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PROFESSOR: All right,
shall we get started?

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00:00:28,350 --> 00:00:35,030
So, today-- well, before I
get started-started-- so,

10
00:00:35,030 --> 00:00:37,066
let me open up to questions.

11
00:00:37,066 --> 00:00:39,856
Do y'all have questions
from the last lecture,

12
00:00:39,856 --> 00:00:41,480
where we finished
off angular momentum?

13
00:00:45,860 --> 00:00:47,995
Or really anything
up to the last exam?

14
00:00:51,849 --> 00:00:53,340
Yeah?

15
00:00:53,340 --> 00:00:56,340
AUDIENCE: So, what exactly
happens with the half l states?

16
00:00:56,340 --> 00:00:57,310
PROFESSOR: Ha, ha, ha!

17
00:00:57,310 --> 00:00:58,890
What happens with
the half l states?

18
00:00:58,890 --> 00:01:00,124
OK, great question!

19
00:01:00,124 --> 00:01:02,540
So, we're gonna talk about
that in some detail in a couple

20
00:01:02,540 --> 00:01:05,960
of weeks, but let me
give you a quick preview.

21
00:01:05,960 --> 00:01:10,920
So, remember that when we
studied the commutation

22
00:01:10,920 --> 00:01:18,880
relations, Lx, Ly
is i h bar Lz .

23
00:01:18,880 --> 00:01:22,029
Without using the representation
in terms of derivatives,

24
00:01:22,029 --> 00:01:23,820
with respect to a
coordinate, without using

25
00:01:23,820 --> 00:01:27,780
the representations, in terms
of translations and rotations

26
00:01:27,780 --> 00:01:29,250
along the sphere, right?

27
00:01:29,250 --> 00:01:31,330
When we just used the
commutation relations,

28
00:01:31,330 --> 00:01:33,430
and nothing else,
what we found was

29
00:01:33,430 --> 00:01:38,850
that the states corresponding
to these guys, came in a tower,

30
00:01:38,850 --> 00:01:41,480
with either one state--
corresponding to little l

31
00:01:41,480 --> 00:01:43,396
equals 0-- or two
states-- with l

32
00:01:43,396 --> 00:01:47,010
equals 1/2-- or three
states-- with little l equals

33
00:01:47,010 --> 00:01:52,740
1-- or four states-- with l
equals 3/2-- and so on, and so

34
00:01:52,740 --> 00:01:53,880
forth.

35
00:01:53,880 --> 00:01:56,140
And we quickly
deduced that it is

36
00:01:56,140 --> 00:02:01,650
impossible to represent the
half integer states with a wave

37
00:02:01,650 --> 00:02:03,450
function which
represents a probability

38
00:02:03,450 --> 00:02:05,099
distribution on a sphere.

39
00:02:05,099 --> 00:02:06,640
We observed that
that was impossible.

40
00:02:06,640 --> 00:02:09,020
And the reason is, if
you did so, then when

41
00:02:09,020 --> 00:02:10,639
you take that wave
function, if you

42
00:02:10,639 --> 00:02:14,080
rotate by 2pi--
in any direction--

43
00:02:14,080 --> 00:02:16,130
if you rotate by 2pi the
wave function comes back

44
00:02:16,130 --> 00:02:17,710
to minus itself.

45
00:02:17,710 --> 00:02:20,050
But the wave function
has to be equal to itself

46
00:02:20,050 --> 00:02:20,910
at that same point.

47
00:02:20,910 --> 00:02:22,120
The value of the wave
function at some point,

48
00:02:22,120 --> 00:02:23,930
is equal to the wave
function at some point.

49
00:02:23,930 --> 00:02:24,790
That means the value
of the wave function

50
00:02:24,790 --> 00:02:26,560
must be equal to minus itself.

51
00:02:26,560 --> 00:02:28,000
That means it must be zero0.

52
00:02:28,000 --> 00:02:29,550
So, you can't write
a wave function--

53
00:02:29,550 --> 00:02:31,864
which is a probability
distribution on a sphere--

54
00:02:31,864 --> 00:02:34,030
if the wave function has
to be equal to minus itself

55
00:02:34,030 --> 00:02:35,950
at any given point.

56
00:02:35,950 --> 00:02:38,150
So, this is a strange thing.

57
00:02:38,150 --> 00:02:42,270
And we sort of said, well, look,
these are some other beasts.

58
00:02:42,270 --> 00:02:45,190
But the question is,
look, these furnish

59
00:02:45,190 --> 00:02:50,000
perfectly reasonable
towers of states respecting

60
00:02:50,000 --> 00:02:51,750
these commutation relations.

61
00:02:51,750 --> 00:02:53,477
So, are they just wrong?

62
00:02:53,477 --> 00:02:54,560
Are they just meaningless?

63
00:02:54,560 --> 00:02:57,810
And what we're going to
discover is the following--

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00:02:57,810 --> 00:03:00,429
and this is really gonna go
back to the very first lecture,

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00:03:00,429 --> 00:03:01,970
and so, we'll do
this in more detail,

66
00:03:01,970 --> 00:03:03,761
but I'm going to quickly
tell you-- imagine

67
00:03:03,761 --> 00:03:08,750
take a magnet, a
little, tiny bar magnet.

68
00:03:08,750 --> 00:03:11,570
In fact, well, imagine you
take a little bar magnet

69
00:03:11,570 --> 00:03:16,560
with some little
magnetization, and you send it

70
00:03:16,560 --> 00:03:21,760
through a region that has a
gradient for magnetic field.

71
00:03:21,760 --> 00:03:24,080
If there's a gradient-- so
you know that a magnet wants

72
00:03:24,080 --> 00:03:26,190
to anti-align with
the nearby magnet,

73
00:03:26,190 --> 00:03:28,590
north-south wants to
go to south-north.

74
00:03:28,590 --> 00:03:30,490
So, you can't put a
force on the magnet,

75
00:03:30,490 --> 00:03:33,160
but if you have a gradient
of a magnetic field,

76
00:03:33,160 --> 00:03:37,810
then one end a dipole--
one end of your magnet--

77
00:03:37,810 --> 00:03:41,230
can feel a stronger effective
torque then the other guy.

78
00:03:41,230 --> 00:03:42,740
And you can get a net force.

79
00:03:45,794 --> 00:03:46,960
So, you can get a net force.

80
00:03:46,960 --> 00:03:48,626
The important thing
here, is that if you

81
00:03:48,626 --> 00:03:55,550
have a magnetic field which
has a gradient, so that you've

82
00:03:55,550 --> 00:03:58,090
got some large B, here, and
some smaller B, here, then

83
00:03:58,090 --> 00:03:59,110
you can get a force.

84
00:03:59,110 --> 00:04:05,260
And that force is going
to be proportional to how

85
00:04:05,260 --> 00:04:06,350
big your magnet is.

86
00:04:06,350 --> 00:04:07,933
But it's also going
to be proportional

87
00:04:07,933 --> 00:04:10,100
to the magnetic field.

88
00:04:10,100 --> 00:04:15,070
And if the force is proportional
to the strength of your magnet,

89
00:04:15,070 --> 00:04:18,540
then how far-- if you send
this magnet through a region,

90
00:04:18,540 --> 00:04:20,790
it'll get deflected in one
direction or the other--

91
00:04:20,790 --> 00:04:23,340
and how far it gets
deflected is determined

92
00:04:23,340 --> 00:04:25,444
by how big of a magnet
you sent through.

93
00:04:25,444 --> 00:04:27,360
You send in a bigger
magnet, it deflects more.

94
00:04:27,360 --> 00:04:28,810
Everyone cool with that?

95
00:04:28,810 --> 00:04:31,930
OK, here's a funny thing.

96
00:04:31,930 --> 00:04:33,740
So, that's fact one.

97
00:04:33,740 --> 00:04:36,780
Fact two, suppose I
have a system which

98
00:04:36,780 --> 00:04:41,310
is a charged particle
moving in a circular orbit.

99
00:04:41,310 --> 00:04:41,810
OK?

100
00:04:41,810 --> 00:04:45,880
A charged particle moving
in a circular orbit.

101
00:04:45,880 --> 00:04:48,810
Or better yet, well,
better yet, imaging

102
00:04:48,810 --> 00:04:52,080
you have a sphere--
this is a better model--

103
00:04:52,080 --> 00:04:55,080
imagine you have a sphere of
uniform charge distribution.

104
00:04:55,080 --> 00:04:55,760
OK?

105
00:04:55,760 --> 00:04:59,160
A little gelatinous sphere of
uniform charge distribution,

106
00:04:59,160 --> 00:05:02,050
and you make it rotate, OK?

107
00:05:02,050 --> 00:05:06,120
So, that's charged, that's
moving, forming a current.

108
00:05:06,120 --> 00:05:08,110
And that current
generates a magnetic field

109
00:05:08,110 --> 00:05:10,160
along the axis of
rotation, right?

110
00:05:10,160 --> 00:05:11,610
Right hand rule.

111
00:05:11,610 --> 00:05:14,510
So, if you have a charged
sphere, and it's rotating,

112
00:05:14,510 --> 00:05:16,342
you get a magnetic moment.

113
00:05:16,342 --> 00:05:17,800
And how big is the
magnetic moment,

114
00:05:17,800 --> 00:05:19,216
it's proportional
to the rotation,

115
00:05:19,216 --> 00:05:21,860
to the angular momentum, OK?

116
00:05:21,860 --> 00:05:26,100
So, you determine that,
for a charged sphere here

117
00:05:26,100 --> 00:05:28,710
which is rotating
with angular momentum,

118
00:05:28,710 --> 00:05:32,440
let's say, l, has a
magnetic moment which

119
00:05:32,440 --> 00:05:34,190
is proportional to l.

120
00:05:37,118 --> 00:05:39,374
OK?

121
00:05:39,374 --> 00:05:40,540
So, let's put this together.

122
00:05:40,540 --> 00:05:41,970
Imagine we take
a charged sphere,

123
00:05:41,970 --> 00:05:43,928
we send it rotating with
same angular momentum,

124
00:05:43,928 --> 00:05:45,460
we send it through
a field gradient,

125
00:05:45,460 --> 00:05:46,710
a gradient for magnetic field.

126
00:05:46,710 --> 00:05:49,900
What we'll see is we can
measure that angular momentum

127
00:05:49,900 --> 00:05:51,804
by measuring the deflection.

128
00:05:51,804 --> 00:05:53,470
Because the bigger
the angular momentum,

129
00:05:53,470 --> 00:05:54,590
the bigger the magnetic
moment, but the

130
00:05:54,590 --> 00:05:56,980
bigger the magnetic moment,
the bigger the deflection.

131
00:05:56,980 --> 00:05:58,392
Cool?

132
00:05:58,392 --> 00:05:59,850
So, now here's the
cool experiment.

133
00:06:03,450 --> 00:06:06,060
Take an electron.

134
00:06:06,060 --> 00:06:08,095
And electron has some charge.

135
00:06:08,095 --> 00:06:09,470
Is it a little,
point-like thing?

136
00:06:09,470 --> 00:06:10,420
Is it a little sphere?

137
00:06:10,420 --> 00:06:15,120
Is it, you know-- Let's not
ask that question just yet.

138
00:06:15,120 --> 00:06:15,829
It's an electron.

139
00:06:15,829 --> 00:06:18,370
The thing you get by ripping a
negative charge off a hydrogen

140
00:06:18,370 --> 00:06:18,930
atom.

141
00:06:18,930 --> 00:06:20,070
So, take your
electron and send it

142
00:06:20,070 --> 00:06:21,130
through a magnetic
field gradient.

143
00:06:21,130 --> 00:06:22,046
Why would you do this?

144
00:06:22,046 --> 00:06:24,177
Because you want to
measure the angular

145
00:06:24,177 --> 00:06:25,260
momentum of this electron.

146
00:06:25,260 --> 00:06:27,635
You want to see whether the
electron is a little rotating

147
00:06:27,635 --> 00:06:28,620
thing or not.

148
00:06:28,620 --> 00:06:30,970
So, you send it through this
magnetic field gradient,

149
00:06:30,970 --> 00:06:33,030
and if it gets
deflected, you will

150
00:06:33,030 --> 00:06:35,090
have measured the
magnetic moment.

151
00:06:35,090 --> 00:06:36,965
And if you have measured
the magnetic moment,

152
00:06:36,965 --> 00:06:39,090
you'll have measured
the angular momentum.

153
00:06:39,090 --> 00:06:39,830
OK?

154
00:06:39,830 --> 00:06:43,332
Here's the funny thing, if
the electron weren't rotating,

155
00:06:43,332 --> 00:06:45,040
it would just go
straight through, right?

156
00:06:45,040 --> 00:06:46,456
It would have no
angular momentum,

157
00:06:46,456 --> 00:06:48,850
and it would have
no magnetic moment,

158
00:06:48,850 --> 00:06:51,130
and thus it would not reflect.

159
00:06:51,130 --> 00:06:51,660
Yeah?

160
00:06:51,660 --> 00:06:54,020
If it's rotating,
it's gonna deflect.

161
00:06:54,020 --> 00:06:56,266
Here's the experiment we do.

162
00:06:56,266 --> 00:06:57,765
And here's the
experimental results.

163
00:06:57,765 --> 00:07:00,100
The experimental results
are every electron

164
00:07:00,100 --> 00:07:02,765
that gets sent through bends.

165
00:07:02,765 --> 00:07:06,080
And it either bends
up a fixed amount,

166
00:07:06,080 --> 00:07:07,910
or it bends down a fixed amount.

167
00:07:07,910 --> 00:07:09,813
It never bends more,
it never bends less,

168
00:07:09,813 --> 00:07:11,730
and it certainly
never been zero.

169
00:07:11,730 --> 00:07:14,720
In fact, it always makes
two spots on the screen.

170
00:07:17,828 --> 00:07:19,250
OK?

171
00:07:19,250 --> 00:07:20,210
Always makes two spots.

172
00:07:20,210 --> 00:07:22,024
It never hits the middle.

173
00:07:22,024 --> 00:07:23,690
No matter how you
build this experiment,

174
00:07:23,690 --> 00:07:25,970
no matter how you rotate
it, no matter what you do,

175
00:07:25,970 --> 00:07:28,150
it always hits one of two spots.

176
00:07:28,150 --> 00:07:30,800
What that tells you is,
the angular momentum--

177
00:07:30,800 --> 00:07:33,690
rather the magnetic moment--
can only take one of two values.

178
00:07:33,690 --> 00:07:36,400
But the angular momentum is just
some geometric constant times

179
00:07:36,400 --> 00:07:37,290
the angular momentum.

180
00:07:37,290 --> 00:07:38,706
So, the angular
momentum must take

181
00:07:38,706 --> 00:07:42,450
one of two possible values.

182
00:07:42,450 --> 00:07:44,020
Everyone cool with that?

183
00:07:44,020 --> 00:07:47,460
So, from this experiment--
glorified as the Stern Gerlach

184
00:07:47,460 --> 00:07:48,880
Experiment-- from
this experiment,

185
00:07:48,880 --> 00:07:52,770
we discover that the
angular momentum, Lz,

186
00:07:52,770 --> 00:07:53,936
takes one of two values.

187
00:07:53,936 --> 00:07:55,310
L, along whatever
direction we're

188
00:07:55,310 --> 00:08:00,280
measuring-- but let's say in
the z direction-- Lz takes

189
00:08:00,280 --> 00:08:07,310
one of two values, plus
some constant and, you know,

190
00:08:07,310 --> 00:08:11,466
plus h bar upon 2, or
minus h bar upon 2.

191
00:08:11,466 --> 00:08:13,390
And you just do
this measurement.

192
00:08:13,390 --> 00:08:15,440
But what this tells
us is, which state?

193
00:08:15,440 --> 00:08:16,290
Which tower?

194
00:08:16,290 --> 00:08:23,100
Which set of states describe
an electron in this apparatus?

195
00:08:23,100 --> 00:08:25,530
L equals 1/2.

196
00:08:25,530 --> 00:08:27,720
But wait, we started
off by talking

197
00:08:27,720 --> 00:08:30,250
about the rotation
of a charged sphere,

198
00:08:30,250 --> 00:08:32,669
and deducing that the magnetic
moment must be proportional

199
00:08:32,669 --> 00:08:33,429
to the angular momentum.

200
00:08:33,429 --> 00:08:34,880
And what we've
just discovered is

201
00:08:34,880 --> 00:08:38,650
that this angular momentum-- the
only sensible angular momentum,

202
00:08:38,650 --> 00:08:42,100
here-- is the two
state tower, which

203
00:08:42,100 --> 00:08:46,920
can't be represented in terms
of rotations on a sphere.

204
00:08:46,920 --> 00:08:48,395
Yeah?

205
00:08:48,395 --> 00:08:50,020
What we've learned
from this experiment

206
00:08:50,020 --> 00:08:53,290
is that electrons carry a
form of angular momentum,

207
00:08:53,290 --> 00:08:55,480
demonstrably.

208
00:08:55,480 --> 00:08:59,920
Which is one of these angular
momentum 1/2 states, which

209
00:08:59,920 --> 00:09:01,544
never doesn't rotate, right?

210
00:09:01,544 --> 00:09:03,210
It always carries
some angular momentum.

211
00:09:03,210 --> 00:09:04,834
However, it can't be
expressed in terms

212
00:09:04,834 --> 00:09:08,231
of rotation of some
spherical electron.

213
00:09:08,231 --> 00:09:09,730
It has nothing to
do with rotations.

214
00:09:09,730 --> 00:09:12,345
If it did, we'd get
this nonsensical thing

215
00:09:12,345 --> 00:09:14,600
of the wave function
identically vanishes.

216
00:09:14,600 --> 00:09:16,700
So, there's some other
form of angular momentum--

217
00:09:16,700 --> 00:09:21,000
a totally different form of
angular momentum-- at least

218
00:09:21,000 --> 00:09:22,380
for electrons.

219
00:09:22,380 --> 00:09:25,240
Which, again, has the
magnetic moment proportional

220
00:09:25,240 --> 00:09:27,630
to this angular momentum
with some coefficient, which

221
00:09:27,630 --> 00:09:28,213
I'll call mu0.

222
00:09:31,580 --> 00:09:35,010
But I don't want to call
it L, because L we usually

223
00:09:35,010 --> 00:09:36,575
use for rotational
angular momentum.

224
00:09:36,575 --> 00:09:38,700
This is a different form
of angular momentum, which

225
00:09:38,700 --> 00:09:41,160
is purely half integer,
and we call that spin.

226
00:09:44,570 --> 00:09:48,540
And the spin satisfies
exactly the same commutation

227
00:09:48,540 --> 00:09:54,280
relations-- it's a vector-- Sx
with Sy is equal to ih bar Sz.

228
00:09:56,817 --> 00:09:59,150
So, it's like an angular
momentum in every possible way,

229
00:09:59,150 --> 00:10:01,250
except it cannot be represented.

230
00:10:01,250 --> 00:10:05,500
Sz does not have
any representation,

231
00:10:05,500 --> 00:10:08,700
in terms of h bar
upon i [INAUDIBLE].

232
00:10:08,700 --> 00:10:11,810
It is not related to a rotation.

233
00:10:11,810 --> 00:10:14,200
It's an intrinsic form
of angular momentum.

234
00:10:14,200 --> 00:10:16,070
An electron just has it.

235
00:10:16,070 --> 00:10:18,490
So, at this point, you
ask me, look, what do you

236
00:10:18,490 --> 00:10:20,430
mean an electron just has it?

237
00:10:20,430 --> 00:10:21,870
And my answer to
that question is,

238
00:10:21,870 --> 00:10:24,330
if you send an electron through
a Stern Gerlach Apparatus,

239
00:10:24,330 --> 00:10:25,663
it always hits one of two spots.

240
00:10:29,160 --> 00:10:30,630
And that's it, right?

241
00:10:30,630 --> 00:10:32,315
It's an experimental fact.

242
00:10:32,315 --> 00:10:34,440
And this is how we describe
that experimental fact.

243
00:10:34,440 --> 00:10:38,577
And the legacy of these
little L equals 1/2 states,

244
00:10:38,577 --> 00:10:40,660
is that they represent an
internal form of angular

245
00:10:40,660 --> 00:10:42,993
momentum that only exists
quantum mechanically, that you

246
00:10:42,993 --> 00:10:46,382
would have never
noticed classically.

247
00:10:46,382 --> 00:10:47,840
That was a very
long answer to what

248
00:10:47,840 --> 00:10:49,280
was initially a simple question.

249
00:10:49,280 --> 00:10:51,330
But we'll come back and
do this in more detail,

250
00:10:51,330 --> 00:10:52,080
this was just a quick intro.

251
00:10:52,080 --> 00:10:52,590
Yeah?

252
00:10:52,590 --> 00:10:54,158
AUDIENCE: So, for
L equals 3/2, does

253
00:10:54,158 --> 00:10:55,750
that mean that there's
4 values of spins?

254
00:10:55,750 --> 00:10:56,430
PROFESSOR: Yeah,
that means there's

255
00:10:56,430 --> 00:10:57,410
[? 4 ?] values of spins.

256
00:10:57,410 --> 00:10:58,909
And so there are
plenty of particles

257
00:10:58,909 --> 00:11:00,810
in the real world that
have L equals 3/2.

258
00:11:00,810 --> 00:11:03,580
They're not fundamental
particles, as far as we know.

259
00:11:03,580 --> 00:11:06,440
There are particles a nuclear
physics that carry spin 3/2.

260
00:11:06,440 --> 00:11:09,190
There are all sorts of
nuclei that carry spin 3/2,

261
00:11:09,190 --> 00:11:11,600
but we don't know of a
fundamental particle.

262
00:11:11,600 --> 00:11:13,480
If super symmetry
is true, then there

263
00:11:13,480 --> 00:11:15,990
must be a particle
called a gravitino, which

264
00:11:15,990 --> 00:11:21,090
would be fundamental, and would
have spin 3/2, and four states,

265
00:11:21,090 --> 00:11:24,150
but that hasn't
been observed, yet.

266
00:11:26,810 --> 00:11:29,333
Other questions?

267
00:11:29,333 --> 00:11:30,776
AUDIENCE: Was the
[? latter of ?]

268
00:11:30,776 --> 00:11:34,784
seemingly nonsensical states
discovered first, and then

269
00:11:34,784 --> 00:11:38,170
the experiment explain it,
or was it the experiment--

270
00:11:38,170 --> 00:11:39,100
PROFESSOR: Oh, no!

271
00:11:39,100 --> 00:11:39,860
Oh, that's a great question.

272
00:11:39,860 --> 00:11:41,140
We'll come back the
that at end of today.

273
00:11:41,140 --> 00:11:43,210
So today, we're gonna do
hydrogen, among other things.

274
00:11:43,210 --> 00:11:45,290
Although, I've taken so
long talking about this,

275
00:11:45,290 --> 00:11:47,654
we might be a little slow.

276
00:11:47,654 --> 00:11:49,320
We'll talk about that
a little more when

277
00:11:49,320 --> 00:11:52,860
we talk about hydrogen, but
it was observed and deduced

278
00:11:52,860 --> 00:11:56,260
from experiment before
it was understood

279
00:11:56,260 --> 00:11:57,960
that there was such
a physical quantity.

280
00:11:57,960 --> 00:12:01,340
However, the observation that
this commutation relation

281
00:12:01,340 --> 00:12:03,460
led to towers of
states with this

282
00:12:03,460 --> 00:12:05,316
pre-existed as a
mathematical statement.

283
00:12:05,316 --> 00:12:06,940
So, that was a
mathematical observation

284
00:12:06,940 --> 00:12:10,440
from long previously, and it
has a beautiful algebraic story,

285
00:12:10,440 --> 00:12:12,220
and all sort of
nice things, but it

286
00:12:12,220 --> 00:12:14,270
hadn't been connected
to the physics.

287
00:12:14,270 --> 00:12:15,990
And so, the observation
that the electron

288
00:12:15,990 --> 00:12:18,073
must carry some intrinsic
form of angular momentum

289
00:12:18,073 --> 00:12:20,930
with one of two values,
neither of which is 0,

290
00:12:20,930 --> 00:12:25,000
was actually an
experimental observation--

291
00:12:25,000 --> 00:12:28,640
quasi-experimental observation--
long before it was understood

292
00:12:28,640 --> 00:12:30,089
exactly how to
connect this stuff.

293
00:12:30,089 --> 00:12:31,130
AUDIENCE: So it wasn't--?

294
00:12:31,130 --> 00:12:32,920
PROFESSOR: I shouldn't say
long, it was like within months,

295
00:12:32,920 --> 00:12:33,300
but whatever.

296
00:12:33,300 --> 00:12:33,680
Sorry.

297
00:12:33,680 --> 00:12:36,013
AUDIENCE: The intent of the
experiment wasn't to solve--

298
00:12:36,013 --> 00:12:36,860
AUDIENCE: No, no.

299
00:12:36,860 --> 00:12:40,750
The experiment was this--
there are the spectrum-- Well,

300
00:12:40,750 --> 00:12:42,890
I'll tell you what the
experiment was in a minute.

301
00:12:42,890 --> 00:12:44,225
OK, yeah?

302
00:12:44,225 --> 00:12:48,960
AUDIENCE: [INAUDIBLE] Z has
to be plus or minus 1/2.

303
00:12:48,960 --> 00:12:51,390
What fixes the direction
in the Z direction?

304
00:12:51,390 --> 00:12:52,382
PROFESSOR: Excellent.

305
00:12:52,382 --> 00:12:54,030
In this experiment,
the thing that

306
00:12:54,030 --> 00:12:55,787
fixed the fact that
I was probing Lz

307
00:12:55,787 --> 00:12:57,370
is that I made the
magnetic field have

308
00:12:57,370 --> 00:12:59,190
a gradient in the Z direction.

309
00:12:59,190 --> 00:13:01,890
So, what I was sensitive to,
since the force is actually

310
00:13:01,890 --> 00:13:05,750
proportionally to mu dot
B-- or, really, mu dot

311
00:13:05,750 --> 00:13:10,510
the gradient of B,
so, we'll do this

312
00:13:10,510 --> 00:13:13,545
in more detail later-- the
direction of the gradient

313
00:13:13,545 --> 00:13:15,670
selects out which component
of the angular momentum

314
00:13:15,670 --> 00:13:16,190
we're looking at.

315
00:13:16,190 --> 00:13:18,000
So, in this experiment, I'm
measuring the angular momentum

316
00:13:18,000 --> 00:13:19,810
along this axis-- which
for fun, I'll call Z,

317
00:13:19,810 --> 00:13:21,476
I could've called it
X-- what I discover

318
00:13:21,476 --> 00:13:23,140
is the angular momentum
along this axis

319
00:13:23,140 --> 00:13:24,560
must take one of two values.

320
00:13:24,560 --> 00:13:27,640
But, the universe is
rotationally invariant.

321
00:13:27,640 --> 00:13:29,300
So, it can't possibly
matter whether I

322
00:13:29,300 --> 00:13:31,240
had done the experiment
in this direction,

323
00:13:31,240 --> 00:13:33,614
or done the experiment in this
direction, what that tells

324
00:13:33,614 --> 00:13:35,710
you is, in any direction
if I measure the angular

325
00:13:35,710 --> 00:13:38,132
momentum of the electron
along that direction,

326
00:13:38,132 --> 00:13:40,131
I will discover that it
takes one of two values.

327
00:13:43,290 --> 00:13:46,800
This is also true of
the L equals 1 states.

328
00:13:46,800 --> 00:13:48,510
Lz takes one of three values.

329
00:13:48,510 --> 00:13:49,360
What about Lx?

330
00:13:49,360 --> 00:13:54,040
Lx also takes one of three
values, those three values.

331
00:13:54,040 --> 00:13:56,936
Is is every system in a
state corresponding to one

332
00:13:56,936 --> 00:13:58,060
of those particular values?

333
00:13:58,060 --> 00:14:00,000
No, it could be in
a superposition.

334
00:14:00,000 --> 00:14:03,540
But the eigenvalues, are
these three eigenvalues,

335
00:14:03,540 --> 00:14:07,460
regardless of whether
it's Lx, or Ly, or Lz.

336
00:14:07,460 --> 00:14:09,111
OK, it's a good thing
to meditate upon.

337
00:14:12,097 --> 00:14:12,680
Anything else?

338
00:14:12,680 --> 00:14:13,180
One more.

339
00:14:13,180 --> 00:14:15,061
Yeah?

340
00:14:15,061 --> 00:14:19,840
AUDIENCE: [INAUDIBLE] the
last problem [INAUDIBLE].

341
00:14:19,840 --> 00:14:20,740
PROFESSOR: Indeed.

342
00:14:20,740 --> 00:14:21,090
Indeed.

343
00:14:21,090 --> 00:14:21,590
OK.

344
00:14:21,590 --> 00:14:24,775
Since some people haven't taken
the-- there will be a conflict

345
00:14:24,775 --> 00:14:30,108
exam later today, so I'm not
going to discuss the exam yet.

346
00:14:30,108 --> 00:14:34,440
But, very good observation,
and not an accident.

347
00:14:34,440 --> 00:14:37,590
OK, so, today we launch into 3D.

348
00:14:37,590 --> 00:14:41,260
We ditch our
tricked-out tricycle,

349
00:14:41,260 --> 00:14:43,740
and we're gonna talk about
real, physical systems in three

350
00:14:43,740 --> 00:14:44,470
dimensions.

351
00:14:44,470 --> 00:14:47,420
And as we'll discover, it's
basically the same as in one

352
00:14:47,420 --> 00:14:50,120
dimension, we just have to
write down more symbols.

353
00:14:50,120 --> 00:14:52,070
But the content is all the same.

354
00:14:52,070 --> 00:14:53,790
So, this will make
obvious the reason

355
00:14:53,790 --> 00:14:56,430
we worked with 1D
up until now, which

356
00:14:56,430 --> 00:14:57,990
is that there's
not a heck of a lot

357
00:14:57,990 --> 00:15:01,230
more to be gained for
the basic principles,

358
00:15:01,230 --> 00:15:03,650
but it's a lot more knowing
to write down the expressions.

359
00:15:03,650 --> 00:15:04,570
So, the first thing
I wanted to do

360
00:15:04,570 --> 00:15:06,400
is write down the
Laplacian in three

361
00:15:06,400 --> 00:15:12,310
dimensions in
spherical coordinates--

362
00:15:12,310 --> 00:15:15,930
And that is a beautiful abuse
of notation-- in spherical

363
00:15:15,930 --> 00:15:17,890
coordinates.

364
00:15:17,890 --> 00:15:19,590
And I want to note
a couple of things.

365
00:15:19,590 --> 00:15:21,420
So, first off, this
Laplacian, this

366
00:15:21,420 --> 00:15:27,650
can be written in the following
form, 1 over r dr r quantity

367
00:15:27,650 --> 00:15:30,164
squared.

368
00:15:30,164 --> 00:15:32,330
OK, that's going to be very
useful for us-- trust me

369
00:15:32,330 --> 00:15:39,710
on this one-- this is also
known as 1 over r dr squared r.

370
00:15:39,710 --> 00:15:43,090
And this, if you look
back at your notes,

371
00:15:43,090 --> 00:15:46,280
this is nothing other
than L squared--

372
00:15:46,280 --> 00:15:48,920
except for the factor of h bar
upon i-- but if it's squared,

373
00:15:48,920 --> 00:15:51,210
it's minus 1 upon h bar squared.

374
00:15:54,500 --> 00:15:57,480
OK, so this horrible
angular derivative,

375
00:15:57,480 --> 00:15:59,030
is nothing but L squared.

376
00:16:02,606 --> 00:16:05,340
OK, and you should remember
the [? dd ?] thetas,

377
00:16:05,340 --> 00:16:07,474
and there are these
funny sines and cosines.

378
00:16:07,474 --> 00:16:09,140
But just go back and
compare your notes.

379
00:16:09,140 --> 00:16:13,220
So, this is an observation
that the Laplacian

380
00:16:13,220 --> 00:16:15,200
in three dimensions and
spherical coordinates

381
00:16:15,200 --> 00:16:16,930
takes this simple form.

382
00:16:16,930 --> 00:16:18,780
A simple radial
derivative, which

383
00:16:18,780 --> 00:16:22,050
is two terms if you write it
out linearly in this fashion,

384
00:16:22,050 --> 00:16:24,560
and one term if you
write it this way,

385
00:16:24,560 --> 00:16:26,550
which is going to turn
out to be useful for us.

386
00:16:26,550 --> 00:16:29,180
And the angular part
can be written as 1

387
00:16:29,180 --> 00:16:34,190
over r squared, times the
angular momentum squared

388
00:16:34,190 --> 00:16:37,666
with a minus 1
over h bar squared.

389
00:16:37,666 --> 00:16:38,165
OK?

390
00:16:41,290 --> 00:16:43,830
So, in just to check,
remember that Lz

391
00:16:43,830 --> 00:16:49,330
is equal to h bar upon i d phi.

392
00:16:49,330 --> 00:16:53,610
So, Lz squared is going to be
equal to minus h bar squared

393
00:16:53,610 --> 00:16:54,622
d phi squared.

394
00:16:54,622 --> 00:16:57,080
And you can see that that's
one contribution to this beast.

395
00:17:01,570 --> 00:17:04,177
But, actually, let me-- I'm
gonna commit a capital sin

396
00:17:04,177 --> 00:17:06,010
and erase what I just
wrote, because I don't

397
00:17:06,010 --> 00:17:10,280
want it to distract you-- OK.

398
00:17:10,280 --> 00:17:14,040
So, with that
useful observation,

399
00:17:14,040 --> 00:17:16,450
I want to think about
central potentials.

400
00:17:16,450 --> 00:17:19,210
I want to think about systems
in 3D, which are spherically

401
00:17:19,210 --> 00:17:21,750
symmetric, because this is
going to be a particularly

402
00:17:21,750 --> 00:17:23,540
simple class of
systems, and it's also

403
00:17:23,540 --> 00:17:24,804
particularly physical.

404
00:17:24,804 --> 00:17:26,470
Simple things like a
harmonic oscillator

405
00:17:26,470 --> 00:17:29,011
In three dimensions, which we
solved in Cartesian coordinates

406
00:17:29,011 --> 00:17:30,480
earlier, we're
gonna solve later,

407
00:17:30,480 --> 00:17:31,790
in spherical coordinates.

408
00:17:31,790 --> 00:17:35,240
Things like the isotropic
harmonic oscillator, things

409
00:17:35,240 --> 00:17:39,840
like hydrogen, where the system
is rotationally independent,

410
00:17:39,840 --> 00:17:44,100
the force of the potential only
depends on the radial distance,

411
00:17:44,100 --> 00:17:45,770
all share a bunch of
common properties,

412
00:17:45,770 --> 00:17:46,950
and I want to explore those.

413
00:17:46,950 --> 00:17:49,960
And along the way, we'll solve
a toy model for hydrogen.

414
00:17:49,960 --> 00:17:54,400
So, the energy for this
is p squared upon 2m,

415
00:17:54,400 --> 00:17:56,400
plus a potential, which
is a function only

416
00:17:56,400 --> 00:17:59,500
of the radial distance.

417
00:17:59,500 --> 00:18:08,426
But now, p squared is equal
to minus h bar squared

418
00:18:08,426 --> 00:18:09,550
times the gradient squared.

419
00:18:12,330 --> 00:18:18,120
But this is gonna be equal to,
from the first term, minus h

420
00:18:18,120 --> 00:18:29,860
bar squared-- let me just
write this out-- times

421
00:18:29,860 --> 00:18:33,550
r dr squared r.

422
00:18:33,550 --> 00:18:36,589
And then from this term, plus
minus h bar squared times minus

423
00:18:36,589 --> 00:18:38,755
1 over h bar squared
[? to L squared ?] [? over ?] r

424
00:18:38,755 --> 00:18:42,595
squared, plus L
squared over r squared.

425
00:18:48,380 --> 00:18:51,080
So, the energy can be
written in a nice form.

426
00:18:51,080 --> 00:18:55,110
This is minus h bar
squared, 1 upon r dr

427
00:18:55,110 --> 00:18:59,435
squared r-- whoops,
sorry-- upon 2m,

428
00:18:59,435 --> 00:19:01,050
because it's p squared upon 2m.

429
00:19:01,050 --> 00:19:04,480
And from the second term, L
squared over r squared upon 2m

430
00:19:04,480 --> 00:19:13,935
plus 1 over 2mr squared
L squared plus u of r.

431
00:19:13,935 --> 00:19:18,400
OK, and this is the energy
operator when the system is

432
00:19:18,400 --> 00:19:21,456
rotational invariant in
spherical coordinates.

433
00:19:21,456 --> 00:19:21,955
Questions?

434
00:19:26,254 --> 00:19:26,754
Yeah?

435
00:19:26,754 --> 00:19:29,994
AUDIENCE: [INAUDIBLE] is
that an equals sign or minus?

436
00:19:29,994 --> 00:19:30,660
PROFESSOR: This?

437
00:19:30,660 --> 00:19:31,350
AUDIENCE: Yeah.

438
00:19:31,350 --> 00:19:32,350
PROFESSOR: Oh, that's
an equals sign.

439
00:19:32,350 --> 00:19:32,880
So, sorry.

440
00:19:32,880 --> 00:19:34,010
This is just quick algebra.

441
00:19:34,010 --> 00:19:35,440
So, it's useful to know it.

442
00:19:35,440 --> 00:19:39,294
So, consider the following
thing, 1 over r dr r.

443
00:19:39,294 --> 00:19:41,085
Why would you ever care
about such a thing?

444
00:19:41,085 --> 00:19:43,260
Well, let's square it.

445
00:19:43,260 --> 00:19:44,557
OK, because I did there.

446
00:19:44,557 --> 00:19:45,640
So, what is this equal to?

447
00:19:45,640 --> 00:19:49,320
Well, this is 1 over r dr r.

448
00:19:49,320 --> 00:19:52,180
1 over r dr r.

449
00:19:52,180 --> 00:19:53,580
These guys cancel, right?

450
00:19:53,580 --> 00:19:59,560
1 over r times dr. So, this is
equal to 1 over r dr squared r.

451
00:19:59,560 --> 00:20:04,459
But, why is this equal to dr
squared plus 2 over r times dr?

452
00:20:04,459 --> 00:20:06,000
And the answer is,
they're operators.

453
00:20:06,000 --> 00:20:08,470
And so, you should ask
how they act on functions.

454
00:20:08,470 --> 00:20:10,220
So, let's ask how
they act on function.

455
00:20:10,220 --> 00:20:15,510
So, dr squared plus 2 over
r dr times a function--

456
00:20:15,510 --> 00:20:18,540
acting as a function-- is
equal to f prime prime--

457
00:20:18,540 --> 00:20:24,400
if this is a function of
r-- plus 2 over r f prime.

458
00:20:24,400 --> 00:20:31,960
On the other hand, 1 over r dr
squared r, acting on f of r,

459
00:20:31,960 --> 00:20:36,096
well, these derivatives can
hit either the r of the f.

460
00:20:36,096 --> 00:20:38,990
So, there's going to be a term
where both derivatives hit

461
00:20:38,990 --> 00:20:42,270
f, in which case the rs cancel,
and I get f prime prime.

462
00:20:42,270 --> 00:20:44,550
There's gonna be two terms
where one of the d's hits

463
00:20:44,550 --> 00:20:47,210
this, one of the d's hits this,
then there's the other term.

464
00:20:47,210 --> 00:20:48,770
So, there're two
terms of that form.

465
00:20:48,770 --> 00:20:51,530
On d hits the r and gives
me one, one d hits the f

466
00:20:51,530 --> 00:20:52,640
and gives me f prime.

467
00:20:52,640 --> 00:20:58,150
And then there's an overall 1
over r plus 2 over r f prime.

468
00:20:58,150 --> 00:21:00,520
And then there's a term
were two d's hit the r,

469
00:21:00,520 --> 00:21:03,720
but if two d's hit
the r, that's 0.

470
00:21:03,720 --> 00:21:04,430
So, that's it.

471
00:21:04,430 --> 00:21:06,650
So, these guys are
equal to each other.

472
00:21:06,650 --> 00:21:09,230
So, why is this a
particularly useful form?

473
00:21:09,230 --> 00:21:10,690
We'll see that in just a minute.

474
00:21:10,690 --> 00:21:13,090
So, I'm cheating a little
bit by just writing this

475
00:21:13,090 --> 00:21:14,680
out and saying, this is
going to be a useful form.

476
00:21:14,680 --> 00:21:16,060
But trust me, it's going
to be a useful form.

477
00:21:16,060 --> 00:21:16,560
Yeah?

478
00:21:16,560 --> 00:21:21,272
AUDIENCE: Do we need to find d
squared [INAUDIBLE] dr squared

479
00:21:21,272 --> 00:21:21,772
r.

480
00:21:21,772 --> 00:21:23,210
Isn't that supposed
to be 1 over r?

481
00:21:23,210 --> 00:21:24,043
PROFESSOR: Oh shoot!

482
00:21:24,043 --> 00:21:26,291
Yes, that's supposed to
be one of our-- Thank you.

483
00:21:26,291 --> 00:21:26,790
Thank you!

484
00:21:26,790 --> 00:21:28,490
Yes, over r.

485
00:21:28,490 --> 00:21:29,340
Thank you.

486
00:21:29,340 --> 00:21:31,300
Yes, thank you for
that typo correction.

487
00:21:31,300 --> 00:21:31,800
Excellent.

488
00:21:34,970 --> 00:21:36,870
Thanks OK.

489
00:21:45,204 --> 00:21:46,620
So, anytime we
have a system which

490
00:21:46,620 --> 00:21:48,780
is rotationally invariant--
whose potential is rotationally

491
00:21:48,780 --> 00:21:50,238
invariant-- we can
write the energy

492
00:21:50,238 --> 00:21:52,290
operator in this fashion.

493
00:21:52,290 --> 00:21:54,160
And now, you see
something really lovely,

494
00:21:54,160 --> 00:21:56,640
which is that this
only depends on r,

495
00:21:56,640 --> 00:21:58,400
this only depends
on r, this depends

496
00:21:58,400 --> 00:21:59,900
on the angular
coordinates, but only

497
00:21:59,900 --> 00:22:03,610
insofar as it
depends on L squared.

498
00:22:03,610 --> 00:22:07,190
So, if we want to find
the eigenfunctions of E,

499
00:22:07,190 --> 00:22:09,000
our life is going to
be a lot easier if we

500
00:22:09,000 --> 00:22:14,611
work in eigenfunctions
of L. Because that's

501
00:22:14,611 --> 00:22:16,860
gonna make this one [? Ex ?]
on an eigenfunction of L,

502
00:22:16,860 --> 00:22:18,700
this is just going
to become a constant.

503
00:22:18,700 --> 00:22:20,949
So, now you have to answer
the question, well, can we?

504
00:22:20,949 --> 00:22:24,450
Can we find functions which are
eigenfunctions of E and of L,

505
00:22:24,450 --> 00:22:25,560
simultaneously?

506
00:22:25,560 --> 00:22:27,040
And so, the answer
to that question

507
00:22:27,040 --> 00:22:28,940
is, well, compute
the commutator.

508
00:22:28,940 --> 00:22:30,550
So, do these guys commute?

509
00:22:30,550 --> 00:22:33,305
In particular, of L squared.

510
00:22:33,305 --> 00:22:35,180
And, well, does L commute
with the derivative

511
00:22:35,180 --> 00:22:36,430
with respect to r, L squared?

512
00:22:41,330 --> 00:22:43,770
Yeah, because L only depends
on angular derivatives.

513
00:22:43,770 --> 00:22:45,440
It doesn't have any rs in it.

514
00:22:45,440 --> 00:22:49,860
And the rs don't care about
the angular variables,

515
00:22:49,860 --> 00:22:51,240
so they commute.

516
00:22:51,240 --> 00:22:52,704
What about with this term?

517
00:22:52,704 --> 00:22:54,620
Well, L squared trivially
commutes with itself

518
00:22:54,620 --> 00:22:56,070
and, again, r doesn't matter.

519
00:22:56,070 --> 00:22:58,490
And ditto, r and
L squared commute.

520
00:22:58,490 --> 00:22:59,297
So, this is 0.

521
00:22:59,297 --> 00:22:59,880
These commute.

522
00:22:59,880 --> 00:23:04,430
So, we can find
common eigenbasis.

523
00:23:04,430 --> 00:23:11,894
We can find a basis
of functions which

524
00:23:11,894 --> 00:23:13,810
are eigenfunctions both
of E and of L squared.

525
00:23:17,117 --> 00:23:18,200
So, now we use separation.

526
00:23:20,960 --> 00:23:23,220
In particular, if we
want to find a function--

527
00:23:23,220 --> 00:23:24,970
an eigenfunction-- of
the energy operator,

528
00:23:24,970 --> 00:23:33,040
E phi E is equal
to E phi E, it's

529
00:23:33,040 --> 00:23:36,280
going to simplify our
lives if we also let phi

530
00:23:36,280 --> 00:23:39,191
be an eigenfunction
of the L squared.

531
00:23:39,191 --> 00:23:41,190
But we know what the
eigenfunctions of L squared

532
00:23:41,190 --> 00:23:41,690
are.

533
00:23:41,690 --> 00:23:45,060
E phi E is equal to--
let me write this--

534
00:23:45,060 --> 00:23:54,440
of r will then be equal
to little phi of r

535
00:23:54,440 --> 00:23:57,890
times yLm of theta and phi.

536
00:24:02,030 --> 00:24:06,870
Now, quickly, because these
are the eigenfunctions of the L

537
00:24:06,870 --> 00:24:08,150
squared operator.

538
00:24:08,150 --> 00:24:11,812
Quick, is little l an
integer or a half integer?

539
00:24:11,812 --> 00:24:13,050
AUDIENCE: [MURMURS] Integer.

540
00:24:13,050 --> 00:24:13,940
PROFESSOR: Why?

541
00:24:13,940 --> 00:24:14,732
AUDIENCE: [MURMURS]

542
00:24:14,732 --> 00:24:17,315
PROFESSOR: Yeah, because we're
working with rotational angular

543
00:24:17,315 --> 00:24:18,380
momentum, right?

544
00:24:18,380 --> 00:24:21,180
And it only makes sense to talk
about integer values of little

545
00:24:21,180 --> 00:24:24,170
l when we have gradients on
a sphere-- when we're talking

546
00:24:24,170 --> 00:24:27,044
about rotations-- on a
spherical coordinates, OK?

547
00:24:27,044 --> 00:24:28,460
So, little l has
to be an integer.

548
00:24:31,980 --> 00:24:35,960
And from this point forward in
the class, any time I write l,

549
00:24:35,960 --> 00:24:39,960
I'll be talking about the
rotational angular momentum

550
00:24:39,960 --> 00:24:42,100
corresponding to integer values.

551
00:24:42,100 --> 00:24:44,330
And when I'm talking about
the half integer values,

552
00:24:44,330 --> 00:24:49,210
I'll write down s, OK?

553
00:24:49,210 --> 00:24:53,430
So, let's use this
separation of variables.

554
00:24:53,430 --> 00:24:55,070
And what does that give us?

555
00:24:55,070 --> 00:24:58,650
Well, l squared
acting on yLm gives us

556
00:24:58,650 --> 00:25:01,390
h bar squared lL plus 1.

557
00:25:01,390 --> 00:25:05,606
So, this tells us that
E, acting on phi E,

558
00:25:05,606 --> 00:25:06,980
takes a particularly
simple form.

559
00:25:06,980 --> 00:25:12,920
If phi E is proportional
to a spherical harmonic,

560
00:25:12,920 --> 00:25:19,860
then this is gonna take the form
minus h bar squared upon 2m 1

561
00:25:19,860 --> 00:25:28,190
over r dr squared r plus
1 over 2mr squared l

562
00:25:28,190 --> 00:25:30,550
squared-- but l squared
acting on the yLm

563
00:25:30,550 --> 00:25:35,840
gives us-- h bar
squared lL plus 1, which

564
00:25:35,840 --> 00:25:53,620
is just a constant over r
squared plus u of r phi E.

565
00:25:53,620 --> 00:25:54,480
Question?

566
00:25:54,480 --> 00:26:00,724
AUDIENCE: Yeah.
[INAUDIBLE] yLm1 and yLm2?

567
00:26:00,724 --> 00:26:01,640
PROFESSOR: Absolutely.

568
00:26:01,640 --> 00:26:03,850
So, can we consider
superpositions of these guys?

569
00:26:03,850 --> 00:26:05,090
Absolutely, we can.

570
00:26:05,090 --> 00:26:07,660
However, we're using separation.

571
00:26:07,660 --> 00:26:09,320
So, we're gonna look
at a single term,

572
00:26:09,320 --> 00:26:12,270
and then after
constructing solutions

573
00:26:12,270 --> 00:26:16,020
with a single
eigenfunction of L squared,

574
00:26:16,020 --> 00:26:19,340
we can then write down
arbitrary superposition of them,

575
00:26:19,340 --> 00:26:21,730
and generate a complete
basis of states.

576
00:26:21,730 --> 00:26:25,600
General statement about
separation of variables.

577
00:26:25,600 --> 00:26:27,850
Other questions?

578
00:26:27,850 --> 00:26:29,160
OK.

579
00:26:29,160 --> 00:26:34,810
So, here's the resulting
energy eigenvalue equation.

580
00:26:34,810 --> 00:26:36,419
But notice that it's
now, really nice.

581
00:26:36,419 --> 00:26:37,710
This is purely a function of r.

582
00:26:37,710 --> 00:26:40,560
We've removed all of
the angular dependence

583
00:26:40,560 --> 00:26:42,390
by making this
proportional to yLm.

584
00:26:42,390 --> 00:26:46,127
So, this has a little phi yLm,
and this has a little phi yLm,

585
00:26:46,127 --> 00:26:47,710
and nothing depends
on the little phi.

586
00:26:50,302 --> 00:26:52,593
Nothing depends on the yLm--
on the angular variables--

587
00:26:52,593 --> 00:26:53,702
I can make this phi of r.

588
00:27:00,540 --> 00:27:03,150
And if I want to make this the
energy eigenvalue equation,

589
00:27:03,150 --> 00:27:05,760
instead of just the action
of the energy operator,

590
00:27:05,760 --> 00:27:08,440
that is now my energy
eigenvalue equation.

591
00:27:08,440 --> 00:27:13,667
This is the result of acting on
phi with the energy operator,

592
00:27:13,667 --> 00:27:15,083
and this is the
energy eigenvalue.

593
00:27:19,840 --> 00:27:21,460
Cool?

594
00:27:21,460 --> 00:27:25,070
So, the upside here is that when
we have a central potential,

595
00:27:25,070 --> 00:27:27,320
when the system is
rotationally invariant,

596
00:27:27,320 --> 00:27:30,840
the potential energy is
invariant under rotations,

597
00:27:30,840 --> 00:27:33,880
then the energy commutes with
the angular momentum squared.

598
00:27:33,880 --> 00:27:36,440
And so, we can find
common eigenfunctions.

599
00:27:36,440 --> 00:27:39,694
When we use separation
of variable,

600
00:27:39,694 --> 00:27:41,360
the resulting energy
eigenvalue equation

601
00:27:41,360 --> 00:27:47,115
becomes nothing but a 1D energy
eigenvalue equation, right?

602
00:27:47,115 --> 00:27:48,240
This is just a 1D equation.

603
00:27:48,240 --> 00:27:49,750
Now, you might look at this
and say, well, it's not quite

604
00:27:49,750 --> 00:27:51,960
a 1D equation, because if
this were a 1D equation,

605
00:27:51,960 --> 00:27:54,620
we wouldn't have this funny
1 over r, and this funny r,

606
00:27:54,620 --> 00:27:55,120
right?

607
00:27:55,120 --> 00:27:57,650
It's not exactly what
we would have got.

608
00:27:57,650 --> 00:28:01,310
It's got the minus h bar
squareds upon 2m-- whoops,

609
00:28:01,310 --> 00:28:05,820
and there's, yeah, OK-- it's got
this funny h bar squareds upon

610
00:28:05,820 --> 00:28:08,010
2m, and it's got these
1 over-- or sorry,--

611
00:28:08,010 --> 00:28:09,900
it's got the correct h
bar squareds upon 2m,

612
00:28:09,900 --> 00:28:11,390
but it's got this
funny r and 1 over r.

613
00:28:11,390 --> 00:28:12,473
So, let's get rid of that.

614
00:28:12,473 --> 00:28:15,330
Let's just quickly dispense
with that funny set of r.

615
00:28:15,330 --> 00:28:17,490
And this comes back
to the sneaky trick

616
00:28:17,490 --> 00:28:21,930
I was referring to earlier,
of writing this expression.

617
00:28:21,930 --> 00:28:23,340
So, rather than
writing this out,

618
00:28:23,340 --> 00:28:25,048
it's convenient to
write it in this form.

619
00:28:25,048 --> 00:28:25,700
Let's see why.

620
00:28:28,290 --> 00:28:35,180
So, if we have the E phi of r
is equal to minus h bar squared

621
00:28:35,180 --> 00:28:43,580
upon 2m, 1 over r d squared
r r, plus-- and now,

622
00:28:43,580 --> 00:28:47,210
what I'm gonna write is-- look,
this is our potential, u of r.

623
00:28:47,210 --> 00:28:50,720
This is some silly,
radial-dependent thing.

624
00:28:50,720 --> 00:28:52,730
I'm gonna write these
two terms together,

625
00:28:52,730 --> 00:28:54,360
rather than writing them over,
and over, and over again, I'm

626
00:28:54,360 --> 00:28:56,830
going to write them together,
and call them V effective.

627
00:28:56,830 --> 00:29:08,040
Plus V effective of r, where V
effective is just these guys,

628
00:29:08,040 --> 00:29:08,540
V effective.

629
00:29:11,260 --> 00:29:13,500
Which has a contribution
from the original potential,

630
00:29:13,500 --> 00:29:15,560
and from the angular
momentum, which,

631
00:29:15,560 --> 00:29:17,950
notice the sign is
plus 1 over r squared.

632
00:29:17,950 --> 00:29:20,820
So, the potential gets really
large as you get to the origin.

633
00:29:24,480 --> 00:29:25,100
Phi of r.

634
00:29:28,200 --> 00:29:30,450
So, this r is annoying, and
this 1 over r is annoying,

635
00:29:30,450 --> 00:29:32,440
but there's a nice
way to get rid of it.

636
00:29:32,440 --> 00:29:35,810
Let phi of r-- well, this r,
we want to get rid of-- so,

637
00:29:35,810 --> 00:29:42,190
let phi of r equals
1 over r u of r.

638
00:29:42,190 --> 00:29:46,600
OK, then 1 over r squared--
or sorry, 1 over r-- dr

639
00:29:46,600 --> 00:29:53,020
squared r phi is equal to 1
over r dr squared r times 1

640
00:29:53,020 --> 00:29:56,030
over r times u, which is just u.

641
00:29:56,030 --> 00:29:59,636
But meanwhile, V on
phi is equal to-- well,

642
00:29:59,636 --> 00:30:01,010
V doesn't have
any r derivatives,

643
00:30:01,010 --> 00:30:06,030
it's just a function-- so, V
of phi is just 1 over r V on u.

644
00:30:06,030 --> 00:30:10,000
So, this equation
becomes E on u,

645
00:30:10,000 --> 00:30:11,920
because this also
picks up a 1 over r,

646
00:30:11,920 --> 00:30:17,950
is equal to minus h bar
squared upon 2m dr squared

647
00:30:17,950 --> 00:30:26,440
plus V effective of r u of r.

648
00:30:26,440 --> 00:30:30,890
And this is exactly the
energy eigenvalue equation

649
00:30:30,890 --> 00:30:34,750
for a 1D problem with
the following potential.

650
00:30:34,750 --> 00:30:41,380
The potential, V effective of r,
does the following two things--

651
00:30:41,380 --> 00:30:48,650
whoops, don't want to
draw it that way-- suppose

652
00:30:48,650 --> 00:30:52,200
we have a potential which
is the Coulomb potential.

653
00:30:52,200 --> 00:30:56,290
So, let's say, u is equal
to minus E squared upon r.

654
00:30:58,899 --> 00:30:59,690
Just as an example.

655
00:31:04,440 --> 00:31:06,850
So, here's r, here
is V effective.

656
00:31:06,850 --> 00:31:13,590
So, u first, so
there's u-- u of r,

657
00:31:13,590 --> 00:31:19,440
so let me draw this-- V
has another term, which

658
00:31:19,440 --> 00:31:25,990
is h bar squared lL
plus 1 over 2mr squared.

659
00:31:25,990 --> 00:31:27,470
This is for any
given value of l.

660
00:31:27,470 --> 00:31:30,700
This is a constant over r
squared, with a plus sign.

661
00:31:30,700 --> 00:31:32,724
So, that's something
that looks like this.

662
00:31:32,724 --> 00:31:34,140
This is falling
off like 1 over r,

663
00:31:34,140 --> 00:31:36,560
this is falling off
like 1 over r squared.

664
00:31:36,560 --> 00:31:38,950
So, it falls off more rapidly.

665
00:31:38,950 --> 00:31:41,045
And finally, can r be negative?

666
00:31:44,130 --> 00:31:44,630
No.

667
00:31:44,630 --> 00:31:47,070
It's defined from 0 to infinity.

668
00:31:47,070 --> 00:31:50,690
So, that's like having
an infinite potential

669
00:31:50,690 --> 00:31:52,530
for negative r.

670
00:31:52,530 --> 00:31:54,130
So, our effective
potential is the sum

671
00:31:54,130 --> 00:31:56,092
of these contributions--
wish I had

672
00:31:56,092 --> 00:32:00,996
colored chalk-- the sum
of these contributions

673
00:32:00,996 --> 00:32:02,120
is going to look like this.

674
00:32:02,120 --> 00:32:05,590
So, that's my V effective.

675
00:32:05,590 --> 00:32:13,360
This is my Ll plus 1 [INAUDIBLE]
squared over 2mr squared.

676
00:32:13,360 --> 00:32:16,920
And this is my u of r.

677
00:32:16,920 --> 00:32:17,910
Question?

678
00:32:17,910 --> 00:32:19,304
AUDIENCE: [INAUDIBLE].

679
00:32:19,304 --> 00:32:19,970
PROFESSOR: Good.

680
00:32:19,970 --> 00:32:23,713
OK, so this is u of r,
the original potential.

681
00:32:23,713 --> 00:32:24,254
AUDIENCE: OK.

682
00:32:24,254 --> 00:32:25,110
PROFESSOR: OK?

683
00:32:25,110 --> 00:32:30,030
This is 1 over L squared-- or
sorry-- lL 1 over 2mr squared.

684
00:32:33,250 --> 00:32:34,166
AUDIENCE: [INAUDIBLE].

685
00:32:36,857 --> 00:32:37,690
PROFESSOR: Oh shoot!

686
00:32:37,690 --> 00:32:38,090
Oh, I'm sorry!

687
00:32:38,090 --> 00:32:38,890
I'm terribly sorry!

688
00:32:38,890 --> 00:32:40,490
I've abused the
notation terribly.

689
00:32:40,490 --> 00:32:41,280
Let's-- Oh!

690
00:32:41,280 --> 00:32:44,320
This is-- Crap!

691
00:32:44,320 --> 00:32:46,930
Sorry.

692
00:32:46,930 --> 00:32:48,180
This is standard notation.

693
00:32:48,180 --> 00:32:51,340
And in text, when I
write this by hand,

694
00:32:51,340 --> 00:32:55,170
the potential is a big
U, and the wave function

695
00:32:55,170 --> 00:32:55,865
is a little u.

696
00:32:55,865 --> 00:32:57,455
So, let this be a little u.

697
00:32:57,455 --> 00:32:59,880
OK, this is my little
u and so, now I'm

698
00:32:59,880 --> 00:33:01,630
gonna have to-- oh
jeez, this is horrible,

699
00:33:01,630 --> 00:33:04,940
sorry-- this is the
potential, capital U

700
00:33:04,940 --> 00:33:07,110
with a bar underneath it.

701
00:33:07,110 --> 00:33:08,990
OK, seriously, so
there's capital U

702
00:33:08,990 --> 00:33:10,780
with a bar underneath it.

703
00:33:10,780 --> 00:33:13,230
And here's V, which is
gonna make my life easier,

704
00:33:13,230 --> 00:33:16,239
and this is the capital U
with the bar underneath it.

705
00:33:16,239 --> 00:33:17,780
Capital U with the
bar underneath it.

706
00:33:17,780 --> 00:33:19,450
Oh, I'm really sorry,
I did not realize

707
00:33:19,450 --> 00:33:21,600
how confusing that would be.

708
00:33:21,600 --> 00:33:23,960
OK, is everyone happy with that?

709
00:33:23,960 --> 00:33:24,791
Yeah?

710
00:33:24,791 --> 00:33:26,835
AUDIENCE: [INAUDIBLE].

711
00:33:26,835 --> 00:33:27,710
PROFESSOR: Which one?

712
00:33:27,710 --> 00:33:28,584
AUDIENCE: Middle.

713
00:33:28,584 --> 00:33:29,460
Middle.

714
00:33:29,460 --> 00:33:30,241
PROFESSOR: Middle.

715
00:33:30,241 --> 00:33:31,665
AUDIENCE: Up, up.

716
00:33:31,665 --> 00:33:32,165
Right there!

717
00:33:32,165 --> 00:33:32,665
Up!

718
00:33:32,665 --> 00:33:33,610
There.

719
00:33:33,610 --> 00:33:34,420
PROFESSOR: Where?

720
00:33:34,420 --> 00:33:37,288
AUDIENCE: To the right.
[CHATTER] Near the eraser mark.

721
00:33:37,288 --> 00:33:39,200
[LAUGHTER]

722
00:33:39,200 --> 00:33:41,560
PROFESSOR: So, these
are the wave function.

723
00:33:41,560 --> 00:33:42,840
AUDIENCE: I know.

724
00:33:42,840 --> 00:33:44,670
PROFESSOR: That's
the wave function.

725
00:33:44,670 --> 00:33:48,519
That is V.

726
00:33:48,519 --> 00:33:51,417
AUDIENCE: [CHATTER]

727
00:33:51,417 --> 00:33:54,464
PROFESSOR: Wait, if I
erased, how can I correct it?

728
00:33:54,464 --> 00:34:04,605
AUDIENCE: [CHATTER] There!

729
00:34:04,605 --> 00:34:06,730
PROFESSOR: Excellent, so
the thing that isn't here,

730
00:34:06,730 --> 00:34:08,410
would have a bar under it.

731
00:34:08,410 --> 00:34:10,151
Oh, oh, oh, oh, sorry!

732
00:34:10,151 --> 00:34:13,040
Ah!

733
00:34:13,040 --> 00:34:17,020
You wouldn't think
it would be so hard.

734
00:34:17,020 --> 00:34:17,560
OK, good.

735
00:34:17,560 --> 00:34:19,684
And this is not [? related ?]
to the wave function.

736
00:34:19,684 --> 00:34:20,719
OK, god, oh!

737
00:34:20,719 --> 00:34:23,260
That's horrible!

738
00:34:23,260 --> 00:34:27,090
Sorry guys, that
notation is not obvious.

739
00:34:27,090 --> 00:34:29,449
My apologies.

740
00:34:29,449 --> 00:34:31,580
Oh, there's a better
way to do this.

741
00:34:31,580 --> 00:34:33,800
OK, here's the better
way to do this.

742
00:34:33,800 --> 00:34:35,729
Instead of calling the
potential-- I'm sorry,

743
00:34:35,729 --> 00:34:37,770
your notes are getting
destroyed now-- so instead

744
00:34:37,770 --> 00:34:41,934
of calling potential capital U,
let's just call this V. Yeah.

745
00:34:41,934 --> 00:34:43,429
AUDIENCE: [LAUGHTER] No!

746
00:34:43,429 --> 00:34:45,520
PROFESSOR: And then
we have V effective.

747
00:34:45,520 --> 00:34:46,020
No, no.

748
00:34:46,020 --> 00:34:46,320
This is good.

749
00:34:46,320 --> 00:34:46,650
This is good.

750
00:34:46,650 --> 00:34:47,858
We can be careful about this.

751
00:34:47,858 --> 00:34:49,630
So, this is V. This
is V effective,

752
00:34:49,630 --> 00:34:52,270
which has V plus the
angular momentum term.

753
00:34:52,270 --> 00:34:54,159
Oh, good Lord!

754
00:34:54,159 --> 00:34:55,280
This is V effective.

755
00:34:55,280 --> 00:35:00,096
This is V. V phi [INAUDIBLE]
U. Good, this is V.

756
00:35:00,096 --> 00:35:02,634
AUDIENCE: [INAUDIBLE]
There's no U--

757
00:35:02,634 --> 00:35:04,050
PROFESSOR: There's
no U underline,

758
00:35:04,050 --> 00:35:09,310
it's now just V, V effective.

759
00:35:09,310 --> 00:35:10,220
Oh!

760
00:35:10,220 --> 00:35:11,670
Good Lord!

761
00:35:11,670 --> 00:35:13,160
OK, wow!

762
00:35:13,160 --> 00:35:14,747
That was an
unnecessary confusion.

763
00:35:14,747 --> 00:35:15,580
AUDIENCE: Top right.

764
00:35:15,580 --> 00:35:16,835
PROFESSOR: Top right.

765
00:35:16,835 --> 00:35:19,617
AUDIENCE: There is no bar.

766
00:35:19,617 --> 00:35:20,450
[? PROFESSOR: Mu. ?]

767
00:35:23,042 --> 00:35:24,910
AUDIENCE: Is that
V or V effective?

768
00:35:24,910 --> 00:35:26,846
PROFESSOR: That's V.
Although, it would've

769
00:35:26,846 --> 00:35:28,220
been just as true
as V effective.

770
00:35:28,220 --> 00:35:29,400
So, we can write V effective.

771
00:35:29,400 --> 00:35:30,191
It's true for both.

772
00:35:33,206 --> 00:35:34,980
Because it's just
a function of r.

773
00:35:34,980 --> 00:35:36,380
Oh, for the love of God!

774
00:35:36,380 --> 00:35:38,090
OK.

775
00:35:38,090 --> 00:35:40,440
Let's check our sanity,
and walk through the logic.

776
00:35:40,440 --> 00:35:43,480
So, the logic here is, we
have some potential, it's

777
00:35:43,480 --> 00:35:45,405
a function only of r, yeah?

778
00:35:45,405 --> 00:35:47,780
As a consequence, since it
doesn't care about the angles,

779
00:35:47,780 --> 00:35:50,113
we can write things in terms
of the spherical harmonics,

780
00:35:50,113 --> 00:35:52,037
we can do separation
of variables.

781
00:35:52,037 --> 00:35:53,620
Here's the energy
eigenvalue equation.

782
00:35:53,620 --> 00:35:57,479
We discover that because we're
working in spherical harmonics,

783
00:35:57,479 --> 00:35:59,770
the angular momentum term
becomes just a function of r,

784
00:35:59,770 --> 00:36:02,080
with no other coefficients.

785
00:36:02,080 --> 00:36:04,500
So, now we have a function
of r plus the potential V,

786
00:36:04,500 --> 00:36:07,310
this looks like an effective
potential, V effective,

787
00:36:07,310 --> 00:36:10,020
which is the sum
of these two terms.

788
00:36:10,020 --> 00:36:11,290
So, there's that equation.

789
00:36:11,290 --> 00:36:13,972
On the other hand,
this is tantalizingly

790
00:36:13,972 --> 00:36:16,180
close to but not quite the
energy eigenvalue equation

791
00:36:16,180 --> 00:36:18,870
for a 1D problem with this
potential, V effective.

792
00:36:18,870 --> 00:36:21,510
To make it obvious that
it's, in fact, a 1D problem,

793
00:36:21,510 --> 00:36:25,490
we do a change of variables,
phi goes to 1 over ru,

794
00:36:25,490 --> 00:36:28,450
and then 1 upon r d
squared r phi becomes 1

795
00:36:28,450 --> 00:36:30,720
over r d squared u,
and V effective phi

796
00:36:30,720 --> 00:36:33,620
becomes 1 over r V effective u.

797
00:36:33,620 --> 00:36:36,340
Plugging that together, gives us
this energy eigenvalue equation

798
00:36:36,340 --> 00:36:41,080
for u, the effective wave
function, which is 1d problem.

799
00:36:41,080 --> 00:36:42,660
So, we can use all
of our intuition

800
00:36:42,660 --> 00:36:44,660
and all of our machinery
to solve this problem.

801
00:36:44,660 --> 00:36:46,160
And now we have to
ask, what exactly

802
00:36:46,160 --> 00:36:47,630
is the effective potential?

803
00:36:47,630 --> 00:36:49,800
And the effective potential
has three contributions.

804
00:36:49,800 --> 00:36:54,321
First, it has the
original V, secondly, it

805
00:36:54,321 --> 00:36:55,820
has the angular
momentum term, which

806
00:36:55,820 --> 00:36:58,440
is a constant over r
squared-- and here is

807
00:36:58,440 --> 00:37:02,480
that, constant over r
squared-- and the sum of these

808
00:37:02,480 --> 00:37:03,502
is the effective.

809
00:37:03,502 --> 00:37:05,710
And this guy dominates
because it's 1 over r squared.

810
00:37:05,710 --> 00:37:08,280
This dominates at small r,
and this dominates at large r

811
00:37:08,280 --> 00:37:10,000
if it's 1 over r.

812
00:37:10,000 --> 00:37:12,750
So, we get an
effective potential--

813
00:37:12,750 --> 00:37:17,870
that I'll check-- there's
the effective potential.

814
00:37:21,950 --> 00:37:24,050
And finally, the
third fact is that r

815
00:37:24,050 --> 00:37:26,269
must be strictly positive,
so as a 1D problem,

816
00:37:26,269 --> 00:37:28,060
that means it can't be
negative, it's gotta

817
00:37:28,060 --> 00:37:29,830
have an infinite
potential on the left.

818
00:37:40,010 --> 00:37:44,170
So, as an example, let's go
ahead and think more carefully

819
00:37:44,170 --> 00:37:47,680
about specifically this problem,
about this Coulomb potential,

820
00:37:47,680 --> 00:37:49,050
and this 1D effective potential.

821
00:37:49,050 --> 00:37:49,883
AUDIENCE: Professor?

822
00:37:49,883 --> 00:37:50,736
PROFESSOR: Yeah?

823
00:37:50,736 --> 00:37:51,652
AUDIENCE: [INAUDIBLE]?

824
00:37:51,652 --> 00:37:52,560
PROFESSOR: Yes?

825
00:37:52,560 --> 00:37:54,674
AUDIENCE: Where does
the 1 over r go?

826
00:37:54,674 --> 00:37:55,340
PROFESSOR: Good.

827
00:37:55,340 --> 00:37:59,660
So, remember the ddr
squared term gave us a 1

828
00:37:59,660 --> 00:38:01,196
over r out front.

829
00:38:01,196 --> 00:38:03,320
So, from this term, there
should be 1 over r, here.

830
00:38:03,320 --> 00:38:05,590
From this term, there
should also be a 1 over r.

831
00:38:05,590 --> 00:38:07,589
And from here, there
should be a 1 over r.

832
00:38:07,589 --> 00:38:08,130
AUDIENCE: Ah!

833
00:38:08,130 --> 00:38:10,171
PROFESSOR: So, then I'm
gonna cancel the 1 over r

834
00:38:10,171 --> 00:38:13,100
by multiplying the
whole equation by r.

835
00:38:13,100 --> 00:38:13,940
Yeah?

836
00:38:13,940 --> 00:38:14,900
Sneaky, sneaky.

837
00:38:14,900 --> 00:38:18,450
So, any time you see-- any
time, this is a general lesson--

838
00:38:18,450 --> 00:38:20,780
anytime you see a
differential equation that

839
00:38:20,780 --> 00:38:23,450
has this form-- two
derivatives, plus 1

840
00:38:23,450 --> 00:38:26,000
over r a derivative--
you know you

841
00:38:26,000 --> 00:38:28,060
can play some game like this.

842
00:38:28,060 --> 00:38:31,400
If you see this, declare in your
mind a brief moment of triumph,

843
00:38:31,400 --> 00:38:33,110
because you know what
technique to use.

844
00:38:33,110 --> 00:38:35,520
You can do this sort of
rescaling by a power of r.

845
00:38:35,520 --> 00:38:38,040
And more generally, if you have
a differential equation that

846
00:38:38,040 --> 00:38:39,727
looks like-- let
me do this here--

847
00:38:39,727 --> 00:38:42,060
if you have a differential
equation that looks something

848
00:38:42,060 --> 00:38:46,010
like a derivative with respect
to r plus a constant over r

849
00:38:46,010 --> 00:38:50,160
times phi, you know
how to solve this.

850
00:38:50,160 --> 00:38:55,160
Let me say plus
dot, dot, dot phi.

851
00:38:55,160 --> 00:38:58,790
You know how to solve this
because ddr plus c over r

852
00:38:58,790 --> 00:39:02,350
means that phi, if there were
nothing else, equals zero.

853
00:39:02,350 --> 00:39:05,210
If there were no other terms
here, then this would say,

854
00:39:05,210 --> 00:39:06,717
ddr plus c over r
is phi, that means

855
00:39:06,717 --> 00:39:08,800
when you take a derivative
it's like dividing by r

856
00:39:08,800 --> 00:39:09,830
and multiplying by c.

857
00:39:09,830 --> 00:39:17,100
That means that phi goes
like r to the minus c, right?

858
00:39:17,100 --> 00:39:19,100
But if phi goes like
r to the minus c,

859
00:39:19,100 --> 00:39:21,870
that's not the exact
solution to the equation,

860
00:39:21,870 --> 00:39:26,880
but I can write phi is equal
to r to the minus c times u.

861
00:39:26,880 --> 00:39:30,730
And then this equation
becomes ddr plus dot, dot,

862
00:39:30,730 --> 00:39:34,320
dot u equals zero.

863
00:39:34,320 --> 00:39:34,980
OK?

864
00:39:34,980 --> 00:39:37,585
Very useful little
trick-- not really

865
00:39:37,585 --> 00:39:41,347
a trick, It's just observation--
and this is the second order

866
00:39:41,347 --> 00:39:42,430
version of the same thing.

867
00:39:42,430 --> 00:39:44,054
Very useful things
to have in your back

868
00:39:44,054 --> 00:39:47,220
pocket for moments of need.

869
00:39:47,220 --> 00:39:48,790
OK?

870
00:39:48,790 --> 00:39:56,459
So, let's pick up with this guy.

871
00:39:56,459 --> 00:39:58,250
So, let me give you a
little name for this.

872
00:39:58,250 --> 00:40:00,333
So, this term that comes
from the angular momentum

873
00:40:00,333 --> 00:40:03,060
[? bit, ?] this originally came
from the kinetic energy, right?

874
00:40:03,060 --> 00:40:04,640
It came from the
L squared over r,

875
00:40:04,640 --> 00:40:06,570
which was from the
gradient squared energy.

876
00:40:06,570 --> 00:40:07,990
This is a kinetic energy term.

877
00:40:07,990 --> 00:40:09,690
Why is there a
kinetic energy term?

878
00:40:09,690 --> 00:40:12,220
Well, what this is telling
you is that if you have some

879
00:40:12,220 --> 00:40:15,040
angular momentum-- if little
l is not equal to 0---

880
00:40:15,040 --> 00:40:17,420
then as you get closer
and closer to the origin,

881
00:40:17,420 --> 00:40:19,510
the potential energy is
getting very, very large.

882
00:40:19,510 --> 00:40:20,290
And this should make sense.

883
00:40:20,290 --> 00:40:22,206
If you're spinning, and
you pull in your arms,

884
00:40:22,206 --> 00:40:23,386
you have to do work, right?

885
00:40:23,386 --> 00:40:25,460
You have to pull those guys in.

886
00:40:25,460 --> 00:40:26,039
You speed up.

887
00:40:26,039 --> 00:40:27,580
You're increasing
your kinetic energy

888
00:40:27,580 --> 00:40:29,930
due to conservation of
angular momentum, right?

889
00:40:29,930 --> 00:40:31,430
If you have
rotationally invariance,

890
00:40:31,430 --> 00:40:35,180
as you bring in your hand you're
increasing the kinetic energy.

891
00:40:35,180 --> 00:40:37,620
And so, this angular
momentum barrier

892
00:40:37,620 --> 00:40:39,030
is just an expression of that.

893
00:40:39,030 --> 00:40:41,742
It's just saying
that as you come

894
00:40:41,742 --> 00:40:43,825
to smaller and smaller
radius, holding the angular

895
00:40:43,825 --> 00:40:46,800
momentum fixed, your velocity--
your angular velocity-- must

896
00:40:46,800 --> 00:40:49,496
increase-- your kinetic
energy must increase--

897
00:40:49,496 --> 00:40:51,120
and we're calling
that a potential term

898
00:40:51,120 --> 00:40:52,985
just because we can.

899
00:40:52,985 --> 00:40:56,036
Because we've worked with
definite angular momentum, OK?

900
00:40:56,036 --> 00:40:58,410
You should have done this in
classical mechanics as well.

901
00:41:02,442 --> 00:41:04,650
Well, you should have done
it in classical mechanics.

902
00:41:04,650 --> 00:41:06,650
So, this is called the
angular momentum barrier.

903
00:41:08,650 --> 00:41:14,300
Quick question, classically,
if you take a charged particle

904
00:41:14,300 --> 00:41:17,790
around in a Coulomb potential,
classically that system decays,

905
00:41:17,790 --> 00:41:18,290
right?

906
00:41:18,290 --> 00:41:20,070
Irradiates away energy.

907
00:41:20,070 --> 00:41:23,870
Does the angular momentum
barrier save us from decaying?

908
00:41:28,140 --> 00:41:29,565
Is that why hydrogen is stable?

909
00:41:37,100 --> 00:41:39,720
No one wants to
stake a claim here?

910
00:41:39,720 --> 00:41:41,890
Is hydrogen stable
because of conversation

911
00:41:41,890 --> 00:41:43,067
of angular momentum?

912
00:41:43,067 --> 00:41:44,380
AUDIENCE: No.

913
00:41:44,380 --> 00:41:45,130
PROFESSOR: No.

914
00:41:45,130 --> 00:41:46,390
Absolutely not, right?

915
00:41:46,390 --> 00:41:48,170
So, first off, in your
first problems set,

916
00:41:48,170 --> 00:41:50,080
when you did that
calculation, that particle

917
00:41:50,080 --> 00:41:52,560
had angular momentum.

918
00:41:52,560 --> 00:41:54,630
So, and if can radiate
that away through

919
00:41:54,630 --> 00:41:56,190
electromagnetic interactions.

920
00:41:56,190 --> 00:41:58,049
So, that didn't save us.

921
00:41:58,049 --> 00:41:59,340
Angular momentum won't save us.

922
00:41:59,340 --> 00:42:01,360
Another way to say this
is that we can construct--

923
00:42:01,360 --> 00:42:02,960
and we just explicitly
see-- we can construct

924
00:42:02,960 --> 00:42:04,870
a state with which
has little l equals 0.

925
00:42:04,870 --> 00:42:06,619
In which case the
angular momentum barrier

926
00:42:06,619 --> 00:42:09,870
is 0 over r squared,
because there's nothing.

927
00:42:09,870 --> 00:42:15,030
Angular momentum barrier's not
what keeps you from decaying.

928
00:42:15,030 --> 00:42:18,930
And the reason is that the
electron can radiate away

929
00:42:18,930 --> 00:42:21,620
energy and angular
momentum, and so l

930
00:42:21,620 --> 00:42:24,660
will decrease and decrease,
and can still fall down.

931
00:42:24,660 --> 00:42:27,680
So, we still need
a reason for why

932
00:42:27,680 --> 00:42:32,284
the hydrogen system, quantum
mechanically, is stable.

933
00:42:32,284 --> 00:42:34,950
[? Why do ?] [? things exist? ?]
So, let's answer that question.

934
00:42:38,440 --> 00:42:40,810
So, what I want to do
now is, I want to solve--

935
00:42:40,810 --> 00:42:43,280
do I really want to do it
that way?-- well, actually,

936
00:42:43,280 --> 00:42:47,080
before we do, let's consider
some last, general conditions.

937
00:42:47,080 --> 00:42:55,730
General facts for
central potentials.

938
00:42:55,730 --> 00:42:58,230
So, let's look at some general
facts for central potentials.

939
00:43:01,930 --> 00:43:06,680
So, the first is, regardless of
what the [? bare ?] potential

940
00:43:06,680 --> 00:43:10,200
was, just due to the
angular momentum barrier,

941
00:43:10,200 --> 00:43:13,190
we have this 1 over r squared
behavior near the origin.

942
00:43:15,810 --> 00:43:20,340
So, we can look at
this, we can ask, look,

943
00:43:20,340 --> 00:43:22,830
what are the boundary
conditions at the origin?

944
00:43:22,830 --> 00:43:26,680
What must be true of u
of r near the origin?

945
00:43:26,680 --> 00:43:31,790
Near u of r-- or
sorry, near r goes

946
00:43:31,790 --> 00:43:42,520
to zero-- what must
be true of u of r?

947
00:43:49,420 --> 00:43:51,414
So, the right way to
ask this question is not

948
00:43:51,414 --> 00:43:54,080
to look at this u of r, which is
not actually the wave function,

949
00:43:54,080 --> 00:43:56,650
but to look at the
actual wave function,

950
00:43:56,650 --> 00:44:01,750
phi sub E, which goes near r
equals 0, like u of r over r.

951
00:44:05,590 --> 00:44:06,910
So, what should be true of u?

952
00:44:10,450 --> 00:44:11,900
Can u diverge?

953
00:44:11,900 --> 00:44:13,335
Is that physical?

954
00:44:13,335 --> 00:44:14,560
Does u have to vanish?

955
00:44:14,560 --> 00:44:17,164
Can it take a constant value?

956
00:44:17,164 --> 00:44:18,830
So, I've given you a
hint by telling you

957
00:44:18,830 --> 00:44:20,590
that I want to think about there
being an infinite potential,

958
00:44:20,590 --> 00:44:21,075
but why?

959
00:44:21,075 --> 00:44:22,491
Why is that the
right thing to do?

960
00:44:28,050 --> 00:44:31,435
Well, imagine u of r went to a
constant value near the origin.

961
00:44:31,435 --> 00:44:33,560
If u of r goes to a constant
value near the origin,

962
00:44:33,560 --> 00:44:36,962
then the wave function
diverges near the origin.

963
00:44:36,962 --> 00:44:41,860
That's maybe not so bad, maybe
it has a 1 over r singularity.

964
00:44:41,860 --> 00:44:44,980
It's not totally obvious
that that's horrible.

965
00:44:44,980 --> 00:44:47,480
What's so bad about having
a 1 over r behavior?

966
00:44:47,480 --> 00:44:51,555
So, suppose u goes
to a constant.

967
00:44:59,700 --> 00:45:03,870
So, phi goes to constant over r.

968
00:45:03,870 --> 00:45:05,090
What's so bad about this?

969
00:45:14,690 --> 00:45:17,850
So, let's look back
at the kinetic energy.

970
00:45:17,850 --> 00:45:19,811
P is equal t--
the kinetic energy

971
00:45:19,811 --> 00:45:22,310
is gonna be minus h bar squared
p squared-- so the energy is

972
00:45:22,310 --> 00:45:26,260
going to go like, p
squared over 2md squared.

973
00:45:26,260 --> 00:45:30,460
But here's an important fact, d
squared-- the Laplacian-- of 1

974
00:45:30,460 --> 00:45:34,380
over r, well, it's
easy to see what

975
00:45:34,380 --> 00:45:36,720
this is at a general point.

976
00:45:36,720 --> 00:45:41,810
At a general point,
d squared has a term

977
00:45:41,810 --> 00:45:45,920
that looks like 1
over rd squared r r.

978
00:45:45,920 --> 00:45:55,880
So, 1 over rd squared
r on 1 over r.

979
00:45:55,880 --> 00:45:58,115
Well, r times 1 over
r, that's just 1.

980
00:46:00,910 --> 00:46:03,890
And this is 0, right?

981
00:46:03,890 --> 00:46:07,970
So, the gradient squared
of 1 over r, is 0.

982
00:46:07,970 --> 00:46:12,930
Except, can that possibly
be true at r equals 0?

983
00:46:12,930 --> 00:46:16,360
No, because what's the
second derivative at 0?

984
00:46:16,360 --> 00:46:19,230
As you approach the
origin from any direction,

985
00:46:19,230 --> 00:46:24,330
the function is going like 1
over r, OK, so it's growing,

986
00:46:24,330 --> 00:46:26,110
but it's growing
in every direction.

987
00:46:26,110 --> 00:46:30,220
So, what's its first
derivative at the origin?

988
00:46:30,220 --> 00:46:33,879
It's actually
ill-defined, because it

989
00:46:33,879 --> 00:46:35,420
depends on the
direction you come in.

990
00:46:35,420 --> 00:46:37,669
The first direction coming
in this way, the derivative

991
00:46:37,669 --> 00:46:40,250
looks like it's becoming
this, from this direction it's

992
00:46:40,250 --> 00:46:42,895
becoming this, it's
actually badly divergent.

993
00:46:42,895 --> 00:46:44,270
So, what's the
second derivative?

994
00:46:44,270 --> 00:46:45,360
Well, the second
derivative has to go

995
00:46:45,360 --> 00:46:46,770
as you go across
this point, it's

996
00:46:46,770 --> 00:46:48,740
telling you how the
first derivative changes.

997
00:46:48,740 --> 00:46:51,220
But it changes from plus
infinity in this direction,

998
00:46:51,220 --> 00:46:53,160
to plus infinity
in this direction.

999
00:46:53,160 --> 00:46:55,250
That's badly singular.

1000
00:46:55,250 --> 00:46:58,140
So, this can't possibly be true,
what I just wrote down here.

1001
00:46:58,140 --> 00:47:02,692
And, in fact, d squared
on 1 over r-- and this

1002
00:47:02,692 --> 00:47:07,450
is a very good exercise
for recitation--

1003
00:47:07,450 --> 00:47:10,340
is equal to delta of r.

1004
00:47:12,910 --> 00:47:18,050
It's 0-- it's clearly 0 for r0
equals 0--- but at the origin,

1005
00:47:18,050 --> 00:47:18,679
it's divergent.

1006
00:47:18,679 --> 00:47:20,220
And it's divergent
in exactly the way

1007
00:47:20,220 --> 00:47:22,986
you need to get
the delta function.

1008
00:47:22,986 --> 00:47:26,070
OK, which is pretty awesome.

1009
00:47:26,070 --> 00:47:29,225
So, what that tells us is that
if we have a wave function that

1010
00:47:29,225 --> 00:47:34,930
goes like 1 over r, then the
energy contribution-- energy

1011
00:47:34,930 --> 00:47:36,640
acting on this wave
function-- gives us

1012
00:47:36,640 --> 00:47:38,404
a delta function at the origin.

1013
00:47:38,404 --> 00:47:40,070
So, unless you have
the potential, which

1014
00:47:40,070 --> 00:47:42,992
is a delta function
at the origin,

1015
00:47:42,992 --> 00:47:44,200
nothing will cancel this off.

1016
00:47:44,200 --> 00:47:48,964
You can't possibly satisfy the
energy eigenvalue equation.

1017
00:47:48,964 --> 00:47:56,790
So, u of r must go
to 0 at r goes to 0.

1018
00:47:59,330 --> 00:48:02,060
Because if it goes to a
constant-- any constant--

1019
00:48:02,060 --> 00:48:06,545
we've got a bad divergence
in the energy, yeah?

1020
00:48:06,545 --> 00:48:08,390
In particular, if we
calculate the energy,

1021
00:48:08,390 --> 00:48:12,254
we'll discover that the
energy is badly divergent.

1022
00:48:12,254 --> 00:48:17,650
It does become divergent if
we don't have u going to 0.

1023
00:48:17,650 --> 00:48:21,930
So, notice, by the way, as a
side note, that since phi goes

1024
00:48:21,930 --> 00:48:23,840
like, phi is equal
to u over r, that

1025
00:48:23,840 --> 00:48:25,820
means that phi
goes to a constant.

1026
00:48:31,680 --> 00:48:33,430
This is good, because
what this is telling

1027
00:48:33,430 --> 00:48:36,980
us is that the wave function--
So, truly, u is vanishing,

1028
00:48:36,980 --> 00:48:38,630
but the probability
density, which

1029
00:48:38,630 --> 00:48:42,970
is the wave function squared,
doesn't have to vanish.

1030
00:48:42,970 --> 00:48:44,370
That's about the
derivative of u,

1031
00:48:44,370 --> 00:48:46,578
as you approach the origin
from [? Lucatau's ?] Rule.

1032
00:48:49,480 --> 00:48:53,890
So, this is the first general
fact about central potential.

1033
00:49:01,520 --> 00:49:07,185
So, the next one-- and this
is really fun one-- good Lord!

1034
00:49:07,185 --> 00:49:12,082
Is that, note---
sorry, two more--

1035
00:49:12,082 --> 00:49:22,350
the energy depends
on l but not on m.

1036
00:49:30,650 --> 00:49:34,290
Just explicitly, in the
energy eigenvalue equation,

1037
00:49:34,290 --> 00:49:37,655
we have the angular
momentum showing up int

1038
00:49:37,655 --> 00:49:39,412
the effective
potential, little l.

1039
00:49:39,412 --> 00:49:42,570
But little m appears absolutely
nowhere except in our choice

1040
00:49:42,570 --> 00:49:43,630
of spherical harmonic.

1041
00:49:43,630 --> 00:49:45,588
For any different m--
and this was pointing out

1042
00:49:45,588 --> 00:49:48,660
before-- for any different m, we
would've got the same equation.

1043
00:49:48,660 --> 00:49:51,400
And that means that the energy
eigenvalue can depend on l,

1044
00:49:51,400 --> 00:49:55,510
but it can't depend on m, right?

1045
00:49:55,510 --> 00:50:03,415
So, that means for each m in
the allowed possible values, l,

1046
00:50:03,415 --> 00:50:09,180
l minus 1, [? i ?]
minus l-- and this

1047
00:50:09,180 --> 00:50:14,060
is 2l plus 1 possible
values-- for each

1048
00:50:14,060 --> 00:50:15,970
of these m's, the
energy is the same.

1049
00:50:20,230 --> 00:50:24,600
And I'll call this E sub
l, because the energy

1050
00:50:24,600 --> 00:50:26,840
can depend on l.

1051
00:50:26,840 --> 00:50:28,350
Why?

1052
00:50:28,350 --> 00:50:32,807
The degeneracy of E sub
L is equal to 2l plus 1.

1053
00:50:47,380 --> 00:50:48,745
Why?

1054
00:50:48,745 --> 00:50:50,160
Why do we have this degeneracy?

1055
00:50:53,698 --> 00:50:55,359
AUDIENCE: [INAUDIBLE].

1056
00:50:55,359 --> 00:50:56,400
PROFESSOR: Yeah, exactly.

1057
00:50:56,400 --> 00:51:00,180
We get the degeneracies when
we have symmetries, right?

1058
00:51:00,180 --> 00:51:02,534
When we have a symmetry,
we get a degeneracy.

1059
00:51:02,534 --> 00:51:03,950
And so, here we
have a degeneracy.

1060
00:51:03,950 --> 00:51:07,060
And this degeneracy isn't
fixed by rotational invariance.

1061
00:51:07,060 --> 00:51:09,040
And why is this the right thing?

1062
00:51:09,040 --> 00:51:10,470
Rotational symmetry?

1063
00:51:10,470 --> 00:51:12,500
So why, did this give
us this degeneracy?

1064
00:51:12,500 --> 00:51:15,020
But what the rotational
degeneracy is saying is,

1065
00:51:15,020 --> 00:51:17,470
look, if you've got some
total angular momentum,

1066
00:51:17,470 --> 00:51:20,780
the energy can't possibly depend
on whether most of it's in Z,

1067
00:51:20,780 --> 00:51:23,886
or most of it's in X,
or most of it's and Y.

1068
00:51:23,886 --> 00:51:25,260
It can't possibly
depend on that,

1069
00:51:25,260 --> 00:51:26,430
but that's what
m is telling you.

1070
00:51:26,430 --> 00:51:27,888
M is just telling
you what fraction

1071
00:51:27,888 --> 00:51:31,580
is contained in a
particular direction.

1072
00:51:31,580 --> 00:51:33,820
So, rotational symmetry
immediately tells you this.

1073
00:51:33,820 --> 00:51:35,870
But there's a nice way to phrase
this, which of the following,

1074
00:51:35,870 --> 00:51:37,250
look, what is
rotational symmetry?

1075
00:51:37,250 --> 00:51:38,750
Rotational symmetry
is the statement

1076
00:51:38,750 --> 00:51:40,840
that the energy doesn't
care about rotations.

1077
00:51:40,840 --> 00:51:48,930
And in particular, it must
commute with Lx, and with Ly,

1078
00:51:48,930 --> 00:51:49,550
and with Lz.

1079
00:51:54,264 --> 00:51:55,680
So, this is
rotationally symmetry.

1080
00:52:00,122 --> 00:52:02,080
And I'm going to interpret
these in a nice way.

1081
00:52:04,730 --> 00:52:10,590
So, this guy tells me I can
find common eigenfunctions.

1082
00:52:13,720 --> 00:52:22,830
And, more to the point, a full
common eigenbasis of E and Lz.

1083
00:52:22,830 --> 00:52:25,590
Can I also find a common
eigenbasis of ELz and Ly?

1084
00:52:29,980 --> 00:52:32,560
Are there common
eigenvectors of ELz and Ly?

1085
00:52:32,560 --> 00:52:36,897
AUDIENCE: [CHATTER]

1086
00:52:36,897 --> 00:52:37,480
PROFESSOR: No.

1087
00:52:37,480 --> 00:52:39,355
Are there common
eigenfunctions of Lz and Ly?

1088
00:52:39,355 --> 00:52:40,130
AUDIENCE: No.

1089
00:52:40,130 --> 00:52:41,610
PROFESSOR: No, because
they don't commute, right?

1090
00:52:41,610 --> 00:52:42,550
E commutes with each of these.

1091
00:52:42,550 --> 00:52:43,870
OK, so, I'm just
going to say, I'm

1092
00:52:43,870 --> 00:52:45,530
gonna pick a common
eigenbasis of E and Lz--

1093
00:52:45,530 --> 00:52:47,613
but I could've picked Lx,
or I could've picked Ly,

1094
00:52:47,613 --> 00:52:49,740
I'm just picking Lz because
that's our convention--

1095
00:52:49,740 --> 00:52:51,722
but what do these two--
Once I've chosen this--

1096
00:52:51,722 --> 00:52:53,930
I'm gonna work with a common
eigenbasis of E and Lz--

1097
00:52:53,930 --> 00:52:55,700
what do these two
commutators tell me?

1098
00:52:55,700 --> 00:52:58,150
These two commutators
tell me that E commutes

1099
00:52:58,150 --> 00:53:02,250
with Lx plus iLy and Lx
minus iLy, L plus/minus.

1100
00:53:06,142 --> 00:53:08,600
So, this tells you that if you
have an eigenfunctions of E,

1101
00:53:08,600 --> 00:53:10,410
and you act with a
raising operator,

1102
00:53:10,410 --> 00:53:13,520
you get another
eigenfunction of E.

1103
00:53:13,520 --> 00:53:15,700
And thus, we get our
2L plus 1 degeneracy,

1104
00:53:15,700 --> 00:53:17,950
because we can walk up and
down the tower using L plus

1105
00:53:17,950 --> 00:53:19,730
and L minus.

1106
00:53:19,730 --> 00:53:21,595
Cool?

1107
00:53:21,595 --> 00:53:23,020
OK.

1108
00:53:23,020 --> 00:53:30,160
So, this is a nice example
that when you have a symmetry

1109
00:53:30,160 --> 00:53:32,860
you get a degeneracy,
and vice versa.

1110
00:53:36,010 --> 00:53:36,510
OK.

1111
00:53:39,470 --> 00:53:44,057
So, let's do some examples of
using these central potentials.

1112
00:53:44,057 --> 00:53:44,890
AUDIENCE: Professor?

1113
00:53:44,890 --> 00:53:45,646
PROFESSOR: Yeah

1114
00:53:45,646 --> 00:53:46,562
AUDIENCE: [INAUDIBLE]?

1115
00:53:52,380 --> 00:53:55,190
PROFESSOR: It's 0.

1116
00:53:55,190 --> 00:53:57,437
So, E with Lx is 0.

1117
00:53:57,437 --> 00:53:59,520
E with Ly-- So, are you
happy with that statement?

1118
00:53:59,520 --> 00:54:00,705
That E with Lx is 0?

1119
00:54:00,705 --> 00:54:01,330
AUDIENCE: Yeah.

1120
00:54:01,330 --> 00:54:01,997
PROFESSOR: Yeah.

1121
00:54:01,997 --> 00:54:02,496
Good.

1122
00:54:02,496 --> 00:54:04,640
OK, and so this 0 because
this is just Lx plus iOi.

1123
00:54:04,640 --> 00:54:06,550
So, E with Lx is
0, and E with Ly

1124
00:54:06,550 --> 00:54:09,224
is 0, so E commutes
with these guys.

1125
00:54:09,224 --> 00:54:10,640
And so, this is
like the statement

1126
00:54:10,640 --> 00:54:15,912
that L squared with L
plus/minus equals 0.

1127
00:54:15,912 --> 00:54:17,710
AUDIENCE: [INAUDIBLE].

1128
00:54:17,710 --> 00:54:18,850
PROFESSOR: Cool.

1129
00:54:18,850 --> 00:54:21,170
OK, so, let's do some examples.

1130
00:54:21,170 --> 00:54:23,890
So, the first example
is gonna be-- actually,

1131
00:54:23,890 --> 00:54:27,480
I'm going to skip this spherical
well example-- because it's

1132
00:54:27,480 --> 00:54:31,750
just not that interesting,
but it's in the notes,

1133
00:54:31,750 --> 00:54:33,625
and you really
need to look at it.

1134
00:54:33,625 --> 00:54:35,000
Oh hell, yes, I'm
going to do it.

1135
00:54:35,000 --> 00:54:36,130
OK, so, the spherical well.

1136
00:54:38,429 --> 00:54:40,220
So, I'm going to do it
in an abridged form,

1137
00:54:40,220 --> 00:54:43,151
and maybe it's a good
thing for recitation.

1138
00:54:43,151 --> 00:54:45,490
AUDIENCE: Professor?

1139
00:54:45,490 --> 00:54:47,370
PROFESSOR: Thank you
recitation leader.

1140
00:54:47,370 --> 00:54:50,390
So, in this spherical
well, what's the potential?

1141
00:54:50,390 --> 00:54:52,830
So, here's v of r.

1142
00:54:52,830 --> 00:54:56,830
Not U bar, and not V
effective, just v or r.

1143
00:54:56,830 --> 00:55:01,120
And the potential is going to
be this, so, here's r equals 0.

1144
00:55:01,120 --> 00:55:04,180
And if it's a spherical infinite
well, then I'm gonna say,

1145
00:55:04,180 --> 00:55:09,030
the potential is infinite
outside of some distance, l.

1146
00:55:09,030 --> 00:55:09,530
OK?

1147
00:55:12,070 --> 00:55:13,540
And it's 0 inside.

1148
00:55:16,650 --> 00:55:19,930
So, what does this give us?

1149
00:55:19,930 --> 00:55:22,394
Well, in order to
solve the system,

1150
00:55:22,394 --> 00:55:23,810
we know that the
first thing we do

1151
00:55:23,810 --> 00:55:26,610
is we separate out
with yLms, and then

1152
00:55:26,610 --> 00:55:29,260
we re-scale by 1 over r
to get the function of u,

1153
00:55:29,260 --> 00:55:33,340
and we get this
equation, which is E on u

1154
00:55:33,340 --> 00:55:40,070
is equal to minus h bar
squared upon 2m dr squared

1155
00:55:40,070 --> 00:55:44,910
and plus v effective--
well, plus [? lL ?] plus 1--

1156
00:55:44,910 --> 00:55:48,445
over r squared with a
2m and an h bar squared.

1157
00:55:52,250 --> 00:55:56,247
And the potential is 0, inside.

1158
00:55:56,247 --> 00:55:57,330
So we can just write this.

1159
00:56:02,700 --> 00:56:05,520
So, if you just--
let me pull out

1160
00:56:05,520 --> 00:56:12,180
the h bar squareds over 2m--
it becomes minus [INAUDIBLE]

1161
00:56:12,180 --> 00:56:13,200
plus 1 over r squared.

1162
00:56:18,090 --> 00:56:20,160
So, this is not a terrible
differential equation.

1163
00:56:20,160 --> 00:56:23,470
And one can do some
good work to solve it,

1164
00:56:23,470 --> 00:56:25,259
but it's a harder
differential equation

1165
00:56:25,259 --> 00:56:27,300
than I want to spend the
time to study right now,

1166
00:56:27,300 --> 00:56:31,920
so I'm just going to consider
the case-- special case-- when

1167
00:56:31,920 --> 00:56:36,580
there's zero angular
momentum, little l equals 0.

1168
00:56:36,580 --> 00:56:39,320
So, in the special
case of a l equals 0,

1169
00:56:39,320 --> 00:56:42,410
E-- and I should
call this u sub l--

1170
00:56:42,410 --> 00:56:46,930
Eu sub 0 is equal to
h bar squared upon 2m.

1171
00:56:46,930 --> 00:56:49,430
And now, this term is gone--
the angular momentum barrier is

1172
00:56:49,430 --> 00:56:51,450
gone-- because there's
no angular momentum,

1173
00:56:51,450 --> 00:56:55,600
dr squared ul.

1174
00:56:55,600 --> 00:56:58,840
Which can be written
succinctly as ul-- or sorry,

1175
00:56:58,840 --> 00:57:03,240
u0-- prime prime, because
this is only a function of r.

1176
00:57:03,240 --> 00:57:05,290
So now, this is a
ridiculously easy equation.

1177
00:57:05,290 --> 00:57:07,410
We know how to solve
this equation, right?

1178
00:57:07,410 --> 00:57:10,360
This is saying that the
energy, a constant, times u

1179
00:57:10,360 --> 00:57:15,150
is two derivatives
times this constant.

1180
00:57:15,150 --> 00:57:24,970
So, u0 can be written as a
cosine of kx-- or sorry-- kr

1181
00:57:24,970 --> 00:57:35,310
plus b sine of kr, where h
bar squared k squared upon 2m

1182
00:57:35,310 --> 00:57:39,460
equals E. And I should
really call this E sub 0,

1183
00:57:39,460 --> 00:57:42,740
because it could depend
on little l, here.

1184
00:57:45,390 --> 00:57:47,000
So, there's our
momentary solution,

1185
00:57:47,000 --> 00:57:49,416
however, we have to satisfy
our boundary conditions, which

1186
00:57:49,416 --> 00:57:51,590
is that it's gotta
vanish at the origin,

1187
00:57:51,590 --> 00:57:54,600
but it's also gotta
vanish at the wall.

1188
00:57:54,600 --> 00:57:59,360
So, the boundary
conditions, u of 0

1189
00:57:59,360 --> 00:58:06,786
equals 0 tells us that a must
be equal to 0, and u of l

1190
00:58:06,786 --> 00:58:11,840
equals 0 tells us that, well,
if this is 0, we've just got B,

1191
00:58:11,840 --> 00:58:15,110
but sine of kr evaluated
at l, which is sine of kl,

1192
00:58:15,110 --> 00:58:16,390
must be equal to 0.

1193
00:58:16,390 --> 00:58:23,250
So, kl must be the 0 of
sine, must be n pi over--

1194
00:58:23,250 --> 00:58:27,290
must be equal to n
pi, a multiple of pi.

1195
00:58:27,290 --> 00:58:30,240
And so, this tells you
what the energy is.

1196
00:58:30,240 --> 00:58:32,930
So, this is just
like the 1D system.

1197
00:58:32,930 --> 00:58:35,840
It's just exactly like
when the 1D system.

1198
00:58:35,840 --> 00:58:38,090
So now, to finally
close this off.

1199
00:58:38,090 --> 00:58:43,360
What does this tell you
that the eigenfunctions are?

1200
00:58:46,440 --> 00:58:48,800
And let me do that here.

1201
00:58:48,800 --> 00:58:57,830
So, therefore, the wave
function phi sub E0 of r theta

1202
00:58:57,830 --> 00:59:09,640
and phi-- oh god, oh jesus,
this is so much easier

1203
00:59:09,640 --> 00:59:12,210
in [INAUDIBLE] so,
phi [INAUDIBLE]

1204
00:59:12,210 --> 00:59:16,062
0 of r theta and
phi is equal to y0m.

1205
00:59:18,975 --> 00:59:21,990
But what must m be?

1206
00:59:21,990 --> 00:59:25,115
0, because m goes from
plus L to minus L, 0.

1207
00:59:31,334 --> 00:59:32,750
I'm just [INAUDIBLE]
the argument.

1208
00:59:32,750 --> 00:59:38,732
Y00 times, not u of r,
times 1 over r times u.

1209
00:59:38,732 --> 00:59:42,860
1 over r times u of r.

1210
00:59:42,860 --> 00:59:47,220
But u of r is a constant
times sine of kr.

1211
00:59:51,080 --> 00:59:59,810
Sine of kr, but k is equal
to n pi over L. N pi over Lr.

1212
00:59:59,810 --> 01:00:00,510
And what's Y00?

1213
01:00:03,350 --> 01:00:04,170
It's a constant.

1214
01:00:04,170 --> 01:00:06,940
And so, there's an overall
normalization constant,

1215
01:00:06,940 --> 01:00:08,780
that I'll call n.

1216
01:00:08,780 --> 01:00:13,180
OK, so, we get that
our wave function

1217
01:00:13,180 --> 01:00:16,900
is 1 over r times
sine of n pi over Lr.

1218
01:00:16,900 --> 01:00:18,930
So, this looks bad.

1219
01:00:18,930 --> 01:00:19,945
There's a 1 over r.

1220
01:00:19,945 --> 01:00:21,035
Why is this not bad?

1221
01:00:24,184 --> 01:00:25,850
At the origin, why
is this not something

1222
01:00:25,850 --> 01:00:26,850
I should worry about it?

1223
01:00:26,850 --> 01:00:27,996
AUDIENCE: [MURMURS]

1224
01:00:27,996 --> 01:00:30,340
PROFESSOR: Yeah, because
sine is linear, first of all,

1225
01:00:30,340 --> 01:00:31,215
[INAUDIBLE] argument.

1226
01:00:31,215 --> 01:00:33,500
So, this goes like,
n pi over L times r.

1227
01:00:33,500 --> 01:00:35,110
That r cancels the 1 over r.

1228
01:00:35,110 --> 01:00:37,950
So, near the origin, this
goes like a constant.

1229
01:00:37,950 --> 01:00:38,870
Yeah?

1230
01:00:38,870 --> 01:00:45,090
So, u has to 0, but the
wave function doesn't.

1231
01:00:45,090 --> 01:00:46,500
Cool?

1232
01:00:46,500 --> 01:00:47,000
OK.

1233
01:00:47,000 --> 01:00:50,510
So, this is a very
nice more general story

1234
01:00:50,510 --> 01:00:58,420
for larger L, which I hope
you see in the recitation.

1235
01:00:58,420 --> 01:01:00,100
OK.

1236
01:01:00,100 --> 01:01:02,860
Questions on the spherical well?

1237
01:01:02,860 --> 01:01:05,089
The whole point
here-- Oh, yeah, go.

1238
01:01:05,089 --> 01:01:07,255
AUDIENCE: What do [INAUDIBLE]
generally [INAUDIBLE]?

1239
01:01:17,550 --> 01:01:18,681
PROFESSOR: That's true.

1240
01:01:18,681 --> 01:01:19,180
So, good.

1241
01:01:19,180 --> 01:01:20,420
So, let me rephrase
the question,

1242
01:01:20,420 --> 01:01:22,128
and tell me if this
is the same question.

1243
01:01:22,128 --> 01:01:23,190
So, this is strange.

1244
01:01:23,190 --> 01:01:25,510
There's nothing special
about the origin.

1245
01:01:25,510 --> 01:01:27,990
So, why do I have
a 0 at the origin?

1246
01:01:27,990 --> 01:01:28,865
Is that the question?

1247
01:01:28,865 --> 01:01:29,490
AUDIENCE: Yeah.

1248
01:01:29,490 --> 01:01:30,110
PROFESSOR: OK.

1249
01:01:30,110 --> 01:01:30,520
It's true.

1250
01:01:30,520 --> 01:01:32,228
There's nothing special
about the origin,

1251
01:01:32,228 --> 01:01:33,582
except for two things.

1252
01:01:33,582 --> 01:01:35,290
One thing that's
special about the origin

1253
01:01:35,290 --> 01:01:36,340
is we're working
in a system which

1254
01:01:36,340 --> 01:01:37,423
has a rotational symmetry.

1255
01:01:37,423 --> 01:01:39,720
But rotational symmetry
is rotational symmetry

1256
01:01:39,720 --> 01:01:41,480
around some particular point.

1257
01:01:41,480 --> 01:01:43,640
So, there's always a special
central point anytime

1258
01:01:43,640 --> 01:01:44,570
you have a rotational symmetry.

1259
01:01:44,570 --> 01:01:46,153
It's the point fixed
by the rotations.

1260
01:01:46,153 --> 01:01:49,180
So, actually, the origin
is a special point here.

1261
01:01:49,180 --> 01:01:52,740
Second, saying that
little u has a 0

1262
01:01:52,740 --> 01:01:56,320
is not the same as saying that
the wave function has a 0.

1263
01:01:56,320 --> 01:01:59,390
Little u has a 0, but it
gets multiplied by 1 over r.

1264
01:01:59,390 --> 01:02:02,440
So, the wave function, in
fact, is non-zero, there.

1265
01:02:02,440 --> 01:02:04,902
So, the physical thing is
the probability distribution,

1266
01:02:04,902 --> 01:02:07,110
which is the [? norm ?]
squared of the wave function.

1267
01:02:07,110 --> 01:02:10,450
And it doesn't have
a 0 at the origin.

1268
01:02:10,450 --> 01:02:12,237
Does that satisfy?

1269
01:02:12,237 --> 01:02:12,820
AUDIENCE: Yes.

1270
01:02:12,820 --> 01:02:13,403
PROFESSOR: OK.

1271
01:02:13,403 --> 01:02:16,860
So, the origin is special when
you have a central potential.

1272
01:02:16,860 --> 01:02:18,990
That's where the
proton is, right?

1273
01:02:18,990 --> 01:02:19,840
Right, OK.

1274
01:02:19,840 --> 01:02:22,970
So, there is something
special about the origin.

1275
01:02:22,970 --> 01:02:25,630
Wow, that was a really
[? anti-caplarian ?] sort

1276
01:02:25,630 --> 01:02:26,130
of argument.

1277
01:02:30,410 --> 01:02:32,637
OK, so that's where
the proton-- so, there

1278
01:02:32,637 --> 01:02:34,220
is something special
about the origin,

1279
01:02:34,220 --> 01:02:38,870
and the wave function doesn't
vanish there, even if u does.

1280
01:02:38,870 --> 01:02:42,310
It may vanish there, but it
doesn't necessarily have to.

1281
01:02:42,310 --> 01:02:44,160
And we'll see that in a minute.

1282
01:02:44,160 --> 01:02:45,940
Other questions?

1283
01:02:45,940 --> 01:02:46,658
Yeah?

1284
01:02:46,658 --> 01:02:48,570
AUDIENCE: So, what again, what's
the reasoning for saying that

1285
01:02:48,570 --> 01:02:50,960
the u of r has to vanish at
[? 0 instead ?] [? of L? ?]

1286
01:02:50,960 --> 01:02:51,626
PROFESSOR: Good.

1287
01:02:51,626 --> 01:02:54,610
The reason that u of r had
to vanish at the origin

1288
01:02:54,610 --> 01:02:56,540
is that if it doesn't
vanish at the origin,

1289
01:02:56,540 --> 01:03:00,200
then the wave function
diverges-- whoops,

1290
01:03:00,200 --> 01:03:06,610
phi goes to constant--
if u doesn't go to 0,

1291
01:03:06,610 --> 01:03:09,020
if it goes to any
constant, non-zero,

1292
01:03:09,020 --> 01:03:10,714
then the wave function diverges.

1293
01:03:10,714 --> 01:03:12,130
And if we calculate
the energy, we

1294
01:03:12,130 --> 01:03:14,170
get a delta function
at the origin.

1295
01:03:14,170 --> 01:03:15,670
So, there's an
infinite contribution

1296
01:03:15,670 --> 01:03:16,670
of energy at the origin.

1297
01:03:16,670 --> 01:03:17,650
That's not physical.

1298
01:03:17,650 --> 01:03:19,760
So, in order to get a
sensible wave function

1299
01:03:19,760 --> 01:03:23,454
with finite energies, we need
to have the u vanishes, because

1300
01:03:23,454 --> 01:03:24,120
of the 1 over u.

1301
01:03:24,120 --> 01:03:26,203
And the reason that we
said it had to vanish at l,

1302
01:03:26,203 --> 01:03:29,062
was because I was considering
this spherical well-- spherical

1303
01:03:29,062 --> 01:03:30,770
infinite well-- where
a particle is stuck

1304
01:03:30,770 --> 01:03:33,860
inside a region of
radius, capital L,

1305
01:03:33,860 --> 01:03:35,860
and that's just what
I mean by saying

1306
01:03:35,860 --> 01:03:37,126
I have an infinite potential.

1307
01:03:37,126 --> 01:03:38,000
AUDIENCE: OK, thanks.

1308
01:03:38,000 --> 01:03:38,666
PROFESSOR: Cool?

1309
01:03:38,666 --> 01:03:39,250
Yeah.

1310
01:03:39,250 --> 01:03:40,630
Others?

1311
01:03:40,630 --> 01:03:41,870
OK.

1312
01:03:41,870 --> 01:03:44,690
So, with all that
done, we can now

1313
01:03:44,690 --> 01:03:47,860
do the hydrogen-- or
the Coulomb-- potential.

1314
01:03:47,860 --> 01:03:51,560
And I want to emphasize that
we often use the following

1315
01:03:51,560 --> 01:03:53,810
words when-- people often
use the following words when

1316
01:03:53,810 --> 01:03:55,670
solving this
problem-- we will now

1317
01:03:55,670 --> 01:03:58,610
solve the problem of hydrogen.

1318
01:03:58,610 --> 01:03:59,990
This is false.

1319
01:03:59,990 --> 01:04:02,900
I am not about to solve for
you the problem of hydrogen.

1320
01:04:02,900 --> 01:04:07,000
I am going to construct for
you a nice toy model, which

1321
01:04:07,000 --> 01:04:10,570
turns out to be an excellent
first pass at explaining

1322
01:04:10,570 --> 01:04:14,360
the properties observed in
hydrogen gases, their emission

1323
01:04:14,360 --> 01:04:15,660
spectra, and their physics.

1324
01:04:15,660 --> 01:04:16,710
This is a model.

1325
01:04:16,710 --> 01:04:18,170
It is a bad model.

1326
01:04:18,170 --> 01:04:20,020
It doesn't fit the data.

1327
01:04:20,020 --> 01:04:21,270
But it's pretty good.

1328
01:04:21,270 --> 01:04:23,000
And we'll be able
to improve it later.

1329
01:04:23,000 --> 01:04:24,710
OK?

1330
01:04:24,710 --> 01:04:27,820
So, it is the solution
of the Coulomb potential.

1331
01:04:27,820 --> 01:04:29,320
And what I want to
emphasize to you,

1332
01:04:29,320 --> 01:04:31,000
I cannot say this
strongly enough,

1333
01:04:31,000 --> 01:04:36,280
physics is a process of building
models that do a good job

1334
01:04:36,280 --> 01:04:36,864
of predicting.

1335
01:04:36,864 --> 01:04:38,863
And the better their
predictions, the better the

1336
01:04:38,863 --> 01:04:39,450
model.

1337
01:04:39,450 --> 01:04:40,790
But they're all wrong.

1338
01:04:40,790 --> 01:04:44,390
Every single model you ever
get from physics is wrong.

1339
01:04:44,390 --> 01:04:47,040
There are just some that
are less stupidly wrong.

1340
01:04:47,040 --> 01:04:49,660
Some are a better
approximation to the data, OK?

1341
01:04:49,660 --> 01:04:51,410
This is not hydrogen.

1342
01:04:51,410 --> 01:04:53,380
This is going to be our
first pass at hydrogen.

1343
01:04:53,380 --> 01:04:56,180
It's the Coulomb potential.

1344
01:04:56,180 --> 01:05:01,700
And the Coulomb potential,
V of r, is equal to minus e

1345
01:05:01,700 --> 01:05:04,475
squared over r.

1346
01:05:04,475 --> 01:05:05,850
This is what you
would get if you

1347
01:05:05,850 --> 01:05:11,740
had a classical particle with
infinite mass and charge plus

1348
01:05:11,740 --> 01:05:12,410
b.

1349
01:05:12,410 --> 01:05:13,910
And then another
particle over here,

1350
01:05:13,910 --> 01:05:17,569
with mass, little m,
and charge, minus e.

1351
01:05:17,569 --> 01:05:19,110
And you didn't pay
too much attention

1352
01:05:19,110 --> 01:05:23,810
to things like relativity,
or spin, or, you know,

1353
01:05:23,810 --> 01:05:24,700
lots of other things.

1354
01:05:24,700 --> 01:05:26,450
And you have no
background magnetic field,

1355
01:05:26,450 --> 01:05:28,330
or electric field,
and anything else.

1356
01:05:28,330 --> 01:05:30,389
And if these are point
particles, and-- All

1357
01:05:30,389 --> 01:05:32,180
of those things are
false that I just said.

1358
01:05:32,180 --> 01:05:33,596
But if all those
things were true,

1359
01:05:33,596 --> 01:05:37,370
in that imaginary universe, this
would be the salient problem

1360
01:05:37,370 --> 01:05:37,870
to solve.

1361
01:05:37,870 --> 01:05:39,109
So, let's solve it.

1362
01:05:39,109 --> 01:05:40,650
Now, are all those
things that I said

1363
01:05:40,650 --> 01:05:43,240
that were false-- the
proton's a point particle,

1364
01:05:43,240 --> 01:05:45,250
the proton is
infinitely massive,

1365
01:05:45,250 --> 01:05:48,660
there's no spin-- are those
preposterously stupid?

1366
01:05:48,660 --> 01:05:49,650
AUDIENCE: No.

1367
01:05:49,650 --> 01:05:51,610
PROFESSOR: No, they're
excellent approximations

1368
01:05:51,610 --> 01:05:52,770
in a lot of situations.

1369
01:05:52,770 --> 01:05:55,010
So, they're not crazy wrong.

1370
01:05:55,010 --> 01:05:58,036
They're just not
exactly correct.

1371
01:05:58,036 --> 01:05:59,410
I want to keep
this in your mind.

1372
01:05:59,410 --> 01:06:02,220
These are gonna be good
models, but they're not exact.

1373
01:06:02,220 --> 01:06:03,761
So, we're not solving
hydrogen, we're

1374
01:06:03,761 --> 01:06:07,680
gonna solve this idealized
Coulomb potential problem.

1375
01:06:07,680 --> 01:06:08,940
OK, so let's solve it.

1376
01:06:08,940 --> 01:06:11,610
So, if V is minus
e over r squared,

1377
01:06:11,610 --> 01:06:15,740
then the equation for the
rescaled wave function, u,

1378
01:06:15,740 --> 01:06:21,570
becomes minus h bar squared
upon 2mu prime prime

1379
01:06:21,570 --> 01:06:24,770
of r plus the effective
potential, which

1380
01:06:24,770 --> 01:06:29,910
is h bar squared
upon 2mll plus 1

1381
01:06:29,910 --> 01:06:41,670
over r squared minus e squared
over r u is equal to e sub l u.

1382
01:06:41,670 --> 01:06:43,540
So, there's the equation
we want to solve.

1383
01:06:43,540 --> 01:06:45,500
We've already used
separation of variables,

1384
01:06:45,500 --> 01:06:46,916
and we know that
the wave function

1385
01:06:46,916 --> 01:06:49,474
is this little u times
1 over r times yLm,

1386
01:06:49,474 --> 01:06:50,390
for some l and some m.

1387
01:06:53,020 --> 01:06:55,340
So, the first thing we
should do any time you're

1388
01:06:55,340 --> 01:06:57,881
solving an interesting problem,
the first thing you should do

1389
01:06:57,881 --> 01:06:59,740
is do dimensional analysis.

1390
01:06:59,740 --> 01:07:02,330
And if you do dimensional
analysis, the units of e

1391
01:07:02,330 --> 01:07:04,720
squared-- well, this
is easy-- e squared

1392
01:07:04,720 --> 01:07:06,850
must be an energy
times a length.

1393
01:07:06,850 --> 01:07:10,350
So, this is an energy
times a length.

1394
01:07:10,350 --> 01:07:13,150
Also known as p
squared l, momentum

1395
01:07:13,150 --> 01:07:16,532
squared over 2m, 2 times
the mass times the length.

1396
01:07:16,532 --> 01:07:18,740
It's useful to put things
in terms of mass, momentum,

1397
01:07:18,740 --> 01:07:20,614
and lengths, because
you can cancel them out.

1398
01:07:20,614 --> 01:07:24,987
H bar has units of p times l.

1399
01:07:24,987 --> 01:07:26,820
And what's the only
other parameter we have?

1400
01:07:26,820 --> 01:07:30,980
We have the mass, which
has units of mass.

1401
01:07:30,980 --> 01:07:31,630
OK.

1402
01:07:31,630 --> 01:07:34,540
And so, from this, we can
build two nice quantities.

1403
01:07:34,540 --> 01:07:36,015
The first, is we can build r0.

1404
01:07:36,015 --> 01:07:40,190
We can build something
with units of a radius.

1405
01:07:40,190 --> 01:07:42,530
And I'm going to choose the
factors of 2 judiciously, h

1406
01:07:42,530 --> 01:07:47,444
bar squared over 2me squared--
whoops, e squared-- so,

1407
01:07:47,444 --> 01:07:49,360
let's just make sure
this has the right units.

1408
01:07:49,360 --> 01:07:52,190
E squared has units of
energy times the length,

1409
01:07:52,190 --> 01:07:55,270
but h bar squared
over 2m has units

1410
01:07:55,270 --> 01:07:58,770
of p squared l squared
over 2m, so that

1411
01:07:58,770 --> 01:08:04,385
has units of energy
times the length squared.

1412
01:08:04,385 --> 01:08:06,760
So, length squared over length,
this has units of length,

1413
01:08:06,760 --> 01:08:08,240
so this is good.

1414
01:08:08,240 --> 01:08:11,215
So, there's a parameter
that has units of length.

1415
01:08:11,215 --> 01:08:12,840
And from this, it's
easy to see that we

1416
01:08:12,840 --> 01:08:15,048
can build a characteristic
energy by taking e squared

1417
01:08:15,048 --> 01:08:17,176
and dividing it by
this length scale.

1418
01:08:17,176 --> 01:08:19,050
And so then, the
energy, which I'll

1419
01:08:19,050 --> 01:08:24,080
call e0, which is equal
to e squared over r0,

1420
01:08:24,080 --> 01:08:29,400
is equal to 2me to the
4th over h bar squared.

1421
01:08:34,350 --> 01:08:37,029
So, before we do anything else,
without solving any problems,

1422
01:08:37,029 --> 01:08:39,065
we immediately can do
a couple of things.

1423
01:08:39,065 --> 01:08:41,439
The first is, if you take the
system and I ask you, look,

1424
01:08:41,439 --> 01:08:42,130
what do you expect?

1425
01:08:42,130 --> 01:08:43,505
If this is a
quantum mechanical--

1426
01:08:43,505 --> 01:08:46,350
a 1d problem in quantum
mechanics-- with a potential,

1427
01:08:46,350 --> 01:08:49,769
and we know something about 1D
quantum mechanical problems--

1428
01:08:49,769 --> 01:08:51,310
I guess, this guy--
we know something

1429
01:08:51,310 --> 01:08:52,851
about 1D quantum
mechanical problems.

1430
01:08:52,851 --> 01:08:55,680
Which is that the ground
state has what energy?

1431
01:08:55,680 --> 01:08:56,582
Some finite energy.

1432
01:08:56,582 --> 01:08:58,290
It doesn't have infinite
negative energy.

1433
01:08:58,290 --> 01:09:00,600
It's got some finite energy.

1434
01:09:00,600 --> 01:09:03,160
What do you expect to be
roughly the ground state energy

1435
01:09:03,160 --> 01:09:06,060
of this system?

1436
01:09:06,060 --> 01:09:06,970
AUDIENCE: [MURMURING]

1437
01:09:06,970 --> 01:09:07,510
PROFESSOR: Yeah.

1438
01:09:07,510 --> 01:09:08,010
Right.

1439
01:09:08,010 --> 01:09:09,090
Roughly minus e0.

1440
01:09:09,090 --> 01:09:10,802
That seems like a
pretty good guess.

1441
01:09:10,802 --> 01:09:12,510
It's the only dimensional
sensible thing.

1442
01:09:12,510 --> 01:09:14,540
Maybe we're off by factors of 2.

1443
01:09:14,540 --> 01:09:18,705
But, maybe it's minus e0.

1444
01:09:18,705 --> 01:09:20,330
So, that's a good
guess, a first thing,

1445
01:09:20,330 --> 01:09:22,890
before we do any calculation.

1446
01:09:22,890 --> 01:09:28,180
And if you actually take mu e
to the 4th over h bar squared,

1447
01:09:28,180 --> 01:09:31,219
this is off by,
unfortunately, a factor of 4.

1448
01:09:31,219 --> 01:09:36,040
This is equal to 4 times
the binding energy, which

1449
01:09:36,040 --> 01:09:39,106
is also called the
Rydberg constant.

1450
01:09:39,106 --> 01:09:41,349
Wanna make sure I get
my factors of two right.

1451
01:09:41,349 --> 01:09:42,874
Yep, I'm off by a factor of 4.

1452
01:09:42,874 --> 01:09:45,180
I'm off by a factor
of 4 from what

1453
01:09:45,180 --> 01:09:51,310
we'll call the Rydberg
energy, which is 13.6 eV.

1454
01:09:51,310 --> 01:09:54,160
And this is observed
binding energy of hydrogen.

1455
01:09:54,160 --> 01:09:56,970
So, before we do anything,
before we solve any equation,

1456
01:09:56,970 --> 01:10:02,570
we have a fabulous estimate of
the binding energy of hydrogen,

1457
01:10:02,570 --> 01:10:03,730
right?

1458
01:10:03,730 --> 01:10:05,790
All the work we're
about to do is

1459
01:10:05,790 --> 01:10:09,070
gonna be to deal with
this factor of 4, right?

1460
01:10:09,070 --> 01:10:11,025
Which, I mean, is
important, but I just

1461
01:10:11,025 --> 01:10:12,650
want to emphasize
how much you get just

1462
01:10:12,650 --> 01:10:14,110
from doing dimensional analysis.

1463
01:10:14,110 --> 01:10:16,515
Immediately upon knowing the
rules of quantum mechanics,

1464
01:10:16,515 --> 01:10:18,640
knowing that this is the
equation you should solve,

1465
01:10:18,640 --> 01:10:20,140
without ever touching
that equation,

1466
01:10:20,140 --> 01:10:22,730
just dimensional analysis
gives you this answer.

1467
01:10:22,730 --> 01:10:23,546
OK?

1468
01:10:23,546 --> 01:10:24,295
Which is fabulous.

1469
01:10:28,020 --> 01:10:31,060
So, with that motivation,
let's solve this problem.

1470
01:10:31,060 --> 01:10:33,730
Oh, by the way,
what do you think

1471
01:10:33,730 --> 01:10:37,330
r0 is a good approximation to?

1472
01:10:37,330 --> 01:10:38,740
Well, it's a length scale.

1473
01:10:38,740 --> 01:10:39,884
AUDIENCE: [INAUDIBLE].

1474
01:10:39,884 --> 01:10:40,550
PROFESSOR: Yeah!

1475
01:10:40,550 --> 01:10:42,633
It's probably something
like the expectation value

1476
01:10:42,633 --> 01:10:44,820
of the radius-- or maybe
of the radius squared--

1477
01:10:44,820 --> 01:10:47,236
because the expectation value
of the radius is probably 0.

1478
01:10:49,660 --> 01:10:52,950
OK, so, let's solve this system.

1479
01:10:52,950 --> 01:10:55,960
And at this point, I'm not
gonna actually solve out

1480
01:10:55,960 --> 01:10:57,820
the differential
equation in detail.

1481
01:10:57,820 --> 01:10:59,980
I'm just gonna tell you
how the solution goes,

1482
01:10:59,980 --> 01:11:04,540
because solving it is a sort
of involved undertaking.

1483
01:11:04,540 --> 01:11:07,630
And so, here's the first thing,
so we look at this equation.

1484
01:11:07,630 --> 01:11:10,310
So, we had this differential
equation-- this guy--

1485
01:11:10,310 --> 01:11:13,150
and we want to solve it.

1486
01:11:13,150 --> 01:11:16,520
So, think back to the
harmonic oscillator

1487
01:11:16,520 --> 01:11:19,910
when we did the brute force
method of solving the hydrogen

1488
01:11:19,910 --> 01:11:21,445
system, OK?

1489
01:11:21,445 --> 01:11:26,830
When we did the brute force
method-- she sells seashells--

1490
01:11:26,830 --> 01:11:30,430
when the brute force method
of solving, what did we do?

1491
01:11:30,430 --> 01:11:33,920
We first did, we did
asymptotic analysis.

1492
01:11:33,920 --> 01:11:36,690
We extracted the
overall asymptotic form,

1493
01:11:36,690 --> 01:11:38,742
at infinity and at
the origin, to get

1494
01:11:38,742 --> 01:11:40,450
a nice regular
differential equation that

1495
01:11:40,450 --> 01:11:42,340
didn't have any
funny singularities,

1496
01:11:42,340 --> 01:11:44,950
and then we did a
series approximation.

1497
01:11:44,950 --> 01:11:46,180
OK?

1498
01:11:46,180 --> 01:11:48,189
Now, do most
differential equations

1499
01:11:48,189 --> 01:11:49,730
have a simple closed
form expression?

1500
01:11:49,730 --> 01:11:50,725
A solution?

1501
01:11:50,725 --> 01:11:53,350
No, most differential equations
of some, maybe if you're lucky,

1502
01:11:53,350 --> 01:11:55,940
it's a special function that
people have studied in detail,

1503
01:11:55,940 --> 01:11:58,148
but most don't have a simple
solution like a Gaussian

1504
01:11:58,148 --> 01:11:59,670
or a power large, or something.

1505
01:11:59,670 --> 01:12:01,941
Most of them just have
some complicated solution.

1506
01:12:01,941 --> 01:12:04,440
This is one of those miraculous
differential equations where

1507
01:12:04,440 --> 01:12:06,773
we can actually exactly write
down the solution by doing

1508
01:12:06,773 --> 01:12:09,510
the series approximation,
having done asymptotic analysis.

1509
01:12:13,190 --> 01:12:15,790
So, the first thing when doing
dimensional analysis too, let's

1510
01:12:15,790 --> 01:12:17,645
make everything dimensionless.

1511
01:12:20,650 --> 01:12:23,834
OK, and it's easy to see what
the right thing to do is.

1512
01:12:23,834 --> 01:12:26,500
Take r and make it dimensionless
by pulling out a factor of rho,

1513
01:12:26,500 --> 01:12:27,540
or of r0.

1514
01:12:27,540 --> 01:12:29,790
So, I'll pick our new variable
is gonna be called rho,

1515
01:12:29,790 --> 01:12:32,420
this is dimensionless.

1516
01:12:32,420 --> 01:12:34,630
And the second thing is I
want to take the energy,

1517
01:12:34,630 --> 01:12:37,600
and I will write it
as minus e0, times

1518
01:12:37,600 --> 01:12:39,720
some dimensionless
energy, epsilon.

1519
01:12:39,720 --> 01:12:43,015
So, these guys are my
dimensionless variables.

1520
01:12:43,015 --> 01:12:45,390
And when you go through and
do that, the equation you get

1521
01:12:45,390 --> 01:12:55,590
is minus d rho squared plus l l
plus 1 over rho squared minus 1

1522
01:12:55,590 --> 01:13:01,670
over rho plus epsilon
u is equal to 0.

1523
01:13:01,670 --> 01:13:03,420
So, the form of this
differential equation

1524
01:13:03,420 --> 01:13:05,450
is, OK, it's not
different in any deep way,

1525
01:13:05,450 --> 01:13:06,955
but it's a little bit easier.

1526
01:13:06,955 --> 01:13:09,455
This is gonna be the easier way
to deal with this, because I

1527
01:13:09,455 --> 01:13:11,840
don't have to deal with
any stupid constant.

1528
01:13:11,840 --> 01:13:14,450
And so now, let's do
the brute force thing.

1529
01:13:14,450 --> 01:13:16,955
Three, asymptotic analysis.

1530
01:13:25,050 --> 01:13:27,220
And here, I'm just going
to write down the answers.

1531
01:13:27,220 --> 01:13:28,791
And the reason is,
first off, this

1532
01:13:28,791 --> 01:13:30,790
is something you should
either do in recitation,

1533
01:13:30,790 --> 01:13:33,030
or see-- go through--
on your own,

1534
01:13:33,030 --> 01:13:35,650
but this is just the mathematics
of solving a differential

1535
01:13:35,650 --> 01:13:36,150
equation.

1536
01:13:36,150 --> 01:13:37,610
This is not the important part.

1537
01:13:37,610 --> 01:13:41,429
So, when rho goes to infinity,
which terms dominate?

1538
01:13:41,429 --> 01:13:42,970
Well, this is not
terribly important.

1539
01:13:42,970 --> 01:13:44,700
This is not terribly important.

1540
01:13:44,700 --> 01:13:46,030
That term is gonna dominate.

1541
01:13:49,470 --> 01:13:54,389
And if we get that d rho squared
plus u, rho goes to infinity,

1542
01:13:54,389 --> 01:13:55,430
these two terms dominate.

1543
01:13:55,430 --> 01:13:57,304
Well, two derivatives
is a constant.

1544
01:13:57,304 --> 01:13:58,720
You know what those
solutions look

1545
01:13:58,720 --> 01:14:00,310
like, they look
like exponentials,

1546
01:14:00,310 --> 01:14:02,250
with the exponential
being brute--

1547
01:14:02,250 --> 01:14:06,830
with the power-- the exponent,
sorry, being root epsilon.

1548
01:14:06,830 --> 01:14:11,550
So, u is going to go like e to
the minus square root epsilon

1549
01:14:11,550 --> 01:14:12,070
rho.

1550
01:14:12,070 --> 01:14:13,820
For normalize-ability,
I picked the minus,

1551
01:14:13,820 --> 01:14:16,195
I could've picked the plus,
that would've been divergent.

1552
01:14:17,990 --> 01:14:22,120
So, as rho goes to
0, what happens?

1553
01:14:22,120 --> 01:14:24,530
Well, as rho goes to 0,
this is insignificant.

1554
01:14:24,530 --> 01:14:26,400
And this totally
dominates over this guy.

1555
01:14:29,800 --> 01:14:31,722
On the other hand,
if l is equal to 0,

1556
01:14:31,722 --> 01:14:33,430
then this is the only
term that survives,

1557
01:14:33,430 --> 01:14:35,680
so we'd better make sure
that that behaves gracefully.

1558
01:14:35,680 --> 01:14:37,980
As rho goes to 0,
asymptotic analysis

1559
01:14:37,980 --> 01:14:40,750
is gonna tell us
that u goes like rho.

1560
01:14:40,750 --> 01:14:43,090
Well, two derivatives, we
pulled down a rho squared,

1561
01:14:43,090 --> 01:14:45,975
and so two derivatives in
this guy, we pulled down an l,

1562
01:14:45,975 --> 01:14:46,797
then an l plus 1.

1563
01:14:46,797 --> 01:14:48,630
So, this should go like
rho to the l plus 1.

1564
01:14:54,800 --> 01:14:56,749
There's also another term.

1565
01:14:56,749 --> 01:14:59,290
So, in the same way that there
were two solutions to this guy

1566
01:14:59,290 --> 01:15:02,160
asymptotically-- one growing,
one decreasing-- here,

1567
01:15:02,160 --> 01:15:04,410
there's another solution,
which is rho to the minus l.

1568
01:15:04,410 --> 01:15:06,410
That also does it, because
we get minus l, then

1569
01:15:06,410 --> 01:15:09,932
minus minus l minus 1, which
gives us the plus l l plus 1.

1570
01:15:09,932 --> 01:15:11,890
But that is also badly
diversion at the origin,

1571
01:15:11,890 --> 01:15:14,090
it goes like 1 over 0 to the l.

1572
01:15:14,090 --> 01:15:14,820
That's bad.

1573
01:15:14,820 --> 01:15:15,945
So, these are my solutions.

1574
01:15:18,750 --> 01:15:20,760
So, this tells us, having
done this in analysis,

1575
01:15:20,760 --> 01:15:24,150
we should write that u is
equal to rho to the l plus

1576
01:15:24,150 --> 01:15:28,170
1 times e to the
minus root epsilon

1577
01:15:28,170 --> 01:15:30,690
rho times some
remaining function,

1578
01:15:30,690 --> 01:15:35,672
which I'll call v, little
v. Little v of rho,

1579
01:15:35,672 --> 01:15:37,130
and this,
asymptotically, should go

1580
01:15:37,130 --> 01:15:40,730
to a constant near the
origin and something that

1581
01:15:40,730 --> 01:15:46,560
vanishes slower than an
exponential at infinity.

1582
01:15:46,560 --> 01:15:50,285
So then, we take this and
we do our series expansion.

1583
01:15:54,759 --> 01:15:56,550
So, we take that
expression, we plug it in.

1584
01:15:56,550 --> 01:15:59,970
At that point, all we're doing
is a change of variables.

1585
01:15:59,970 --> 01:16:02,690
We plug it in, and we get
a resulting differential

1586
01:16:02,690 --> 01:16:03,190
equation.

1587
01:16:05,810 --> 01:16:13,670
Rho v prime prime plus 2 1
plus l minus root epsilon rho

1588
01:16:13,670 --> 01:16:24,437
v prime plus 1 minus 2 root
epsilon l plus 1 v equals 0.

1589
01:16:24,437 --> 01:16:26,395
So, this is the resulting
differential equation

1590
01:16:26,395 --> 01:16:29,840
for the little v guy.

1591
01:16:29,840 --> 01:16:33,400
And we do a series expansion.

1592
01:16:33,400 --> 01:16:39,285
V is equal to sum
over, sum from j

1593
01:16:39,285 --> 01:16:46,260
equals 0 to infinity,
of a sub j rho to the j.

1594
01:16:46,260 --> 01:16:49,600
Plug this guy in here, just
like in the case of the harmonic

1595
01:16:49,600 --> 01:16:52,850
oscillator equation, and
get a series expansion.

1596
01:16:52,850 --> 01:16:57,010
Now, OK, let me write
it out this way.

1597
01:17:02,670 --> 01:17:04,470
And the series expansion
has a solution,

1598
01:17:04,470 --> 01:17:05,646
which is a sub j plus 1.

1599
01:17:05,646 --> 01:17:07,520
And this is, actually,
kind of a fun process.

1600
01:17:07,520 --> 01:17:11,824
So, if you, you know, like
quick little calculations,

1601
01:17:11,824 --> 01:17:13,240
this is a sweet
little calculation

1602
01:17:13,240 --> 01:17:14,530
to take this expression.

1603
01:17:14,530 --> 01:17:17,510
Plug it in and derive
this recursion relation,

1604
01:17:17,510 --> 01:17:23,100
which is root-- or 2 root--
epsilon times j plus l

1605
01:17:23,100 --> 01:17:33,340
plus 1 minus 1 over j plus
1 j plus 2 l plus 2 aj.

1606
01:17:38,570 --> 01:17:40,990
So, here's our series expansion.

1607
01:17:40,990 --> 01:17:49,130
And in order for
this terminate, we

1608
01:17:49,130 --> 01:17:54,120
must have that some aj
max plus 1 is equal to 0.

1609
01:17:54,120 --> 01:17:56,210
So, one of these guys
must eventually vanish.

1610
01:17:56,210 --> 01:17:58,260
And the only thing's that's
changing is little j.

1611
01:17:58,260 --> 01:18:01,700
So, what that tells us is
that for some maximum value

1612
01:18:01,700 --> 01:18:05,750
of little j, root 2 epsilon
times j maximum plus l plus 1

1613
01:18:05,750 --> 01:18:07,700
is equal to minus 1.

1614
01:18:07,700 --> 01:18:10,140
But that gives us a
relationship between overall j

1615
01:18:10,140 --> 01:18:13,074
max, little l, and the energy.

1616
01:18:13,074 --> 01:18:14,740
And if you go through,
what you discover

1617
01:18:14,740 --> 01:18:21,460
is that the energy is equal
to 1 over 4 n squared, where

1618
01:18:21,460 --> 01:18:27,370
n is equal to j
max plus l plus 1.

1619
01:18:30,010 --> 01:18:35,700
And what this tells is
that the energy is labeled

1620
01:18:35,700 --> 01:18:40,010
by an integer, n, and an
integer, l, and an integer,

1621
01:18:40,010 --> 01:18:42,450
m-- these are from the
spherical harmonics,

1622
01:18:42,450 --> 01:18:44,640
and n came from the
series expansion--

1623
01:18:44,640 --> 01:18:51,895
and it's equal to minus e0 over
4 n squared, independent of l

1624
01:18:51,895 --> 01:18:52,395
and m.

1625
01:18:58,680 --> 01:19:01,310
And so, by solving the
differential equation exactly,

1626
01:19:01,310 --> 01:19:05,400
which in this case we kind of
amazingly can, what we discover

1627
01:19:05,400 --> 01:19:09,420
is that the energy eigenvalues
are, indeed, exactly 1/4 of e0.

1628
01:19:12,229 --> 01:19:14,020
And they're spaced with
a 1 over n squared,

1629
01:19:14,020 --> 01:19:15,040
which does two things.

1630
01:19:15,040 --> 01:19:17,150
Not only does that
explain-- so, let's think

1631
01:19:17,150 --> 01:19:19,790
about the consequence of this
very briefly-- not only does

1632
01:19:19,790 --> 01:19:25,440
that explain the minus 13.6
eV, not only does that explain

1633
01:19:25,440 --> 01:19:28,650
the binding energy of
hydrogen as is observed,

1634
01:19:28,650 --> 01:19:29,460
that it does more.

1635
01:19:29,460 --> 01:19:31,960
Remember in the very beginning
one of the experimental facts

1636
01:19:31,960 --> 01:19:33,584
we wanted to explain
about the universe

1637
01:19:33,584 --> 01:19:43,320
was that the spectrum
of light of hydrogen

1638
01:19:43,320 --> 01:19:47,530
went like 30 over 4 n squared.

1639
01:19:51,550 --> 01:19:53,040
This was the Rydberg relation.

1640
01:19:56,400 --> 01:19:58,600
And now we see explicitly.

1641
01:19:58,600 --> 01:20:00,730
So, we've solved
for that expansion.

1642
01:20:00,730 --> 01:20:02,180
But there's a real puzzle here.

1643
01:20:05,350 --> 01:20:07,512
Purely on very
general grounds, we

1644
01:20:07,512 --> 01:20:09,970
derived earlier that when you
have a rotationally invariant

1645
01:20:09,970 --> 01:20:11,664
potential-- a
central potential--

1646
01:20:11,664 --> 01:20:13,080
every energy should
be degenerate,

1647
01:20:13,080 --> 01:20:14,860
with degeneracy 2l plus 1.

1648
01:20:14,860 --> 01:20:18,530
It can depend on l, but it
must be independent of m.

1649
01:20:18,530 --> 01:20:20,190
But here, we've
discovered-- first off,

1650
01:20:20,190 --> 01:20:21,840
we've fit a nice bit
of experimental data,

1651
01:20:21,840 --> 01:20:24,048
but we've discovered the
energy is, in fact, not just

1652
01:20:24,048 --> 01:20:27,370
independent of m, but it's
independent of l, too.

1653
01:20:27,370 --> 01:20:28,530
Why?

1654
01:20:28,530 --> 01:20:33,260
What symmetry is explaining
this extra degeneracy?

1655
01:20:33,260 --> 01:20:35,720
We'll pick that up next time.