1
00:00:01,260 --> 00:00:03,540
PROFESSOR: So we'll do the
relativistic corrections.

2
00:00:03,540 --> 00:00:06,790
And all the corrections
that I'll do today,

3
00:00:06,790 --> 00:00:12,860
I'll skip the easy but sometimes
a little tedious algebra.

4
00:00:12,860 --> 00:00:14,550
It's not very tedious.

5
00:00:14,550 --> 00:00:17,470
Nothing that is pages
and pages of algebra.

6
00:00:17,470 --> 00:00:19,650
It's lines of algebra.

7
00:00:19,650 --> 00:00:23,070
But why would I
do it in lecture?

8
00:00:23,070 --> 00:00:25,620
No point for that.

9
00:00:25,620 --> 00:00:30,240
So let's see what we can do.

10
00:00:30,240 --> 00:00:33,000
This is the
relativistic correction,

11
00:00:33,000 --> 00:00:34,500
the minus p squared.

12
00:00:34,500 --> 00:00:38,320
So could we write this for
the relativistic correction?

13
00:00:38,320 --> 00:00:42,390
We're going to do first order
correction, relativistic,

14
00:00:42,390 --> 00:00:46,188
of the levels n l ml.

15
00:00:50,780 --> 00:00:52,190
Let's put a question mark.

16
00:00:52,190 --> 00:01:04,305
Minus 1 over 8 m cubed c squared
psi n l ml p to the fourth.

17
00:01:12,450 --> 00:01:17,160
Now recall that p to the
fourth, the way it was given,

18
00:01:17,160 --> 00:01:20,685
is really p squared
times p squared.

19
00:01:24,020 --> 00:01:27,480
You have four things that
have to be multiplied.

20
00:01:27,480 --> 00:01:31,590
So it's not px to the fourth
plus py to the fourth plus

21
00:01:31,590 --> 00:01:36,870
pz to the fourth is px
squared plus py squared

22
00:01:36,870 --> 00:01:41,950
plus pz squared, all
squared, just in case

23
00:01:41,950 --> 00:01:44,430
there's an ambiguity.

24
00:01:44,430 --> 00:01:46,050
That seems reasonable.

25
00:01:46,050 --> 00:01:48,210
The first order
corrections should

26
00:01:48,210 --> 00:01:53,810
be found by taking the
states and finding this.

27
00:01:53,810 --> 00:01:56,684
But there is a
big question mark.

28
00:01:56,684 --> 00:01:58,100
And this kind of
question is going

29
00:01:58,100 --> 00:02:01,560
to come up every time you
think about these things.

30
00:02:05,020 --> 00:02:07,450
This formula, where
I said the shift

31
00:02:07,450 --> 00:02:11,920
of the energy of this state
is that state evaluated here,

32
00:02:11,920 --> 00:02:17,360
applies for nondegenerate
perturbation theory.

33
00:02:17,360 --> 00:02:19,820
And if the hydrogen
atom is anything,

34
00:02:19,820 --> 00:02:22,670
it's a system with a
lot of degeneracies.

35
00:02:22,670 --> 00:02:29,510
So why can I use that,
or can I use that?

36
00:02:29,510 --> 00:02:31,620
We have the hydrogen atom.

37
00:02:31,620 --> 00:02:33,405
I just deleted it here.

38
00:02:37,610 --> 00:02:46,710
So here, if you have n,
for degeneracies you fix n.

39
00:02:46,710 --> 00:02:49,870
For degeneracies, you
fix some value of n.

40
00:02:49,870 --> 00:02:52,170
And now you have the
degeneracies between

41
00:02:52,170 --> 00:02:57,680
the various l's, for each
l between the various m's.

42
00:02:57,680 --> 00:03:00,210
A gigantic amount of degeneracy.

43
00:03:00,210 --> 00:03:03,770
Who allows me to do that?

44
00:03:03,770 --> 00:03:10,480
I'm supposed to take that
level three has nine states,

45
00:03:10,480 --> 00:03:11,140
remember?

46
00:03:11,140 --> 00:03:12,060
n square states.

47
00:03:12,060 --> 00:03:18,680
Well, we should do a 9 by 9
matrix here and calculate this.

48
00:03:18,680 --> 00:03:20,790
Nine sounds awful.

49
00:03:20,790 --> 00:03:24,880
We don't want to do
that so we better think.

50
00:03:24,880 --> 00:03:27,520
So this is the situation
you find yourself.

51
00:03:27,520 --> 00:03:29,830
Technically speaking,
this is a problem

52
00:03:29,830 --> 00:03:31,760
in the degenerate
perturbation theory.

53
00:03:31,760 --> 00:03:34,320
We should do that.

54
00:03:34,320 --> 00:03:38,350
And you better think about
this every time you face this

55
00:03:38,350 --> 00:03:42,410
problem because sometimes you
can get away without doing

56
00:03:42,410 --> 00:03:45,935
the degenerate analysis,
but sometimes you can't.

57
00:03:45,935 --> 00:03:46,435
Yes?

58
00:03:46,435 --> 00:03:48,226
AUDIENCE: It's like
rotationally symmetric.

59
00:03:48,226 --> 00:03:52,277
So you can mix terms with
different [INAUDIBLE]..

60
00:03:52,277 --> 00:03:52,860
PROFESSOR: OK.

61
00:03:55,690 --> 00:04:02,180
So you're saying, basically,
that this thing in this basis--

62
00:04:02,180 --> 00:04:05,030
so we have nine states here.

63
00:04:05,030 --> 00:04:05,810
n equal 3.

64
00:04:05,810 --> 00:04:08,570
In these nine states,
it doesn't mix them.

65
00:04:08,570 --> 00:04:11,180
So this is diagonal here.

66
00:04:11,180 --> 00:04:14,540
And what one is
claiming by doing

67
00:04:14,540 --> 00:04:18,890
that is that this
is a good basis,

68
00:04:18,890 --> 00:04:23,630
that delta H is
already diagonal there.

69
00:04:23,630 --> 00:04:27,120
And don't worry, we can do it.

70
00:04:27,120 --> 00:04:29,100
In fact, that is true.

71
00:04:29,100 --> 00:04:33,770
And the argument goes like that.

72
00:04:33,770 --> 00:04:40,010
We know that p to the
fourth, the perturbation,

73
00:04:40,010 --> 00:04:42,380
commutes with l squared.

74
00:04:47,090 --> 00:04:49,100
We'll discuss it a little more.

75
00:04:49,100 --> 00:04:57,480
And p to the fourth
commutes with lz as well.

76
00:04:57,480 --> 00:04:59,450
So these are two claims.

77
00:04:59,450 --> 00:05:01,220
Very important claims.

78
00:05:01,220 --> 00:05:04,730
Remember, we had a
remark that I told you

79
00:05:04,730 --> 00:05:07,130
few times few lectures ago.

80
00:05:07,130 --> 00:05:08,750
Very important.

81
00:05:08,750 --> 00:05:11,660
If you have a
Hermitian operator that

82
00:05:11,660 --> 00:05:15,320
commutes with your
perturbation for which

83
00:05:15,320 --> 00:05:18,500
the states of your
bases are eigenstates

84
00:05:18,500 --> 00:05:24,460
with different eigenvalues,
then the basis is good.

85
00:05:24,460 --> 00:05:26,970
So here it is.

86
00:05:26,970 --> 00:05:32,110
l squared commutes
with p to the fourth.

87
00:05:32,110 --> 00:05:33,430
Why?

88
00:05:33,430 --> 00:05:38,380
Because, in fact, p
to the fourth commutes

89
00:05:38,380 --> 00:05:42,100
with any angular momentum
because p to the fourth

90
00:05:42,100 --> 00:05:44,830
is p squared times p squared.

91
00:05:44,830 --> 00:05:48,340
And p squared is
rotational invariant.

92
00:05:48,340 --> 00:05:51,850
p squared commutes with any l.

93
00:05:51,850 --> 00:05:55,850
If that's not
obvious intuitively,

94
00:05:55,850 --> 00:05:58,840
which it should become
something you trust--

95
00:05:58,840 --> 00:06:03,700
this is rotational invariant.
p squared dot product doesn't

96
00:06:03,700 --> 00:06:05,020
depend on rotation.

97
00:06:05,020 --> 00:06:08,145
If you have a p and you square
it or you have a rotated p

98
00:06:08,145 --> 00:06:09,520
and you square
it, it's the same.

99
00:06:09,520 --> 00:06:12,940
So p to the fourth
commutes with any component

100
00:06:12,940 --> 00:06:14,350
of angular momentum.

101
00:06:14,350 --> 00:06:16,960
So these two are written
like great facts,

102
00:06:16,960 --> 00:06:23,560
but the basic fact is that
p squared with any li is 0.

103
00:06:23,560 --> 00:06:27,270
And all this follows from here.

104
00:06:27,270 --> 00:06:30,990
But this is a
Hermitian operator.

105
00:06:30,990 --> 00:06:32,910
This is a Hermitian operator.

106
00:06:32,910 --> 00:06:37,480
And the various states,
when you have fixed n,

107
00:06:37,480 --> 00:06:40,060
you can have different l's.

108
00:06:40,060 --> 00:06:42,475
But when you have
different l's, there

109
00:06:42,475 --> 00:06:45,950
are different
eigenvalues of l squared.

110
00:06:45,950 --> 00:06:50,560
So in those cases, the
matrix element will vanish.

111
00:06:50,560 --> 00:06:55,630
When you have the same l's
but different m's, these

112
00:06:55,630 --> 00:06:59,010
are different eigenvalues of lz.

113
00:06:59,010 --> 00:07:01,390
So the matrix element
should also vanish.

114
00:07:01,390 --> 00:07:08,030
So this establishes rigorously
that that perturbation,

115
00:07:08,030 --> 00:07:13,390
p to the fourth, is
diagonal in that subspace.

116
00:07:13,390 --> 00:07:18,270
So the subspace relevant
here is this whole thing.

117
00:07:18,270 --> 00:07:23,310
And in this subspace,
it's completely diagonal.

118
00:07:23,310 --> 00:07:24,340
Good.

119
00:07:24,340 --> 00:07:30,720
So generally, this kind of point
is not emphasized too much.

120
00:07:30,720 --> 00:07:34,580
But it's, in fact, the most
important and more interesting

121
00:07:34,580 --> 00:07:39,000
and more difficult point
in this calculations.

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00:07:39,000 --> 00:07:42,780
We'll have one more
thing to say about this.

123
00:07:42,780 --> 00:07:47,810
But let's continue with this.

124
00:07:47,810 --> 00:07:51,870
I'll say the following.

125
00:07:51,870 --> 00:07:59,930
We use the Hermiticity
of p squared

126
00:07:59,930 --> 00:08:03,180
to move one p squared
to the other side.

127
00:08:03,180 --> 00:08:12,530
So Enl ml 1 is equal to
minus 1 over 8m cubed

128
00:08:12,530 --> 00:08:22,375
c squared p squared psi
nlm p squared psi nml--

129
00:08:26,917 --> 00:08:27,417
nlm.

130
00:08:30,810 --> 00:08:32,429
OK.

131
00:08:32,429 --> 00:08:34,230
We move this p to the fourth.

132
00:08:34,230 --> 00:08:35,789
It was p squared
times p squared.

133
00:08:35,789 --> 00:08:38,130
One p squared is Hermitian.

134
00:08:38,130 --> 00:08:39,480
We move it here.

135
00:08:39,480 --> 00:08:45,420
And then, instead of calculating
a billion derivatives here,

136
00:08:45,420 --> 00:08:53,330
you use the fact that p squared
over 2m plus v of r on the wave

137
00:08:53,330 --> 00:08:58,370
function is equal to the energy
of that wave function that

138
00:08:58,370 --> 00:09:00,650
depends on n times
the wave function.

139
00:09:04,730 --> 00:09:05,960
These are eigenstates.

140
00:09:05,960 --> 00:09:09,730
So p squared-- we don't
want to take derivatives,

141
00:09:09,730 --> 00:09:14,290
and those expectation values can
be replaced by a simpler thing.

142
00:09:14,290 --> 00:09:26,535
P squared on psi is just
2m En minus v of r psi.

143
00:09:31,120 --> 00:09:44,700
So Enlm 1 is equal to minus
1 over 8m cubed c squared.

144
00:09:44,700 --> 00:09:47,705
Here we have, well, the m's.

145
00:09:47,705 --> 00:09:54,360
Two m's are out, so we'll
put a 2 and an mc squared.

146
00:09:54,360 --> 00:09:56,730
Yep.

147
00:09:56,730 --> 00:10:14,370
En minus v of r psi En
minus v of r psi nlm nlm.

148
00:10:14,370 --> 00:10:16,540
OK.

149
00:10:16,540 --> 00:10:20,080
We got it to the point
where I think you can all

150
00:10:20,080 --> 00:10:22,990
agree this is doable.

151
00:10:22,990 --> 00:10:23,540
Why?

152
00:10:23,540 --> 00:10:27,460
Because, again, this
term is Hermitian,

153
00:10:27,460 --> 00:10:29,230
so you can put it
to the other side.

154
00:10:29,230 --> 00:10:32,800
And you'll have terms in which
you compute the expectation

155
00:10:32,800 --> 00:10:36,150
value on this
state of E squared.

156
00:10:36,150 --> 00:10:40,540
E squared is a number, so
it goes out, times 1, easy.

157
00:10:40,540 --> 00:10:44,840
En cross terms with vr
is the expectation of v

158
00:10:44,840 --> 00:10:46,810
of r in this state,
is the expectation

159
00:10:46,810 --> 00:10:49,540
of 1 over r in a state.

160
00:10:49,540 --> 00:10:50,140
That's easy.

161
00:10:50,140 --> 00:10:52,930
It comes from the
Virial theorem.

162
00:10:52,930 --> 00:10:56,830
Then you'll have the expectation
of v squared in a state,

163
00:10:56,830 --> 00:11:00,520
and that's the expectation of
1 over r squared in a state.

164
00:11:00,520 --> 00:11:02,770
You've also done it.

165
00:11:02,770 --> 00:11:07,570
So yes, getting all together,
getting the factors right

166
00:11:07,570 --> 00:11:13,360
would take you 15 minutes
or 20 minutes or whatever.

167
00:11:13,360 --> 00:11:17,930
But the answer is already clear.

168
00:11:17,930 --> 00:11:19,705
So let's write the answer.

169
00:11:33,080 --> 00:11:45,700
And the answer is that
Enl ml 1 relativistic is

170
00:11:45,700 --> 00:11:49,690
minus 1/8 alpha to the fourth.

171
00:11:49,690 --> 00:11:55,650
That's our very
recognizable factor.

172
00:11:55,650 --> 00:12:03,830
mc squared 4n over
l plus 1/2 minus 3.

173
00:12:03,830 --> 00:12:07,210
Now, fine structure
is something all of us

174
00:12:07,210 --> 00:12:09,860
must do at least
once in our life.

175
00:12:09,860 --> 00:12:13,300
So I do encourage you to
read the notes carefully

176
00:12:13,300 --> 00:12:15,160
and just do it.

177
00:12:15,160 --> 00:12:17,680
Just become familiar with it.

178
00:12:17,680 --> 00:12:21,640
It's a very nice subject,
and it's something

179
00:12:21,640 --> 00:12:25,180
you should understand.

180
00:12:25,180 --> 00:12:31,460
So here, again, I have to
do a comment about basis,

181
00:12:31,460 --> 00:12:35,080
and those comments keep
coming because it's

182
00:12:35,080 --> 00:12:36,760
an important subject.

183
00:12:36,760 --> 00:12:42,790
And I want to emphasize it.

184
00:12:42,790 --> 00:12:43,950
So what is the reason?

185
00:12:43,950 --> 00:12:46,630
The reason I wanted to comment
is because in a second,

186
00:12:46,630 --> 00:12:49,720
I'm going to do the
spin orbit term.

187
00:12:49,720 --> 00:12:55,800
And in that case, I would like
to work with a coupled basis.

188
00:12:55,800 --> 00:12:59,430
Here, I'm working with
the uncoupled basis.

189
00:12:59,430 --> 00:13:03,730
And really, this thing
is the expectation value

190
00:13:03,730 --> 00:13:24,130
of Hl relativistic in nl ml ms
nl ml ms. This is really that.

191
00:13:27,350 --> 00:13:33,350
I wrote psi nl ml, so you should
trust the first three labels.

192
00:13:33,350 --> 00:13:36,170
And ms goes for the ride.

193
00:13:36,170 --> 00:13:37,870
It's this spin.

194
00:13:37,870 --> 00:13:41,300
The operator you're putting
here, delta H relativistic,

195
00:13:41,300 --> 00:13:42,800
has nothing to do with spin.

196
00:13:42,800 --> 00:13:45,110
Could not change the
spin of the states.

197
00:13:45,110 --> 00:13:47,600
This has to be diagonal in spin.

198
00:13:47,600 --> 00:13:53,690
So this number you've
computed is nothing else

199
00:13:53,690 --> 00:14:00,920
than this overlap in
the uncoupled basis.

200
00:14:00,920 --> 00:14:11,460
So this calculation was
uncoupled basis matrix element.

201
00:14:15,180 --> 00:14:16,950
And we saw that it's diagonal.

202
00:14:21,980 --> 00:14:24,800
In fact, this whole
thing is nothing

203
00:14:24,800 --> 00:14:28,250
but the function of n and l.

204
00:14:31,110 --> 00:14:32,230
n and l.

205
00:14:32,230 --> 00:14:45,860
And independent
of ml and ms. OK.

206
00:14:45,860 --> 00:14:47,210
That's what we've calculated.

207
00:14:47,210 --> 00:14:48,965
So here is the question.

208
00:14:51,800 --> 00:15:05,190
We could consider this in
the coupled bases nlj mj.

209
00:15:08,160 --> 00:15:13,235
nlj mj.

210
00:15:21,120 --> 00:15:26,490
And the question is, do I
have to recalculate this

211
00:15:26,490 --> 00:15:28,155
in the coupled basis or not?

212
00:15:32,470 --> 00:15:37,290
And here is an argument that I
don't have to recalculate it.

213
00:15:37,290 --> 00:15:42,560
So I'm going to claim that
this is really equal to that.

214
00:15:42,560 --> 00:15:43,290
Just the same.

215
00:15:45,950 --> 00:15:48,520
It's the kind of thing that
makes you a little uneasy,

216
00:15:48,520 --> 00:15:50,700
but bear with me.

217
00:15:50,700 --> 00:15:53,310
Why should it be the same?

218
00:15:53,310 --> 00:16:01,080
Think of this as fixed
n and l because this

219
00:16:01,080 --> 00:16:02,790
depends on n and l.

220
00:16:02,790 --> 00:16:09,620
If we have the
hydrogen atom here,

221
00:16:09,620 --> 00:16:17,660
you'd take one of these
elements, one of these states--

222
00:16:17,660 --> 00:16:21,740
this is a fixed n, fixed l.

223
00:16:21,740 --> 00:16:25,861
And we're looking
at fixed n, fixed l.

224
00:16:25,861 --> 00:16:26,360
Yes.

225
00:16:26,360 --> 00:16:31,730
There are lots of states here
that have different ml and ms.

226
00:16:31,730 --> 00:16:37,070
But the answer doesn't depend
on ml and ms. In this basis,

227
00:16:37,070 --> 00:16:40,920
we are also looking at that
subspace, that multiplet,

228
00:16:40,920 --> 00:16:42,710
nl fixed.

229
00:16:42,710 --> 00:16:48,170
And they have reorganized
the states with j and mj.

230
00:16:48,170 --> 00:16:54,830
In fact, with two values of
j and several values of mj.

231
00:16:54,830 --> 00:16:57,860
But at the end of the
day, the coupled basis

232
00:16:57,860 --> 00:17:01,250
is another way to describe
these states coming

233
00:17:01,250 --> 00:17:05,750
from tensoring the l
multiplet with a spin 1/2.

234
00:17:05,750 --> 00:17:08,450
So it gives you two multiplets,
but they are the same states.

235
00:17:10,980 --> 00:17:17,460
So the fact that
every state here

236
00:17:17,460 --> 00:17:21,660
is some linear combination
of states in the uncoupled

237
00:17:21,660 --> 00:17:27,210
bases with different values of
ml and ms that add up to mj.

238
00:17:27,210 --> 00:17:30,280
But this answer doesn't
depend on ml and ms.

239
00:17:30,280 --> 00:17:33,480
So whatever linear
combination you need,

240
00:17:33,480 --> 00:17:36,330
it doesn't change because
the answer doesn't

241
00:17:36,330 --> 00:17:44,680
depend on ml and ms. So this
must be the same as that.

242
00:17:44,680 --> 00:17:46,390
I'll give another argument.

243
00:17:46,390 --> 00:17:50,950
Maybe a little more abstract,
but clearer perhaps.

244
00:17:50,950 --> 00:17:52,780
Think of this.

245
00:17:52,780 --> 00:17:54,640
So this can be--

246
00:17:54,640 --> 00:17:59,230
in the notes, I explain that by
changing basis and explaining

247
00:17:59,230 --> 00:18:02,170
why exactly
everything works out.

248
00:18:02,170 --> 00:18:04,870
But there is no need
for that argument

249
00:18:04,870 --> 00:18:07,600
if you think a little
more abstractly.

250
00:18:07,600 --> 00:18:09,460
Think of this subspace.

251
00:18:09,460 --> 00:18:14,200
Because with fixed n and
l, we have this subspace.

252
00:18:14,200 --> 00:18:20,720
In this subspace,
the uncoupled basis

253
00:18:20,720 --> 00:18:23,700
makes the perturbation diagonal.

254
00:18:23,700 --> 00:18:28,520
But more than diagonal, it makes
the perturbation proportional

255
00:18:28,520 --> 00:18:34,010
to the unit matrix, because
every eigenvalue is the same.

256
00:18:34,010 --> 00:18:37,780
Because in this subspace,
n and l is fixed.

257
00:18:37,780 --> 00:18:41,450
And yes, m and ms
change, but the answer

258
00:18:41,450 --> 00:18:43,100
doesn't depend on that.

259
00:18:43,100 --> 00:18:48,410
So this matrix, delta
H, in this subspace

260
00:18:48,410 --> 00:18:52,730
is proportional to
the unit matrix.

261
00:18:52,730 --> 00:18:55,820
And when a matrix is
proportional to the unit

262
00:18:55,820 --> 00:18:59,030
matrix, it is
proportional to the unit

263
00:18:59,030 --> 00:19:02,390
matrix in any orthogonal basis.

264
00:19:02,390 --> 00:19:05,990
A unit matrix
doesn't get rotated.

265
00:19:05,990 --> 00:19:09,440
So it should be a unit
matrix here as well,

266
00:19:09,440 --> 00:19:11,820
and it should be
the same matrix.

267
00:19:11,820 --> 00:19:14,290
So this is the same pair.