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PROFESSOR: All right.

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00:00:26,380 --> 00:00:28,050
Welcome, everyone.

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00:00:28,050 --> 00:00:29,710
Hi.

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00:00:29,710 --> 00:00:32,850
So today we're going
to pick up where

12
00:00:32,850 --> 00:00:36,430
we left off last time in our
study of angular momentum

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00:00:36,430 --> 00:00:38,865
and rotations in
quantum mechanics.

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00:00:38,865 --> 00:00:40,240
Before I get
started, let me open

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00:00:40,240 --> 00:00:43,520
up for questions, pragmatic
and physics related.

16
00:00:48,350 --> 00:00:49,786
Yeah?

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00:00:49,786 --> 00:00:52,744
AUDIENCE: When we were solving
for the 3D harmonic oscillator

18
00:00:52,744 --> 00:00:55,866
we solved for the energy
eigenfunction that

19
00:00:55,866 --> 00:00:59,153
was a product of phi
x, phi y, and phi z.

20
00:00:59,153 --> 00:01:02,604
We made an assumption
that phi e was

21
00:01:02,604 --> 00:01:05,972
equal to phi sub n
of x plus that sum.

22
00:01:05,972 --> 00:01:08,433
How did you get
from [INAUDIBLE]??

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00:01:08,433 --> 00:01:09,100
PROFESSOR: Good.

24
00:01:09,100 --> 00:01:11,100
So what we had was that
we had that the energy--

25
00:01:11,100 --> 00:01:15,590
we wanted to find the energy
of the 3D harmonic oscillator.

26
00:01:15,590 --> 00:01:18,345
And we wanted to find the energy
eigenfunctions and eigenvalues.

27
00:01:18,345 --> 00:01:19,970
And they way we did
this was by saying,

28
00:01:19,970 --> 00:01:22,540
look, the energy of the 3D
harmonic oscillator, which

29
00:01:22,540 --> 00:01:24,730
I can think of as a
function of x and px

30
00:01:24,730 --> 00:01:28,480
and y and py and z and
pz, has this nice form.

31
00:01:28,480 --> 00:01:31,540
We could write it as
the energy operator

32
00:01:31,540 --> 00:01:35,080
purely in terms of x,
p squared x upon 2m

33
00:01:35,080 --> 00:01:37,710
plus m omega squared
upon 2x squared.

34
00:01:37,710 --> 00:01:42,500
Plus-- so this is a single
1D harmonic oscillator energy

35
00:01:42,500 --> 00:01:44,150
operator in the x direction.

36
00:01:44,150 --> 00:01:48,730
Plus E 1D in the y
direction, plus a harmonic

37
00:01:48,730 --> 00:01:52,510
oscillator energy 1D
in the z direction.

38
00:01:52,510 --> 00:01:54,110
So that was the
first observation.

39
00:01:54,110 --> 00:01:58,430
And then we said that given that
this splits in this fashion,

40
00:01:58,430 --> 00:02:03,230
I'm going to write my
energy eigenfunction, phi

41
00:02:03,230 --> 00:02:08,710
of x, y, and z in separated
form as a product.

42
00:02:08,710 --> 00:02:16,620
Phi x of x, phi y of
y, and phi z of z.

43
00:02:16,620 --> 00:02:19,830
And we used this and deduced
that in order for this

44
00:02:19,830 --> 00:02:22,280
to be an eigenfunction of
the 3D harmonic oscillator

45
00:02:22,280 --> 00:02:25,140
it must be true that
5x of x was itself

46
00:02:25,140 --> 00:02:28,520
an eigenfunction of the
x harmonic oscillator

47
00:02:28,520 --> 00:02:32,716
equals with some energy
epsilon, which I will

48
00:02:32,716 --> 00:02:35,870
call epsilon sub x, phi sub x.

49
00:02:35,870 --> 00:02:37,450
And ditto for y and z.

50
00:02:37,450 --> 00:02:40,770
And if this is true, if phi sub
x is an eigenfunction of the 1D

51
00:02:40,770 --> 00:02:43,780
harmonic oscillator, then
this is an eigenfunction

52
00:02:43,780 --> 00:02:45,272
of the 3D harmonic oscillator.

53
00:02:45,272 --> 00:02:46,730
But we know what
the eigenfunctions

54
00:02:46,730 --> 00:02:47,850
are of the harmonic oscillator.

55
00:02:47,850 --> 00:02:49,490
The eigenfunctions
of the 1D harmonic

56
00:02:49,490 --> 00:02:51,130
oscillator we gave a name.

57
00:02:51,130 --> 00:02:53,790
We call them pi sub n.

58
00:02:53,790 --> 00:02:56,850
So what we then said
was, look, if this

59
00:02:56,850 --> 00:02:59,790
is an eigenfunction of
the 1D harmonic oscillator

60
00:02:59,790 --> 00:03:04,930
in the x direction, then
it's labeled by an n.

61
00:03:04,930 --> 00:03:06,900
And if this one is
in the y direction,

62
00:03:06,900 --> 00:03:08,150
it's labeled by an l.

63
00:03:08,150 --> 00:03:10,630
And if this one is in the z
direction it's labeled by a z.

64
00:03:10,630 --> 00:03:11,980
But on top of that we know more.

65
00:03:11,980 --> 00:03:13,970
We know what these
energy eigenvalues are.

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00:03:13,970 --> 00:03:16,270
The energy eigenvalues
corresponding to these guys

67
00:03:16,270 --> 00:03:19,750
are if this is
phi n, this is En.

68
00:03:19,750 --> 00:03:23,460
I can simply write this as En
for the 1D harmonic oscillator.

69
00:03:23,460 --> 00:03:27,570
And taking this, using the fact
that they're 1D eigenfunctions

70
00:03:27,570 --> 00:03:29,320
and plugging it into
the energy eigenvalue

71
00:03:29,320 --> 00:03:31,040
equation for the 3D
harmonic oscillator

72
00:03:31,040 --> 00:03:33,510
tells us that the energy
eigenvalues for the 3D case

73
00:03:33,510 --> 00:03:40,680
are of the form E1d
x plus E1d in the y

74
00:03:40,680 --> 00:03:44,750
plus E1d in the z, which is
equal to, since these were

75
00:03:44,750 --> 00:03:50,600
the same frequency, h bar
omega times n plus l plus m,

76
00:03:50,600 --> 00:03:53,820
from each of them
a 1/2, so plus 3/2.

77
00:03:53,820 --> 00:03:54,320
Cool?

78
00:03:54,320 --> 00:03:55,460
That answer your question?

79
00:03:55,460 --> 00:03:56,310
Excellent.

80
00:03:56,310 --> 00:03:57,541
Other questions?

81
00:03:57,541 --> 00:03:59,505
Yeah?

82
00:03:59,505 --> 00:04:01,469
AUDIENCE: Energy
and angular momentum

83
00:04:01,469 --> 00:04:03,257
have to be related
somehow, right?

84
00:04:03,257 --> 00:04:03,924
PROFESSOR: Yeah.

85
00:04:03,924 --> 00:04:05,397
AUDIENCE: I mean, of course.

86
00:04:05,397 --> 00:04:06,390
Because it's both.

87
00:04:06,390 --> 00:04:07,140
PROFESSOR: Indeed.

88
00:04:07,140 --> 00:04:09,340
AUDIENCE: [INAUDIBLE].

89
00:04:09,340 --> 00:04:13,741
The thing is, we have a ladder.

90
00:04:13,741 --> 00:04:14,829
Is there limits on them?

91
00:04:14,829 --> 00:04:16,079
PROFESSOR: Very good question.

92
00:04:16,079 --> 00:04:17,420
So this is where we're
going to pick up.

93
00:04:17,420 --> 00:04:18,380
Let me rephrase this.

94
00:04:18,380 --> 00:04:19,588
This is really two questions.

95
00:04:19,588 --> 00:04:20,523
Question number one.

96
00:04:20,523 --> 00:04:22,690
Look, there should be a
relationship between angular

97
00:04:22,690 --> 00:04:23,780
momentum and energy.

98
00:04:23,780 --> 00:04:25,697
But we're just talking
about angular momentum.

99
00:04:25,697 --> 00:04:26,510
Why?

100
00:04:26,510 --> 00:04:28,410
Second question, look,
we've got a ladder.

101
00:04:28,410 --> 00:04:29,910
But is the ladder infinite?

102
00:04:29,910 --> 00:04:31,280
So let me come back to
the second question.

103
00:04:31,280 --> 00:04:32,850
That's going to be the
beginning of the lecture.

104
00:04:32,850 --> 00:04:34,900
On the first question,
yes angular momentum

105
00:04:34,900 --> 00:04:37,108
is going to play a role when
we calculate the energy.

106
00:04:37,108 --> 00:04:38,723
But two quick things to note.

107
00:04:38,723 --> 00:04:40,140
First off, consider
a system which

108
00:04:40,140 --> 00:04:42,637
is spherically symmetric,
rotationally invariant.

109
00:04:42,637 --> 00:04:44,970
That means that the energy
doesn't depend on a rotation.

110
00:04:44,970 --> 00:04:47,137
If I rotate the system I
haven't changed the energy.

111
00:04:47,137 --> 00:04:50,250
So if the system is
rotationally invariant,

112
00:04:50,250 --> 00:04:52,550
that's going to imply some
constraints on the energy

113
00:04:52,550 --> 00:04:55,390
eigenvalues and how they
depend on the angular momentum,

114
00:04:55,390 --> 00:04:56,880
as we discussed last time.

115
00:04:56,880 --> 00:04:58,422
Let me say that
slightly differently.

116
00:05:00,630 --> 00:05:03,620
When we talk about the free
particle, 1D free particle--

117
00:05:03,620 --> 00:05:05,250
we've talked about
this one to death.

118
00:05:05,250 --> 00:05:06,850
Take the 1D free particle.

119
00:05:06,850 --> 00:05:08,870
We can write the
energy eigenfunctions

120
00:05:08,870 --> 00:05:11,750
as momentum eigenfunctions,
because the momentum commutes

121
00:05:11,750 --> 00:05:13,765
with the energy.

122
00:05:13,765 --> 00:05:15,723
And so the way the
eigenfunctions of the energy

123
00:05:15,723 --> 00:05:18,240
operate are indeed e to
iKX there are plane waves,

124
00:05:18,240 --> 00:05:20,980
they're eigenfunctions of the
momentum operator as well.

125
00:05:20,980 --> 00:05:23,085
Similarly, when we
talk about a 3D system

126
00:05:23,085 --> 00:05:25,210
it's going to be useful in
talking about the energy

127
00:05:25,210 --> 00:05:29,690
eigenvalues to know a basis of
eigenfunctions of the angular

128
00:05:29,690 --> 00:05:30,600
momentum operator.

129
00:05:30,600 --> 00:05:32,320
Knowing the angular
momentum operator

130
00:05:32,320 --> 00:05:35,713
is going to allow us to
write energy eigenfunctions

131
00:05:35,713 --> 00:05:37,130
in a natural way
and a simply way,

132
00:05:37,130 --> 00:05:39,530
in the same way that knowing
the momentum operator allowed

133
00:05:39,530 --> 00:05:40,905
us to write energy
eigenfunctions

134
00:05:40,905 --> 00:05:43,700
in a simple way in the 1D case.

135
00:05:43,700 --> 00:05:44,717
That make sense?

136
00:05:44,717 --> 00:05:47,300
We're going to have a glorified
version of the Fourier theorum

137
00:05:47,300 --> 00:05:49,090
where instead of
something over e to iKX,

138
00:05:49,090 --> 00:05:50,150
we're going to have something
over angular momentum

139
00:05:50,150 --> 00:05:51,072
eigenstates.

140
00:05:51,072 --> 00:05:53,130
And those are called
the spherical harmonics.

141
00:05:53,130 --> 00:05:55,750
And they are the analog
of Fourier expansion

142
00:05:55,750 --> 00:05:56,912
for this year.

143
00:05:56,912 --> 00:05:57,620
But you're right.

144
00:05:57,620 --> 00:05:58,960
We're going to have
to understand how

145
00:05:58,960 --> 00:06:00,252
that interacts with the energy.

146
00:06:00,252 --> 00:06:02,160
And that'll be the topic
of the next lecture.

147
00:06:02,160 --> 00:06:04,160
We're going to finish up
angular momentum today.

148
00:06:04,160 --> 00:06:06,950
Other questions?

149
00:06:06,950 --> 00:06:11,360
OK, so from last time, these
are the commutation relations

150
00:06:11,360 --> 00:06:13,450
which we partially
derived in lecture

151
00:06:13,450 --> 00:06:16,630
and which you will be
driving on your problem set.

152
00:06:16,630 --> 00:06:18,140
It's a really good exercise.

153
00:06:18,140 --> 00:06:19,230
Commit these to memory.

154
00:06:19,230 --> 00:06:20,840
They're your friends.

155
00:06:20,840 --> 00:06:23,130
Key thing here to keep in mind.

156
00:06:23,130 --> 00:06:26,270
h bar has units of angular
momentum, so this makes sense.

157
00:06:26,270 --> 00:06:29,270
Angular momentum, angular
momentum, angular momentum.

158
00:06:29,270 --> 00:06:34,220
So when you see an h
bar in this setting,

159
00:06:34,220 --> 00:06:37,520
its job, in some sense, is to
make everything dimensionally

160
00:06:37,520 --> 00:06:39,330
sensible.

161
00:06:39,330 --> 00:06:42,545
So the important things here are
that lx and ly do not commute.

162
00:06:42,545 --> 00:06:44,840
They commute to lz.

163
00:06:44,840 --> 00:06:47,070
Can you have a state with
definite angular momentum

164
00:06:47,070 --> 00:06:49,445
in the x direction and definite
angular momentum in the y

165
00:06:49,445 --> 00:06:50,622
direction simultaneously?

166
00:06:50,622 --> 00:06:52,330
No, because of this
commutation relation.

167
00:06:52,330 --> 00:06:55,760
It would have to vanish.

168
00:06:55,760 --> 00:06:58,065
This is, say, the x component
of the angular momentum.

169
00:06:58,065 --> 00:07:00,190
Can you have a state with
definite angular momentum

170
00:07:00,190 --> 00:07:03,850
in the x direction and total
angular momentum all squared?

171
00:07:03,850 --> 00:07:04,380
Yes.

172
00:07:04,380 --> 00:07:04,880
OK, great.

173
00:07:04,880 --> 00:07:06,570
That's going to be
important for us.

174
00:07:06,570 --> 00:07:08,410
So in order to construct
the eigenfunctions

175
00:07:08,410 --> 00:07:09,993
it turns out to be
useful to construct

176
00:07:09,993 --> 00:07:12,140
these so-called raising
and lowering operators,

177
00:07:12,140 --> 00:07:15,167
which are Lx plus iLy.

178
00:07:15,167 --> 00:07:16,750
They have a couple
of nice properties.

179
00:07:16,750 --> 00:07:19,260
The first is, since these are
built up out of Lx and Ly,

180
00:07:19,260 --> 00:07:20,802
both of which commute
with L squared,

181
00:07:20,802 --> 00:07:24,280
the L plus minuses
commute with L squared.

182
00:07:24,280 --> 00:07:25,370
So these guys commute.

183
00:07:25,370 --> 00:07:29,990
So if we have an
eigenfunction of L squared,

184
00:07:29,990 --> 00:07:34,700
acting with L plus does
not change its eigenvalue.

185
00:07:34,700 --> 00:07:37,130
Similarly, L plus
commuting with Lz

186
00:07:37,130 --> 00:07:42,360
gives us h bar L plus or minus,
with a plus or minus out front.

187
00:07:42,360 --> 00:07:47,570
This is just like the raising
and lowering operators

188
00:07:47,570 --> 00:07:49,060
for the harmonic oscillator.

189
00:07:49,060 --> 00:07:52,850
But instead of the energy we
have the angular momentum.

190
00:07:52,850 --> 00:07:57,260
So this is going to tell us
that the angular momentum

191
00:07:57,260 --> 00:07:59,970
eigenvalues, the
eigenvalues of Lz,

192
00:07:59,970 --> 00:08:02,330
are shifted by
plus or minus h bar

193
00:08:02,330 --> 00:08:04,140
when we raise or
lower with L plus or L

194
00:08:04,140 --> 00:08:07,200
minus, just like the
energy was shifted--

195
00:08:07,200 --> 00:08:12,130
for the 1d harmonic oscillator
the energy was shifted,

196
00:08:12,130 --> 00:08:15,990
plus h bar omega a
dagger, was shifted

197
00:08:15,990 --> 00:08:17,800
by h bar omega when
we acted with a plus

198
00:08:17,800 --> 00:08:18,860
on an energy eigenstate.

199
00:08:18,860 --> 00:08:19,370
Same thing.

200
00:08:22,580 --> 00:08:24,930
Questions on the commutators
before we get going?

201
00:08:24,930 --> 00:08:27,090
In some sense, we're
going to just to just take

202
00:08:27,090 --> 00:08:28,757
advantage of these
commutation relations

203
00:08:28,757 --> 00:08:30,820
and explore their
consequences today.

204
00:08:30,820 --> 00:08:40,360
So our goal is going to be
to build the eigenfunctions

205
00:08:40,360 --> 00:08:47,347
and eigenvalues of the
angular momentum operators,

206
00:08:47,347 --> 00:08:49,680
and in particular of the most
angular momentum operators

207
00:08:49,680 --> 00:08:51,980
we [INAUDIBLE] complete set
of commuting observables,

208
00:08:51,980 --> 00:08:53,310
L squared and Lz.

209
00:08:53,310 --> 00:08:54,810
You might complain,
look, why Lz?

210
00:08:54,810 --> 00:08:55,310
Whoops.

211
00:08:55,310 --> 00:08:56,550
I don't mean a commutator.

212
00:08:56,550 --> 00:08:57,710
I mean the set.

213
00:08:57,710 --> 00:08:59,480
You might say, why Lz?

214
00:08:59,480 --> 00:09:00,620
Why not Lx?

215
00:09:00,620 --> 00:09:03,247
And if you call this
the z direction then

216
00:09:03,247 --> 00:09:04,830
I will simply choose
a new basis where

217
00:09:04,830 --> 00:09:06,380
this is called the x direction.

218
00:09:06,380 --> 00:09:08,110
So it makes no
difference whatsoever.

219
00:09:08,110 --> 00:09:08,950
It's just a name.

220
00:09:08,950 --> 00:09:10,490
The reason we're
going to choose Lz

221
00:09:10,490 --> 00:09:13,480
is because that coordinate
system plays nicely

222
00:09:13,480 --> 00:09:16,710
with spherical coordinates,
just the conventional choice

223
00:09:16,710 --> 00:09:19,260
of spherical coordinates where
theta equals 0 is the up axis.

224
00:09:19,260 --> 00:09:20,760
But there's nothing
deep about that.

225
00:09:20,760 --> 00:09:24,670
We could have
taken any of these.

226
00:09:24,670 --> 00:09:26,220
OK.

227
00:09:26,220 --> 00:09:27,740
So this is our goal.

228
00:09:27,740 --> 00:09:28,670
So let's get started.

229
00:09:28,670 --> 00:09:32,902
So first, because of these
commutation relations

230
00:09:32,902 --> 00:09:34,360
and in particular
this one, we know

231
00:09:34,360 --> 00:09:36,485
that we can find common
eigenfunctions of L squared

232
00:09:36,485 --> 00:09:37,480
and Lz.

233
00:09:37,480 --> 00:09:41,610
Let us call those common
eigenfunctions by a name,

234
00:09:41,610 --> 00:09:47,720
Y sub lm such that--
so let these guys

235
00:09:47,720 --> 00:09:56,460
be the common eigenfunctions
of L squared and Lz.

236
00:09:56,460 --> 00:10:02,120
I.e., L squared Ylm is equal
to-- well first off, units.

237
00:10:02,120 --> 00:10:04,220
This has units of
angular momentum squared,

238
00:10:04,220 --> 00:10:05,040
h bar squared.

239
00:10:05,040 --> 00:10:06,590
So that got rid of the units.

240
00:10:06,590 --> 00:10:08,490
And we want our eigenvalue, lm.

241
00:10:08,490 --> 00:10:11,495
And because I know
the answer, I'm

242
00:10:11,495 --> 00:10:13,120
going to give--
instead of calling this

243
00:10:13,120 --> 00:10:16,210
a random dimensionless number,
which would be the eigenvalue,

244
00:10:16,210 --> 00:10:19,950
I'm going to call it a very
specific thing, l, l plus 1.

245
00:10:19,950 --> 00:10:21,880
This is a slightly
grotesque thing to do,

246
00:10:21,880 --> 00:10:24,120
but it will make the
algebra much easier.

247
00:10:24,120 --> 00:10:28,080
So similarly, so that's
what the little l is.

248
00:10:28,080 --> 00:10:30,080
Little l is labeling the
eigenvalue of L squared

249
00:10:30,080 --> 00:10:31,747
and the actual value
of that eigenvalue.

250
00:10:31,747 --> 00:10:33,668
I'm just calling h
bar squared ll plus 1.

251
00:10:33,668 --> 00:10:35,460
That doesn't tell you
anything interesting.

252
00:10:35,460 --> 00:10:37,430
This was already a
real positive number.

253
00:10:37,430 --> 00:10:40,660
So this could have been any
real positive number as well,

254
00:10:40,660 --> 00:10:43,260
by tuning L.

255
00:10:43,260 --> 00:10:48,140
Similarly, Lz Ylm-- I want
this to be an eigenfunction--

256
00:10:48,140 --> 00:10:49,795
this has units of
angular momentum.

257
00:10:49,795 --> 00:10:50,670
So I'll put an h bar.

258
00:10:50,670 --> 00:10:52,640
Now we have a dimensionless
coefficient Ylm.

259
00:10:52,640 --> 00:10:55,870
And I'll simply call that m.

260
00:10:55,870 --> 00:10:57,690
So if you will, these
are the definitions

261
00:10:57,690 --> 00:11:02,440
of the symbols m and little l.

262
00:11:02,440 --> 00:11:05,600
And Ylm are just the names I'm
giving to the angular momentum

263
00:11:05,600 --> 00:11:06,820
eigenfunctions.

264
00:11:06,820 --> 00:11:07,830
Cool?

265
00:11:07,830 --> 00:11:09,330
So I haven't actually
done anything.

266
00:11:09,330 --> 00:11:11,120
I've just told you that
these are the eigenfunctions.

267
00:11:11,120 --> 00:11:13,493
What we want to know is what
properties do they have,

268
00:11:13,493 --> 00:11:15,910
and what are the actual allowed
values of the eigenvalues.

269
00:11:18,680 --> 00:11:30,470
So the two key things to note
are first that L plus and minus

270
00:11:30,470 --> 00:11:32,160
leave the eigenvalue
of L squared alone.

271
00:11:32,160 --> 00:11:33,077
So they leave l alone.

272
00:11:36,350 --> 00:11:40,890
So this is the statement
that L squared on L plus Ylm

273
00:11:40,890 --> 00:11:45,230
is equal to L plus
on L squared Ylm

274
00:11:45,230 --> 00:11:50,180
is equal to h bar
squared ll plus 1,

275
00:11:50,180 --> 00:11:56,760
the eigenvalue of L squared
acting on Ylm, L plus Ylm.

276
00:11:56,760 --> 00:12:00,120
And so L plus Ylm is just as
much an eigenfunction of L

277
00:12:00,120 --> 00:12:05,960
squared as Ylm was itself,
with the same eigenvalue.

278
00:12:05,960 --> 00:12:06,850
Two.

279
00:12:06,850 --> 00:12:07,932
So that came from--

280
00:12:10,710 --> 00:12:14,200
where are we--- this
commutation relation.

281
00:12:14,200 --> 00:12:23,620
OK, so similarly, L plus
minus raise or lower m by one.

282
00:12:26,420 --> 00:12:29,290
And the way to see that is to
do exactly the same computation,

283
00:12:29,290 --> 00:12:37,530
Lz on L plus, for example,
Ylm is equal to L plus--

284
00:12:37,530 --> 00:12:39,750
now we can write
the commutator--

285
00:12:39,750 --> 00:12:50,110
Lz, L plus, plus L plus Lz Ylm.

286
00:12:50,110 --> 00:12:53,160
But the commutator of Lz
with L plus we already have,

287
00:12:53,160 --> 00:12:54,820
is plus h bar L plus.

288
00:12:54,820 --> 00:12:58,340
So this is equal
to h bar L plus.

289
00:12:58,340 --> 00:13:01,210
And from this term, L plus
Lz of Ylm, Lz acting on Ylm

290
00:13:01,210 --> 00:13:08,360
gives us h bar m, plus h bar
m L plus Ylm is equal to,

291
00:13:08,360 --> 00:13:11,800
pulling this out, this is h bar
times m plus 1 times L plus.

292
00:13:11,800 --> 00:13:16,810
h bar m plus 1 L plus Ylm.

293
00:13:16,810 --> 00:13:21,390
So L plus has raised
the eigenvalue m by one.

294
00:13:21,390 --> 00:13:24,270
This state, what we get by
acting on Ylm with the raising

295
00:13:24,270 --> 00:13:26,987
operator is a thing
with m greater by one.

296
00:13:26,987 --> 00:13:28,820
And that came from the
commutation relation.

297
00:13:28,820 --> 00:13:30,530
And if we had done the
same thing with minus,

298
00:13:30,530 --> 00:13:33,060
if you go through the minus
signs, it just gives us this.

299
00:13:36,660 --> 00:13:39,460
What does that tell us?

300
00:13:39,460 --> 00:13:41,460
What this tells us is we
get a ladder of states.

301
00:13:44,773 --> 00:13:46,190
Let's look at what
they look like.

302
00:13:46,190 --> 00:13:50,110
Each ladder, for a given value
of L, if you raise with L plus

303
00:13:50,110 --> 00:13:52,180
and lower with L minus,
you don't change L

304
00:13:52,180 --> 00:13:53,670
but you do change m.

305
00:13:53,670 --> 00:13:56,890
So we get ladders that are
labeled by L. So for example,

306
00:13:56,890 --> 00:13:59,910
if I have some value L1,
this is going to give me

307
00:13:59,910 --> 00:14:03,330
some state labeled by m.

308
00:14:03,330 --> 00:14:04,790
Let me put the m to the side.

309
00:14:04,790 --> 00:14:08,590
I can raise it to get
m plus 1 by L plus.

310
00:14:08,590 --> 00:14:13,930
And I can lower it to
get m minus 1 by L minus.

311
00:14:13,930 --> 00:14:15,855
So I got a tower.

312
00:14:18,500 --> 00:14:21,240
And if we have another value,
a different value of L,

313
00:14:21,240 --> 00:14:25,370
I'll call it L2, we
got another tower.

314
00:14:25,370 --> 00:14:28,050
You get m, m plus 1, m plus
2, dot, dot, dot, minus

315
00:14:28,050 --> 00:14:29,890
1, dot, dot, dot.

316
00:14:29,890 --> 00:14:34,270
So we have separated towers with
different values of L squared.

317
00:14:34,270 --> 00:14:37,310
And within each tower we can
raise and lower by L plus,

318
00:14:37,310 --> 00:14:38,110
skipping by one.

319
00:14:41,840 --> 00:14:43,830
OK, questions?

320
00:14:43,830 --> 00:14:47,260
So that was basically the
end of the last lecture,

321
00:14:47,260 --> 00:14:48,710
said slightly differently.

322
00:14:52,500 --> 00:14:54,767
Now here's the question.

323
00:14:54,767 --> 00:14:56,850
So this is the question
that a student asked right

324
00:14:56,850 --> 00:14:59,102
at the beginning.

325
00:14:59,102 --> 00:15:00,060
Is this tower infinite?

326
00:15:05,225 --> 00:15:05,850
Or does it end?

327
00:15:10,890 --> 00:15:12,140
So I pose to you the question.

328
00:15:12,140 --> 00:15:15,410
Is the tower infinite,
or does it end?

329
00:15:15,410 --> 00:15:16,010
And why?

330
00:15:19,209 --> 00:15:21,040
AUDIENCE: [INAUDIBLE] direction.

331
00:15:21,040 --> 00:15:23,498
PROFESSOR: OK, so it's tempting
to say it's infinite in one

332
00:15:23,498 --> 00:15:25,290
direction, because--?

333
00:15:25,290 --> 00:15:28,020
AUDIENCE: There are no bounds
to the angular momentum.

334
00:15:28,020 --> 00:15:29,520
PROFESSOR: OK, so
it's because there

335
00:15:29,520 --> 00:15:31,800
are no bounds to the angular
momentum one can have.

336
00:15:31,800 --> 00:15:32,467
That's tempting.

337
00:15:34,950 --> 00:15:37,557
AUDIENCE: But at
the same time, when

338
00:15:37,557 --> 00:15:40,735
you act a raising or
lowering operator on L,

339
00:15:40,735 --> 00:15:44,550
the eigenvalue of L
squared remains the same.

340
00:15:44,550 --> 00:15:48,327
So then you can't
raise the z [INAUDIBLE]

341
00:15:48,327 --> 00:15:50,410
of the angular momentum
above the actual momentum.

342
00:15:50,410 --> 00:15:50,950
PROFESSOR: Thank you.

343
00:15:50,950 --> 00:15:51,520
Exactly.

344
00:15:51,520 --> 00:15:52,320
So here's the statement.

345
00:15:52,320 --> 00:15:53,153
Let me restate that.

346
00:15:53,153 --> 00:15:54,380
That's exactly right.

347
00:15:54,380 --> 00:15:57,450
So look, L squared is the
eigenvalue of the total angular

348
00:15:57,450 --> 00:15:57,950
momentum.

349
00:15:57,950 --> 00:16:00,870
Roughly speaking, it's giving
you precisely in the state Ylm

350
00:16:00,870 --> 00:16:03,730
it tells you the expected value
of the total angular momentum,

351
00:16:03,730 --> 00:16:04,900
L squared.

352
00:16:04,900 --> 00:16:06,510
That's some number.

353
00:16:06,510 --> 00:16:08,890
Now, if you act with L
plus you keep increasing

354
00:16:08,890 --> 00:16:10,525
the expected value of Lz.

355
00:16:10,525 --> 00:16:12,692
But if you keep increasing
it and keep increasing it

356
00:16:12,692 --> 00:16:16,010
and keep increasing
it, Lz will eventually

357
00:16:16,010 --> 00:16:19,505
get much larger than the
square root of L squared.

358
00:16:19,505 --> 00:16:21,990
That probably isn't true.

359
00:16:21,990 --> 00:16:23,502
That sounds wrong.

360
00:16:23,502 --> 00:16:24,460
That was the statement.

361
00:16:24,460 --> 00:16:24,960
Excellent.

362
00:16:24,960 --> 00:16:25,720
Exactly right.

363
00:16:25,720 --> 00:16:27,400
So let's make that precise.

364
00:16:27,400 --> 00:16:28,930
So is the tower infinite?

365
00:16:28,930 --> 00:16:29,540
No.

366
00:16:29,540 --> 00:16:31,810
It's probably not, for
precisely that reason.

367
00:16:31,810 --> 00:16:32,935
So let's make that precise.

368
00:16:32,935 --> 00:16:34,560
So here's the way
we're going to do it.

369
00:16:34,560 --> 00:16:35,980
This is a useful
trick in general.

370
00:16:35,980 --> 00:16:37,940
This will outlive
angular momentum

371
00:16:37,940 --> 00:16:41,400
and be a useful trick throughout
quantum mechanics for you.

372
00:16:41,400 --> 00:16:43,090
I used it in a paper once.

373
00:16:43,090 --> 00:16:45,630
So here's the nice observation.

374
00:16:45,630 --> 00:16:47,365
Suppose it's true
that the tower ends.

375
00:16:47,365 --> 00:16:49,365
Just like for the raising
and lowering operators

376
00:16:49,365 --> 00:16:51,852
for the harmonic oscillator
in one dimension,

377
00:16:51,852 --> 00:16:53,560
that tells us that in
order for the power

378
00:16:53,560 --> 00:16:57,210
to end that state must
be 0 once we raise it.

379
00:16:57,210 --> 00:17:02,360
The last state, so Yl, and I'll
call this m plus, must be 0.

380
00:17:02,360 --> 00:17:03,740
There must be a max.

381
00:17:03,740 --> 00:17:06,150
Oh, sorry.

382
00:17:06,150 --> 00:17:08,480
Let me actually, before
I walk through exactly

383
00:17:08,480 --> 00:17:09,859
this statement--

384
00:17:09,859 --> 00:17:11,859
So let me make,
first, this, the no,

385
00:17:11,859 --> 00:17:13,560
slightly more
obvious and precise.

386
00:17:13,560 --> 00:17:16,420
So let's turn that argument
into a precise statement.

387
00:17:16,420 --> 00:17:18,420
L squared is equal to,
just from the definition,

388
00:17:18,420 --> 00:17:22,780
Lx squared plus Ly
squared plus Lz squared.

389
00:17:22,780 --> 00:17:24,630
Now let's take the
expectation value

390
00:17:24,630 --> 00:17:29,940
in this state Ylm of both
sides of this equation.

391
00:17:29,940 --> 00:17:34,130
So on the left-hand side we
get h bar squared ll plus 1.

392
00:17:34,130 --> 00:17:36,770
And on the right-hand side we
get the expectation value of Lx

393
00:17:36,770 --> 00:17:40,470
squared plus the expectation
value of Ly squared

394
00:17:40,470 --> 00:17:42,653
plus the expectation
value of Lz squared.

395
00:17:42,653 --> 00:17:44,820
But we know that the
expectation value of Lz squared

396
00:17:44,820 --> 00:17:46,520
is h bar squared, m squared.

397
00:17:46,520 --> 00:17:49,850
But the expectation of
Lx squared and Ly squared

398
00:17:49,850 --> 00:17:52,070
are strictly positive.

399
00:17:52,070 --> 00:17:54,410
Because this can
be written as a sum

400
00:17:54,410 --> 00:17:57,932
over all possible
eigenvalues of Lx squared,

401
00:17:57,932 --> 00:17:59,890
which is the square of
the possible eigenvalues

402
00:17:59,890 --> 00:18:03,170
of Lx times the
probability distribution.

403
00:18:03,170 --> 00:18:05,570
That's a sum of positive,
strictly positive [INAUDIBLE]..

404
00:18:05,570 --> 00:18:07,830
These are positive [INAUDIBLE].

405
00:18:07,830 --> 00:18:10,580
So ll plus 1 is
equal to positive

406
00:18:10,580 --> 00:18:12,700
plus positive plus h
bar squared m squared.

407
00:18:12,700 --> 00:18:14,283
In particular that
tells you that it's

408
00:18:14,283 --> 00:18:18,690
greater than or equal to
h bar squared m squared.

409
00:18:18,690 --> 00:18:20,620
Maybe these are 0.

410
00:18:20,620 --> 00:18:22,350
So the least it can be is--

411
00:18:22,350 --> 00:18:24,270
so the most m
squared can possibly

412
00:18:24,270 --> 00:18:26,340
be is the square
root of L plus 1.

413
00:18:26,340 --> 00:18:28,310
Or the most m can be.

414
00:18:28,310 --> 00:18:30,120
So m is bounded.

415
00:18:30,120 --> 00:18:33,800
There must be a maximum m,
and there must be a minimum m.

416
00:18:33,800 --> 00:18:34,800
Because this is squared.

417
00:18:34,800 --> 00:18:36,160
The sign doesn't matter.

418
00:18:36,160 --> 00:18:37,160
Everyone cool with that?

419
00:18:39,920 --> 00:18:42,770
OK, so let's turn this into,
now, a precise argument.

420
00:18:42,770 --> 00:18:44,710
What are the values
of m plus and m minus?

421
00:18:44,710 --> 00:18:47,770
What is the top of the
value of each tower?

422
00:18:47,770 --> 00:18:50,388
Probably it's going to
depend on total L, right?

423
00:18:50,388 --> 00:18:52,930
So it's going to depend on each
value of L. Let's check that.

424
00:18:52,930 --> 00:18:54,030
So here's the nice trick.

425
00:18:54,030 --> 00:18:57,300
Suppose we really do
have a maximum m plus.

426
00:18:57,300 --> 00:19:01,190
That means that if I try to
raise the state Ylm plus,

427
00:19:01,190 --> 00:19:05,420
I should get the state
0, which ends the tower.

428
00:19:05,420 --> 00:19:07,120
So suppose this is true.

429
00:19:07,120 --> 00:19:10,210
In that case, in particular
here's the nice trick,

430
00:19:10,210 --> 00:19:12,830
L plus Ylm, if we
take its magnitude,

431
00:19:12,830 --> 00:19:15,450
if we take the magnitude of
this state, the state is 0.

432
00:19:15,450 --> 00:19:16,450
It's the zero function.

433
00:19:16,450 --> 00:19:18,690
So what's its magnitude?

434
00:19:18,690 --> 00:19:19,360
Zero.

435
00:19:19,360 --> 00:19:22,370
You might not think that's all
that impressive an observation.

436
00:19:22,370 --> 00:19:24,660
OK, but note what this is.

437
00:19:24,660 --> 00:19:26,125
We know how to work with this.

438
00:19:26,125 --> 00:19:27,625
And in particular
this is equal to--

439
00:19:27,625 --> 00:19:30,860
I should have done
this, dot, dot, dot--

440
00:19:30,860 --> 00:19:33,140
I'm now going to use
the Hermitian adjoint

441
00:19:33,140 --> 00:19:35,110
and pull this over to
the right-hand side.

442
00:19:35,110 --> 00:19:36,970
The adjoint of L
plus is L minus.

443
00:19:36,970 --> 00:19:43,330
So this gives us Ylm,
L minus, L plus Ylm.

444
00:19:43,330 --> 00:19:45,517
This also doesn't look like
much of an improvement,

445
00:19:45,517 --> 00:19:47,600
until you notice from the
definition of L plus and

446
00:19:47,600 --> 00:19:50,140
L minus that what's
L minus L plus?

447
00:19:50,140 --> 00:19:53,870
Well, L minus L plus, we're
going to get an Lx squared.

448
00:19:53,870 --> 00:20:01,470
So Ylm, we get an Lx
squared plus an Ly squared.

449
00:20:01,470 --> 00:20:04,890
I'm going to get my sign
right, if I'm not careful.

450
00:20:04,890 --> 00:20:06,560
Plus-- well, let's just do it.

451
00:20:06,560 --> 00:20:08,240
So we have L minus L plus.

452
00:20:08,240 --> 00:20:12,770
So we're going to get a
Lx iLy minus i, iLyLx.

453
00:20:12,770 --> 00:20:15,410
So plus i commutator
of Lx with Ly.

454
00:20:18,930 --> 00:20:19,980
Good.

455
00:20:19,980 --> 00:20:23,050
So that's progress, Ylm.

456
00:20:23,050 --> 00:20:25,820
But it's still not progress,
because the natural operators

457
00:20:25,820 --> 00:20:28,900
with which to act on the
Ylm's are L squared and Lz.

458
00:20:28,900 --> 00:20:31,350
So can we put this in the
form L squared and Lz?

459
00:20:31,350 --> 00:20:31,850
Sure.

460
00:20:31,850 --> 00:20:35,970
This is L squared
minus Lz squared.

461
00:20:35,970 --> 00:20:41,790
Ylm L squared minus Lz squared.

462
00:20:41,790 --> 00:20:47,430
And this is i h bar Lz
times i is minus h bar Lz.

463
00:20:47,430 --> 00:20:51,270
So minus h bar Lz, Ylm.

464
00:20:53,920 --> 00:20:56,750
And this must be equal to 0.

465
00:20:56,750 --> 00:20:58,510
But this is equal to--

466
00:20:58,510 --> 00:21:04,160
from L squared we get h
bar squared, ll plus 1.

467
00:21:04,160 --> 00:21:08,100
From the Lz squared we get
minus m squared h bar squared.

468
00:21:08,100 --> 00:21:10,530
And from here we get
minus h bar, h bar m,

469
00:21:10,530 --> 00:21:13,395
so minus m each bar squared.

470
00:21:13,395 --> 00:21:14,770
Notice that the
units worked out.

471
00:21:14,770 --> 00:21:17,970
And all of this was
multiplying Ylm, Ylm.

472
00:21:17,970 --> 00:21:20,850
But Ylm, Ylm, if it's a properly
normalized eigenstate, which

473
00:21:20,850 --> 00:21:24,310
is what we were assuming at
the beginning, is just 1.

474
00:21:24,310 --> 00:21:27,310
So we get this times 1 is 0.

475
00:21:27,310 --> 00:21:28,080
Aha.

476
00:21:28,080 --> 00:21:29,730
And notice that in all
of this, this was m plus.

477
00:21:29,730 --> 00:21:31,313
We were assuming,
we were working here

478
00:21:31,313 --> 00:21:35,740
with the assumption that L plus
annihilated this top state.

479
00:21:35,740 --> 00:21:38,488
And so grouping this together,
this says therefore h bar

480
00:21:38,488 --> 00:21:40,780
squared times-- pulling out
a common h bar squared from

481
00:21:40,780 --> 00:21:41,620
of all this--

482
00:21:41,620 --> 00:21:52,740
times l, l plus 1 minus m plus,
m plus, plus 1 is equal to 0.

483
00:21:52,740 --> 00:21:57,120
And this tells us that
m plus is equal to l.

484
00:21:57,120 --> 00:22:01,870
And if you want to be
strict, put a plus sign.

485
00:22:01,870 --> 00:22:03,160
Everyone cool with that?

486
00:22:03,160 --> 00:22:04,480
This is a very useful trick.

487
00:22:04,480 --> 00:22:06,130
If you know something
is 0 as function,

488
00:22:06,130 --> 00:22:07,220
you know its norm is 0.

489
00:22:07,220 --> 00:22:09,500
And now you can use things
like Hermitian adjoints.

490
00:22:09,500 --> 00:22:10,300
Very, very useful.

491
00:22:13,400 --> 00:22:14,925
OK questions about that?

492
00:22:14,925 --> 00:22:16,410
Yeah.

493
00:22:16,410 --> 00:22:20,755
AUDIENCE: [INAUDIBLE]

494
00:22:20,755 --> 00:22:21,630
PROFESSOR: Excellent.

495
00:22:21,630 --> 00:22:23,463
Where this came from
is I was just literally

496
00:22:23,463 --> 00:22:26,600
taking L plus and L minus,
taking the definitions,

497
00:22:26,600 --> 00:22:27,680
and plugging them in.

498
00:22:27,680 --> 00:22:30,320
So if I have L minus
L plus, L minus

499
00:22:30,320 --> 00:22:31,820
is, just to write
it out explicitly,

500
00:22:31,820 --> 00:22:33,090
L minus is equal to Lx.

501
00:22:33,090 --> 00:22:34,990
Because the [INAUDIBLE]
are observables,

502
00:22:34,990 --> 00:22:37,115
and so they're Hermitians,
so they're self-adjoint,

503
00:22:37,115 --> 00:22:38,470
the minus iLy.

504
00:22:38,470 --> 00:22:41,270
So L minus L plus
gives me an LxLx.

505
00:22:41,270 --> 00:22:43,020
It gives me an LxiLy.

506
00:22:43,020 --> 00:22:45,020
It gives you a minus iLyLx.

507
00:22:45,020 --> 00:22:49,370
And it gives me a minus ii,
which is plus 1 Ly squared.

508
00:22:49,370 --> 00:22:50,140
Cool?

509
00:22:50,140 --> 00:22:51,400
Excellent.

510
00:22:51,400 --> 00:22:53,650
I'm in a state of
serious desperation here.

511
00:22:58,720 --> 00:22:59,290
Good.

512
00:22:59,290 --> 00:23:00,468
Other questions?

513
00:23:00,468 --> 00:23:01,782
Yeah.

514
00:23:01,782 --> 00:23:04,640
AUDIENCE: [INAUDIBLE] How do
you get from that to that?

515
00:23:04,640 --> 00:23:05,640
PROFESSOR: This to here?

516
00:23:05,640 --> 00:23:07,010
OK, good.

517
00:23:07,010 --> 00:23:08,463
Sorry, I jumped a step here.

518
00:23:08,463 --> 00:23:09,880
So here what I
said is, look, I've

519
00:23:09,880 --> 00:23:12,210
got this nice set of
operators acting on Ylm.

520
00:23:12,210 --> 00:23:15,240
I know how each of these
operators acts on Ylm.

521
00:23:15,240 --> 00:23:19,060
Lz gives me an h bar m,
minus some h bar squared.

522
00:23:19,060 --> 00:23:20,360
Lz squared, L squared.

523
00:23:20,360 --> 00:23:23,290
And then overall that was a just
some number times Ylm, which

524
00:23:23,290 --> 00:23:24,950
I can pull out of
the inner product.

525
00:23:24,950 --> 00:23:25,450
Cool?

526
00:23:25,450 --> 00:23:26,498
Yeah?

527
00:23:26,498 --> 00:23:28,513
AUDIENCE: [INAUDIBLE]

528
00:23:28,513 --> 00:23:29,180
PROFESSOR: Good.

529
00:23:29,180 --> 00:23:30,750
This subscript plus
meant that, look,

530
00:23:30,750 --> 00:23:32,250
there was a maximum value of m.

531
00:23:32,250 --> 00:23:34,990
So m squared had to be
less than or equal to this.

532
00:23:34,990 --> 00:23:37,630
There's a maximum value
and a minimum value.

533
00:23:37,630 --> 00:23:38,630
Maybe they're different.

534
00:23:38,630 --> 00:23:39,172
I don't know.

535
00:23:39,172 --> 00:23:42,470
I'm just going to be
open-minded about that.

536
00:23:42,470 --> 00:23:42,970
Others?

537
00:23:42,970 --> 00:23:43,850
Yeah?

538
00:23:43,850 --> 00:23:45,975
AUDIENCE: Will that mean,
then, that you can't ever

539
00:23:45,975 --> 00:23:48,758
have all the angular
momentum [INAUDIBLE]??

540
00:23:48,758 --> 00:23:50,300
PROFESSOR: Yeah,
awesome observation.

541
00:23:50,300 --> 00:23:51,490
Exactly.

542
00:23:51,490 --> 00:23:53,282
We were going to get
there in a little bit.

543
00:23:53,282 --> 00:23:53,782
I like that.

544
00:23:53,782 --> 00:23:54,700
That's exactly right.

545
00:23:57,920 --> 00:24:00,100
Let me go through a couple
more steps, and then I'll

546
00:24:00,100 --> 00:24:01,400
come back to that observation.

547
00:24:01,400 --> 00:24:02,655
And I promise I will say so.

548
00:24:05,200 --> 00:24:07,560
So if we go through
exactly a similar argument,

549
00:24:07,560 --> 00:24:10,710
let's see what happens if
we did L minus on Ylm minus,

550
00:24:10,710 --> 00:24:12,330
just to walk through the logic.

551
00:24:12,330 --> 00:24:15,660
So if this were-- you lower
the lowest one, you get 0.

552
00:24:15,660 --> 00:24:18,170
Then we'd get the same
story, L minus L minus.

553
00:24:18,170 --> 00:24:21,800
When we take the adjoint
we'd get L plus, L minus.

554
00:24:21,800 --> 00:24:23,200
And L plus L minus,
what changes?

555
00:24:23,200 --> 00:24:24,742
The only thing that's
going to change

556
00:24:24,742 --> 00:24:27,790
is that you get this
commutator the other direction.

557
00:24:27,790 --> 00:24:29,720
And there are minus
signs in various places.

558
00:24:29,720 --> 00:24:33,470
The upshot of which is that
m minus is equal to minus l.

559
00:24:35,990 --> 00:24:37,240
So this is quite parsimonious.

560
00:24:37,240 --> 00:24:38,440
It's symmetric.

561
00:24:38,440 --> 00:24:40,118
If you take z to
minus z, if you switch

562
00:24:40,118 --> 00:24:42,660
the sign of the angular momentum
you get the same thing back.

563
00:24:42,660 --> 00:24:43,940
That's satisfying, perhaps.

564
00:24:48,580 --> 00:24:50,790
But it's way more than that.

565
00:24:50,790 --> 00:24:53,460
This tells us a lot about
the possible eigenvalues,

566
00:24:53,460 --> 00:24:55,330
in the following way.

567
00:24:55,330 --> 00:24:56,980
Look back at our towers.

568
00:24:56,980 --> 00:25:01,242
In our towers we have that
the angular momentum is

569
00:25:01,242 --> 00:25:02,450
raised and lowered by L plus.

570
00:25:02,450 --> 00:25:05,450
So the Lz angular momentum is
raised and lowered by L plus.

571
00:25:05,450 --> 00:25:06,970
l remains the same.

572
00:25:06,970 --> 00:25:10,230
But there's a maximum
state now, which is plus l.

573
00:25:10,230 --> 00:25:12,450
And there's a minimum
stay in here, l1.

574
00:25:12,450 --> 00:25:14,527
And here there's a
minimum state, minus l1.

575
00:25:14,527 --> 00:25:16,110
Similarly for the
other tower, there's

576
00:25:16,110 --> 00:25:20,000
a minimum state minus l2
and a maximum state, l2.

577
00:25:20,000 --> 00:25:23,127
So how big each tower is depends
on the total angular momentum.

578
00:25:23,127 --> 00:25:24,460
That kind of makes sense, right?

579
00:25:24,460 --> 00:25:26,150
If you've got more
angular momentum

580
00:25:26,150 --> 00:25:28,080
and you can only step
Lz by one, you've

581
00:25:28,080 --> 00:25:30,535
got more room to move
with a large value of l

582
00:25:30,535 --> 00:25:32,500
than with a small value of l.

583
00:25:32,500 --> 00:25:33,000
OK.

584
00:25:35,600 --> 00:25:38,370
So what does that tell
us about the values of m?

585
00:25:38,370 --> 00:25:47,530
Notice that m spans the
values from its minimal value,

586
00:25:47,530 --> 00:25:53,220
minus l to l in integer
steps, in unit steps.

587
00:25:58,140 --> 00:26:04,895
So if you think about m
is l, then m is l minus 1.

588
00:26:04,895 --> 00:26:09,490
And if I keep lowering I get
down to m is minus l plus 1.

589
00:26:09,490 --> 00:26:11,976
And then m is equal to minus l.

590
00:26:15,240 --> 00:26:18,435
So the difference in the Lz
eigenvalue between these guys,

591
00:26:18,435 --> 00:26:19,310
the difference is 2l.

592
00:26:23,070 --> 00:26:24,930
But the number of
unit steps in here

593
00:26:24,930 --> 00:26:30,230
is one fewer, because one,
dot, dot, dot 2l minus 1.

594
00:26:30,230 --> 00:26:33,090
So this is some
integer, which is

595
00:26:33,090 --> 00:26:35,330
the number of states minus 1.

596
00:26:35,330 --> 00:26:40,940
So if these are n states, and
I'll call this N sub l states,

597
00:26:40,940 --> 00:26:44,570
then this difference twice
l is N sub l minus 1,

598
00:26:44,570 --> 00:26:45,590
because it's unit steps.

599
00:26:45,590 --> 00:26:47,440
Cool?

600
00:26:47,440 --> 00:26:51,690
So for example, if there
were two states, so N is 2,

601
00:26:51,690 --> 00:26:54,530
2l is equal to--

602
00:26:54,530 --> 00:26:58,575
well, that's 1,
which is 2 minus 1,

603
00:26:58,575 --> 00:27:00,492
the number of states
minus 1, also known as 1.

604
00:27:00,492 --> 00:27:02,700
So l is 1/2.

605
00:27:02,700 --> 00:27:04,420
And more generally
we find that l

606
00:27:04,420 --> 00:27:06,107
must be, in order
for this process

607
00:27:06,107 --> 00:27:07,690
to make sense, in
order for the m plus

608
00:27:07,690 --> 00:27:11,380
and the m minus to
match up, we need

609
00:27:11,380 --> 00:27:17,860
that l is of the form an
integer N sub l minus 1 upon 2.

610
00:27:17,860 --> 00:27:20,670
So this tells us that
where Nl is an integer--

611
00:27:20,670 --> 00:27:23,780
Nl is just the number
of states in this tower,

612
00:27:23,780 --> 00:27:25,580
and it's a strictly
positive integer.

613
00:27:25,580 --> 00:27:27,122
If you have zero
states in the tower,

614
00:27:27,122 --> 00:27:29,440
then that's not
very interesting.

615
00:27:29,440 --> 00:27:40,500
So this tells us that l is
an integer a half integer,

616
00:27:40,500 --> 00:27:42,810
but nothing else.

617
00:27:42,810 --> 00:27:43,310
Cool?

618
00:27:49,106 --> 00:27:52,940
In particular, if
it's a half integer,

619
00:27:52,940 --> 00:27:59,837
that means 1/2, 3/2, 5/2, if
it's an integer or it can be 0.

620
00:27:59,837 --> 00:28:01,420
There's nothing wrong
with it being 0.

621
00:28:01,420 --> 00:28:03,376
But then 1 on.

622
00:28:07,750 --> 00:28:12,010
So that means we can plot our
system in the following way.

623
00:28:12,010 --> 00:28:22,320
If we have l equals 0, how
many states do we have?

624
00:28:22,320 --> 00:28:24,690
If little l is 0,
how many states are?

625
00:28:24,690 --> 00:28:28,990
What's the largest value of Lz?

626
00:28:28,990 --> 00:28:32,230
What's the largest allowed
value of Lz or of m

627
00:28:32,230 --> 00:28:34,160
if little l is equal to 0?

628
00:28:34,160 --> 00:28:34,660
AUDIENCE: 0.

629
00:28:34,660 --> 00:28:35,202
PROFESSOR: 0.

630
00:28:35,202 --> 00:28:37,350
Because it goes from
plus 0 to minus 0.

631
00:28:37,350 --> 00:28:38,760
So that's pretty much 0.

632
00:28:38,760 --> 00:28:43,930
So we have a single state with
m equals 0 and l equals 0.

633
00:28:43,930 --> 00:28:46,470
If l is equal to-- what's
the next possible value?--

634
00:28:46,470 --> 00:28:52,130
1/2, then m can be
either 1/2 or we lower it

635
00:28:52,130 --> 00:28:53,740
by 1, which is minus 1/2.

636
00:28:53,740 --> 00:28:57,460
So there are two
states, m equals

637
00:28:57,460 --> 00:28:59,400
1/2 and m equals minus 1/2.

638
00:29:02,030 --> 00:29:03,040
So there's one power.

639
00:29:03,040 --> 00:29:04,040
It's a very short tower.

640
00:29:04,040 --> 00:29:07,400
This is the shortest
possible tower.

641
00:29:07,400 --> 00:29:11,670
Then we also have the state l
equals 1, which has m equals 0.

642
00:29:11,670 --> 00:29:15,050
It has m equals 1.

643
00:29:15,050 --> 00:29:18,120
And it has m equals minus 1.

644
00:29:18,120 --> 00:29:20,900
And that's it.

645
00:29:20,900 --> 00:29:22,030
And so on and so forth.

646
00:29:25,570 --> 00:29:33,190
Four states for l equals 3/2,
with states 3/2, 1/2, this is m

647
00:29:33,190 --> 00:29:38,710
equals 1/2, m equals
minus 1/2, and minus 3/2.

648
00:29:38,710 --> 00:29:41,390
The values of m span from minus
l to l with integer steps.

649
00:29:46,480 --> 00:29:51,260
And this is possible for
all values of l which are

650
00:29:51,260 --> 00:30:02,920
of the form all integer
or half integer l's.

651
00:30:02,920 --> 00:30:05,040
So for every different
value of the total angular

652
00:30:05,040 --> 00:30:06,390
momentum, the total
amount of angular momentum

653
00:30:06,390 --> 00:30:08,300
you have, you have
a tower of states

654
00:30:08,300 --> 00:30:10,660
labeled by Lz's eigenvalue
in this fashion.

655
00:30:13,960 --> 00:30:14,460
Questions?

656
00:30:17,508 --> 00:30:21,510
Does anyone notice anything
troubling about these?

657
00:30:21,510 --> 00:30:23,458
Something physically
a little discomfiting

658
00:30:23,458 --> 00:30:24,250
about any of these?

659
00:30:27,262 --> 00:30:29,470
What does it mean to say
I'm in the state l equals 0,

660
00:30:29,470 --> 00:30:29,970
m equals 0?

661
00:30:29,970 --> 00:30:33,050
What is that telling you?

662
00:30:33,050 --> 00:30:34,060
Zero angular momentum.

663
00:30:34,060 --> 00:30:36,540
What's the expectation
value of l squared?

664
00:30:36,540 --> 00:30:37,040
Zero.

665
00:30:37,040 --> 00:30:39,400
The expectation value of Lz?

666
00:30:39,400 --> 00:30:44,050
And by rotational invariance,
the expected value of Lx or Ly?

667
00:30:44,050 --> 00:30:45,500
That thing is not rotating.

668
00:30:45,500 --> 00:30:47,580
There's no angular momentum.

669
00:30:47,580 --> 00:30:49,990
So the no angular momentum
state is one with l equals 0,

670
00:30:49,990 --> 00:30:50,490
m equals 0.

671
00:30:50,490 --> 00:30:52,840
So this is not spinning.

672
00:30:52,840 --> 00:30:54,590
What about this guy?

673
00:30:54,590 --> 00:30:56,900
L equals 1, m equals 0.

674
00:30:56,900 --> 00:31:00,300
Does this thing carry
angular momentum?

675
00:31:00,300 --> 00:31:01,965
Yeah, absolutely.

676
00:31:01,965 --> 00:31:03,840
So it just doesn't carry
any angular momentum

677
00:31:03,840 --> 00:31:04,680
in the z direction.

678
00:31:04,680 --> 00:31:06,660
But its total angular
momentum on average,

679
00:31:06,660 --> 00:31:09,180
its expected total
angular momentum

680
00:31:09,180 --> 00:31:13,940
is ll plus 1, which is
2, times h bar squared.

681
00:31:13,940 --> 00:31:18,295
So if you measure Lx and Ly
what do you expect to get?

682
00:31:18,295 --> 00:31:20,170
Well, if you measure Lx
squared or Ly squared

683
00:31:20,170 --> 00:31:22,087
you probably expect to
get something non-zero.

684
00:31:22,087 --> 00:31:24,260
We'll come back to
that in just a second.

685
00:31:24,260 --> 00:31:28,073
And what about the state
l equals 1, m equals 1?

686
00:31:28,073 --> 00:31:29,490
Your angular
momentum-- you've got

687
00:31:29,490 --> 00:31:30,850
as much angular momentum
in the z direction as you

688
00:31:30,850 --> 00:31:31,970
possibly can.

689
00:31:31,970 --> 00:31:33,890
So that definitely
carries angular momentum.

690
00:31:33,890 --> 00:31:35,580
So there's a state
that has no angular

691
00:31:35,580 --> 00:31:36,860
momentum in the z direction.

692
00:31:36,860 --> 00:31:38,277
And there's a state
that has some,

693
00:31:38,277 --> 00:31:40,210
and there's a state
that has less.

694
00:31:40,210 --> 00:31:41,190
That make sense.

695
00:31:41,190 --> 00:31:42,628
Yeah?

696
00:31:42,628 --> 00:31:45,048
AUDIENCE: If m equals
1 and l equals 1,

697
00:31:45,048 --> 00:31:47,952
that means that the angular
momentum for the z direction

698
00:31:47,952 --> 00:31:50,173
is the total angular momentum?

699
00:31:50,173 --> 00:31:50,881
PROFESSOR: Is it?

700
00:31:50,881 --> 00:31:54,180
AUDIENCE: And then
Lx and Ly was 0?

701
00:31:54,180 --> 00:31:56,998
PROFESSOR: OK, that's
an excellent question.

702
00:31:56,998 --> 00:31:58,540
Let me answer that
now, and then I'll

703
00:31:58,540 --> 00:32:00,470
come back to the point
I wanted to make.

704
00:32:00,470 --> 00:32:00,730
Hold on a second.

705
00:32:00,730 --> 00:32:02,105
Let me just answer
that question.

706
00:32:05,610 --> 00:32:06,270
I'll work here.

707
00:32:11,390 --> 00:32:14,120
So here's the crucial thing.

708
00:32:14,120 --> 00:32:16,860
Even in this state, so you
were asking about the state l

709
00:32:16,860 --> 00:32:18,110
equals 1, m equals 1.

710
00:32:18,110 --> 00:32:21,170
And the question that was asked,
a very good question, is look,

711
00:32:21,170 --> 00:32:23,930
does that mean all the angular
momentum is in the z direction,

712
00:32:23,930 --> 00:32:26,247
and Lx and Ly are 0?

713
00:32:26,247 --> 00:32:27,830
But let me just ask
this more broadly.

714
00:32:27,830 --> 00:32:29,830
Suppose we have a state
with angular momentum l,

715
00:32:29,830 --> 00:32:31,840
and m is equal to l.

716
00:32:31,840 --> 00:32:33,590
The most angular
momentum you can possibly

717
00:32:33,590 --> 00:32:35,215
have in the z direction,
same question.

718
00:32:35,215 --> 00:32:38,025
Is all the angular momentum
in the z direction?

719
00:32:38,025 --> 00:32:39,650
And what I want to
emphasize you is no,

720
00:32:39,650 --> 00:32:41,040
that's absolutely not the case.

721
00:32:41,040 --> 00:32:42,730
So two arguments for that.

722
00:32:42,730 --> 00:32:45,950
The first is, suppose it's true
that Lx and Ly are identically

723
00:32:45,950 --> 00:32:48,512
0.

724
00:32:48,512 --> 00:32:50,220
Can that satisfy the
uncertainty relation

725
00:32:50,220 --> 00:32:51,430
due to those commutators?

726
00:32:51,430 --> 00:32:51,930
No.

727
00:32:51,930 --> 00:32:53,670
There must be
uncertainty in Lx and Ly,

728
00:32:53,670 --> 00:32:57,540
because Lz has a non-zero
expectation value.

729
00:32:57,540 --> 00:32:59,873
So it can't be that Lx
squared and Ly squared

730
00:32:59,873 --> 00:33:01,040
have zero expectation value.

731
00:33:01,040 --> 00:33:03,100
But let's be more
precise about this.

732
00:33:03,100 --> 00:33:05,440
The expectation value of L
squared is easy to calculate.

733
00:33:05,440 --> 00:33:09,050
It's h bar squared l, plus 1.

734
00:33:09,050 --> 00:33:10,570
Because we're in the state Ylm.

735
00:33:10,570 --> 00:33:15,660
This is the state
Y l sub l, or ll.

736
00:33:15,660 --> 00:33:21,490
The expectation of Lz squared
is equal to h bar squared,

737
00:33:21,490 --> 00:33:22,492
m squared.

738
00:33:22,492 --> 00:33:23,950
And m squared now,
m is equal to l.

739
00:33:23,950 --> 00:33:25,880
So h bar squared, l squared.

740
00:33:25,880 --> 00:33:26,425
Aha.

741
00:33:26,425 --> 00:33:27,800
So the expected
value of l square

742
00:33:27,800 --> 00:33:29,425
is not the same as
Lx squared, but this

743
00:33:29,425 --> 00:33:33,510
is equal to the expected value
of Lx squared plus Ly squared

744
00:33:33,510 --> 00:33:34,340
plus Lz squared.

745
00:33:41,790 --> 00:33:44,260
Therefore the
expected value of Lx

746
00:33:44,260 --> 00:33:49,160
squared plus the expected
value of Ly squared

747
00:33:49,160 --> 00:33:51,498
is equal to the difference
between this and this.

748
00:33:51,498 --> 00:33:54,040
We just subtract this off h bar
squared ll plus 1 minus h bar

749
00:33:54,040 --> 00:33:57,000
squared, l squared,
h bar squared l.

750
00:33:57,000 --> 00:33:58,900
And by symmetry you
don't expect the symmetry

751
00:33:58,900 --> 00:34:00,932
to be broken between Lx and Ly.

752
00:34:00,932 --> 00:34:02,390
You can actually
do the calculation

753
00:34:02,390 --> 00:34:03,450
and not just be glib about it.

754
00:34:03,450 --> 00:34:05,408
But both arguments give
you the correct answer.

755
00:34:05,408 --> 00:34:08,110
The expectation
value of Lx squared

756
00:34:08,110 --> 00:34:11,960
is equal to 1/2 h bar squared l.

757
00:34:11,960 --> 00:34:16,690
And ditto for y, in this state.

758
00:34:16,690 --> 00:34:17,667
So notice two things.

759
00:34:17,667 --> 00:34:20,000
First off is we make the total
angular momentum little l

760
00:34:20,000 --> 00:34:21,719
large.

761
00:34:21,719 --> 00:34:24,460
The amount by which we fail
to have all the angular

762
00:34:24,460 --> 00:34:26,860
momentum in the z direction
is getting larger and larger.

763
00:34:26,860 --> 00:34:30,410
We're increasing the crappiness
of putting all the angular

764
00:34:30,410 --> 00:34:32,010
momentum in the z direction.

765
00:34:32,010 --> 00:34:37,080
However, as a ratio of the total
angular momentum divided by L

766
00:34:37,080 --> 00:34:40,630
squared, so this
divided by L squared,

767
00:34:40,630 --> 00:34:43,280
and this is h bar
squared, l, l plus 1,

768
00:34:43,280 --> 00:34:45,690
and in particular this
is l squared plus l,

769
00:34:45,690 --> 00:34:48,578
so if we took the ratio,
the rational mismatch

770
00:34:48,578 --> 00:34:49,870
is getting smaller and smaller.

771
00:34:49,870 --> 00:34:50,449
And that's good.

772
00:34:50,449 --> 00:34:52,280
Because as we go to very
large angular momentums

773
00:34:52,280 --> 00:34:53,719
where things should
start getting classical

774
00:34:53,719 --> 00:34:56,250
in some sense, we should get
back the familiar intuition

775
00:34:56,250 --> 00:34:59,290
that you can put all the angular
momentum in the z direction.

776
00:34:59,290 --> 00:35:00,228
Yeah?

777
00:35:00,228 --> 00:35:02,520
AUDIENCE: Why are we imposing
the [INAUDIBLE] condition

778
00:35:02,520 --> 00:35:05,440
that the expectation values of
Lx and Ly should be identical?

779
00:35:05,440 --> 00:35:05,600
PROFESSOR: Excellent.

780
00:35:05,600 --> 00:35:07,030
That's why I was saying
this is a glib argument.

781
00:35:07,030 --> 00:35:08,190
You don't have to impose--

782
00:35:08,190 --> 00:35:10,130
that needs to be done a
little bit more delicately.

783
00:35:10,130 --> 00:35:11,400
But we can just
directly compute this.

784
00:35:11,400 --> 00:35:12,830
And you do so on
your problem set.

785
00:35:15,610 --> 00:35:16,140
Yeah?

786
00:35:16,140 --> 00:35:18,250
AUDIENCE: Why
doesn't the existence

787
00:35:18,250 --> 00:35:21,770
of the l equals 0
eigenfunction where the angular

788
00:35:21,770 --> 00:35:26,095
momentum the L squared
is definitely 0

789
00:35:26,095 --> 00:35:27,870
violate the
uncertainty principle?

790
00:35:27,870 --> 00:35:28,250
PROFESSOR: Awesome.

791
00:35:28,250 --> 00:35:30,250
On your problem set you're
going to answer that,

792
00:35:30,250 --> 00:35:33,030
but let me give you
a quick preview.

793
00:35:33,030 --> 00:35:34,520
This is such a great response.

794
00:35:34,520 --> 00:35:37,140
Just Let me give
you a quick preview.

795
00:35:37,140 --> 00:35:38,420
So from that Lx--

796
00:35:38,420 --> 00:35:40,090
OK, this is a
really fun question.

797
00:35:40,090 --> 00:35:43,060
Let me go into it
in some detail.

798
00:35:43,060 --> 00:35:44,610
Wow, I'm going fast today.

799
00:35:44,610 --> 00:35:46,320
Am I going way too fast?

800
00:35:46,320 --> 00:35:47,550
No?

801
00:35:47,550 --> 00:35:48,110
A little?

802
00:35:48,110 --> 00:35:49,210
Little too fast?

803
00:35:49,210 --> 00:35:50,960
OK, ask me more questions
to slow me down.

804
00:35:50,960 --> 00:35:51,520
I'm excited.

805
00:35:51,520 --> 00:35:52,978
I didn't get much
sleep last night.

806
00:35:57,090 --> 00:35:58,840
One of the great joys
of being a physicist

807
00:35:58,840 --> 00:36:01,280
is working with
other physicists.

808
00:36:01,280 --> 00:36:03,210
So yesterday one of
my very good friends

809
00:36:03,210 --> 00:36:06,870
and a collaborator I really
delight in talking with

810
00:36:06,870 --> 00:36:07,790
came to visit.

811
00:36:07,790 --> 00:36:09,210
And we had a late night dinner.

812
00:36:09,210 --> 00:36:11,835
And this led to me late at
night, not doing my work,

813
00:36:11,835 --> 00:36:14,210
but reading papers about what
our conversation was about.

814
00:36:14,210 --> 00:36:17,552
And only then at the very end,
when I was just about to die

815
00:36:17,552 --> 00:36:19,760
did I write the response to
the reviewer on the paper

816
00:36:19,760 --> 00:36:22,135
that I was supposed to be
doing by last night's deadline.

817
00:36:22,135 --> 00:36:24,750
So I'm kind of tired, but
I'm in a really good mood.

818
00:36:28,610 --> 00:36:29,610
It's a really good job.

819
00:36:34,580 --> 00:36:36,330
Now I've totally lost
my train of thought.

820
00:36:36,330 --> 00:36:37,992
What was the question again?

821
00:36:37,992 --> 00:36:39,760
AUDIENCE: The existence
of the l equals

822
00:36:39,760 --> 00:36:43,250
0 state where the total angular
momentum is definitely--

823
00:36:43,250 --> 00:36:45,000
PROFESSOR: Excellent,
and the uncertainty.

824
00:36:45,000 --> 00:36:47,910
So the question is, why don't
we violate the uncertainty when

825
00:36:47,910 --> 00:36:49,530
we know that L
equals-- where am I;

826
00:36:49,530 --> 00:36:53,608
I just covered it-- when we know
that L equals 0 and m equals 0,

827
00:36:53,608 --> 00:36:55,150
doesn't that destroy
our uncertainty?

828
00:36:55,150 --> 00:36:57,233
Because we know that the
angular momentum Lz is 0.

829
00:36:57,233 --> 00:36:58,442
Angular momentum for Lx is 0.

830
00:36:58,442 --> 00:36:59,665
Angular momentum for Ly is 0.

831
00:36:59,665 --> 00:37:00,902
All of them vanish.

832
00:37:00,902 --> 00:37:02,860
Doesn't that violate the
uncertainty principle?

833
00:37:02,860 --> 00:37:04,980
So let's remind ourselves what
the form of that uncertainty

834
00:37:04,980 --> 00:37:05,750
relation is.

835
00:37:05,750 --> 00:37:08,680
The form of the uncertainty
relation following from Lx Ly

836
00:37:08,680 --> 00:37:10,200
is i h bar Lz.

837
00:37:10,200 --> 00:37:13,260
Recall the general statement.

838
00:37:13,260 --> 00:37:16,710
The uncertainty in A times
the uncertainty in B, squared,

839
00:37:16,710 --> 00:37:18,700
squared, is equal to--

840
00:37:18,700 --> 00:37:23,430
let me just write
it as h bar upon 2--

841
00:37:23,430 --> 00:37:25,290
sorry, 1/2.

842
00:37:25,290 --> 00:37:27,230
The absolute value of
the expectation value

843
00:37:27,230 --> 00:37:33,550
of the commutator,
A with B. Good lord.

844
00:37:33,550 --> 00:37:35,230
Dimensionally, does this work?

845
00:37:35,230 --> 00:37:36,090
Yes.

846
00:37:36,090 --> 00:37:36,590
OK, good.

847
00:37:36,590 --> 00:37:39,570
Because units of A, units of
B. Unites of A, units of B.

848
00:37:39,570 --> 00:37:41,602
Triumph.

849
00:37:41,602 --> 00:37:43,810
And these are going to be
quantum-mechanically small,

850
00:37:43,810 --> 00:37:45,660
because commutators have h bars.

851
00:37:45,660 --> 00:37:47,120
And commutators have h bars why?

852
00:37:49,850 --> 00:37:51,020
Not because God hates us.

853
00:37:51,020 --> 00:37:52,460
Why do commutators have h bars?

854
00:37:55,810 --> 00:37:58,950
What happens classically?

855
00:37:58,950 --> 00:38:01,670
In classical mechanics,
do things commute?

856
00:38:01,670 --> 00:38:02,250
Yes.

857
00:38:02,250 --> 00:38:04,470
Why are the h bars
and commutators

858
00:38:04,470 --> 00:38:06,020
physical observables?

859
00:38:06,020 --> 00:38:08,133
Because we exist.

860
00:38:08,133 --> 00:38:09,550
Because there's a
classical limit.

861
00:38:09,550 --> 00:38:11,630
OK so this is going to make
quantum-mechanically small,

862
00:38:11,630 --> 00:38:13,505
so we expect the
uncertainty relation to also

863
00:38:13,505 --> 00:38:15,398
be quantum-mechanically small.

864
00:38:15,398 --> 00:38:16,440
Just important intuition.

865
00:38:16,440 --> 00:38:20,730
So let's look at the specific
example of Lx, Ly, and Lz.

866
00:38:20,730 --> 00:38:23,290
So the uncertainty in
Lx, in any particular--

867
00:38:23,290 --> 00:38:24,550
remember that this is
defined as the uncertainty

868
00:38:24,550 --> 00:38:26,770
in a particular state psi,
in a particular state psi.

869
00:38:26,770 --> 00:38:29,145
And this expectation value is
taken in a particular state

870
00:38:29,145 --> 00:38:32,010
psi, that same state.

871
00:38:32,010 --> 00:38:37,470
So the uncertainty of
Lx, in some state Ylm,

872
00:38:37,470 --> 00:38:41,610
times the uncertainty of
Ly in that same state Ylm

873
00:38:41,610 --> 00:38:47,171
should be greater
than or equal to 1/2

874
00:38:47,171 --> 00:38:48,963
the absolute value of
the expectation value

875
00:38:48,963 --> 00:38:50,330
of the commutator of Lx and Ly.

876
00:38:50,330 --> 00:38:53,390
But the commutator of
Lx and Ly is i h bar Lz.

877
00:38:53,390 --> 00:38:56,140
And i, when we pull it
through this absolute value,

878
00:38:56,140 --> 00:38:58,710
is going to give me just 1. h
bar is going to give me h bar.

879
00:38:58,710 --> 00:39:04,330
So h bar upon 2, expectation
value of Lz, absolute value.

880
00:39:04,330 --> 00:39:06,420
Yeah?

881
00:39:06,420 --> 00:39:08,480
Can Lx and Ly have
zero uncertainty?

882
00:39:16,340 --> 00:39:18,450
When?

883
00:39:18,450 --> 00:39:21,197
Expectation value of Lz is 0.

884
00:39:21,197 --> 00:39:22,030
So that sounds good.

885
00:39:22,030 --> 00:39:27,080
It sounds like if Lz has
expectation value of 0,

886
00:39:27,080 --> 00:39:29,590
then we can have Lx
and Ly, definite.

887
00:39:29,590 --> 00:39:31,430
But that's bad.

888
00:39:31,430 --> 00:39:32,700
Really?

889
00:39:32,700 --> 00:39:34,460
Really, can we do that?

890
00:39:34,460 --> 00:39:35,790
Why not?

891
00:39:35,790 --> 00:39:37,690
AUDIENCE: [INAUDIBLE]

892
00:39:37,690 --> 00:39:40,065
[LAUGHTER]

893
00:39:43,532 --> 00:39:44,740
PROFESSOR: Bless you, my son.

894
00:39:51,060 --> 00:39:54,520
Can we have Lx and Ly both
take definite values, just

895
00:39:54,520 --> 00:39:57,050
because Lz?

896
00:39:57,050 --> 00:39:59,210
Why?

897
00:39:59,210 --> 00:40:03,070
What else do we have to satisfy?

898
00:40:03,070 --> 00:40:05,750
What other uncertainty
relations must we satisfy?

899
00:40:08,390 --> 00:40:10,820
There are two more.

900
00:40:10,820 --> 00:40:15,230
And I invite you to go look
at what those two more are

901
00:40:15,230 --> 00:40:19,520
and deduce that this is
only possible if Lx squared,

902
00:40:19,520 --> 00:40:23,110
Ly squared, and Lz squared all
have zero expectation value.

903
00:40:23,110 --> 00:40:26,770
In fact, I think it's just
a great question that I

904
00:40:26,770 --> 00:40:30,358
think it's on your problem set.

905
00:40:30,358 --> 00:40:31,650
So thank you for that question.

906
00:40:31,650 --> 00:40:32,460
It's a really good question.

907
00:40:32,460 --> 00:40:34,040
There was another
question in here.

908
00:40:34,040 --> 00:40:35,160
Yeah?

909
00:40:35,160 --> 00:40:40,944
AUDIENCE: Something that
you said earlier [INAUDIBLE]

910
00:40:40,944 --> 00:40:41,812
half integer.

911
00:40:41,812 --> 00:40:43,354
So were you deriving
this by counting

912
00:40:43,354 --> 00:40:46,728
the number of equations
and somehow asserting--

913
00:40:46,728 --> 00:40:47,792
Why does--?

914
00:40:47,792 --> 00:40:48,500
PROFESSOR: Great.

915
00:40:48,500 --> 00:40:50,570
So the question
is, wait, really?

916
00:40:50,570 --> 00:40:52,273
Why is L and integer
a half integer.

917
00:40:52,273 --> 00:40:53,440
That was a little too quick.

918
00:40:53,440 --> 00:40:55,017
Is that roughly the
right statement?

919
00:40:55,017 --> 00:40:56,503
AUDIENCE: [INAUDIBLE].

920
00:40:56,503 --> 00:40:57,170
PROFESSOR: Good.

921
00:40:57,170 --> 00:40:57,660
Excellent.

922
00:40:57,660 --> 00:40:58,827
Let me go through the logic.

923
00:40:58,827 --> 00:41:01,590
So the logic goes like this.

924
00:41:01,590 --> 00:41:02,870
I know that the Ylm's--

925
00:41:02,870 --> 00:41:06,640
if the Ylm's are eigenfunctions
of L squared and Lz,

926
00:41:06,640 --> 00:41:09,178
and I've constructed this
tower of them using the raising

927
00:41:09,178 --> 00:41:10,720
and lowering operators,
we've already

928
00:41:10,720 --> 00:41:13,300
shown that the largest
possible value of m is l

929
00:41:13,300 --> 00:41:15,470
and the least possible
value is minus l.

930
00:41:15,470 --> 00:41:17,000
And these states
must be separated

931
00:41:17,000 --> 00:41:19,370
by integer steps in m.

932
00:41:19,370 --> 00:41:19,870
OK, good.

933
00:41:19,870 --> 00:41:23,960
So pick a value of l,
a particular tower.

934
00:41:23,960 --> 00:41:27,530
And let the number of states
in that tower be N sub l.

935
00:41:27,530 --> 00:41:28,990
So there's N sub l of them.

936
00:41:28,990 --> 00:41:29,700
Great.

937
00:41:29,700 --> 00:41:34,355
And what's the distance
between these guys?

938
00:41:34,355 --> 00:41:35,980
We haven't assumed
l is an integer yet.

939
00:41:35,980 --> 00:41:38,610
We haven't assumed that.

940
00:41:38,610 --> 00:41:42,431
So this N sub l is an integer.

941
00:41:42,431 --> 00:41:43,848
Because it's the
number of states.

942
00:41:43,848 --> 00:41:45,938
And the number of
states can't be a pi.

943
00:41:50,313 --> 00:41:51,230
Now let's count that--

944
00:41:51,230 --> 00:41:53,693
that so how many
states are there?

945
00:41:53,693 --> 00:41:56,195
There are Nl.

946
00:41:56,195 --> 00:41:59,360
But if I count one,
two, three, four,

947
00:41:59,360 --> 00:42:02,320
the total angular
momentum down here is 2l.

948
00:42:04,830 --> 00:42:11,500
I had some pithy way of
giving this a fancy name.

949
00:42:11,500 --> 00:42:13,820
But I can't remember
what it was.

950
00:42:13,820 --> 00:42:15,670
So the length of
the tower in units

951
00:42:15,670 --> 00:42:22,140
of h bar, the height
of this tower is 2l.

952
00:42:22,140 --> 00:42:27,030
But the number of steps I
took in here was Nl minus 1.

953
00:42:27,030 --> 00:42:29,550
And that number of
steps is times 1.

954
00:42:29,550 --> 00:42:32,435
So we get that 2l is N minus 1.

955
00:42:32,435 --> 00:42:33,560
There's nothing fancy here.

956
00:42:33,560 --> 00:42:36,128
I'm just saying if L is 0
we go from here to here.

957
00:42:36,128 --> 00:42:37,170
There's just one element.

958
00:42:37,170 --> 00:42:39,400
So number states is 1, L is 0.

959
00:42:39,400 --> 00:42:46,288
So it's that same logic just
repeated for every value of L.

960
00:42:46,288 --> 00:42:46,955
Other questions?

961
00:42:52,890 --> 00:42:59,030
Coming back to this,
something on this board

962
00:42:59,030 --> 00:43:01,825
should cause you some
serious physical discomfort.

963
00:43:05,525 --> 00:43:07,650
We've talked about the l
equals 0 m equals 0 state.

964
00:43:07,650 --> 00:43:10,025
This is a state which has no
angular momentum whatsoever,

965
00:43:10,025 --> 00:43:11,490
in any direction at all.

966
00:43:11,490 --> 00:43:13,867
We've talked about the l
equals 1 m equals 0 state.

967
00:43:13,867 --> 00:43:15,950
It carries no angular
momentum in the z direction,

968
00:43:15,950 --> 00:43:18,540
but it presumably has non-zero
expectation value for L

969
00:43:18,540 --> 00:43:22,830
squared x and a Ly squared.

970
00:43:22,830 --> 00:43:23,980
These guys are also fine.

971
00:43:23,980 --> 00:43:26,980
What about these guys?

972
00:43:26,980 --> 00:43:29,920
AUDIENCE: [INAUDIBLE].

973
00:43:29,920 --> 00:43:32,290
PROFESSOR: Yes!

974
00:43:32,290 --> 00:43:34,180
That's disconcerting.

975
00:43:34,180 --> 00:43:38,034
Do I have to have angular
momentum in the z direction?

976
00:43:38,034 --> 00:43:40,022
AUDIENCE: [INAUDIBLE]

977
00:43:40,022 --> 00:43:42,010
[LAUGHTER]

978
00:43:45,000 --> 00:43:48,620
PROFESSOR: That should
go on a shirt somewhere.

979
00:43:48,620 --> 00:43:50,420
Let me ask the question
more precisely,

980
00:43:50,420 --> 00:43:54,080
or in a way that's a little
less threatening to me.

981
00:43:54,080 --> 00:43:58,420
Do you have to have angular
momentum in the z direction?

982
00:43:58,420 --> 00:43:59,630
I'm sorry, what's your name?

983
00:43:59,630 --> 00:44:02,587
Does David need to have angular
momentum in the z direction?

984
00:44:02,587 --> 00:44:03,170
AUDIENCE: Yes.

985
00:44:03,170 --> 00:44:04,710
PROFESSOR: Does this chalk
need to have angular dimension

986
00:44:04,710 --> 00:44:05,240
in the--

987
00:44:05,240 --> 00:44:06,760
well, the chalk's--

988
00:44:06,760 --> 00:44:10,163
OK, classically no.

989
00:44:10,163 --> 00:44:12,330
It can have some total
angular momentum, which is 0,

990
00:44:12,330 --> 00:44:14,840
and it can be rotating
not in the z direction.

991
00:44:14,840 --> 00:44:17,720
It can be rotating
in the zx plane.

992
00:44:17,720 --> 00:44:19,430
If it's rotating
in the zx plane,

993
00:44:19,430 --> 00:44:22,670
it's got total angular
momentum L squared as non-zero.

994
00:44:22,670 --> 00:44:25,410
But its angular momentum
in the z direction is 0.

995
00:44:25,410 --> 00:44:28,610
Its axis is exactly
along the y direction.

996
00:44:28,610 --> 00:44:30,360
And so it's got no
angular moment-- that's

997
00:44:30,360 --> 00:44:32,020
perfectly possible classically.

998
00:44:32,020 --> 00:44:34,690
And that's perfectly possible
when L is an integer.

999
00:44:34,690 --> 00:44:36,600
Similarly when L is 2.

1000
00:44:36,600 --> 00:44:39,200
This is the particular
tower that I love the most.

1001
00:44:39,200 --> 00:44:43,790
2, 1, 0, minus 1, minus 2.

1002
00:44:43,790 --> 00:44:45,900
The reason I love
this the most is

1003
00:44:45,900 --> 00:44:49,300
that it's related to gravity,
which is pretty awesome.

1004
00:44:52,035 --> 00:44:53,160
That's a whole other story.

1005
00:44:56,050 --> 00:44:59,760
I really shouldn't
have said that.

1006
00:44:59,760 --> 00:45:01,740
That's only going
to confuse you.

1007
00:45:01,740 --> 00:45:04,980
So for any integer, it's
possible to have no angular

1008
00:45:04,980 --> 00:45:06,573
momentum in the z direction.

1009
00:45:06,573 --> 00:45:07,990
That means it's
possible to rotate

1010
00:45:07,990 --> 00:45:11,543
around the x-axis or the y-axis,
orthogonal to the z-axis.

1011
00:45:11,543 --> 00:45:12,210
That make sense?

1012
00:45:12,210 --> 00:45:15,690
But for these half integer guys,
you are inescapably spinning.

1013
00:45:15,690 --> 00:45:18,160
There is no such thing as
a state of this guy that

1014
00:45:18,160 --> 00:45:19,880
carries no angular momentum.

1015
00:45:19,880 --> 00:45:23,780
Anything well described
by these quantum states

1016
00:45:23,780 --> 00:45:26,830
is perpetually
rotating or spinning.

1017
00:45:26,830 --> 00:45:29,380
It carries angular
momentum, we say precisely.

1018
00:45:29,380 --> 00:45:31,797
Perpetually carries angular
momentum in the z direction.

1019
00:45:31,797 --> 00:45:34,130
Any time you measure it, it
carries an angular momentum,

1020
00:45:34,130 --> 00:45:36,920
either plus a half integer
or minus a half integer.

1021
00:45:36,920 --> 00:45:40,700
But never, ever zero.

1022
00:45:40,700 --> 00:45:43,390
AUDIENCE: But in the
classical limit where

1023
00:45:43,390 --> 00:45:47,610
you have very large L, the
m equals one half state,

1024
00:45:47,610 --> 00:45:49,534
or the m equals
minus a half state is

1025
00:45:49,534 --> 00:45:51,784
going to get arbitrarily
small compared to the angular

1026
00:45:51,784 --> 00:45:52,284
momentum.

1027
00:45:52,284 --> 00:45:53,880
So isn't it just
like where we said,

1028
00:45:53,880 --> 00:45:56,196
well, OK, you can never
have your angular momentum

1029
00:45:56,196 --> 00:45:58,710
only in the z direction,
but we don't care?

1030
00:45:58,710 --> 00:46:01,985
Because in the classical
limit it gets arbitrarily

1031
00:46:01,985 --> 00:46:02,610
close to there.

1032
00:46:02,610 --> 00:46:03,920
PROFESSOR: See, one of the
nice things about writing

1033
00:46:03,920 --> 00:46:06,462
lectures like this is that you
get to leave little landlines.

1034
00:46:06,462 --> 00:46:07,970
So this is exactly one of those.

1035
00:46:07,970 --> 00:46:09,700
Thank you for asking
this question.

1036
00:46:09,700 --> 00:46:11,470
Let me rephrase that question.

1037
00:46:11,470 --> 00:46:14,250
Look, we all took
high school chemistry.

1038
00:46:14,250 --> 00:46:16,080
We all know about spin.

1039
00:46:16,080 --> 00:46:17,577
The nuclei have spin.

1040
00:46:17,577 --> 00:46:18,910
They have some angular momentum.

1041
00:46:18,910 --> 00:46:20,327
But if you build
up a lot of them,

1042
00:46:20,327 --> 00:46:21,830
you build up a piece
of chalk, look,

1043
00:46:21,830 --> 00:46:23,560
as we said before,
while it's true

1044
00:46:23,560 --> 00:46:26,385
that there's some
mismatch in the angular

1045
00:46:26,385 --> 00:46:28,510
momentum in the z direction
for some large L state,

1046
00:46:28,510 --> 00:46:29,802
it's not only angular momentum.

1047
00:46:29,802 --> 00:46:32,025
Some is in Lz, Lx,
and Ly as well.

1048
00:46:32,025 --> 00:46:34,700
It's preposterously small
for a macroscopic object

1049
00:46:34,700 --> 00:46:36,200
where L is macroscopic.

1050
00:46:36,200 --> 00:46:38,030
It's the angular
momentum in Planck units,

1051
00:46:38,030 --> 00:46:41,010
in units of the Planck constant.

1052
00:46:41,010 --> 00:46:42,220
10 to the 26th--

1053
00:46:42,220 --> 00:46:43,960
something huge.

1054
00:46:43,960 --> 00:46:45,410
Why would we even notice?

1055
00:46:45,410 --> 00:46:46,700
But here's the real statement.

1056
00:46:46,700 --> 00:46:48,230
The statement isn't
just that this

1057
00:46:48,230 --> 00:46:49,837
is true of macroscopic objects.

1058
00:46:49,837 --> 00:46:51,420
But imagine you take
a small particle.

1059
00:46:51,420 --> 00:46:54,500
Imagine you take a single atom.

1060
00:46:54,500 --> 00:46:56,690
We're deep in the quantum
mechanical regime.

1061
00:46:56,690 --> 00:46:59,220
We're not in the
classical regime.

1062
00:46:59,220 --> 00:47:01,540
We take that single atom.

1063
00:47:01,540 --> 00:47:03,280
And if it carries
angular momentum

1064
00:47:03,280 --> 00:47:05,240
and it's described by
L equals 1/2 state,

1065
00:47:05,240 --> 00:47:09,000
that atom will never, ever, ever
be measured to have its angular

1066
00:47:09,000 --> 00:47:13,600
momentum in the z direction,
or indeed any direction, be 0.

1067
00:47:13,600 --> 00:47:15,278
You will never,
in any direction,

1068
00:47:15,278 --> 00:47:16,820
measure its angular
momentum to be 0.

1069
00:47:16,820 --> 00:47:20,110
That atom perpetually
carries angular momentum.

1070
00:47:20,110 --> 00:47:22,010
And that is weird.

1071
00:47:22,010 --> 00:47:23,798
OK, maybe you don't
find it weird.

1072
00:47:23,798 --> 00:47:24,340
This is good.

1073
00:47:24,340 --> 00:47:26,673
You've grown up in an era
when that's not a weird thing.

1074
00:47:26,673 --> 00:47:29,230
But I find this
deeply disconcerting.

1075
00:47:29,230 --> 00:47:32,680
And you might say, look, we
never actually measure an atom.

1076
00:47:32,680 --> 00:47:33,640
But we do.

1077
00:47:33,640 --> 00:47:34,803
We do all the time.

1078
00:47:34,803 --> 00:47:36,970
Because we measure things
like the spectre of light,

1079
00:47:36,970 --> 00:47:40,960
as we'll study when we study
atoms in a week, a couple

1080
00:47:40,960 --> 00:47:42,580
weeks because of the exam--

1081
00:47:42,580 --> 00:47:44,650
sorry guys, there
has to be an exam--

1082
00:47:44,650 --> 00:47:46,400
as we will find
when we study atoms

1083
00:47:46,400 --> 00:47:50,210
in more detail,
or indeed, at all,

1084
00:47:50,210 --> 00:47:52,140
we'll be sensitive
to the angular

1085
00:47:52,140 --> 00:47:54,780
momentum of the
constituents of the atom.

1086
00:47:54,780 --> 00:47:56,190
And we'll see different spectra.

1087
00:47:56,190 --> 00:48:01,730
So it's an observable property
when you shine light on gases.

1088
00:48:01,730 --> 00:48:05,380
This is something we
can really observe.

1089
00:48:05,380 --> 00:48:09,390
So what this suggests
is one of two things.

1090
00:48:09,390 --> 00:48:13,005
Either these are just
crazy and ridiculous

1091
00:48:13,005 --> 00:48:15,305
and we should ignore
them, or there's

1092
00:48:15,305 --> 00:48:16,930
something interesting
and intrinsically

1093
00:48:16,930 --> 00:48:19,580
quantum-mechanical about
them that's not so familiar.

1094
00:48:19,580 --> 00:48:21,038
And the answer is
going to turn out

1095
00:48:21,038 --> 00:48:24,500
to be the second,
the latter of those.

1096
00:48:24,500 --> 00:48:25,030
The ladder?

1097
00:48:28,210 --> 00:48:31,120
That was not intentional.

1098
00:48:31,120 --> 00:48:32,300
Maybe it was subconscious.

1099
00:48:32,300 --> 00:48:35,300
OK, so I want to think
about some more consequences

1100
00:48:35,300 --> 00:48:37,630
of the structure of the
Ylm's and the eigenvalues,

1101
00:48:37,630 --> 00:48:39,698
in particular of
this tower structure.

1102
00:48:39,698 --> 00:48:40,990
I want to understand some more.

1103
00:48:40,990 --> 00:48:43,925
What other physics can we
extract from this story?

1104
00:48:48,540 --> 00:48:50,290
First, a very
useful thing is just

1105
00:48:50,290 --> 00:48:52,370
to get a picture of
these guys in your head.

1106
00:48:52,370 --> 00:48:55,940
Let's draw the angular
momentum eigenfunctions.

1107
00:48:58,232 --> 00:48:59,190
So what does that mean?

1108
00:48:59,190 --> 00:49:02,350
Well first, when we talked about
the eigenfunctions of momentum,

1109
00:49:02,350 --> 00:49:04,730
linear momentum
in one dimension,

1110
00:49:04,730 --> 00:49:06,740
we immediately went
to the wave function.

1111
00:49:06,740 --> 00:49:09,260
We talked about
how the amplitude

1112
00:49:09,260 --> 00:49:11,450
to beat a particular
spot varied in space.

1113
00:49:11,450 --> 00:49:13,820
And the amplitude was
just e to the iKX.

1114
00:49:13,820 --> 00:49:15,660
So the amplitude
was an oscillating--

1115
00:49:15,660 --> 00:49:17,010
the phase rotated.

1116
00:49:17,010 --> 00:49:19,060
And the absolute value of
the probability density

1117
00:49:19,060 --> 00:49:19,940
was completely constant.

1118
00:49:19,940 --> 00:49:21,040
Everybody cool with that?

1119
00:49:21,040 --> 00:49:22,537
That was the 1D plane wave.

1120
00:49:22,537 --> 00:49:25,120
So the variables there, we had
an angular momentum eigenstate.

1121
00:49:25,120 --> 00:49:27,913
And that's a function
of the position.

1122
00:49:27,913 --> 00:49:29,580
Or, sorry, a linear
momentum eigenstate.

1123
00:49:29,580 --> 00:49:30,340
And that's a function
of the position.

1124
00:49:30,340 --> 00:49:32,110
Angular momentum
eigenstates are going

1125
00:49:32,110 --> 00:49:35,920
to be functions of
angular position.

1126
00:49:35,920 --> 00:49:39,770
So I want to know what these
wave functions look like, not

1127
00:49:39,770 --> 00:49:40,658
just the eigenvalues.

1128
00:49:40,658 --> 00:49:42,950
But I want to know what is
the wave function associated

1129
00:49:42,950 --> 00:49:44,980
to eigenvalues
little l and little m

1130
00:49:44,980 --> 00:49:48,910
look like, y sub lm of
the angles theta and phi.

1131
00:49:48,910 --> 00:49:52,118
What do these guys look like?

1132
00:49:52,118 --> 00:49:52,910
How do we get them?

1133
00:49:54,358 --> 00:49:56,650
This is going to be your goal
for the next two minutes.

1134
00:49:59,160 --> 00:50:03,970
So the first thing
to notice is that we

1135
00:50:03,970 --> 00:50:06,340
know what the form of the
eigenvalues and eigenfunctions

1136
00:50:06,340 --> 00:50:06,840
are.

1137
00:50:06,840 --> 00:50:09,610
If we act with Lz
we get h bar m back.

1138
00:50:09,610 --> 00:50:13,510
If we act with L squared we get
h bar squared ll plus 1 back.

1139
00:50:13,510 --> 00:50:16,910
But we also have other
expressions for Lz and L

1140
00:50:16,910 --> 00:50:17,410
squared.

1141
00:50:17,410 --> 00:50:19,210
In particular-- I wrote
them down last time

1142
00:50:19,210 --> 00:50:20,430
in spherical coordinates.

1143
00:50:20,430 --> 00:50:23,140
So I'm going to working in the
spherical coordinates where

1144
00:50:23,140 --> 00:50:26,440
the declination from the
vertical is an angle theta,

1145
00:50:26,440 --> 00:50:31,510
and the angle around the
equator is an angle phi.

1146
00:50:31,510 --> 00:50:34,523
And theta equals 0 is going
to be up in the z direction.

1147
00:50:34,523 --> 00:50:35,940
It's just a choice
of coordinates.

1148
00:50:35,940 --> 00:50:37,942
There's nothing deep.

1149
00:50:37,942 --> 00:50:39,150
There's nothing even shallow.

1150
00:50:39,150 --> 00:50:40,653
It's just definitions.

1151
00:50:40,653 --> 00:50:42,820
So we're going to work in
the spherical coordinates.

1152
00:50:42,820 --> 00:50:44,620
And in spherical
coordinates we observe

1153
00:50:44,620 --> 00:50:46,370
that this angular
momentum, just following

1154
00:50:46,370 --> 00:50:49,850
the definition from
r cross p, takes

1155
00:50:49,850 --> 00:50:51,040
a particularly simple form.

1156
00:50:54,580 --> 00:50:55,790
That's a typo.

1157
00:50:55,790 --> 00:50:59,310
h bar upon i, dd phi.

1158
00:50:59,310 --> 00:51:00,810
And instead of
writing L squared I'm

1159
00:51:00,810 --> 00:51:02,980
going to write L plus
minus, because it's shorter

1160
00:51:02,980 --> 00:51:05,397
and also because it's going
to turn out to be more useful.

1161
00:51:05,397 --> 00:51:08,260
So this takes the form h
bar, e to the plus minus i

1162
00:51:08,260 --> 00:51:19,060
phi, d theta, plus or minus
cotangent of theta, d phi.

1163
00:51:23,652 --> 00:51:25,110
So these are the
expressions for Lz

1164
00:51:25,110 --> 00:51:27,943
and L plus minus in
spherical coordinates,

1165
00:51:27,943 --> 00:51:29,235
in these spherical coordinates.

1166
00:51:34,900 --> 00:51:36,900
I want to know how Ylm
depends on theta and phi.

1167
00:51:36,900 --> 00:51:38,358
And it's clearly
going to be easier

1168
00:51:38,358 --> 00:51:40,660
to ask about the
Lz eigenequation.

1169
00:51:40,660 --> 00:51:41,960
So let's look at that.

1170
00:51:41,960 --> 00:51:48,530
Lz on Ylm gives me h--

1171
00:51:48,530 --> 00:51:49,040
Oh yeah?

1172
00:51:49,040 --> 00:51:51,255
AUDIENCE: Is it h or h bar?

1173
00:51:51,255 --> 00:51:52,130
PROFESSOR: Oh, Jesus.

1174
00:51:56,140 --> 00:51:59,580
When I write letters
by hand, which

1175
00:51:59,580 --> 00:52:03,610
is basically when I write to
my mom, all my h's are crossed.

1176
00:52:03,610 --> 00:52:04,283
I can't help it.

1177
00:52:04,283 --> 00:52:05,450
So this is like the inverse.

1178
00:52:10,550 --> 00:52:12,550
It's been a long time
since I made that mistake.

1179
00:52:12,550 --> 00:52:14,120
It's usually the other.

1180
00:52:14,120 --> 00:52:22,122
So Lz acting on Ylm gives
me h bar m, acting on Ylm.

1181
00:52:22,122 --> 00:52:23,830
But that's what I get
when I act with Lz,

1182
00:52:23,830 --> 00:52:25,955
so let's just write out
the differential equation .

1183
00:52:25,955 --> 00:52:29,150
So h bar m Ylm where m is an
integer, or a half integer,

1184
00:52:29,150 --> 00:52:31,660
depending on whether l is
an integer or half integer.

1185
00:52:31,660 --> 00:52:37,000
h bar m Ylm is equal to h
bar upin i, d phi of Ylm.

1186
00:52:39,360 --> 00:52:41,610
Using the awesome power of
division and multiplication

1187
00:52:41,610 --> 00:52:43,860
I will divide both
sides by h bar.

1188
00:52:43,860 --> 00:52:47,730
And I will multiply
both sides by i.

1189
00:52:47,730 --> 00:52:53,707
And we now have the equation
for the eigenfunctions of Lz,

1190
00:52:53,707 --> 00:52:55,290
which we actually
worked on last time.

1191
00:52:55,290 --> 00:52:57,410
And we can solve
this very simply.

1192
00:52:57,410 --> 00:52:59,973
This says that, remember, Ylm
is a function of theta and phi.

1193
00:52:59,973 --> 00:53:01,890
Here we're only looking
at the phi dependence,

1194
00:53:01,890 --> 00:53:04,015
because that's all that
showed up in this equation.

1195
00:53:04,015 --> 00:53:06,820
So this tells us that the
eigenfunctions Ylm are

1196
00:53:06,820 --> 00:53:12,080
of the form of theta and phi,
are of the form e to the im

1197
00:53:12,080 --> 00:53:17,540
phi times some
remaining dependence

1198
00:53:17,540 --> 00:53:21,080
on theta, which I'll
write as p of theta.

1199
00:53:21,080 --> 00:53:24,510
And that p could
depend on l and m.

1200
00:53:24,510 --> 00:53:25,010
Cool?

1201
00:53:34,820 --> 00:53:36,980
Already, before we even
ask about that dependence

1202
00:53:36,980 --> 00:53:41,230
on theta, the p dependence, we
learn something pretty awesome.

1203
00:53:41,230 --> 00:53:43,880
Look at this wave function.

1204
00:53:43,880 --> 00:53:48,450
Whatever else we know,
its dependence on phi

1205
00:53:48,450 --> 00:53:50,500
is e to the im phi.

1206
00:53:50,500 --> 00:53:52,260
Now, remember what phi is.

1207
00:53:52,260 --> 00:53:54,670
Phi is the angle
around the equator.

1208
00:53:54,670 --> 00:53:57,180
So it goes from 0 to 2 pi.

1209
00:53:57,180 --> 00:53:59,790
And when it comes back to
2 pi it's the same point.

1210
00:53:59,790 --> 00:54:01,250
Phi equals 0 and
phi equals pi are

1211
00:54:01,250 --> 00:54:03,120
two names for the same point.

1212
00:54:03,120 --> 00:54:03,620
Yes?

1213
00:54:06,750 --> 00:54:08,520
But that should worry you.

1214
00:54:08,520 --> 00:54:11,540
Because note that as
a consequence of this,

1215
00:54:11,540 --> 00:54:18,630
Ylm of theta 0 is equal
to, well, whatever it is.

1216
00:54:21,600 --> 00:54:31,120
Sorry, theta of 2 pi is equal
to e to the im 2 pi times Plm

1217
00:54:31,120 --> 00:54:31,620
to theta.

1218
00:54:35,370 --> 00:54:36,720
But this is equal to--

1219
00:54:36,720 --> 00:54:39,230
oh, now we're in trouble.

1220
00:54:39,230 --> 00:54:45,710
If m is an integer, this is
equal to e to the i integer 2

1221
00:54:45,710 --> 00:54:50,240
pi 1.

1222
00:54:50,240 --> 00:54:51,090
Yes, OK.

1223
00:54:51,090 --> 00:54:52,715
You're supposed to
cheer at that point.

1224
00:54:52,715 --> 00:54:54,548
It's like the coolest
identity in the world.

1225
00:54:54,548 --> 00:54:58,040
So e to the i 2 pi, that's one.

1226
00:54:58,040 --> 00:55:05,400
So if m is an integer
then this is just

1227
00:55:05,400 --> 00:55:10,180
Plm of theta, which is also
what we get by putting in phi

1228
00:55:10,180 --> 00:55:11,210
equals 0.

1229
00:55:11,210 --> 00:55:17,930
So this is equal to
Ylm of theta, comma, 0.

1230
00:55:17,930 --> 00:55:21,350
But if m is a half integer
then e to the i half

1231
00:55:21,350 --> 00:55:26,010
integer times 2 pi is minus 1.

1232
00:55:26,010 --> 00:55:31,650
And so that gives us
minus Ylm of theta and 0,

1233
00:55:31,650 --> 00:55:34,010
if m is a half integer.

1234
00:55:38,330 --> 00:55:43,730
So let me say that again,
in the same words, actually.

1235
00:55:43,730 --> 00:55:47,760
But let me just say it again
with different emphasis.

1236
00:55:47,760 --> 00:55:53,180
What this tells us
is that Ylm at 0,

1237
00:55:53,180 --> 00:55:55,805
as function of the coordinates
theta and phi, Ylm--

1238
00:55:58,390 --> 00:56:02,240
so let's take m as an integer.

1239
00:56:02,240 --> 00:56:11,530
Ylm at theta and 0 is equal
to Ylm at theta and 0.

1240
00:56:11,530 --> 00:56:12,630
That's good.

1241
00:56:12,630 --> 00:56:20,310
But if Y is a half integer,
then Ylm at theta and 2 pi,

1242
00:56:20,310 --> 00:56:23,760
which is the same point
as Ylm at theta and 0,

1243
00:56:23,760 --> 00:56:30,000
is equal to minus
Ylm at theta and 0.

1244
00:56:30,000 --> 00:56:31,660
That's less good.

1245
00:56:31,660 --> 00:56:34,494
What must be true of
Ylm, of theta and 0?

1246
00:56:34,494 --> 00:56:36,150
0.

1247
00:56:36,150 --> 00:56:38,470
And was there anything
special about the point 0?

1248
00:56:38,470 --> 00:56:38,970
No.

1249
00:56:38,970 --> 00:56:40,345
I could have just
taken any point

1250
00:56:40,345 --> 00:56:41,690
and rotated it around by pi.

1251
00:56:41,690 --> 00:56:47,610
So this tells us that
Ylm of theta and phi

1252
00:56:47,610 --> 00:56:51,990
is identically equal to
0 if m is a half integer.

1253
00:56:57,410 --> 00:56:58,700
Huh.

1254
00:56:58,700 --> 00:57:00,387
That's bad.

1255
00:57:00,387 --> 00:57:01,970
Because what's the
probability density

1256
00:57:01,970 --> 00:57:05,030
of being found at any
particular angular position?

1257
00:57:05,030 --> 00:57:06,390
0.

1258
00:57:06,390 --> 00:57:07,814
Can you normalize that state?

1259
00:57:07,814 --> 00:57:08,603
No.

1260
00:57:08,603 --> 00:57:10,520
That is not a state that
describes a particle.

1261
00:57:10,520 --> 00:57:12,895
That is a state that describes
the absence of a particle.

1262
00:57:12,895 --> 00:57:14,400
That is not what we want.

1263
00:57:14,400 --> 00:57:20,060
So these states cannot describe
these values of l and m,

1264
00:57:20,060 --> 00:57:22,480
which seem like perfectly
reasonable values of l and m,

1265
00:57:22,480 --> 00:57:25,740
perfectly reasonable
eigenvalues of L squared and lm.

1266
00:57:29,930 --> 00:57:35,570
They cannot be used to label
wave functions of physical

1267
00:57:35,570 --> 00:57:38,670
states corresponding to
wave functions on a sphere.

1268
00:57:38,670 --> 00:57:40,070
You can't do it.

1269
00:57:40,070 --> 00:57:42,560
Because if you try, you find
that those wave functions

1270
00:57:42,560 --> 00:57:44,750
identically vanish.

1271
00:57:44,750 --> 00:57:45,840
OK?

1272
00:57:45,840 --> 00:57:47,310
So these cannot be used.

1273
00:57:47,310 --> 00:57:56,755
These do not describe wave
functions of a particle

1274
00:57:56,755 --> 00:57:57,630
in quantum mechanics.

1275
00:57:57,630 --> 00:57:58,463
They cannot be used.

1276
00:57:58,463 --> 00:58:01,598
Those values, those towers
cannot be used to describe

1277
00:58:01,598 --> 00:58:03,140
particles moving in
three dimensions.

1278
00:58:06,950 --> 00:58:07,830
Questions about that?

1279
00:58:07,830 --> 00:58:11,550
This is a slightly
subtle argument.

1280
00:58:11,550 --> 00:58:12,050
Yeah?

1281
00:58:12,050 --> 00:58:13,800
AUDIENCE: You said
something earlier about

1282
00:58:13,800 --> 00:58:17,484
how atoms would never have
any zero angular momentum.

1283
00:58:17,484 --> 00:58:20,760
And so the ones that have
no zero angular momentum,

1284
00:58:20,760 --> 00:58:22,177
we just said they're
not possible.

1285
00:58:22,177 --> 00:58:23,170
So [INAUDIBLE]?

1286
00:58:23,170 --> 00:58:24,420
PROFESSOR: Excellent question.

1287
00:58:24,420 --> 00:58:26,295
I said it slightly diff--
so the question is,

1288
00:58:26,295 --> 00:58:28,555
look, earlier you were
saying, yeah, yeah, yeah.

1289
00:58:28,555 --> 00:58:31,180
There are atoms in the world and
they have half integer angular

1290
00:58:31,180 --> 00:58:31,690
momentum.

1291
00:58:31,690 --> 00:58:33,065
And you can shine
a light on them

1292
00:58:33,065 --> 00:58:35,620
and you can tell and stuff.

1293
00:58:35,620 --> 00:58:38,490
But you just said
these can't exist.

1294
00:58:38,490 --> 00:58:41,143
So how can those two
things both be true?

1295
00:58:41,143 --> 00:58:42,310
Thank you for this question.

1296
00:58:42,310 --> 00:58:43,393
It's a very good question.

1297
00:58:43,393 --> 00:58:45,490
I actually said a
slightly different thing.

1298
00:58:45,490 --> 00:58:49,660
What I said was, states where
angular momentum lm are half

1299
00:58:49,660 --> 00:58:53,070
integers cannot be
described by a wave function

1300
00:58:53,070 --> 00:58:54,500
of the coordinates.

1301
00:58:54,500 --> 00:58:56,790
We're going to need some
different description.

1302
00:58:56,790 --> 00:58:58,420
And in particular, we're going
to need a different description

1303
00:58:58,420 --> 00:58:59,440
that does what?

1304
00:58:59,440 --> 00:59:06,650
Well, as we take phi from 0 to
2 pi, as we rotate the system,

1305
00:59:06,650 --> 00:59:08,750
we're going to pick
up a minus sign.

1306
00:59:08,750 --> 00:59:11,020
So in order to describe
an object with lm

1307
00:59:11,020 --> 00:59:13,710
being a half integer, we
can't use the wave function.

1308
00:59:13,710 --> 00:59:17,125
We need something that is
allowed to be doubly valued.

1309
00:59:17,125 --> 00:59:19,500
And in particular, we need
something that behaves nicely.

1310
00:59:19,500 --> 00:59:21,830
When you rotate
around by 2 pi, we

1311
00:59:21,830 --> 00:59:25,330
need to come back to a
minus sign, not itself.

1312
00:59:25,330 --> 00:59:29,910
So at some object
that's not a function,

1313
00:59:29,910 --> 00:59:31,420
it's called a spinner.

1314
00:59:31,420 --> 00:59:35,080
So we'll talk about it later.

1315
00:59:35,080 --> 00:59:36,880
We need some object
that does that.

1316
00:59:36,880 --> 00:59:39,347
So there's this
classic demonstration

1317
00:59:39,347 --> 00:59:41,680
at this point, which is
supposed to be done in a quantum

1318
00:59:41,680 --> 00:59:42,675
mechanics class.

1319
00:59:42,675 --> 00:59:44,050
So at this point
the lecturist is

1320
00:59:44,050 --> 00:59:45,650
obliged to do the
following thing.

1321
00:59:45,650 --> 00:59:48,700
You say, blah, blah, blah,
if things rotate by 2 pi

1322
00:59:48,700 --> 00:59:50,480
they have to come
back to themselves.

1323
00:59:50,480 --> 00:59:52,790
And then you do this.

1324
00:59:52,790 --> 00:59:59,170
I'm going to rotate my
hand like a record by 2 pi.

1325
00:59:59,170 --> 01:00:00,760
And it will not
come back to itself.

1326
01:00:00,760 --> 01:00:01,260
OK?

1327
01:00:04,250 --> 01:00:07,080
And it quite uncomfortably
has not come back to itself.

1328
01:00:07,080 --> 01:00:08,970
But I can do a further
rotation to show you

1329
01:00:08,970 --> 01:00:09,928
that it's a minus sign.

1330
01:00:09,928 --> 01:00:11,640
I can do a further
the rotation by 2 pi

1331
01:00:11,640 --> 01:00:12,973
and have it come back to itself.

1332
01:00:15,390 --> 01:00:16,970
And I kept the axis
vertical, yeah?

1333
01:00:16,970 --> 01:00:19,512
OK, so at this point you're all
supposed to go like, oh, yes,

1334
01:00:19,512 --> 01:00:22,070
uh-huh, mmm.

1335
01:00:22,070 --> 01:00:28,460
So now that we've got that
out of the way, I have an arm.

1336
01:00:28,460 --> 01:00:30,687
So the story is a little
more complicated than that.

1337
01:00:30,687 --> 01:00:32,270
This is actually a
fair demonstration,

1338
01:00:32,270 --> 01:00:33,645
but it's a slightly
subtle story.

1339
01:00:33,645 --> 01:00:36,145
If you want to understand it,
ask your recitation instructor

1340
01:00:36,145 --> 01:00:37,383
or come to my office hours.

1341
01:00:37,383 --> 01:00:40,100
AUDIENCE: What about USB sticks?

1342
01:00:40,100 --> 01:00:41,342
PROFESSOR: USB sticks?

1343
01:00:41,342 --> 01:00:43,800
AUDIENCE: You insert them here
and they don't go and insert

1344
01:00:43,800 --> 01:00:45,045
the other way.

1345
01:00:45,045 --> 01:00:46,905
[LAUGHTER]

1346
01:00:48,805 --> 01:00:50,180
PROFESSOR: What
about USB sticks?

1347
01:00:50,180 --> 01:00:51,410
You insert them this
way, they don't work.

1348
01:00:51,410 --> 01:00:53,160
You insert them this
way, they don't work.

1349
01:00:53,160 --> 01:00:55,450
But if you do it
again, then they do.

1350
01:00:55,450 --> 01:00:57,410
[LAUGHTER]

1351
01:00:59,860 --> 01:01:01,820
[APPLAUSE]

1352
01:01:02,707 --> 01:01:03,290
PROFESSOR: OK.

1353
01:01:07,220 --> 01:01:10,200
That's pretty good.

1354
01:01:10,200 --> 01:01:13,245
So for the moment, as long as
we want to describe our system

1355
01:01:13,245 --> 01:01:14,620
with a wave function
of position,

1356
01:01:14,620 --> 01:01:16,870
which means we're thinking
about where will we find it

1357
01:01:16,870 --> 01:01:19,230
as a function of angle, we
cannot use the half integer l

1358
01:01:19,230 --> 01:01:20,932
or m.

1359
01:01:20,932 --> 01:01:23,015
So if we can't use the
half interger l or m, fine.

1360
01:01:23,015 --> 01:01:24,230
We'll just throw them
out for the moment

1361
01:01:24,230 --> 01:01:26,050
and we'll use the
integer l and m.

1362
01:01:26,050 --> 01:01:28,860
And let's keep going.

1363
01:01:28,860 --> 01:01:30,500
What we need to
determine now is the P.

1364
01:01:30,500 --> 01:01:32,000
We've determined
the phi dependence,

1365
01:01:32,000 --> 01:01:33,430
but we need the
theta dependence.

1366
01:01:33,430 --> 01:01:36,940
We can get the theta
dependence in a sneaky fashion.

1367
01:01:36,940 --> 01:01:39,470
Remember the harmonic
oscillator in 1D.

1368
01:01:39,470 --> 01:01:41,720
When we wanted to find the
ground state wave function,

1369
01:01:41,720 --> 01:01:44,230
we could either solve the energy
eigenvalue equation, which

1370
01:01:44,230 --> 01:01:47,380
is a second order differential
equation and kind of horrible,

1371
01:01:47,380 --> 01:01:50,450
or we could solve the
ground state equation that

1372
01:01:50,450 --> 01:01:53,020
said that the ground state is
annihilated by the annihilation

1373
01:01:53,020 --> 01:01:55,458
operator, which is a first
order difference equation

1374
01:01:55,458 --> 01:01:56,500
and much easier to solve.

1375
01:01:56,500 --> 01:01:57,140
Yeah?

1376
01:01:57,140 --> 01:01:58,140
Let's do the same thing.

1377
01:01:58,140 --> 01:01:59,620
We have an
annihilation condition.

1378
01:01:59,620 --> 01:02:04,047
If we have the top state, L
plus on Y ll is equal to 0.

1379
01:02:04,047 --> 01:02:06,130
But L plus is a first order
differential operator.

1380
01:02:06,130 --> 01:02:07,370
This is going to be easier.

1381
01:02:07,370 --> 01:02:09,490
So we need to find a
solution to this equation.

1382
01:02:09,490 --> 01:02:11,327
And do I want to
go through this?

1383
01:02:11,327 --> 01:02:11,910
Yeah, why not.

1384
01:02:11,910 --> 01:02:13,440
OK, so I want this
to be equal to 0.

1385
01:02:13,440 --> 01:02:16,340
But L plus Yll is
equal to-- well,

1386
01:02:16,340 --> 01:02:25,460
it's h bar, e to the plus i phi,
d theta, plus cotangent theta,

1387
01:02:25,460 --> 01:02:28,560
d phi on Yll.

1388
01:02:28,560 --> 01:02:36,390
And Yll is e to the i l
phi times Plm of theta.

1389
01:02:36,390 --> 01:02:37,720
Cool?

1390
01:02:37,720 --> 01:02:41,760
So dd phi on e to the il phi,
there's no phi dependence here.

1391
01:02:41,760 --> 01:02:44,970
It's just going to
give us a vector of il.

1392
01:02:44,970 --> 01:02:46,336
Did I get the--

1393
01:02:46,336 --> 01:02:47,210
yeah.

1394
01:02:47,210 --> 01:02:52,030
So that's going to give us
a plus il and no dd phi.

1395
01:02:58,860 --> 01:03:01,500
And this e to the il
phi we can pull out.

1396
01:03:01,500 --> 01:03:02,840
But this has to be equal to 0.

1397
01:03:02,840 --> 01:03:10,670
So this says that 0 is equal to
d theta plus il cotangent theta

1398
01:03:10,670 --> 01:03:13,090
P lm of theta.

1399
01:03:15,843 --> 01:03:17,510
And this is actually
much better than it

1400
01:03:17,510 --> 01:03:18,802
seems for the following reason.

1401
01:03:18,802 --> 01:03:21,420
dd theta-- find the dd theta.

1402
01:03:21,420 --> 01:03:25,350
Cotangent of theta
is cosine over sine.

1403
01:03:25,350 --> 01:03:28,030
That's what you get if you
take the derivative of lots

1404
01:03:28,030 --> 01:03:30,270
of sine functions--

1405
01:03:30,270 --> 01:03:32,210
sine to the l, say.

1406
01:03:32,210 --> 01:03:32,960
Take a derivative.

1407
01:03:32,960 --> 01:03:35,418
You lose a power of sine and
you pick up a power of cosine.

1408
01:03:35,418 --> 01:03:37,840
So multiplying by cotangent
gets rid of a power of sign

1409
01:03:37,840 --> 01:03:43,450
and gives you a power of cosine,
which is a derivative of sine.

1410
01:03:43,450 --> 01:03:45,830
So Plm, noticing the l--

1411
01:03:45,830 --> 01:03:49,400
and I screwed up an i somewhere.

1412
01:03:49,400 --> 01:03:52,330
I think I wanted an i up here.

1413
01:03:52,330 --> 01:03:53,980
Let's see.

1414
01:03:53,980 --> 01:03:54,480
i.

1415
01:04:06,290 --> 01:04:06,790
Sorry.

1416
01:04:10,280 --> 01:04:12,490
Yes, I want an i cotangent.

1417
01:04:12,490 --> 01:04:14,280
OK, that's much better.

1418
01:04:14,280 --> 01:04:16,510
So i cotangent-- good, good.

1419
01:04:16,510 --> 01:04:20,970
And then the i squareds
give me a minus l.

1420
01:04:20,970 --> 01:04:25,260
And this tells us that Plm,
so if this is sine to the l,

1421
01:04:25,260 --> 01:04:29,690
then d theta gives us an l sine
to the l minus 1 cosine, which

1422
01:04:29,690 --> 01:04:31,777
is what I get by
taking sine to the l

1423
01:04:31,777 --> 01:04:33,610
and multiplying by
cosine, dividing by sine,

1424
01:04:33,610 --> 01:04:35,200
and multplying by l.

1425
01:04:35,200 --> 01:04:38,120
So this gives me Pll.

1426
01:04:38,120 --> 01:04:40,847
This is for the
particular state ll.

1427
01:04:40,847 --> 01:04:43,180
We're looking at the top state
and we're annhilating it.

1428
01:04:43,180 --> 01:04:47,570
So Pll is equal to
some coefficient,

1429
01:04:47,570 --> 01:04:52,150
so I'll just say proportional
to sine to the l of theta.

1430
01:04:55,450 --> 01:05:02,710
So this tells us that Yll is
equal to some normalization,

1431
01:05:02,710 --> 01:05:06,730
sub ll, just some number,
times from the phi dependence

1432
01:05:06,730 --> 01:05:10,400
e to the il phi, and from
the theta dependence,

1433
01:05:10,400 --> 01:05:12,640
sine to the l of theta.

1434
01:05:16,580 --> 01:05:19,570
So this is the form.

1435
01:05:19,570 --> 01:05:20,846
Sorry, go ahead.

1436
01:05:20,846 --> 01:05:22,980
AUDIENCE: What's
the symbol there?

1437
01:05:22,980 --> 01:05:25,250
PROFESSOR: Oh, this
twisted horrible thing?

1438
01:05:25,250 --> 01:05:26,610
It's proportional to.

1439
01:05:26,610 --> 01:05:29,930
But it was a long night.

1440
01:05:33,350 --> 01:05:36,160
So now we have the wave function
explicitly as a function phi

1441
01:05:36,160 --> 01:05:38,870
and as a function of theta
completely understood

1442
01:05:38,870 --> 01:05:41,640
for the top state in any tower.

1443
01:05:41,640 --> 01:05:43,740
This is for any L. The
top state in any tower

1444
01:05:43,740 --> 01:05:45,070
is e to the il phi.

1445
01:05:45,070 --> 01:05:46,050
Does that make sense?

1446
01:05:46,050 --> 01:05:48,495
Well Lz is h bar upon id phi.

1447
01:05:48,495 --> 01:05:51,760
So that gives us
h bar as m as l.

1448
01:05:51,760 --> 01:05:52,400
So that's good.

1449
01:05:52,400 --> 01:05:53,340
That's the top state.

1450
01:05:53,340 --> 01:05:55,007
And from the sine
theta we just checked.

1451
01:05:55,007 --> 01:05:58,490
We constructed that
this indeed has the--

1452
01:05:58,490 --> 01:06:00,860
well, if you then check you
will find that the L squared

1453
01:06:00,860 --> 01:06:02,818
eigenvalue, which you'll
do on the problem set,

1454
01:06:02,818 --> 01:06:05,990
the l squared eigenvalue
is h bar squared ll plus 1.

1455
01:06:05,990 --> 01:06:07,720
AUDIENCE: Quick question.

1456
01:06:07,720 --> 01:06:10,530
The expression we have for
the L plus minus operators,

1457
01:06:10,530 --> 01:06:12,970
how did we construct the
expressions for Lx and Ly?

1458
01:06:12,970 --> 01:06:14,130
PROFESSOR: Good.

1459
01:06:14,130 --> 01:06:15,800
It's much easier than you think.

1460
01:06:15,800 --> 01:06:18,380
So Lx is equal to--

1461
01:06:18,380 --> 01:06:19,500
L is r cross b, right?

1462
01:06:19,500 --> 01:06:22,460
So this is going to
be yPz minus zPy.

1463
01:06:28,200 --> 01:06:37,391
And this is equal to h
bar upon i, ydz minus zdy.

1464
01:06:37,391 --> 01:06:40,010
But you know what y is
in spherical coordinates.

1465
01:06:40,010 --> 01:06:40,990
And you know what
derivitive with respect to z

1466
01:06:40,990 --> 01:06:41,750
is in spherical coordinates.

1467
01:06:41,750 --> 01:06:43,810
Because you know what z is,
and you know the chain rule.

1468
01:06:43,810 --> 01:06:45,940
So taking this and just
plugging in explicit expression

1469
01:06:45,940 --> 01:06:48,107
for the change of variables
to spherical coordinates

1470
01:06:48,107 --> 01:06:49,550
takes care of it.

1471
01:06:49,550 --> 01:06:50,050
Yeah?

1472
01:06:50,050 --> 01:06:51,855
AUDIENCE: What does the
superscript of the sine

1473
01:06:51,855 --> 01:06:52,465
indicate?

1474
01:06:52,465 --> 01:06:53,960
Is that sine to the power of l?

1475
01:06:53,960 --> 01:06:54,668
PROFESSOR: Sorry.

1476
01:06:54,668 --> 01:06:56,090
This is bad notation.

1477
01:06:56,090 --> 01:06:59,410
It's not bad notation, it's
just not familiar notation.

1478
01:06:59,410 --> 01:07:02,075
It's notation that is used
throughout theoretical physics.

1479
01:07:02,075 --> 01:07:04,310
It means this, sine
theta to the l.

1480
01:07:04,310 --> 01:07:08,090
The lth power of sine.

1481
01:07:08,090 --> 01:07:10,250
For typesetting reasons
we often put the power

1482
01:07:10,250 --> 01:07:12,130
before the argument.

1483
01:07:12,130 --> 01:07:13,630
Yeah, no, it's a
very good question.

1484
01:07:13,630 --> 01:07:15,505
Thank you for asking,
because it was unclear.

1485
01:07:15,505 --> 01:07:16,610
I appreciate that.

1486
01:07:16,610 --> 01:07:19,122
Other questions?

1487
01:07:19,122 --> 01:07:22,220
AUDIENCE: [INAUDIBLE]
L hat [INAUDIBLE]??

1488
01:07:22,220 --> 01:07:24,860
PROFESSOR: How did we come
up with the L hat plus minus?

1489
01:07:24,860 --> 01:07:27,030
That was from this.

1490
01:07:27,030 --> 01:07:29,645
So we know what the components
of the angular momentum

1491
01:07:29,645 --> 01:07:31,600
are in Cartesian coordinates.

1492
01:07:31,600 --> 01:07:33,910
And you know how,
because it's coordinates,

1493
01:07:33,910 --> 01:07:36,640
to change variables from
Cartesian to spherical.

1494
01:07:36,640 --> 01:07:38,290
So you just plug this in for Lx.

1495
01:07:38,290 --> 01:07:41,200
But L plus is Lx plus
ioy, and so you just

1496
01:07:41,200 --> 01:07:43,130
take these guys
in spherical form

1497
01:07:43,130 --> 01:07:44,880
and add them together
with the relative i.

1498
01:07:44,880 --> 01:07:48,008
And that gives you
that expression.

1499
01:07:48,008 --> 01:07:49,966
AUDIENCE: Maybe I missed
this, but can you just

1500
01:07:49,966 --> 01:07:53,173
explain the distinction
between Y sub ll and Y sub lm?

1501
01:07:53,173 --> 01:07:54,340
PROFESSOR: Yeah, absolutely.

1502
01:07:54,340 --> 01:08:01,756
So Y sub ll, it means Y sub
lm where m is equal to l.

1503
01:08:01,756 --> 01:08:02,464
AUDIENCE: Oh, OK.

1504
01:08:02,464 --> 01:08:04,780
It was just the
generic [INAUDIBLE]..

1505
01:08:04,780 --> 01:08:07,030
PROFESSOR: It's a generic
eigenfunction of the angular

1506
01:08:07,030 --> 01:08:10,600
momentum, with the angular
momentum in the z direction

1507
01:08:10,600 --> 01:08:13,140
being equal to the angular
momentum in the total angular

1508
01:08:13,140 --> 01:08:14,200
momentum.

1509
01:08:14,200 --> 01:08:16,680
At least for these numbers.

1510
01:08:16,680 --> 01:08:17,380
OK?

1511
01:08:17,380 --> 01:08:19,490
Cool.

1512
01:08:19,490 --> 01:08:22,609
Good, so now, if we know this,
how do we just as a side note--

1513
01:08:22,609 --> 01:08:24,609
suppose we know-- well,
suppose we know this?

1514
01:08:24,609 --> 01:08:25,170
We know this.

1515
01:08:25,170 --> 01:08:26,760
We know what the top state
in the tower looks like.

1516
01:08:26,760 --> 01:08:28,677
How do I get the next
state down in the tower?

1517
01:08:28,677 --> 01:08:30,923
how do I get Yl l minus 1?

1518
01:08:30,923 --> 01:08:31,715
AUDIENCE: Lower it.

1519
01:08:31,715 --> 01:08:32,548
PROFESSOR: Lower it.

1520
01:08:32,548 --> 01:08:33,130
Exactly.

1521
01:08:33,130 --> 01:08:35,643
So this is easy, L minus on Yll.

1522
01:08:35,643 --> 01:08:37,560
And we have to be careful
about normalization.

1523
01:08:37,560 --> 01:08:41,262
So again, it's proportional to.

1524
01:08:41,262 --> 01:08:41,970
But this is easy.

1525
01:08:41,970 --> 01:08:43,260
We don't have to solve
any difference equations.

1526
01:08:43,260 --> 01:08:44,490
We just have to
take derivatives.

1527
01:08:44,490 --> 01:08:45,819
So it's just like
the raising operator

1528
01:08:45,819 --> 01:08:47,040
for a harmonic oscillator.

1529
01:08:47,040 --> 01:08:48,787
We can raise and
lower along the tower

1530
01:08:48,787 --> 01:08:50,120
and get the right wave function.

1531
01:08:54,920 --> 01:08:57,924
To give you some examples--

1532
01:08:57,924 --> 01:08:59,043
yeah, let's do that here.

1533
01:08:59,043 --> 01:09:01,340
Let me just quickly
give you a few examples

1534
01:09:01,340 --> 01:09:05,125
of the first few
spherical harmonics.

1535
01:09:05,125 --> 01:09:06,750
Sorry, I should give
these guys a name.

1536
01:09:06,750 --> 01:09:08,939
These functions, Ylm
of theta and phi,

1537
01:09:08,939 --> 01:09:10,970
they're called the
spherical harmonics.

1538
01:09:10,970 --> 01:09:13,630
They're called these because
they solve the Laplacian

1539
01:09:13,630 --> 01:09:16,750
equation on the sphere, which
is just the eigenvalue equation.

1540
01:09:16,750 --> 01:09:19,500
L squared on them is equal to
a constant times those things

1541
01:09:19,500 --> 01:09:20,000
back.

1542
01:09:24,240 --> 01:09:28,279
Just to tabulate a couple of
examples for you concretely,

1543
01:09:28,279 --> 01:09:31,529
consider the l equals 0 states.

1544
01:09:31,529 --> 01:09:34,500
What are the allowed values
of m for little l equals 0?

1545
01:09:34,500 --> 01:09:35,000
0.

1546
01:09:35,000 --> 01:09:36,710
So Y0,0 is the only state.

1547
01:09:36,710 --> 01:09:41,020
And if you properly normalize
it, it's 1 over root 4 pi.

1548
01:09:41,020 --> 01:09:41,660
OK, good.

1549
01:09:41,660 --> 01:09:43,460
what about l equals 1?

1550
01:09:43,460 --> 01:09:50,520
Then we have Y1, 0 and
we have Y1 minus 1.

1551
01:09:50,520 --> 01:09:52,960
And we have Y1,1.

1552
01:09:52,960 --> 01:09:56,370
So these guys take a
particularly simple form.

1553
01:09:56,370 --> 01:09:59,805
Root 3-- I'm not even going to
worry about the coefficient.

1554
01:09:59,805 --> 01:10:00,680
They're in the notes.

1555
01:10:00,680 --> 01:10:03,500
You can look them up anywhere.

1556
01:10:03,500 --> 01:10:08,110
So first off let's
look at Y1, 1.

1557
01:10:08,110 --> 01:10:09,000
So non-linear today.

1558
01:10:09,000 --> 01:10:11,045
So Y1, 1, it's going to
be some normalization.

1559
01:10:11,045 --> 01:10:12,130
And what is the form?

1560
01:10:15,940 --> 01:10:19,490
It's just e to the il phi
sine theta to the l. l is 1.

1561
01:10:19,490 --> 01:10:27,040
So this is some constant times
e to the i phi, sine theta.

1562
01:10:27,040 --> 01:10:27,740
Who 1, 0?

1563
01:10:27,740 --> 01:10:32,560
Well, it's got angular momentum
0 in the z direction, in Lz.

1564
01:10:32,560 --> 01:10:35,540
So that means how
does it depend on phi?

1565
01:10:35,540 --> 01:10:36,590
It doesn't.

1566
01:10:36,590 --> 01:10:39,112
And you can easily
see that, because when

1567
01:10:39,112 --> 01:10:42,500
we lower we get an e
to the i minus i phi.

1568
01:10:42,500 --> 01:10:45,780
Anyway, so this gives
us a constant times

1569
01:10:45,780 --> 01:10:47,760
no e to the i phi,
no phi dependents,

1570
01:10:47,760 --> 01:10:50,380
and cosine of theta.

1571
01:10:50,380 --> 01:10:51,990
And if you get a
cosine of theta,

1572
01:10:51,990 --> 01:10:53,760
the d theta and
the cotangent d phi

1573
01:10:53,760 --> 01:10:55,270
will give you the same thing.

1574
01:10:55,270 --> 01:10:58,350
And Y1 minus 1 is again
a constant, times e

1575
01:10:58,350 --> 01:10:59,790
to the minus i phi.

1576
01:10:59,790 --> 01:11:04,530
So it's got m equals minus
1 and sine theta again.

1577
01:11:04,530 --> 01:11:07,170
Notice a pleasing
parsimony here.

1578
01:11:07,170 --> 01:11:10,795
The theta dependence is the
same for plus m and minus m.

1579
01:11:13,660 --> 01:11:14,560
So what about Y2, 2?

1580
01:11:17,740 --> 01:11:25,970
Some constant e to the i 2 phi,
sine squared theta, dot, dot,

1581
01:11:25,970 --> 01:11:27,957
dot, Y0, 0.

1582
01:11:27,957 --> 01:11:29,040
And here it's interesting.

1583
01:11:29,040 --> 01:11:31,850
Here we just got one term
from taking the derivative.

1584
01:11:31,850 --> 01:11:33,190
They both give you cosine.

1585
01:11:33,190 --> 01:11:36,780
But now there are two ways to
act with the two derivatives.

1586
01:11:36,780 --> 01:11:38,190
And this gives you a constant.

1587
01:11:38,190 --> 01:11:39,440
Now what's the phi dependence?

1588
01:11:39,440 --> 01:11:41,560
It's nothing, because
it's got m equals 0.

1589
01:11:41,560 --> 01:11:43,250
And so the only
dependence is on theta.

1590
01:11:43,250 --> 01:11:46,470
And we get a cos squared
whoops, there's a 3--

1591
01:11:46,470 --> 01:11:49,150
3 cos squared theta minus 1.

1592
01:11:49,150 --> 01:11:50,895
AUDIENCE: Do you mean Y2, 0?

1593
01:11:50,895 --> 01:11:51,770
PROFESSOR: Oh, shoot.

1594
01:11:51,770 --> 01:11:52,270
Thank you.

1595
01:11:52,270 --> 01:11:53,080
Yes, I mean Y2, 0.

1596
01:11:53,080 --> 01:11:53,980
Thank you.

1597
01:11:53,980 --> 01:11:58,600
And then if we continue
lowering to Y2 minus 2,

1598
01:11:58,600 --> 01:12:00,100
this is equal to,
again, a constant.

1599
01:12:00,100 --> 01:12:02,350
And the it's going to be the
same dependence on theta,

1600
01:12:02,350 --> 01:12:05,890
but a different dependence of
phi. e to the minus 2i phi,

1601
01:12:05,890 --> 01:12:07,850
sine squared theta.

1602
01:12:07,850 --> 01:12:08,350
OK?

1603
01:12:08,350 --> 01:12:09,348
Yeah.

1604
01:12:09,348 --> 01:12:11,340
AUDIENCE: Aren't these
not normalizable?

1605
01:12:14,325 --> 01:12:14,950
PROFESSOR: Why?

1606
01:12:14,950 --> 01:12:16,550
AUDIENCE: Oh, never mind.

1607
01:12:16,550 --> 01:12:17,480
PROFESSOR: Good.

1608
01:12:17,480 --> 01:12:18,980
So let me turn that
into a question.

1609
01:12:18,980 --> 01:12:21,790
The question is, are
these normalizable?

1610
01:12:21,790 --> 01:12:23,590
Yeah, so how would
we normalize them?

1611
01:12:23,590 --> 01:12:25,320
What's the check for if
they're normalizable or not?

1612
01:12:25,320 --> 01:12:26,270
AUDIENCE: [INAUDIBLE]

1613
01:12:26,270 --> 01:12:27,687
PROFESSOR: Yeah,
we integrate them

1614
01:12:27,687 --> 01:12:30,890
norm squared over a sphere.

1615
01:12:30,890 --> 01:12:32,800
Not over a volume,
just over a sphere.

1616
01:12:32,800 --> 01:12:35,150
Because they're only wave
functions on the sphere.

1617
01:12:35,150 --> 01:12:36,360
We haven't dealt with
the radial function.

1618
01:12:36,360 --> 01:12:37,485
We'll deal with that later.

1619
01:12:37,485 --> 01:12:39,403
That will come in
the next lecture.

1620
01:12:39,403 --> 01:12:40,070
Other questions?

1621
01:12:40,070 --> 01:12:40,630
Yeah?

1622
01:12:40,630 --> 01:12:43,020
AUDIENCE: Can you explain
one more time why m equals

1623
01:12:43,020 --> 01:12:44,915
0 doesn't have [INAUDIBLE]?

1624
01:12:44,915 --> 01:12:46,540
PROFESSOR: Yeah, why
m equals 0 doesn't

1625
01:12:46,540 --> 01:12:49,070
have any phi dependence?

1626
01:12:49,070 --> 01:12:51,570
If m equals 0 had
phi dependence,

1627
01:12:51,570 --> 01:12:53,990
then we know that
the eigenvalue of Llz

1628
01:12:53,990 --> 01:12:56,540
is what we get when we take a
derivative with respect to phi.

1629
01:12:56,540 --> 01:12:59,110
But if the Lz
eigenvalue is 0, that

1630
01:12:59,110 --> 01:13:01,190
means that when we act
with dd phi we get 0.

1631
01:13:01,190 --> 01:13:03,220
That means it can't
depend on phi.

1632
01:13:03,220 --> 01:13:05,950
Cool?

1633
01:13:05,950 --> 01:13:07,280
Other questions?

1634
01:13:07,280 --> 01:13:09,280
What I'm going to do next
is I'm going show you,

1635
01:13:09,280 --> 01:13:10,863
walk you through
some of these angular

1636
01:13:10,863 --> 01:13:14,280
momentum eigenstates,
graphically on the computer.

1637
01:13:14,280 --> 01:13:18,700
Before we do that, any questions
about the calculation so far?

1638
01:13:18,700 --> 01:13:19,200
OK.

1639
01:13:30,620 --> 01:13:34,690
So this mathematical package
I'll post on the web page.

1640
01:13:34,690 --> 01:13:37,050
And at the moment I think
it's doing the real part.

1641
01:13:37,050 --> 01:13:38,610
So what we're looking
at now is the real part.

1642
01:13:38,610 --> 01:13:40,402
Actually, let's look
at the absolute value.

1643
01:13:43,350 --> 01:13:44,300
Good.

1644
01:13:44,300 --> 01:13:50,490
So here we are, looking
at the absolute value of--

1645
01:13:50,490 --> 01:13:54,180
that's not what I wanted to do.

1646
01:13:54,180 --> 01:13:56,465
So what we're looking
at in this notation

1647
01:13:56,465 --> 01:13:57,840
is some horrible
parametric plot.

1648
01:13:57,840 --> 01:14:00,090
You don't really need to see
the mathemat-- oh, shoot.

1649
01:14:02,330 --> 01:14:02,830
Sorry.

1650
01:14:05,605 --> 01:14:07,480
You don't need to see
the code, particularly.

1651
01:14:07,480 --> 01:14:10,240
So I'm not going to worry about
it, but it will be posted.

1652
01:14:10,240 --> 01:14:13,230
So here we're looking at
the absolute value, the norm

1653
01:14:13,230 --> 01:14:19,810
squared of the spherical
harmonic Y. And the lower

1654
01:14:19,810 --> 01:14:22,926
eigenvalue here, lower
coordinate is the l.

1655
01:14:22,926 --> 01:14:25,440
And the upper is m.

1656
01:14:25,440 --> 01:14:29,210
So when l is 0, what we get--

1657
01:14:29,210 --> 01:14:32,000
here's what this
plot is indicating.

1658
01:14:32,000 --> 01:14:34,570
The distance away from the
origin in a particular angular

1659
01:14:34,570 --> 01:14:37,430
direction is the absolute
value of the wave function.

1660
01:14:37,430 --> 01:14:39,330
So the further away
from the origin

1661
01:14:39,330 --> 01:14:42,320
the colored point you see is,
the larger the absolute value.

1662
01:14:45,720 --> 01:14:48,560
And the color here is just to
indicate depth and position.

1663
01:14:48,560 --> 01:14:50,110
It's not terribly meaningful.

1664
01:14:50,110 --> 01:14:52,520
So here we see that
we get a sphere.

1665
01:14:52,520 --> 01:14:54,330
So the probability
density or the norm

1666
01:14:54,330 --> 01:14:57,280
squared of the wave function
of the spherical harmonic

1667
01:14:57,280 --> 01:14:57,997
is constant.

1668
01:14:57,997 --> 01:14:58,830
So that makes sense.

1669
01:14:58,830 --> 01:14:59,960
It's spherically symmetric.

1670
01:14:59,960 --> 01:15:01,400
It has no angular momentum.

1671
01:15:01,400 --> 01:15:03,400
As we start increasing
the angular momentum,

1672
01:15:03,400 --> 01:15:06,778
let's take the angular
momentum l is 1, m is 0 state,

1673
01:15:06,778 --> 01:15:08,195
now something
interesting happens.

1674
01:15:11,037 --> 01:15:12,370
The total angular momentum is 1.

1675
01:15:12,370 --> 01:15:15,260
And we see that there
are two spheres.

1676
01:15:15,260 --> 01:15:16,940
Let me sort of rotate this.

1677
01:15:16,940 --> 01:15:19,700
So there are two spheres,
and there's the z-axis

1678
01:15:19,700 --> 01:15:20,720
passing through them.

1679
01:15:20,720 --> 01:15:23,430
And so the probability
is much larger

1680
01:15:23,430 --> 01:15:25,980
around this lobe on the top
or the lobe on the bottom.

1681
01:15:25,980 --> 01:15:29,070
And it's 0 on the plane.

1682
01:15:29,070 --> 01:15:31,280
Now, that's kind
of non-intuitive

1683
01:15:31,280 --> 01:15:37,060
if you think, well,
Lz is large, so why

1684
01:15:37,060 --> 01:15:38,220
is it along the vertical?

1685
01:15:38,220 --> 01:15:39,300
Why is that true?

1686
01:15:39,300 --> 01:15:41,990
So here, this is the
Lz equals 0 state.

1687
01:15:41,990 --> 01:15:46,550
That means it carries no angular
momentum along the z-axis.

1688
01:15:46,550 --> 01:15:50,740
That means it's not rotating
far out in the xy plane.

1689
01:15:50,740 --> 01:15:52,950
So your probability of
finding it in the xy plane

1690
01:15:52,950 --> 01:15:53,550
is very small.

1691
01:15:53,550 --> 01:15:55,450
Because if it was
rotating in the xy plane

1692
01:15:55,450 --> 01:15:57,367
it would carry a large
angular momentum in Lz.

1693
01:15:57,367 --> 01:15:58,110
But Lz is 0.

1694
01:15:58,110 --> 01:16:00,560
So it can't be extended
out in the xy plane.

1695
01:16:00,560 --> 01:16:01,580
Cool?

1696
01:16:01,580 --> 01:16:03,656
On the other hand, it can
carry angular momentum

1697
01:16:03,656 --> 01:16:05,063
in the x or the y direction.

1698
01:16:05,063 --> 01:16:06,480
But if it carries
angular momentum

1699
01:16:06,480 --> 01:16:07,990
in the x direction
for example, that

1700
01:16:07,990 --> 01:16:09,920
means the system is
rotating around the z--

1701
01:16:09,920 --> 01:16:10,420
sorry.

1702
01:16:10,420 --> 01:16:12,503
If it carries angular
momentum in the x direction,

1703
01:16:12,503 --> 01:16:15,020
it's rotating in the zy plane.

1704
01:16:15,020 --> 01:16:17,600
So there's some
probability to find it out

1705
01:16:17,600 --> 01:16:20,090
of the plane in y and z.

1706
01:16:20,090 --> 01:16:22,290
But it can't be in the xy plane.

1707
01:16:22,290 --> 01:16:24,690
Hence it's got to be
in the lobes up above.

1708
01:16:24,690 --> 01:16:25,710
That cool?

1709
01:16:25,710 --> 01:16:28,230
So it's very useful to develop
an intuition for this stuff

1710
01:16:28,230 --> 01:16:30,342
if you're going to do
chemistry or crystallography

1711
01:16:30,342 --> 01:16:31,800
or any condensed
manner of physics.

1712
01:16:31,800 --> 01:16:32,717
It's just very useful.

1713
01:16:32,717 --> 01:16:35,330
So I encourage you to play
with these little applets.

1714
01:16:35,330 --> 01:16:38,490
And I'll post this
mathematics package.

1715
01:16:38,490 --> 01:16:41,170
But let's looked at what happens
now if we crank up the angular

1716
01:16:41,170 --> 01:16:41,670
momentum.

1717
01:16:41,670 --> 01:16:43,897
So as we crank up the
l angular momentum,

1718
01:16:43,897 --> 01:16:45,980
now we're getting this
sort of lobe-y thing, which

1719
01:16:45,980 --> 01:16:49,290
looks like some sort of
'50s sci-fi apparatus.

1720
01:16:49,290 --> 01:16:52,460
So what's going on there?

1721
01:16:52,460 --> 01:16:54,360
This is the 2, 0 state.

1722
01:16:54,360 --> 01:16:57,495
And the 2, 0 state has a 3
cos squared theta minus 1.

1723
01:16:57,495 --> 01:16:58,870
But cos squared
theta, that means

1724
01:16:58,870 --> 01:17:02,910
it's got two periods as it
goes from vertical to negative.

1725
01:17:02,910 --> 01:17:04,630
And if you take that
and you square it,

1726
01:17:04,630 --> 01:17:06,670
you get exactly this.

1727
01:17:06,670 --> 01:17:07,540
OK?

1728
01:17:07,540 --> 01:17:10,200
So they're using the cos squared
as a function of the angle

1729
01:17:10,200 --> 01:17:12,080
of declination from vertical.

1730
01:17:12,080 --> 01:17:16,885
And it's m equals 0, so you're
saying no dependence on phi.

1731
01:17:16,885 --> 01:17:18,260
Of course, that's
a little cheap.

1732
01:17:18,260 --> 01:17:19,970
Because the angular
dependence on phi

1733
01:17:19,970 --> 01:17:21,870
is just an overall phase.

1734
01:17:21,870 --> 01:17:24,416
So we're not going to see
it in the absolute value.

1735
01:17:24,416 --> 01:17:25,570
Everyone agree on that?

1736
01:17:25,570 --> 01:17:26,810
We're not going to
see the absolute--

1737
01:17:26,810 --> 01:17:27,090
OK.

1738
01:17:27,090 --> 01:17:28,140
So let's check that.

1739
01:17:28,140 --> 01:17:30,880
Let's take l2 and m2.

1740
01:17:30,880 --> 01:17:34,520
So now when l is 2 and m is
2, we just get this donut.

1741
01:17:34,520 --> 01:17:37,470
So what that's saying is, we've
got some angular momentum.

1742
01:17:37,470 --> 01:17:37,970
l is 2.

1743
01:17:37,970 --> 01:17:40,390
But all the angular momentum,
almost all of it, anyway,

1744
01:17:40,390 --> 01:17:41,932
is in the z direction.

1745
01:17:41,932 --> 01:17:43,390
And is that what
we're seeing here?

1746
01:17:43,390 --> 01:17:44,070
Well, yeah.

1747
01:17:44,070 --> 01:17:46,340
It seems like it's most
likely to find the particle,

1748
01:17:46,340 --> 01:17:49,980
the probability is greatest, out
in this donut around the plane.

1749
01:17:49,980 --> 01:17:53,890
Now, if it were Lz is equal
to l, it would be flat.

1750
01:17:53,890 --> 01:17:55,955
It would be a strictly
0 thickness pancake.

1751
01:17:55,955 --> 01:17:59,750
But we have some uncertainty
in what Lx and Ly are,

1752
01:17:59,750 --> 01:18:01,510
which is why we got
this fattened donut.

1753
01:18:01,510 --> 01:18:02,760
Everyone cool with that?

1754
01:18:02,760 --> 01:18:07,630
And if we crank up l and
we make-- yeah, right?

1755
01:18:07,630 --> 01:18:10,460
So you can see that you've
got some complicated shapes.

1756
01:18:10,460 --> 01:18:12,010
But as we crank up
l and crank up m,

1757
01:18:12,010 --> 01:18:13,690
we just get a thinner
and thinner donut.

1758
01:18:13,690 --> 01:18:16,065
And the fact that the donut's
getting thinner and thinner

1759
01:18:16,065 --> 01:18:18,050
is that l over L squared
that we did earlier.

1760
01:18:18,050 --> 01:18:18,590
Cool?

1761
01:18:18,590 --> 01:18:20,210
It's still a donut,
but it's getting

1762
01:18:20,210 --> 01:18:22,660
relatively thinner
and thinner, by virtue

1763
01:18:22,660 --> 01:18:24,940
of getting wider and wider.

1764
01:18:24,940 --> 01:18:30,060
And a last thing to show you
is let's take a look now at--

1765
01:18:30,060 --> 01:18:32,780
in fact, let's go to
the l equals 0 state.

1766
01:18:32,780 --> 01:18:35,620
Let's take a look
at the real part.

1767
01:18:39,490 --> 01:18:42,240
So now we're looking
at the real part.

1768
01:18:42,240 --> 01:18:45,990
And nothing much
changed for the Y0, 0.

1769
01:18:45,990 --> 01:18:49,620
But for Y2, 0, well,
still not much changed.

1770
01:18:49,620 --> 01:18:52,280
For Y2, 0 let's now--

1771
01:18:52,280 --> 01:18:53,540
this is sort of disheartening.

1772
01:18:53,540 --> 01:18:56,490
Nothing really has changed.

1773
01:18:56,490 --> 01:18:59,410
Why?

1774
01:18:59,410 --> 01:19:00,850
Because it's real, exactly.

1775
01:19:00,850 --> 01:19:04,530
So for Y2, 0, as long as m
is equal to 0, this is real.

1776
01:19:04,530 --> 01:19:06,210
There's no phase.

1777
01:19:06,210 --> 01:19:08,380
The phase information
contains information

1778
01:19:08,380 --> 01:19:10,710
about the Lz eigenvalue.

1779
01:19:10,710 --> 01:19:19,540
So we can correct this by
changing the angular momentum.

1780
01:19:19,540 --> 01:19:22,113
Let's-- oh shoot,
how do I do that?

1781
01:19:22,113 --> 01:19:23,530
I can't turn it
off at the moment.

1782
01:19:23,530 --> 01:19:24,600
OK, whatever.

1783
01:19:24,600 --> 01:19:29,930
So here we have a
large M. And now

1784
01:19:29,930 --> 01:19:31,750
we've got this very
funny lobe-y structure.

1785
01:19:31,750 --> 01:19:34,250
So this is the Y2, 2, which a
minute ago looked rotationally

1786
01:19:34,250 --> 01:19:34,873
symmetric.

1787
01:19:34,873 --> 01:19:36,540
And now it's not
rotationally symmetric.

1788
01:19:36,540 --> 01:19:40,340
It's this lobe-y structure,
where the lobes are-- remember,

1789
01:19:40,340 --> 01:19:41,430
previously we had a donut.

1790
01:19:41,430 --> 01:19:44,580
Now we have these lobes when
we look at the real part.

1791
01:19:44,580 --> 01:19:45,850
How does that make sense?

1792
01:19:45,850 --> 01:19:53,050
Well, we've got an
e to the i 2 phi.

1793
01:19:53,050 --> 01:19:56,420
And if we look at the real
part, that's cosine of 2 pi.

1794
01:19:56,420 --> 01:19:59,095
And so we're getting a cosine
function modulating the donut.

1795
01:19:59,095 --> 01:20:01,470
And if we look at the real
part, let's do the same thing.

1796
01:20:01,470 --> 01:20:03,550
Let's look at the
imaginary part.

1797
01:20:06,420 --> 01:20:10,360
And the imaginary part
of Y2, 0, we get nothing.

1798
01:20:10,360 --> 01:20:13,098
That's good, because
Y2, 0 was real.

1799
01:20:13,098 --> 01:20:14,140
That would have been bad.

1800
01:20:14,140 --> 01:20:16,140
But if we look at the
imaginary part of Y2, 2,

1801
01:20:16,140 --> 01:20:19,380
we get the corresponding
lobes, the other lobes,

1802
01:20:19,380 --> 01:20:22,473
so that cos squared
plus sine squared is 1.

1803
01:20:22,473 --> 01:20:23,140
Play with these.

1804
01:20:23,140 --> 01:20:24,440
Develop some intuition.

1805
01:20:24,440 --> 01:20:26,023
They're going to be
very useful for us

1806
01:20:26,023 --> 01:20:28,640
when we talk about hydrogen
and the structure of solids.

1807
01:20:28,640 --> 01:20:30,640
And I will see you next week.

1808
01:20:30,640 --> 01:20:33,390
[APPLAUSE]