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00:00:25,840 --> 00:00:27,730
PROFESSOR: --afternoon.

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00:00:27,730 --> 00:00:31,490
Our main subject for today
is the fine structure

10
00:00:31,490 --> 00:00:39,450
of atoms, when spin couples and
we bring spin into the picture.

11
00:00:39,450 --> 00:00:41,750
And if time permits,
a Lamb shift.

12
00:00:41,750 --> 00:00:43,650
But before I get
started, I just want

13
00:00:43,650 --> 00:00:49,512
to finish up our topic of the
last class, which was helium.

14
00:00:49,512 --> 00:00:53,020
And I said for many,
many years, we only

15
00:00:53,020 --> 00:00:57,240
discussed single electron atoms,
but I introduced the helium

16
00:00:57,240 --> 00:01:00,840
atom into the curriculum because
there are two aspects which

17
00:01:00,840 --> 00:01:03,680
are really new and
which we can learn

18
00:01:03,680 --> 00:01:07,780
from helium with a
more simple example.

19
00:01:07,780 --> 00:01:17,080
One was the fact of
how the spin leads

20
00:01:17,080 --> 00:01:21,540
to interactions even if nothing
couples directly to the spin.

21
00:01:21,540 --> 00:01:25,275
And we discussed it last
class that because we

22
00:01:25,275 --> 00:01:29,610
have singlet and triplet
states, they impose symmetry

23
00:01:29,610 --> 00:01:32,440
or anti-symmetry onto the
spatial wave function.

24
00:01:32,440 --> 00:01:34,910
And that makes a big
difference in the energy.

25
00:01:34,910 --> 00:01:38,370
And it is sort of this
big Coulomb energy, which

26
00:01:38,370 --> 00:01:40,810
is connected to spin,
which is actually

27
00:01:40,810 --> 00:01:44,590
responsible for
magnetism in nature.

28
00:01:44,590 --> 00:01:48,190
The second aspect, the
second new feature,

29
00:01:48,190 --> 00:01:50,230
which is, of course,
also related to spin

30
00:01:50,230 --> 00:01:55,400
is the presence of singlet
and triplet states.

31
00:01:55,400 --> 00:01:57,390
And it gives me an
opportunity-- and this

32
00:01:57,390 --> 00:01:59,830
was the last point we
discussed on Wednesday--

33
00:01:59,830 --> 00:02:03,490
to raise the question,
what can cause

34
00:02:03,490 --> 00:02:08,400
a transition from
triplet to singlet?

35
00:02:08,400 --> 00:02:19,330
And at the end of
Wednesday's lecture,

36
00:02:19,330 --> 00:02:23,600
we discussed the question,
what kind of couplings

37
00:02:23,600 --> 00:02:27,185
would allow you want to drive
a transition from triplet

38
00:02:27,185 --> 00:02:28,860
to singlet?

39
00:02:28,860 --> 00:02:33,060
And at least based on the
clicker result, for most of you

40
00:02:33,060 --> 00:02:36,080
the surprising answer was none.

41
00:02:36,080 --> 00:02:37,320
Optical fields, not.

42
00:02:37,320 --> 00:02:40,670
Rotating fields, not.

43
00:02:40,670 --> 00:02:43,980
Within the approximation
we have so far-- namely,

44
00:02:43,980 --> 00:02:48,060
just a Hamiltonian with
Coulomb interaction--

45
00:02:48,060 --> 00:02:50,620
there is nothing
in the system which

46
00:02:50,620 --> 00:02:55,100
connects the singlet
to the triplet world.

47
00:02:55,100 --> 00:02:57,050
Well, there is a
very weak transition,

48
00:02:57,050 --> 00:02:58,960
but the lifetime of
the triplet counts

49
00:02:58,960 --> 00:03:01,750
it in helium, as I will
tell you a little bit later,

50
00:03:01,750 --> 00:03:03,220
is 8,000 seconds.

51
00:03:03,220 --> 00:03:04,760
It's extremely long life.

52
00:03:04,760 --> 00:03:07,770
We have an extreme form
of metastability here.

53
00:03:07,770 --> 00:03:09,300
Just an anecdote.

54
00:03:09,300 --> 00:03:11,920
In the early days
when people learned

55
00:03:11,920 --> 00:03:15,380
about the structure of atoms,
they looked at spectra.

56
00:03:15,380 --> 00:03:19,050
And they never
found any line which

57
00:03:19,050 --> 00:03:22,370
was connecting the singlet
states with the triplet states.

58
00:03:22,370 --> 00:03:27,780
So it looked like if there were
two different forms of helium,

59
00:03:27,780 --> 00:03:29,750
orthohelium and parahelium.

60
00:03:29,750 --> 00:03:31,690
So people even
believed for a while

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00:03:31,690 --> 00:03:33,880
there were two kinds
of helium in nature

62
00:03:33,880 --> 00:03:37,290
because they seemed to be
completely disconnected.

63
00:03:37,290 --> 00:03:39,910
But as we know now,
of course, this simply

64
00:03:39,910 --> 00:03:42,485
reflects the high
degree of metastability.

65
00:03:55,220 --> 00:03:58,750
I want to emphasize
before I give you

66
00:03:58,750 --> 00:04:02,020
the general framework and the
symmetry, spatial symmetry

67
00:04:02,020 --> 00:04:03,890
and spin symmetry,
and we discuss all

68
00:04:03,890 --> 00:04:05,680
that in exchange
symmetry-- I want

69
00:04:05,680 --> 00:04:09,960
to emphasize what I said
at the end of the lecture

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00:04:09,960 --> 00:04:10,900
on Wednesday.

71
00:04:10,900 --> 00:04:16,550
If you try to drive a transition
between singlet and triplet,

72
00:04:16,550 --> 00:04:20,440
if you use a rotating magnetic
field, you can't do that.

73
00:04:20,440 --> 00:04:25,380
Because a rotating magnetic
field couples to the total

74
00:04:25,380 --> 00:04:26,130
spin.

75
00:04:26,130 --> 00:04:30,375
It just makes the
total spin precess

76
00:04:30,375 --> 00:04:34,080
around the axis of
the magnetic field.

77
00:04:34,080 --> 00:04:37,460
But what the magnetic
field will never do

78
00:04:37,460 --> 00:04:41,570
is break up the relative
angle between the two spins.

79
00:04:41,570 --> 00:04:44,900
And in a triplet state,
the two spins are aligned.

80
00:04:44,900 --> 00:04:47,250
In the singlet states,
they are anti-aligned.

81
00:04:47,250 --> 00:04:49,590
And with a magnetic
field, you address

82
00:04:49,590 --> 00:04:51,520
both spins simultaneously.

83
00:04:51,520 --> 00:04:53,840
They precess together,
and you can never

84
00:04:53,840 --> 00:04:56,240
change the angle
between the two.

85
00:04:56,240 --> 00:04:58,120
So a singlet-triplet
transition really

86
00:04:58,120 --> 00:05:00,630
requires that you go deeper.

87
00:05:00,630 --> 00:05:06,590
You really have a handle on the
spins relative to each other.

88
00:05:06,590 --> 00:05:09,530
Let me make a second comment
about the possibility

89
00:05:09,530 --> 00:05:11,253
of doing an optical transition.

90
00:05:15,350 --> 00:05:19,610
An optical transition, at least
in the dipole approximation,

91
00:05:19,610 --> 00:05:23,530
has an electric field
acting on the dipole moment

92
00:05:23,530 --> 00:05:25,710
of the electron.

93
00:05:25,710 --> 00:05:28,190
There is no spin involved.

94
00:05:28,190 --> 00:05:31,860
And let me sort of accentuate
it by saying, you should really

95
00:05:31,860 --> 00:05:35,520
have in your mind that, at
least in dipole approximation,

96
00:05:35,520 --> 00:05:41,350
the laser, the light, optical
beams, do not couple to spin.

97
00:05:41,350 --> 00:05:45,350
The only couple to the spatial
part of the wave function.

98
00:05:45,350 --> 00:05:50,830
And that has important
consequences.

99
00:05:50,830 --> 00:05:54,900
In a lot of our research,
we do Raman transitions.

100
00:05:54,900 --> 00:05:58,700
And with a Raman transition,
we do spin flips.

101
00:05:58,700 --> 00:06:02,780
So how can a laser, or two
laser beams through a Raman

102
00:06:02,780 --> 00:06:07,140
transition, flip the
spin if the laser is not

103
00:06:07,140 --> 00:06:08,820
talking to the spin
degree of freedom?

104
00:06:12,370 --> 00:06:18,255
Well, if you have
a Raman transition

105
00:06:18,255 --> 00:06:22,730
and you excite a state which
has spin orbit coupling,

106
00:06:22,730 --> 00:06:24,360
you excite the atom.

107
00:06:24,360 --> 00:06:26,850
The spin does nothing
during the excitation.

108
00:06:26,850 --> 00:06:31,270
But then in the
intermediate state,

109
00:06:31,270 --> 00:06:35,410
spin and angular momentum may be
coupled by spin orbit coupling.

110
00:06:35,410 --> 00:06:37,710
And the spin is
starting to precess

111
00:06:37,710 --> 00:06:39,600
due to spin orbit coupling.

112
00:06:39,600 --> 00:06:42,160
And then the spin
rotates, and then

113
00:06:42,160 --> 00:06:44,620
the second leg of
the Raman transition

114
00:06:44,620 --> 00:06:46,810
transfers the atoms back
to the ground state,

115
00:06:46,810 --> 00:06:49,450
but now the spin has rotated.

116
00:06:49,450 --> 00:06:52,410
But it is never the laser
which rotates the spin.

117
00:06:55,000 --> 00:06:56,880
What should be immediately
clear from that

118
00:06:56,880 --> 00:06:58,910
is if you try to
have a spin flip

119
00:06:58,910 --> 00:07:01,820
Raman transition in an atom
which has no spin orbit

120
00:07:01,820 --> 00:07:04,440
coupling don't
even look for that.

121
00:07:04,440 --> 00:07:07,230
It doesn't exist.

122
00:07:07,230 --> 00:07:09,330
Anyway, we just
encountered it here

123
00:07:09,330 --> 00:07:12,660
and I want to tell
you that there is even

124
00:07:12,660 --> 00:07:15,130
more general
consequences of that.

125
00:07:15,130 --> 00:07:18,750
But let's now go back
to the helium atom

126
00:07:18,750 --> 00:07:24,310
and look at the
spatial symmetry which

127
00:07:24,310 --> 00:07:27,250
comes, of course,
because operators

128
00:07:27,250 --> 00:07:29,715
are symmetric with regard
to the two electrons.

129
00:07:33,610 --> 00:07:39,860
So what we have is we
have the total spin

130
00:07:39,860 --> 00:07:45,650
is the sum of the spins of
electron 1 and electron 2.

131
00:07:48,350 --> 00:08:01,970
And all spin operators--
the spin operator s

132
00:08:01,970 --> 00:08:09,566
and its components-- are
symmetric against particle

133
00:08:09,566 --> 00:08:10,065
exchange.

134
00:08:14,720 --> 00:08:17,150
In other words, if
you couple to the spin

135
00:08:17,150 --> 00:08:20,860
through a magnetic field, you
couple to the two electrons

136
00:08:20,860 --> 00:08:22,670
symmetrically.

137
00:08:22,670 --> 00:08:26,360
And therefore, you
have the selection

138
00:08:26,360 --> 00:08:31,780
rule that only symmetric and
symmetric and anti-symmetric

139
00:08:31,780 --> 00:08:33,559
and anti-symmetric
states couple.

140
00:08:36,200 --> 00:08:47,110
Or, to express the same result
in different words, both

141
00:08:47,110 --> 00:08:56,540
the spatial and the spin
symmetry, each of them

142
00:08:56,540 --> 00:08:59,249
can be symmetric
and anti-symmetric.

143
00:08:59,249 --> 00:09:01,540
Of course, the product of
them has to be anti-symmetric

144
00:09:01,540 --> 00:09:04,600
because we're talking
about fermions.

145
00:09:04,600 --> 00:09:09,910
So both of these symmetries
are good quantum numbers.

146
00:09:18,530 --> 00:09:25,396
Or more formally, all
observables commute

147
00:09:25,396 --> 00:09:26,895
with the particle
exchange operator.

148
00:09:29,820 --> 00:09:35,230
So any arbitrary
observable commute.

149
00:09:53,870 --> 00:09:58,540
So the commutator of the
particle exchange operator

150
00:09:58,540 --> 00:09:59,040
vanishes.

151
00:10:04,255 --> 00:10:04,755
OK.

152
00:10:07,840 --> 00:10:10,530
But now, what is special
to the approximation

153
00:10:10,530 --> 00:10:12,970
in which we have treated helium?

154
00:10:12,970 --> 00:10:17,130
As long as wave
functions and operators

155
00:10:17,130 --> 00:10:22,200
separate into spin-dependent
and space-dependent parts.

156
00:10:22,200 --> 00:10:24,930
The wave function
had a spatial part,

157
00:10:24,930 --> 00:10:28,180
and then we multiplied
with a spin wave function.

158
00:10:28,180 --> 00:10:32,180
So the wave function
factor-wise separated

159
00:10:32,180 --> 00:10:33,810
into a spin part
and a spatial part.

160
00:10:40,020 --> 00:10:40,921
Also, the operators.

161
00:10:40,921 --> 00:10:42,420
If you have operator
for total spin,

162
00:10:42,420 --> 00:10:45,200
if you have operator
for dipole moment,

163
00:10:45,200 --> 00:10:47,120
the operators are
either addressing

164
00:10:47,120 --> 00:10:51,790
the spatial degree of freedom
or the spin degree of freedom.

165
00:10:51,790 --> 00:10:55,520
So as long as this
is the case, we

166
00:10:55,520 --> 00:10:58,420
have an even extended
conservation law

167
00:10:58,420 --> 00:11:03,800
for both spin exchange for
an exchange of the particles

168
00:11:03,800 --> 00:11:07,030
in the spatial domain
and in the spin domain.

169
00:11:07,030 --> 00:11:21,560
So as long as wave
functions and operators

170
00:11:21,560 --> 00:11:36,910
separate into spin and
space-dependent parts,

171
00:11:36,910 --> 00:11:41,080
then the particle
exchange operator

172
00:11:41,080 --> 00:11:47,180
can be regarded as the product
of an exchange operator which

173
00:11:47,180 --> 00:11:49,926
only acts on the
spatial wave function

174
00:11:49,926 --> 00:11:51,675
and on the spin part
of the wave function.

175
00:11:59,030 --> 00:12:04,450
And both quantum numbers
for spin exchange--

176
00:12:04,450 --> 00:12:07,730
symmetric/anti-symmetric--
and for exchange of particles

177
00:12:07,730 --> 00:12:09,960
in the spatial part
of the wave function,

178
00:12:09,960 --> 00:12:13,220
symmetric or anti-symmetric--
are conserved.

179
00:12:27,530 --> 00:12:30,750
So often, we discuss
symmetries and the reason

180
00:12:30,750 --> 00:12:33,550
why something is
conserved because then we

181
00:12:33,550 --> 00:12:36,770
can think about
how to evaluate it.

182
00:12:36,770 --> 00:12:39,480
Often, you can say,
conservation laws

183
00:12:39,480 --> 00:12:42,350
are often written down just
as a way to think about.

184
00:12:42,350 --> 00:12:44,900
How can we break them?

185
00:12:44,900 --> 00:12:53,020
And in this case, we can
break this symmetries

186
00:12:53,020 --> 00:12:56,050
which doesn't allow transitions
between singlet and triplet.

187
00:12:56,050 --> 00:12:59,660
We have to break them to
get intercombination lines.

188
00:12:59,660 --> 00:13:08,490
So in order to get a singlet
to triplet transition,

189
00:13:08,490 --> 00:13:10,401
it's only possible
when we violate

190
00:13:10,401 --> 00:13:12,025
the assumptions we
have just discussed.

191
00:13:28,330 --> 00:13:31,590
And of course, an assumption was
that the wave function factors

192
00:13:31,590 --> 00:13:34,580
into spin part and spatial part.

193
00:13:34,580 --> 00:13:36,930
But when spin and
spatial wave functions

194
00:13:36,930 --> 00:13:38,320
are mixed-- for
instance, by spin

195
00:13:38,320 --> 00:13:41,630
orbit coupling-- then the
assumptions we have discussed

196
00:13:41,630 --> 00:13:42,630
are no longer valid.

197
00:13:45,230 --> 00:13:47,000
So what we need
is, for instance,

198
00:13:47,000 --> 00:13:55,770
a mixing of spin and
spatial wave functions

199
00:13:55,770 --> 00:13:56,905
by spin orbit coupling.

200
00:14:04,510 --> 00:14:07,800
Well, as we will
learn later today,

201
00:14:07,800 --> 00:14:13,270
spin orbit coupling
is, actually,

202
00:14:13,270 --> 00:14:14,600
a relativistic effect.

203
00:14:14,600 --> 00:14:17,010
It naturally arises
in the Dirac equation.

204
00:14:17,010 --> 00:14:23,690
And it scales with-- the
nuclear charge is z to the 4.

205
00:14:23,690 --> 00:14:26,230
So therefore in helium,
it is very, very weak.

206
00:14:32,150 --> 00:14:46,000
Therefore, the triplet ground
state is extremely long-lived.

207
00:14:46,000 --> 00:14:49,860
The lifetime in helium
is 8,000 seconds.

208
00:14:49,860 --> 00:14:53,480
It's one of the most
longest-lived metastable states

209
00:14:53,480 --> 00:14:57,250
you can imagine and which
you find in atomic physics.

210
00:14:57,250 --> 00:15:03,730
Now, since spin orbit coupling
becomes rapidly stronger

211
00:15:03,730 --> 00:15:09,010
with nuclear charge, you would
expect the metastable lifetime

212
00:15:09,010 --> 00:15:12,390
to be much shorter
for the analogous

213
00:15:12,390 --> 00:15:15,100
state in the other rare gases.

214
00:15:15,100 --> 00:15:17,500
And indeed, the
other rare gasses

215
00:15:17,500 --> 00:15:19,300
are on the order of 40 seconds.

216
00:15:27,270 --> 00:15:30,510
You have other atoms where you
have singlet and triplet lines.

217
00:15:34,832 --> 00:15:38,100
We need two electrons
and the two spins

218
00:15:38,100 --> 00:15:39,920
can form singlet or triplet.

219
00:15:39,920 --> 00:15:44,670
So that naturally
also happens in atoms

220
00:15:44,670 --> 00:15:48,470
which are in the second
column of the periodic table.

221
00:15:48,470 --> 00:15:54,360
Magnesium, calcium, strontium
have intercombination lines.

222
00:15:58,800 --> 00:16:05,910
And they are actually very
relevant in current research

223
00:16:05,910 --> 00:16:11,625
because these are candidates
for atomic clocks.

224
00:16:17,440 --> 00:16:20,040
In those atoms--
magnesium, calcium,

225
00:16:20,040 --> 00:16:24,130
strontium-- physics
is more complicated.

226
00:16:24,130 --> 00:16:26,660
The line width is
typically kilohertz.

227
00:16:30,960 --> 00:16:33,250
But that's probably
just what you want.

228
00:16:33,250 --> 00:16:37,270
If the line width is kilohertz,
the lifetime is millisecond.

229
00:16:37,270 --> 00:16:39,850
You still have a
matrix element that you

230
00:16:39,850 --> 00:16:42,180
can drive the transition
and observe it.

231
00:16:42,180 --> 00:16:44,580
For an atomic clock, you
don't want a transition

232
00:16:44,580 --> 00:16:50,270
which is thousands of seconds
or half the age of the universe

233
00:16:50,270 --> 00:16:54,330
because it's too weak to be
observed, to weak to be driven.

234
00:16:54,330 --> 00:16:57,402
So intercombination
lines of kilohertz,

235
00:16:57,402 --> 00:16:59,110
line widths of few
hundred Hertz or such,

236
00:16:59,110 --> 00:17:03,200
are almost ideal for
optical atomic clocks.

237
00:17:07,230 --> 00:17:08,619
Any questions?

238
00:17:14,670 --> 00:17:19,619
So maybe then I should just
entertain you for a few minutes

239
00:17:19,619 --> 00:17:26,086
about, how do you measure
an 8,000 second lifetime?

240
00:17:26,086 --> 00:17:33,818
Well, there is a recent paper.

241
00:17:33,818 --> 00:17:34,460
Insert Picture.

242
00:17:34,460 --> 00:17:34,960
Desktop.

243
00:17:46,610 --> 00:17:48,730
There's a recent
paper a few years ago,

244
00:17:48,730 --> 00:17:50,900
Physical Review
Letters, determination

245
00:17:50,900 --> 00:17:54,570
of the lifetime of
metastable helium.

246
00:17:54,570 --> 00:17:56,970
Traditionally, people
have measured the lifetime

247
00:17:56,970 --> 00:17:59,330
of metastable helium
in the following way.

248
00:17:59,330 --> 00:18:02,550
No, they haven't observed
helium for 8,000 seconds.

249
00:18:02,550 --> 00:18:05,400
Because if you have helium
in the metastable state,

250
00:18:05,400 --> 00:18:08,200
maybe after a few seconds there
is some collisions-- collision

251
00:18:08,200 --> 00:18:09,410
or deactivation.

252
00:18:09,410 --> 00:18:11,080
8,000 seconds, you
would just have

253
00:18:11,080 --> 00:18:12,940
many, many systematic effects.

254
00:18:12,940 --> 00:18:19,960
What people did is
they put some atoms

255
00:18:19,960 --> 00:18:23,420
into the metastable state,
measured the number of excited

256
00:18:23,420 --> 00:18:28,550
atoms, and then-- maybe
during 100 milliseconds,

257
00:18:28,550 --> 00:18:32,000
1 out of 80,000
atoms should decay.

258
00:18:32,000 --> 00:18:35,070
And by determining what
small fraction of the atoms

259
00:18:35,070 --> 00:18:37,695
has decayed, or how
many photons you observe

260
00:18:37,695 --> 00:18:41,350
in 100 milliseconds divided
by the total number of atoms,

261
00:18:41,350 --> 00:18:43,690
that's how you can
measure the lifetime.

262
00:18:43,690 --> 00:18:45,600
Of course, the
problem is, yes, you

263
00:18:45,600 --> 00:18:47,020
can measure the
number of photons.

264
00:18:47,020 --> 00:18:49,061
But if you want to divide
by the number of atoms,

265
00:18:49,061 --> 00:18:52,310
you have to know very accurately
how many atoms you have.

266
00:18:52,310 --> 00:18:59,430
And this Australian group
used the very clever trick

267
00:18:59,430 --> 00:19:02,570
to eliminate this uncertainty.

268
00:19:14,580 --> 00:19:16,680
And this is as follows.

269
00:19:16,680 --> 00:19:21,400
We have this extremely
long-lived 8,000 seconds

270
00:19:21,400 --> 00:19:22,580
lifetime here.

271
00:19:22,580 --> 00:19:24,450
And this is the transition.

272
00:19:24,450 --> 00:19:33,920
But you can now excite the atom
to the triplet P0 state, which

273
00:19:33,920 --> 00:19:36,195
is also sort of--
it's a triplet state.

274
00:19:39,720 --> 00:19:43,980
Well, I assume because
it is a P state,

275
00:19:43,980 --> 00:19:45,430
it has been orbit coupling.

276
00:19:45,430 --> 00:19:48,150
The lifetime is-- well,
for an atom is very long.

277
00:19:48,150 --> 00:19:50,000
Typically, atoms
decay in nanoseconds.

278
00:19:50,000 --> 00:19:52,180
But this lifetime
is milliseconds.

279
00:19:52,180 --> 00:19:55,690
So what they did is they
just excited all the atoms

280
00:19:55,690 --> 00:19:59,870
with a strongly saturating laser
to the triplet P0 transition.

281
00:19:59,870 --> 00:20:01,890
And then within a
few milliseconds,

282
00:20:01,890 --> 00:20:03,850
all the atoms emitted.

283
00:20:03,850 --> 00:20:06,380
And so all they had to
do is measure accurately

284
00:20:06,380 --> 00:20:12,360
the ratio of the VUV photons
emitted by the triplet ground

285
00:20:12,360 --> 00:20:16,330
state and by the
triplet P0 state.

286
00:20:16,330 --> 00:20:19,950
And then the number
of atoms canceled out.

287
00:20:19,950 --> 00:20:26,830
All they had to do is measure
the ratio of 2 VUV intensities.

288
00:20:26,830 --> 00:20:30,180
So based on that research
now, we have higher accuracy

289
00:20:30,180 --> 00:20:36,930
on the lifetime of
metastable helium.

290
00:20:36,930 --> 00:20:44,860
The question is
when I said we have

291
00:20:44,860 --> 00:20:49,710
to violate the above
assumptions to get

292
00:20:49,710 --> 00:20:51,850
any decay between
singlet and triplet.

293
00:20:54,480 --> 00:20:56,450
Metastable helium,
pretty much everything

294
00:20:56,450 --> 00:20:58,690
you can think about
it is forbidden.

295
00:20:58,690 --> 00:21:03,360
So for a long time,
people believed

296
00:21:03,360 --> 00:21:04,690
it was a two-photon transition.

297
00:21:07,260 --> 00:21:08,830
One-photon transition
was forbidden,

298
00:21:08,830 --> 00:21:11,960
but a two-photon
transition was allowed.

299
00:21:11,960 --> 00:21:15,790
We know now that
the decay path is

300
00:21:15,790 --> 00:21:18,970
one photon in one transition.

301
00:21:18,970 --> 00:21:20,880
Magnetic dipole transition.

302
00:21:20,880 --> 00:21:26,050
If you don't know what M1 is, we
discuss it later in the course.

303
00:21:26,050 --> 00:21:33,480
But you may find it interesting
that until rather recently,

304
00:21:33,480 --> 00:21:36,320
it was not clear what
the decay path is.

305
00:21:36,320 --> 00:21:41,260
Well, recently means
probably a few decades ago.

306
00:21:41,260 --> 00:21:52,010
But for a long time,
it wasn't even clear

307
00:21:52,010 --> 00:21:54,110
and the calculations
were not accurate enough

308
00:21:54,110 --> 00:21:56,740
what was the mechanism by
which the metastable count

309
00:21:56,740 --> 00:21:58,380
state of helium could decay.

310
00:22:01,890 --> 00:22:04,780
When I prepared the class,
I wanted to sort of give you

311
00:22:04,780 --> 00:22:06,260
a simple picture
and say, hey, it

312
00:22:06,260 --> 00:22:10,230
is this mechanism which
causes the transition

313
00:22:10,230 --> 00:22:13,080
from the triplet ground state
to the singlet ground state,

314
00:22:13,080 --> 00:22:15,700
but I couldn't find
a simple operator.

315
00:22:15,700 --> 00:22:18,600
It seems a little
bit more complicated.

316
00:22:18,600 --> 00:22:22,630
The best I can tell you is
that the decay path requires

317
00:22:22,630 --> 00:22:26,610
higher-order terms using the
Dirac equation with coupling

318
00:22:26,610 --> 00:22:27,850
to the electromagnetic field.

319
00:22:33,770 --> 00:22:34,600
Any questions?

320
00:22:34,600 --> 00:22:36,515
Yes, Nancy.

321
00:22:36,515 --> 00:22:43,445
AUDIENCE: In the measurement
of the lifetime of that state,

322
00:22:43,445 --> 00:22:48,395
what is the limiting factor?

323
00:22:48,395 --> 00:22:51,860
What makes this measurement
to be not more precise?

324
00:22:54,830 --> 00:22:57,804
PROFESSOR: Here
is the reference.

325
00:22:57,804 --> 00:22:59,720
I mean, this was a
highly-accurate measurement

326
00:22:59,720 --> 00:23:02,250
for that it improved all
previous measurements.

327
00:23:02,250 --> 00:23:05,250
What the current limit of
accuracy is, I don't remember.

328
00:23:05,250 --> 00:23:14,610
But for instance, they had to
put all atoms into the excited

329
00:23:14,610 --> 00:23:19,460
state or by strong saturation.

330
00:23:19,460 --> 00:23:21,245
Of course, saturation
is never complete.

331
00:23:21,245 --> 00:23:23,270
Maybe there's a
limitation to that.

332
00:23:23,270 --> 00:23:28,240
Finally, you have to
measure the ratio of photons

333
00:23:28,240 --> 00:23:31,190
emitted by that state and
emitted by that state.

334
00:23:31,190 --> 00:23:33,240
Well, they're both
in the vacuum UV,

335
00:23:33,240 --> 00:23:35,220
but you may have a
systematic effect

336
00:23:35,220 --> 00:23:38,190
if your photo-detector has a
slightly different detection

337
00:23:38,190 --> 00:23:42,180
efficiency for 58
nanometer and 62 nanometer.

338
00:23:42,180 --> 00:23:44,170
Nothing is perfect,
but I really have

339
00:23:44,170 --> 00:23:46,751
to refer you to the
reference for details.

340
00:23:49,950 --> 00:23:50,510
OK.

341
00:23:50,510 --> 00:23:56,480
So that's pretty
much what I wanted

342
00:23:56,480 --> 00:23:59,640
to tell you about electronic
structure, the hydrogen

343
00:23:59,640 --> 00:24:04,050
atom and special features
about the helium atom.

344
00:24:04,050 --> 00:24:15,450
So we can now proceed from
electronic energies, which

345
00:24:15,450 --> 00:24:18,510
were on the order of Rydberg.

346
00:24:18,510 --> 00:24:23,250
Or remember atomic units, fine
structure constant squared

347
00:24:23,250 --> 00:24:25,780
times rest mass of the electron.

348
00:24:25,780 --> 00:24:28,260
So we've taken care
of those energies

349
00:24:28,260 --> 00:24:33,210
and now we want to take
care of smaller corrections.

350
00:24:33,210 --> 00:24:43,200
And what we discuss next is
the fine structure and the Lamb

351
00:24:43,200 --> 00:24:43,700
shift.

352
00:24:49,616 --> 00:24:56,480
So this is now the title of our
next chapter, Fine Structure

353
00:24:56,480 --> 00:24:57,270
and Lamb Shift.

354
00:25:05,300 --> 00:25:14,010
The fine structure energies
are alpha square times smaller

355
00:25:14,010 --> 00:25:16,030
than the Rydberg.

356
00:25:16,030 --> 00:25:21,330
So they're on the order of
alpha to the 4 mc squared.

357
00:25:21,330 --> 00:25:27,850
And well, if I write that as
alpha square Rydberg, which

358
00:25:27,850 --> 00:25:32,090
is the same, but it may--
this explains to you

359
00:25:32,090 --> 00:25:35,180
why alpha is called the
fine structure constant.

360
00:25:35,180 --> 00:25:40,280
People had learned about the
hydrogen spectrum, Bohr's model

361
00:25:40,280 --> 00:25:43,780
and all that, and then they
found finer corrections.

362
00:25:43,780 --> 00:25:47,140
And those finer
corrections were on a scale

363
00:25:47,140 --> 00:25:49,270
alpha square times smaller.

364
00:25:49,270 --> 00:25:51,090
And the constant
which appears here,

365
00:25:51,090 --> 00:25:54,040
alpha, was called the
fine structure constant

366
00:25:54,040 --> 00:25:59,780
because it described the
structure of the fine structure

367
00:25:59,780 --> 00:26:03,390
corrections and the fine
structure splittings.

368
00:26:03,390 --> 00:26:05,900
Of course if you want, you
can make the fine structure

369
00:26:05,900 --> 00:26:09,060
constant already appear
for electronic energies.

370
00:26:09,060 --> 00:26:10,970
But this simply
reflects that I have

371
00:26:10,970 --> 00:26:12,910
used the rest mass
of the electron

372
00:26:12,910 --> 00:26:16,660
as a reference energy,
which unless you want

373
00:26:16,660 --> 00:26:18,710
to discuss fundamental
units doesn't make

374
00:26:18,710 --> 00:26:22,765
any sense because c,
the speed of light,

375
00:26:22,765 --> 00:26:25,370
it does not appear in
the electronic structure

376
00:26:25,370 --> 00:26:26,830
as we discussed earlier.

377
00:26:26,830 --> 00:26:29,100
So therefore, the fine
structure constant

378
00:26:29,100 --> 00:26:35,996
does enter the picture only when
we now discuss fine structure

379
00:26:35,996 --> 00:26:37,620
corrections, fine
structure splittings.

380
00:26:41,120 --> 00:26:45,630
The Lamb shift, which
we will then discuss,

381
00:26:45,630 --> 00:26:48,710
is even higher order.

382
00:26:48,710 --> 00:26:49,775
It's alpha to the fifth.

383
00:26:52,860 --> 00:26:57,290
But just to tell
you, the Lamb shift

384
00:26:57,290 --> 00:26:59,930
is on the order of a gigahertz.

385
00:26:59,930 --> 00:27:03,750
And you know that standard
precision in all of your labs

386
00:27:03,750 --> 00:27:05,400
is now megahertz or better.

387
00:27:05,400 --> 00:27:08,310
So the lamb shift, also
it's alpha to the fifth,

388
00:27:08,310 --> 00:27:12,742
is a big effect on the scale
of how we understand and prove

389
00:27:12,742 --> 00:27:13,950
the structure of atoms today.

390
00:27:18,440 --> 00:27:21,900
OK, so let's talk about
the fine structure.

391
00:27:34,270 --> 00:27:39,460
You will actually find
a complete discussion

392
00:27:39,460 --> 00:27:41,720
of the fine structure
in several textbooks

393
00:27:41,720 --> 00:27:43,430
on quantum mechanics--
[INAUDIBLE].

394
00:27:46,010 --> 00:27:51,150
What I want to do here is
to have a careful discussion

395
00:27:51,150 --> 00:27:54,890
of the physical origin and
provide a clear understanding

396
00:27:54,890 --> 00:27:59,170
what is responsible
for the fine structure.

397
00:27:59,170 --> 00:28:04,230
So if you just want to
get an accurate result,

398
00:28:04,230 --> 00:28:08,190
there's only one
thing you have to do--

399
00:28:08,190 --> 00:28:09,620
solve the Dirac equation.

400
00:28:12,760 --> 00:28:16,580
The solution of the Dirac
equation for the hydrogen atom

401
00:28:16,580 --> 00:28:19,830
gives us the
electronic structure,

402
00:28:19,830 --> 00:28:22,030
the same as comes out
of the Bohr model,

403
00:28:22,030 --> 00:28:25,450
but now including
the fine structure.

404
00:28:25,450 --> 00:28:26,200
So it's nice.

405
00:28:26,200 --> 00:28:27,260
It just comes out.

406
00:28:27,260 --> 00:28:29,920
You don't have to put in the
spin of the electron by hand.

407
00:28:29,920 --> 00:28:31,470
It comes out automatically.

408
00:28:31,470 --> 00:28:33,430
But it also comes
out as a result

409
00:28:33,430 --> 00:28:36,930
and you don't fully understand
what is the origin of that.

410
00:28:36,930 --> 00:28:39,682
Because you've just
treated everything fully

411
00:28:39,682 --> 00:28:40,390
relativistically.

412
00:28:43,110 --> 00:28:46,760
So what I want to
show you here is

413
00:28:46,760 --> 00:28:51,680
I want to show you
that we can distinguish

414
00:28:51,680 --> 00:28:52,940
three contributions.

415
00:28:52,940 --> 00:28:55,390
There are three
physical effects which

416
00:28:55,390 --> 00:29:00,450
all come in at the level
alpha to the 4 mc squared.

417
00:29:00,450 --> 00:29:08,410
One is relativistic corrections
to the kinetic energy.

418
00:29:08,410 --> 00:29:13,040
The second one is
spin orbit coupling.

419
00:29:13,040 --> 00:29:15,480
And then there is
a third one, which

420
00:29:15,480 --> 00:29:17,130
goes by the name Darwin term.

421
00:29:19,810 --> 00:29:27,500
And we can get this
physical insight

422
00:29:27,500 --> 00:29:36,160
and obtain those
three terms separately

423
00:29:36,160 --> 00:29:41,410
when we use what
is called the Pauli

424
00:29:41,410 --> 00:29:43,315
approximation to
the Dirac equation.

425
00:29:47,230 --> 00:29:49,760
If you do the so-called
Pauli approximation

426
00:29:49,760 --> 00:29:55,470
to the Dirac
equation, you simply

427
00:29:55,470 --> 00:30:01,820
expand the Dirac equation
in powers of v/c.

428
00:30:01,820 --> 00:30:05,390
And then you get the Schrodinger
equations plus correction

429
00:30:05,390 --> 00:30:06,660
terms.

430
00:30:06,660 --> 00:30:10,770
And what you obtain
then is a Hamiltonian,

431
00:30:10,770 --> 00:30:15,480
which is the rest energy,
which is constant;

432
00:30:15,480 --> 00:30:22,020
the non-relativistic kinetic
energy, the Coulomb energy.

433
00:30:22,020 --> 00:30:24,820
But then, we have
three more terms.

434
00:30:24,820 --> 00:30:26,790
And these are the
three contributions

435
00:30:26,790 --> 00:30:28,420
to the fine structure.

436
00:30:28,420 --> 00:30:33,890
One is kinetic
energy is p squared,

437
00:30:33,890 --> 00:30:37,050
but now there is a higher-order
relativistic contribution,

438
00:30:37,050 --> 00:30:38,030
p to the 4.

439
00:30:44,280 --> 00:30:49,610
Then, there is a term,
spin orbit coupling term.

440
00:30:58,880 --> 00:31:02,630
And then there is
the third term,

441
00:31:02,630 --> 00:31:07,546
which will we identify
as the Darwin term.

442
00:31:07,546 --> 00:31:10,634
Laplace operator
e squared over r.

443
00:31:13,420 --> 00:31:17,470
So we have a kinetic
energy correction.

444
00:31:17,470 --> 00:31:23,590
We have a spin orbit correction
and we a Darwin term.

445
00:31:23,590 --> 00:31:27,360
And what we want to do in
the next 45 minutes or so is

446
00:31:27,360 --> 00:31:31,430
we want to look
at each of those.

447
00:31:31,430 --> 00:31:32,556
Any questions?

448
00:31:46,660 --> 00:31:56,850
The kinetic energy contribution
is the simple expansion.

449
00:31:56,850 --> 00:31:59,430
You can get it from the
Pauli approximation.

450
00:31:59,430 --> 00:32:02,010
You take the Dirac
creation and deal

451
00:32:02,010 --> 00:32:04,540
with all the operators
in the Dirac equation

452
00:32:04,540 --> 00:32:06,050
and do an expansion.

453
00:32:06,050 --> 00:32:09,940
But you can also just
lean back and say, OK,

454
00:32:09,940 --> 00:32:11,600
what is the kinetic energy?

455
00:32:11,600 --> 00:32:16,070
And use the relativistic form
of the kinetic energy, which

456
00:32:16,070 --> 00:32:18,980
is rest energy and pc squared.

457
00:32:18,980 --> 00:32:22,000
The square root of it.

458
00:32:22,000 --> 00:32:26,720
And if you say the kinetic
energy is the energy minus mc

459
00:32:26,720 --> 00:32:41,620
squared, and then you take the
full relativistic expression

460
00:32:41,620 --> 00:32:46,770
for the total energy minus 1.

461
00:32:46,770 --> 00:32:50,930
And now you do a Taylor
expansion of the square root.

462
00:32:50,930 --> 00:32:55,970
You find the non-relativistic
kinetic energy.

463
00:32:55,970 --> 00:33:01,270
And the first correction term
with all the correct prefactors

464
00:33:01,270 --> 00:33:01,870
is this one.

465
00:33:08,910 --> 00:33:11,680
So this is the relativistic
energy correction

466
00:33:11,680 --> 00:33:12,730
in the Pauli equation.

467
00:33:17,510 --> 00:33:19,860
That's all we have to say.

468
00:33:22,740 --> 00:33:27,730
The second term is
spin orbit interaction.

469
00:33:32,460 --> 00:33:36,440
And as such, it is, I would say,
more interesting because it now

470
00:33:36,440 --> 00:33:40,240
really addresses the spin.

471
00:33:40,240 --> 00:33:44,850
So yes, the term comes
from the Dirac equation

472
00:33:44,850 --> 00:33:48,900
and appears separately
as an identifiable term

473
00:33:48,900 --> 00:33:50,600
if you do the
Pauli approximation

474
00:33:50,600 --> 00:33:51,800
to the Dirac equation.

475
00:33:51,800 --> 00:33:54,920
So I could just stop there and
say, here, that's what it is.

476
00:33:54,920 --> 00:33:59,190
But I think it's insightful if
I re-derive this term for you

477
00:33:59,190 --> 00:34:01,100
by using a physical picture.

478
00:34:01,100 --> 00:34:06,250
What do we have to add to the
Schrodinger equation to get it?

479
00:34:06,250 --> 00:34:08,770
What we have to add to the
Schrodinger equation is,

480
00:34:08,770 --> 00:34:13,500
of course, we have to put in
that the electron has a spin.

481
00:34:13,500 --> 00:34:15,639
The Dirac equation is
formulated in spin.

482
00:34:15,639 --> 00:34:18,502
It's automatically formulated
for spin 1/2 particles.

483
00:34:18,502 --> 00:34:20,460
But if you start with
the Schrodinger equation,

484
00:34:20,460 --> 00:34:23,199
you can first write it down
for a spin-less particle,

485
00:34:23,199 --> 00:34:26,639
and now you have to
add the spin by hand.

486
00:34:26,639 --> 00:34:30,380
So we have to add to the
Schrodinger equation,

487
00:34:30,380 --> 00:34:36,130
or to the Bohr
model, that we have

488
00:34:36,130 --> 00:34:40,360
an electron with
an intrinsic spin.

489
00:34:40,360 --> 00:34:46,590
And the spin will couple
to the rest of the world

490
00:34:46,590 --> 00:34:52,350
because it has a magnetic
moment which is the g factor.

491
00:34:52,350 --> 00:34:54,570
We'll talk about g factors
later in more detail.

492
00:34:58,540 --> 00:35:02,910
The Bohr magneton
times the spin.

493
00:35:05,900 --> 00:35:09,460
If you add it to the
Schrodinger equation,

494
00:35:09,460 --> 00:35:13,170
we have to assume the
experimental value

495
00:35:13,170 --> 00:35:14,720
or you can see the
value of which--

496
00:35:14,720 --> 00:35:16,810
the approximate value
of the exact value

497
00:35:16,810 --> 00:35:22,750
from the Dirac equation,
which is a g factor of 2.

498
00:35:22,750 --> 00:35:26,850
So if you put in by hand
that the electron has a spin,

499
00:35:26,850 --> 00:35:30,370
the spin is associated
with the magnetic moment.

500
00:35:30,370 --> 00:35:36,600
Now we have a coupling
because an electron which

501
00:35:36,600 --> 00:35:40,970
moves in the Coulomb
field of the nucleus

502
00:35:40,970 --> 00:35:45,290
sees in its moving
frame a magnetic field.

503
00:35:45,290 --> 00:35:47,440
And this magnetic field
couples to the spin.

504
00:35:52,330 --> 00:36:12,040
So the moving electron "sees"
a motional magnetic field.

505
00:36:12,040 --> 00:36:16,830
Well, you know when you
have an electric field

506
00:36:16,830 --> 00:36:23,140
and you look at the-- you have
an electric field in one frame

507
00:36:23,140 --> 00:36:26,400
and you have a second frame,
spatial relativity, which

508
00:36:26,400 --> 00:36:31,060
moves with a velocity v.
Then, in the moving frame,

509
00:36:31,060 --> 00:36:36,040
you observe a motional
magnetic field.

510
00:36:36,040 --> 00:36:38,365
Well, our moving frame
is now the electron.

511
00:36:41,410 --> 00:36:47,290
So that looks like coupling
to the Coulomb field,

512
00:36:47,290 --> 00:36:51,745
but we can immediately
write it in a form

513
00:36:51,745 --> 00:36:54,800
where we recognize
angular momentum.

514
00:36:59,730 --> 00:37:02,660
So this is the Coulomb field.

515
00:37:02,660 --> 00:37:07,020
And it involves now v cross r.

516
00:37:07,020 --> 00:37:11,088
And v cross r is just minus L,
the orbital angular momentum.

517
00:37:20,290 --> 00:37:24,530
So this is now our
motional magnetic field.

518
00:37:24,530 --> 00:37:28,740
And then the spin couples to it
and we get spin orbit coupling.

519
00:37:28,740 --> 00:37:31,600
By the way, an
equivalent picture

520
00:37:31,600 --> 00:37:33,990
because you will
read about it or hear

521
00:37:33,990 --> 00:37:38,390
about it is you can get the
magnetic field which couples

522
00:37:38,390 --> 00:37:41,780
to the spin by simply
transforming the Coulomb

523
00:37:41,780 --> 00:37:44,750
field into the frame
of the moving electron.

524
00:37:44,750 --> 00:37:48,740
But there is a more naive
picture which is correct, too.

525
00:37:48,740 --> 00:37:51,090
If you are in the
frame of the electron,

526
00:37:51,090 --> 00:37:54,570
then the proton is
orbiting around you.

527
00:37:54,570 --> 00:37:58,670
And the orbiting proton
is a ring current,

528
00:37:58,670 --> 00:38:01,180
which creates a magnetic field.

529
00:38:01,180 --> 00:38:03,430
And this magnetic
field is or it has

530
00:38:03,430 --> 00:38:06,460
to be the same as
we just got by doing

531
00:38:06,460 --> 00:38:09,217
the relativistic transformation
of the Coulomb field

532
00:38:09,217 --> 00:38:10,800
into the frame of
the moving electron.

533
00:38:20,350 --> 00:38:23,487
So therefore, let me
just leave some room

534
00:38:23,487 --> 00:38:24,820
because I have to add something.

535
00:38:27,800 --> 00:38:32,022
If we want the spin
orbit Hamiltonian,

536
00:38:32,022 --> 00:38:39,211
it's now the magnetic moment
of the electron times this B

537
00:38:39,211 --> 00:38:39,710
field.

538
00:38:42,500 --> 00:38:47,360
And just collecting the terms,
e squared, h bar squared,

539
00:38:47,360 --> 00:38:53,220
m squared, c squared,
1 over r cubed.

540
00:38:53,220 --> 00:38:56,790
The magnetic moment is
proportional to s, the spin.

541
00:38:56,790 --> 00:38:59,604
The motional B field
is proportional to L.

542
00:38:59,604 --> 00:39:01,770
And now we have the famous
spin orbit coupling term.

543
00:39:06,060 --> 00:39:08,100
OK, that would be too simple.

544
00:39:08,100 --> 00:39:11,810
So we have to now
consider one more thing.

545
00:39:11,810 --> 00:39:14,870
And this is-- I told you
that we can transform

546
00:39:14,870 --> 00:39:17,740
the electric field into the
moving frame of the electron,

547
00:39:17,740 --> 00:39:21,150
but those transformations
from one Lorentz

548
00:39:21,150 --> 00:39:25,780
frame into the other are simple
when you go from one frame

549
00:39:25,780 --> 00:39:28,930
to another and they move
with a constant velocity

550
00:39:28,930 --> 00:39:30,140
relative to each other.

551
00:39:30,140 --> 00:39:33,940
The electron is not moving
with a constant velocity.

552
00:39:33,940 --> 00:39:35,980
It actually has a
constant acceleration

553
00:39:35,980 --> 00:39:38,500
because it's on
a circular orbit.

554
00:39:38,500 --> 00:39:50,370
So therefore, we have to use
the relativistic transformation

555
00:39:50,370 --> 00:39:55,080
between coordinate frames.

556
00:39:55,080 --> 00:39:59,360
And it is in textbooks
on spatial relativity.

557
00:39:59,360 --> 00:40:07,920
What happens is if
you have-- if you

558
00:40:07,920 --> 00:40:16,260
do relativistic transformations
between different moving

559
00:40:16,260 --> 00:40:21,640
coordinate systems,
then you find

560
00:40:21,640 --> 00:40:25,140
that if you have a velocity
like this, velocity like this,

561
00:40:25,140 --> 00:40:27,160
and you do subsequent
transformations,

562
00:40:27,160 --> 00:40:30,900
you are not just going
into a Lorentz frame which

563
00:40:30,900 --> 00:40:35,110
has very solid and relativistic
addition of the velocities.

564
00:40:35,110 --> 00:40:38,090
In addition, your
final Lorentz frame

565
00:40:38,090 --> 00:40:40,990
is rotated against
the first frame.

566
00:40:40,990 --> 00:40:46,715
So adding a number of velocities
which are not collinear--

567
00:40:46,715 --> 00:40:48,840
and this is what we have
to do if the electron goes

568
00:40:48,840 --> 00:40:49,340
in a circle.

569
00:40:49,340 --> 00:40:51,150
We have to sort of
follow the electron,

570
00:40:51,150 --> 00:40:53,900
follow different
velocities and do

571
00:40:53,900 --> 00:40:56,880
relativistic transformations.

572
00:40:56,880 --> 00:41:00,280
Several relativistic
Lorentz transformation

573
00:41:00,280 --> 00:41:06,140
executed in sequence are not a
simple Lorentz transformation,

574
00:41:06,140 --> 00:41:10,010
they are Lorentz
transformation plus rotation.

575
00:41:10,010 --> 00:41:18,160
And that means that if
we go-- and the result

576
00:41:18,160 --> 00:41:20,270
can be summarized as follows.

577
00:41:20,270 --> 00:41:28,250
There is a rotational
angular velocity

578
00:41:28,250 --> 00:41:38,630
which can be written as follows.

579
00:41:38,630 --> 00:41:40,630
You see immediately
that this term

580
00:41:40,630 --> 00:41:46,270
is 0 when a, acceleration,
and velocity are the same.

581
00:41:46,270 --> 00:41:48,820
So if you have a moving frame
and you accelerate the moving

582
00:41:48,820 --> 00:41:51,470
frame, you don't get
this term if everything

583
00:41:51,470 --> 00:41:52,690
is in one dimension.

584
00:41:52,690 --> 00:41:57,160
But if the acceleration
is not in one dimension--

585
00:41:57,160 --> 00:42:00,020
and that's the case when the
electron goes on the circle,

586
00:42:00,020 --> 00:42:02,680
you get an additional
contribution.

587
00:42:02,680 --> 00:42:08,050
And if the acceleration
is constant,

588
00:42:08,050 --> 00:42:10,050
you get-- and this is
on the left-hand side--

589
00:42:10,050 --> 00:42:12,720
a constant angular velocity.

590
00:42:12,720 --> 00:42:14,540
A precession velocity.

591
00:42:14,540 --> 00:42:21,180
And the letter T indicates that
it's the Thomas precession.

592
00:42:24,810 --> 00:42:26,840
Now, we have
learned-- and that's

593
00:42:26,840 --> 00:42:29,550
nice to come back to the
beginning of this course.

594
00:42:29,550 --> 00:42:34,480
We have discussed that if
you have a magnetic field,

595
00:42:34,480 --> 00:42:37,270
we can go into a rotating
frame and the magnetic field

596
00:42:37,270 --> 00:42:38,670
disappears.

597
00:42:38,670 --> 00:42:41,700
Well now, due to
relativistic kinematics,

598
00:42:41,700 --> 00:42:43,690
we have an additional rotation.

599
00:42:43,690 --> 00:42:45,600
But using the same
argument we used

600
00:42:45,600 --> 00:42:48,430
at the beginning of the course,
this additional rotation

601
00:42:48,430 --> 00:42:50,320
corresponds to an
additional magnetic field.

602
00:42:52,850 --> 00:42:57,300
So therefore, this precession
is identical to the effect

603
00:42:57,300 --> 00:43:01,430
of an additional magnetic field.

604
00:43:01,430 --> 00:43:04,970
So there is an
additional magnetic field

605
00:43:04,970 --> 00:43:09,370
with the letter T, the
Thomas magnetic field.

606
00:43:09,370 --> 00:43:13,510
Those magnetic fields are
related to the precession

607
00:43:13,510 --> 00:43:18,780
frequency by the
gyromagnetic ratio.

608
00:43:18,780 --> 00:43:26,930
And if we use the value for the
electron's gyromagnetic ratio--

609
00:43:26,930 --> 00:43:30,980
and in addition, we use the
fact that the acceleration

610
00:43:30,980 --> 00:43:35,080
term for this
relativistic kinematics

611
00:43:35,080 --> 00:43:44,620
is nothing else than the
Coulomb acceleration,

612
00:43:44,620 --> 00:43:54,000
then we find that the
Thomas magnetic field is

613
00:43:54,000 --> 00:43:58,940
just minus 1/2 the
motional magnetic field we

614
00:43:58,940 --> 00:43:59,671
derived earlier.

615
00:44:03,910 --> 00:44:07,800
So therefore, if
we go back and want

616
00:44:07,800 --> 00:44:11,870
to find an expression for
the spin orbit coupling,

617
00:44:11,870 --> 00:44:15,710
we have to use the
magnetic field, which

618
00:44:15,710 --> 00:44:20,210
is the sum of the two
contributions we discussed,

619
00:44:20,210 --> 00:44:25,300
the motional contribution for--
as if the electron would move

620
00:44:25,300 --> 00:44:28,730
in a linear way, plus
the Thomas precession

621
00:44:28,730 --> 00:44:33,590
due to the rotational motion
due to relativistic kinematics.

622
00:44:33,590 --> 00:44:38,000
And this will just
reduce the final result

623
00:44:38,000 --> 00:44:39,988
by a factor of 1/2.

624
00:44:47,180 --> 00:44:47,908
Yes.

625
00:44:47,908 --> 00:44:49,580
AUDIENCE: Two questions.

626
00:44:49,580 --> 00:44:55,380
Small one, what letter is the
gyromagnetic ratio denoted by?

627
00:44:55,380 --> 00:44:57,930
PROFESSOR: Sorry
for the handwriting.

628
00:44:57,930 --> 00:45:02,370
We have usually used
lowercase gamma.

629
00:45:02,370 --> 00:45:04,210
And the e means the electron.

630
00:45:04,210 --> 00:45:06,330
So gamma electron is
the gyromagnetic ratio

631
00:45:06,330 --> 00:45:07,876
of the electron.

632
00:45:07,876 --> 00:45:08,822
AUDIENCE: OK.

633
00:45:08,822 --> 00:45:11,832
And then my other question, or
more substantively concern--

634
00:45:11,832 --> 00:45:13,580
or confused, I guess.

635
00:45:13,580 --> 00:45:16,410
I wasn't sure where
the h bar comes

636
00:45:16,410 --> 00:45:19,890
from in the expression
for angular momentum.

637
00:45:19,890 --> 00:45:21,375
One line up.

638
00:45:25,890 --> 00:45:30,338
You go from v cross
r to h bar over mL.

639
00:45:30,338 --> 00:45:33,320
And I wasn't sure where
the h bar came from.

640
00:45:37,296 --> 00:45:38,780
PROFESSOR: Oh.

641
00:45:38,780 --> 00:45:39,405
It now depends.

642
00:45:44,850 --> 00:45:47,470
You're correct, angular
momentum is r cross

643
00:45:47,470 --> 00:45:50,440
v. r cross v gives
angular momentum.

644
00:45:50,440 --> 00:45:52,610
And because it's r cross
v and not v cross r,

645
00:45:52,610 --> 00:45:54,750
the minus sign is
absorbed into it.

646
00:45:54,750 --> 00:45:57,290
Now, the angular
momentum-- it depends now.

647
00:45:57,290 --> 00:45:58,730
It's just purely definitional.

648
00:45:58,730 --> 00:46:00,760
If you wanted, the
angular momentum

649
00:46:00,760 --> 00:46:03,550
has eigenvalues of
L times L plus 1,

650
00:46:03,550 --> 00:46:06,240
which are dimensionless, then
we have to account for h bar

651
00:46:06,240 --> 00:46:07,290
explicitly.

652
00:46:07,290 --> 00:46:10,930
So question is, do we want
the angular momentum operator

653
00:46:10,930 --> 00:46:15,910
to have this kind of
integer, or L times L plus 1?

654
00:46:15,910 --> 00:46:16,420
Yeah.

655
00:46:16,420 --> 00:46:19,230
This integer spectrum,
or do we want

656
00:46:19,230 --> 00:46:21,290
to measure-- or, do we
want to use operators

657
00:46:21,290 --> 00:46:23,050
which have units of h bar?

658
00:46:23,050 --> 00:46:26,434
That's where the
h bar comes from.

659
00:46:26,434 --> 00:46:27,100
Other questions?

660
00:46:29,660 --> 00:46:30,160
OK.

661
00:46:30,160 --> 00:46:33,540
So this is spin orbit coupling.

662
00:46:33,540 --> 00:46:47,640
Finally, the last contribution
to the fine structure

663
00:46:47,640 --> 00:46:48,910
is the Darwin term.

664
00:46:55,070 --> 00:46:58,920
The Darwin term is
sort of peculiar,

665
00:46:58,920 --> 00:47:03,480
but there is sort of
an intuitive picture

666
00:47:03,480 --> 00:47:04,640
in which I can derive it.

667
00:47:09,360 --> 00:47:17,920
Well, it starts by saying
that an electron-- well, we

668
00:47:17,920 --> 00:47:22,490
know an electron is, as far
as we know, a point object.

669
00:47:22,490 --> 00:47:25,330
But in order to derive
the Darwin term,

670
00:47:25,330 --> 00:47:29,350
we assume that the
electron is not

671
00:47:29,350 --> 00:47:33,250
arbitrarily
accurately localized,

672
00:47:33,250 --> 00:47:34,570
but it is smeared out.

673
00:47:40,410 --> 00:47:43,720
And you may remember
when I discussed

674
00:47:43,720 --> 00:47:48,900
fundamental units and the
fine structure constant alpha,

675
00:47:48,900 --> 00:47:53,110
and I told you that the
fact that alpha is small

676
00:47:53,110 --> 00:47:55,500
leads to the fact
that people say

677
00:47:55,500 --> 00:47:57,580
electromagnetic
interactions are weak.

678
00:47:57,580 --> 00:48:00,200
I discussed with
you that if you want

679
00:48:00,200 --> 00:48:03,090
to do single-particle physics
and do single-particle pictures

680
00:48:03,090 --> 00:48:05,090
and single-particle
equations, you

681
00:48:05,090 --> 00:48:10,190
can never assume that a
particle is localized better

682
00:48:10,190 --> 00:48:11,970
than its Compton wavelengths.

683
00:48:11,970 --> 00:48:14,260
If you would localize
the particle better

684
00:48:14,260 --> 00:48:16,240
than the Compton
wavelengths, it would

685
00:48:16,240 --> 00:48:18,260
have a momentum
uncertainty, which

686
00:48:18,260 --> 00:48:20,540
would lead to an energy
uncertainty, which

687
00:48:20,540 --> 00:48:22,310
is larger than its rest mass.

688
00:48:22,310 --> 00:48:27,355
And you get into quantum field
theory and pair production.

689
00:48:27,355 --> 00:48:30,200
So at some point,
you have to be aware

690
00:48:30,200 --> 00:48:33,480
that single-particle
pictures break down

691
00:48:33,480 --> 00:48:36,910
when you assume a
localization of particles

692
00:48:36,910 --> 00:48:40,240
better than the
Compton wavelengths.

693
00:48:40,240 --> 00:48:42,690
So therefore-- and
I'm waving my hands

694
00:48:42,690 --> 00:48:44,740
because it's a
hand-waving argument.

695
00:48:44,740 --> 00:48:47,360
But we can derive
the Darwin term

696
00:48:47,360 --> 00:48:52,390
by assuming that the electron
is sort of smeared out

697
00:48:52,390 --> 00:48:56,200
over a dimension.

698
00:48:56,200 --> 00:49:07,920
And this dimension can only
be the Compton wavelengths,

699
00:49:07,920 --> 00:49:12,640
which is h bar over mc.

700
00:49:12,640 --> 00:49:15,040
And there is a nice
German word for it,

701
00:49:15,040 --> 00:49:19,790
how you can imagine that
the electron is smeared out.

702
00:49:19,790 --> 00:49:21,720
It goes by the word
[SPEAKING GERMAN].

703
00:49:26,840 --> 00:49:29,880
I think the German word is much
nicer than its translation,

704
00:49:29,880 --> 00:49:30,970
which is trembling motion.

705
00:49:34,080 --> 00:49:36,630
So at least in this
picture you assume

706
00:49:36,630 --> 00:49:39,200
that the electron is
sort of smeared out.

707
00:49:39,200 --> 00:49:41,440
It's not because the
electron has a size.

708
00:49:41,440 --> 00:49:42,650
It just trembles.

709
00:49:42,650 --> 00:49:47,670
It just trembles and that leads
to a smear-out over the Compton

710
00:49:47,670 --> 00:49:49,850
wavelengths.

711
00:49:49,850 --> 00:49:53,110
But that implies now that
if this trembling motion is

712
00:49:53,110 --> 00:49:57,620
sufficiently fast, we should not
use in our Schrodinger equation

713
00:49:57,620 --> 00:49:59,040
the Coulomb potential.

714
00:49:59,040 --> 00:50:02,242
You should use the Coulomb
potential which is spatially

715
00:50:02,242 --> 00:50:03,825
averaged over the
Compton wavelengths.

716
00:50:15,910 --> 00:50:21,430
So therefore, the
phenomenological derivation

717
00:50:21,430 --> 00:50:25,430
of the Darwin term goes by
replacing the Coulomb potential

718
00:50:25,430 --> 00:50:29,100
in the Schrodinger equation by
the spatially-averaged Coulomb

719
00:50:29,100 --> 00:50:30,570
potential.

720
00:50:30,570 --> 00:50:34,410
So let me now derive
what this means.

721
00:50:38,010 --> 00:50:42,370
So if we assume there is a
very small displacement s,

722
00:50:42,370 --> 00:50:44,620
now we can take the
Coulomb potential

723
00:50:44,620 --> 00:50:55,670
and expand it into its Coulomb
potential plus derivative times

724
00:50:55,670 --> 00:51:05,580
s plus the second-order
Taylor expansion

725
00:51:05,580 --> 00:51:09,040
the xi, the xj component.

726
00:51:09,040 --> 00:51:15,070
And then we need the
second-order derivative xi

727
00:51:15,070 --> 00:51:18,801
xj of the Coulomb potential.

728
00:51:21,690 --> 00:51:28,750
So the correction is now--
well, if you ever reach it

729
00:51:28,750 --> 00:51:32,015
in an isotropic way,
the gradient term,

730
00:51:32,015 --> 00:51:33,650
a linear term
doesn't contribute.

731
00:51:33,650 --> 00:51:35,580
It just averages to 0.

732
00:51:35,580 --> 00:51:41,500
So the leading correction
comes from the curvature term.

733
00:51:41,500 --> 00:51:50,720
So therefore, we have to take
the curvature or the Laplacian

734
00:51:50,720 --> 00:51:53,310
of the Coulomb potential.

735
00:51:53,310 --> 00:51:58,870
The scale of sx squared,
sy squared, sz squared

736
00:51:58,870 --> 00:52:05,450
is the Compton
wavelength squared.

737
00:52:05,450 --> 00:52:08,810
And well, maybe I'm taking
the argument of the Compton

738
00:52:08,810 --> 00:52:10,060
wavelengths now too literally.

739
00:52:10,060 --> 00:52:13,460
But at least if I want to
carry it through, sx squared,

740
00:52:13,460 --> 00:52:21,120
sy squared, sz squared, each
of them 1/3 of s absolute value

741
00:52:21,120 --> 00:52:21,620
squared.

742
00:52:26,260 --> 00:52:35,030
So if I now use the
Coulomb potential,

743
00:52:35,030 --> 00:52:46,680
I obtain the result where I
have the Laplacian of 1/r.

744
00:52:46,680 --> 00:52:50,910
And as you, of course,
know is the 1/r function

745
00:52:50,910 --> 00:52:54,150
has a non-vanishing
Laplacian only at the origin.

746
00:52:54,150 --> 00:52:56,560
So therefore, the
correction is only valid

747
00:52:56,560 --> 00:53:01,550
for s electrons which have
a non-vanishing probability

748
00:53:01,550 --> 00:53:04,534
to feel the origin of
the Coulomb potential.

749
00:53:13,260 --> 00:53:16,020
OK, I did a lot of
hand-waving here.

750
00:53:16,020 --> 00:53:19,700
And yes, those
factors of 1/2 are

751
00:53:19,700 --> 00:53:24,990
the fact that the electron is
smeared out over the Compton

752
00:53:24,990 --> 00:53:27,010
radius and maybe prefactors.

753
00:53:27,010 --> 00:53:29,680
But OK, this is the
physical picture.

754
00:53:29,680 --> 00:53:33,170
You can get an exact
result by simply taking

755
00:53:33,170 --> 00:53:36,660
the Dirac equation, doing the
non-relativistic approximation,

756
00:53:36,660 --> 00:53:42,170
and the term which appears
in the Pauli equation

757
00:53:42,170 --> 00:53:44,430
is actually the same.

758
00:53:44,430 --> 00:53:47,805
The only thing
which is different

759
00:53:47,805 --> 00:53:50,840
is that the
prefactor is not 1/6.

760
00:53:50,840 --> 00:53:53,980
The exact prefactor is 1/8.

761
00:53:53,980 --> 00:53:56,810
So the exact results
actually validates

762
00:53:56,810 --> 00:53:58,800
that the physical
picture we have used

763
00:53:58,800 --> 00:54:01,520
is at least a reasonable
approximation.

764
00:54:11,940 --> 00:54:19,040
OK, so we have now--
let us summarize.

765
00:54:19,040 --> 00:54:29,090
For the fine structure, we
have three different terms.

766
00:54:29,090 --> 00:54:32,895
And they all have a
different signature.

767
00:54:32,895 --> 00:54:36,330
We have the relativistic
kinetic energy correction,

768
00:54:36,330 --> 00:54:40,289
we have the spin orbit coupling,
and we have the Darwin term.

769
00:54:43,070 --> 00:54:49,430
We can now ask, which term
contributes for an s electron?

770
00:54:52,210 --> 00:54:55,970
Well, relative kinetic
energy, of course.

771
00:54:55,970 --> 00:54:58,950
The Darwin term is
only for s electrons.

772
00:54:58,950 --> 00:55:04,076
LS does not contribute because
an s electron has L equal 0.

773
00:55:11,020 --> 00:55:16,190
So if we go to p and d states,
to states with finite angular

774
00:55:16,190 --> 00:55:20,555
momentum, then we get
contribution of LS.

775
00:55:20,555 --> 00:55:24,580
We always get a relativistic
kinetic energy contribution.

776
00:55:24,580 --> 00:55:29,500
But for finite angular momentum,
we don't have the Darwin term.

777
00:55:29,500 --> 00:55:31,580
And finally, we
can also classify

778
00:55:31,580 --> 00:55:35,980
those different contribution
by asking, what is the sign?

779
00:55:35,980 --> 00:55:43,770
Do those corrections lower
or increase the total energy?

780
00:55:43,770 --> 00:55:51,000
Well, the Darwin term
reduces the binding energy

781
00:55:51,000 --> 00:55:53,410
because it sort of
takes a cusp away

782
00:55:53,410 --> 00:55:57,020
from the Coulomb interaction.

783
00:55:57,020 --> 00:56:01,190
The kinetic energy has the
opposite sign and spin orbit

784
00:56:01,190 --> 00:56:06,110
coupling is L dot S. And S can
be parallel or anti-parallel

785
00:56:06,110 --> 00:56:09,530
to L. So therefore, the
spin orbit interaction has

786
00:56:09,530 --> 00:56:10,540
plus or minus sign.

787
00:56:20,470 --> 00:56:23,095
Any questions about
those three terms?

788
00:56:28,190 --> 00:56:30,200
OK.

789
00:56:30,200 --> 00:56:34,110
The fact that we
have spin and we

790
00:56:34,110 --> 00:56:39,720
have spin orbit interaction
means that the spin orbit

791
00:56:39,720 --> 00:56:57,250
interaction, L dot S is
not diagonal in L or S.

792
00:56:57,250 --> 00:57:00,980
So what we have to do is
we have to introduce now

793
00:57:00,980 --> 00:57:16,150
the total angular momentum
J. And since L plus S squared

794
00:57:16,150 --> 00:57:24,630
equals J squared, we
can write the product LS

795
00:57:24,630 --> 00:57:32,660
as 1/2 J squared minus L
squared minus S squared.

796
00:57:32,660 --> 00:57:42,030
So therefore, the LS interaction
is diagonal in the J basis.

797
00:57:42,030 --> 00:57:49,730
So if you couple L and S to J
a given state of the operator

798
00:57:49,730 --> 00:57:52,650
J has a definite value for LS.

799
00:57:56,480 --> 00:57:56,980
OK.

800
00:58:03,150 --> 00:58:06,810
Let's put in things together.

801
00:58:06,810 --> 00:58:10,190
So at the beginning
of this class,

802
00:58:10,190 --> 00:58:13,020
we started out with
a hydrogen atom

803
00:58:13,020 --> 00:58:19,390
where-- let me just focus on
n equals 2, where we had n

804
00:58:19,390 --> 00:58:20,840
equals 2.

805
00:58:20,840 --> 00:58:23,840
All n equals 2 states have
the same energy, that's

806
00:58:23,840 --> 00:58:25,980
what comes out of the
non-relativistic Schrodinger

807
00:58:25,980 --> 00:58:28,000
equation with a
Coulomb potential.

808
00:58:28,000 --> 00:58:32,880
And that means we had a
degeneracy for the 2p 1/2,

809
00:58:32,880 --> 00:58:37,940
for the 2p 3/2 and
the 2s 1/2 state.

810
00:58:48,390 --> 00:58:58,890
If you now put in spin
orbit interaction, well,

811
00:58:58,890 --> 00:59:01,460
nothing of course happens
to the s 1/2 state

812
00:59:01,460 --> 00:59:04,700
because it has no
orbital angular momentum.

813
00:59:04,700 --> 00:59:09,950
And then in the P
3/2 state, L equals

814
00:59:09,950 --> 00:59:14,540
1, 1 unit of orbital angular
momentum, and the spin aligned.

815
00:59:14,540 --> 00:59:17,810
In the p 1/2 state,
they are anti-aligned.

816
00:59:17,810 --> 00:59:22,870
So therefore, the two
terms have opposite shifts.

817
00:59:22,870 --> 00:59:30,170
So the p 3/2 state is shifted
up and the p 1/2 state

818
00:59:30,170 --> 00:59:31,655
is shifted down.

819
00:59:35,620 --> 00:59:39,150
But then in the
second step, we want

820
00:59:39,150 --> 00:59:44,850
to add in now the two
other contributions

821
00:59:44,850 --> 00:59:49,185
to the fine structure, the
kinetic and the Darwin term.

822
00:59:52,910 --> 00:59:56,740
And then, something happens
which is really remarkable.

823
00:59:56,740 --> 00:59:59,300
We have split the three levels.

824
00:59:59,300 --> 01:00:01,780
We have lifted the degeneracy
between the three levels.

825
01:00:01,780 --> 01:00:05,310
But if you now add the
kinetic and the Darwin term,

826
01:00:05,310 --> 01:00:21,560
it turns out that the s
1/2 and the p 1/2 state

827
01:00:21,560 --> 01:00:24,180
become exactly degenerate again.

828
01:00:29,360 --> 01:00:31,670
So there is still a
degeneracy in the spectrum.

829
01:00:34,380 --> 01:00:36,540
If we would use a
non-relativistic approach

830
01:00:36,540 --> 01:00:39,260
and derive the Darwin
term, the spin orbit

831
01:00:39,260 --> 01:00:41,939
term, and the relativistic
correction separately,

832
01:00:41,939 --> 01:00:42,980
there would be no reason.

833
01:00:42,980 --> 01:00:46,820
It would just look like a
freak accident in nature

834
01:00:46,820 --> 01:00:49,190
that those two levels
come out equal.

835
01:00:49,190 --> 01:00:51,390
However, it's not
a freak accident.

836
01:00:51,390 --> 01:00:54,070
It's the symmetry of
the Dirac equation.

837
01:00:54,070 --> 01:00:59,140
So all those corrections
have a deep connection

838
01:00:59,140 --> 01:01:01,100
in relativistic physics.

839
01:01:01,100 --> 01:01:07,910
And relativistic physics
preserves the degeneracy in J.

840
01:01:07,910 --> 01:01:09,650
So let me write that down.

841
01:01:09,650 --> 01:01:26,930
So the fine structure does
not lift the degeneracy

842
01:01:26,930 --> 01:01:30,960
between s 1/2 and p 1/2.

843
01:01:34,230 --> 01:01:39,450
It turns out that when we
use the Dirac equation,

844
01:01:39,450 --> 01:01:49,320
we can get an exact expression
for the fine structure

845
01:01:49,320 --> 01:01:50,370
into the [INAUDIBLE].

846
01:01:50,370 --> 01:01:53,430
And it's a principal
quantum number.

847
01:01:53,430 --> 01:02:01,831
And then the fine structure only
depends on J and not on L and S

848
01:02:01,831 --> 01:02:02,331
separately.

849
01:02:07,520 --> 01:02:14,510
So that tells us that
eventually the spin

850
01:02:14,510 --> 01:02:17,940
of the electron and
the fine structure

851
01:02:17,940 --> 01:02:21,720
really have deep origins
in the relativistic nature

852
01:02:21,720 --> 01:02:22,860
of the underlying physics.

853
01:02:30,140 --> 01:02:32,620
Any questions about
fine structure?

854
01:02:39,820 --> 01:02:46,240
Let me get a white
page and draw you

855
01:02:46,240 --> 01:02:50,510
sort of an energy-level diagram,
put in a few numbers which then

856
01:02:50,510 --> 01:02:53,330
leads us to the discussion
of the Lamb shift.

857
01:02:56,795 --> 01:03:02,160
So we have hydrogenic
energy levels with n

858
01:03:02,160 --> 01:03:04,570
equals 1 and n equals 2.

859
01:03:08,640 --> 01:03:16,240
We have s states and
we have p states.

860
01:03:16,240 --> 01:03:28,060
We just discussed the fine
structure which shifts down

861
01:03:28,060 --> 01:03:37,810
the energy of all those levels
in the ground-- I mentioned

862
01:03:37,810 --> 01:03:43,370
that the fine structure is
alpha to the 4 mc squared.

863
01:03:43,370 --> 01:03:47,550
And for the ground state,
the prefector is minus 1/8.

864
01:03:50,440 --> 01:03:56,266
And this is the one-- the
term designation is s 1/2.

865
01:03:56,266 --> 01:04:09,610
Here, we get the 2s 1/2 level
and the shift is minus 5/128

866
01:04:09,610 --> 01:04:16,570
in the same unit alpha
to the 4 mc squared.

867
01:04:16,570 --> 01:04:21,220
We have the exact
degeneracy between the s 1/2

868
01:04:21,220 --> 01:04:25,150
and the 2p 1/2 state.

869
01:04:25,150 --> 01:04:47,780
Whereas, the p 3/2 state is
only shifted by minus 1/128.

870
01:04:47,780 --> 01:04:55,130
To put in some numbers
in electron volt,

871
01:04:55,130 --> 01:04:58,490
the fine structure in the ground
state is on the order of 10

872
01:04:58,490 --> 01:05:00,560
to the minus 4 electron volt.

873
01:05:04,080 --> 01:05:07,680
The splitting here, which is
the fine structure splitting

874
01:05:07,680 --> 01:05:12,600
of the p 1/2 and the
p 3/2 state is 4 times

875
01:05:12,600 --> 01:05:16,620
10 to the minus 5 electron volt.

876
01:05:16,620 --> 01:05:22,370
So that means at the level of
10 to the minus 4 electron volt,

877
01:05:22,370 --> 01:05:24,284
we understand atomic structure.

878
01:05:28,180 --> 01:05:30,990
Well, but we want to go further.

879
01:05:30,990 --> 01:05:36,180
And the next thing
we want to discuss

880
01:05:36,180 --> 01:05:42,270
is that-- what is the exact
degeneracy between s 1/2

881
01:05:42,270 --> 01:05:50,890
and p 1/2 is actually lifted
when we introduce photons.

882
01:05:50,890 --> 01:05:53,410
When we allow the
electrons to couple

883
01:05:53,410 --> 01:05:55,460
to the electromagnetic
field, that's

884
01:05:55,460 --> 01:05:58,360
QED, Quantum Electrodynamics.

885
01:05:58,360 --> 01:06:03,140
And that introduces the
Lamb shift, which in the n

886
01:06:03,140 --> 01:06:07,640
equals 2 state is
10 to the minus 6

887
01:06:07,640 --> 01:06:11,790
electron volt, which is
approximately 1 gigahertz.

888
01:06:11,790 --> 01:06:14,430
And that's what we
want to derive next.

889
01:06:17,460 --> 01:06:20,200
Since I have this
nice, level diagram

890
01:06:20,200 --> 01:06:26,150
I want to also indicate to
you what we do next week.

891
01:06:26,150 --> 01:06:32,570
And this is when we
bring in the proton.

892
01:06:32,570 --> 01:06:34,910
So far we have said
there is a point charge,

893
01:06:34,910 --> 01:06:38,310
but the proton is a particle
which has finite size.

894
01:06:38,310 --> 01:06:41,270
But also, it has finite
angular momentum.

895
01:06:41,270 --> 01:06:44,380
And then we have to
bring in fine structure.

896
01:06:44,380 --> 01:06:47,990
And we will learn
that the s 1/2 ground

897
01:06:47,990 --> 01:06:53,190
state splits into two
hyperfine components.

898
01:06:53,190 --> 01:06:59,310
And the hyperfine structure
will be at a similar energy

899
01:06:59,310 --> 01:07:03,400
scale as the Lamb shift.

900
01:07:03,400 --> 01:07:07,170
And actually, this fine
structure splitting

901
01:07:07,170 --> 01:07:12,350
is the famous 14--
it's about 1 gigahertz.

902
01:07:12,350 --> 01:07:14,950
It's the famous
1,420 megahertz line,

903
01:07:14,950 --> 01:07:18,150
which everybody who does
astrophysics knows about it.

904
01:07:18,150 --> 01:07:22,100
It's also, in other units,
the famous hydrogen measure,

905
01:07:22,100 --> 01:07:24,330
which has an emission
line at 21 centimeter.

906
01:07:29,040 --> 01:07:31,650
Anyway, this is
sort of our roadmap.

907
01:07:31,650 --> 01:07:34,360
We have taken care of
everything down to 10

908
01:07:34,360 --> 01:07:36,760
to the minus 5 electron volts.

909
01:07:36,760 --> 01:07:43,470
And now we want to proceed
today and in the next lecture

910
01:07:43,470 --> 01:07:45,170
to 10 to the minus 6.

911
01:07:45,170 --> 01:07:48,830
First, the Lamb shift, where
we are still holding on

912
01:07:48,830 --> 01:07:53,120
that the nucleus is a point
charge, has no structure.

913
01:07:53,120 --> 01:07:55,560
Then, we have done
everything we can do.

914
01:07:55,560 --> 01:07:58,140
What happens in the Coulomb
field of a point charge?

915
01:07:58,140 --> 01:08:00,000
We have the spin
of the electron.

916
01:08:00,000 --> 01:08:00,550
We have QED.

917
01:08:00,550 --> 01:08:02,000
We have done everything.

918
01:08:02,000 --> 01:08:05,400
And then what remains is to put
in the structure of the proton,

919
01:08:05,400 --> 01:08:07,150
which will lead us to
hyperfine structure.

920
01:08:10,850 --> 01:08:11,405
Any question?

921
01:08:28,696 --> 01:08:38,859
OK, then let's get started
with the Lamb shift.

922
01:08:46,450 --> 01:08:51,069
Just to make sure that
you're not getting confused,

923
01:08:51,069 --> 01:08:54,880
there is also a lamp shift.

924
01:08:54,880 --> 01:08:57,660
This is actually an
old-fashioned word

925
01:08:57,660 --> 01:09:00,569
for an AC Stark effect.

926
01:09:00,569 --> 01:09:03,399
People in the '50s
and '60s found

927
01:09:03,399 --> 01:09:06,540
that spectral lines
in a mercury discharge

928
01:09:06,540 --> 01:09:10,279
shifted when you took
another mercury lamp

929
01:09:10,279 --> 01:09:14,810
and illuminated the mercury with
a strong light of another lamp.

930
01:09:14,810 --> 01:09:17,620
So a lamp shift
is simply a shift

931
01:09:17,620 --> 01:09:21,950
due to the electromagnetic
radiation emitted by a lamp.

932
01:09:21,950 --> 01:09:23,819
But that's what we
are talking about.

933
01:09:23,819 --> 01:09:25,700
We want to talk about
the Lamb shift, which

934
01:09:25,700 --> 01:09:28,660
is named after Willis Lamb,
who received the Nobel

935
01:09:28,660 --> 01:09:33,630
Prize for the discovery that
the 2s 1/2 and the 2p 1/2

936
01:09:33,630 --> 01:09:36,109
level in hydrogen
are not degenerate.

937
01:09:36,109 --> 01:09:38,220
And he wouldn't have
gotten the Nobel Prize

938
01:09:38,220 --> 01:09:40,890
if it would have been
simply of one little feature

939
01:09:40,890 --> 01:09:42,580
in the spectrum of hydrogen.

940
01:09:42,580 --> 01:09:47,180
But the discovery of this
lifting of the degeneracy

941
01:09:47,180 --> 01:09:50,560
actually opened up the
field for the development

942
01:09:50,560 --> 01:09:52,620
of quantum electrodynamics.

943
01:09:52,620 --> 01:09:54,860
So this was the
experimental discovery

944
01:09:54,860 --> 01:09:56,510
which led to quantum
electrodynamics.

945
01:09:56,510 --> 01:09:59,940
And this is why it's a
very famous effect done

946
01:09:59,940 --> 01:10:02,185
by a now very famous
man, Willis Lamb.

947
01:10:16,890 --> 01:10:21,110
The full description of the
Lamb shift would require QED.

948
01:10:21,110 --> 01:10:24,830
And later in this course,
we are developing the tools

949
01:10:24,830 --> 01:10:25,555
to do that.

950
01:10:25,555 --> 01:10:26,680
I don't want to do it here.

951
01:10:26,680 --> 01:10:30,530
I want to give you a simplified
physical picture which actually

952
01:10:30,530 --> 01:10:34,780
nicely relates the Lamb shift
to the Darwin term which

953
01:10:34,780 --> 01:10:35,890
we have just discussed.

954
01:10:35,890 --> 01:10:39,130
But sort of just in full
disclosure, what you have to do

955
01:10:39,130 --> 01:10:42,440
is, if you fully quantize
the electromagnetic field,

956
01:10:42,440 --> 01:10:44,250
you have a vector
potential which

957
01:10:44,250 --> 01:10:46,320
describes the vacuum mode.

958
01:10:46,320 --> 01:10:47,910
And you have a vector
potential, which

959
01:10:47,910 --> 01:10:50,630
is the operator of the
fully-quantized field.

960
01:10:50,630 --> 01:10:58,670
And if you now carry out
second-order perturbation

961
01:10:58,670 --> 01:11:02,700
theory in this operator A of
the quantized electromagnetic

962
01:11:02,700 --> 01:11:03,530
field.

963
01:11:03,530 --> 01:11:06,360
In other words, you allow
the atom or the electron

964
01:11:06,360 --> 01:11:10,440
in the atom to couple to all
the empty modes of the vacuum,

965
01:11:10,440 --> 01:11:15,910
then you obtain the Lamb
shift in its full beauty.

966
01:11:15,910 --> 01:11:19,380
This is discussed nicely in
"Atom Photon Interaction"

967
01:11:19,380 --> 01:11:21,700
by Claude
Cohen-Tannoudji, et al.

968
01:11:21,700 --> 01:11:22,800
So this is the nature.

969
01:11:22,800 --> 01:11:26,430
The nature of the Lamb
shift is the coupling

970
01:11:26,430 --> 01:11:28,370
to the vacuum modes.

971
01:11:28,370 --> 01:11:32,470
But I want to capture that now
in a semi-classical picture.

972
01:11:32,470 --> 01:11:43,130
So the picture I
am presenting here

973
01:11:43,130 --> 01:11:45,058
is due to Welton
and Viki Weisskopf.

974
01:11:49,550 --> 01:11:54,920
And yes, it has to address
the nature of the Lamb shift.

975
01:11:54,920 --> 01:12:00,900
And this is the coupling of
the electron to the vacuum,

976
01:12:00,900 --> 01:12:03,670
but the vacuum is not empty.

977
01:12:03,670 --> 01:12:06,280
The vacuum is filled
with a zero point

978
01:12:06,280 --> 01:12:08,140
energy of the
electromagnetic field.

979
01:12:12,920 --> 01:12:20,530
What we need is that we
have electromagnetic modes

980
01:12:20,530 --> 01:12:22,823
and they have a
zero point energy.

981
01:12:26,280 --> 01:12:29,360
And each mode has a zero
point energy of h bar omega.

982
01:12:32,520 --> 01:12:36,305
And maybe before I do
any equations, which

983
01:12:36,305 --> 01:12:38,890
will take five minutes
and about one page,

984
01:12:38,890 --> 01:12:41,080
I should just give you
the physical picture.

985
01:12:41,080 --> 01:12:43,500
I think it's much clearer
to understand that.

986
01:12:43,500 --> 01:12:46,310
So the moment each
mode has an energy,

987
01:12:46,310 --> 01:12:50,740
then each mode has a
fluctuating electric field.

988
01:12:50,740 --> 01:12:52,610
Now what we have
to do is we have

989
01:12:52,610 --> 01:12:58,440
to multiply the contribution--
the energy of each mode

990
01:12:58,440 --> 01:12:59,920
by the density of modes.

991
01:12:59,920 --> 01:13:01,770
We have to sum over all modes.

992
01:13:01,770 --> 01:13:04,950
But then in the end what
we get is an expression

993
01:13:04,950 --> 01:13:08,040
for the fluctuating
electric field.

994
01:13:08,040 --> 01:13:10,390
And remember, in
the Darwin term,

995
01:13:10,390 --> 01:13:14,210
we assumed that the electron
is trembling by itself

996
01:13:14,210 --> 01:13:17,970
and is smeared out over
the Compton wavelengths.

997
01:13:17,970 --> 01:13:21,250
But now in addition to
its own trembling motion,

998
01:13:21,250 --> 01:13:25,490
the electron is now shaken
by the electric field

999
01:13:25,490 --> 01:13:26,610
of the vacuum.

1000
01:13:26,610 --> 01:13:30,280
And this leads to an
additional smear out.

1001
01:13:30,280 --> 01:13:32,990
And this is the Lamb shift.

1002
01:13:32,990 --> 01:13:35,160
So that's what we
want to put together.

1003
01:13:35,160 --> 01:13:38,690
So we want to calculate
an additional contribution

1004
01:13:38,690 --> 01:13:41,180
to a trembling motion
of the electron.

1005
01:13:41,180 --> 01:13:46,440
But this time, it is driven
by the electric field which

1006
01:13:46,440 --> 01:13:49,910
persists even in the vacuum.

1007
01:13:49,910 --> 01:13:52,830
So that's the Lamb shift.

1008
01:13:52,830 --> 01:13:58,620
OK, so it's 1/2 h
bar omega per mode.

1009
01:13:58,620 --> 01:14:02,930
We have to use the
density of modes.

1010
01:14:02,930 --> 01:14:08,360
And the density is, of
course, per unit volume

1011
01:14:08,360 --> 01:14:14,130
and frequency interval.

1012
01:14:14,130 --> 01:14:15,410
So we have a density.

1013
01:14:15,410 --> 01:14:20,510
mu is the frequency
and you probably

1014
01:14:20,510 --> 01:14:27,196
remember that the density of
modes scales with frequency

1015
01:14:27,196 --> 01:14:27,696
squared.

1016
01:14:34,990 --> 01:14:46,440
So that means the density
of the zero point energy

1017
01:14:46,440 --> 01:14:54,080
is 1/2 h mu per mode
times the mode density.

1018
01:14:54,080 --> 01:14:59,239
And this means it scales
now with mu cubed,

1019
01:14:59,239 --> 01:15:00,616
the frequency cubed.

1020
01:15:04,290 --> 01:15:07,290
And since we want
to introduce now

1021
01:15:07,290 --> 01:15:12,460
the electric field, the
[INAUDIBLE] electric field.

1022
01:15:14,980 --> 01:15:17,210
If you go from electric
field to energy,

1023
01:15:17,210 --> 01:15:19,950
the energy density
of the electric field

1024
01:15:19,950 --> 01:15:23,740
is proportional to e Squared.

1025
01:15:23,740 --> 01:15:27,270
So therefore, what we
derive from this picture

1026
01:15:27,270 --> 01:15:32,790
that the vacuum is filled with
an oscillating electric field.

1027
01:15:32,790 --> 01:15:35,060
And this oscillating
electric field

1028
01:15:35,060 --> 01:15:40,840
is characterized by a value, by
a spectral density E squared,

1029
01:15:40,840 --> 01:15:44,730
which is proportional
to mu cubed.

1030
01:15:44,730 --> 01:15:52,554
And if I collect the constants
h and speed of light, it's this.

1031
01:15:55,250 --> 01:15:58,010
So the question is, what
does this electric field

1032
01:15:58,010 --> 01:16:00,250
do to the electron?

1033
01:16:00,250 --> 01:16:08,000
Well, let's not discuss the
electron in a hydrogenic orbit.

1034
01:16:08,000 --> 01:16:12,260
Let's, rather, discuss
in a simplified picture,

1035
01:16:12,260 --> 01:16:17,230
what is the effect of such
a field on a free electron?

1036
01:16:20,520 --> 01:16:24,170
And we will later discuss that
for very high-frequencies,

1037
01:16:24,170 --> 01:16:27,910
an electron can be
regarded as free.

1038
01:16:27,910 --> 01:16:35,170
So therefore, if s is the
coordinate of the electron,

1039
01:16:35,170 --> 01:16:41,230
and you drive it with an
oscillating electric field

1040
01:16:41,230 --> 01:16:50,670
at frequency mu, then the
driven amplitude of the electron

1041
01:16:50,670 --> 01:16:56,758
is the electric field.

1042
01:16:56,758 --> 01:16:59,830
I just want to make sure we
keep track of the powers of mu.

1043
01:17:03,080 --> 01:17:06,180
Because we take the
second-derivative acceleration,

1044
01:17:06,180 --> 01:17:08,160
we get two powers of mu.

1045
01:17:08,160 --> 01:17:10,576
So we have an extra
mu squared term.

1046
01:17:21,010 --> 01:17:23,640
Of course, the phase
is random, so we're not

1047
01:17:23,640 --> 01:17:25,690
interested in the amplitude.

1048
01:17:25,690 --> 01:17:29,530
We are interested in sort
of an average amplitude

1049
01:17:29,530 --> 01:17:33,700
square or an RMS
amplitude, which

1050
01:17:33,700 --> 01:17:37,120
is-- the amplitude
was proportional

1051
01:17:37,120 --> 01:17:38,710
to the electric field.

1052
01:17:38,710 --> 01:17:41,820
The amplitude squared is
proportional to the square

1053
01:17:41,820 --> 01:17:44,590
of the electric field.

1054
01:17:44,590 --> 01:17:54,460
And our prefactor
involves now mu to the 4.

1055
01:17:59,460 --> 01:18:04,080
Just in the equation
at the top, we

1056
01:18:04,080 --> 01:18:06,770
had an expression for
the spectral density

1057
01:18:06,770 --> 01:18:11,060
of the electric field E
squared which goes mu cubed.

1058
01:18:11,060 --> 01:18:13,400
So that means now that
the spectral density

1059
01:18:13,400 --> 01:18:21,390
of the motion of the electron
s mu squared goes as 1 over mu.

1060
01:18:25,590 --> 01:18:34,780
And yup, e squared h bar pi
squared m squared e cubed.

1061
01:18:34,780 --> 01:18:37,330
And now, yes, I wanted to
do it in the same lecture

1062
01:18:37,330 --> 01:18:41,260
as the Darwin term
because we know

1063
01:18:41,260 --> 01:18:44,270
that there was a change to the
Coulomb potential in the Darwin

1064
01:18:44,270 --> 01:18:49,380
term which came because the
electron was smeared out

1065
01:18:49,380 --> 01:18:51,130
over an area s squared.

1066
01:18:51,130 --> 01:18:55,120
Remember, this was the
Compton [INAUDIBLE] squared.

1067
01:18:55,120 --> 01:19:00,710
And it involved the Laplacian of
the potential Taylor expansion

1068
01:19:00,710 --> 01:19:03,760
second order, second
derivative in three dimension

1069
01:19:03,760 --> 01:19:06,070
is the Laplacian.

1070
01:19:06,070 --> 01:19:09,250
And for the Coulomb
potential, this

1071
01:19:09,250 --> 01:19:12,604
is non-vanishing, of
course, only at the origin.

1072
01:19:16,300 --> 01:19:20,970
So therefore, we
have now executed

1073
01:19:20,970 --> 01:19:22,690
what I discussed before.

1074
01:19:22,690 --> 01:19:27,380
We calculated the
sort of displacement

1075
01:19:27,380 --> 01:19:29,880
of the free electron
driven by the vacuum

1076
01:19:29,880 --> 01:19:32,400
fluctuations of
the electric field.

1077
01:19:32,400 --> 01:19:40,190
And these smear
out of the electron

1078
01:19:40,190 --> 01:19:45,980
leads to a change of the-- of
the average Coulomb potential.

1079
01:19:45,980 --> 01:19:48,883
We average the Coulomb
potential over the first motion

1080
01:19:48,883 --> 01:19:51,340
of the electron.

1081
01:19:51,340 --> 01:20:01,270
And that means now that we
get a change of the binding

1082
01:20:01,270 --> 01:20:11,630
energy of the electron, which
is nothing else in perturbation

1083
01:20:11,630 --> 01:20:17,770
theory than the matrix element
of the perturbation operator.

1084
01:20:17,770 --> 01:20:23,340
But this perturbation
operator was a delta function.

1085
01:20:28,440 --> 01:20:32,240
And we have the prefactor,
which is the spectral density

1086
01:20:32,240 --> 01:20:35,250
of the displacement
s mu squared.

1087
01:20:35,250 --> 01:20:37,840
So let's wrap up.

1088
01:20:37,840 --> 01:20:41,260
So we know because of
the delta function,

1089
01:20:41,260 --> 01:20:45,249
it only affects s electrons.

1090
01:20:51,476 --> 01:20:54,930
And remember I discussed
at great lengths

1091
01:20:54,930 --> 01:20:59,360
that for an electron with
principal quantum number n,

1092
01:20:59,360 --> 01:21:05,150
the probability to be at
the center was n cubed.

1093
01:21:05,150 --> 01:21:07,650
So here again, we have
the n cubed factor,

1094
01:21:07,650 --> 01:21:10,440
which we discussed for the
hydrogenic wave function.

1095
01:21:14,530 --> 01:21:19,520
And the only thing which is
non non-trivial we have to do

1096
01:21:19,520 --> 01:21:28,990
is we have to integrate
over all modes mu.

1097
01:21:28,990 --> 01:21:32,850
So therefore, what we
had so far was the shift

1098
01:21:32,850 --> 01:21:36,540
caused by one frequency of
the electromagnetic spectrum.

1099
01:21:36,540 --> 01:21:40,360
But now we have to integrate
over all frequencies.

1100
01:21:40,360 --> 01:21:56,940
So what we have to do is-- our
spectrum s squared goes as 1

1101
01:21:56,940 --> 01:21:57,840
over mu.

1102
01:21:57,840 --> 01:22:01,730
And if you integrate 1 over
mu over all frequencies,

1103
01:22:01,730 --> 01:22:03,240
you get a logarithm.

1104
01:22:03,240 --> 01:22:07,250
You get logarithmic divergences.

1105
01:22:07,250 --> 01:22:13,910
So therefore, our result is--
and I'm using now atomic units,

1106
01:22:13,910 --> 01:22:20,460
alpha cubed z to the
4 the n cubed factor.

1107
01:22:20,460 --> 01:22:24,480
Everything is in
units of atomic units.

1108
01:22:24,480 --> 01:22:28,990
The atomic unit of energy
is 1 Hartree or 2 Rydberg.

1109
01:22:28,990 --> 01:22:32,270
But the factor which
is now non-trivial

1110
01:22:32,270 --> 01:22:37,160
is from the logarithm,
we integrate.

1111
01:22:37,160 --> 01:22:40,310
And we have divergences
at both ends.

1112
01:22:40,310 --> 01:22:44,035
So we need a cutoff at a minimum
and at a maximum frequency.

1113
01:22:49,710 --> 01:22:51,910
For the maximum
frequency-- well,

1114
01:22:51,910 --> 01:22:54,480
there is a natural cutoff.

1115
01:22:54,480 --> 01:22:57,980
We have to cutoff at the
rest energy of the electron,

1116
01:22:57,980 --> 01:23:02,620
otherwise we are no longer
doing single-particle theory.

1117
01:23:02,620 --> 01:23:06,080
And the cutoff, the lower
cutoff, is the following.

1118
01:23:10,570 --> 01:23:14,610
We have used a picture
here for free electron,

1119
01:23:14,610 --> 01:23:18,550
but we can regard the
orbiting electron only as

1120
01:23:18,550 --> 01:23:20,800
a free electron on
time scales which

1121
01:23:20,800 --> 01:23:23,420
are much faster than
one orbital period.

1122
01:23:23,420 --> 01:23:28,220
So therefore, we have
to use as a lower cutoff

1123
01:23:28,220 --> 01:23:38,330
the frequency of the
orbiting electron, which

1124
01:23:38,330 --> 01:23:46,120
is z squared n cubed
in atomic units.

1125
01:23:46,120 --> 01:23:52,880
So with that, then I should
stop because time is over.

1126
01:23:52,880 --> 01:23:58,290
We have a little
bit hand-wavingly,

1127
01:23:58,290 --> 01:24:01,360
but it appears in
a logarithm anyway.

1128
01:24:01,360 --> 01:24:07,890
We have gotten for the ratio of
the upper and the lower cutoff

1129
01:24:07,890 --> 01:24:15,670
an expression which is n cubed
over z squared times alpha

1130
01:24:15,670 --> 01:24:17,480
squared.

1131
01:24:17,480 --> 01:24:20,227
And if we apply it
to s state where

1132
01:24:20,227 --> 01:24:21,770
it's most important
because there

1133
01:24:21,770 --> 01:24:25,810
is the degeneracy between p
1/2 and p 3/2 to be lifted,

1134
01:24:25,810 --> 01:24:29,390
we get a result that
the energy splitting is

1135
01:24:29,390 --> 01:24:37,410
now-- and with the
assumptions we have made,

1136
01:24:37,410 --> 01:24:41,630
the logarithm is 8 over
a squared, alpha squared.

1137
01:24:41,630 --> 01:24:44,310
And from that, we
would have derived

1138
01:24:44,310 --> 01:24:48,100
a Lamb shift of 1,600 megahertz.

1139
01:24:48,100 --> 01:25:04,314
Well, the exact value
is 1,058 megahertz.

1140
01:25:04,314 --> 01:25:07,400
I'm missing five minutes to
discuss the result in terms

1141
01:25:07,400 --> 01:25:09,430
of vacuum polarization,
but that's

1142
01:25:09,430 --> 01:25:11,970
something we do on Monday.

1143
01:25:11,970 --> 01:25:15,380
Any quick question?

1144
01:25:15,380 --> 01:25:16,100
OK, great.

1145
01:25:16,100 --> 01:25:17,950
See you on Monday.