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PROFESSOR: Good afternoon.

9
00:00:23,590 --> 00:00:30,780
So last week, we started
our discussion of atoms.

10
00:00:30,780 --> 00:00:33,210
So these are, of
course, the key players

11
00:00:33,210 --> 00:00:35,740
in a course of atomic physics.

12
00:00:35,740 --> 00:00:40,120
And we will reveal the structure
of atoms going from the larger

13
00:00:40,120 --> 00:00:43,120
energies to the
smaller energies.

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00:00:43,120 --> 00:00:46,940
That means we start with
electronic energies mainly

15
00:00:46,940 --> 00:00:48,960
determined by Coulomb energy.

16
00:00:48,960 --> 00:00:50,820
And then we go to
finer levels which

17
00:00:50,820 --> 00:00:53,780
are fine structure, hyperfine
structure, Lamb shift, and all

18
00:00:53,780 --> 00:00:55,300
that.

19
00:00:55,300 --> 00:01:01,670
So we have started last week
to discuss the Schrodinger

20
00:01:01,670 --> 00:01:04,970
equation, discuss
hydrogenic energy levels,

21
00:01:04,970 --> 00:01:11,440
and I pointed out some important
results on typical length

22
00:01:11,440 --> 00:01:14,450
scales and the scaling
of the wave function.

23
00:01:14,450 --> 00:01:17,060
We will meet that later today.

24
00:01:17,060 --> 00:01:20,250
But before I continue
with atomic structure,

25
00:01:20,250 --> 00:01:26,060
I want to start today
to discuss the units.

26
00:01:26,060 --> 00:01:30,860
The atomic units we are
using to describe the atom.

27
00:01:30,860 --> 00:01:38,640
So the nat-- and for
every problem in physics,

28
00:01:38,640 --> 00:01:42,240
you have what one may
call natural units.

29
00:01:44,850 --> 00:01:51,680
And for us these are also
called the atomic units.

30
00:01:51,680 --> 00:01:53,200
So what are atomic units?

31
00:01:53,200 --> 00:01:57,850
Well, atomic units are the
units for length, for energy,

32
00:01:57,850 --> 00:02:00,310
for velocity, for
electric field.

33
00:02:00,310 --> 00:02:03,850
And all of these
units, you should

34
00:02:03,850 --> 00:02:07,960
be able to construct them
out of fundamental constants.

35
00:02:07,960 --> 00:02:11,440
The fundamental constants
which appear in the Schrodinger

36
00:02:11,440 --> 00:02:15,610
equation for the
electron within the atom

37
00:02:15,610 --> 00:02:22,190
are the charge of the electron,
the mass of the electron,

38
00:02:22,190 --> 00:02:22,690
and h-bar.

39
00:02:25,430 --> 00:02:28,760
Now, is there any other
fundamental constant

40
00:02:28,760 --> 00:02:33,395
we should include here to
construct our natural units?

41
00:02:37,330 --> 00:02:40,110
c, good question.

42
00:02:40,110 --> 00:02:42,368
What about c?

43
00:02:42,368 --> 00:02:44,910
The speed of light.

44
00:02:44,910 --> 00:02:48,090
Should the speed
of light be part

45
00:02:48,090 --> 00:02:50,930
of our system of atomic units?

46
00:03:00,130 --> 00:03:01,060
Let's not go there.

47
00:03:01,060 --> 00:03:02,520
[LAUGHTER]

48
00:03:02,520 --> 00:03:06,150
Because if you set it to
one, you have made a choice.

49
00:03:06,150 --> 00:03:08,440
You have constrained
your system.

50
00:03:08,440 --> 00:03:12,030
And you're almost obscuring the
fact whether it should appear

51
00:03:12,030 --> 00:03:13,330
or not.

52
00:03:13,330 --> 00:03:15,710
Well, the strong message
I want to give you

53
00:03:15,710 --> 00:03:19,232
is at the level of
electronic structure,

54
00:03:19,232 --> 00:03:20,940
at the level of the
Schrodinger equation,

55
00:03:20,940 --> 00:03:22,600
it should not be there.

56
00:03:22,600 --> 00:03:25,230
Because we are
talking at this point

57
00:03:25,230 --> 00:03:27,680
about solutions
of the Schrodinger

58
00:03:27,680 --> 00:03:30,190
equation, the hydrogenic levels.

59
00:03:30,190 --> 00:03:33,720
And there is no c, no speed
of light in the Schrodinger

60
00:03:33,720 --> 00:03:34,980
equation.

61
00:03:34,980 --> 00:03:39,830
So c is not part of the
fundamental constants

62
00:03:39,830 --> 00:03:42,780
we have to consider now.

63
00:03:42,780 --> 00:03:45,870
It will later come in the
fine-structure constant,

64
00:03:45,870 --> 00:03:47,870
but this is a different story.

65
00:03:47,870 --> 00:03:51,750
So we have three
units, e, m, and c.

66
00:03:51,750 --> 00:03:55,360
And you can just play
the game of combinatorics

67
00:03:55,360 --> 00:03:59,480
and see can you
find a length which

68
00:03:59,480 --> 00:04:01,520
consists of those three units.

69
00:04:01,520 --> 00:04:06,030
And, well, it's h
squared, m over e squared.

70
00:04:06,030 --> 00:04:09,200
This is how you get
the unit of length.

71
00:04:09,200 --> 00:04:13,930
And this length is
called the Bohr radius.

72
00:04:13,930 --> 00:04:17,860
And indeed, this is
sort of the RMS size

73
00:04:17,860 --> 00:04:21,165
of the electron in the 1s state.

74
00:04:21,165 --> 00:04:24,830
You can play the
game again and ask,

75
00:04:24,830 --> 00:04:29,460
can we construct an energy?

76
00:04:29,460 --> 00:04:35,050
Well, you find that if
you take e to the four,

77
00:04:35,050 --> 00:04:39,560
if you take the mass and
finally divide by h square,

78
00:04:39,560 --> 00:04:42,910
then you have a unit of energy.

79
00:04:42,910 --> 00:04:47,310
And this unit of
energy turns out

80
00:04:47,310 --> 00:04:55,930
to be one Hartree or 27.2
electron volts or two Rydbergs,

81
00:04:55,930 --> 00:04:59,690
twice the binding energy of
the electron in the 1s state.

82
00:04:59,690 --> 00:05:02,900
And we had some discussion last
week that the factor of two

83
00:05:02,900 --> 00:05:05,600
reflects the virial theorem.

84
00:05:05,600 --> 00:05:08,380
This is actually that one
Hartree is the Coulomb energy

85
00:05:08,380 --> 00:05:12,340
of the electron in the 1s state,
which is the binding energy.

86
00:05:12,340 --> 00:05:14,690
But then half of it
is kinetic energy.

87
00:05:14,690 --> 00:05:18,010
And, therefore, the total
energy of the 1s electron

88
00:05:18,010 --> 00:05:20,950
is half the Coulomb energy.

89
00:05:20,950 --> 00:05:23,570
And it's that.

90
00:05:23,570 --> 00:05:25,350
But no.

91
00:05:25,350 --> 00:05:28,150
OK, so, so far no c.

92
00:05:28,150 --> 00:05:32,190
The energy, the energy
levels, the wave function.

93
00:05:32,190 --> 00:05:34,810
If there is no c, no speed
of light in the Schrodinger

94
00:05:34,810 --> 00:05:38,070
equation, there is no c in the
solution of the Schrodinger

95
00:05:38,070 --> 00:05:40,540
equation.

96
00:05:40,540 --> 00:05:43,510
And if you set it 1, sure,
it wouldn't-- I mean,

97
00:05:43,510 --> 00:05:45,790
if your relativity
equation said c equals 1,

98
00:05:45,790 --> 00:05:46,740
you obscure the fact.

99
00:05:46,740 --> 00:05:50,070
But here, it's
definitely not there.

100
00:05:50,070 --> 00:05:52,485
But now we can also
see, well, there

101
00:05:52,485 --> 00:05:56,120
are other important energies.

102
00:05:56,120 --> 00:05:59,350
One energy, and now
I bring in the c

103
00:05:59,350 --> 00:06:01,570
just because I want to
compare two energies which

104
00:06:01,570 --> 00:06:05,820
include the speed of light, the
rest energy of the electron.

105
00:06:05,820 --> 00:06:12,820
Or a very fundamental unit
of length is h-bar over mc.

106
00:06:12,820 --> 00:06:16,460
Which is the
Compton wavelengths.

107
00:06:16,460 --> 00:06:19,230
So that's lambda Compton,
the Compton wavelengths

108
00:06:19,230 --> 00:06:21,540
of the electron.

109
00:06:21,540 --> 00:06:26,320
And, well, if we
try to figure out

110
00:06:26,320 --> 00:06:30,440
what is the ratio of the
atomic unit of lengths.

111
00:06:30,440 --> 00:06:37,310
And the Compton radius,
we have to multiply

112
00:06:37,310 --> 00:06:41,420
with a dimensionless unit
which is hc over e squared.

113
00:06:41,420 --> 00:06:46,430
Or if I take the reverse,
e squared over hc, h-bar c.

114
00:06:46,430 --> 00:06:52,130
And similarly, here the
dimensionless quantity

115
00:06:52,130 --> 00:06:55,110
to multiply is that.

116
00:06:55,110 --> 00:07:02,690
So what do we find now here
is we find that what we get

117
00:07:02,690 --> 00:07:05,944
is a quantity which
I want to call alpha,

118
00:07:05,944 --> 00:07:07,110
the fine-structure constant.

119
00:07:09,620 --> 00:07:17,210
And what I find here is the same
constant, alpha to the minus 1,

120
00:07:17,210 --> 00:07:18,795
times the Compton wavelengths.

121
00:07:24,092 --> 00:07:26,470
Let me discuss the
fine-structure constant

122
00:07:26,470 --> 00:07:30,750
in a second, but there's
still two more atomic units

123
00:07:30,750 --> 00:07:32,330
we want to discuss.

124
00:07:32,330 --> 00:07:36,280
There is the velocity and
there is the electric field.

125
00:07:40,910 --> 00:07:44,290
I can simply get the
velocity by saying, well,

126
00:07:44,290 --> 00:07:47,720
the velocity enters the
kinetic energy in mv square.

127
00:07:47,720 --> 00:07:51,670
And if I said mv square, I want
to skip all factors of unity.

128
00:07:51,670 --> 00:07:54,530
So that's not one half,
it's just mv square.

129
00:07:54,530 --> 00:08:02,280
And if I said mv square
equal to one Hartree,

130
00:08:02,280 --> 00:08:07,260
then I find that the
atomic unit of velocity

131
00:08:07,260 --> 00:08:09,080
is e squared over h-bar.

132
00:08:12,350 --> 00:08:18,960
But this turns out
to be alpha times c.

133
00:08:18,960 --> 00:08:22,780
So again, what we find is alpha,
the fine-structure constant.

134
00:08:26,690 --> 00:08:30,630
Alpha is, of course,
dimensionless.

135
00:08:30,630 --> 00:08:37,809
It is 1/137.

136
00:08:37,809 --> 00:08:45,300
So therefore, we see that if the
velocity-- and this is actually

137
00:08:45,300 --> 00:08:48,200
the orbital velocity
of the 1s electron.

138
00:08:48,200 --> 00:08:52,190
If this is alpha
times c that confirms

139
00:08:52,190 --> 00:08:56,379
that the electron
non-relativistic.

140
00:08:56,379 --> 00:08:58,337
We have solved the
non-relativistic Schrodinger

141
00:08:58,337 --> 00:09:00,050
equation for it.

142
00:09:00,050 --> 00:09:05,800
And consistently, we find that
the velocity of the electron

143
00:09:05,800 --> 00:09:11,072
is 1/137 of the speed of light.

144
00:09:11,072 --> 00:09:12,280
It's actually physics trivia.

145
00:09:12,280 --> 00:09:14,820
If somebody asks you
how fast is the electron

146
00:09:14,820 --> 00:09:19,640
in the hydrogen atom, about
1% of the speed of light.

147
00:09:19,640 --> 00:09:23,825
Of course, if you had solved
the non-relativistic Schrodinger

148
00:09:23,825 --> 00:09:25,170
equation.

149
00:09:25,170 --> 00:09:31,050
And you solve it for, let's
say, a naked uranium nucleus

150
00:09:31,050 --> 00:09:37,500
where z, the charge
of the nucleus, is 92,

151
00:09:37,500 --> 00:09:42,390
then you find that this
fine-structure constant times

152
00:09:42,390 --> 00:09:43,930
z is on the order of unity.

153
00:09:43,930 --> 00:09:48,490
You would find that the electron
moves at the speed of light.

154
00:09:48,490 --> 00:09:51,190
And then you realize, gosh,
I've solved the wrong equation.

155
00:09:51,190 --> 00:09:53,230
Because a non-relativistic
Schrodinger equation

156
00:09:53,230 --> 00:09:54,885
when the solution
is that something

157
00:09:54,885 --> 00:09:56,620
moves at the speed
of light, I'd better

158
00:09:56,620 --> 00:09:59,270
start with a different equation.

159
00:09:59,270 --> 00:10:01,590
But here, we find
we are consistent.

160
00:10:01,590 --> 00:10:11,830
An electron in the hydrogen
atom for low nuclear charges

161
00:10:11,830 --> 00:10:16,650
of a few hydrogen, helium,
and so on is non-relativistic.

162
00:10:16,650 --> 00:10:18,720
So let's just finish that.

163
00:10:18,720 --> 00:10:22,610
The electric field
is the electric field

164
00:10:22,610 --> 00:10:26,360
felt by the electron
which orbits

165
00:10:26,360 --> 00:10:30,210
the nucleus on the 1s shell.

166
00:10:30,210 --> 00:10:38,900
And this is 5.1 times 10 to
the 9 volts per centimeter.

167
00:10:38,900 --> 00:10:43,080
So everything I've constructed
here out of the three

168
00:10:43,080 --> 00:10:45,990
fundamental
constants, e, m, and h

169
00:10:45,990 --> 00:10:48,720
are typical for the 1s
electron for the ground

170
00:10:48,720 --> 00:10:50,280
state of hydrogen.

171
00:10:50,280 --> 00:10:53,460
We got the typical
lengths, the Bohr radius,

172
00:10:53,460 --> 00:10:57,710
the typical energy, the
Hartree, the typical velocity,

173
00:10:57,710 --> 00:11:04,770
which doesn't have a name,
and the electric field

174
00:11:04,770 --> 00:11:08,490
experienced by this electron.

175
00:11:08,490 --> 00:11:11,955
So let's now talk about alpha.

176
00:11:14,590 --> 00:11:17,720
So alpha is dimensionless.

177
00:11:21,520 --> 00:11:25,990
It is, you know, if a constant
has dimension like h-bar,

178
00:11:25,990 --> 00:11:30,860
like c, actually the
value of it reflects

179
00:11:30,860 --> 00:11:32,550
our system of metrology.

180
00:11:32,550 --> 00:11:35,080
If you define the second
in a different way,

181
00:11:35,080 --> 00:11:36,790
h-bar will change.

182
00:11:36,790 --> 00:11:38,920
c will change.

183
00:11:38,920 --> 00:11:44,290
So a lot of constants are
not fundamental constants,

184
00:11:44,290 --> 00:11:46,910
being fundamental to
the physics at hand.

185
00:11:46,910 --> 00:11:49,300
They are more kind
of translating

186
00:11:49,300 --> 00:11:52,730
our metrological system
into the equations

187
00:11:52,730 --> 00:11:55,330
we use to describe our system.

188
00:11:55,330 --> 00:11:58,030
But if something
has no dimension,

189
00:11:58,030 --> 00:12:02,920
it is not related to a unit
like the kilogram or the second,

190
00:12:02,920 --> 00:12:07,000
it has really
fundamental importance.

191
00:12:07,000 --> 00:12:11,550
So, therefore, alpha is
the fundamental constant

192
00:12:11,550 --> 00:12:12,445
in atomic physics.

193
00:12:22,650 --> 00:12:27,660
And if you have a
fundamental constant,

194
00:12:27,660 --> 00:12:33,490
ultimately, there should
be a theory of everything

195
00:12:33,490 --> 00:12:39,680
which should ultimately
predict the value of alpha.

196
00:12:39,680 --> 00:12:49,880
Which, ultimately, we
predicted by a complete theory.

197
00:12:54,840 --> 00:13:03,900
So alpha, the fine-structure
constant, is smaller than 1.

198
00:13:03,900 --> 00:13:06,910
It's 1/137.

199
00:13:06,910 --> 00:13:10,840
And the fact that
alpha is smaller like 1

200
00:13:10,840 --> 00:13:17,230
is often phrased in these words.

201
00:13:17,230 --> 00:13:20,990
That since alpha is
much smaller than 1

202
00:13:20,990 --> 00:13:29,230
that this implies the
electromagnetic interactions

203
00:13:29,230 --> 00:13:29,750
are weak.

204
00:13:34,820 --> 00:13:36,960
OK, I want to explain that.

205
00:13:36,960 --> 00:13:39,240
I mean, I've heard
this many, many times.

206
00:13:39,240 --> 00:13:42,240
But what does it mean
interactions are weak?

207
00:13:42,240 --> 00:13:46,300
So let me give you sort
of my, in 90 seconds,

208
00:13:46,300 --> 00:13:47,840
sort of my spiel on it.

209
00:13:47,840 --> 00:13:50,735
Why does alpha mean that the
electromagnetic interactions

210
00:13:50,735 --> 00:13:52,570
are weak?

211
00:13:52,570 --> 00:13:56,190
Well, we have to compare the
electromagnetic interaction

212
00:13:56,190 --> 00:13:57,210
to something else.

213
00:13:57,210 --> 00:14:01,120
And the result will be for
this situation I create,

214
00:14:01,120 --> 00:14:03,010
the electromagnetic
interactions are weak.

215
00:14:13,070 --> 00:14:15,160
Electromagnetic interactions
of the Coulomb field

216
00:14:15,160 --> 00:14:17,410
gets, of course,
stronger, and stronger,

217
00:14:17,410 --> 00:14:20,960
and stronger the closer you
move two charges together.

218
00:14:20,960 --> 00:14:24,580
So what does it mean that the
Coulomb interaction is weak?

219
00:14:24,580 --> 00:14:27,750
Weaker compared to
strong interactions

220
00:14:27,750 --> 00:14:29,370
or other interactions.

221
00:14:29,370 --> 00:14:34,680
Well, let me try to
justify it as follows.

222
00:14:34,680 --> 00:14:41,210
We cannot go to epitary
small distances.

223
00:14:41,210 --> 00:14:42,870
We can do it in
classical physics,

224
00:14:42,870 --> 00:14:45,250
but not in quantum physics.

225
00:14:45,250 --> 00:14:47,390
So if you go to very
small distances,

226
00:14:47,390 --> 00:14:50,740
if you localize
particles very tightly,

227
00:14:50,740 --> 00:14:52,520
they have a lot of
momentum uncertainty.

228
00:14:52,520 --> 00:14:55,430
The momentum uncertainty
means energy uncertainty.

229
00:14:55,430 --> 00:14:57,650
And the energy
uncertainty may mean

230
00:14:57,650 --> 00:15:01,040
that we can create electrons
and positron pairs.

231
00:15:01,040 --> 00:15:04,230
So the moment we have
an energy uncertainty,

232
00:15:04,230 --> 00:15:08,390
by our definition of bringing
two particles close together,

233
00:15:08,390 --> 00:15:12,130
and this energy uncertainty
is larger than the rest mass,

234
00:15:12,130 --> 00:15:13,380
we have to be very careful.

235
00:15:13,380 --> 00:15:16,630
We can no longer use a
single particle description,

236
00:15:16,630 --> 00:15:18,980
or our concept of
single particle physics

237
00:15:18,980 --> 00:15:22,560
breaks down if you have
seen the particles prepared

238
00:15:22,560 --> 00:15:25,210
with an energy uncertainty,
which would spontaneously

239
00:15:25,210 --> 00:15:27,100
create more particles.

240
00:15:27,100 --> 00:15:35,740
So therefore, let me
postulate that our picture

241
00:15:35,740 --> 00:15:38,050
how we think about
those interactions,

242
00:15:38,050 --> 00:15:41,390
arranging two charges
and writing down

243
00:15:41,390 --> 00:15:43,430
what the Coulomb energy is.

244
00:15:43,430 --> 00:15:57,730
That if energy uncertainties
become on the order of the rest

245
00:15:57,730 --> 00:16:09,470
energy, then the concept of
single particles breaks down.

246
00:16:16,500 --> 00:16:18,490
Of course, that's not
the end of physics.

247
00:16:18,490 --> 00:16:22,420
You need now a field theory
for particles where particles

248
00:16:22,420 --> 00:16:23,923
are just excitations
in your field.

249
00:16:26,620 --> 00:16:28,290
But here in atomic
physics, you want

250
00:16:28,290 --> 00:16:31,550
to describe an electron
bound to a nucleus.

251
00:16:31,550 --> 00:16:33,870
And we want to use
those concepts.

252
00:16:33,870 --> 00:16:37,650
So let's just sort of say,
what does it mean with delta e

253
00:16:37,650 --> 00:16:39,680
is mc squared?

254
00:16:39,680 --> 00:16:42,620
Well, that means the
momentum uncertainty

255
00:16:42,620 --> 00:16:45,140
is on the order of mc.

256
00:16:45,140 --> 00:16:48,250
And with this
momentum uncertainty,

257
00:16:48,250 --> 00:16:56,140
I can localize particles
to within h-bar over mc.

258
00:16:56,140 --> 00:16:58,685
And this just turns
out to be the Compton

259
00:16:58,685 --> 00:17:01,490
radius of the electron.

260
00:17:01,490 --> 00:17:03,960
So, therefore, I
should be careful

261
00:17:03,960 --> 00:17:07,069
when I talk about the Coulomb
energy between particles

262
00:17:07,069 --> 00:17:10,030
if I would go closer
than the Compton radius.

263
00:17:10,030 --> 00:17:15,920
So, therefore, let me compare
now the Coulomb interaction

264
00:17:15,920 --> 00:17:20,109
at the Compton radius
to something else.

265
00:17:20,109 --> 00:17:24,170
So these Coulomb energy is
e squared over the radius.

266
00:17:24,170 --> 00:17:26,270
The Compton radius.

267
00:17:26,270 --> 00:17:34,810
And this turns out to be
e squared, mc over h-bar.

268
00:17:34,810 --> 00:17:39,370
So unless I want to
get into quantum field

269
00:17:39,370 --> 00:17:49,230
theory of particles, before I
need a different description,

270
00:17:49,230 --> 00:17:52,210
the strongest
Coulomb interaction

271
00:17:52,210 --> 00:18:04,550
I can create by putting two
particles at the Compton radius

272
00:18:04,550 --> 00:18:06,080
is that.

273
00:18:06,080 --> 00:18:09,780
And I can now compare
this Coulomb interaction,

274
00:18:09,780 --> 00:18:16,020
the Coulomb energy, to the
rest energy, mc squared.

275
00:18:16,020 --> 00:18:22,660
And, well, if I take what I had
above, e squared mc over h-bar,

276
00:18:22,660 --> 00:18:25,150
I divide by mc squared.

277
00:18:25,150 --> 00:18:29,200
I find that e square, that
the result, the ratio,

278
00:18:29,200 --> 00:18:33,880
is e squared over h-bar c.

279
00:18:33,880 --> 00:18:37,500
And this is just alpha.

280
00:18:37,500 --> 00:18:40,750
So in other words, what
I've shown to you, if you

281
00:18:40,750 --> 00:18:43,450
try to bring two
charges as close as

282
00:18:43,450 --> 00:18:47,900
possible before spontaneous
pair production sets in, then

283
00:18:47,900 --> 00:18:51,410
you find that the
Coulomb energy is not

284
00:18:51,410 --> 00:18:53,390
in the dominant
energy in the system.

285
00:18:53,390 --> 00:18:56,000
The dominant energy
is the rest energy,

286
00:18:56,000 --> 00:19:01,060
is the mass of the
electron itself.

287
00:19:01,060 --> 00:19:04,820
And the ratio of those
two energies at this point

288
00:19:04,820 --> 00:19:07,420
is, of course, completely
independent of what

289
00:19:07,420 --> 00:19:10,670
metrological system you use
for energy lengths and such.

290
00:19:10,670 --> 00:19:13,390
It's really something which
says something fundamental

291
00:19:13,390 --> 00:19:15,720
about the nature of interaction.

292
00:19:15,720 --> 00:19:22,650
And what I just presented to
you leads to the statement

293
00:19:22,650 --> 00:19:25,870
that the Coulomb interaction,
the electromagnetic

294
00:19:25,870 --> 00:19:27,160
interaction, is weak.

295
00:19:27,160 --> 00:19:29,475
Because the
fine-structure constant

296
00:19:29,475 --> 00:19:30,600
is much smaller than unity.

297
00:19:35,280 --> 00:19:42,680
Of course, if you use a nucleus
of uranium, naked uranium,

298
00:19:42,680 --> 00:19:47,610
and people have ion traps where
they create uranium 92 plus

299
00:19:47,610 --> 00:19:49,880
and then they add
an electron, you're

300
00:19:49,880 --> 00:19:51,970
really studying very
interesting physics.

301
00:19:51,970 --> 00:19:54,920
You're studying the physics
of an electron for which

302
00:19:54,920 --> 00:19:57,130
the effective
fine-structure constant is

303
00:19:57,130 --> 00:19:58,640
on the order of unity.

304
00:19:58,640 --> 00:20:00,640
And that's why people are
very interested in it.

305
00:20:00,640 --> 00:20:04,210
And that's one area
of current research.

306
00:20:09,070 --> 00:20:09,690
Any questions?

307
00:20:15,360 --> 00:20:20,330
OK, so that's our little
excursion about units.

308
00:20:23,990 --> 00:20:32,010
Let's talk now about
some general properties

309
00:20:32,010 --> 00:20:36,485
of one electron
atoms with cores.

310
00:20:43,380 --> 00:20:46,820
A lot of research in our field
is done with alkali atoms.

311
00:20:46,820 --> 00:20:52,070
Alkali atoms are not
the hydrogen atom.

312
00:20:52,070 --> 00:20:54,450
But they have one
outer electron.

313
00:20:54,450 --> 00:20:56,500
So they are hydrogen-like.

314
00:20:56,500 --> 00:21:01,230
And now we want to
sort of figure out

315
00:21:01,230 --> 00:21:03,800
what is the main
difference because

316
00:21:03,800 --> 00:21:06,650
for one outer electron
in rubidium and sodium.

317
00:21:06,650 --> 00:21:11,030
And the electron in hydrogen.

318
00:21:11,030 --> 00:21:15,740
Well, if we have an electron.

319
00:21:15,740 --> 00:21:19,560
And this electron orbits around.

320
00:21:19,560 --> 00:21:29,700
In the alkali atom,
there is an ionic core

321
00:21:29,700 --> 00:21:34,670
which has a charge of a z plus.

322
00:21:34,670 --> 00:21:38,370
But then in this
sort of compact core,

323
00:21:38,370 --> 00:21:53,630
there are also c
minus 1 electron.

324
00:21:53,630 --> 00:21:56,830
So the electron, the
outer electron pretty much

325
00:21:56,830 --> 00:22:00,760
feels the electric field
of a single charge,

326
00:22:00,760 --> 00:22:04,170
but there is a the of the core.

327
00:22:04,170 --> 00:22:09,960
So what I want to
discuss with you is now

328
00:22:09,960 --> 00:22:15,410
what is the leading correction
to the properties of this atom?

329
00:22:15,410 --> 00:22:17,840
What is the leading correction
to the hydrogen-like wave

330
00:22:17,840 --> 00:22:22,356
function due to the fact that
we have an ionic core and not

331
00:22:22,356 --> 00:22:22,855
a proton.

332
00:22:25,700 --> 00:22:33,850
So for hydrogen, we would
have the Rydberg formula.

333
00:22:33,850 --> 00:22:47,520
that for principal quantum
number n, the energy is-- oh,

334
00:22:47,520 --> 00:22:48,310
just one second.

335
00:22:57,460 --> 00:22:57,960
Yes.

336
00:23:12,480 --> 00:23:20,410
OK, the hydrogenic
energy would be z squared

337
00:23:20,410 --> 00:23:25,375
times the Rydberg constant
divided by n squared.

338
00:23:48,821 --> 00:23:52,980
I'm confused about the factor
of z squared, which I clearly

339
00:23:52,980 --> 00:23:53,750
have in my notes.

340
00:24:09,565 --> 00:24:12,270
I have to read up
something about it.

341
00:24:12,270 --> 00:24:15,840
Let me take it out here.

342
00:24:15,840 --> 00:24:18,500
You know, we have a charge of z.

343
00:24:18,500 --> 00:24:20,940
But what I'm talking
about is one electron

344
00:24:20,940 --> 00:24:23,030
which is very far
away from the nucleus.

345
00:24:23,030 --> 00:24:28,200
And this one electron fields,
in effect, only a single charge.

346
00:24:28,200 --> 00:24:30,696
And so, ultimately,
you would expect

347
00:24:30,696 --> 00:24:32,070
that the Rydberg
spectrum, if you

348
00:24:32,070 --> 00:24:34,410
go to higher and higher
end, would actually

349
00:24:34,410 --> 00:24:38,600
converge to the Rydberg spectrum
of hydrogen which would not

350
00:24:38,600 --> 00:24:40,130
have the factor of z squared.

351
00:24:42,890 --> 00:24:46,830
I hope I'm not
overlooking something,

352
00:24:46,830 --> 00:24:49,920
but I'm just correcting
my notes on the fly.

353
00:24:49,920 --> 00:24:53,710
So this would simply be
what an electron would

354
00:24:53,710 --> 00:24:56,420
do in the Coulomb field
of a single charge.

355
00:24:56,420 --> 00:25:01,770
And the question is, what
is the leading correction

356
00:25:01,770 --> 00:25:03,900
to this formula?

357
00:25:03,900 --> 00:25:07,670
And I want to ask it
as a clicker question.

358
00:25:07,670 --> 00:25:18,540
So that's the effect, that
we have an ionic core.

359
00:25:18,540 --> 00:25:22,020
Does it make a constant
offset to the binding energy?

360
00:25:25,000 --> 00:25:31,805
Or does it make the
correction which,

361
00:25:31,805 --> 00:25:33,590
of course, is 1 over n squared.

362
00:25:33,590 --> 00:25:37,540
This would just scale
the overall spectrum.

363
00:25:37,540 --> 00:25:40,275
It would actually lead to
a modified rescaled Rydberg

364
00:25:40,275 --> 00:25:40,775
constant.

365
00:25:43,410 --> 00:25:54,940
Or is the correction
higher order in n?

366
00:25:54,940 --> 00:26:02,060
Or finally, is it
a correction which

367
00:26:02,060 --> 00:26:05,812
changes the effective
principle quantum number n?

368
00:26:09,590 --> 00:26:11,533
Maybe you know the
answer or maybe you

369
00:26:11,533 --> 00:26:13,875
want to try to guess the answer.

370
00:26:13,875 --> 00:26:15,125
So these are the four choices.

371
00:26:32,130 --> 00:26:34,100
Of course delta has
different units.

372
00:26:34,100 --> 00:26:39,590
I just used the same symbol
for the correction term.

373
00:26:39,590 --> 00:26:44,490
But depending whether it
appears it has units of energy

374
00:26:44,490 --> 00:26:47,580
or if it appears in
the denominator with n,

375
00:26:47,580 --> 00:26:51,648
it is dimensionless.

376
00:26:51,648 --> 00:26:55,676
AUDIENCE: Aren't these questions
also C and D [INAUDIBLE]?

377
00:26:55,676 --> 00:26:57,300
PROFESSOR: Yes, this
is the first thing

378
00:26:57,300 --> 00:27:01,820
I wanted to tell you, that
answers C and D are the same.

379
00:27:01,820 --> 00:27:04,380
Because if we assume
delta is small

380
00:27:04,380 --> 00:27:07,220
and you do a Taylor
expansion of the denominator,

381
00:27:07,220 --> 00:27:09,760
you just get this.

382
00:27:09,760 --> 00:27:11,670
So c and d are
actually equivalent.

383
00:27:16,320 --> 00:27:21,160
And so if I add
up 20 people voted

384
00:27:21,160 --> 00:27:26,430
for c and d, which
is equivalent.

385
00:27:26,430 --> 00:27:31,550
So this is equivalent
by Taylor expansion.

386
00:27:37,980 --> 00:27:43,990
And, indeed, this
is also-- this time

387
00:27:43,990 --> 00:27:46,080
there's two correct answers.

388
00:27:46,080 --> 00:27:48,470
So let me quickly derive it.

389
00:27:48,470 --> 00:27:49,840
The derivation is short.

390
00:27:49,840 --> 00:27:53,270
And it adds some insight.

391
00:27:53,270 --> 00:27:56,005
We want to do
perturbation theory.

392
00:27:58,750 --> 00:28:08,120
And in perturbation
theory, we simply take

393
00:28:08,120 --> 00:28:12,660
the wave function of the
simplified Hamiltonian

394
00:28:12,660 --> 00:28:14,495
and ask what is the
energy correction

395
00:28:14,495 --> 00:28:19,840
due to the fact that we
have a finite core size?

396
00:28:19,840 --> 00:28:29,230
Since the finite core is near
r equals 0 at the origin,

397
00:28:29,230 --> 00:28:33,230
we only need the scaling
of the wave function close

398
00:28:33,230 --> 00:28:35,410
to the origin.

399
00:28:35,410 --> 00:28:38,840
And this is r to the l.

400
00:28:38,840 --> 00:28:45,120
And we have n to the power
3/2, as we discussed last week.

401
00:28:45,120 --> 00:28:47,900
So our Hamiltonian is
now the Hamiltonian

402
00:28:47,900 --> 00:28:53,260
of the hydrogen atom
plus a perturbation term.

403
00:28:53,260 --> 00:28:56,520
And the perturbation
term is the derivation

404
00:28:56,520 --> 00:29:01,480
of the potentially experienced
by the outer electron,

405
00:29:01,480 --> 00:29:05,960
the deviation from a
pure Coulomb potential.

406
00:29:05,960 --> 00:29:15,020
So the energy correction,
en, is the expectation value

407
00:29:15,020 --> 00:29:22,860
of the hydrogenic wave
function with the perturbation

408
00:29:22,860 --> 00:29:23,930
Hamiltonian.

409
00:29:23,930 --> 00:29:26,550
And the only thing we have to
know about the perturbation

410
00:29:26,550 --> 00:29:31,600
Hamiltonian is that
this Hamiltonian is

411
00:29:31,600 --> 00:29:38,500
localized around the origin.

412
00:29:38,500 --> 00:29:43,610
And then we immediately
find because of the scaling

413
00:29:43,610 --> 00:29:49,590
of the wave function with
n that this is 1/n cubed.

414
00:29:52,550 --> 00:29:58,320
And it's proportional.

415
00:29:58,320 --> 00:30:04,990
And by factoring out
the Rydberg constant,

416
00:30:04,990 --> 00:30:08,480
I can parameterize
this matrix element

417
00:30:08,480 --> 00:30:14,840
with a quantity, delta l,
which is dimensionless.

418
00:30:14,840 --> 00:30:20,190
So that means that
the binding energy

419
00:30:20,190 --> 00:30:30,140
is the hydrogenic binding
energy plus this correction.

420
00:30:30,140 --> 00:30:33,510
And then to leading
order in delta

421
00:30:33,510 --> 00:30:40,980
l, it's identical
to this result.

422
00:30:40,980 --> 00:30:45,230
And this parameter,
delta l, which

423
00:30:45,230 --> 00:30:50,080
is characterizing a whole
Rydberg series for all n

424
00:30:50,080 --> 00:30:53,940
values for given l, this is
called the quantum defect.

425
00:30:56,730 --> 00:30:59,260
So people realized early
on in the early days

426
00:30:59,260 --> 00:31:01,600
of quantum mechanics
before they understood it

427
00:31:01,600 --> 00:31:05,480
that the spectrum of
many atoms followed

428
00:31:05,480 --> 00:31:09,790
a formula which was not
1/n squared is hydrogen.

429
00:31:09,790 --> 00:31:14,070
It was 1n minus delta l squared.

430
00:31:14,070 --> 00:31:15,470
They didn't understand it.

431
00:31:15,470 --> 00:31:17,490
But this is, of course,
now the derivation.

432
00:31:26,620 --> 00:31:33,560
There are many other
derivations which

433
00:31:33,560 --> 00:31:38,040
you may enjoy reading about
it in our atomic physics wiki.

434
00:31:38,040 --> 00:31:42,340
There's a derivation using
the semi-classical approach

435
00:31:42,340 --> 00:31:44,810
and using the very
nice physical picture

436
00:31:44,810 --> 00:31:50,130
that an electron, when it
comes close to the core,

437
00:31:50,130 --> 00:31:52,590
experiences scattering
phase shift.

438
00:31:52,590 --> 00:31:55,020
And this scattering phase
shift is directly related

439
00:31:55,020 --> 00:31:56,850
to the quantum defect.

440
00:31:56,850 --> 00:32:01,020
Or you can make a
model Hamiltonian

441
00:32:01,020 --> 00:32:06,930
which is exactly solvable where
the perturbation Hamiltonian is

442
00:32:06,930 --> 00:32:10,580
not completely
localized at the core.

443
00:32:10,580 --> 00:32:14,100
But it's proportional
to 1/r squared.

444
00:32:14,100 --> 00:32:16,230
And then you can
exactly solve it

445
00:32:16,230 --> 00:32:18,200
because you have
already a term which

446
00:32:18,200 --> 00:32:20,860
is 1/r squared in your
Schrodinger equation.

447
00:32:20,860 --> 00:32:22,730
Which is the centrifugal term.

448
00:32:22,730 --> 00:32:25,790
So therefore, the
perturbation is only

449
00:32:25,790 --> 00:32:28,580
redefining the centrifugal term.

450
00:32:28,580 --> 00:32:31,246
It's redefining what l is.

451
00:32:31,246 --> 00:32:32,995
And eventually, you
can solve it directly.

452
00:32:39,135 --> 00:32:39,635
Questions?

453
00:32:46,180 --> 00:32:46,840
OK.

454
00:32:46,840 --> 00:32:49,840
Then let me spend
five to 10 minutes

455
00:32:49,840 --> 00:32:52,160
on spectroscopic notation.

456
00:33:06,700 --> 00:33:08,790
So the next five
or 10 minutes, how

457
00:33:08,790 --> 00:33:14,570
we describe the
configuration of an atom,

458
00:33:14,570 --> 00:33:17,080
well, I don't particularly
like to teach it.

459
00:33:17,080 --> 00:33:20,970
Because it's more nomenclature
about old-fashioned symbols.

460
00:33:20,970 --> 00:33:24,070
On the other hand, if
you're working with atoms,

461
00:33:24,070 --> 00:33:26,960
you have to learn the language
how to describe atoms.

462
00:33:26,960 --> 00:33:31,630
And I also know that
an appreciable fraction

463
00:33:31,630 --> 00:33:34,840
of oral exams there will be
one person in the community

464
00:33:34,840 --> 00:33:36,870
and say, what is
your favorite atom?

465
00:33:36,870 --> 00:33:40,550
And what is the
configuration of your atom?

466
00:33:40,550 --> 00:33:43,880
So it's something if
you're an atomic physicist

467
00:33:43,880 --> 00:33:46,990
you're supposed to know.

468
00:33:46,990 --> 00:33:52,270
So the spectroscopic
notation, the term designation

469
00:33:52,270 --> 00:33:58,170
focuses on the fact that if
you have an isolated atom,

470
00:33:58,170 --> 00:34:02,850
we have angular
momentum conservation.

471
00:34:02,850 --> 00:34:10,927
And so we have at least
two quantum numbers.

472
00:34:10,927 --> 00:34:13,010
Which are sometimes also
two good quantum numbers.

473
00:34:13,010 --> 00:34:14,940
We have some approximate
quantum numbers

474
00:34:14,940 --> 00:34:19,540
where we have additional terms
which break certain symmetries.

475
00:34:19,540 --> 00:34:24,179
But an isolated atom
lives in isotropic space.

476
00:34:24,179 --> 00:34:30,070
The total angular momentum
of this atom is conserved.

477
00:34:30,070 --> 00:34:31,830
It's absolutely conserved.

478
00:34:31,830 --> 00:34:34,150
It's absolutely
good quantum number.

479
00:34:34,150 --> 00:34:37,739
And the good quantum numbers is
the total angular momentum, j,

480
00:34:37,739 --> 00:34:39,510
and its protection energy.

481
00:34:43,989 --> 00:34:54,040
So in the language of atomic
physics, we call j a level.

482
00:34:54,040 --> 00:34:55,800
It's different from states.

483
00:34:55,800 --> 00:35:05,050
So one level has now 2j
plus 1 sub-levels or states.

484
00:35:08,760 --> 00:35:10,850
So usually when we
talk about a level,

485
00:35:10,850 --> 00:35:14,950
we assume the level
has degeneracies

486
00:35:14,950 --> 00:35:17,340
because there is the
Mj quantum number.

487
00:35:21,060 --> 00:35:27,030
So j, you're talking about
electronic structure.

488
00:35:27,030 --> 00:35:36,700
So j can have, when we
have an isolated atom,

489
00:35:36,700 --> 00:35:40,275
can have contributions
from several electrons.

490
00:35:51,040 --> 00:35:53,930
It can have contributions
and these electrons

491
00:35:53,930 --> 00:36:04,990
can contribute through
spin, s, and orbital angular

492
00:36:04,990 --> 00:36:06,128
momentum, l.

493
00:36:13,950 --> 00:36:17,990
In many situations,
especially with alkali atoms,

494
00:36:17,990 --> 00:36:21,050
the inner core is
completely field shell.

495
00:36:21,050 --> 00:36:23,780
There is no s, no l
from the inner electrons

496
00:36:23,780 --> 00:36:24,857
which contribute.

497
00:36:24,857 --> 00:36:26,940
And all the contribution
with the angular momentum

498
00:36:26,940 --> 00:36:28,190
comes from the outer electron.

499
00:36:31,220 --> 00:36:38,420
Especially for
the lighter atoms.

500
00:36:38,420 --> 00:36:41,800
The non-relativistic atoms.

501
00:36:41,800 --> 00:36:49,390
The different electrons
undergo ls copying.

502
00:36:49,390 --> 00:36:53,080
In other words, if you
have multiple electrons,

503
00:36:53,080 --> 00:36:58,325
their orbital angular
momentum couple up

504
00:36:58,325 --> 00:37:01,760
to the total orbital
angular momentum, l.

505
00:37:01,760 --> 00:37:08,290
And all the spins couple
up to the total spin, s.

506
00:37:08,290 --> 00:37:14,120
So therefore, before we
introduce spin orbit copying,

507
00:37:14,120 --> 00:37:16,070
l is a good quantum number.

508
00:37:16,070 --> 00:37:17,800
s is a good quantum number.

509
00:37:17,800 --> 00:37:22,500
And then they couple to
a good quantum number, j.

510
00:37:22,500 --> 00:37:26,500
Of course, once l and s
couple to j, orbital and spin

511
00:37:26,500 --> 00:37:31,260
angular momentum precess around
the total angular momentum.

512
00:37:31,260 --> 00:37:32,910
Anyway, I just
want to say when I

513
00:37:32,910 --> 00:37:36,326
talk about j, what are
possible ingredients?

514
00:37:40,030 --> 00:37:45,460
So let's assume we have an
atom which has total angular

515
00:37:45,460 --> 00:37:47,520
momentum, j.

516
00:37:47,520 --> 00:37:52,337
And which is the sum of orbital
angular momentum and spin

517
00:37:52,337 --> 00:37:53,045
angular momentum.

518
00:37:55,800 --> 00:38:10,020
And then in this case, we
use a term designation.

519
00:38:10,020 --> 00:38:21,250
A level is designated
by a term which

520
00:38:21,250 --> 00:38:28,030
is written as l, the value
of orbital angular momentum.

521
00:38:28,030 --> 00:38:34,420
The spin multiplicity, 2s
plus 1 is an upper left index.

522
00:38:34,420 --> 00:38:36,780
And a lower right index is j.

523
00:38:41,130 --> 00:38:50,190
And of course if l is 0,
we use the letter s, p, d.

524
00:38:50,190 --> 00:38:54,000
This is sort of the
historic letter designation

525
00:38:54,000 --> 00:38:56,680
for l equals 0, 1 and 2.

526
00:39:04,140 --> 00:39:07,040
So in other words, if you have
an atom where the total angular

527
00:39:07,040 --> 00:39:10,310
momentum is composed of orbital
angular momentum and spin,

528
00:39:10,310 --> 00:39:12,310
you can always
write this symbol.

529
00:39:12,310 --> 00:39:15,700
And this symbol is
the term designation

530
00:39:15,700 --> 00:39:20,190
which characterizes the state,
the ground state or an excited

531
00:39:20,190 --> 00:39:21,250
state of your atom.

532
00:39:27,130 --> 00:39:49,270
If you have the hydrogenic
atom, often you precede the term

533
00:39:49,270 --> 00:39:58,000
by the principal
quantum number, n.

534
00:40:01,910 --> 00:40:05,920
So let me give you an example.

535
00:40:05,920 --> 00:40:15,060
If you have the sodium atom, the
outer electron has n equals 3.

536
00:40:15,060 --> 00:40:18,670
It has 0 orbital
angular momentum.

537
00:40:18,670 --> 00:40:20,420
It has spin 1/2.

538
00:40:20,420 --> 00:40:23,360
And 2s plus 1 is 2.

539
00:40:23,360 --> 00:40:30,710
And the total angular
momentum is 1/2.

540
00:40:30,710 --> 00:40:34,860
If you go to the first excited
state, you're still in n

541
00:40:34,860 --> 00:40:36,080
equals 3.

542
00:40:36,080 --> 00:40:38,970
But you have promoted the
electron from an s state

543
00:40:38,970 --> 00:40:40,690
to p state.

544
00:40:40,690 --> 00:40:43,260
So, therefore, the
orbital angular momentum

545
00:40:43,260 --> 00:40:47,330
is now 1 designated by p.

546
00:40:47,330 --> 00:40:50,190
The spin is still spin 1/2.

547
00:40:50,190 --> 00:40:54,780
But now orbital angular
momentum of 1 and spin 1/2

548
00:40:54,780 --> 00:40:58,450
can form a total
angular momentum

549
00:40:58,450 --> 00:41:02,915
which can either be 1/2 or 3/2.

550
00:41:07,240 --> 00:41:14,100
So if you're asked what is the
state you prepare your atom in,

551
00:41:14,100 --> 00:41:18,520
you would give it a
symbol 3 doublet p 1/2.

552
00:41:18,520 --> 00:41:20,220
And I've explained
to you what it means.

553
00:41:23,920 --> 00:41:26,090
There is one addition.

554
00:41:26,090 --> 00:41:36,120
And sometimes you want
to not just mention

555
00:41:36,120 --> 00:41:39,570
what is the principle quantum
number of the outer electrons.

556
00:41:39,570 --> 00:41:43,675
Sometimes you want to specify
the whole configuration.

557
00:41:54,760 --> 00:41:59,570
So this would mean you want to
sort of build up the electron

558
00:41:59,570 --> 00:42:04,750
shell and say that I have
two electrons in 1s, two

559
00:42:04,750 --> 00:42:11,280
electrons in 2s, one
electron in 2p, and so on.

560
00:42:11,280 --> 00:42:18,910
So you use, I think this
would now be beryllium atom?

561
00:42:23,400 --> 00:42:25,870
1s.

562
00:42:25,870 --> 00:42:29,920
So 1s is hydrogen,
1s2 is helium.

563
00:42:29,920 --> 00:42:32,876
Then we go to lithium.

564
00:42:32,876 --> 00:42:34,727
AUDIENCE: [INAUDIBLE] is boron.

565
00:42:34,727 --> 00:42:35,810
PROFESSOR: It's boron, no?

566
00:42:35,810 --> 00:42:37,660
OK, so this would be boron.

567
00:42:37,660 --> 00:42:42,010
So what we use is here
we use the products.

568
00:42:42,010 --> 00:42:52,120
We use products of
symbols n, l, m.

569
00:42:52,120 --> 00:42:56,600
So to come back to
the example of sodium,

570
00:42:56,600 --> 00:43:00,220
so sodium is filled up.

571
00:43:00,220 --> 00:43:03,690
As in the first shells.

572
00:43:03,690 --> 00:43:05,840
So it's 2p6.

573
00:43:05,840 --> 00:43:10,230
And then we have one electron,
the outer electron in 3s.

574
00:43:14,350 --> 00:43:17,470
However, let me point
out that this way

575
00:43:17,470 --> 00:43:20,610
to specify the
configuration strongly

576
00:43:20,610 --> 00:43:23,420
depends on a hydrogenic model.

577
00:43:23,420 --> 00:43:27,380
It assumes that the electrons
are non-interacting.

578
00:43:27,380 --> 00:43:30,170
And is, therefore,
an approximation.

579
00:43:30,170 --> 00:43:33,290
In contrast, the
term designation

580
00:43:33,290 --> 00:43:38,342
with a total angular
momentum is always exact.

581
00:43:38,342 --> 00:43:40,050
Well, at least the
total angular momentum

582
00:43:40,050 --> 00:43:41,270
is an exact quantum number.

583
00:43:41,270 --> 00:43:43,500
Whereas the
configuration is based

584
00:43:43,500 --> 00:43:47,630
on the independent
electron approximation

585
00:43:47,630 --> 00:43:49,540
in hydrogenic orbits.

586
00:43:49,540 --> 00:43:52,680
And usually when
you have a real atom

587
00:43:52,680 --> 00:43:55,770
and you calculate
with high precision

588
00:43:55,770 --> 00:43:58,530
what the electronic
wave function is,

589
00:43:58,530 --> 00:44:02,640
you find actually that total
many-body wave function

590
00:44:02,640 --> 00:44:05,610
is a superposition of
many such configurations.

591
00:44:05,610 --> 00:44:08,780
But as long as one
configuration is dominant,

592
00:44:08,780 --> 00:44:11,000
this configuration
designation makes sense.

593
00:44:16,650 --> 00:44:17,330
Any questions?

594
00:44:21,910 --> 00:44:26,060
OK, so we come back
to the hydrogen atom

595
00:44:26,060 --> 00:44:32,370
when we discuss smaller
features of the energy levels.

596
00:44:32,370 --> 00:44:35,530
Fine structure, hyperfine
structure, and so on.

597
00:44:35,530 --> 00:44:39,890
But in our discussion of
electronic energies, that's all

598
00:44:39,890 --> 00:44:44,010
I want to say about
one electron atoms.

599
00:44:44,010 --> 00:44:49,710
So let's now proceed and
discuss the helium atom.

600
00:44:54,600 --> 00:45:00,580
So we want to understand now
what are the new effects when

601
00:45:00,580 --> 00:45:03,090
we have not only one
electron, but two electrons.

602
00:45:10,150 --> 00:45:11,030
And don't worry.

603
00:45:11,030 --> 00:45:13,700
We are not proceeding to
three, four, five electrons.

604
00:45:13,700 --> 00:45:16,340
I think to go from one
to two, we actually

605
00:45:16,340 --> 00:45:18,730
capture the most
important state.

606
00:45:18,730 --> 00:45:22,100
Namely, the interaction between
the two electrons and what

607
00:45:22,100 --> 00:45:23,345
the results of that is.

608
00:45:26,930 --> 00:45:31,550
For the helium atom, there
are some excellent treatments

609
00:45:31,550 --> 00:45:35,330
in standard textbooks
of quantum mechanics.

610
00:45:35,330 --> 00:45:42,410
One is the famous
quantum mechanic

611
00:45:42,410 --> 00:45:44,930
text by Cohen-Tannoudji et al.

612
00:45:47,690 --> 00:45:54,700
But also the text
of Gasiorowicz.

613
00:46:03,410 --> 00:46:07,870
The reason why I have added the
helium atom to the curriculum

614
00:46:07,870 --> 00:46:11,930
is because it is a
simple system where

615
00:46:11,930 --> 00:46:16,180
we can discuss singlet
and triplet states.

616
00:46:16,180 --> 00:46:19,220
And singlet and
triplet configuration

617
00:46:19,220 --> 00:46:24,810
is important for population
of inoptical lattices

618
00:46:24,810 --> 00:46:26,690
for quantum magnetism and such.

619
00:46:26,690 --> 00:46:29,510
Actually, you can say if
you have two electrons

620
00:46:29,510 --> 00:46:33,080
and they align in a triplet
state or a singlet state,

621
00:46:33,080 --> 00:46:34,910
one you can say
is ferromagnetic.

622
00:46:34,910 --> 00:46:37,430
The other one is paired
and antiferromagnetic.

623
00:46:37,430 --> 00:46:41,740
It's the simplest example
where we can discuss magnetism.

624
00:46:41,740 --> 00:46:45,130
So that's my motivation why
I want you to know something

625
00:46:45,130 --> 00:46:46,410
about the helium atom.

626
00:46:49,630 --> 00:46:58,810
So, therefore, let's now
discuss energy levels of helium.

627
00:47:09,740 --> 00:47:20,220
And let's just start with
the most basic model.

628
00:47:20,220 --> 00:47:24,400
The helium atom has two charges.

629
00:47:24,400 --> 00:47:28,740
So if you regard it
as a hydrogen problem

630
00:47:28,740 --> 00:47:36,700
and we put two electrons
into the 1s state,

631
00:47:36,700 --> 00:47:43,220
we would expect that based
on the hydrogenic model

632
00:47:43,220 --> 00:47:49,870
that the binding energy
of that is, per electron,

633
00:47:49,870 --> 00:47:53,450
is the Rydberg energy
as in hydrogen.

634
00:47:53,450 --> 00:47:58,520
But now we have to scale it with
z squared, the nuclear charge.

635
00:47:58,520 --> 00:48:00,890
And this gives us a factor of 4.

636
00:48:00,890 --> 00:48:05,530
So we would expect that per
electron, the binding energy

637
00:48:05,530 --> 00:48:10,095
in the more simple hydrogenic
model is 54 electron volt.

638
00:48:12,750 --> 00:48:22,770
So that would mean that the
binding energy of the ground

639
00:48:22,770 --> 00:48:29,220
state is minus
108 electron volt.

640
00:48:37,050 --> 00:48:45,420
However, the
experimental result is

641
00:48:45,420 --> 00:48:50,610
that it's only 79 electron volt.

642
00:48:50,610 --> 00:49:01,140
So we find that there is a
big discrepancy of 29 electron

643
00:49:01,140 --> 00:49:02,230
volt.

644
00:49:02,230 --> 00:49:05,390
Which is really huge.

645
00:49:05,390 --> 00:49:10,480
So what is responsible
for this big discrepancy?

646
00:49:10,480 --> 00:49:12,810
Well, what we have
neglected, of course,

647
00:49:12,810 --> 00:49:15,250
is the interaction
between the two electrons.

648
00:49:34,730 --> 00:49:39,810
So we can fix that
in the simplest way

649
00:49:39,810 --> 00:49:43,380
by keeping the wave function
from the hydrogenic model.

650
00:49:43,380 --> 00:49:48,920
But now calculating the
electronic energies,

651
00:49:48,920 --> 00:49:54,200
the electron-electron
energy, by using

652
00:49:54,200 --> 00:49:59,066
the electron-electron
interaction as a perturbation

653
00:49:59,066 --> 00:49:59,565
operator.

654
00:50:03,510 --> 00:50:08,530
So we still use as the wave
function for the ground

655
00:50:08,530 --> 00:50:15,890
state, electron 1
in the 1s state.

656
00:50:15,890 --> 00:50:20,460
So 1-0-0 is a designation for n,
l, m for the hydrogenic quantum

657
00:50:20,460 --> 00:50:21,550
numbers.

658
00:50:21,550 --> 00:50:24,640
And we assume that the
ground state is simply

659
00:50:24,640 --> 00:50:29,220
the product of two
electrons in the 1s state.

660
00:50:29,220 --> 00:50:34,280
So if I calculate for
this perturbation operator

661
00:50:34,280 --> 00:50:38,970
the expectation value
with this ground state,

662
00:50:38,970 --> 00:50:46,470
we find that there is an
energy correction which

663
00:50:46,470 --> 00:50:52,010
is 34 electron volt.

664
00:50:52,010 --> 00:51:00,680
So this removes most
of the discrepancy.

665
00:51:10,610 --> 00:51:16,390
You can improve on it by
a variation wave function

666
00:51:16,390 --> 00:51:20,270
if you use hydrogenic
eigenfunctions as your tile

667
00:51:20,270 --> 00:51:21,710
wave function.

668
00:51:21,710 --> 00:51:25,520
But you are now calculating
those hydrogenic wave functions

669
00:51:25,520 --> 00:51:31,640
not for nuclear charge 2, but
in nuclear charge c star, which

670
00:51:31,640 --> 00:51:33,900
you keep as a
variation of parameter.

671
00:51:33,900 --> 00:51:39,890
You find that you find
even better wave functions.

672
00:51:39,890 --> 00:51:42,830
And you can remove
2/3 of these remaining

673
00:51:42,830 --> 00:51:45,570
discrepancy of five
electron volts.

674
00:51:45,570 --> 00:51:49,879
This variational wave
function is left to you

675
00:51:49,879 --> 00:51:50,920
as a homework assignment.

676
00:52:05,540 --> 00:52:09,155
So where z is z
equals 2 is replaced

677
00:52:09,155 --> 00:52:11,530
by a variation of parameter.

678
00:52:11,530 --> 00:52:13,540
Anyway, that's all
I want to tell you

679
00:52:13,540 --> 00:52:17,550
about the ground
state of helium.

680
00:52:17,550 --> 00:52:20,280
It's pretty much
finding a wave function

681
00:52:20,280 --> 00:52:23,460
which correctly captures
the Coulomb energies.

682
00:52:23,460 --> 00:52:28,900
The interaction between
the two electrons.

683
00:52:28,900 --> 00:52:31,370
But what is much
more interesting,

684
00:52:31,370 --> 00:52:33,310
and this is what
I want to focus on

685
00:52:33,310 --> 00:52:35,370
for the rest of this
lecture, is what

686
00:52:35,370 --> 00:52:39,200
happens in the excited state.

687
00:52:39,200 --> 00:52:40,800
Before I do that,
let me just tell you

688
00:52:40,800 --> 00:52:42,540
what we just discussed.

689
00:52:42,540 --> 00:52:45,300
So we have to this
hydrogenic estimate

690
00:52:45,300 --> 00:52:51,840
and eventually the Coulomb
energy raises the energy level

691
00:52:51,840 --> 00:52:55,520
to what we have just discussed.

692
00:52:55,520 --> 00:52:58,120
But, you know, as the ground
state has no degeneracy.

693
00:52:58,120 --> 00:53:01,970
And so we all talk about
quantitative shifts.

694
00:53:01,970 --> 00:53:05,240
However, when we go
to the excited state,

695
00:53:05,240 --> 00:53:07,930
we will find the degeneracies.

696
00:53:07,930 --> 00:53:09,800
And degeneracies are
much more interesting

697
00:53:09,800 --> 00:53:12,960
because something which
was degenerate can split.

698
00:53:12,960 --> 00:53:14,410
You have two different terms.

699
00:53:14,410 --> 00:53:16,910
So suddenly there
is richer physics.

700
00:53:16,910 --> 00:53:21,930
So, therefore, we want to
discuss now the excited state.

701
00:53:21,930 --> 00:53:28,830
So starting again with
the hydrogenic model.

702
00:53:28,830 --> 00:53:32,170
In hydrogen, the
2s and 2p state are

703
00:53:32,170 --> 00:53:37,420
degenerate so we have two
configurations contributing

704
00:53:37,420 --> 00:53:38,540
to the same energy.

705
00:53:38,540 --> 00:53:40,880
1s 2s and 1s 2p.

706
00:53:46,200 --> 00:53:49,620
The binding energy in
the hydrogenic model

707
00:53:49,620 --> 00:53:51,900
is a quarter of a Rydberg.

708
00:53:51,900 --> 00:53:55,450
Rydberg over n square and ns2.

709
00:53:55,450 --> 00:53:57,900
We have to scale it by z square.

710
00:53:57,900 --> 00:54:04,480
And we find 13.6 electron volt.

711
00:54:04,480 --> 00:54:06,880
But now what happens
is that we have

712
00:54:06,880 --> 00:54:14,040
to introduce the Coulomb
energy between the electrons.

713
00:54:14,040 --> 00:54:22,250
And if you do that, it shifts
up the levels in different ways.

714
00:54:22,250 --> 00:54:24,860
So this is 1s, 2s.

715
00:54:24,860 --> 00:54:26,050
This is 1s, 2p.

716
00:54:31,000 --> 00:54:32,110
So why is this different?

717
00:54:54,870 --> 00:54:56,780
Well, you can say the following.

718
00:55:08,970 --> 00:55:10,370
You have two electrons.

719
00:55:10,370 --> 00:55:13,400
And you have the helium nucleus.

720
00:55:13,400 --> 00:55:18,630
And you first put
in the 1s electron.

721
00:55:18,630 --> 00:55:22,410
And now the second electron
when it is in a p state,

722
00:55:22,410 --> 00:55:25,490
in a 2p state, it's further out.

723
00:55:25,490 --> 00:55:28,110
And it pretty much
experiences out

724
00:55:28,110 --> 00:55:31,240
there the charge of
the helium nucleus

725
00:55:31,240 --> 00:55:33,840
shielded by the 1s electron.

726
00:55:33,840 --> 00:55:37,330
And, therefore, it sees
in effect a smaller

727
00:55:37,330 --> 00:55:39,450
nuclear charge.

728
00:55:39,450 --> 00:55:43,950
Whereas, the two s
electron penetrates deeper.

729
00:55:43,950 --> 00:55:46,450
Gets closer to the
nucleus and will still

730
00:55:46,450 --> 00:55:49,990
realize that the nucleus
has a charge of 2.

731
00:55:49,990 --> 00:55:51,860
And not a shielded charge.

732
00:55:51,860 --> 00:55:56,190
So, therefore, you would expect
that the shielding effect

733
00:55:56,190 --> 00:55:59,580
due to the innermost
electron is more important,

734
00:55:59,580 --> 00:56:02,530
has a bigger effect
for the 2p electron

735
00:56:02,530 --> 00:56:03,665
then for the 2s electron.

736
00:56:07,610 --> 00:56:10,000
So let me just write that down.

737
00:56:10,000 --> 00:56:17,830
So the 2p electron it
sees a shielded nucleus.

738
00:56:22,700 --> 00:56:26,260
In other words, what
it experiences more

739
00:56:26,260 --> 00:56:31,500
as an inner core, the Coulomb
potential of helium plus.

740
00:56:31,500 --> 00:56:33,945
And not so much
of helium 2 plus.

741
00:56:41,280 --> 00:56:45,060
This is due to the 1s electron.

742
00:56:45,060 --> 00:56:53,000
And, therefore, the 2p electron
has a smaller binding energy.

743
00:56:55,610 --> 00:57:07,900
It's actually comparable
to the binding energy

744
00:57:07,900 --> 00:57:09,855
of the 2p state in hydrogen.

745
00:57:12,840 --> 00:57:15,348
Which is on the order
of 4 electron volt.

746
00:57:19,220 --> 00:57:32,690
And this effect is
smaller for the 2s state.

747
00:57:36,050 --> 00:57:37,900
So let's go back to
the energy diagram.

748
00:57:42,980 --> 00:57:51,020
So we have now the situation
which I just described.

749
00:57:51,020 --> 00:57:54,110
We have two degenerate
configurations

750
00:57:54,110 --> 00:57:56,020
for in the hydrogen model.

751
00:57:56,020 --> 00:58:01,320
And when we add in Coulomb
energy between the electrons,

752
00:58:01,320 --> 00:58:04,400
there is quite a big splitting
of several electron volts.

753
00:58:08,780 --> 00:58:13,832
But now, each label
undergoes further splitting.

754
00:58:13,832 --> 00:58:15,290
And this is what
I want to discuss.

755
00:58:17,800 --> 00:58:20,010
So we are still sorting
out just the preliminaries.

756
00:58:20,010 --> 00:58:22,120
What I really want
to discuss with you

757
00:58:22,120 --> 00:58:23,920
is the singlet
and triplet thing.

758
00:58:23,920 --> 00:58:27,090
But now we are there
where we can do it.

759
00:58:27,090 --> 00:58:32,420
So if you have a
configuration with 1s, 2s,

760
00:58:32,420 --> 00:58:37,330
there are two possibilities
for the total angular momentum.

761
00:58:37,330 --> 00:58:39,500
Two electrons in
an s state, there

762
00:58:39,500 --> 00:58:41,090
is no orbital angular momentum.

763
00:58:41,090 --> 00:58:43,200
But there are two spins, 1/2.

764
00:58:43,200 --> 00:58:47,130
And they can add up to
1 or can add up to 0.

765
00:58:47,130 --> 00:58:53,120
So, therefore, we will
have two different terms.

766
00:58:53,120 --> 00:58:57,760
One is singlet s0.

767
00:58:57,760 --> 00:59:01,750
And one is triplet s1.

768
00:59:01,750 --> 00:59:05,840
And this splitting
is 0.8 electron volt.

769
00:59:05,840 --> 00:59:08,800
And this is what we want to
discuss in the following.

770
00:59:08,800 --> 00:59:12,080
For completeness, but
the physics is similar,

771
00:59:12,080 --> 00:59:18,270
let me mention that the
1s, 2p state also gives

772
00:59:18,270 --> 00:59:19,780
rise to two terms.

773
00:59:19,780 --> 00:59:24,406
We have now the total
orbital angular momentum, p.

774
00:59:24,406 --> 00:59:26,610
Orbital angular momentum of 1.

775
00:59:26,610 --> 00:59:30,350
The spin angular
momentum is 0 or 1.

776
00:59:30,350 --> 00:59:33,080
So we have singlet and triplet.

777
00:59:33,080 --> 00:59:43,400
And the total angular
momentum is 1 in this case.

778
00:59:43,400 --> 00:59:49,410
Or in this case, 2, 1, or 0.

779
00:59:49,410 --> 00:59:53,240
And the splitting
in this situation

780
00:59:53,240 --> 00:59:57,240
is also on the order of a
fraction of an electron volt.

781
01:00:00,400 --> 01:00:02,840
So what we want
to understand now

782
01:00:02,840 --> 01:00:10,090
is why do we observe a splitting
between those two levels

783
01:00:10,090 --> 01:00:12,990
which seems to
depend on the spin?

784
01:00:12,990 --> 01:00:16,290
How can the spin
cause a splitting?

785
01:00:16,290 --> 01:00:23,230
Because the spin so far has not
appeared in our Hamiltonian.

786
01:00:23,230 --> 01:00:24,920
We rarely have a
Hamiltonian which

787
01:00:24,920 --> 01:00:26,970
has only the Coulomb energy.

788
01:00:26,970 --> 01:00:30,050
And the spin is not part of it.

789
01:00:30,050 --> 01:00:31,660
So we don't have
a magnetic field

790
01:00:31,660 --> 01:00:34,640
to which the magnetic momentum
of the spin would couple.

791
01:00:34,640 --> 01:00:38,680
And also we have not yet
introduced spin orbit coupling.

792
01:00:38,680 --> 01:00:41,620
But if this is on your
mind, take it off your mind.

793
01:00:41,620 --> 01:00:45,710
Spin orbit coupling is a
much, much smaller effect.

794
01:00:45,710 --> 01:00:48,350
Energies on the order
of 1 electron volt,

795
01:00:48,350 --> 01:00:51,510
you just cannot get from
spin orbit coupling.

796
01:00:51,510 --> 01:00:54,820
Spin orbit coupling is smaller
than electronic energies,

797
01:00:54,820 --> 01:00:58,050
as I will explain
to you on Friday,

798
01:00:58,050 --> 01:01:00,470
is smaller by the fine
structure constant.

799
01:01:00,470 --> 01:01:05,010
So the typical scale
for spin orbit coupling

800
01:01:05,010 --> 01:01:10,600
is maybe 10 or 100
million electron volt.

801
01:01:10,600 --> 01:01:12,880
It's much smaller.

802
01:01:12,880 --> 01:01:15,610
So, therefore, we
want to understand now

803
01:01:15,610 --> 01:01:18,890
why do we have a spin
dependent energy.

804
01:01:18,890 --> 01:01:23,140
Also we haven't coupled at this
point the spin to any field.

805
01:01:29,910 --> 01:01:35,680
OK, so we are focusing
now on the splitting.

806
01:01:44,030 --> 01:01:50,980
So we have the 1s,
2s configuration.

807
01:01:50,980 --> 01:01:53,650
We get two terms,
as I just discussed.

808
01:01:57,540 --> 01:02:02,180
One is singlet and
one is triplet.

809
01:02:04,720 --> 01:02:11,550
And still using the
hydrogenic model

810
01:02:11,550 --> 01:02:14,240
of non-interacting
electrons, we want

811
01:02:14,240 --> 01:02:17,290
to write down the wave function.

812
01:02:17,290 --> 01:02:22,010
So the wave function
of the twin electrons

813
01:02:22,010 --> 01:02:28,870
is we have one electron
in the 1s state.

814
01:02:28,870 --> 01:02:36,140
We have one electron
in the 2s state.

815
01:02:36,140 --> 01:02:40,720
But now since we
have fermionic atoms,

816
01:02:40,720 --> 01:02:46,030
we have to correctly
symmetrize it.

817
01:02:46,030 --> 01:02:54,390
So whether we want the symmetric
or antisymmetric combination,

818
01:02:54,390 --> 01:02:58,300
we exchange the two electrons.

819
01:02:58,300 --> 01:03:03,375
So now we have r2 and
r2 in reverse order.

820
01:03:07,410 --> 01:03:17,060
Of course, the total wave
function for two fermions

821
01:03:17,060 --> 01:03:19,880
has to be antisymmetric.

822
01:03:19,880 --> 01:03:23,740
But the total wave function is
the product of the spatial wave

823
01:03:23,740 --> 01:03:28,770
function which I just wrote down
times the spin wave function.

824
01:03:28,770 --> 01:03:36,330
The spin wave function can be
antisymmetric and symmetric.

825
01:03:36,330 --> 01:03:41,460
And the antisymmetric
spin wave function

826
01:03:41,460 --> 01:03:44,270
has to combine with the
symmetric spatial wave

827
01:03:44,270 --> 01:03:47,010
function, and vice
versa, to make sure

828
01:03:47,010 --> 01:03:50,080
that the total wave
function is antisymmetric.

829
01:03:50,080 --> 01:03:54,120
And the correct
description for fermions.

830
01:03:54,120 --> 01:03:56,940
The designation here,
symmetric and antisymmetric

831
01:03:56,940 --> 01:04:02,100
for the total wave function
reflects the spatial part.

832
01:04:02,100 --> 01:04:04,690
The total wave function, of
course, including the spin wave

833
01:04:04,690 --> 01:04:06,065
function is always
antisymmetric.

834
01:04:12,900 --> 01:04:20,250
OK, so we have two
wave functions.

835
01:04:20,250 --> 01:04:27,210
One has a symmetric
spatial wave function.

836
01:04:31,410 --> 01:04:34,960
The other one, an antisymmetric.

837
01:04:34,960 --> 01:04:38,280
The symmetric
spatial wave function

838
01:04:38,280 --> 01:04:44,120
has an antisymmetric
spin wave function.

839
01:04:48,070 --> 01:04:53,650
And that means we
have s equals 0.

840
01:04:53,650 --> 01:04:57,040
That's up, down, minus
down, up is antisymmetric.

841
01:04:57,040 --> 01:04:59,730
Where as the
antisymmetric spatial

842
01:04:59,730 --> 01:05:04,550
wave function goes together
with the symmetric spin wave

843
01:05:04,550 --> 01:05:08,040
function s equals 1.

844
01:05:08,040 --> 01:05:10,900
So this is the situation
which gives rise

845
01:05:10,900 --> 01:05:19,720
to that triplet s1 term.

846
01:05:19,720 --> 01:05:22,810
And this here is
the singlet s0 term.

847
01:05:27,390 --> 01:05:28,330
OK.

848
01:05:28,330 --> 01:05:31,996
As long as we have
non-interacting electron,

849
01:05:31,996 --> 01:05:36,300
the two wave functions
are degenerate.

850
01:05:36,300 --> 01:05:44,200
But now we want to bring in the
Coulomb energy between the two

851
01:05:44,200 --> 01:05:47,460
electrons which we had
already discussed before.

852
01:06:17,590 --> 01:06:27,720
And if you calculate the energy
using this as a perturbation

853
01:06:27,720 --> 01:06:32,880
operator, well, remember the
wave function had two parts.

854
01:06:32,880 --> 01:06:37,670
It was 100, r1, 200, r2.

855
01:06:37,670 --> 01:06:41,570
And the part where our 1
and our 2 were flipped.

856
01:06:41,570 --> 01:06:46,730
And now if we have the wave
function, the perturbation

857
01:06:46,730 --> 01:06:51,500
operator, we get a
total of 4 terms.

858
01:06:51,500 --> 01:06:52,310
2 times 2.

859
01:06:52,310 --> 01:06:55,220
We have to sort of x it out.

860
01:06:55,220 --> 01:07:01,770
And we will then have the
sort of diagonal parts.

861
01:07:01,770 --> 01:07:04,450
And we have the parts
which are off diagonal.

862
01:07:04,450 --> 01:07:13,720
And for the off
diagonal parts, it

863
01:07:13,720 --> 01:07:16,120
matters whether we had
the plus or minus sign.

864
01:07:16,120 --> 01:07:19,350
You know, if you have plus,
plus and minus, minus,

865
01:07:19,350 --> 01:07:20,880
you give a positive
contribution.

866
01:07:20,880 --> 01:07:24,590
But if you connect
the wave function

867
01:07:24,590 --> 01:07:32,700
before and after the
operator, plus with minus,

868
01:07:32,700 --> 01:07:33,880
we get minus signs.

869
01:07:33,880 --> 01:07:36,920
So in other words, we
have one contribution

870
01:07:36,920 --> 01:07:38,600
where it doesn't
matter whether we

871
01:07:38,600 --> 01:07:40,690
have the symmetric or
antisymmetric spatial wave

872
01:07:40,690 --> 01:07:41,460
function.

873
01:07:41,460 --> 01:07:44,094
But then we have
another term where

874
01:07:44,094 --> 01:07:46,510
it matters whether we have the
symmetric and antisymmetric

875
01:07:46,510 --> 01:07:47,120
wave function.

876
01:07:47,120 --> 01:07:49,940
That's where the plus or minus
sign from the symmetrized wave

877
01:07:49,940 --> 01:07:51,710
function appears.

878
01:07:51,710 --> 01:07:55,350
So we have two contributions
now to the energy correction.

879
01:07:55,350 --> 01:07:59,010
One is independent of
the spin wave function

880
01:07:59,010 --> 01:08:01,930
whether we have the symmetric
or antisymmetric configuration.

881
01:08:01,930 --> 01:08:03,430
The other one is not.

882
01:08:03,430 --> 01:08:06,190
The first term is called
the Coulomb energy.

883
01:08:06,190 --> 01:08:09,090
The second term is called
the exchange energy.

884
01:08:20,439 --> 01:08:24,630
So what we find is-- we
have to trace back the sign,

885
01:08:24,630 --> 01:08:34,100
but you find that the triplet
state symmetric in spin.

886
01:08:34,100 --> 01:08:38,720
And antisymmetric in the
spatial wave function.

887
01:08:38,720 --> 01:08:47,710
Has the lower energy,
is more strongly bound

888
01:08:47,710 --> 01:09:06,244
because the antisymmetric
spatial wave function reduces

889
01:09:06,244 --> 01:09:08,410
the repulsive interaction
between the two electrons.

890
01:09:15,660 --> 01:09:17,870
So in other words,
we do not have

891
01:09:17,870 --> 01:09:21,300
any spin term in
the Hamiltonian.

892
01:09:21,300 --> 01:09:24,700
It's just that whether
the spin wave function is

893
01:09:24,700 --> 01:09:28,990
symmetric or antisymmetric,
it requires the spatial wave

894
01:09:28,990 --> 01:09:31,760
function to be the opposite.

895
01:09:31,760 --> 01:09:36,040
And now when we calculate
the Coulomb energy

896
01:09:36,040 --> 01:09:39,420
for the spatial or antisymmetric
spatial wave function,

897
01:09:39,420 --> 01:09:41,370
we find a big difference.

898
01:09:41,370 --> 01:09:44,950
And the big difference
is the exchange energy.

899
01:09:44,950 --> 01:09:51,529
So it is a spin dependent
term for the total energy.

900
01:09:51,529 --> 01:09:56,810
But what is behind this energy
is simply the Coulomb energy.

901
01:09:56,810 --> 01:10:01,270
So the spin through the
symmetry of the wave function

902
01:10:01,270 --> 01:10:05,626
leads to a difference
in the Coulomb energy.

903
01:10:05,626 --> 01:10:07,000
And actually what
I'm telling you

904
01:10:07,000 --> 01:10:12,140
is the explanation why we have
magnetism at room temperature.

905
01:10:12,140 --> 01:10:14,530
It was Heisenberg's
idea when he realized

906
01:10:14,530 --> 01:10:18,050
for the first time what
can cause ferromagnetism.

907
01:10:18,050 --> 01:10:23,160
It was pretty much the model
of the helium atom expanded

908
01:10:23,160 --> 01:10:26,620
to many, many
electrons in a lattice.

909
01:10:26,620 --> 01:10:30,260
So the result is that the
Curie temperature of thousands

910
01:10:30,260 --> 01:10:33,790
of degrees, we have magnetism
below 1,000 degrees.

911
01:10:33,790 --> 01:10:37,780
This energy scale is
an electronic energy,

912
01:10:37,780 --> 01:10:41,050
is in Coulomb energy scale
and not an energy scale

913
01:10:41,050 --> 01:10:43,190
where spin interactions
comes into play.

914
01:10:43,190 --> 01:10:45,530
If it were not for
the exchange energy,

915
01:10:45,530 --> 01:10:50,190
we would not have magnetic
materials above 1 calorie.

916
01:10:50,190 --> 01:10:52,310
So this is what you see
here in the helium atom.

917
01:10:52,310 --> 01:10:57,650
How speed leads to an
energy splitting which

918
01:10:57,650 --> 01:11:01,160
is an electronic Coulombic
energy splitting.

919
01:11:05,530 --> 01:11:10,590
OK, let's make it maybe
even more obvious.

920
01:11:10,590 --> 01:11:15,360
The above equation
can be rewritten.

921
01:11:24,690 --> 01:11:28,780
So this energy
splitting can be written

922
01:11:28,780 --> 01:11:39,390
as a constant, alpha,
plus a constant beta,

923
01:11:39,390 --> 01:11:41,580
times the product of s1 and s2.

924
01:11:44,390 --> 01:11:56,030
Well, what happens is-- let
me just give you as a sidebar,

925
01:11:56,030 --> 01:12:01,080
the product of s1
and s2 can be written

926
01:12:01,080 --> 01:12:08,190
as minus s1 square, minus
s2 square, plus s square.

927
01:12:13,110 --> 01:12:18,890
s1 square is one half times 3/2,
it's 3/4. s2 squared is 3/4.

928
01:12:18,890 --> 01:12:25,282
And s square is either
0 or 2 depending

929
01:12:25,282 --> 01:12:27,240
whether you're in the
singlet or triplet state.

930
01:12:27,240 --> 01:12:30,710
In other words, I just
want to with the sidebar,

931
01:12:30,710 --> 01:12:34,270
which you have seen many,
many times, just remind you

932
01:12:34,270 --> 01:12:38,320
that the product of s1 and
s2 has only two values.

933
01:12:41,230 --> 01:12:45,990
One for the singlet state,
one for the triplet state.

934
01:12:45,990 --> 01:12:48,790
So, therefore, if I
have a singlet level

935
01:12:48,790 --> 01:12:51,110
and a triplet
level, I can always

936
01:12:51,110 --> 01:12:54,410
parametrize it like this.

937
01:12:54,410 --> 01:13:01,500
And I can make it more
obvious by showing

938
01:13:01,500 --> 01:13:07,690
what this formula on the right
hand side has two values.

939
01:13:07,690 --> 01:13:09,480
One for s equals 0.

940
01:13:09,480 --> 01:13:11,820
And one for s equals 1.

941
01:13:11,820 --> 01:13:17,520
And this can be
rewritten as alpha.

942
01:13:17,520 --> 01:13:19,930
You find a more explicit
calculation in the [? video, ?]

943
01:13:19,930 --> 01:13:22,890
but it's also just one
more line of algebra.

944
01:13:22,890 --> 01:13:29,320
You can write it in
the following way.

945
01:13:29,320 --> 01:13:32,180
So, therefore, the
alpha and beta parameter

946
01:13:32,180 --> 01:13:35,920
are just a way of, you know,
you have two energy levels.

947
01:13:35,920 --> 01:13:38,620
And with two constants,
you can always

948
01:13:38,620 --> 01:13:40,030
describe two energy levels.

949
01:13:40,030 --> 01:13:42,500
Here, I've done it
with alpha and beta.

950
01:13:42,500 --> 01:13:46,960
But before, I did it
with the Coulomb energy.

951
01:13:50,690 --> 01:13:58,896
And the exchange energy which
we obtained in this equation

952
01:13:58,896 --> 01:14:01,580
when we perturbatively
calculated the integral.

953
01:14:12,040 --> 01:14:16,740
So by writing it as
s1 dot s2, I even

954
01:14:16,740 --> 01:14:19,990
suggest that the
two spins interact

955
01:14:19,990 --> 01:14:22,410
like a dipole-dipole
interaction.

956
01:14:22,410 --> 01:14:23,947
But they don't.

957
01:14:23,947 --> 01:14:25,530
It comes from the
Coulomb interaction,

958
01:14:25,530 --> 01:14:29,410
but it is equivalent to
a gigantic dipole-dipole

959
01:14:29,410 --> 01:14:31,510
interaction.

960
01:14:31,510 --> 01:14:33,420
So, therefore, the
conclusion of this

961
01:14:33,420 --> 01:14:37,310
is that what I have derived
for you for the helium atom,

962
01:14:37,310 --> 01:14:50,936
it looks like a ferromagnetic
spin-spin interaction.

963
01:14:54,290 --> 01:14:58,015
And, well, it looks
like it, it is actually

964
01:14:58,015 --> 01:15:02,770
an effective ferromagnetic
spin-spin interaction.

965
01:15:02,770 --> 01:15:07,635
However, the coupling
is purely electrostatic

966
01:15:07,635 --> 01:15:14,581
here and not magnetic.

967
01:15:28,494 --> 01:15:28,994
Questions?

968
01:15:35,000 --> 01:15:37,108
OK, yes?

969
01:15:37,108 --> 01:15:40,100
AUDIENCE: So since we're still
using the hydrogenic wave

970
01:15:40,100 --> 01:15:41,980
function as a basis,
how closely does this

971
01:15:41,980 --> 01:15:48,290
get to the actual
measured values?

972
01:15:48,290 --> 01:15:51,080
PROFESSOR: The question
is, how close do we

973
01:15:51,080 --> 01:15:53,930
get with hydrogenic
wave functions

974
01:15:53,930 --> 01:15:55,215
to the actual measured energy?

975
01:15:58,460 --> 01:15:59,912
I don't know.

976
01:15:59,912 --> 01:16:03,130
I don't know the exact
numbers for the excited state.

977
01:16:03,130 --> 01:16:05,650
I assume that it's similar
to the ground state.

978
01:16:05,650 --> 01:16:07,260
You saw that for
the ground state,

979
01:16:07,260 --> 01:16:08,790
we had a big discrepancy.

980
01:16:08,790 --> 01:16:12,860
Most of that was closed by
using hydrogenic wave functions.

981
01:16:12,860 --> 01:16:15,920
And just calculating
the perturbative terms.

982
01:16:15,920 --> 01:16:19,040
So you pretty much
get qualitatively or

983
01:16:19,040 --> 01:16:22,280
semi-quantitatively, you
get the picture out of it.

984
01:16:22,280 --> 01:16:23,980
And unless you're
really interested

985
01:16:23,980 --> 01:16:26,340
in the absolute values,
you can stop there.

986
01:16:26,340 --> 01:16:30,250
But one way to go further and
reduce the discrepancy by 75%

987
01:16:30,250 --> 01:16:32,840
is to use this variational
wave function where

988
01:16:32,840 --> 01:16:35,020
you use hydrogenic
wave function,

989
01:16:35,020 --> 01:16:38,040
but you use z, the
nuclear charge,

990
01:16:38,040 --> 01:16:40,210
as a variation of parameter.

991
01:16:40,210 --> 01:16:42,770
And I mean, this is,
I mean, it's amazing.

992
01:16:42,770 --> 01:16:44,270
I mean, you have a
two electron atom

993
01:16:44,270 --> 01:16:46,230
and you use a
hydrogenic wave function

994
01:16:46,230 --> 01:16:47,230
with one free parameter.

995
01:16:47,230 --> 01:16:50,380
And you get binding
energies which

996
01:16:50,380 --> 01:16:53,920
are on the order of 70
electron volt accurate to

997
01:16:53,920 --> 01:16:57,250
within better than 3%, 4%, 5%.

998
01:16:57,250 --> 01:16:59,560
But by adding other terms
or using a little bit

999
01:16:59,560 --> 01:17:01,670
more fancy wave
functions, I'm sure you

1000
01:17:01,670 --> 01:17:03,494
can get further and further.

1001
01:17:06,840 --> 01:17:16,320
OK, so the last thing
I wanted to discuss

1002
01:17:16,320 --> 01:17:24,010
is the new feature
of two electrons

1003
01:17:24,010 --> 01:17:30,380
is that we have singlet
and triplet levels.

1004
01:17:30,380 --> 01:17:42,550
So we have a letter of states
which are singlet states.

1005
01:17:47,210 --> 01:17:49,875
And then, and this is
what we just discussed,

1006
01:17:49,875 --> 01:17:52,360
let me just make dashed lines.

1007
01:17:52,360 --> 01:17:57,520
Because of this ferromagnetic
spin-spin interaction,

1008
01:17:57,520 --> 01:18:02,400
we have triplets states
which have lower energy.

1009
01:18:02,400 --> 01:18:05,610
So these are n equals
2 triplet states.

1010
01:18:12,250 --> 01:18:16,100
n equals 2 triplet states.

1011
01:18:16,100 --> 01:18:17,400
j equals 1.

1012
01:18:17,400 --> 01:18:21,500
j equals 0, 1, 2.

1013
01:18:21,500 --> 01:18:27,710
So of course, there
are transitions

1014
01:18:27,710 --> 01:18:29,800
between those levels.

1015
01:18:29,800 --> 01:18:34,276
And form a p state, you can have
transitions down to an s state.

1016
01:18:38,130 --> 01:18:44,230
The question I have is what
about possible transitions

1017
01:18:44,230 --> 01:18:47,380
between triplet and singlet?

1018
01:18:51,160 --> 01:18:58,560
So what I want to ask you know
with a clicker question is

1019
01:18:58,560 --> 01:19:00,910
what kind?

1020
01:19:00,910 --> 01:19:05,720
So those transitions
here are transitions

1021
01:19:05,720 --> 01:19:09,530
between singlet and triplet.

1022
01:19:09,530 --> 01:19:12,980
And the technical term for
transitions between singlet

1023
01:19:12,980 --> 01:19:16,320
and triplet are
intercombination lines.

1024
01:19:19,550 --> 01:19:35,880
So the question I have for you
is what fields or couplings

1025
01:19:35,880 --> 01:19:40,308
drive singlet
triplet transitions?

1026
01:19:48,220 --> 01:19:54,280
So I want you to think about the
model we have discussed so far.

1027
01:19:54,280 --> 01:19:57,060
All we have is Coulomb energy.

1028
01:19:57,060 --> 01:20:00,290
Coulomb energy between the
nucleus and the electrons.

1029
01:20:00,290 --> 01:20:02,530
And between the electrons.

1030
01:20:02,530 --> 01:20:03,240
That's it.

1031
01:20:03,240 --> 01:20:06,730
We do not put any other
terms into the Hamiltonian.

1032
01:20:06,730 --> 01:20:09,820
And now we have obtained
those wave functions.

1033
01:20:09,820 --> 01:20:12,020
We have obtained those
energies splittings.

1034
01:20:12,020 --> 01:20:16,300
And now we want to ask are
there transitions possible?

1035
01:20:16,300 --> 01:20:21,040
So one possibility is that
we can drive the transition

1036
01:20:21,040 --> 01:20:22,070
with optical fields.

1037
01:20:26,420 --> 01:20:29,100
Let's say our dipole operator.

1038
01:20:29,100 --> 01:20:32,330
Which many of you have
encountered in life.

1039
01:20:32,330 --> 01:20:39,934
We have already discussed
rotating magnetic fields

1040
01:20:39,934 --> 01:20:43,940
in the first part of the course.

1041
01:20:43,940 --> 01:20:46,460
Is it possible to use
both magnetic fields

1042
01:20:46,460 --> 01:20:48,390
and optical fields?

1043
01:20:48,390 --> 01:20:53,160
Or the last answer
is none of the above.

1044
01:20:53,160 --> 01:20:56,070
It's not possible with
any of those fields

1045
01:20:56,070 --> 01:21:00,674
to create any kind of transition
between singlet and triplet.

1046
01:21:28,562 --> 01:21:36,330
OK, all right.

1047
01:21:39,130 --> 01:21:42,660
We have to spend the first 10
minutes on the class on Friday

1048
01:21:42,660 --> 01:21:44,400
to discuss that.

1049
01:21:44,400 --> 01:21:47,560
But the answer is none.

1050
01:21:47,560 --> 01:21:52,375
There is no way how you can get
a transition between singlet

1051
01:21:52,375 --> 01:21:55,200
and triplet using
the approximations

1052
01:21:55,200 --> 01:21:57,200
for the description
of the helium atom

1053
01:21:57,200 --> 01:21:58,350
we have done so far.

1054
01:22:01,980 --> 01:22:05,720
Since I don't want to end
with such a cliffhanger,

1055
01:22:05,720 --> 01:22:09,290
let me just say, we have
actually a selection rule

1056
01:22:09,290 --> 01:22:13,780
which says, we are not
changing the total spin.

1057
01:22:13,780 --> 01:22:15,510
And you would say, hey, come on.

1058
01:22:15,510 --> 01:22:18,850
Can't I just take a magnetic
field and flip one of the spin,

1059
01:22:18,850 --> 01:22:22,190
go from a triplet
to a singlet state?

1060
01:22:22,190 --> 01:22:27,490
Well, the answer is no.

1061
01:22:27,490 --> 01:22:29,540
And let me just give
it to you qualitatively

1062
01:22:29,540 --> 01:22:32,850
and deformalize it on Friday.

1063
01:22:32,850 --> 01:22:38,240
First, as far as transverse
magnetic fields, rotating

1064
01:22:38,240 --> 01:22:42,450
magnetic fields, are
concerned. transverse B fields.

1065
01:22:42,450 --> 01:22:47,050
Remember, transverse B fields
create some f of the spin.

1066
01:22:47,050 --> 01:22:50,490
But both spins precess equally.

1067
01:22:50,490 --> 01:22:52,800
So you can never
even classically,

1068
01:22:52,800 --> 01:22:55,400
you cannot change the angle
between the two spins.

1069
01:22:55,400 --> 01:22:57,020
When the two spins
are interparallel,

1070
01:22:57,020 --> 01:22:58,192
they stay interparallel.

1071
01:22:58,192 --> 01:23:00,150
When they stay parallel
when they are parallel,

1072
01:23:00,150 --> 01:23:01,400
they stay parallel.

1073
01:23:01,400 --> 01:23:05,860
Or to say the
transverse B field means

1074
01:23:05,860 --> 01:23:09,650
we have a coupling, sx, sy.

1075
01:23:09,650 --> 01:23:16,190
And sx and sy can be
written as letter operators,

1076
01:23:16,190 --> 01:23:18,355
s plus and s minus.

1077
01:23:21,010 --> 01:23:24,820
And as you know, s
plus and s minus only

1078
01:23:24,820 --> 01:23:27,330
change the magnetic
quantum number.

1079
01:23:30,340 --> 01:23:35,530
But they do not change
the value of the total s.

1080
01:23:35,530 --> 01:23:37,480
In other words, if
you have a spin which

1081
01:23:37,480 --> 01:23:39,790
is s equals 1 triplet
state, you can

1082
01:23:39,790 --> 01:23:43,490
change the angle, how
the spin equals 1 points.

1083
01:23:43,490 --> 01:23:45,560
But it still will
be spin 1 where

1084
01:23:45,560 --> 01:23:48,340
both electrons are
aligned parallel.

1085
01:23:48,340 --> 01:23:51,640
There is no magnetic field
which can selectively

1086
01:23:51,640 --> 01:23:57,390
talk to one spin and rotate
spin 1 with respect to spin 2.

1087
01:23:57,390 --> 01:23:59,090
And coming to the
other questions

1088
01:23:59,090 --> 01:24:03,250
about optical fields
with a dipole operator,

1089
01:24:03,250 --> 01:24:06,510
the answer is also
a resounding no.

1090
01:24:06,510 --> 01:24:12,130
Because the dipole operator
of the electromagnetic field,

1091
01:24:12,130 --> 01:24:15,150
the dipole operator acts only
on the spatial wave function.

1092
01:24:25,560 --> 01:24:29,340
Not on the spin part.

1093
01:24:29,340 --> 01:24:31,550
So a laser beam through
the dipole operator

1094
01:24:31,550 --> 01:24:33,440
can never ever flip the spin.

1095
01:24:33,440 --> 01:24:35,640
It only acts on the
spatial wave function.

1096
01:24:35,640 --> 01:24:39,780
So anyway, I wanted to just give
you the answer why you cannot

1097
01:24:39,780 --> 01:24:43,470
go from singlet to triplet
state with sort of our standard

1098
01:24:43,470 --> 01:24:46,785
operators, with the electric
dipole operator or with

1099
01:24:46,785 --> 01:24:47,790
the rotating field.

1100
01:24:47,790 --> 01:24:51,230
And I hope it was worthwhile
to explain in detail why each

1101
01:24:51,230 --> 01:24:53,020
of them cannot do it.

1102
01:24:53,020 --> 01:24:54,940
But on Friday, I
explained to you

1103
01:24:54,940 --> 01:24:57,540
that there is a general
symmetry behind it

1104
01:24:57,540 --> 01:24:59,300
that even more fancy
combinations will not

1105
01:24:59,300 --> 01:25:00,435
be able to do that.