cannot measure it at
low angular momentum.

131
00:09:57,570 --> 00:09:59,970
So then you would say in
the body-fixed system,

132
00:09:59,970 --> 00:10:03,480
there is a deformation, but
it only manifests itself

133
00:10:03,480 --> 00:10:07,980
in the lab frame if I
add angular momentum.

134
00:10:07,980 --> 00:10:12,450
However, you may have an
object, let's say a molecule,

135
00:10:12,450 --> 00:10:16,880
which is so weakly bound
that one quantum of angular

136
00:10:16,880 --> 00:10:21,490
momentum due to centrifugal
forces rips it apart.

137
00:10:21,490 --> 00:10:22,670
That exists.

138
00:10:22,670 --> 00:10:27,560
Extremely bound state which
cannot provide enough binding

139
00:10:27,560 --> 00:10:32,130
force to withstand even one
unit of angular momentum.

140
00:10:32,130 --> 00:10:35,000
So that's an object
which you cannot rotate,

141
00:10:35,000 --> 00:10:38,695
you cannot transfer any angular
momentum without destroying it,

142
00:10:38,695 --> 00:10:41,130
without ripping it apart.

143
00:10:41,130 --> 00:10:45,850
And now to say that this
molecule has a dipole moment

144
00:10:45,850 --> 00:10:51,010
or has some anisotropy, you're
making a statement which cannot

145
00:10:51,010 --> 00:10:52,210
be tested at all.

146
00:10:52,210 --> 00:10:54,374
So at that point, you
should rather say,

147
00:10:54,374 --> 00:10:56,290
what matters is what I
can measure in the lab,

148
00:10:56,290 --> 00:10:57,748
and I will never
be able to measure

149
00:10:57,748 --> 00:11:01,140
any deformation in the lab.

150
00:11:01,140 --> 00:11:06,560
So the first example
with the stick

151
00:11:06,560 --> 00:11:09,170
will apply to a very
stable molecule which

152
00:11:09,170 --> 00:11:11,844
may have a dipole
moment, and you

153
00:11:11,844 --> 00:11:13,885
assume it has the same
dipole moment whether it's

154
00:11:13,885 --> 00:11:15,450
rotating or not.

155
00:11:15,450 --> 00:11:18,160
And then you would say
at zero angular momentum,

156
00:11:18,160 --> 00:11:21,910
I cannot measure the dipole
moment but I know it exists

157
00:11:21,910 --> 00:11:22,910
in the body-fixed frame.

158
00:11:25,640 --> 00:11:30,330
If you have a
nucleus, and you have

159
00:11:30,330 --> 00:11:33,930
sort of a wave function of
the protons and neutrons,

160
00:11:33,930 --> 00:11:38,640
and it's I equals 0, you
will not find any moment.

161
00:11:38,640 --> 00:11:40,690
At least for the
ground state of nuclei,

162
00:11:40,690 --> 00:11:42,880
when you add angular
momentum, you really

163
00:11:42,880 --> 00:11:44,720
change the internal structure.

164
00:11:44,720 --> 00:11:47,706
You have to promote
nucleons to higher orbits,

165
00:11:47,706 --> 00:11:50,715
so you cannot add angular
momentum and still have

166
00:11:50,715 --> 00:11:51,410
the same object.

167
00:11:51,410 --> 00:11:53,835
So therefore, you have
to say the ground state

168
00:11:53,835 --> 00:11:56,595
with I equals 0 has no
deformation, because there

169
00:11:56,595 --> 00:11:59,541
is no way to ever find
[INAUDIBLE] deformation,

170
00:11:59,541 --> 00:12:02,010
so it doesn't exist.

171
00:12:02,010 --> 00:12:04,370
But there are excited
states of nuclei

172
00:12:04,370 --> 00:12:08,370
which have a deformation,
and these deformed nuclei

173
00:12:08,370 --> 00:12:12,110
can be put into a multiplex
of angular momentum states.

174
00:12:12,110 --> 00:12:16,270
So then you would see, I have
the same kind of nucleus,

175
00:12:16,270 --> 00:12:19,490
but at different
angular momentum states.

176
00:12:19,490 --> 00:12:21,910
At higher angular
momentum states,

177
00:12:21,910 --> 00:12:24,814
you can define [INAUDIBLE]
determine quadrupole moments

178
00:12:24,814 --> 00:12:26,950
and things like this.

179
00:12:26,950 --> 00:12:30,730
And then you may say the
same internal state has now

180
00:12:30,730 --> 00:12:33,990
a non-rotating state, and
you would still be tempted,

181
00:12:33,990 --> 00:12:37,770
and you are correct with that,
to associate a deformation even

182
00:12:37,770 --> 00:12:39,010
in the non-rotating state.

183
00:12:42,380 --> 00:12:45,850
I hope those remarks help
you reconcile the two aspects

184
00:12:45,850 --> 00:12:50,020
whether you have an object which
is stable enough to be spun up,

185
00:12:50,020 --> 00:12:51,530
and then I think
you can always talk

186
00:12:51,530 --> 00:12:53,175
about the body-fixed frame.

187
00:12:53,175 --> 00:12:56,240
But if you have an object
where you change the internal

188
00:12:56,240 --> 00:12:59,930
structure when you add
angular momentum, I think,

189
00:12:59,930 --> 00:13:03,072
for fundamental reasons,
you cannot associate any

190
00:13:03,072 --> 00:13:06,270
deformation with it.

191
00:13:06,270 --> 00:13:07,746
Any questions about that?

192
00:13:16,720 --> 00:13:20,920
Let me just write
down the summary.

193
00:13:20,920 --> 00:13:36,270
The definition of a deformation
in a body-fixed frame

194
00:13:36,270 --> 00:13:55,850
makes sense only if you can
add or change angular momentum

195
00:13:55,850 --> 00:14:05,756
without significantly changing
the internal structure.

196
00:14:29,176 --> 00:14:29,675
Questions?

197
00:14:58,440 --> 00:15:00,310
I think the computer is set up.

198
00:15:00,310 --> 00:15:02,584
I cannot go backward
in Presentation mode.

199
00:15:17,385 --> 00:15:22,860
What I want to discuss
next is give you

200
00:15:22,860 --> 00:15:26,315
a little bit of an historic
summary how spectroscopy

201
00:15:26,315 --> 00:15:30,960
of hydrogen is developed,
in particular also focusing

202
00:15:30,960 --> 00:15:34,335
on one important discovery,
the discovery of QED

203
00:15:34,335 --> 00:15:37,100
through the Lamb shift.

204
00:15:37,100 --> 00:15:42,340
Of course, you all love hydrogen
because it's the simplest atom

205
00:15:42,340 --> 00:15:45,580
but it has so much
interesting physics in it.

206
00:15:45,580 --> 00:15:49,160
I've summarized for you here
some papers on hydrogen,

207
00:15:49,160 --> 00:15:53,030
and I used them to
illustrate several points.

208
00:15:53,030 --> 00:15:57,040
I will show you that actually,
the discovery of the Lamb shift

209
00:15:57,040 --> 00:15:58,580
had precursors.

210
00:15:58,580 --> 00:16:01,700
10 years before the Lamb
shift was discovered,

211
00:16:01,700 --> 00:16:03,740
people had even some
idea that something

212
00:16:03,740 --> 00:16:05,910
may be wrong with
the understanding

213
00:16:05,910 --> 00:16:07,360
of the structure of hydrogen.

214
00:16:07,360 --> 00:16:10,690
So you can say they came
so close, people 10 years

215
00:16:10,690 --> 00:16:14,367
before, in realizing
the Lamb shift,

216
00:16:14,367 --> 00:16:16,200
there were people who
maybe missed the Nobel

217
00:16:16,200 --> 00:16:18,560
prize by just a tiny little bit.

218
00:16:18,560 --> 00:16:20,510
They had all the
insight that there

219
00:16:20,510 --> 00:16:22,801
may be QED correction
in hydrogen.

220
00:16:22,801 --> 00:16:24,300
They just didn't
have the technology

221
00:16:24,300 --> 00:16:27,150
to measure it accurately enough.

222
00:16:27,150 --> 00:16:29,120
The second example
I want to show

223
00:16:29,120 --> 00:16:32,980
is that we always talk about
fundamental limitations,

224
00:16:32,980 --> 00:16:35,450
but fundamental limitations
can disappear in time

225
00:16:35,450 --> 00:16:39,710
because they may not be as
fundamental as they appear.

226
00:16:39,710 --> 00:16:51,266
So for instance,
there were limitations

227
00:16:51,266 --> 00:16:52,970
with the limitations
of the Lamb shift

228
00:16:52,970 --> 00:16:55,720
because you had a short
lifetime of p states,

229
00:16:55,720 --> 00:16:58,330
but with the event of
two photon transition,

230
00:16:58,330 --> 00:17:02,380
you can go from s to
s and s to d states,

231
00:17:02,380 --> 00:17:04,450
and therefore map
out Lamb shifts

232
00:17:04,450 --> 00:17:06,980
with much, much higher
precision not limited

233
00:17:06,980 --> 00:17:12,060
by the finite
lifetime of p states.

234
00:17:12,060 --> 00:17:16,640
Finally, you would say Lamb
shifts are small splittings,

235
00:17:16,640 --> 00:17:19,152
and for many, many
years, Lamb shifts

236
00:17:19,152 --> 00:17:21,855
were measured by
making radio frequency

237
00:17:21,855 --> 00:17:24,665
transitions between
two s and two p states.

238
00:17:24,665 --> 00:17:28,590
Well, today, the most accurate
measurement of the Lamb shift

239
00:17:28,590 --> 00:17:30,156
is with an optical
transition where

240
00:17:30,156 --> 00:17:32,290
you need a much, much
higher relative precision

241
00:17:32,290 --> 00:17:34,760
to see the tiny Lamb shift.

242
00:17:34,760 --> 00:17:39,666
But optical metrology with
direct frequency measurements

243
00:17:39,666 --> 00:17:43,380
and frequency combs has so
much improved in precision

244
00:17:43,380 --> 00:17:46,126
that now, an optical
measurement, even if it comes

245
00:17:46,126 --> 00:17:49,070
to a tiny difference, is
more accurate than a direct

246
00:17:49,070 --> 00:17:50,350
[INAUDIBLE].

247
00:17:50,350 --> 00:17:52,310
So I think the
history of hydrogen

248
00:17:52,310 --> 00:17:55,067
shows you that
technology can completely

249
00:17:55,067 --> 00:17:57,990
change the paradigm how
measurements are made.

250
00:17:57,990 --> 00:18:00,280
Fundamental limits
disappear because new tools

251
00:18:00,280 --> 00:18:02,720
or new insight is available.

252
00:18:02,720 --> 00:18:06,360
And also, I find it interesting
that discoveries often

253
00:18:06,360 --> 00:18:10,925
have precursors, and people
have a hunch, know about it,

254
00:18:10,925 --> 00:18:13,297
and then finally,
it is discovered.

255
00:18:13,297 --> 00:18:15,380
Let me just take you through
some historic papers.

256
00:18:32,640 --> 00:18:39,160
This paper is 1933, 15
years before the Lamb shift,

257
00:18:39,160 --> 00:18:42,170
and it says that one
possible explanation

258
00:18:42,170 --> 00:18:45,250
for some discrepancy of the
structure of the Balmer lines

259
00:18:45,250 --> 00:18:48,060
is that the effect
of the interaction

260
00:18:48,060 --> 00:18:50,680
between the radiation field and
the atom has been neglected.

261
00:18:50,680 --> 00:18:52,015
That's QED.

262
00:18:52,015 --> 00:18:55,758
You cannot just calculate the
structure of the hydrogen atom

263
00:18:55,758 --> 00:18:57,150
from the Coulomb field.

264
00:18:57,150 --> 00:19:00,660
You have to allow the
radiation field, all the modes

265
00:19:00,660 --> 00:19:02,400
of the vacuum to be included.

266
00:19:02,400 --> 00:19:04,830
So this insight
is not due to Lamb

267
00:19:04,830 --> 00:19:10,450
It was there already, 1933.

268
00:19:10,450 --> 00:19:14,910
Same year, look at the
title, "On the Breakdown

269
00:19:14,910 --> 00:19:18,500
of the Coulomb Law for
the Hydrogen Atom."

270
00:19:18,500 --> 00:19:24,360
People speculated or discussed
that the Coulomb law will not

271
00:19:24,360 --> 00:19:27,090
be valid at very
small distances.

272
00:19:27,090 --> 00:19:30,640
This is ultimately what QED,
raided correction, the Lamb

273
00:19:30,640 --> 00:19:33,480
shift, vacuum
polarization all is about.

274
00:19:39,950 --> 00:19:42,740
Finally, people had an
understanding of the hydrogen

275
00:19:42,740 --> 00:19:45,060
atom, and they
measure-- I want you

276
00:19:45,060 --> 00:19:47,900
to keep that in mind--
they did optical.

277
00:19:47,900 --> 00:19:55,350
They measured the Balmer lines
of hydrogen and deuterium,

278
00:19:55,350 --> 00:19:59,270
and they couldn't
fully resolve it

279
00:19:59,270 --> 00:20:02,400
because of the finite
lifetime of the peak state,

280
00:20:02,400 --> 00:20:05,180
but there was some
hunch when you

281
00:20:05,180 --> 00:20:10,030
try to get the envelope from
the underlying structure

282
00:20:10,030 --> 00:20:11,780
that there was a discrepancy.

283
00:20:11,780 --> 00:20:14,600
It was just not significant
enough to say for sure,

284
00:20:14,600 --> 00:20:16,370
there is an
additional line shift

285
00:20:16,370 --> 00:20:18,758
which is not accounted
for by theory.

286
00:20:27,670 --> 00:20:29,129
There was a
discussion that there

287
00:20:29,129 --> 00:20:32,560
is a deviation of the Coulomb
law, but here is the insight.

288
00:20:32,560 --> 00:20:35,390
As was indicated by
previous authors,

289
00:20:35,390 --> 00:20:38,465
the interaction required
to change the Coulomb law

290
00:20:38,465 --> 00:20:41,340
at small distances
is much too large

291
00:20:41,340 --> 00:20:43,410
to be accounted for
by the assumption

292
00:20:43,410 --> 00:20:46,180
of a finite size of
electron and proton.

293
00:20:46,180 --> 00:20:48,155
So the Coulomb field
has to be modified

294
00:20:48,155 --> 00:20:53,485
at short distances in a
much stronger way than just

295
00:20:53,485 --> 00:20:55,710
the finite size of the proton.

296
00:20:55,710 --> 00:20:58,690
We'll talk about the finite
size of the proton in a minute.

297
00:20:58,690 --> 00:21:04,180
And then, of course,
1947, UIF oscillators

298
00:21:04,180 --> 00:21:07,360
have been developed in
the pursuit of radar,

299
00:21:07,360 --> 00:21:11,060
experimental tools are there
now-- high power IF sources,

300
00:21:11,060 --> 00:21:13,680
cumulative sources,
and such, and then

301
00:21:13,680 --> 00:21:15,834
Lamb and Retherford
in his landmark paper

302
00:21:15,834 --> 00:21:20,025
look at the fine structure
of the hydrogen atom,

303
00:21:20,025 --> 00:21:22,310
and this is the famous result.

304
00:21:22,310 --> 00:21:28,510
They measured transitions as
a function of magnetic field,

305
00:21:28,510 --> 00:21:30,860
and you see the solid
line, which I think

306
00:21:30,860 --> 00:21:33,430
was the theory without
the Lamb shift.

307
00:21:33,430 --> 00:21:38,460
The dashed line is the hyper
fine structure of hydrogen,

308
00:21:38,460 --> 00:21:41,630
and the lines converge,
and the difference

309
00:21:41,630 --> 00:21:46,214
is 1,000 megacycles, the first
determination of the Lamb

310
00:21:46,214 --> 00:21:48,150
shift.

311
00:21:48,150 --> 00:21:50,950
So this was 1947.

312
00:21:50,950 --> 00:21:55,450
It's interesting
that it's just one

313
00:21:55,450 --> 00:21:58,450
or two weeks later, there is
a theoretical paper by Hans

314
00:21:58,450 --> 00:22:03,000
Bethe providing an explanation
for the Lamb shift,

315
00:22:03,000 --> 00:22:06,510
so already coming up
with the first model

316
00:22:06,510 --> 00:22:10,770
how to account for QED.

317
00:22:10,770 --> 00:22:12,910
I didn't look it
up in detail, but I

318
00:22:12,910 --> 00:22:15,450
thought the spirit was similar
to what I presented you

319
00:22:15,450 --> 00:22:17,335
in class, that the
electromagnetic zero point

320
00:22:17,335 --> 00:22:23,210
energy is shaping the electron
and leading to corrections.

321
00:22:23,210 --> 00:22:26,840
But it's amazing that
within weeks, theorists

322
00:22:26,840 --> 00:22:29,770
figured nearly out, yes,
this is the explanation.

323
00:22:29,770 --> 00:22:32,120
This is how we have
to explain the theory.

324
00:22:36,300 --> 00:22:40,930
For a number of years,
people pursued measurements

325
00:22:40,930 --> 00:22:44,250
of the Lamb shifts with
higher and higher accuracy.

326
00:22:44,250 --> 00:22:47,390
I just like the
last sentence here.

327
00:22:47,390 --> 00:22:52,330
This is now the next paper by
Willis Lamb and Retherford,

328
00:22:52,330 --> 00:22:55,150
and they are sort of saying that
they wanted to measure the Lamb

329
00:22:55,150 --> 00:22:56,900
shift with higher
precision, but then they

330
00:22:56,900 --> 00:22:59,630
said, "the program was
large and encountered

331
00:22:59,630 --> 00:23:03,260
unexpected difficulties
which required much more time

332
00:23:03,260 --> 00:23:04,290
to surmount.

333
00:23:04,290 --> 00:23:08,070
As a result, the paper promised
two years ago was delayed."

334
00:23:08,070 --> 00:23:10,930
I think this applies to many,
many papers to be written,

335
00:23:10,930 --> 00:23:14,300
but here, the authors
even say that upfront,

336
00:23:14,300 --> 00:23:16,905
it took us two years
longer to do the research

337
00:23:16,905 --> 00:23:18,234
than we initially anticipated.

338
00:23:20,990 --> 00:23:23,990
You see now the
growing accuracy.

339
00:23:23,990 --> 00:23:26,770
We have the Lamb shift, which
is on the order of 1,000

340
00:23:26,770 --> 00:23:29,255
megacycles, one gigahertz,
and the precision

341
00:23:29,255 --> 00:23:31,900
is now in the 100
kilohertz range.

342
00:23:31,900 --> 00:23:36,731
This was the technology
of the original discovery.

343
00:23:36,731 --> 00:23:40,995
Then there was a next
generation of experiments

344
00:23:40,995 --> 00:23:45,490
on the Lamb shift using
separated [? oscilloatomic ?]

345
00:23:45,490 --> 00:23:49,260
fields, [INAUDIBLE] techniques.

346
00:23:49,260 --> 00:23:51,340
We'll talk about that
later in the course.

347
00:23:51,340 --> 00:23:54,840
And with these techniques,
the accuracy of the Lamb shift

348
00:23:54,840 --> 00:24:03,378
is now one digit further
in the 10 kilohertz region.

349
00:24:06,560 --> 00:24:10,590
There is a nice feature that
we'll also discuss later,

350
00:24:10,590 --> 00:24:14,400
that it was possible to
obtain line weights, which

351
00:24:14,400 --> 00:24:15,930
is [INAUDIBLE].

352
00:24:15,930 --> 00:24:18,260
We'll talk about
it later, but it's

353
00:24:18,260 --> 00:24:25,010
possible to do spectroscopy on
unstable states, which provides

354
00:24:25,010 --> 00:24:29,083
line widths narrower than
the actual line widths.

355
00:24:29,083 --> 00:24:32,200
If you want to get one
sentence as an appetizer,

356
00:24:32,200 --> 00:24:35,970
you just look at the
atoms which have not

357
00:24:35,970 --> 00:24:38,930
decayed for a long time,
and if you play some tricks,

358
00:24:38,930 --> 00:24:43,210
you can then get line
widths which covers points

359
00:24:43,210 --> 00:24:46,970
to several lifetimes, and not
just to the one [INAUDIBLE]

360
00:24:46,970 --> 00:24:48,380
lifetime.

361
00:24:48,380 --> 00:24:50,520
Some conditions have to
be met, and those authors

362
00:24:50,520 --> 00:24:54,220
used it here to
advantage to narrow

363
00:24:54,220 --> 00:24:56,360
the line for the measurement
of the Lamb shift.

364
00:24:56,360 --> 00:24:56,860
Yes?

365
00:24:56,860 --> 00:24:59,180
AUDIENCE: The abstract
says that the result is not

366
00:24:59,180 --> 00:25:00,530
in good agreement with theory.

367
00:25:00,530 --> 00:25:06,040
What is the theoretical--
how far had it gone?

368
00:25:06,040 --> 00:25:08,870
PROFESSOR: I don't
know it at this point.

369
00:25:08,870 --> 00:25:10,900
I will later give you
comparison with theory

370
00:25:10,900 --> 00:25:14,090
which is highly accurate.

371
00:25:14,090 --> 00:25:18,811
I assume at this
point-- I'm not sure

372
00:25:18,811 --> 00:25:20,540
if it was instrumental
difficulty.

373
00:25:20,540 --> 00:25:21,528
I didn't [INAUDIBLE].

374
00:25:29,790 --> 00:25:33,340
So these are the same authors
just a few years later.

375
00:25:33,340 --> 00:25:35,320
The agreement between
theory and experiment

376
00:25:35,320 --> 00:25:37,986
is two standard deviations, so I
think this problem disappeared.

377
00:25:37,986 --> 00:25:42,810
I don't know if it was the
fault of theory or experiment.

378
00:25:42,810 --> 00:25:47,680
But now we go a step forward
to optical spectroscopy.

379
00:25:47,680 --> 00:25:51,480
You remember originally, and
this is why the Lamb shift was

380
00:25:51,480 --> 00:25:56,960
not jumping into people's eye
when they did spectroscopy

381
00:25:56,960 --> 00:25:59,810
of the Balmer
spectrum of hydrogen,

382
00:25:59,810 --> 00:26:02,680
you cannot resolve
the structure.

383
00:26:02,680 --> 00:26:05,540
The Lamb shift is
there, but it's

384
00:26:05,540 --> 00:26:09,650
hidden in the envelope
of the unresolved lines.

385
00:26:09,650 --> 00:26:14,550
And now the advent
of lasers, and people

386
00:26:14,550 --> 00:26:17,952
immediately developed
saturation spectroscopy.

387
00:26:17,952 --> 00:26:19,410
That's how most of
our laboratories

388
00:26:19,410 --> 00:26:22,010
stabilize lasers using
Doppler-free saturation

389
00:26:22,010 --> 00:26:23,220
spectroscopy.

390
00:26:23,220 --> 00:26:25,460
And when saturation
spectroscopy was invented

391
00:26:25,460 --> 00:26:28,950
by Hansch and Schawlow
for the first time,

392
00:26:28,950 --> 00:26:31,230
you can break through
the Doppler [INAUDIBLE],

393
00:26:31,230 --> 00:26:33,510
and now you see
the lines resolved.

394
00:26:33,510 --> 00:26:36,120
And here, I think
for the first time,

395
00:26:36,120 --> 00:26:39,065
you see two peaks
separated and the splitting

396
00:26:39,065 --> 00:26:43,140
is the Lamb shift, which
until then was only accessible

397
00:26:43,140 --> 00:26:46,370
through [INAUDIBLE]
frequency methods.

398
00:26:46,370 --> 00:26:50,496
Of course, these were the first
lasers, just pulsed lasers,

399
00:26:50,496 --> 00:26:52,880
and we couldn't even
think about precision.

400
00:26:55,640 --> 00:26:58,400
But then, of course,
using metrology,

401
00:26:58,400 --> 00:27:04,750
using frequency chains, people
could do precision measurements

402
00:27:04,750 --> 00:27:09,100
in the optical domain.

403
00:27:09,100 --> 00:27:11,930
These are now papers in the
'90s, "Optical Measurement

404
00:27:11,930 --> 00:27:20,110
on the Lamb Shift in the Ground
State or in the Excited State."

405
00:27:20,110 --> 00:27:23,380
Talking about the comparison
with experiment and theory,

406
00:27:23,380 --> 00:27:26,670
experimental is pushed to
higher and higher precision,

407
00:27:26,670 --> 00:27:29,800
and suddenly, there
was a discrepancy,

408
00:27:29,800 --> 00:27:33,300
and it was a discrepancy
in the 1s Lamb shift.

409
00:27:33,300 --> 00:27:35,344
In the 1s state, the
Lamb shift is much bigger

410
00:27:35,344 --> 00:27:41,630
than in the 2s state because
the electron interacts much more

411
00:27:41,630 --> 00:27:46,225
intimately with the [INAUDIBLE]
than the Coulomb potential is

412
00:27:46,225 --> 00:27:47,600
on this.

413
00:27:47,600 --> 00:27:50,270
People found that the
experiment did no longer

414
00:27:50,270 --> 00:27:52,490
agree with theory.

415
00:27:52,490 --> 00:27:56,630
But then the theorists had to
check all their assumptions,

416
00:27:56,630 --> 00:28:00,565
and it was found that there
were two new binding corrections

417
00:28:00,565 --> 00:28:02,505
which were surprisingly large.

418
00:28:02,505 --> 00:28:06,030
Often, you make an estimate
that those terms are small.

419
00:28:06,030 --> 00:28:08,917
You say it's higher
order, but you may not

420
00:28:08,917 --> 00:28:10,200
know the pre-factor.

421
00:28:10,200 --> 00:28:12,350
And here, something
was surprisingly large,

422
00:28:12,350 --> 00:28:14,440
and by now you're
proving the theory.

423
00:28:14,440 --> 00:28:18,520
There was again agreement
between experiment and theory.

424
00:28:18,520 --> 00:28:21,780
It's getting now down
to the kilohertz level.

425
00:28:21,780 --> 00:28:28,610
And at least as a few years
ago, this was state of the art.

426
00:28:28,610 --> 00:28:32,090
Remember the Lamb shift is about
1,000 cycles one gigahertz,

427
00:28:32,090 --> 00:28:34,988
and now the precision is
in the single kilohertz.

428
00:28:39,930 --> 00:28:42,620
As I pointed out,
precision was reached

429
00:28:42,620 --> 00:28:46,860
by directly measuring the
frequency of laser, frequency

430
00:28:46,860 --> 00:28:48,850
metrology.

431
00:28:48,850 --> 00:28:52,950
This was actually, for historic
interest, frequency metrology,

432
00:28:52,950 --> 00:29:00,320
they used peak nodes
between the laser used

433
00:29:00,320 --> 00:29:04,380
to measure the hydrogen
line and some other lasers.

434
00:29:04,380 --> 00:29:08,620
This is just a few years
before comb generators.

435
00:29:08,620 --> 00:29:10,926
Frequency combs completely
changed things again.

436
00:29:10,926 --> 00:29:13,594
But they had already the
precision of a direct frequency

437
00:29:13,594 --> 00:29:15,330
measurement.

438
00:29:15,330 --> 00:29:18,090
You can read about
it when I paused it.

439
00:29:18,090 --> 00:29:19,510
It shows you what
the [INAUDIBLE]

440
00:29:19,510 --> 00:29:22,810
is in those installations.

441
00:29:22,810 --> 00:29:25,150
This is a slide I
borrowed from Ted Hansch,

442
00:29:25,150 --> 00:29:27,545
"Optical Spectroscopy
of Hydrogen."

443
00:29:27,545 --> 00:29:32,440
It just shows the advances
in frequency metrology.

444
00:29:36,530 --> 00:29:41,060
It shows how caesium clocks
and optical spectroscopy,

445
00:29:41,060 --> 00:29:43,960
how they have
changed in precision,

446
00:29:43,960 --> 00:29:47,485
and eventually, we
are now past the time

447
00:29:47,485 --> 00:29:52,470
where optical spectroscopy is
more accurate than microwave

448
00:29:52,470 --> 00:29:54,895
and radio frequency
spectroscopy.

449
00:29:54,895 --> 00:29:58,290
Optical clocks are more accurate
than [INAUDIBLE] frequency

450
00:29:58,290 --> 00:29:59,760
standards.

451
00:29:59,760 --> 00:30:02,400
I'm sure in your
lifetime, you will

452
00:30:02,400 --> 00:30:04,620
experience the
redefinition of this

453
00:30:04,620 --> 00:30:08,870
again because the caesium clock
is no longer accurate enough

454
00:30:08,870 --> 00:30:11,870
compared with the most
precise optical measurements.

455
00:30:14,870 --> 00:30:18,090
And I think there was a peak
gap, a change in slope here.

456
00:30:18,090 --> 00:30:23,156
In the old days, you measured
the wavelengths of light

457
00:30:23,156 --> 00:30:27,870
by making a measurement of
the wavelengths using maybe

458
00:30:27,870 --> 00:30:30,860
a grading or interferometry,
but when you now

459
00:30:30,860 --> 00:30:33,560
started to measure
frequencies directly,

460
00:30:33,560 --> 00:30:36,990
then there was a change in
slope and major improvements

461
00:30:36,990 --> 00:30:38,376
in precision.

462
00:30:38,376 --> 00:30:40,686
And today, of course, the
most precise measurement

463
00:30:40,686 --> 00:30:45,180
of laser frequencies is not
through the wavelengths,

464
00:30:45,180 --> 00:30:49,284
that they can count the number
of cycles in a peak node

465
00:30:49,284 --> 00:30:51,679
with an optical comb generator.

466
00:30:54,560 --> 00:30:57,244
So what can you do with
ever increasing precision?

467
00:31:00,450 --> 00:31:03,880
This is also a slide I
borrowed from Ted Hansch.

468
00:31:03,880 --> 00:31:08,960
If you measure very, very
accurately an atomic line,

469
00:31:08,960 --> 00:31:13,555
let's say the 1s 2s transition
in hydrogen using two photon

470
00:31:13,555 --> 00:31:17,258
spectroscopy, what you can
do is you can measure it,

471
00:31:17,258 --> 00:31:19,960
and a few years later,
you can measure it again.

472
00:31:19,960 --> 00:31:22,165
And now it becomes an
interesting question

473
00:31:22,165 --> 00:31:23,290
if you have this precision.

474
00:31:23,290 --> 00:31:26,734
Will the result be the
same as a function of time?

475
00:31:26,734 --> 00:31:33,550
If there were a small change,
which there wasn't, you

476
00:31:33,550 --> 00:31:35,040
could only come
to one conclusion,

477
00:31:35,040 --> 00:31:38,460
and this is fundamental
common sense in nature,

478
00:31:38,460 --> 00:31:40,710
change is a function of time.

479
00:31:40,710 --> 00:31:43,190
So this precision
of metrology is now

480
00:31:43,190 --> 00:31:46,828
being used to test the
fundamental constants, again,

481
00:31:46,828 --> 00:31:49,238
constant as a function of time.

482
00:31:57,790 --> 00:32:00,380
Precision is even
more improving.

483
00:32:00,380 --> 00:32:04,575
The latest in the development
in the spectroscopy of hydrogen

484
00:32:04,575 --> 00:32:09,026
is what is called the
size of the proton atom.

485
00:32:09,026 --> 00:32:11,805
In your homework
assignment, you are actually

486
00:32:11,805 --> 00:32:16,300
calculating what is the
correction due to transition

487
00:32:16,300 --> 00:32:19,330
frequencies in
hydrogen because you

488
00:32:19,330 --> 00:32:22,438
don't have a proton field
of a point particle.

489
00:32:22,438 --> 00:32:24,310
The proton has a finite size.

490
00:32:24,310 --> 00:32:27,520
Or vice versa, if you have
sufficient prediction,

491
00:32:27,520 --> 00:32:31,020
if you have sufficient
accuracy of the measurement,

492
00:32:31,020 --> 00:32:33,632
you can determine the
size of the proton

493
00:32:33,632 --> 00:32:35,340
from the measured
transition frequencies.

494
00:32:37,950 --> 00:32:44,160
This was done, and in 2010,
there was a big surprise

495
00:32:44,160 --> 00:32:48,415
that the size of the proton
determined from hydrogen

496
00:32:48,415 --> 00:32:52,290
spectroscopy did not agree
with scattering measurements

497
00:32:52,290 --> 00:32:56,592
where you scatter electrons and
protons to measure the proton

498
00:32:56,592 --> 00:32:58,330
size.

499
00:32:58,330 --> 00:33:00,970
This is still a puzzle.

500
00:33:00,970 --> 00:33:02,835
It's called the
proton radius puzzle,

501
00:33:02,835 --> 00:33:07,705
and it is not clear
what is causing it.

502
00:33:07,705 --> 00:33:11,952
What happened is in 2010,
there was a big improvement

503
00:33:11,952 --> 00:33:15,110
in measuring the
size of the proton,

504
00:33:15,110 --> 00:33:20,440
and this was done by replacing
the electron with a muon, which

505
00:33:20,440 --> 00:33:25,425
is a heavy electron, but since
the muon is so much heavier,

506
00:33:25,425 --> 00:33:29,470
the bore orbit of the muon,
the negative particle going

507
00:33:29,470 --> 00:33:32,960
around the proton,
it is much smaller.

508
00:33:35,880 --> 00:33:41,790
Therefore, the Lamb
shift also, but also

509
00:33:41,790 --> 00:33:44,830
the correction due to the
finite size of the proton,

510
00:33:44,830 --> 00:33:46,865
is much, much
larger because there

511
00:33:46,865 --> 00:33:49,410
is much more overlay
of the muonic wave

512
00:33:49,410 --> 00:33:54,790
function with a proton
than for an electron.

513
00:33:54,790 --> 00:33:56,850
So there was a huge
improvement in the precision

514
00:33:56,850 --> 00:33:58,810
of the measurement
of the proton size,

515
00:33:58,810 --> 00:34:01,180
and this has really
led to what's

516
00:34:01,180 --> 00:34:03,750
called the proton radius puzzle.

517
00:34:03,750 --> 00:34:08,244
It's not sure if that is at the
same level of the Lamb shift,

518
00:34:08,244 --> 00:34:10,714
which gave rise to
fundamental new physics.

519
00:34:10,714 --> 00:34:13,750
Maybe this is the discovery
of the new Lamb shift in 2010,

520
00:34:13,750 --> 00:34:18,070
and it changes our understanding
of fundamental physics,

521
00:34:18,070 --> 00:34:20,488
but maybe it's something else.

522
00:34:20,488 --> 00:34:24,080
The answer is not [INAUDIBLE].

523
00:34:24,080 --> 00:34:25,620
At least this is 2010.

524
00:34:25,620 --> 00:34:28,280
A few years of
checking the theory

525
00:34:28,280 --> 00:34:32,175
and checking the experiment has
not removed the discrepancy.

526
00:34:32,175 --> 00:34:34,495
Rather to the contrary.

527
00:34:34,495 --> 00:34:38,541
It has hardened that there
is some discrepancy which

528
00:34:38,541 --> 00:34:42,219
needs to be resolved.

529
00:34:42,219 --> 00:34:44,612
So this was just a
little shot of excursion,

530
00:34:44,612 --> 00:34:47,920
a little bit of summary of
spectroscopy of hydrogen

531
00:34:47,920 --> 00:34:52,190
over 80 years, from
precursors to the Lamb shift

532
00:34:52,190 --> 00:34:56,440
to the proton radius puzzle.

533
00:34:56,440 --> 00:34:58,600
AUDIENCE: So I
know the same group

534
00:34:58,600 --> 00:35:04,390
was using the same technique to
measure deuterium, maybe helium

535
00:35:04,390 --> 00:35:07,015
PROFESSOR: I think they want to
do it but they haven't done it.

536
00:35:07,015 --> 00:35:09,490
That's what they plan to do.

537
00:35:09,490 --> 00:35:10,975
AUDIENCE: That was my question.

538
00:35:31,850 --> 00:35:33,674
PROFESSOR: So our
next topic are now

539
00:35:33,674 --> 00:35:36,144
atoms in external
magnetic fields.

540
00:36:00,090 --> 00:36:14,510
The first chapter is on fine
structure and the lambda g

541
00:36:14,510 --> 00:36:15,010
factor.

542
00:36:18,380 --> 00:36:21,685
But maybe more colloquially,
atoms in external fields

543
00:36:21,685 --> 00:36:26,296
means that we add one
more vector to the mix.

544
00:36:26,296 --> 00:36:29,390
In fine structure, we have
orbital angular momentum

545
00:36:29,390 --> 00:36:32,390
and spin angular
momentum, and we

546
00:36:32,390 --> 00:36:35,437
discussed how spin orbit
coupling eventually

547
00:36:35,437 --> 00:36:40,742
couples L and S to J and so
on, but now, we extend the game

548
00:36:40,742 --> 00:36:46,850
by one more vector, B, an
external magnetic field.

549
00:36:46,850 --> 00:36:49,750
It really becomes a
player in the game

550
00:36:49,750 --> 00:36:53,665
because you know that if you
have spin orbit coupling,

551
00:36:53,665 --> 00:36:58,770
we use the vector model that
L and S couple and precess

552
00:36:58,770 --> 00:37:04,850
around the axis of J, the
total angular momentum.

553
00:37:04,850 --> 00:37:08,386
So the game we play when we
couple angular momentum is

554
00:37:08,386 --> 00:37:10,506
that angular
momentum couple, they

555
00:37:10,506 --> 00:37:13,210
precess around an axis
that involves some quantum

556
00:37:13,210 --> 00:37:15,100
numbers and so on.

557
00:37:15,100 --> 00:37:19,460
But now, you can add a
new quantization axis

558
00:37:19,460 --> 00:37:22,690
with a magnetic field,
and then angular momenta

559
00:37:22,690 --> 00:37:26,340
will precess around
the magnetic field.

560
00:37:26,340 --> 00:37:28,310
And there may be
actually a conflict,

561
00:37:28,310 --> 00:37:31,157
that for strong magnetic
field, the precession is

562
00:37:31,157 --> 00:37:33,652
different than in
weak magnetic fields,

563
00:37:33,652 --> 00:37:36,150
and this is what we
want to discuss now.

564
00:37:36,150 --> 00:37:47,050
So it is the game
of L, S, and B.

565
00:37:47,050 --> 00:37:52,090
So what we are adding
with magnetic fields

566
00:37:52,090 --> 00:37:55,880
is we are adding one term
to the Hamiltonian, which

567
00:37:55,880 --> 00:38:05,360
is the Zeeman term, where we
have an external magnetic field

568
00:38:05,360 --> 00:38:08,142
and we couple with
a magnetic moment.

569
00:38:11,550 --> 00:38:14,970
The one thing which
makes it interesting

570
00:38:14,970 --> 00:38:18,850
when we discuss fine
structure is the following,

571
00:38:18,850 --> 00:38:23,560
that we have actually two
components to the angular

572
00:38:23,560 --> 00:38:26,942
momentum of the atom,
which is the spin

573
00:38:26,942 --> 00:38:29,720
and the orbital
angular momentum,

574
00:38:29,720 --> 00:38:33,215
and the two have
different g factors.

575
00:38:36,000 --> 00:38:38,706
So it's not that
the magnetic moment

576
00:38:38,706 --> 00:38:41,171
is just proportional to
the angular momentum.

577
00:38:41,171 --> 00:38:43,841
If the angular momentum
comes from spin,

578
00:38:43,841 --> 00:38:47,160
it has a different weight than
when the angular momentum comes

579
00:38:47,160 --> 00:38:49,860
from orbital angular
momentum, and this

580
00:38:49,860 --> 00:38:52,850
is what we want
to now understand.

581
00:38:56,490 --> 00:39:00,260
We want to determine what
is the magnetic moment

582
00:39:00,260 --> 00:39:03,843
of the atom when the
angular momentum has

583
00:39:03,843 --> 00:39:06,150
two different sources?

584
00:39:06,150 --> 00:39:09,640
What we will find is we will
find that there is a lambda g

585
00:39:09,640 --> 00:39:12,520
factor which is sometimes
2, which is sometimes 1,

586
00:39:12,520 --> 00:39:15,350
or which is sometimes
somewhere in between,

587
00:39:15,350 --> 00:39:18,550
depending how S
and L are arranged

588
00:39:18,550 --> 00:39:19,994
with respect to each other.

589
00:39:26,130 --> 00:39:30,264
So our Hamiltonian is the
Hamiltonian for the atom.

590
00:39:30,264 --> 00:39:32,685
We know that we
have fine structure,

591
00:39:32,685 --> 00:39:39,580
we discussed that,
which couples L and S.

592
00:39:39,580 --> 00:40:14,050
And now have a magnetic
moment due to the spin

593
00:40:14,050 --> 00:40:17,540
and due to the orbital
angular momentum,

594
00:40:17,540 --> 00:40:22,363
and then couples to the
external magnetic field.

595
00:40:22,363 --> 00:40:26,686
In other words, if we had
no fine structure coupling,

596
00:40:26,686 --> 00:40:30,200
if S and L would not
be coupled, the answer

597
00:40:30,200 --> 00:40:31,970
would be very simple.

598
00:40:31,970 --> 00:40:34,970
S just couples to
the magnetic field,

599
00:40:34,970 --> 00:40:37,462
gives the same [INAUDIBLE],
the g factor of 2,

600
00:40:37,462 --> 00:40:42,362
and L couples to the magnetic
field with a g factor of 1.

601
00:40:42,362 --> 00:40:45,870
But now the two are coupled
with respect to each other,

602
00:40:45,870 --> 00:40:51,205
and if you have L-S coupling,
the projection of S and L

603
00:40:51,205 --> 00:40:54,680
on the z-axis is not a good
quantum number anymore.

604
00:40:54,680 --> 00:40:58,620
So therefore, we have
two different terms

605
00:40:58,620 --> 00:41:01,670
which are diagonal in
two different bases,

606
00:41:01,670 --> 00:41:05,380
and that's what we
want to discuss.

607
00:41:05,380 --> 00:41:11,870
So the g factor of orbital
angular momentum is 1.

608
00:41:11,870 --> 00:41:19,000
The g factor of the
spin is 2, or if you

609
00:41:19,000 --> 00:41:24,100
want to include the leading
correction here from QED,

610
00:41:24,100 --> 00:41:26,980
it's the fine structure
constant over 2 pi.

611
00:41:37,910 --> 00:41:44,120
We did discuss that
the fine structure can

612
00:41:44,120 --> 00:41:49,650
be related as the Zeeman
energy of the spin

613
00:41:49,650 --> 00:41:52,180
in a magnetic field
which is created

614
00:41:52,180 --> 00:41:55,650
by the electron
due to its motion.

615
00:41:55,650 --> 00:41:57,620
Or, if you take the
frame of the electron,

616
00:41:57,620 --> 00:41:59,935
the electron sees
the proton orbiting

617
00:41:59,935 --> 00:42:02,300
around that creates
a magnetic field,

618
00:42:02,300 --> 00:42:05,980
and this magnetic field
couples to the state.

619
00:42:05,980 --> 00:42:09,760
So therefore, we can
associate fine structure

620
00:42:09,760 --> 00:42:13,272
with an internal magnetic
field inside the atom.

621
00:42:22,620 --> 00:42:25,190
And this internal magnetic
field is rather large.

622
00:42:25,190 --> 00:42:29,270
It's on the order of 1 Tesla.

623
00:42:29,270 --> 00:42:35,170
So therefore, for our discussion
of the lambda g factor and fine

624
00:42:35,170 --> 00:42:36,880
structure in applied
magnetic fields,

625
00:42:36,880 --> 00:42:45,250
we will assume that we are
in the weak field limit

626
00:42:45,250 --> 00:42:48,950
where the fine structure
term, the first term,

627
00:42:48,950 --> 00:42:54,170
is much larger than
the Zeeman term.

628
00:42:54,170 --> 00:42:55,995
Of course, if you use
very strong magnets,

629
00:42:55,995 --> 00:42:58,520
you can go to the
high field case,

630
00:42:58,520 --> 00:43:01,990
but I will discuss explicitly
the transition from weak field

631
00:43:01,990 --> 00:43:04,810
to high field for
hyperfine structure,

632
00:43:04,810 --> 00:43:07,573
and the phenomenon
for fine structure

633
00:43:07,573 --> 00:43:08,862
is completely analogous.

634
00:43:08,862 --> 00:43:11,760
It just happens at
much higher fields.

635
00:43:11,760 --> 00:43:15,268
So anyway, I will discuss the
high field case and transition

636
00:43:15,268 --> 00:43:18,928
with the high field case with
a much more elegant example

637
00:43:18,928 --> 00:43:22,100
of hyperfine structure
that you can immediately

638
00:43:22,100 --> 00:43:24,052
apply to fine
structure if you like.

639
00:43:32,896 --> 00:43:41,210
If I want to solve the problem,
calculate the lambda g factor,

640
00:43:41,210 --> 00:43:51,080
I could directly calculate
just one matrix element

641
00:43:51,080 --> 00:43:52,888
and it would be done.

642
00:43:55,600 --> 00:43:59,240
So all I want to know
is what is the Zeeman

643
00:43:59,240 --> 00:44:02,830
energy because Zeeman energy
divided by the magnetic field

644
00:44:02,830 --> 00:44:04,270
is the magnetic moment.

645
00:44:07,210 --> 00:44:11,960
And since I assume that I'm
in the weak field limit,

646
00:44:11,960 --> 00:44:15,665
I can simply use the
quantum numbers S, L, and S

647
00:44:15,665 --> 00:44:22,552
and L coupled to J, and the
magnetic quantum number is nJ.

648
00:44:29,890 --> 00:44:41,560
So by simply calculating this
expectation value, I'm done

649
00:44:41,560 --> 00:44:43,935
and I've solved the problem.

650
00:44:56,190 --> 00:45:00,080
However, I want to do the
derivation using the vector

651
00:45:00,080 --> 00:45:02,865
model because it provides
some additional insight.

652
00:45:07,980 --> 00:45:19,730
So in the vector model, we
have L and S coupled to J

653
00:45:19,730 --> 00:45:21,510
to the total angular momentum.

654
00:45:25,950 --> 00:45:32,355
And in the vector model, we
assume that L and S rapidly

655
00:45:32,355 --> 00:45:41,590
precess around J. And therefore,
the only thing which matters

656
00:45:41,590 --> 00:45:45,528
are the projections.

657
00:45:45,528 --> 00:45:51,090
Only the projections of L and
S on the J-axis are important.

658
00:45:53,905 --> 00:45:56,280
So you can say if you have an
rapid precession of L and S

659
00:45:56,280 --> 00:45:59,220
around J, the transverse
components rapidly average

660
00:45:59,220 --> 00:46:00,190
out and [INAUDIBLE].

661
00:46:05,040 --> 00:46:09,075
So therefore, our
Zeeman Hamiltonian

662
00:46:09,075 --> 00:46:12,750
has to be rewritten
in the following way.

663
00:46:12,750 --> 00:46:15,290
The Zeeman Hamiltonian
was the magnetic moment

664
00:46:15,290 --> 00:46:18,930
times the external magnetic
field with a minus sign.

665
00:46:28,860 --> 00:46:39,420
But what matters is the
projection on the J direction,

666
00:46:39,420 --> 00:46:44,490
so we do the
projection in this way.

667
00:46:44,490 --> 00:46:46,640
And also, in the
end, what matters

668
00:46:46,640 --> 00:46:50,230
is, since the magnetic
moments are aligned with J,

669
00:46:50,230 --> 00:46:59,880
it is now the scalar product
of the magnetic field with J.

670
00:46:59,880 --> 00:47:05,950
So in the vector model, we
calculate the Zeeman energies

671
00:47:05,950 --> 00:47:11,060
in that way, but just to
mention that if you don't

672
00:47:11,060 --> 00:47:14,450
like the vector model and the
assumption of rapid precession,

673
00:47:14,450 --> 00:47:16,140
just take this matrix element.

674
00:47:16,140 --> 00:47:18,190
It's exactly the same.

675
00:47:18,190 --> 00:47:21,740
In other words, I give
you the intuitive picture

676
00:47:21,740 --> 00:47:23,440
what is inside those
matrix elements.

677
00:47:26,480 --> 00:47:29,070
So let's evaluate that.

678
00:47:36,110 --> 00:47:37,800
Let me factor out
the Bohr magneton.

679
00:47:42,533 --> 00:47:48,990
We have J squared, taking
one of each bracket.

680
00:47:48,990 --> 00:47:55,890
And now assuming that the
g factor of the spin is 2,

681
00:47:55,890 --> 00:48:01,370
the magnetic moment is the
Bohr magneton times L--

682
00:48:01,370 --> 00:48:09,370
the g factor of L is 1-- plus
S, but the g factor of S is 2.

683
00:48:09,370 --> 00:48:11,972
So this is now the
magnetic moment accounting

684
00:48:11,972 --> 00:48:18,140
for the two different g factors
we projected on the J-axis.

685
00:48:18,140 --> 00:48:23,080
And the second racket,
B dot J becomes

686
00:48:23,080 --> 00:48:24,710
the value of the magnetic field.

687
00:48:24,710 --> 00:48:28,160
We assume the magnetic field
points in the z direction,

688
00:48:28,160 --> 00:48:31,465
so therefore, it is the z
component of the total angular

689
00:48:31,465 --> 00:48:31,965
momentum.

690
00:48:40,414 --> 00:48:44,900
Let me collect the simple terms.

691
00:48:44,900 --> 00:48:54,250
Now L plus 2S,
because L plus S is J,

692
00:48:54,250 --> 00:49:01,130
can be written as J plus S.

693
00:49:01,130 --> 00:49:09,630
So now we have the product
of J with J, which gives us

694
00:49:09,630 --> 00:49:21,140
J squared, and then we need
the product of S and J.

695
00:49:21,140 --> 00:49:27,310
And as usual, we can get
an expression for that

696
00:49:27,310 --> 00:49:31,210
by using the summation
of angular momenta.

697
00:49:31,210 --> 00:49:35,190
If we square it on
the right hand side,

698
00:49:35,190 --> 00:49:40,510
we have the scalar
product of J and S,

699
00:49:40,510 --> 00:49:43,005
but we have now the
scalar product of J and S

700
00:49:43,005 --> 00:49:45,122
expressed by L
squared J squared.

701
00:49:51,940 --> 00:49:59,416
J squared plus S squared minus
L squared divided by J squared.

702
00:50:08,820 --> 00:50:18,085
Now, we're just one line
away from the final result.

703
00:50:24,520 --> 00:50:29,100
Jz is a good quantum number.

704
00:50:29,100 --> 00:50:32,346
It's MJ, the projection
of the total angular

705
00:50:32,346 --> 00:50:33,834
momentum on the z-axis.

706
00:50:40,835 --> 00:50:45,970
The bracket here is
now the famous result

707
00:50:45,970 --> 00:50:47,360
for the lambda g factor.

708
00:50:52,800 --> 00:50:56,440
So we have J squared over J
squared, which gives us 1.

709
00:51:02,730 --> 00:51:08,390
And then I simply put
in the quantum numbers

710
00:51:08,390 --> 00:51:10,980
for J squared, S
squared, L squared, which

711
00:51:10,980 --> 00:51:16,020
is J times J plus 1
plus S times S plus 1

712
00:51:16,020 --> 00:51:23,050
minus L times L plus 1, and
we divide by 2 times J plus 1.

713
00:51:28,980 --> 00:51:33,930
So therefore, the Zeeman
structure in a magnetic field

714
00:51:33,930 --> 00:51:37,298
is the Bohr magneton
times the magnetic field

715
00:51:37,298 --> 00:51:40,100
times the angular momentum
in the J direction,

716
00:51:40,100 --> 00:51:41,984
but then we multiply
with the g factor.

717
00:51:48,510 --> 00:51:50,285
These are now limiting cases.

718
00:51:53,240 --> 00:52:06,452
If we do not have spin,
then the only ingredients,

719
00:52:06,452 --> 00:52:08,400
the only [INAUDIBLE]
angular momentum

720
00:52:08,400 --> 00:52:12,110
is orbital angular momentum
and we have a g factor of 1.

721
00:52:12,110 --> 00:52:15,706
So the lambda g factor
simply becomes gL.

722
00:52:15,706 --> 00:52:19,190
In the case we don't
have angular momentum,

723
00:52:19,190 --> 00:52:23,940
you can just evaluate this
expression for L equals 0.

724
00:52:23,940 --> 00:52:29,780
You find indeed that
the g factor is 2.

725
00:52:34,235 --> 00:52:37,474
But it can have
different values.

726
00:52:37,474 --> 00:52:39,904
It depends on the
atomic structure.

727
00:52:48,550 --> 00:52:50,730
That's all I want to
say about fine structure

728
00:52:50,730 --> 00:52:51,827
in a magnetic field.

729
00:52:54,960 --> 00:52:57,080
The next step is now
hyperfine structure.

730
00:53:14,760 --> 00:53:17,904
We are adding one more
vector to the game.

731
00:53:17,904 --> 00:53:25,380
So we add angular
momentum of the nucleus,

732
00:53:25,380 --> 00:53:29,300
so now the game being
played is not only L and S.

733
00:53:29,300 --> 00:53:36,000
We have I and B. It's a
game of the four vectors

734
00:53:36,000 --> 00:53:38,794
and eventually how they
precess around each other,

735
00:53:38,794 --> 00:53:43,715
and that gives rise to the
structure of hyperfine levels

736
00:53:43,715 --> 00:53:45,432
in an external magnetic field.

737
00:53:49,150 --> 00:53:54,700
We assume that L and
S have coupled to J,

738
00:53:54,700 --> 00:53:57,375
so we have actually
a coupling of J, I,

739
00:53:57,375 --> 00:54:00,953
but now we have an external
quantization axis in Zeeman.

740
00:54:00,953 --> 00:54:05,770
It has to be due to the
external magnetic field.

741
00:54:05,770 --> 00:54:07,800
And of course, in
hyperfine structure,

742
00:54:07,800 --> 00:54:13,710
we discussed that I and J are
no longer conserved angular

743
00:54:13,710 --> 00:54:17,231
momenta because they couple to
a total angular momentum, which

744
00:54:17,231 --> 00:54:20,880
is F.

745
00:54:20,880 --> 00:54:30,100
So our Hamiltonian
is the Hamiltonian

746
00:54:30,100 --> 00:54:34,780
without any kind of
hyperfine and fine structure

747
00:54:34,780 --> 00:54:38,350
when we have the hyperfine
coupling, which couples I

748
00:54:38,350 --> 00:54:44,190
and J with the product
I dot J. And then we

749
00:54:44,190 --> 00:54:49,050
have an external magnetic
field, which couples

750
00:54:49,050 --> 00:54:56,646
to the magnetic moment
of the electron.

751
00:54:56,646 --> 00:55:01,760
This may be a smaller term, but
you can easily carry with us.

752
00:55:01,760 --> 00:55:05,760
There's also a coupling with the
magnetic moment of the nucleus.

753
00:55:16,560 --> 00:55:19,480
So in this case, because
the hyperfine structure

754
00:55:19,480 --> 00:55:21,990
is smaller than
the fine structure,

755
00:55:21,990 --> 00:55:25,240
I want to discuss both the
weak field and the strong field

756
00:55:25,240 --> 00:55:25,800
case.

757
00:55:25,800 --> 00:55:27,966
Because magnetic fields of
a few hundred [INAUDIBLE]

758
00:55:27,966 --> 00:55:32,020
may actually take into
the high field momentum.

759
00:55:32,020 --> 00:55:38,324
I want to discuss both the low
field and the high field limit.

760
00:55:38,324 --> 00:55:43,865
The low field limit implies
that the Zeeman energies

761
00:55:43,865 --> 00:55:47,124
are much smaller than
the hyperfine splittings.

762
00:55:53,377 --> 00:56:05,410
And then the way we describe the
system is that J and I couple.

763
00:56:05,410 --> 00:56:10,650
So J, which is responsible
for the magnetic moment,

764
00:56:10,650 --> 00:56:16,290
precesses around F, the
total angular momentum,

765
00:56:16,290 --> 00:56:18,325
but then the total
angular momentum

766
00:56:18,325 --> 00:56:21,070
precesses around
the magnetic field.

767
00:56:21,070 --> 00:56:24,830
In other words, you assume that
the coupling between J and I

768
00:56:24,830 --> 00:56:27,296
is so strong and
they couple to F,

769
00:56:27,296 --> 00:56:29,060
the magnetic field
is not breaking up

770
00:56:29,060 --> 00:56:34,700
the coupling between J and
I. J and I together form F,

771
00:56:34,700 --> 00:56:37,560
and this hyperfine state,
this magnetic moment,

772
00:56:37,560 --> 00:56:41,669
precesses around B.

773
00:56:41,669 --> 00:56:42,835
This is sort of the picture.

774
00:56:42,835 --> 00:56:44,220
You have to get used to it.

775
00:56:44,220 --> 00:56:46,450
J precesses around
F and F precesses

776
00:56:46,450 --> 00:56:50,030
around the external
magnetic field, B0.

777
00:56:55,830 --> 00:56:58,960
But again, if you don't
like the precession model,

778
00:56:58,960 --> 00:57:01,855
just calculate the
quantum mechanical energy

779
00:57:01,855 --> 00:57:03,770
that will stabilize
the Hamiltonian.

780
00:57:03,770 --> 00:57:05,718
The answer is identical.

781
00:57:22,540 --> 00:57:28,866
The Zeeman Hamiltonian
couples to the magnetic field,

782
00:57:28,866 --> 00:57:34,850
and we have two contributions
to the magnetic moment,

783
00:57:34,850 --> 00:57:36,978
the electron of the nucleus.

784
00:57:36,978 --> 00:57:42,510
And in the weak field
limit, we use a treatment

785
00:57:42,510 --> 00:57:46,984
which is almost completely
analogous to the treatment we

786
00:57:46,984 --> 00:57:50,442
used when we derived the
[INAUDIBLE] g factor.

787
00:57:59,334 --> 00:58:02,166
We can treat the
same as Hamiltonian

788
00:58:02,166 --> 00:58:10,680
perturbation theory, and
it's exactly analogous

789
00:58:10,680 --> 00:58:15,810
when we added a weak magnetic
field to the fine structure.

790
00:58:25,050 --> 00:58:33,150
In the vector model, we
have the coupling of J and I

791
00:58:33,150 --> 00:58:40,170
to F. The relevant
term in the Hamiltonian

792
00:58:40,170 --> 00:58:46,800
is we have the magnetic
moment of the electron, which

793
00:58:46,800 --> 00:58:52,040
is proportionate to J
and it couples to B.

794
00:58:52,040 --> 00:58:56,020
So this relevant term, this
is fully analogous to what

795
00:58:56,020 --> 00:58:58,350
I did five or 10
minutes ago, has

796
00:58:58,350 --> 00:59:02,610
to be replaced in the presence
of the nuclear angular

797
00:59:02,610 --> 00:59:03,550
momentum.

798
00:59:03,550 --> 00:59:12,093
We have to project everything
on the axis of the total angular

799
00:59:12,093 --> 00:59:22,960
momentum, F.

800
00:59:22,960 --> 00:59:32,210
Therefore, the
Zeeman Hamiltonian

801
00:59:32,210 --> 00:59:40,495
had the contribution
to the magnetic moment

802
00:59:40,495 --> 00:59:45,790
due to the electron
and due to the nucleus.

803
00:59:45,790 --> 00:59:49,100
This is proportionate
to J, but now we

804
00:59:49,100 --> 00:59:56,620
have to project it onto F. And
similarly, the magnetic moment

805
00:59:56,620 --> 01:00:01,100
of the nucleus is
proportionate to I,

806
01:00:01,100 --> 01:00:05,140
but what matters is
the projection on F.

807
01:00:05,140 --> 01:00:10,650
And since I factored
out the Bohr magneton,

808
01:00:10,650 --> 01:00:12,950
the magnetic moment
of the nucleus

809
01:00:12,950 --> 01:00:16,610
is proportionate to
the nuclear magneton.

810
01:00:16,610 --> 01:00:18,200
I have to account for the ratio.

811
01:00:24,720 --> 01:00:30,470
What matters now is the
projection of F on B0.

812
01:00:30,470 --> 01:00:37,020
So therefore, collecting
all the terms,

813
01:00:37,020 --> 01:00:40,670
we have the Bohr
magneton, which is

814
01:00:40,670 --> 01:00:43,250
setting the scale
of the interaction.

815
01:00:43,250 --> 01:00:46,770
The last term is
the magnetic field,

816
01:00:46,770 --> 01:00:49,950
but the projection of F
onto the magnetic field

817
01:00:49,950 --> 01:00:56,540
gives us the mF quantum
number, and all the rest

818
01:00:56,540 --> 01:00:59,707
is called the g factor of
the hyperfine structure.

819
01:01:04,200 --> 01:01:07,256
And the g factor of
the hyperfine structure

820
01:01:07,256 --> 01:01:19,090
is-- let me just
simplify and neglect

821
01:01:19,090 --> 01:01:21,590
the small contribution--
it's 1,000 times smaller--

822
01:01:21,590 --> 01:01:25,610
of the nuclear magnetic
moment, but if you want,

823
01:01:25,610 --> 01:01:29,265
you can easily include it.

824
01:01:29,265 --> 01:01:37,522
With this approximation,
the g factor

825
01:01:37,522 --> 01:01:47,660
of the hyperfine
structure is this.

826
01:01:47,660 --> 01:01:53,220
It's proportionate to the
lambda g factor we just derived.

827
01:01:53,220 --> 01:01:55,120
And then using exactly
the same thing,

828
01:01:55,120 --> 01:01:58,920
you have J dot F. You
can express it now

829
01:01:58,920 --> 01:02:02,930
by the quantum numbers of F
squared, J squared, I squared.

830
01:02:14,964 --> 01:02:21,280
You find the final
result, what are

831
01:02:21,280 --> 01:02:24,060
the g factors of the
hyperfine levels,

832
01:02:24,060 --> 01:02:25,980
of the hyperfine states.

833
01:02:30,310 --> 01:02:35,300
So this is the hyperfine
structure of atoms [INAUDIBLE]

834
01:02:35,300 --> 01:02:35,970
magnetic fields.

835
01:02:39,775 --> 01:02:42,568
Let's immediately go to
the high field limit.

836
01:02:46,800 --> 01:02:54,870
The high field limit means that
the electronic Zeeman energy

837
01:02:54,870 --> 01:02:59,554
is much larger than
the hyperfine coupling.

838
01:03:05,130 --> 01:03:17,357
And that means always,
when we treat the problem,

839
01:03:17,357 --> 01:03:20,766
we first take care of
the big contributions

840
01:03:20,766 --> 01:03:21,740
to the Hamiltonian.

841
01:03:21,740 --> 01:03:24,175
We try to solve it, if
possible, [INAUDIBLE].

842
01:03:24,175 --> 01:03:28,419
And then the weaker term
can be any perturbative.

843
01:03:28,419 --> 01:03:30,710
So now we are in the situation
that the Zeeman coupling

844
01:03:30,710 --> 01:03:34,246
is the big term and
the hyperfine coupling

845
01:03:34,246 --> 01:03:35,980
is the weaker term.

846
01:03:38,550 --> 01:03:47,060
So in other words, what comes
first now is the Zeeman energy,

847
01:03:47,060 --> 01:03:51,180
so we are not coupling the
electronic angular momentum

848
01:03:51,180 --> 01:03:55,084
and the nuclear angular momentum
to total angular momentum

849
01:03:55,084 --> 01:03:57,820
because this coupling is weak.

850
01:03:57,820 --> 01:04:02,230
We rather say that
the electronic

851
01:04:02,230 --> 01:04:05,620
and the nuclear angular momentum
align with the magnetic field.

852
01:04:05,620 --> 01:04:11,450
We quantize along the direction
of the magnetic field,

853
01:04:11,450 --> 01:04:14,103
and then later we add
the hyperfine coupling

854
01:04:14,103 --> 01:04:16,070
in a determinative way.

855
01:04:16,070 --> 01:04:26,380
So B0 now quantizes
J along the direction

856
01:04:26,380 --> 01:04:31,449
of the magnetic field,
and therefore we use,

857
01:04:31,449 --> 01:04:35,345
as a good quantum
number, J, the projection

858
01:04:35,345 --> 01:04:37,780
of J on the external
magnetic field axis.

859
01:04:43,530 --> 01:04:51,180
So this takes care of J. J and
mJ are good quantum numbers.

860
01:04:51,180 --> 01:04:52,180
What about I?

861
01:04:54,790 --> 01:04:58,070
Does the nuclear
angular momentum

862
01:04:58,070 --> 01:05:00,645
and the nuclear magnetic
moment strongly couple

863
01:05:00,645 --> 01:05:03,580
to the magnetic field?

864
01:05:03,580 --> 01:05:07,315
Well, the answer is
yes, but the argument

865
01:05:07,315 --> 01:05:09,416
is a little bit more subtle.

866
01:05:09,416 --> 01:05:13,225
The direct coupling
of the magnetic moment

867
01:05:13,225 --> 01:05:16,232
of the nucleus with
the magnetic field

868
01:05:16,232 --> 01:05:24,590
may be smaller than the
hyperfine interaction.

869
01:05:31,310 --> 01:05:37,480
So then you would say, the
nuclear angular momentum

870
01:05:37,480 --> 01:05:39,190
should not couple to
the magnetic field.

871
01:05:39,190 --> 01:05:42,510
It should first be coupled
to the hyperfine interaction.

872
01:05:45,850 --> 01:05:52,310
The hyperfine interaction
is I dot J. However,

873
01:05:52,310 --> 01:05:56,290
J, which couples strongly
to the magnetic field

874
01:05:56,290 --> 01:05:58,576
because it couples
to the Bohr magneton,

875
01:05:58,576 --> 01:06:01,930
has already been coupled
to the magnetic field.

876
01:06:01,930 --> 01:06:13,470
So therefore, the hyperfine
interaction, which is I dot J,

877
01:06:13,470 --> 01:06:17,620
is now modified
because J couples

878
01:06:17,620 --> 01:06:19,805
to the magnetic
field, which means

879
01:06:19,805 --> 01:06:23,900
we have to project it onto
the magnetic field axis.

880
01:06:23,900 --> 01:06:28,379
So therefore, the
nucleus now experiences

881
01:06:28,379 --> 01:06:30,920
an electronic magnetic moment
or electronic angular momentum,

882
01:06:30,920 --> 01:06:35,420
which has already been
coupled to the z-axis.

883
01:06:35,420 --> 01:06:37,610
And therefore, the
hyperfine interaction

884
01:06:37,610 --> 01:06:43,602
is also coupling the nuclear
angular momentum to the z-axis.

885
01:06:47,570 --> 01:06:53,150
So therefore, the
result is that it is now

886
01:06:53,150 --> 01:06:54,945
this indirect coupling.

887
01:06:54,945 --> 01:06:57,975
You couple the electron
angular momentum to the z-axis

888
01:06:57,975 --> 01:06:59,835
and the electron
angular momentum

889
01:06:59,835 --> 01:07:04,600
couples the nuclear angular
momentum to the z-axis.

890
01:07:04,600 --> 01:07:11,090
So now this quantizes the
nuclear angular momentum

891
01:07:11,090 --> 01:07:14,600
along the z-axis,
which means that n

892
01:07:14,600 --> 01:07:17,225
sub I becomes a
good quantum number.

893
01:07:19,820 --> 01:07:23,910
Anyway, maybe the result is even
simpler than the explanation.

894
01:07:23,910 --> 01:07:27,980
Our Zeeman
Hamiltonian now simply

895
01:07:27,980 --> 01:07:34,170
means that we have an
external magnetic field

896
01:07:34,170 --> 01:07:41,320
and you the electron couples
to the magnetic field,

897
01:07:41,320 --> 01:07:45,140
so what matters is
the projection, mJ.

898
01:07:45,140 --> 01:07:54,950
The same happens for the
nuclear magnetic moment,

899
01:07:54,950 --> 01:07:58,330
and now we have to add the
hyperfine interaction, which

900
01:07:58,330 --> 01:08:07,340
was originally I dot J, but
since I and J are projected

901
01:08:07,340 --> 01:08:14,132
on the z-axis, what is really
left over are only mI and mJ.

902
01:08:17,260 --> 01:08:19,619
I could have gotten this
expression immediately

903
01:08:19,619 --> 01:08:24,609
by just telling you, J and
I no longer couple to F.

904
01:08:24,609 --> 01:08:27,440
This is destroyed by a
strong magnetic field,

905
01:08:27,440 --> 01:08:32,126
and the good quantum numbers are
J and I and their projection,

906
01:08:32,126 --> 01:08:33,779
mJ and mI.

907
01:08:33,779 --> 01:08:36,450
And then just writing
down the expectation value

908
01:08:36,450 --> 01:08:38,490
of the Hamiltonian
in these spaces

909
01:08:38,490 --> 01:08:41,630
would have immediately
given me this result.

910
01:08:41,630 --> 01:08:46,510
I wanted to give you the more
mechanistic explanation what's

911
01:08:46,510 --> 01:08:49,932
going on inside the atom and
what leads to this result.

912
01:08:57,700 --> 01:09:05,979
I have discussed for you the two
limiting cases, the weak field

913
01:09:05,979 --> 01:09:11,080
and the strong field case,
but you can solve it also

914
01:09:11,080 --> 01:09:12,670
for intermediate fields.

915
01:09:12,670 --> 01:09:23,148
You simply have to do
an exact diagonalization

916
01:09:23,148 --> 01:09:29,176
of the Hamiltonian,
which involves

917
01:09:29,176 --> 01:09:30,134
the hyperfine coupling.

918
01:09:32,760 --> 01:09:36,580
And the hyperfine
coupling, if you want,

919
01:09:36,580 --> 01:09:41,986
can be diagonalized
as eigenfunctions

920
01:09:41,986 --> 01:09:48,580
where the quantums numbers
are J, I coupled to F,

921
01:09:48,580 --> 01:09:57,070
and the projection of F, the
magnetic quantum number is mF.

922
01:09:57,070 --> 01:10:06,700
But now we have the Zeeman
Hamiltonian, where everything

923
01:10:06,700 --> 01:10:11,115
is projected on the z-axis,
so we have mJ and mI.

924
01:10:15,870 --> 01:10:25,250
And the Zeeman term
can be diagonalized

925
01:10:25,250 --> 01:10:33,156
in a different basis, which is
the basis of J, I, mJ, and mI.

926
01:10:39,370 --> 01:10:45,950
So I've shown you the weak
field limit, where we simply

927
01:10:45,950 --> 01:10:48,700
assume those quantum
numbers and calculate

928
01:10:48,700 --> 01:10:51,850
this term determinatively, and
I've shown you the high field

929
01:10:51,850 --> 01:10:53,850
limit, where we used
those quantum numbers

930
01:10:53,850 --> 01:10:57,040
and calculated this
field determinatively.

931
01:10:57,040 --> 01:11:01,860
But in general, you
just have to write down

932
01:11:01,860 --> 01:11:04,950
the matrix element of this
Hamiltonian in whatever basis

933
01:11:04,950 --> 01:11:05,620
you choose.

934
01:11:05,620 --> 01:11:08,420
You can use the
weak field basis.

935
01:11:08,420 --> 01:11:09,610
This term is diagonal.

936
01:11:09,610 --> 01:11:10,540
This is off diagonal.

937
01:11:10,540 --> 01:11:12,100
Or you can use the
strong field basis

938
01:11:12,100 --> 01:11:15,849
where this is the diagonal
and this is off diagonal

939
01:11:15,849 --> 01:11:17,900
and simply diagonalize
your Hamiltonian.

940
01:11:17,900 --> 01:11:21,070
Find the wave function,
find the I energies.

941
01:11:21,070 --> 01:11:26,960
And since, for cases
where S equals 1/2,

942
01:11:26,960 --> 01:11:30,295
it's only a two by two matrix
which has to be diagonalized,

943
01:11:30,295 --> 01:11:32,654
you can do it
analytically, and this

944
01:11:32,654 --> 01:11:34,320
leads to the famous
[INAUDIBLE] formula.

945
01:11:42,060 --> 01:11:59,720
So the solution is
analytic for J equals 1/2,

946
01:11:59,720 --> 01:12:02,262
and it's a beautiful
example which

947
01:12:02,262 --> 01:12:04,742
you should solve in your
homework assignment.

948
01:12:09,210 --> 01:12:10,830
Let me just sketch the solution.

949
01:12:21,390 --> 01:12:26,350
When you go from the weak field
to the strong field limit,

950
01:12:26,350 --> 01:12:31,750
the z component of the total
angular momentum in one case

951
01:12:31,750 --> 01:12:33,516
is mF.

952
01:12:33,516 --> 01:12:38,560
In the other case,
it is mI plus mJ.

953
01:12:38,560 --> 01:12:42,850
So when you go from
one limit to the next,

954
01:12:42,850 --> 01:12:47,682
you connect only states
where mI plus mJ equals mF.

955
01:12:54,760 --> 01:13:05,160
So the structure of
the general solution

956
01:13:05,160 --> 01:13:22,360
can be explained by repulsion
and anti-crossings of states

957
01:13:22,360 --> 01:13:29,796
with the same
[INAUDIBLE] number.

958
01:13:39,160 --> 01:13:44,180
So let me show you the weak
field, the strong field limit,

959
01:13:44,180 --> 01:13:46,011
and do a graphic interpolation.

960
01:13:50,566 --> 01:13:59,410
What I've chosen as an example
is the case of a 2 duplet S 1/2

961
01:13:59,410 --> 01:14:06,900
ground state and a nuclear
angular momentum of 3/2.

962
01:14:14,300 --> 01:14:17,340
Examples for that is
sodium and rubidium 87.

963
01:14:34,780 --> 01:14:35,895
This is magnetic field.

964
01:14:38,510 --> 01:14:44,053
At weak magnetic field
or zero magnetic field,

965
01:14:44,053 --> 01:14:50,160
the spin 1/2 couples
to the nucleus in 3/2,

966
01:14:50,160 --> 01:14:58,035
and that gives to
hyperfine states F

967
01:14:58,035 --> 01:15:04,965
equals 1 and F equals 2.

968
01:15:09,940 --> 01:15:15,200
The splitting is given by
the hyperfine constant, a,

969
01:15:15,200 --> 01:15:23,700
and the hyperfine interaction
is A times H times I times S.

970
01:15:23,700 --> 01:15:27,010
The structure is such that the
center of mass of the energy

971
01:15:27,010 --> 01:15:31,700
levels is preserved,
so therefore, one state

972
01:15:31,700 --> 01:15:33,490
is moved up by 3/4 a.

973
01:15:33,490 --> 01:15:36,800
The other state is
moved down by 5/4.

974
01:15:39,720 --> 01:15:41,950
And since F equals 1
has three components,

975
01:15:41,950 --> 01:15:44,930
F equals 2 has five
components, the center of mass

976
01:15:44,930 --> 01:15:46,230
is preserved.

977
01:15:46,230 --> 01:15:51,696
We have calculated the g
factor for those states,

978
01:15:51,696 --> 01:15:55,150
and the g factor
tells us what is

979
01:15:55,150 --> 01:15:57,900
the structure in
weak magnetic fields.

980
01:16:01,730 --> 01:16:07,130
So this is the weak
magnetic field solution.

981
01:16:07,130 --> 01:16:10,020
At high magnetic
field, you know what

982
01:16:10,020 --> 01:16:12,180
you have in high magnetic field.

983
01:16:12,180 --> 01:16:16,060
You have a single electron
which can spin up and spin down.

984
01:16:16,060 --> 01:16:20,610
So if you have an electron
which is spin up and spin down,

985
01:16:20,610 --> 01:16:24,560
it pretty much is linear
Zeeman shift for spin down

986
01:16:24,560 --> 01:16:26,760
and linear Zeeman
shift for spin up.

987
01:16:34,740 --> 01:16:36,760
This is sort of what we expect.

988
01:16:36,760 --> 01:16:50,780
So what will happen is
that the energy levels

989
01:16:50,780 --> 01:16:54,335
will evolve like this.

990
01:17:02,070 --> 01:17:04,900
So in other words,
we have eight levels.

991
01:17:04,900 --> 01:17:07,966
We have the structure
at weak magnetic fields.

992
01:17:07,966 --> 01:17:10,130
At high magnetic
fields, of course,

993
01:17:10,130 --> 01:17:13,745
we also have eight levels, but
they pretty much group into

994
01:17:13,745 --> 01:17:17,450
spin up of the electron,
spin down of the electron.

995
01:17:17,450 --> 01:17:20,400
And then there is a
smaller hyperfine structure

996
01:17:20,400 --> 01:17:23,470
on top of it because
now the nuclear spin

997
01:17:23,470 --> 01:17:25,100
can have various orientations.

998
01:17:25,100 --> 01:17:28,980
And I equals 3/2 state
has four orientations,

999
01:17:28,980 --> 01:17:31,785
so therefore, electron spin
up and electron spin down

1000
01:17:31,785 --> 01:17:34,830
will obtain four sub-levels.

1001
01:17:34,830 --> 01:17:38,430
And if you ask how
did I connected,

1002
01:17:38,430 --> 01:17:46,950
I've connected the quantum
numbers as such what is here,

1003
01:17:46,950 --> 01:17:50,030
the states are
labeled by mI and mS,

1004
01:17:50,030 --> 01:17:52,995
and here, they're
labeled by mF, but mF

1005
01:17:52,995 --> 01:17:57,095
equals mI plus mS. This
is how you correlate

1006
01:17:57,095 --> 01:17:59,972
the states in the
high field case

1007
01:17:59,972 --> 01:18:02,442
to the states in
the low field case.

1008
01:18:10,660 --> 01:18:20,160
I'm running out of time, but
here, we have mJ equals 1/2.

1009
01:18:20,160 --> 01:18:22,860
Here we have the
electron spin minus 1/2.

1010
01:18:40,420 --> 01:18:45,250
These four levels are now
four different quantum numbers

1011
01:18:45,250 --> 01:18:49,460
for the nuclear
angular momentum, which

1012
01:18:49,460 --> 01:18:57,990
are minus 3/2, minus 1/2,
plus 1/2, and plus 3/2.

1013
01:19:03,970 --> 01:19:06,610
And this is what I meant
by avoided crossing.

1014
01:19:10,942 --> 01:19:12,650
At some point, I think,
draw it yourself,

1015
01:19:12,650 --> 01:19:16,760
put the quantum numbers
on it, and you'll

1016
01:19:16,760 --> 01:19:18,000
learn a lot by doing it.

1017
01:19:18,000 --> 01:19:20,895
What you realize also when
you solve the Hamiltonian

1018
01:19:20,895 --> 01:19:26,480
that this structure can be
explained the following way.

1019
01:19:26,480 --> 01:19:31,346
You will always find you
have some states which

1020
01:19:31,346 --> 01:19:35,450
are stretched, where there
is only one state which

1021
01:19:35,450 --> 01:19:39,980
has the maximum angular
momentum inside all

1022
01:19:39,980 --> 01:19:41,890
the stretched states.

1023
01:19:41,890 --> 01:19:44,080
And then the other
states, you always

1024
01:19:44,080 --> 01:19:50,260
need to find two states
which have the same total mF,

1025
01:19:50,260 --> 01:19:53,630
and those two states
avoid each other.

1026
01:19:53,630 --> 01:19:56,860
You can say, just pointing on
two states, those two states,

1027
01:19:56,860 --> 01:19:59,920
let's just assume
they have the same mF.

1028
01:19:59,920 --> 01:20:01,955
They undergo an
avoided crossing,

1029
01:20:01,955 --> 01:20:04,220
and that's exactly
what you get out

1030
01:20:04,220 --> 01:20:06,730
of the diagonalization
of the two by two matrix.

1031
01:20:06,730 --> 01:20:09,240
So this whole diagram
can be understood

1032
01:20:09,240 --> 01:20:13,090
by you have stretched states
which form a one by one matrix.

1033
01:20:13,090 --> 01:20:16,260
There is no recoupling
taking place.

1034
01:20:16,260 --> 01:20:23,300
And then you have three pairs
of states which form two

1035
01:20:23,300 --> 01:20:27,920
by two matrix, and in each pair,
if you would now focus on it,

1036
01:20:27,920 --> 01:20:31,418
you really see the avoided
crossing typical for a two

1037
01:20:31,418 --> 01:20:32,666
by two matrix.

1038
01:20:36,159 --> 01:20:38,155
Any questions about that?

1039
01:20:41,160 --> 01:20:44,470
The next thing would be to go
through some bigger questions

1040
01:20:44,470 --> 01:20:46,740
and review atomic
structure, including

1041
01:20:46,740 --> 01:20:48,270
external magnetic fields.

1042
01:20:48,270 --> 01:20:52,770
But we'll do that at the
beginning of the next class.