1
00:00:01,390 --> 00:00:06,170
PROFESSOR: OK, degenerate
perturbation theory.

2
00:00:06,170 --> 00:00:11,420
OK, we've done nicely
our non-degenerate case.

3
00:00:11,420 --> 00:00:19,450
So we'll get degenerate done
in a very clear way, I think.

4
00:00:32,060 --> 00:00:34,170
One more blackboard.

5
00:00:34,170 --> 00:00:37,950
So the first thing I want
to say about degenerate

6
00:00:37,950 --> 00:00:42,060
perturbation theories, when
it fails and why it fails

7
00:00:42,060 --> 00:00:43,350
and what goes wrong.

8
00:00:43,350 --> 00:00:49,170
So degenerate
perturbation theory.

9
00:01:04,480 --> 00:01:09,580
Trivial example to begin
with, why not, h of lambda,

10
00:01:09,580 --> 00:01:15,880
again, is equal to h0
plus lambda delta h.

11
00:01:15,880 --> 00:01:22,300
And for h0, we'll take original
Hamiltonian with energies 1

12
00:01:22,300 --> 00:01:28,755
and 1, and for
delta h, I'll take

13
00:01:28,755 --> 00:01:34,340
an off-diagonal
Hamiltonian, a Pauli matrix,

14
00:01:34,340 --> 00:01:36,305
and that's our delta h.

15
00:01:42,750 --> 00:01:51,270
And then we say OK, blindly,
not worry about this degeneracy,

16
00:01:51,270 --> 00:01:54,810
and use the formulas that we
have for non-degenerate states.

17
00:01:54,810 --> 00:01:58,410
And we would say well,
for the first state

18
00:01:58,410 --> 00:02:00,330
and the second state
what do we have?

19
00:02:04,420 --> 00:02:13,110
E, I'll call them 1
and 2, not 0 and 1.

20
00:02:13,110 --> 00:02:19,310
So let's call them
1 of lambda will

21
00:02:19,310 --> 00:02:27,080
be equal to the first value, 1,
plus, if the formula is right,

22
00:02:27,080 --> 00:02:32,540
delta h times lambda, 1, 1.

23
00:02:32,540 --> 00:02:38,250
The matrix element of the
perturbation in the 1, 1 state.

24
00:02:38,250 --> 00:02:42,020
Similarly, E2, the
other eigenvalue,

25
00:02:42,020 --> 00:02:49,070
should be 1, there's the
second eigenvalue, plus lambda,

26
00:02:49,070 --> 00:02:51,260
delta h 2, 2.

27
00:02:54,530 --> 00:02:58,080
That's what the
formula would say.

28
00:02:58,080 --> 00:03:02,430
But this is
completely incorrect.

29
00:03:02,430 --> 00:03:02,940
Why?

30
00:03:02,940 --> 00:03:12,150
Because delta h 1, 1 is the 1, 1
element of delta h, and it's 0.

31
00:03:12,150 --> 00:03:18,720
And the 2, 2 element
of delta h is also 0,

32
00:03:18,720 --> 00:03:22,770
and therefore this says 1 and
1, that the eigenvalues haven't

33
00:03:22,770 --> 00:03:25,260
changed.

34
00:03:25,260 --> 00:03:31,650
On the other hand, we know that
the matrix, the total matrix,

35
00:03:31,650 --> 00:03:39,660
which is 1 lambda, lambda
1 has eigenvectors,

36
00:03:39,660 --> 00:03:44,670
1 over square root of 2, 1, 1,
and 1 over square root of 2,

37
00:03:44,670 --> 00:03:49,600
1 minus 1, with eigenvalues--

38
00:03:49,600 --> 00:03:54,730
this is our eigenvectors
and eigenvalues--

39
00:03:54,730 --> 00:03:57,955
1 plus lambda and
1 minus lambda.

40
00:04:02,370 --> 00:04:05,660
So the eigenvalues of the
matrix are 1 plus lambda

41
00:04:05,660 --> 00:04:07,050
and 1 minus lambda.

42
00:04:07,050 --> 00:04:11,880
And this formula, to first order
in lambda, gave us nothing.

43
00:04:11,880 --> 00:04:16,430
It failed badly.

44
00:04:16,430 --> 00:04:17,184
It's a disaster.

45
00:04:20,630 --> 00:04:25,920
So you can say,
well, what happened?

46
00:04:25,920 --> 00:04:32,120
And maybe a clue is on the
fact of the eigenvectors.

47
00:04:37,500 --> 00:04:44,450
These are the eigenvectors
of the total matrix.

48
00:04:44,450 --> 00:04:47,140
These are in fact,
also the eigenvectors

49
00:04:47,140 --> 00:04:48,180
of the perturbation.

50
00:04:53,230 --> 00:04:56,440
On the other hand,
when we said OK, this

51
00:04:56,440 --> 00:05:00,880
is the first state,
the first eigenvector,

52
00:05:00,880 --> 00:05:04,840
we thought of it as 1,
0 on the Hamiltonian,

53
00:05:04,840 --> 00:05:07,840
on that eigenvector
gave you the energy 1.

54
00:05:07,840 --> 00:05:13,660
The second energy corresponds
to the eigenvector 0, 1.

55
00:05:13,660 --> 00:05:16,240
That's what we usually
use for our matrices.

56
00:05:18,800 --> 00:05:22,450
So here is the strange thing
that seems to have happened.

57
00:05:22,450 --> 00:05:25,510
For the original
Hamiltonian, [INAUDIBLE]

58
00:05:25,510 --> 00:05:28,690
we had this eigenvector
and this eigenvector,

59
00:05:28,690 --> 00:05:32,170
and then suddenly you
turn on the perturbation

60
00:05:32,170 --> 00:05:36,370
and lambda can be 10
to the minus 1,000,

61
00:05:36,370 --> 00:05:42,280
and already the eigenvectors
jump and become this one.

62
00:05:42,280 --> 00:05:44,860
They go from this
one to that one.

63
00:05:44,860 --> 00:05:48,870
You just add a perturbation
that you even cannot measure,

64
00:05:48,870 --> 00:05:51,859
and the eigenvectors change.

65
00:05:51,859 --> 00:05:52,400
That's crazy.

66
00:05:55,850 --> 00:06:02,570
The explanation is this
thing that we usually,

67
00:06:02,570 --> 00:06:08,180
sometimes forget to say in a
mathematically precise way.

68
00:06:08,180 --> 00:06:15,050
These are not the eigenvectors
of the original matrix.

69
00:06:15,050 --> 00:06:18,050
Since the original
matrix is degenerate,

70
00:06:18,050 --> 00:06:22,220
this is one possible
choice of eigenvectors.

71
00:06:22,220 --> 00:06:26,610
Any linear combination of
them is an eigenvector.

72
00:06:26,610 --> 00:06:30,720
So this space of
eigenvectors of this matrix

73
00:06:30,720 --> 00:06:34,100
is the span of these things.

74
00:06:34,100 --> 00:06:36,590
So there was no reason
to say, oh, these

75
00:06:36,590 --> 00:06:41,510
were the eigenvectors before,
and now they've changed.

76
00:06:41,510 --> 00:06:42,680
They're this.

77
00:06:42,680 --> 00:06:43,670
No.

78
00:06:43,670 --> 00:06:47,280
Before, you don't know
what are the eigenvectors.

79
00:06:47,280 --> 00:06:48,330
They're ambiguous.

80
00:06:48,330 --> 00:06:52,070
There's no way to
decide who they are.

81
00:06:52,070 --> 00:06:54,590
This, this, or any
linear combination.

82
00:06:54,590 --> 00:06:57,380
Remember, when you have
a degenerate eigenstate,

83
00:06:57,380 --> 00:07:00,980
any superposition of them
is a degenerate eigenstate.

84
00:07:00,980 --> 00:07:05,896
So the explanation is
that, in some sense,

85
00:07:05,896 --> 00:07:11,570
these eigenvectors of
the perturbed Hamiltonian

86
00:07:11,570 --> 00:07:15,740
is one of the possible choices
of a basis of eigenvectors

87
00:07:15,740 --> 00:07:19,730
for the original space,
and what the perturbation

88
00:07:19,730 --> 00:07:23,000
does is break the degeneracy.

89
00:07:23,000 --> 00:07:27,410
All those equivalent
eigenvectors suddenly

90
00:07:27,410 --> 00:07:29,110
are not all equivalent.

91
00:07:29,110 --> 00:07:30,680
There are some preferred ones.

92
00:07:34,420 --> 00:07:38,690
So the fact is that the
formulas are not working right,

93
00:07:38,690 --> 00:07:43,530
and we have to do this again.

94
00:07:43,530 --> 00:07:45,210
So here is what
we're going to do.

95
00:07:45,210 --> 00:07:52,100
We're going to set this up
for a systematic analysis

96
00:07:52,100 --> 00:07:55,280
to get this right.

97
00:07:55,280 --> 00:07:57,230
So systematic analysis.

98
00:08:02,920 --> 00:08:11,310
So again, we say h0 is known
and it will have an E of, say,

99
00:08:11,310 --> 00:08:23,220
1, 0, an E 2, 0, like that,
until you encounter an E n, 0

100
00:08:23,220 --> 00:08:28,680
that happens to be equal to
the next one, E, n plus 1, 0,

101
00:08:28,680 --> 00:08:30,750
and happens to be
equal to the next one,

102
00:08:30,750 --> 00:08:34,650
all the way to an
E n plus capital

103
00:08:34,650 --> 00:08:40,950
N minus 1, 0, which
is then smaller

104
00:08:40,950 --> 00:08:49,320
than the next one, which is
E n plus N and then again.

105
00:08:49,320 --> 00:08:54,090
So I'm saying in
funny ways the fact

106
00:08:54,090 --> 00:08:59,550
that, yes, you had some
states and suddenly you

107
00:08:59,550 --> 00:09:09,700
hit a collection of N energy
eigenvalues that are identical.

108
00:09:09,700 --> 00:09:13,500
They're all equal to En0.

109
00:09:13,500 --> 00:09:23,990
All n states have energy en0.

110
00:09:23,990 --> 00:09:28,940
So it is as if you
have this matrix h0

111
00:09:28,940 --> 00:09:30,690
and there are all
kinds of numbers.

112
00:09:30,690 --> 00:09:35,240
And then there's a whole block
where all the entries of size

113
00:09:35,240 --> 00:09:39,900
n by n where all the
entries are the same.

114
00:09:39,900 --> 00:09:48,540
So n, n plus 1, n plus 2, up
to N minus 1 is n entries,

115
00:09:48,540 --> 00:09:50,685
and they all have
the same energy.

116
00:09:53,790 --> 00:09:58,140
And the corresponding
eigenstates

117
00:09:58,140 --> 00:10:01,260
are going to be called n--

118
00:10:01,260 --> 00:10:04,420
we're going to use the
label n for all of them,

119
00:10:04,420 --> 00:10:07,800
even though the energies
are all the same.

120
00:10:07,800 --> 00:10:10,330
So there's just an E n energy.

121
00:10:10,330 --> 00:10:13,440
So we have N state,
so there will

122
00:10:13,440 --> 00:10:24,460
be an n1, an n2, up to an nN.

123
00:10:28,620 --> 00:10:39,210
All those states are going
to form an orthonormal basis

124
00:10:39,210 --> 00:10:43,080
for a space that we're
going to call vn.

125
00:10:45,870 --> 00:10:52,110
So the space spanned by this
state, we're going to go vn,

126
00:10:52,110 --> 00:10:55,620
with the n reminding you
it has to do with the n

127
00:10:55,620 --> 00:11:04,890
states, states with energy
described by the label n.

128
00:11:04,890 --> 00:11:08,580
So this is the
degenerate subspace.

129
00:11:08,580 --> 00:11:14,760
This is the space of all those
states that have energy En0.

130
00:11:14,760 --> 00:11:16,905
So I'll state that here.

131
00:11:20,970 --> 00:11:25,380
So it's quite important to
get our notation right here.

132
00:11:27,890 --> 00:11:30,200
Otherwise we don't
understand what's going on.

133
00:11:30,200 --> 00:11:42,290
So h0, n0k, they all
have energy En0, n0k.

134
00:11:45,440 --> 00:11:54,940
And this is for k equals
1 up to N. Now OK,

135
00:11:54,940 --> 00:11:58,159
this is your
degenerate subspace.

136
00:11:58,159 --> 00:11:59,200
These are all the states.

137
00:11:59,200 --> 00:12:03,270
And we now want to know
what happens to this space,

138
00:12:03,270 --> 00:12:07,780
to all these states,
when the energy turns on.

139
00:12:07,780 --> 00:12:11,890
This is not just a
theoretical construct.

140
00:12:11,890 --> 00:12:16,120
When you start asking what
happens to the first excited

141
00:12:16,120 --> 00:12:18,640
states in the hydrogen
atom, you already

142
00:12:18,640 --> 00:12:24,710
have four states there, l
equals 1, l equals 0 states,

143
00:12:24,710 --> 00:12:26,880
n equals 2.

144
00:12:26,880 --> 00:12:30,510
And you have degenerate
states and immediately you're

145
00:12:30,510 --> 00:12:34,360
stuck in this situation.

146
00:12:34,360 --> 00:12:46,240
So I will also use the
notation that the total space,

147
00:12:46,240 --> 00:12:56,920
a curly h, is the sum
of vn plus a v hat.

148
00:12:56,920 --> 00:12:59,420
This is called the direct sum.

149
00:12:59,420 --> 00:13:03,380
At this point it just
doesn't matter too much.

150
00:13:03,380 --> 00:13:08,940
These are linearly independent
subspaces, vn and v hat,

151
00:13:08,940 --> 00:13:14,320
and basically you think as v
hat as all the other states.

152
00:13:14,320 --> 00:13:17,470
Yes, there were these
degenerate subspace

153
00:13:17,470 --> 00:13:23,110
and all the other
states of our v hat.

154
00:13:23,110 --> 00:13:32,140
So v hat is the span
of all other states.

155
00:13:32,140 --> 00:13:37,790
And we'll call them
p0, or what letter?

156
00:13:37,790 --> 00:13:44,620
Yeah, p0, with p
equals some numbers.

157
00:13:44,620 --> 00:13:47,050
So this is a good notation.

158
00:13:47,050 --> 00:13:49,510
It allows you to distinguish.

159
00:13:49,510 --> 00:13:54,040
The degenerate
states use 2 labels.

160
00:13:54,040 --> 00:13:56,560
Maybe you only needed
1 label, but you

161
00:13:56,560 --> 00:13:58,660
will get confused
with the notation

162
00:13:58,660 --> 00:14:00,350
if you just use 1 label.

163
00:14:00,350 --> 00:14:03,130
So if that's the case,
degenerate state,

164
00:14:03,130 --> 00:14:05,560
but by putting
the n here you now

165
00:14:05,560 --> 00:14:07,870
know that you're talking
about degenerate states.

166
00:14:07,870 --> 00:14:09,520
When you have a
single label you're

167
00:14:09,520 --> 00:14:14,110
talking about the other
states in the space.

168
00:14:14,110 --> 00:14:19,390
So it's very good
when your notation

169
00:14:19,390 --> 00:14:22,000
makes it easy to
think and recognize

170
00:14:22,000 --> 00:14:24,890
the equations that you have.

171
00:14:24,890 --> 00:14:30,400
And v hat in this,
are orthogonal spaces.

172
00:14:30,400 --> 00:14:34,900
v hat is perpendicular to vn.

173
00:14:34,900 --> 00:14:39,700
The inner product of any state
outside of degenerate space,

174
00:14:39,700 --> 00:14:42,090
with a state in the
general space, is 0.

175
00:14:46,130 --> 00:14:46,630
OK.

176
00:14:51,030 --> 00:14:51,750
Very good.

177
00:14:51,750 --> 00:14:56,350
So let's now do the
important thing.

178
00:15:00,380 --> 00:15:02,560
I can do it here, I think.

179
00:15:02,560 --> 00:15:04,690
I'll do it here.

180
00:15:04,690 --> 00:15:06,360
So what do we want to do?

181
00:15:06,360 --> 00:15:08,310
What is the question here?

182
00:15:08,310 --> 00:15:10,660
The question is the following.

183
00:15:10,660 --> 00:15:14,370
We have a state n0k.

184
00:15:14,370 --> 00:15:19,350
We have N of those
states, and we

185
00:15:19,350 --> 00:15:23,250
want to figure out
what they become as you

186
00:15:23,250 --> 00:15:25,200
turn on the perturbation.

187
00:15:25,200 --> 00:15:30,030
So using the old notation,
we'll say they become this.

188
00:15:30,030 --> 00:15:32,340
And what is that going to be?

189
00:15:32,340 --> 00:15:37,650
It is going to be n0k,
the original state, what

190
00:15:37,650 --> 00:15:42,270
it was, plus order lambda,
the first correction,

191
00:15:42,270 --> 00:15:51,010
so n1 for this k state,
plus dot dot dot.

192
00:15:51,010 --> 00:15:55,210
For the energies, we have En0.

193
00:15:55,210 --> 00:15:58,780
That was the energy
of every state

194
00:15:58,780 --> 00:16:00,340
in the degenerate subspace.

195
00:16:00,340 --> 00:16:05,230
Now it's going to
become En lambda.

196
00:16:11,740 --> 00:16:16,590
But if I say En lambda I'm
already making a mistake,

197
00:16:16,590 --> 00:16:20,860
because I'm looking
at the fixed k.

198
00:16:20,860 --> 00:16:25,240
And a fixed k means we
chose one of these states,

199
00:16:25,240 --> 00:16:29,800
and any single one of
the states, the energy

200
00:16:29,800 --> 00:16:31,720
can grow in a different way.

201
00:16:31,720 --> 00:16:40,140
So I should call this
Enk, because we're

202
00:16:40,140 --> 00:16:46,210
talking about this state,
will be equal to En0--

203
00:16:46,210 --> 00:16:51,130
yes, they all have the same
first zeroth order energy.

204
00:16:51,130 --> 00:17:07,520
But then Enk1 plus lambda
squared Enk2 and so on.

205
00:17:07,520 --> 00:17:16,730
So this is the fate
of the kth state.

206
00:17:22,900 --> 00:17:28,569
And the degeneracy will
be broken to first order

207
00:17:28,569 --> 00:17:34,450
if these Enk's become different
numbers, because if they become

208
00:17:34,450 --> 00:17:37,120
different numbers in
lambda they're going

209
00:17:37,120 --> 00:17:39,410
to start splitting the states.

210
00:17:39,410 --> 00:17:43,810
Remember this picture we had
last time of several states

211
00:17:43,810 --> 00:17:45,100
at one point?

212
00:17:45,100 --> 00:17:48,610
If the order lambda
corrections are different,

213
00:17:48,610 --> 00:17:50,560
the states split.

214
00:17:50,560 --> 00:17:53,620
The degeneracy is resolved.

215
00:17:53,620 --> 00:17:56,680
And we can calculate
things more easily.

216
00:17:56,680 --> 00:18:01,620
If it's not resolved to
first order, it's harder.

217
00:18:01,620 --> 00:18:04,210
It will be all of next
lecture to figure out

218
00:18:04,210 --> 00:18:06,910
what happens in that case.

219
00:18:06,910 --> 00:18:10,300
We discussed last time
that the n1 correction

220
00:18:10,300 --> 00:18:14,980
didn't have any component
along the n0 vector.

221
00:18:14,980 --> 00:18:18,550
We could always
arrange that to happen.

222
00:18:18,550 --> 00:18:23,420
Here, we can arrange
something similar to happen.

223
00:18:23,420 --> 00:18:29,780
So we will still
assume, and we can

224
00:18:29,780 --> 00:18:36,350
check that's always
possible, that vnp,

225
00:18:36,350 --> 00:18:50,060
the corrections to the state k
at order p for p equals 1, 2,

226
00:18:50,060 --> 00:18:54,960
3, because it's not 0,
are orthogonal to n0k.

227
00:18:59,800 --> 00:19:05,830
So this state doesn't have
any component along n0k,

228
00:19:05,830 --> 00:19:07,960
and this state doesn't
have any component

229
00:19:07,960 --> 00:19:10,010
along the original one.

230
00:19:10,010 --> 00:19:13,300
It's always possible to do
that, just like we did it

231
00:19:13,300 --> 00:19:15,580
in the non-degenerate case.

232
00:19:18,220 --> 00:19:22,950
Moreover, this is something
I want you to notice.

233
00:19:22,950 --> 00:19:27,450
We're saying that
the state n1 doesn't

234
00:19:27,450 --> 00:19:31,860
have any component along
the state n0 for a given k,

235
00:19:31,860 --> 00:19:38,280
but the state n1 can
have components along n0

236
00:19:38,280 --> 00:19:41,910
for a different k.

237
00:19:41,910 --> 00:19:46,980
So while they've nth order
or a pth order correction

238
00:19:46,980 --> 00:19:49,410
doesn't have
component along this,

239
00:19:49,410 --> 00:19:54,240
it can have component
for an l here,

240
00:19:54,240 --> 00:19:56,430
where l is different from k.

241
00:19:56,430 --> 00:20:16,040
So it means that npk still
can have a component in vn.

242
00:20:16,040 --> 00:20:19,020
Remember, vn is
this whole thing.

243
00:20:19,020 --> 00:20:22,850
So if you're doing
the kth one, well, n1k

244
00:20:22,850 --> 00:20:24,770
doesn't have a component
along that one,

245
00:20:24,770 --> 00:20:28,320
but it may have a component
along all the others.

246
00:20:28,320 --> 00:20:32,120
So it may still have a
component, that correction,

247
00:20:32,120 --> 00:20:33,110
in vn.

248
00:20:33,110 --> 00:20:35,990
In fact, that's what will
make the problem a little hard

249
00:20:35,990 --> 00:20:46,040
to do, but it's a good thing to
try to figure out this thing.

250
00:20:46,040 --> 00:20:50,560
So what is the equation
we want to solve?

251
00:20:50,560 --> 00:20:57,250
The usual equation, h
of lambda on nk lambda

252
00:20:57,250 --> 00:21:05,860
is equal to Enk of
lambda nk of lambda.

253
00:21:05,860 --> 00:21:08,830
So we produce the
general state that's

254
00:21:08,830 --> 00:21:11,380
its energy for the
full Hamiltonian.

255
00:21:17,360 --> 00:21:19,350
That's our Schrodinger equation.

256
00:21:19,350 --> 00:21:20,920
So what do we have to do?

257
00:21:20,920 --> 00:21:25,640
We have to do exactly
what we did before,

258
00:21:25,640 --> 00:21:29,930
plug in that series, separate
the terms with various lambdas,

259
00:21:29,930 --> 00:21:33,150
and see what we get.

260
00:21:33,150 --> 00:21:34,580
So here is what we get.

261
00:21:34,580 --> 00:21:48,830
To order lambda to the 0,
you get h0 minus En0 on n0k

262
00:21:48,830 --> 00:21:49,750
equals 0.

263
00:21:53,670 --> 00:21:59,340
And that equation, as it was
the case for the non-degenerate

264
00:21:59,340 --> 00:22:01,770
case--

265
00:22:01,770 --> 00:22:04,620
sorry for the redundancy--

266
00:22:04,620 --> 00:22:06,420
it's trivially satisfied.

267
00:22:06,420 --> 00:22:10,560
We've stated that those are the
energies in the top equation

268
00:22:10,560 --> 00:22:11,490
on that blackboard.

269
00:22:11,490 --> 00:22:13,980
So this equation we don't
need to worry about.

270
00:22:13,980 --> 00:22:17,150
That's not too difficult.

271
00:22:17,150 --> 00:22:31,180
Lambda 1, h0 minus
En0 and 1k will

272
00:22:31,180 --> 00:22:40,495
be equal to Enk1
minus delta h n0k.

273
00:22:50,180 --> 00:22:51,910
Last equation.

274
00:22:51,910 --> 00:22:58,980
I'm starting to get lazy to
write them out, but we must.

275
00:23:02,190 --> 00:23:25,790
n2k is equal to minus delta
h and 1k plus Enk2 and 0k.

276
00:23:29,040 --> 00:23:38,100
OK and these equations
are equations for k fixed,

277
00:23:38,100 --> 00:23:42,600
but every term in the
equation has the same k,

278
00:23:42,600 --> 00:23:46,560
but k then can run
from 1 up to n.

279
00:23:46,560 --> 00:23:52,675
So three equations times
n times, there we go.

280
00:23:55,520 --> 00:23:58,115
OK, these are our
equations this time.

281
00:24:01,570 --> 00:24:03,060
And we need to understand them.

282
00:24:05,620 --> 00:24:08,430
In fact, we need to solve them.

283
00:24:08,430 --> 00:24:13,920
What we've gained experience
with the non-degenerate case

284
00:24:13,920 --> 00:24:16,020
is going to come
very useful here,

285
00:24:16,020 --> 00:24:19,440
although some things are not
going to be exactly the same.

286
00:24:19,440 --> 00:24:25,310
We're going to try to find
the energy corrections here,

287
00:24:25,310 --> 00:24:29,780
but calculating the state n1
is going to be a little harder.

288
00:24:29,780 --> 00:24:31,250
We won't finish today--

289
00:24:31,250 --> 00:24:33,350
we won't have much time left--

290
00:24:33,350 --> 00:24:37,900
but what's going to
happen is that you

291
00:24:37,900 --> 00:24:47,400
can calculate the state n1,
the part in the space v hat.

292
00:24:47,400 --> 00:24:50,550
But the part in the
degenerate subspace,

293
00:24:50,550 --> 00:24:54,500
where I said that npk
still can have a component

294
00:24:54,500 --> 00:24:57,870
in the degenerate subspace,
cannot be calculated from this

295
00:24:57,870 --> 00:24:59,460
equation.

296
00:24:59,460 --> 00:25:03,570
So even when we're finished
doing this first equation,

297
00:25:03,570 --> 00:25:05,160
you're going to
find this equation,

298
00:25:05,160 --> 00:25:09,720
you still have not calculated
all of the states n1.

299
00:25:09,720 --> 00:25:13,080
You're going to have to go
through the second equation

300
00:25:13,080 --> 00:25:16,380
to find the missing
part of the state n1.

301
00:25:16,380 --> 00:25:20,720
So it's going to be
pretty interesting.