1
00:00:00,050 --> 00:00:01,670
The following
content is provided

2
00:00:01,670 --> 00:00:03,810
under a Creative
Commons license.

3
00:00:03,810 --> 00:00:06,540
Your support will help MIT
OpenCourseWare continue

4
00:00:06,540 --> 00:00:10,120
to offer high quality
educational resources for free.

5
00:00:10,120 --> 00:00:12,700
To make a donation or to
view additional materials

6
00:00:12,700 --> 00:00:16,570
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:16,570 --> 00:00:17,205
at ocw.mit.edu.

8
00:00:21,220 --> 00:00:25,880
PROFESSOR: So today,
let me remind you,

9
00:00:25,880 --> 00:00:28,000
for the convenience
of also the people

10
00:00:28,000 --> 00:00:29,690
that weren't here
last time, we don't

11
00:00:29,690 --> 00:00:32,600
need too much of what
we did last time, except

12
00:00:32,600 --> 00:00:34,970
to know, more or
less, what's going on.

13
00:00:34,970 --> 00:00:38,130
We were solving central
potential problems

14
00:00:38,130 --> 00:00:42,600
in which you have a potential
that just depends on r.

15
00:00:42,600 --> 00:00:46,230
And at the end of the
day, the wave functions

16
00:00:46,230 --> 00:00:51,080
were shown to take the
form of a radial part

17
00:00:51,080 --> 00:00:56,600
and an angular part with
the spherical harmonic here.

18
00:00:56,600 --> 00:00:59,980
The radial part was
very conveniently

19
00:00:59,980 --> 00:01:03,890
presented as a U
function divided by r.

20
00:01:03,890 --> 00:01:07,940
That's another function, but
the differential equation for U

21
00:01:07,940 --> 00:01:08,980
is nice.

22
00:01:08,980 --> 00:01:13,050
It takes the form of a
1-dimensional Schrodinger

23
00:01:13,050 --> 00:01:16,980
equation for a particle
under the influence

24
00:01:16,980 --> 00:01:19,030
of an effective potential.

25
00:01:19,030 --> 00:01:22,280
This potential,
effective potential,

26
00:01:22,280 --> 00:01:25,890
has the potential that you
have in your Hamiltonian

27
00:01:25,890 --> 00:01:29,630
plus an extra term, a barrier.

28
00:01:29,630 --> 00:01:33,190
It's a potential that
grows as r goes to 0,

29
00:01:33,190 --> 00:01:38,830
so it's a barrier that
explodes at r equals 0.

30
00:01:38,830 --> 00:01:42,110
And this being the
effective potential that

31
00:01:42,110 --> 00:01:45,710
enters into this 1-dimensional
Schrodinger equation,

32
00:01:45,710 --> 00:01:51,070
we made some observations
about this function U.

33
00:01:51,070 --> 00:01:53,530
The normalization of
this wave function

34
00:01:53,530 --> 00:01:59,920
is guaranteed if the integral of
U squared over r is equal to 1.

35
00:01:59,920 --> 00:02:02,320
So that's a pretty nice thing.

36
00:02:02,320 --> 00:02:07,300
U squared meaning absolute
value squared of U.

37
00:02:07,300 --> 00:02:15,550
And we also noticed that U
must go like r to the l plus 1

38
00:02:15,550 --> 00:02:18,540
near r going to 0.

39
00:02:18,540 --> 00:02:23,300
So those were the
general properties of U.

40
00:02:23,300 --> 00:02:25,890
I'm trying to catch
up with notes.

41
00:02:25,890 --> 00:02:30,360
I hope to put some
notes out today.

42
00:02:30,360 --> 00:02:33,680
But this material, in fact,
you can find parts of it

43
00:02:33,680 --> 00:02:35,440
in almost any book.

44
00:02:35,440 --> 00:02:37,970
It will just be presented
a little differently.

45
00:02:37,970 --> 00:02:42,860
But this is not
very unusual stuff.

46
00:02:42,860 --> 00:02:47,580
Now, the diagram that I wanted
to emphasize to you last time

47
00:02:47,580 --> 00:02:50,410
was that if you're
trying to discuss

48
00:02:50,410 --> 00:02:54,160
the spectrum of a
central state potential,

49
00:02:54,160 --> 00:02:56,090
you do it with a
diagram in which you

50
00:02:56,090 --> 00:02:59,641
list the energies
as a function of l.

51
00:02:59,641 --> 00:03:04,560
And it's like a histogram
in which for l equals 0,

52
00:03:04,560 --> 00:03:08,350
you have to solve some
1-dimensional Schrodinger

53
00:03:08,350 --> 00:03:09,770
equation.

54
00:03:09,770 --> 00:03:12,070
This 1-dimensional
Schrodinger equation

55
00:03:12,070 --> 00:03:17,050
will have a bound state
spectrum that is non-degenerate.

56
00:03:17,050 --> 00:03:20,090
So for l equal 0, there
will be one solution,

57
00:03:20,090 --> 00:03:21,450
two solutions, three.

58
00:03:21,450 --> 00:03:25,640
I don't know how many before
the continuous spectrum sets in,

59
00:03:25,640 --> 00:03:27,700
or if there is a
continuous spectrum.

60
00:03:27,700 --> 00:03:30,150
But there are some solutions.

61
00:03:30,150 --> 00:03:33,100
For l equal 1, there will
be some other solutions.

62
00:03:33,100 --> 00:03:36,060
For l equal 2, there might
be some other solutions.

63
00:03:36,060 --> 00:03:41,000
And that depends on which
problem you are solving.

64
00:03:41,000 --> 00:03:45,830
In general, there's no rhyme
or reason in this diagram,

65
00:03:45,830 --> 00:03:50,470
except that the lowest energy
state for each level goes up.

66
00:03:50,470 --> 00:03:54,560
And that's because the
potential goes up and up

67
00:03:54,560 --> 00:03:55,830
as you increase l.

68
00:03:55,830 --> 00:03:58,340
Notice this is totally positive.

69
00:03:58,340 --> 00:04:00,810
So whatever potential
you have, it's

70
00:04:00,810 --> 00:04:03,530
just going up as you increase l.

71
00:04:03,530 --> 00:04:08,620
So the ground
state should go up.

72
00:04:08,620 --> 00:04:10,980
The ground state
energy should go up.

73
00:04:10,980 --> 00:04:13,320
So this diagram looks like this.

74
00:04:13,320 --> 00:04:17,170
We also emphasized
that for every l,

75
00:04:17,170 --> 00:04:22,260
there are 2l plus 1 solutions
obtained by varying M,

76
00:04:22,260 --> 00:04:25,990
because M goes
from l to minus l.

77
00:04:25,990 --> 00:04:29,530
Therefore, this
bar here represents

78
00:04:29,530 --> 00:04:35,470
a single multiplate of l equals
1, therefore three states.

79
00:04:35,470 --> 00:04:39,990
This is a single multiplate of
l equals 1, three more states.

80
00:04:39,990 --> 00:04:43,240
Here is five
states, five states,

81
00:04:43,240 --> 00:04:49,100
but only one l equal 1
multiplate, one l equal

82
00:04:49,100 --> 00:04:51,730
1 multiplate, 1, 1.

83
00:04:51,730 --> 00:04:53,640
There are no cases
in which you have

84
00:04:53,640 --> 00:04:58,440
two multiplates because that
would contradict our known

85
00:04:58,440 --> 00:05:05,770
statement that the spectrum of
the potential of bound states

86
00:05:05,770 --> 00:05:08,990
in one dimensions
is non-degenerate.

87
00:05:08,990 --> 00:05:13,220
So that was one thing we did.

88
00:05:13,220 --> 00:05:15,250
And the other thing
that we concluded

89
00:05:15,250 --> 00:05:18,400
that ties up with what
I want to talk now

90
00:05:18,400 --> 00:05:25,495
was a discussion of the free
particle, free particle.

91
00:05:28,850 --> 00:05:30,870
And in the case
of a free particle

92
00:05:30,870 --> 00:05:32,830
you say, well, so
what are you solving?

93
00:05:32,830 --> 00:05:34,980
Well, we're solving
for solutions

94
00:05:34,980 --> 00:05:37,530
that have radial symmetry.

95
00:05:37,530 --> 00:05:41,200
So they are functions
of r [INAUDIBLE]

96
00:05:41,200 --> 00:05:43,150
angular distribution.

97
00:05:43,150 --> 00:05:51,450
So what do you find is UEl
of r is equal to rJl of kr,

98
00:05:51,450 --> 00:05:55,190
as we explained, where these
were the spherical Bessel

99
00:05:55,190 --> 00:05:55,690
functions.

100
00:05:58,860 --> 00:06:05,570
And those are not as bad as
the usual Bessel functions,

101
00:06:05,570 --> 00:06:06,870
not that complicated.

102
00:06:06,870 --> 00:06:10,220
They're finite
series constructed

103
00:06:10,220 --> 00:06:16,590
with sines and cosines, so
these are quite tractable.

104
00:06:16,590 --> 00:06:18,380
And that was for
a free particle.

105
00:06:18,380 --> 00:06:20,620
So we decided that
we would solve

106
00:06:20,620 --> 00:06:33,310
the case of an infinite
spherical well, which

107
00:06:33,310 --> 00:06:44,492
is a potential V of r, which is
equal to 0 if r is less than a,

108
00:06:44,492 --> 00:06:48,846
and infinity if r is
greater or equal than a.

109
00:06:48,846 --> 00:06:53,400
It's a small-- well, a
is whatever size it is.

110
00:06:53,400 --> 00:06:58,810
It's a cavity, spherical
cavity where you can live.

111
00:06:58,810 --> 00:07:01,775
And outside you can't be there.

112
00:07:01,775 --> 00:07:04,980
This is the analog of
the infinite square

113
00:07:04,980 --> 00:07:08,090
well in one dimension.

114
00:07:08,090 --> 00:07:09,610
But this is in three dimensions.

115
00:07:09,610 --> 00:07:12,420
An infinite
spherical well should

116
00:07:12,420 --> 00:07:20,320
be imagined as some sort of hole
in the material and electrons

117
00:07:20,320 --> 00:07:26,380
or particles can move inside
and nothing can escape this.

118
00:07:26,380 --> 00:07:34,660
So this is a hollow thing.

119
00:07:34,660 --> 00:07:39,450
So this is a classic problem.

120
00:07:39,450 --> 00:07:42,480
You would say this must
be as simple to solve

121
00:07:42,480 --> 00:07:45,950
as the infinite square well.

122
00:07:45,950 --> 00:07:49,320
And no, it's more complicated.

123
00:07:49,320 --> 00:07:54,390
Not conceptually much more
complicated, but mathematically

124
00:07:54,390 --> 00:07:55,145
more work.

125
00:07:57,710 --> 00:08:03,200
You will consider some aspects
of the finite spherical

126
00:08:03,200 --> 00:08:05,790
well in the homework.

127
00:08:05,790 --> 00:08:09,270
The finite square
well, you remember,

128
00:08:09,270 --> 00:08:11,050
is a bit more complicated.

129
00:08:11,050 --> 00:08:13,020
You can't solve it exactly.

130
00:08:13,020 --> 00:08:16,710
The finite spherical
well, of course,

131
00:08:16,710 --> 00:08:19,850
you can't solve exactly either.

132
00:08:19,850 --> 00:08:23,260
But you will look at
some aspects of it,

133
00:08:23,260 --> 00:08:27,870
the most famous result
of which is the statement

134
00:08:27,870 --> 00:08:32,430
that while any attractive
potential in one dimension

135
00:08:32,430 --> 00:08:36,679
has a bound state
in three dimensions.

136
00:08:36,679 --> 00:08:42,530
An attractive potential,
so a finite spherical well,

137
00:08:42,530 --> 00:08:46,220
may not have a bound state,
even a single bound state.

138
00:08:46,220 --> 00:08:48,990
So that's a very
interesting thing

139
00:08:48,990 --> 00:08:54,360
that you will understand in
the homework in several ways.

140
00:08:54,360 --> 00:08:57,520
You will also understand some
things about delta functions,

141
00:08:57,520 --> 00:08:58,600
that they're important.

142
00:08:58,600 --> 00:09:04,320
So we'll touch base with that.

143
00:09:04,320 --> 00:09:08,840
So that's as far as I got
last time and just a review.

144
00:09:08,840 --> 00:09:12,350
If there are any
questions, don't be shy

145
00:09:12,350 --> 00:09:15,765
if you weren't here and
you have a question.

146
00:09:15,765 --> 00:09:16,265
Yes.

147
00:09:16,265 --> 00:09:21,601
AUDIENCE: Is there any reason to
expect [INAUDIBLE] intuitively

148
00:09:21,601 --> 00:09:25,740
should be like [INAUDIBLE]?

149
00:09:25,740 --> 00:09:29,720
PROFESSOR: Well, the reason,
intuitively the reason

150
00:09:29,720 --> 00:09:37,810
is basically the conspiracy
between this UEl,

151
00:09:37,810 --> 00:09:47,810
as I was saying,
UEl as r goes to 0

152
00:09:47,810 --> 00:09:51,080
goes like r to the l plus 1.

153
00:09:51,080 --> 00:09:58,650
So first of all, this
potential is very repulsive.

154
00:09:58,650 --> 00:09:59,700
Is that right?

155
00:09:59,700 --> 00:10:02,130
So that tends to ruin things.

156
00:10:02,130 --> 00:10:06,260
So you could say, oh, well,
this thing is probably not

157
00:10:06,260 --> 00:10:08,770
going to get anything
because near r equal 0,

158
00:10:08,770 --> 00:10:09,830
you're being repelled.

159
00:10:09,830 --> 00:10:13,480
But you cay say, no, let's
look at that l equal 0.

160
00:10:13,480 --> 00:10:16,950
So you don't have
that, so just V of r.

161
00:10:16,950 --> 00:10:25,630
But we take l equals 0--
I'm sorry, U here, U of El

162
00:10:25,630 --> 00:10:27,270
has to go like that.

163
00:10:27,270 --> 00:10:31,810
So actually, U will
vanish for r equals 0.

164
00:10:31,810 --> 00:10:37,500
So the effective potential
for the 1-dimensional problem

165
00:10:37,500 --> 00:10:41,850
may look like a finite square
well, that is like that.

166
00:10:41,850 --> 00:10:47,140
But the wave function has
to vanish on this side.

167
00:10:47,140 --> 00:10:53,970
Even though you would say,
it's a finite spherical well,

168
00:10:53,970 --> 00:10:56,280
why does it have to
vanish Here well,

169
00:10:56,280 --> 00:11:00,090
it's the unusual behavior
of this U function.

170
00:11:00,090 --> 00:11:03,790
So the wave function that
you can sort of imagine

171
00:11:03,790 --> 00:11:05,640
must vanish here.

172
00:11:05,640 --> 00:11:07,710
So in order to
get a bound state,

173
00:11:07,710 --> 00:11:10,080
it has to have enough
time to sort of curve

174
00:11:10,080 --> 00:11:14,090
so that it can fall, and it's
sometimes difficult to do it.

175
00:11:14,090 --> 00:11:16,680
So basically, it's the fact
that the wave function has

176
00:11:16,680 --> 00:11:21,370
to vanish at the origin, the
U wave function has to vanish.

177
00:11:21,370 --> 00:11:23,200
Now, the whole wave
function doesn't

178
00:11:23,200 --> 00:11:26,380
vanish because
it's divided by r.

179
00:11:26,380 --> 00:11:27,620
But the U does.

180
00:11:27,620 --> 00:11:34,980
So it's the reason why you don't
have bound states in general.

181
00:11:34,980 --> 00:11:38,470
And then there's also funny
things like a delta function.

182
00:11:38,470 --> 00:11:41,850
You would say, well, a
3-dimensional delta function,

183
00:11:41,850 --> 00:11:45,160
how many bound states do
you get, or what's going on?

184
00:11:45,160 --> 00:11:47,820
With a 1-dimensional
delta function,

185
00:11:47,820 --> 00:11:51,290
you have one bound
state, and that's it.

186
00:11:51,290 --> 00:11:54,930
With a 3-dimensional delta
function, as you will find,

187
00:11:54,930 --> 00:11:59,690
it's [INAUDIBLE]
is rather singular,

188
00:11:59,690 --> 00:12:03,610
and you tend to get
infinitely many bound states.

189
00:12:03,610 --> 00:12:08,110
And you cannot even calculate
them because they fall off all

190
00:12:08,110 --> 00:12:14,510
the way through r and go
to minus infinity energy.

191
00:12:14,510 --> 00:12:17,230
It's a rather strange situation.

192
00:12:17,230 --> 00:12:18,030
All right.

193
00:12:18,030 --> 00:12:19,045
Any other questions?

194
00:12:26,740 --> 00:12:31,105
So let's do this
infinite spherical well.

195
00:12:33,650 --> 00:12:38,590
Now, the reason we did
the free particle first

196
00:12:38,590 --> 00:12:41,740
was that inside here,
this is all free,

197
00:12:41,740 --> 00:12:45,830
so the solutions will
be sort of simple.

198
00:12:45,830 --> 00:12:52,820
Nevertheless, we can begin with
looking at the differential

199
00:12:52,820 --> 00:12:57,280
equation directly for inside.

200
00:12:57,280 --> 00:13:09,470
So r less than a, you would
have minus d second UEl

201
00:13:09,470 --> 00:13:15,240
over d rho squared,
actually, plus l times l

202
00:13:15,240 --> 00:13:21,310
plus 1 over rho
squared UEl equals

203
00:13:21,310 --> 00:13:26,690
UEl, where rho is equal to kr.

204
00:13:26,690 --> 00:13:30,740
And k-- I'm sorry, I
didn't write it there--

205
00:13:30,740 --> 00:13:35,330
is 2mE over h squared as usual.

206
00:13:38,110 --> 00:13:41,660
So here I didn't say what k was.

207
00:13:41,660 --> 00:13:46,430
That was 2mE over h squared.

208
00:13:49,460 --> 00:13:53,760
And this doesn't quite look
like the differential equation

209
00:13:53,760 --> 00:13:54,570
you have here.

210
00:13:58,930 --> 00:14:02,010
Well, V of r is 0
for r less than a,

211
00:14:02,010 --> 00:14:04,230
so you just have this term.

212
00:14:04,230 --> 00:14:07,570
The h squared's
over 2m and the E

213
00:14:07,570 --> 00:14:12,430
have been rescaled
by changing r to rho.

214
00:14:12,430 --> 00:14:14,320
So the differential
equation becomes

215
00:14:14,320 --> 00:14:18,460
simple and looking like this.

216
00:14:18,460 --> 00:14:22,540
So that was a manipulation that
was done in detail last time,

217
00:14:22,540 --> 00:14:24,530
but you can redo it.

218
00:14:24,530 --> 00:14:29,550
Now, this, as I mentioned,
is not a simple differential

219
00:14:29,550 --> 00:14:30,240
equation.

220
00:14:30,240 --> 00:14:33,540
If you didn't have this, it
would have a power solution.

221
00:14:33,540 --> 00:14:38,440
If you don't have this,
it's just a sine or cosines.

222
00:14:38,440 --> 00:14:41,040
But if you have
both, it's Bessel.

223
00:14:41,040 --> 00:14:43,040
So having a
differential with two

224
00:14:43,040 --> 00:14:48,170
derivatives, 1 over
rho squared and 1,

225
00:14:48,170 --> 00:14:50,880
brings you into
Bessel territory.

226
00:14:50,880 --> 00:14:53,450
Anyway, this is
the equation that,

227
00:14:53,450 --> 00:14:56,530
in fact, is solved
by these functions

228
00:14:56,530 --> 00:15:00,920
because it's a free
Schrodinger equation,

229
00:15:00,920 --> 00:15:05,260
and you can take
it for l equal 0.

230
00:15:05,260 --> 00:15:08,550
This is the only
case we can do easily

231
00:15:08,550 --> 00:15:11,100
without looking up
any Bessel functions

232
00:15:11,100 --> 00:15:12,710
or anything like that.

233
00:15:12,710 --> 00:15:20,256
You then have d second UE0
d rho squared is equal UE0.

234
00:15:22,910 --> 00:15:39,060
And therefore, UE0 goes like A
sine of rho plus B cosine rho.

235
00:15:39,060 --> 00:15:42,290
Rho is kr.

236
00:15:42,290 --> 00:15:46,570
UEl must behave like
r to the l plus 1,

237
00:15:46,570 --> 00:15:49,700
so UE0 must behave like r.

238
00:15:49,700 --> 00:15:54,270
So for this thing to
behave, must behave like r.

239
00:15:54,270 --> 00:15:59,980
So it must behave like
rho as rho goes to 0.

240
00:15:59,980 --> 00:16:03,510
Therefore, this term
cannot be there.

241
00:16:03,510 --> 00:16:10,920
The only solution is
UE0 is equal to sine

242
00:16:10,920 --> 00:16:14,260
of rho, which is kr.

243
00:16:14,260 --> 00:16:21,990
So UE0 of r must
be of this form.

244
00:16:21,990 --> 00:16:28,060
Then in order to have a
solution of the 1-dimensional

245
00:16:28,060 --> 00:16:30,610
Schrodinger equation, it's
true that the potential

246
00:16:30,610 --> 00:16:33,070
becomes infinite for r equal a.

247
00:16:33,070 --> 00:16:34,880
So that is familiar.

248
00:16:34,880 --> 00:16:38,380
It's not the point r
equal 0 that is unusual.

249
00:16:38,380 --> 00:16:41,520
r equal a, this must vanish.

250
00:16:41,520 --> 00:16:50,280
So we need that UE0
of a will equal to 0.

251
00:16:50,280 --> 00:17:04,040
So this requires k equal some
kn so that kna is equal to n pi.

252
00:17:04,040 --> 00:17:10,599
So for k is equal to kn,
where kn,a is equal to n pi,

253
00:17:10,599 --> 00:17:17,010
a multiple of pi, then the
wave function will vanish at r

254
00:17:17,010 --> 00:17:17,990
equals a.

255
00:17:20,829 --> 00:17:23,050
So easy enough.

256
00:17:23,050 --> 00:17:25,220
We've found the values of k.

257
00:17:25,220 --> 00:17:31,150
This is quite analogous to
the infinite square well.

258
00:17:31,150 --> 00:17:37,690
And now the energies
from this formula En

259
00:17:37,690 --> 00:17:44,060
will be equal to h squared
kn squared over 2m.

260
00:17:44,060 --> 00:17:46,950
And it's convenient,
of course, to divide

261
00:17:46,950 --> 00:17:55,240
by ma squared so that
you have kna squared.

262
00:17:55,240 --> 00:17:59,636
So the energies are h
squared over 2ma squared.

263
00:18:03,790 --> 00:18:08,870
Here we have n pi squared.

264
00:18:08,870 --> 00:18:10,370
I'll put them like this.

265
00:18:10,370 --> 00:18:19,360
En,0 for l equal
0, En,l's energies.

266
00:18:19,360 --> 00:18:23,020
Now, if you want to
remember something

267
00:18:23,020 --> 00:18:25,470
about this, of course,
all these constants

268
00:18:25,470 --> 00:18:28,340
are kind of irrelevant.

269
00:18:28,340 --> 00:18:33,070
But the good thing
is that this carries

270
00:18:33,070 --> 00:18:35,230
the full units of energy.

271
00:18:35,230 --> 00:18:38,950
And you know in a system
with length scale a,

272
00:18:38,950 --> 00:18:41,140
this is the typical energy.

273
00:18:41,140 --> 00:18:45,560
So the energies are essentially
that typical energy times

274
00:18:45,560 --> 00:18:47,620
n squared pi squared.

275
00:18:47,620 --> 00:18:56,530
So it's convenient to define,
in general, En,l to be En,l,

276
00:18:56,530 --> 00:18:59,490
for any l that you
may be solving,

277
00:18:59,490 --> 00:19:03,693
divided by h squared
over 2ma squared.

278
00:19:07,030 --> 00:19:09,970
So that this thing has no units.

279
00:19:09,970 --> 00:19:14,260
And it tells you for any
level, the calligraphic E,

280
00:19:14,260 --> 00:19:19,300
roughly how much bigger it
is than the natural energy

281
00:19:19,300 --> 00:19:22,020
scale of your problem.

282
00:19:22,020 --> 00:19:24,230
So it's a nice definition.

283
00:19:24,230 --> 00:19:32,660
And in this way, we've learned
that En,0 is equal to n pi

284
00:19:32,660 --> 00:19:35,510
squared.

285
00:19:35,510 --> 00:19:54,640
And a few values of this are
E1,0 about 9,869 [INAUDIBLE],

286
00:19:54,640 --> 00:20:12,940
E2,0 equal 39,478, and
E3,0 is equal 88,826.

287
00:20:12,940 --> 00:20:17,940
Not very dramatic
numbers, but they're

288
00:20:17,940 --> 00:20:21,700
still kind of interesting.

289
00:20:21,700 --> 00:20:25,930
So what else about this problem?

290
00:20:25,930 --> 00:20:29,740
Well, we can do
the general case.

291
00:20:29,740 --> 00:20:36,520
Let me erase a little here
so that we can proceed.

292
00:20:36,520 --> 00:20:40,470
The general case
is based on knowing

293
00:20:40,470 --> 00:20:44,710
the zeroes of this
spherical Bessel function.

294
00:20:44,710 --> 00:20:48,140
So this is something that the
first one you can do easily.

295
00:20:51,238 --> 00:20:57,200
The zeroes of J1
of rho are points

296
00:20:57,200 --> 00:21:01,370
at which tan rho
is equal to rho.

297
00:21:01,370 --> 00:21:06,520
That is a short calculation
if you ever want to do it.

298
00:21:06,520 --> 00:21:08,615
That's not that
difficult, of course,

299
00:21:08,615 --> 00:21:11,210
but you have to
do it numerically.

300
00:21:11,210 --> 00:21:14,840
So the zeroes of
the Bessel functions

301
00:21:14,840 --> 00:21:17,100
are known and are tabulated.

302
00:21:17,100 --> 00:21:22,440
You can find them on the web,
little programs that do it

303
00:21:22,440 --> 00:21:24,760
on the web and give you
[? directly those ?] zeroes.

304
00:21:24,760 --> 00:21:27,000
So how are they defined?

305
00:21:27,000 --> 00:21:39,850
Basically, people define Zn,l to
be the n-th zero with n equals

306
00:21:39,850 --> 00:21:45,770
1 like that of Jl.

307
00:21:45,770 --> 00:21:54,050
So more precisely, Jl
of Zn,l is equal to 0.

308
00:21:54,050 --> 00:21:57,590
And all the Z and l's
are different from 0.

309
00:21:57,590 --> 00:22:02,000
There's a trivial zero at 0.

310
00:22:02,000 --> 00:22:05,160
And nevertheless,
that is not counted.

311
00:22:05,160 --> 00:22:09,170
It's just too trivial
for it to be interesting.

312
00:22:09,170 --> 00:22:16,070
So these numbers, Z and
l, are basically it.

313
00:22:16,070 --> 00:22:16,620
Why?

314
00:22:16,620 --> 00:22:19,600
Because what you
need is, if you're

315
00:22:19,600 --> 00:22:28,000
looking for the l-th solution,
you need UEl of a equal 0.

316
00:22:30,880 --> 00:22:44,260
And UEa of that equal 0 means
that you need kn,l times a be

317
00:22:44,260 --> 00:22:45,300
equal to Zn,l.

318
00:22:48,110 --> 00:22:54,960
So kn,l is the value of k.

319
00:22:54,960 --> 00:23:00,310
And just like we quantized
here, we had kn, well,

320
00:23:00,310 --> 00:23:03,980
if you have various
l's, put the kn,l.

321
00:23:03,980 --> 00:23:11,410
So for every value of l, you
have kn,l's that are given

322
00:23:11,410 --> 00:23:12,360
by this.

323
00:23:12,360 --> 00:23:17,905
And the energy's like this.

324
00:23:21,620 --> 00:23:24,310
Let me copy what this would be.

325
00:23:27,070 --> 00:23:40,290
En,l would be En,l
over this ratio.

326
00:23:40,290 --> 00:23:47,880
And En,l h squared, well,
let me do it this way.

327
00:23:47,880 --> 00:23:49,530
I'm sorry.

328
00:23:49,530 --> 00:24:02,190
En,l would be h squared
kn,l over 2ma squared,

329
00:24:02,190 --> 00:24:04,200
over 2m like that.

330
00:24:04,200 --> 00:24:07,110
Then you multiply
by a squared again.

331
00:24:07,110 --> 00:24:16,900
So you get kn,l a
squared over 2ma squared.

332
00:24:16,900 --> 00:24:25,020
So what you learn from
this is that En,l,

333
00:24:25,020 --> 00:24:32,206
you divide this by that, is
just kn,l times a, which is Zm,l

334
00:24:32,206 --> 00:24:32,706
squared.

335
00:24:36,160 --> 00:24:38,440
So that's the simple result.

336
00:24:38,440 --> 00:24:44,740
The En,l's are just the squares
of the zeroes of the Bessel

337
00:24:44,740 --> 00:24:45,240
function.

338
00:24:49,080 --> 00:24:52,200
So you divide it again
by h squared over 2ma,

339
00:24:52,200 --> 00:24:54,540
and that's all that was left.

340
00:24:54,540 --> 00:24:57,450
So you need to know the
zeroes of the Bessel function.

341
00:24:57,450 --> 00:25:01,050
And there's one,
you might say, well,

342
00:25:01,050 --> 00:25:02,960
what for do I care about this?

343
00:25:02,960 --> 00:25:07,830
But it's kind of
nice to see them.

344
00:25:07,830 --> 00:25:20,360
So Z1,1 is equal to 4.49
Z2,1 is equal to 7.72,

345
00:25:20,360 --> 00:25:32,420
and Z3,1 is 10.90, numbers that
may have no rhyme or reason.

346
00:25:32,420 --> 00:25:35,330
Now, you've done
here l equals 1.

347
00:25:35,330 --> 00:25:38,890
Of course, it continues
down, down, down.

348
00:25:38,890 --> 00:25:43,140
You can continue with the first
zero, first nontrivial zero,

349
00:25:43,140 --> 00:25:45,990
second nontrivial zero,
third nontrivial zero,

350
00:25:45,990 --> 00:25:48,070
and it goes on.

351
00:25:48,070 --> 00:25:50,290
The energies the squares.

352
00:25:50,290 --> 00:26:01,790
So the squared goes like 20.19.

353
00:26:01,790 --> 00:26:08,580
This goes like 59.7.

354
00:26:08,580 --> 00:26:12,320
And this goes like 119 roughly.

355
00:26:16,500 --> 00:26:19,810
Then you have the other zeroes.

356
00:26:19,810 --> 00:26:28,680
First zero for l equals
2, that is 5.76 roughly.

357
00:26:28,680 --> 00:26:37,670
Second zero for l equals
to 2 is 9.1 roughly.

358
00:26:40,800 --> 00:26:45,780
And if you square those to
see those other energies,

359
00:26:45,780 --> 00:26:59,750
you would get, by
squaring, 33.21 and 82.72.

360
00:26:59,750 --> 00:27:04,320
And finally, let me do one more.

361
00:27:04,320 --> 00:27:11,640
Z1,3, the first zero of the
l equal 3, and the Z2,3,

362
00:27:11,640 --> 00:27:21,450
the second zero,
are 6.99 and 10.4,

363
00:27:21,450 --> 00:27:33,890
which when squared give
you 48.83 and 108.5.

364
00:27:33,890 --> 00:27:34,390
OK.

365
00:27:37,180 --> 00:27:40,240
Why do you want to
see those numbers?

366
00:27:40,240 --> 00:27:42,120
I think the reason
you want to see

367
00:27:42,120 --> 00:27:45,710
them is to just look at
the diagram of energies,

368
00:27:45,710 --> 00:27:48,660
which is kind of interesting.

369
00:27:48,660 --> 00:27:51,910
So let's do that.

370
00:27:51,910 --> 00:27:55,960
So here I'll plot
energies, and here I put l.

371
00:27:58,750 --> 00:28:02,740
And now I need a big diagram.

372
00:28:02,740 --> 00:28:05,710
Here I'll put the
curly energies.

373
00:28:05,710 --> 00:28:18,350
And here is 10, 20, 30,
40, 50, 60-- and now

374
00:28:18,350 --> 00:28:31,382
I need the next blackboard,
let's see, we're 60,

375
00:28:31,382 --> 00:28:35,420
let's see, more or less,
here is about right--

376
00:28:35,420 --> 00:28:48,110
70, 80, 90, 100, 110.

377
00:28:48,110 --> 00:28:49,255
How far do I need?

378
00:28:51,810 --> 00:28:57,840
120, ooh, OK, 120.

379
00:28:57,840 --> 00:28:59,590
There we go.

380
00:28:59,590 --> 00:29:10,330
So just for the fun of it.

381
00:29:10,330 --> 00:29:13,110
Look at them to
see how they look,

382
00:29:13,110 --> 00:29:14,820
if you can see any pattern.

383
00:29:14,820 --> 00:29:24,070
So the first energy was
986, so that's roughly here.

384
00:29:24,070 --> 00:29:27,360
That's l equals 0
is the first state.

385
00:29:27,360 --> 00:29:32,395
Second is 39.47, so it's
a little below here.

386
00:29:34,990 --> 00:29:44,680
Next is 88.82, so we
are here, roughly.

387
00:29:47,550 --> 00:29:50,670
Then we go l equals 1.

388
00:29:50,670 --> 00:29:53,010
What are the values?

389
00:29:53,010 --> 00:30:00,580
This one's 20.19.

390
00:30:00,580 --> 00:30:06,950
L equals 1, 20.19,
so we're around here.

391
00:30:06,950 --> 00:30:13,410
Then 59.7 is almost 60.

392
00:30:13,410 --> 00:30:19,043
And then 119, so that's why
we needed to go that high.

393
00:30:24,610 --> 00:30:25,720
So here we are.

394
00:30:33,360 --> 00:30:45,760
And then l equals 3, you
have 48.83, so that's 50.

395
00:30:45,760 --> 00:30:49,888
I'm sorry, l equals 2.

396
00:30:49,888 --> 00:30:52,590
48.83.

397
00:30:52,590 --> 00:30:54,140
A little lower than that.

398
00:31:00,190 --> 00:31:00,900
No, I'm sorry.

399
00:31:00,900 --> 00:31:03,340
It's 33.21.

400
00:31:03,340 --> 00:31:06,490
I'm misreading that.

401
00:31:06,490 --> 00:31:11,310
33 over here.

402
00:31:11,310 --> 00:31:23,910
And then 82.72, so we are here.

403
00:31:23,910 --> 00:31:30,470
And then l equals
3, we have 48.83,

404
00:31:30,470 --> 00:31:51,640
so that was the one
I wanted, and 108.5.

405
00:31:51,640 --> 00:31:57,980
That's it, and there's
no pattern whatsoever.

406
00:31:57,980 --> 00:32:00,120
The zeroes never match.

407
00:32:00,120 --> 00:32:07,140
The only thing that is
true is that 0, 1, 2, 3,

408
00:32:07,140 --> 00:32:09,900
they were ascending
as we predicted.

409
00:32:09,900 --> 00:32:15,300
But no level matches
with any other level.

410
00:32:15,300 --> 00:32:18,700
If you were trying to say, OK,
this potential is interesting,

411
00:32:18,700 --> 00:32:24,110
is special, it has magic to
it, a spherical square well,

412
00:32:24,110 --> 00:32:27,030
it doesn't seem to have
anything to it, in fact.

413
00:32:27,030 --> 00:32:30,480
It's totally random.

414
00:32:30,480 --> 00:32:35,550
I cannot prove for you,
but it's probably true,

415
00:32:35,550 --> 00:32:38,860
and probably not
impossible to prove,

416
00:32:38,860 --> 00:32:44,010
that these zeroes
are never the same.

417
00:32:44,010 --> 00:32:47,990
No l and l prime will
have the same zero.

418
00:32:52,920 --> 00:32:59,190
No degeneracy ever occurs
that needs an explanation.

419
00:32:59,190 --> 00:33:05,010
For example, this state could
have ended up equal to this one

420
00:33:05,010 --> 00:33:08,020
or equal to this one,
and it doesn't happen.

421
00:33:08,020 --> 00:33:10,760
And that's OK,
because at this level,

422
00:33:10,760 --> 00:33:13,430
we would not be able to
predict why it happened.

423
00:33:13,430 --> 00:33:20,980
We actually, apart from the fact
that this a round, nice box,

424
00:33:20,980 --> 00:33:23,620
what symmetries does
it have, that box,

425
00:33:23,620 --> 00:33:25,340
except rotational symmetry?

426
00:33:25,340 --> 00:33:28,590
Nothing all that dramatic.

427
00:33:28,590 --> 00:33:33,700
So you would say, OK, let's
look for a problem, which we'll

428
00:33:33,700 --> 00:33:40,290
deal now, that does have a
more surprising structure,

429
00:33:40,290 --> 00:33:41,965
and let's try to figure it out.

430
00:33:45,570 --> 00:33:52,500
Let's try the three dimensional
harmonic oscillator.

431
00:33:52,500 --> 00:33:56,980
So 3D SHO.

432
00:33:59,804 --> 00:34:00,303
Isotropic.

433
00:34:04,510 --> 00:34:06,180
What is the potential?

434
00:34:06,180 --> 00:34:10,689
It's 1/2 m omega
squared x squared

435
00:34:10,689 --> 00:34:15,750
for x plus y squared plus z
squared, all the same constant.

436
00:34:15,750 --> 00:34:21,532
So it's 1/2 m omega
squared r squared.

437
00:34:21,532 --> 00:34:24,690
You would say, this
potential may or may not

438
00:34:24,690 --> 00:34:26,880
be nicer than the
spherical well,

439
00:34:26,880 --> 00:34:30,820
but actually, it
is extraordinarily

440
00:34:30,820 --> 00:34:34,949
symmetric in a way that
the spherical well is not.

441
00:34:34,949 --> 00:34:36,640
So we'll see why is that.

442
00:34:39,980 --> 00:34:41,880
Let's look at the
states of this.

443
00:34:41,880 --> 00:34:45,120
Now, we're going to
do it with numerology.

444
00:34:45,120 --> 00:34:47,239
Everything will be
kind of numerology

445
00:34:47,239 --> 00:34:50,050
here because I don't
want to calculate

446
00:34:50,050 --> 00:34:52,639
things for this problem.

447
00:34:52,639 --> 00:34:57,010
So first thing, how
you build the spectrum?

448
00:34:57,010 --> 00:35:11,810
H is equal to h bar omega
N1 plus N2 plus N3 plus 3/2,

449
00:35:11,810 --> 00:35:18,850
where these are the three
number operators, and 0.

450
00:35:18,850 --> 00:35:23,920
Now, just for you to realize,
in the language of things

451
00:35:23,920 --> 00:35:27,850
that we've been doing,
what is the state space?

452
00:35:27,850 --> 00:35:42,370
If we call H1 the state space
of a one dimensional SHO,

453
00:35:42,370 --> 00:35:47,550
what is the state space of
the three dimensional SHO?

454
00:35:51,630 --> 00:36:00,410
Well, conceptually, how do you
build a three dimensional SHO?

455
00:36:00,410 --> 00:36:05,185
Well, you have the creation
annihilation operators

456
00:36:05,185 --> 00:36:09,040
that you had for
the x, y, and z.

457
00:36:09,040 --> 00:36:17,450
So you have the ax dagger, the
ay dagger, and the az dagger,

458
00:36:17,450 --> 00:36:21,640
and you could act on the vacuum.

459
00:36:21,640 --> 00:36:26,460
So the way you can think of
the state space of the one

460
00:36:26,460 --> 00:36:30,250
dimensional oscillator is this
is one dimensional oscillator

461
00:36:30,250 --> 00:36:32,200
and I have all these things.

462
00:36:32,200 --> 00:36:33,850
Here is the other
one dimensional,

463
00:36:33,850 --> 00:36:35,900
here is the last
one dimensional.

464
00:36:35,900 --> 00:36:40,570
But if I want to build a
state of the three dimensional

465
00:36:40,570 --> 00:36:45,272
oscillator, I have to say how
many ax's, how many ay's, how

466
00:36:45,272 --> 00:36:46,890
many az's.

467
00:36:46,890 --> 00:36:52,350
So you're really
multiplying the states

468
00:36:52,350 --> 00:36:55,110
in the sense of tensor products.

469
00:36:55,110 --> 00:37:03,616
So the H, for a 3D SHO,
is the tensor product

470
00:37:03,616 --> 00:37:11,900
of three H1's, the H1
x, the 1y, and the z.

471
00:37:11,900 --> 00:37:14,190
You're really multiplying
all the states together.

472
00:37:14,190 --> 00:37:16,358
Yes?

473
00:37:16,358 --> 00:37:18,024
AUDIENCE: So this is
generalized to when

474
00:37:18,024 --> 00:37:19,360
you have a wave function
that's separable into products

475
00:37:19,360 --> 00:37:20,420
of different coordinates.

476
00:37:20,420 --> 00:37:22,087
Can you express those
as tensor products

477
00:37:22,087 --> 00:37:23,544
of the different
states, basically?

478
00:37:23,544 --> 00:37:25,410
BARTON ZWIEBACH: You
see, the separable

479
00:37:25,410 --> 00:37:29,170
into different coordinates,
it's yet another thing

480
00:37:29,170 --> 00:37:33,990
because it would be the
question of whether the state is

481
00:37:33,990 --> 00:37:37,880
separable or is entangled.

482
00:37:37,880 --> 00:37:40,610
If you choose, for example,
one term like that,

483
00:37:40,610 --> 00:37:52,710
a1, x, ax dagger, ay dagger, az
dagger, with two of those here,

484
00:37:52,710 --> 00:37:57,667
the wave function is the product
of an x wave function, a y wave

485
00:37:57,667 --> 00:37:59,000
function, and a z wave function.

486
00:37:59,000 --> 00:38:05,610
But if you add to this ax
dagger squared plus ay plus az,

487
00:38:05,610 --> 00:38:10,130
it will also be factorable,
but the sum is not factorable.

488
00:38:10,130 --> 00:38:13,700
So you get the
entanglement kind of thing.

489
00:38:13,700 --> 00:38:18,800
So this is the general
thing, and the basis vectors

490
00:38:18,800 --> 00:38:23,940
of this tensor product are
basis vectors of one times basis

491
00:38:23,940 --> 00:38:25,680
vectors of the
other basis vector.

492
00:38:25,680 --> 00:38:28,600
So basically, one
basis vector here,

493
00:38:28,600 --> 00:38:31,880
you pick some number of ax's,
some number of ay's, some

494
00:38:31,880 --> 00:38:33,840
number of az's.

495
00:38:33,840 --> 00:38:35,910
So this shows,
and it's a point I

496
00:38:35,910 --> 00:38:38,250
want to emphasize
at this moment,

497
00:38:38,250 --> 00:38:42,610
it's very important, that even
though we started thinking

498
00:38:42,610 --> 00:38:45,890
of tensor products of
two particles, here,

499
00:38:45,890 --> 00:38:49,585
there are no two particles in
the three dimensional harmonic

500
00:38:49,585 --> 00:38:52,130
oscillator, no three particles.

501
00:38:52,130 --> 00:38:56,610
There's just one particle where
there's one kind of attribute

502
00:38:56,610 --> 00:38:58,800
that that's doing in
the x direction, one

503
00:38:58,800 --> 00:39:00,670
kind of attribute
that it's doing

504
00:39:00,670 --> 00:39:03,920
in the y, one kind of attribute
that it's doing in the z.

505
00:39:03,920 --> 00:39:07,260
And therefore, you
need data for each one,

506
00:39:07,260 --> 00:39:10,680
and the right data is
the tensor product.

507
00:39:10,680 --> 00:39:13,150
You're just combining
them together.

508
00:39:13,150 --> 00:39:17,260
We mentioned that the basis
vectors of a tensor product

509
00:39:17,260 --> 00:39:21,010
are the products of
those basis vectors,

510
00:39:21,010 --> 00:39:25,440
of each one, so that's exactly
how you build states here.

511
00:39:25,440 --> 00:39:28,700
So I think, actually, you
probably have this intuition.

512
00:39:28,700 --> 00:39:36,140
I just wanted to make it a
little more explicit for you.

513
00:39:36,140 --> 00:39:39,320
So you don't need to
have two particles

514
00:39:39,320 --> 00:39:41,730
to get a tensor product.

515
00:39:41,730 --> 00:39:44,920
It can happen in simpler cases.

516
00:39:44,920 --> 00:39:49,240
So here's the thing that
I want to do with you.

517
00:39:49,240 --> 00:39:52,260
I would like to find that
diagram for the three

518
00:39:52,260 --> 00:39:54,370
dimensional SHO.

519
00:39:54,370 --> 00:39:57,740
That's our goal that we're
going to spend the next 15

520
00:39:57,740 --> 00:39:59,770
minutes probably doing.

521
00:39:59,770 --> 00:40:01,200
How does that diagram look?

522
00:40:03,730 --> 00:40:08,340
So I'll put it somewhere
here maybe, or maybe here.

523
00:40:08,340 --> 00:40:11,345
I won't need such a big diagram.

524
00:40:17,700 --> 00:40:19,842
So I'll have it here.

525
00:40:19,842 --> 00:40:24,260
Here is l, and here
are the energies.

526
00:40:24,260 --> 00:40:27,050
So ground states.

527
00:40:27,050 --> 00:40:32,565
The ground state, you can think
of it as a state like that.

528
00:40:35,440 --> 00:40:37,380
How should I write it?

529
00:40:37,380 --> 00:40:40,220
A state like that.

530
00:40:40,220 --> 00:40:46,300
No oscillators acting on it
whatsoever, so the N's are N1

531
00:40:46,300 --> 00:40:50,990
equals N2 equals N3
equals 0, and you

532
00:40:50,990 --> 00:40:57,660
get E equals h bar
omega times 3/2.

533
00:40:57,660 --> 00:40:59,620
So 3/2 h bar omega.

534
00:40:59,620 --> 00:41:05,275
So actually, we got one state,
and it's the lowest energy

535
00:41:05,275 --> 00:41:05,775
state.

536
00:41:08,320 --> 00:41:10,480
Energy lowest possible.

537
00:41:10,480 --> 00:41:18,010
So let me write here, energy
equals 3/2 h bar omega.

538
00:41:18,010 --> 00:41:24,445
We got one state over here.

539
00:41:27,520 --> 00:41:32,980
Now, can it be an l
equals 1 state or an l

540
00:41:32,980 --> 00:41:35,900
equals 2 state or
an l equals 3 state?

541
00:41:40,290 --> 00:41:43,530
How much is l for that state?

542
00:41:43,530 --> 00:41:46,770
You see, if it's a
spherically symmetric problem,

543
00:41:46,770 --> 00:41:48,810
it has to give you
a table like that.

544
00:41:48,810 --> 00:41:53,480
It's guaranteed by angular
momentum, so we must find.

545
00:41:53,480 --> 00:42:01,480
My question is whether it's l
equals 0, 1, 2, 3, or whatever.

546
00:42:01,480 --> 00:42:04,566
Anybody would like to say
what do they think it is?

547
00:42:04,566 --> 00:42:05,065
Kevin?

548
00:42:05,065 --> 00:42:06,204
AUDIENCE: It's 0, right?

549
00:42:06,204 --> 00:42:06,995
BARTON ZWIEBACH: 0.

550
00:42:06,995 --> 00:42:08,180
And why?

551
00:42:08,180 --> 00:42:13,270
AUDIENCE: Because we
wrote the operator for l

552
00:42:13,270 --> 00:42:18,035
in terms of ax, ay, and az, and
you need one to be non-zero.

553
00:42:18,035 --> 00:42:22,490
You need a difference between
them to generate a rotation.

554
00:42:22,490 --> 00:42:24,140
BARTON ZWIEBACH:
OK, that's true.

555
00:42:24,140 --> 00:42:26,110
It's a good answer.

556
00:42:26,110 --> 00:42:28,680
It's very explicit.

557
00:42:28,680 --> 00:42:32,400
Let me say it some other
way, why it couldn't possibly

558
00:42:32,400 --> 00:42:36,270
be l equals 1.

559
00:42:36,270 --> 00:42:37,234
Yes?

560
00:42:37,234 --> 00:42:39,162
AUDIENCE: Because
the ground state

561
00:42:39,162 --> 00:42:42,466
decreases for l decreasing.

562
00:42:42,466 --> 00:42:44,590
BARTON ZWIEBACH: The ground
state energy does what?

563
00:42:44,590 --> 00:42:46,482
AUDIENCE: It's
smaller for smaller l,

564
00:42:46,482 --> 00:42:49,740
and so for l equals 0, you
have to have a smaller ground

565
00:42:49,740 --> 00:42:51,850
state than for l equals 1.

566
00:42:51,850 --> 00:42:54,080
BARTON ZWIEBACH: That's true.

567
00:42:54,080 --> 00:42:55,330
Absolutely true.

568
00:42:55,330 --> 00:42:58,480
The energy increases so
it cannot be l equals 1,

569
00:42:58,480 --> 00:43:00,500
because then there will
be something below which

570
00:43:00,500 --> 00:43:01,580
doesn't exist.

571
00:43:01,580 --> 00:43:04,220
But there may be a
more plain answer.

572
00:43:04,220 --> 00:43:06,226
AUDIENCE: The state
is non-degenerative.

573
00:43:06,226 --> 00:43:07,100
BARTON ZWIEBACH: Yes.

574
00:43:07,100 --> 00:43:09,440
There's just one state here.

575
00:43:09,440 --> 00:43:12,510
We built one state.

576
00:43:12,510 --> 00:43:17,180
If it would be l equals 1,
there should be three states

577
00:43:17,180 --> 00:43:22,450
because l equals 1 comes with
m equals 1, 0, and minus 1.

578
00:43:22,450 --> 00:43:26,610
So unless there are three
states, you cannot have that.

579
00:43:26,610 --> 00:43:27,370
All right.

580
00:43:27,370 --> 00:43:31,280
So then we go to the next level.

581
00:43:31,280 --> 00:43:35,420
So I can build a state with
ax dagger on the vacuum,

582
00:43:35,420 --> 00:43:38,380
a state with ay
dagger on the vacuum,

583
00:43:38,380 --> 00:43:41,180
and a state with az
dagger on the vacuum

584
00:43:41,180 --> 00:43:43,210
using one oscillator.

585
00:43:43,210 --> 00:43:46,870
Here, the N's are
1, different ones,

586
00:43:46,870 --> 00:43:55,320
and the energy is h bar
omega 1 plus 3/2, so 5/2.

587
00:43:55,320 --> 00:43:58,880
And I got three states.

588
00:43:58,880 --> 00:44:01,920
What can that be?

589
00:44:01,920 --> 00:44:11,360
Well, could it be three
states of l equals 0?

590
00:44:11,360 --> 00:44:11,860
No.

591
00:44:11,860 --> 00:44:14,800
We said there's never
a degeneracy here.

592
00:44:14,800 --> 00:44:18,740
There's always one thing, so
there would be one state here,

593
00:44:18,740 --> 00:44:21,160
one state here, one
state here maybe.

594
00:44:21,160 --> 00:44:24,370
We don't know, but they would
not have the same energy,

595
00:44:24,370 --> 00:44:27,140
so it cannot be l equals 0.

596
00:44:27,140 --> 00:44:33,000
Now, you probably remember that
l equals 1 has three states.

597
00:44:33,000 --> 00:44:37,220
So without doing
any computation,

598
00:44:37,220 --> 00:44:40,980
I think I can argue that
this must be l equals 1.

599
00:44:40,980 --> 00:44:43,590
That cannot be any other thing.

600
00:44:43,590 --> 00:44:47,180
It cannot be l equals 2
because you need five states.

601
00:44:47,180 --> 00:44:49,860
Cannot be anything
with l equals 0.

602
00:44:49,860 --> 00:44:52,520
So it must be l equals 1.

603
00:44:52,520 --> 00:44:59,150
So here is l equals 0,
and here is l equals 1,

604
00:44:59,150 --> 00:45:04,770
and there's no state here, but
there's one at 5/2 h bar omega.

605
00:45:04,770 --> 00:45:07,160
So we obtain one state here.

606
00:45:09,950 --> 00:45:13,650
And this corresponds
to a degeneracy.

607
00:45:13,650 --> 00:45:18,580
This must correspond to l equals
1 because it's three states.

608
00:45:18,580 --> 00:45:21,740
And that degeneracy
is totally explained

609
00:45:21,740 --> 00:45:25,630
by angular momentum's
central potential.

610
00:45:25,630 --> 00:45:29,870
It has to group in that way.

611
00:45:29,870 --> 00:45:33,860
Of course, if my oscillator
had not been isotopic,

612
00:45:33,860 --> 00:45:35,860
it would not group that way.

613
00:45:35,860 --> 00:45:40,640
So we've got that one and we're,
I think, reasonably happy.

614
00:45:40,640 --> 00:45:45,580
Now, let's list the various l's.

615
00:45:45,580 --> 00:45:50,860
l equals 0, l equals 1,
l equals 2, l equals 3,

616
00:45:50,860 --> 00:45:53,720
l equals 4, l equals 5.

617
00:45:53,720 --> 00:45:54,750
How many states?

618
00:45:54,750 --> 00:46:01,660
1, 3, 5, 7, 9, 11.

619
00:46:01,660 --> 00:46:04,150
OK, good enough.

620
00:46:04,150 --> 00:46:08,360
So we succeeded, so let's
proceed to one more level.

621
00:46:08,360 --> 00:46:10,180
Let's see how we do.

622
00:46:10,180 --> 00:46:14,450
Here, I would have
ax dagger squared

623
00:46:14,450 --> 00:46:20,640
on the vacuum, ay dagger
squared on the vacuum, az

624
00:46:20,640 --> 00:46:23,890
dagger squared on the vacuum.

625
00:46:23,890 --> 00:46:26,800
Three states, but then
I have three more,

626
00:46:26,800 --> 00:46:39,130
ax ay, both dagger on the
vacuum, ax az, and ay az,

627
00:46:39,130 --> 00:46:41,385
for a total of six states.

628
00:46:48,040 --> 00:46:57,710
So at N equals 2,
the next level,

629
00:46:57,710 --> 00:47:05,350
let's call N equals
N1 plus N2 plus N3.

630
00:47:05,350 --> 00:47:09,690
So this is N equals 2.

631
00:47:09,690 --> 00:47:11,740
This is N equals 1.

632
00:47:11,740 --> 00:47:13,255
You've got six states.

633
00:47:16,710 --> 00:47:20,510
They must organize themselves
into representations

634
00:47:20,510 --> 00:47:27,120
of angular momentum, so they
must be billed by these things.

635
00:47:27,120 --> 00:47:29,240
So I cannot have l equals 3.

636
00:47:29,240 --> 00:47:32,790
I don't have that many states.

637
00:47:32,790 --> 00:47:38,340
I could have two l equals
1 states, three and three.

638
00:47:38,340 --> 00:47:40,925
That would give six states,
or a five and a one.

639
00:47:45,020 --> 00:47:48,410
So what are we looking at?

640
00:47:48,410 --> 00:47:52,990
Let's see what we could have.

641
00:47:52,990 --> 00:47:59,850
Well, we're trying to figure
out the next level, which

642
00:47:59,850 --> 00:48:04,050
is 7/2 h bar omega.

643
00:48:04,050 --> 00:48:11,690
If I say this is built
by two l equals 1's, I

644
00:48:11,690 --> 00:48:17,350
would have to put two things
here, and that's wrong.

645
00:48:17,350 --> 00:48:22,320
There cannot be two
multiplates at the same energy.

646
00:48:22,320 --> 00:48:27,490
So even though it looks like you
could build it with two l equal

647
00:48:27,490 --> 00:48:29,270
1's, you cannot.

648
00:48:29,270 --> 00:48:33,470
So it must be an l equals
2 and an l equals 0.

649
00:48:33,470 --> 00:48:39,810
So l equals 2 plus l equals
0, this one giving you

650
00:48:39,810 --> 00:48:43,240
five states and this
giving you one state.

651
00:48:43,240 --> 00:48:48,600
So at the next level,
this cannot be,

652
00:48:48,600 --> 00:48:52,170
but what you get
instead, l equals 2.

653
00:48:52,170 --> 00:48:56,590
You get one state here
and one state there.

654
00:49:05,330 --> 00:49:10,440
This is already something a
little strange and unexpected.

655
00:49:10,440 --> 00:49:13,380
For the first time,
you've got things

656
00:49:13,380 --> 00:49:17,000
in different columns that
are matching together.

657
00:49:17,000 --> 00:49:19,840
Why would these ones
match with these ones?

658
00:49:19,840 --> 00:49:22,580
That requires an explanation.

659
00:49:22,580 --> 00:49:26,140
You will see that explanation
a little later in the course,

660
00:49:26,140 --> 00:49:29,530
and that's something
we need to understand.

661
00:49:29,530 --> 00:49:31,060
So far, so good.

662
00:49:31,060 --> 00:49:33,530
We seem to be making
good progress.

663
00:49:33,530 --> 00:49:35,680
Let's do one more.

664
00:49:35,680 --> 00:49:39,074
In fact, we need to
do maybe a couple more

665
00:49:39,074 --> 00:49:40,115
to see the whole pattern.

666
00:49:55,650 --> 00:49:59,055
Let's do the next
one, N total equals 3.

667
00:50:02,570 --> 00:50:11,110
And now you have-- I'll be very
brief-- ax cubed, ay cubed, az

668
00:50:11,110 --> 00:50:28,094
cubed, ax squared times ay or
az, ay squared times ax or az,

669
00:50:28,094 --> 00:50:41,080
and az squared times ay or ax,
and ax ay az, all different.

670
00:50:41,080 --> 00:50:48,340
And that builds for three
states here, two states here,

671
00:50:48,340 --> 00:50:52,480
two states here, two states
here, and one state here.

672
00:50:52,480 --> 00:50:54,595
So that's 10 states.

673
00:51:01,820 --> 00:51:04,238
Yes?

674
00:51:04,238 --> 00:51:05,660
AUDIENCE: [INAUDIBLE]?

675
00:51:05,660 --> 00:51:06,493
BARTON ZWIEBACH: No.

676
00:51:06,493 --> 00:51:08,540
It's just laziness.

677
00:51:08,540 --> 00:51:10,890
I just should have
put ax squared

678
00:51:10,890 --> 00:51:14,461
ay dagger or ax
squared az squared.

679
00:51:14,461 --> 00:51:15,377
AUDIENCE: [INAUDIBLE]?

680
00:51:18,510 --> 00:51:20,710
BARTON ZWIEBACH:
No, this is a sum.

681
00:51:20,710 --> 00:51:25,740
This is what we used to call
the direct sum of vector spaces.

682
00:51:25,740 --> 00:51:28,840
This is not the product.

683
00:51:28,840 --> 00:51:31,630
That's pretty important.

684
00:51:31,630 --> 00:51:33,060
Here, it's a sum.

685
00:51:33,060 --> 00:51:38,310
We're saying 6 is 5
plus 1, basically.

686
00:51:38,310 --> 00:51:42,990
Six states are five
vectors plus one vector.

687
00:51:42,990 --> 00:51:44,790
Now, it can seem
a little confusing

688
00:51:44,790 --> 00:51:47,750
because-- well,
it's not confusing.

689
00:51:47,750 --> 00:51:51,760
If it would be a product, it
would be 1 times 5, which is 5.

690
00:51:51,760 --> 00:51:53,230
So here, it's 6.

691
00:51:53,230 --> 00:51:54,540
It's a direct sum.

692
00:51:54,540 --> 00:51:59,250
It's saying the space
of states at this level

693
00:51:59,250 --> 00:52:00,750
is six dimensional.

694
00:52:00,750 --> 00:52:03,290
This is a five
dimensional vector space,

695
00:52:03,290 --> 00:52:05,650
this is a one
dimensional vector space.

696
00:52:05,650 --> 00:52:09,270
This is a direct sum,
something we defined

697
00:52:09,270 --> 00:52:14,220
a month ago or two
months ago, direct sums.

698
00:52:14,220 --> 00:52:17,100
So this is funny how
this is happening.

699
00:52:17,100 --> 00:52:21,750
This tensor product is giving
you direct sums of states.

700
00:52:21,750 --> 00:52:25,550
Anyway, 10 states here.

701
00:52:25,550 --> 00:52:31,690
And now it does look like we
finally have an ambiguity.

702
00:52:31,690 --> 00:52:37,970
We could have l equals 4, which
is nine states, plus l equals

703
00:52:37,970 --> 00:52:38,470
0.

704
00:52:41,750 --> 00:52:45,100
You cannot use any
one more than once.

705
00:52:45,100 --> 00:52:48,075
We've learned that
for any energy level,

706
00:52:48,075 --> 00:52:52,050
we cannot have some l appear
more than once because it would

707
00:52:52,050 --> 00:52:53,320
imply degeneracy.

708
00:52:53,320 --> 00:52:57,180
So I cannot build
this with 10 singlets,

709
00:52:57,180 --> 00:53:01,860
or three l equal 1's
and one l equals 0.

710
00:53:01,860 --> 00:53:03,930
I have to build it
with different things,

711
00:53:03,930 --> 00:53:08,390
but I can build it as 9
plus 1, or I can build it

712
00:53:08,390 --> 00:53:13,190
as l equals 3 plus l equals 1.

713
00:53:17,880 --> 00:53:20,570
And the question
is, which one is it?

714
00:53:24,674 --> 00:53:25,890
AUDIENCE: [INAUDIBLE].

715
00:53:25,890 --> 00:53:28,570
BARTON ZWIEBACH: 3
and 1, is that right?

716
00:53:28,570 --> 00:53:31,190
How would you see that?

717
00:53:31,190 --> 00:53:34,100
AUDIENCE: Because the
lowest energy with l3

718
00:53:34,100 --> 00:53:37,565
has to be lower than the
lowest energy with l4.

719
00:53:37,565 --> 00:53:38,440
BARTON ZWIEBACH: Yes.

720
00:53:41,180 --> 00:53:43,310
Indeed, it would
be very strange.

721
00:53:43,310 --> 00:53:45,220
It shouldn't happen.

722
00:53:45,220 --> 00:53:47,170
The energies are
sort of in units,

723
00:53:47,170 --> 00:53:53,150
so here is l3 and
here is l equals 4.

724
00:53:53,150 --> 00:53:56,880
If l4 would be here,
where could be l3?

725
00:53:56,880 --> 00:53:59,350
It cannot be at a lower energy.

726
00:53:59,350 --> 00:54:01,650
We've accounted
for all of those.

727
00:54:01,650 --> 00:54:07,295
This is terribly unlikely,
and it must be this.

728
00:54:10,060 --> 00:54:19,180
And therefore, you found here
next level, 9/2 h bar omega,

729
00:54:19,180 --> 00:54:25,950
you got l equals 3, l equals 1.

730
00:54:25,950 --> 00:54:28,710
It's possible to count.

731
00:54:28,710 --> 00:54:31,810
You start to get bored
counting these things.

732
00:54:31,810 --> 00:54:35,100
So if you had to count,
for example, the number

733
00:54:35,100 --> 00:54:40,680
of states with 4, how would
you count them a little easier?

734
00:54:40,680 --> 00:54:50,500
Well, you say, I need ax dagger
to the nx, ay dagger to the ny,

735
00:54:50,500 --> 00:54:54,860
and az dagger to the nz.

736
00:54:54,860 --> 00:54:55,980
That's the state.

737
00:54:55,980 --> 00:55:03,410
And you must have nx
plus ny plus nz equals 4.

738
00:55:03,410 --> 00:55:06,820
And you can plot this,
make a little diagram

739
00:55:06,820 --> 00:55:12,890
like this, in which
you put nx, ny, and nz.

740
00:55:12,890 --> 00:55:16,470
And you say, well, this
can be as far as 4,

741
00:55:16,470 --> 00:55:20,310
this can be as high as 4,
this can be as high as 4,

742
00:55:20,310 --> 00:55:23,090
so you have triangle,
but you only

743
00:55:23,090 --> 00:55:25,430
have the integer solutions.

744
00:55:25,430 --> 00:55:31,200
nx plus ny plus nz equals 4 is
that whole hyperplane, but only

745
00:55:31,200 --> 00:55:33,320
integers and positive one.

746
00:55:33,320 --> 00:55:36,970
So you have here, for
example, a solution.

747
00:55:36,970 --> 00:55:41,500
This line is when nz
plus ny is equal to 4.

748
00:55:41,500 --> 00:55:46,595
So here's nz equals 4,
nz equals 3, 2, 1, 0.

749
00:55:49,450 --> 00:55:51,150
These are solutions.

750
00:55:51,150 --> 00:55:54,300
Here, you have
just one solution.

751
00:55:54,300 --> 00:55:57,750
Then you would have
two solutions here,

752
00:55:57,750 --> 00:56:03,050
three solutions here,
four here, and five there.

753
00:56:03,050 --> 00:56:06,910
So the number of states
is actually 1 plus 2

754
00:56:06,910 --> 00:56:10,930
plus 3 plus 4 plus 5.

755
00:56:10,930 --> 00:56:14,590
The number of states is
1 plus 2 plus 3 plus 4

756
00:56:14,590 --> 00:56:18,480
plus 5, which is 15.

757
00:56:21,030 --> 00:56:24,790
And you don't have
to write them.

758
00:56:24,790 --> 00:56:30,720
So 15 states, what could it be?

759
00:56:30,720 --> 00:56:33,120
Well, you go through
the numerology

760
00:56:33,120 --> 00:56:36,390
and there seem to be several
options, but not too many

761
00:56:36,390 --> 00:56:37,240
that make sense.

762
00:56:45,450 --> 00:56:47,780
You could have something
with l equals 5,

763
00:56:47,780 --> 00:56:50,990
but by the same
argument, it's unlikely.

764
00:56:50,990 --> 00:56:53,790
But you could have
something with l equals 4

765
00:56:53,790 --> 00:56:54,920
and begin with it.

766
00:56:54,920 --> 00:57:00,160
So it must be an l equals 4,
which gives me already nine

767
00:57:00,160 --> 00:57:04,550
states, and there are
left with six states.

768
00:57:04,550 --> 00:57:07,670
But you know that with
six states, pretty much

769
00:57:07,670 --> 00:57:12,900
the only thing you can do is
l equals 2 and l equals 0,

770
00:57:12,900 --> 00:57:15,860
so that must be it.

771
00:57:15,860 --> 00:57:20,390
The next state here,
l equals 4, is here.

772
00:57:20,390 --> 00:57:30,240
This was 11/2 h bar omega,
and then it goes 4, 2, 0.

773
00:57:30,240 --> 00:57:34,180
Enough to see the
pattern, I think.

774
00:57:34,180 --> 00:57:36,620
You could do the next one.

775
00:57:36,620 --> 00:57:40,420
Now it's quick because you
just need to add 6 here.

776
00:57:40,420 --> 00:57:43,310
It adds one more,
so it's 21 states,

777
00:57:43,310 --> 00:57:45,620
and you can see
what can you build.

778
00:57:45,620 --> 00:57:49,220
But it does look like
you have this, this,

779
00:57:49,220 --> 00:57:52,940
and that you jump by two units.

780
00:57:52,940 --> 00:57:56,750
So you have 0, then
1, and nothing.

781
00:57:56,750 --> 00:57:59,970
Then 2, and you
jump the next to 0.

782
00:57:59,970 --> 00:58:04,250
And then 3 is the next one, and
then you jump 2, and that's it.

783
00:58:04,250 --> 00:58:07,340
And here, jump 2 and jump 2.

784
00:58:07,340 --> 00:58:12,750
So in jumps of 2, you go to the
angular momentum that you need.

785
00:58:12,750 --> 00:58:16,390
So how can you
understand a little more

786
00:58:16,390 --> 00:58:17,840
of what's going on here?

787
00:58:17,840 --> 00:58:19,950
Why these things?

788
00:58:19,950 --> 00:58:28,820
Well, as you may recall, we
used to have this discussion

789
00:58:28,820 --> 00:58:32,970
in which you have an a x and ay.

790
00:58:32,970 --> 00:58:39,550
You could trade for
a right and a left.

791
00:58:39,550 --> 00:58:43,710
And with those, the angular
momentum in the z-direction

792
00:58:43,710 --> 00:58:48,750
was h bar N right minus N left.

793
00:58:48,750 --> 00:58:51,140
This is for a
two-dimensional oscillator,

794
00:58:51,140 --> 00:58:55,750
but the x and y of the
three-dimensional oscillator

795
00:58:55,750 --> 00:58:57,830
works exactly the same way.

796
00:58:57,830 --> 00:59:05,660
So Lz is nicely written in
terms of these variables.

797
00:59:05,660 --> 00:59:10,360
And it takes a little more
work to get the other-- the Lx

798
00:59:10,360 --> 00:59:13,760
and Ly, but they
can be calculated.

799
00:59:13,760 --> 00:59:16,260
And they correspond
to the true angular

800
00:59:16,260 --> 00:59:18,460
momentum of this particle.

801
00:59:18,460 --> 00:59:20,800
It's the real angular momentum.

802
00:59:20,800 --> 00:59:22,770
It's not the angular
momentum that you

803
00:59:22,770 --> 00:59:26,150
found for the two-dimensional
harmonic oscillator.

804
00:59:26,150 --> 00:59:27,280
It's the real one.

805
00:59:27,280 --> 00:59:33,670
So here we go with
a little analysis.

806
00:59:33,670 --> 00:59:38,310
How would you build now
states in this language?

807
00:59:38,310 --> 00:59:40,960
You can understand things
better in this case

808
00:59:40,960 --> 00:59:45,520
because, for example,
for N equals 1,

809
00:59:45,520 --> 00:59:49,520
you could have a
state a right dagger

810
00:59:49,520 --> 01:00:00,230
on the vacuum, a z dagger,
a left dagger on the vacuum.

811
01:00:00,230 --> 01:00:04,220
And then you can say, what
is the Lz of this state?

812
01:00:06,920 --> 01:00:14,030
Well, a right dagger on the
vacuum has Lz equal h bar.

813
01:00:14,030 --> 01:00:17,750
This has 0 because
Lz doesn't depend

814
01:00:17,750 --> 01:00:21,130
on the z-component
of the oscillator.

815
01:00:21,130 --> 01:00:24,020
And this has minus h bar.

816
01:00:24,020 --> 01:00:27,740
So here you see actually,
the whole structure of the L

817
01:00:27,740 --> 01:00:29,610
equal 1 multiplet.

818
01:00:29,610 --> 01:00:33,180
We said that we have at
this level L equals 1.

819
01:00:33,180 --> 01:00:36,640
And indeed, for L equals
1, you expect the state

820
01:00:36,640 --> 01:00:40,640
with Lz equal plus
1, 0, and minus 1.

821
01:00:40,640 --> 01:00:42,890
So you see the whole thing.

822
01:00:42,890 --> 01:00:48,260
For n equals 2, what do you get?

823
01:00:48,260 --> 01:00:54,570
Well, you see a state, for
example, of a right dagger

824
01:00:54,570 --> 01:00:58,800
a right dagger on the vacuum.

825
01:00:58,800 --> 01:01:07,740
And that has Lz equals 2 h bar.

826
01:01:07,740 --> 01:01:12,900
And therefore, you must have--
since you cannot have states

827
01:01:12,900 --> 01:01:16,880
with higher Lz, you cannot
have a state, for example,

828
01:01:16,880 --> 01:01:21,460
here with Lz equal 3.

829
01:01:21,460 --> 01:01:23,950
So you cannot have an L equal 3.

830
01:01:23,950 --> 01:01:27,300
In fact, for any N
that you build states,

831
01:01:27,300 --> 01:01:31,440
you can only get
states with whatever

832
01:01:31,440 --> 01:01:35,540
N is is the maximum
value that Lz can have,

833
01:01:35,540 --> 01:01:38,540
which is something I want to
illustrate just generically

834
01:01:38,540 --> 01:01:39,980
for a second.

835
01:01:39,980 --> 01:01:48,790
So in order to show that, let
me go back here and exhibit

836
01:01:48,790 --> 01:01:51,395
for you a little of
the general structure.

837
01:01:54,820 --> 01:02:01,450
So suppose you're building
now with N equal n.

838
01:02:01,450 --> 01:02:05,170
The total number
is N. So you have

839
01:02:05,170 --> 01:02:10,820
a state with a right dagger
to the n on the vacuum.

840
01:02:10,820 --> 01:02:15,240
And this is the state
with highest possible Lz

841
01:02:15,240 --> 01:02:19,840
because all the
oscillators are aR dagger.

842
01:02:19,840 --> 01:02:21,590
So Lz is the highest.

843
01:02:26,980 --> 01:02:32,965
And highest Lz is,
in fact, n h bar.

844
01:02:35,510 --> 01:02:40,150
Now, let's try to build a state
with a little bit less Lz.

845
01:02:40,150 --> 01:02:43,470
You see, if this is
a multiplet, this

846
01:02:43,470 --> 01:02:47,960
has to be a multiplet with some
amount of angular momentum.

847
01:02:47,960 --> 01:02:52,260
So it's going to go from
Lz equal n, n minus 1,

848
01:02:52,260 --> 01:02:53,900
up to minus n.

849
01:02:53,900 --> 01:02:59,190
There are going to be 2n plus
1 states of this much angular

850
01:02:59,190 --> 01:03:01,560
momentum because this
has to be a multiplet.

851
01:03:01,560 --> 01:03:05,190
So here you have a
state with one unit

852
01:03:05,190 --> 01:03:11,450
less of angular momentum, a
right dagger to the n minus 1,

853
01:03:11,450 --> 01:03:13,000
times an az dagger.

854
01:03:15,700 --> 01:03:18,390
I claim that's the
only state that you

855
01:03:18,390 --> 01:03:22,020
can build with one
unit less of angular

856
01:03:22,020 --> 01:03:25,560
momentum in the
z-direction because I've

857
01:03:25,560 --> 01:03:32,640
traded this aR for an az.

858
01:03:32,640 --> 01:03:35,930
So this must be the second
state in the multiplet.

859
01:03:35,930 --> 01:03:39,720
This multiplet with
highest value of L,

860
01:03:39,720 --> 01:03:45,630
which is equal to n, corresponds
to an angular momentum l,

861
01:03:45,630 --> 01:03:47,400
little l, equals n.

862
01:03:50,190 --> 01:03:53,990
And then, it must have
this 2n plus 1 states.

863
01:03:53,990 --> 01:03:55,510
And here is the second state.

864
01:03:55,510 --> 01:04:00,090
So this is Lz equals nh bar.

865
01:04:00,090 --> 01:04:03,050
And here, n minus 1 h bar.

866
01:04:03,050 --> 01:04:05,680
And I don't think there's any
other state at that level.

867
01:04:05,680 --> 01:04:09,870
Let's lower the angular
momentum once more.

868
01:04:09,870 --> 01:04:12,720
So what do we get?

869
01:04:12,720 --> 01:04:20,073
a right dagger n minus
2 az dagger squared.

870
01:04:24,210 --> 01:04:30,780
That's another state with
one less angular momentum

871
01:04:30,780 --> 01:04:31,440
than this.

872
01:04:31,440 --> 01:04:35,220
This, in fact, has n
minus 2 times h bar.

873
01:04:35,220 --> 01:04:38,270
Now, is that the
unique state that I

874
01:04:38,270 --> 01:04:41,510
can have with two units
less of angular momentum?

875
01:04:41,510 --> 01:04:42,360
No.

876
01:04:42,360 --> 01:04:44,282
What is the other one?

877
01:04:44,282 --> 01:04:47,270
AUDIENCE: aR to
the n minus 1 a l?

878
01:04:47,270 --> 01:04:49,110
PROFESSOR: Correct,
that lowers it.

879
01:04:49,110 --> 01:04:59,820
third so here you have aR to the
n minus 1 a left on the vacuum.

880
01:04:59,820 --> 01:05:05,270
That's another state with two
units less of angular momentum.

881
01:05:05,270 --> 01:05:11,160
So in this situation, a
funny thing has happened.

882
01:05:11,160 --> 01:05:17,620
And here's why you
understand the jump of 2.

883
01:05:17,620 --> 01:05:19,540
This state, you
actually-- if you're

884
01:05:19,540 --> 01:05:22,240
trying to build this
multiplet, now you

885
01:05:22,240 --> 01:05:26,670
have two states that have
the same value of Lz.

886
01:05:26,670 --> 01:05:30,590
And you actually don't
know whether the next state

887
01:05:30,590 --> 01:05:32,650
in the multiplet
is this, or that,

888
01:05:32,650 --> 01:05:34,770
or some linear combination.

889
01:05:34,770 --> 01:05:38,420
It better be some
linear combination.

890
01:05:38,420 --> 01:05:43,220
But the fact is that at this
level, you found another state.

891
01:05:43,220 --> 01:05:46,720
So this multiplet
will go on and it

892
01:05:46,720 --> 01:05:48,880
will be some linear combination.

893
01:05:48,880 --> 01:05:51,360
Maybe this diagram
doesn't illustrate that.

894
01:05:51,360 --> 01:05:54,810
But then you will have
another state here.

895
01:05:54,810 --> 01:05:59,280
So some other linear combination
that builds another multiplet.

896
01:05:59,280 --> 01:06:04,840
And this multiplet has two
units less of angular momentum.

897
01:06:04,840 --> 01:06:10,422
And that explains why
this diagram always jumps.

898
01:06:10,422 --> 01:06:13,010
It always jumps 2.

899
01:06:13,010 --> 01:06:15,590
And you could do that here.

900
01:06:15,590 --> 01:06:19,170
If you tried to write
the next things here,

901
01:06:19,170 --> 01:06:21,750
you will find two states
that you can write.

902
01:06:21,750 --> 01:06:24,190
But if you go one lower,
you will find three states.

903
01:06:24,190 --> 01:06:26,400
Which means that
at the next level,

904
01:06:26,400 --> 01:06:30,550
you built another-- you
need another state with two

905
01:06:30,550 --> 01:06:34,820
units less of angular
momentum each time.

906
01:06:34,820 --> 01:06:37,190
So pretty much that's it.

907
01:06:37,190 --> 01:06:41,132
That illustrates how
this diagram happens.

908
01:06:41,132 --> 01:06:42,590
The only thing we
haven't answered,

909
01:06:42,590 --> 01:06:48,770
and you will see
that in an exercise,

910
01:06:48,770 --> 01:06:53,010
how could I have understood
from the beginning

911
01:06:53,010 --> 01:06:56,010
that this would have happened
rather than building it

912
01:06:56,010 --> 01:06:58,880
this way that
there's this thing?

913
01:06:58,880 --> 01:07:00,500
And what you will
find is that there's

914
01:07:00,500 --> 01:07:04,200
some operators that commute
with the Hamiltonian that

915
01:07:04,200 --> 01:07:06,220
move you from here to here.

916
01:07:06,220 --> 01:07:07,960
And that explains it all.

917
01:07:07,960 --> 01:07:10,420
Because if you have an
operator that commutes with

918
01:07:10,420 --> 01:07:13,040
the Hamiltonian, it
cannot change the energy.

919
01:07:13,040 --> 01:07:18,020
And if it changes the value of
L, it explains why it happened.

920
01:07:18,020 --> 01:07:20,770
So that's something that
you need to discover,

921
01:07:20,770 --> 01:07:22,730
what are these operators.

922
01:07:22,730 --> 01:07:24,670
I can give you a hint.

923
01:07:24,670 --> 01:07:32,960
An operator for the
type ax dagger ay

924
01:07:32,960 --> 01:07:37,680
destroys a y oscillator,
creates an x one.

925
01:07:37,680 --> 01:07:41,480
It doesn't change the energy
because it adds one thing

926
01:07:41,480 --> 01:07:42,740
and loses one.

927
01:07:42,740 --> 01:07:46,290
So this kind of thing must
commute with the Hamiltonian.

928
01:07:46,290 --> 01:07:48,370
And these are the
kind of objects--

929
01:07:48,370 --> 01:07:49,880
there are lots of them.

930
01:07:49,880 --> 01:07:53,720
So surprising new things that
commute with the Hamiltonian.

931
01:07:53,720 --> 01:07:57,220
And there's a whole
hidden symmetry in here

932
01:07:57,220 --> 01:08:00,370
that is generated by
operators of this form.

933
01:08:00,370 --> 01:08:03,160
So it's something you will see.

934
01:08:03,160 --> 01:08:05,960
Now, the last 15 minutes
I want to show you

935
01:08:05,960 --> 01:08:09,780
what happens with hydrogen.

936
01:08:09,780 --> 01:08:12,000
There's a similar
phenomenon there

937
01:08:12,000 --> 01:08:15,670
that we're going to try
to explain in detail.

938
01:08:15,670 --> 01:08:20,210
So a couple of things
about hydrogen.

939
01:08:24,000 --> 01:08:32,290
So hydrogen H is
equal to p squared

940
01:08:32,290 --> 01:08:38,450
over 2m minus e squared over r.

941
01:08:38,450 --> 01:08:43,979
There's a natural length scale
that people many times struggle

942
01:08:43,979 --> 01:08:47,569
to find it, the Bohr radius.

943
01:08:47,569 --> 01:08:48,740
This does it come from here?

944
01:08:48,740 --> 01:08:52,149
How do you see immediately
what is the Bohr radius?

945
01:08:52,149 --> 01:08:56,260
Well, the Bohr radius can
be estimated by saying,

946
01:08:56,260 --> 01:09:01,350
well, this is an energy but it
must be similar to this energy.

947
01:09:01,350 --> 01:09:08,040
So p is like h over a distance.

948
01:09:08,040 --> 01:09:12,050
So let's call it a0.

949
01:09:12,050 --> 01:09:14,050
So that's p squared.

950
01:09:14,050 --> 01:09:20,740
m is the reduced mass of the
proton electron system roughly

951
01:09:20,740 --> 01:09:23,080
equal to the electron mass.

952
01:09:23,080 --> 01:09:26,300
And then you set it
equal to this one

953
01:09:26,300 --> 01:09:29,200
because you're just doing units.

954
01:09:29,200 --> 01:09:32,359
You want to find the quantity
that has units of length

955
01:09:32,359 --> 01:09:33,910
and there you got it.

956
01:09:33,910 --> 01:09:38,560
That's the famous Bohr radius.

957
01:09:38,560 --> 01:09:42,069
p is h bar over a
distance, therefore

958
01:09:42,069 --> 01:09:43,899
this thing must be an energy.

959
01:09:43,899 --> 01:09:45,510
It must be equal to this.

960
01:09:45,510 --> 01:09:49,750
And from this one,
you can solve for a0.

961
01:09:49,750 --> 01:09:54,710
It's h squared over m e squared.

962
01:09:54,710 --> 01:09:58,970
The 1 over e squared is
very famous and important.

963
01:09:58,970 --> 01:10:02,170
It reflects the
fact that if you had

964
01:10:02,170 --> 01:10:03,970
the interaction
between the electron

965
01:10:03,970 --> 01:10:08,140
and the proton go to 0, the
radius would be infinite.

966
01:10:08,140 --> 01:10:10,660
As it becomes weaker and
weaker the interaction,

967
01:10:10,660 --> 01:10:13,890
the radius of the
hydrogen atom blows up.

968
01:10:13,890 --> 01:10:20,330
So this is about
0.529 angstroms,

969
01:10:20,330 --> 01:10:23,210
where an angstrom is 10
to the minus 10 meters.

970
01:10:25,750 --> 01:10:28,016
And what is the energy scale?

971
01:10:34,980 --> 01:10:43,820
Well, e squared over a0 is
the energy scale, roughly.

972
01:10:43,820 --> 01:10:48,740
And in fact, e squared
over 2a 0, if you wish,

973
01:10:48,740 --> 01:10:49,885
is a famous number.

974
01:10:49,885 --> 01:10:57,302
It's about 13.6 ev.

975
01:10:57,302 --> 01:11:00,440
So how about the spectrum?

976
01:11:00,440 --> 01:11:03,600
And how do you find that?

977
01:11:03,600 --> 01:11:08,090
Well, there's one
nice way of doing

978
01:11:08,090 --> 01:11:10,570
this, which you will
see in the problem,

979
01:11:10,570 --> 01:11:12,670
to find at least
the ground state.

980
01:11:12,670 --> 01:11:18,330
And it's a very elegant
way based on factorization.

981
01:11:18,330 --> 01:11:20,670
Let we mention it.

982
01:11:20,670 --> 01:11:22,340
It is called Hamiltonian.

983
01:11:22,340 --> 01:11:26,580
It can be written as a
constant gamma plus 1

984
01:11:26,580 --> 01:11:42,645
over 2m sum over k pk plus i
beta xk over r times pk minus i

985
01:11:42,645 --> 01:11:48,194
beta xk over r.

986
01:11:48,194 --> 01:11:52,130
It's a factorized version of the
Hamiltonian of a hydrogen atom.

987
01:11:52,130 --> 01:11:55,710
Apparently, not a
well-known result.

988
01:11:55,710 --> 01:11:57,910
Professor [? Jackiw ?]
mentioned it to me.

989
01:11:57,910 --> 01:12:00,990
I don't think I've
seen it in any book.

990
01:12:04,200 --> 01:12:08,350
So there's a constant
beta and a constant gamma

991
01:12:08,350 --> 01:12:11,370
for which this becomes
exactly that one.

992
01:12:11,370 --> 01:12:13,410
So gamma and beta
to be determined.

993
01:12:20,100 --> 01:12:22,720
And you have to be a
little careful here

994
01:12:22,720 --> 01:12:28,060
when you expand that this term
and this term don't commute.

995
01:12:28,060 --> 01:12:29,920
And this and this don't commute.

996
01:12:29,920 --> 01:12:33,230
But after you're
done, it comes out.

997
01:12:33,230 --> 01:12:37,440
And then, the ground
state wave function

998
01:12:37,440 --> 01:12:41,940
is-- since this is an
operator and here it's dagger,

999
01:12:41,940 --> 01:12:46,690
the ground state wave function
is-- the lowest possible energy

1000
01:12:46,690 --> 01:12:48,580
wave function is
one in which this

1001
01:12:48,580 --> 01:12:50,380
would kill the wave function.

1002
01:12:50,380 --> 01:12:58,540
So pk minus i beta xk over r
should kill the ground state

1003
01:12:58,540 --> 01:13:01,270
wave function.

1004
01:13:01,270 --> 01:13:04,570
And then the energy,
E ground, would

1005
01:13:04,570 --> 01:13:08,850
be equal to precisely
this constant gamma.

1006
01:13:08,850 --> 01:13:13,200
And you will show, in fact,
that yes, this has a solution.

1007
01:13:13,200 --> 01:13:15,090
And that's the
ground state energy

1008
01:13:15,090 --> 01:13:20,380
of the oscillator of the
hydrogen atom, of course.

1009
01:13:20,380 --> 01:13:28,690
So this looks like three
equations, pk with k

1010
01:13:28,690 --> 01:13:30,155
equals 1 to 3.

1011
01:13:33,140 --> 01:13:37,690
But it reduces to 1 if the
state is spherically symmetric.

1012
01:13:37,690 --> 01:13:43,070
So it's a nice thing and
it gives you the answer.

1013
01:13:43,070 --> 01:13:46,280
Now, the whole spectrum
of the hydrogen atom

1014
01:13:46,280 --> 01:13:51,240
is as interestingly
degenerate as one

1015
01:13:51,240 --> 01:13:54,570
of the three-dimensional
harmonic oscillator.

1016
01:13:54,570 --> 01:14:03,180
And a reminder of it is
that-- should I go here?

1017
01:14:03,180 --> 01:14:05,880
Yes.

1018
01:14:05,880 --> 01:14:10,510
You have here energies
and here l's. l

1019
01:14:10,510 --> 01:14:13,310
equals 0 you have one state.

1020
01:14:13,310 --> 01:14:17,730
l equals 1 you have
another state that's here.

1021
01:14:17,730 --> 01:14:21,090
But actually, l equals 0
will have another state.

1022
01:14:21,090 --> 01:14:24,270
And then it goes on like
that, another state here,

1023
01:14:24,270 --> 01:14:28,240
state here, state
here for l equals 2.

1024
01:14:31,000 --> 01:14:33,300
And the first state is here.

1025
01:14:33,300 --> 01:14:35,690
The first state of this
one aligns with this one.

1026
01:14:35,690 --> 01:14:38,180
The first state of
that aligns with that.

1027
01:14:38,180 --> 01:14:44,170
So they go like
that, the states just

1028
01:14:44,170 --> 01:14:48,750
continue to go exactly
with this symmetry.

1029
01:14:48,750 --> 01:14:54,180
So let me use label
that is common,

1030
01:14:54,180 --> 01:14:58,173
to call this the state nu
equals 0 for L equals 0.

1031
01:14:58,173 --> 01:14:59,735
Nu equals 1.

1032
01:14:59,735 --> 01:15:01,000
Nu equals 2.

1033
01:15:01,000 --> 01:15:02,970
Nu equals 3.

1034
01:15:02,970 --> 01:15:06,620
This is the first state
with L equals 1 is here.

1035
01:15:06,620 --> 01:15:08,900
So we'll call it nu equals 0.

1036
01:15:08,900 --> 01:15:10,460
Nu equals 1.

1037
01:15:10,460 --> 01:15:12,210
New equals 2.

1038
01:15:12,210 --> 01:15:18,070
The first state here is
nu equals 0, nu equals 1.

1039
01:15:18,070 --> 01:15:21,190
And then the energies.

1040
01:15:21,190 --> 01:15:28,170
You define n to be
nu plus l plus 1.

1041
01:15:28,170 --> 01:15:33,170
Therefore, this
corresponds to n equals 1.

1042
01:15:33,170 --> 01:15:36,000
This corresponds to n equals 2.

1043
01:15:36,000 --> 01:15:40,920
That corresponds to nu can be
1 and l equals 0 or nu can be 0

1044
01:15:40,920 --> 01:15:43,370
and l equals 1.

1045
01:15:43,370 --> 01:15:47,850
This is n equal 3,
and things like that.

1046
01:15:47,850 --> 01:15:54,380
And then the energies
of those states, nl

1047
01:15:54,380 --> 01:15:58,450
is, in fact, minus z squared.

1048
01:15:58,450 --> 01:16:00,840
Well, forget the z squared.

1049
01:16:00,840 --> 01:16:05,145
e squared over 2 a0
1 over n squared.

1050
01:16:09,960 --> 01:16:11,650
So the only thing
that happens is

1051
01:16:11,650 --> 01:16:15,840
that there's a degeneracy,
complete degeneracy.

1052
01:16:15,840 --> 01:16:18,200
Very powerful degeneracy.

1053
01:16:18,200 --> 01:16:30,730
And then, l can only run up
to-- from 0, 1, up to n minus 1

1054
01:16:30,730 --> 01:16:32,350
in these variables.

1055
01:16:32,350 --> 01:16:35,450
So this is the
picture of hydrogen.

1056
01:16:35,450 --> 01:16:40,320
So you've seen several pictures
already-- the square well,

1057
01:16:40,320 --> 01:16:43,530
the three-dimensional
harmonic oscillator,

1058
01:16:43,530 --> 01:16:44,860
and the hydrogen one.

1059
01:16:44,860 --> 01:16:48,200
Each one has a
different picture.

1060
01:16:48,200 --> 01:16:52,510
Now, in order to
understand this one--

1061
01:16:52,510 --> 01:16:54,470
this one is not that difficult.

1062
01:16:54,470 --> 01:16:57,790
But the one of the hydrogen
is really more interesting.

1063
01:16:57,790 --> 01:17:00,780
It all originates
with the idea of what

1064
01:17:00,780 --> 01:17:04,830
is called the
Runge-Lenz vector, which

1065
01:17:04,830 --> 01:17:08,720
I'm going to use the last
five minutes to introduce.

1066
01:17:08,720 --> 01:17:10,745
And think about it a little.

1067
01:17:17,660 --> 01:17:20,680
So it comes from
classical mechanics.

1068
01:17:20,680 --> 01:17:26,180
So we have an elliptical
orbit, orbits,

1069
01:17:26,180 --> 01:17:28,910
and people figured out
there was something

1070
01:17:28,910 --> 01:17:32,620
very funny about characterizing
elliptical orbit.

1071
01:17:32,620 --> 01:17:35,600
So consider a
Hamiltonian, which is p

1072
01:17:35,600 --> 01:17:40,780
squared over 2m plus
v of r, a potential.

1073
01:17:40,780 --> 01:17:44,530
The force, classically,
would be minus the gradient

1074
01:17:44,530 --> 01:17:49,320
of the potential, which is minus
the derivative of the potential

1075
01:17:49,320 --> 01:17:52,445
with respect to r times
the r unit vector.

1076
01:17:55,850 --> 01:18:00,385
Now classically-- this
all begins classically.

1077
01:18:04,950 --> 01:18:09,260
Except for spin 1/2
systems, classical physics

1078
01:18:09,260 --> 01:18:13,680
really tells you a lot
of what's going on.

1079
01:18:13,680 --> 01:18:25,490
So classically, dp dt is
the force and it's minus v

1080
01:18:25,490 --> 01:18:30,220
prime over r r vector over r.

1081
01:18:30,220 --> 01:18:37,330
And dl dt, the angular momentum,
it's a central potential.

1082
01:18:37,330 --> 01:18:41,040
The angular momentum is 0.

1083
01:18:41,040 --> 01:18:42,720
It's rate of change is 0.

1084
01:18:42,720 --> 01:18:48,710
There's no torque on the
particle, so this should be 0.

1085
01:18:48,710 --> 01:18:51,560
Now, the interesting
thing that happens

1086
01:18:51,560 --> 01:18:54,690
is that this doesn't
exhaust the kind of things

1087
01:18:54,690 --> 01:18:58,490
that are, in fact, conserved.

1088
01:18:58,490 --> 01:19:06,900
So there is something
more that is conserved.

1089
01:19:06,900 --> 01:19:08,825
And it's a very
surprising quantity.

1090
01:19:08,825 --> 01:19:11,780
It's so surprising
that people have

1091
01:19:11,780 --> 01:19:16,960
a hard time
imagining what it is.

1092
01:19:16,960 --> 01:19:25,430
I will write it down and show
you how it sort of happens.

1093
01:19:25,430 --> 01:19:30,750
Well, you have to
begin with p cross L.

1094
01:19:30,750 --> 01:19:34,570
Why you would think of p cross
l is a little bit of a mystery,

1095
01:19:34,570 --> 01:19:38,530
but it's an interesting thing.

1096
01:19:38,530 --> 01:19:42,900
Now, here is a computation
that will be in the notes

1097
01:19:42,900 --> 01:19:45,030
that you can try doing.

1098
01:19:45,030 --> 01:19:52,740
And it takes a little bit
of work, but it's algebra.

1099
01:19:52,740 --> 01:19:57,490
If you compute this and
do a fair amount of work,

1100
01:19:57,490 --> 01:20:01,070
like five, six lines--
I would suspect it's

1101
01:20:01,070 --> 01:20:03,130
fairly non-trivial
to do it if you

1102
01:20:03,130 --> 01:20:06,170
don't see how it's
being done, but it

1103
01:20:06,170 --> 01:20:09,720
will be in the notes-- you
get the following thing.

1104
01:20:13,980 --> 01:20:18,110
Just by manipulating the
time derivative of p cross L,

1105
01:20:18,110 --> 01:20:21,280
you get this.

1106
01:20:21,280 --> 01:20:25,420
Which is equal to m times
the potential differentiator

1107
01:20:25,420 --> 01:20:28,040
times r squared times the
time derivative of this.

1108
01:20:28,040 --> 01:20:30,410
So time derivative,
time derivative.

1109
01:20:30,410 --> 01:20:35,840
You can get the conservation
if this is a constant.

1110
01:20:35,840 --> 01:20:37,800
So when is this a constant?

1111
01:20:37,800 --> 01:20:42,390
If this is some
constant, say, e squared,

1112
01:20:42,390 --> 01:20:44,190
you would get a conservation.

1113
01:20:44,190 --> 01:20:52,550
But what is v prime equals
e squared over r squared?

1114
01:20:52,550 --> 01:20:57,180
It would give you that v
of r is essentially minus

1115
01:20:57,180 --> 01:20:59,830
e squared over r.

1116
01:20:59,830 --> 01:21:02,050
That's the potential
of hydrogen.

1117
01:21:02,050 --> 01:21:08,870
Or the 1 over r potential, 1
over r squared force field.

1118
01:21:08,870 --> 01:21:14,730
So in 1 over r potentials,
this is a number.

1119
01:21:14,730 --> 01:21:19,360
And then you get an incredible
conservation law, d dt

1120
01:21:19,360 --> 01:21:31,810
of p cross L minus m e squared
r hat over r is equal to 0.

1121
01:21:31,810 --> 01:21:36,420
So something fairly unexpected
that something like this

1122
01:21:36,420 --> 01:21:38,290
could be conserved.

1123
01:21:38,290 --> 01:21:42,625
So actually, you can try
to figure out what this is.

1124
01:21:45,990 --> 01:21:49,360
And there's two neat--
first, one thing

1125
01:21:49,360 --> 01:21:52,920
that people do,
which is convenient,

1126
01:21:52,920 --> 01:21:56,050
is to make this into
unit-free vector.

1127
01:21:56,050 --> 01:22:05,450
So define R to be p
cross L over m e squared

1128
01:22:05,450 --> 01:22:09,150
minus r vector over r.

1129
01:22:09,150 --> 01:22:10,460
This has no units.

1130
01:22:13,450 --> 01:22:14,990
And it's supposed
to be conserved.

1131
01:22:21,230 --> 01:22:25,570
Now, one thing you will
check in the homework

1132
01:22:25,570 --> 01:22:28,980
is that this is conserved
quantum mechanically as well.

1133
01:22:28,980 --> 01:22:31,190
That is, this is
an operator that

1134
01:22:31,190 --> 01:22:33,370
commutes with a Hamiltonian.

1135
01:22:33,370 --> 01:22:36,800
Very interesting calculation.

1136
01:22:36,800 --> 01:22:38,945
This is a Hermitian
operator, so you

1137
01:22:38,945 --> 01:22:43,400
will have to Hermiticize
the p cross L to do that.

1138
01:22:43,400 --> 01:22:46,020
But it will commute
with the Hamiltonian.

1139
01:22:46,020 --> 01:22:49,870
But what I want to finish
now is with your intuition

1140
01:22:49,870 --> 01:22:52,530
as to what this is.

1141
01:22:52,530 --> 01:22:56,640
And this was a very interesting
discovery, this vector.

1142
01:22:56,640 --> 01:22:59,170
In fact, people
didn't appreciate

1143
01:22:59,170 --> 01:23:03,680
what was the role of this
vector for quite some time.

1144
01:23:03,680 --> 01:23:08,270
So apparently, it was discovered
and forgotten, and discovered

1145
01:23:08,270 --> 01:23:11,490
and forgotten like
two or three times.

1146
01:23:11,490 --> 01:23:15,700
And for us, it's going to
be quite crucial because I

1147
01:23:15,700 --> 01:23:19,170
said to you that this operator
commutes with the Hamiltonian.

1148
01:23:22,920 --> 01:23:25,660
So actually, you will
get conservation laws

1149
01:23:25,660 --> 01:23:28,911
and will help us explain the
degeneracy of the hydrogen

1150
01:23:28,911 --> 01:23:29,410
atom.

1151
01:23:29,410 --> 01:23:32,390
So it will be very
important for us.

1152
01:23:32,390 --> 01:23:35,850
Now, how does this look?

1153
01:23:35,850 --> 01:23:39,750
First of all, if you
had a circular orbit,

1154
01:23:39,750 --> 01:23:42,500
how does it work?

1155
01:23:42,500 --> 01:23:45,050
Have a circular orbit.

1156
01:23:45,050 --> 01:23:51,355
Let's see, p is here,
L is out of the board.

1157
01:23:51,355 --> 01:23:59,400
p cross L is here
over m e squared.

1158
01:23:59,400 --> 01:24:08,190
And the radial vector
is here, the hat vector.

1159
01:24:08,190 --> 01:24:13,870
So the sum of these two vectors
p cross L and the radial vector

1160
01:24:13,870 --> 01:24:14,685
must be conserved.

1161
01:24:17,250 --> 01:24:19,820
But how could it be?

1162
01:24:19,820 --> 01:24:24,950
If they don't cancel, it
either points in or points out.

1163
01:24:24,950 --> 01:24:29,140
And then it would just rotate
and it would not be conserved.

1164
01:24:29,140 --> 01:24:33,650
So actually, for a circular
orbit, you can calculate it.

1165
01:24:33,650 --> 01:24:35,070
See the notes.

1166
01:24:35,070 --> 01:24:36,880
Actually, it's an
easy calculation.

1167
01:24:36,880 --> 01:24:40,960
And you can verify that this
vector is, in fact, precisely

1168
01:24:40,960 --> 01:24:42,260
opposite this.

1169
01:24:42,260 --> 01:24:44,080
And it's 0.

1170
01:24:44,080 --> 01:24:45,160
So you say, great.

1171
01:24:45,160 --> 01:24:48,495
You discover something that
is conserved, but it's 0.

1172
01:24:48,495 --> 01:24:50,010
No.

1173
01:24:50,010 --> 01:24:54,935
The thing is that this thing is
not 0 for an elliptical orbit.

1174
01:24:57,970 --> 01:24:59,830
So how can you see that?

1175
01:24:59,830 --> 01:25:05,330
Well here at this
point, p is up here.

1176
01:25:05,330 --> 01:25:06,450
L is out.

1177
01:25:06,450 --> 01:25:13,680
And p cross L, just like
before, is out and r hat is in.

1178
01:25:16,662 --> 01:25:18,890
And you say, well, OK.

1179
01:25:18,890 --> 01:25:20,420
Now the same problem.

1180
01:25:20,420 --> 01:25:23,820
If they don't cancel,
it's going to be a vector

1181
01:25:23,820 --> 01:25:25,160
and going to rotate.

1182
01:25:25,160 --> 01:25:27,540
But it has to be conserved.

1183
01:25:27,540 --> 01:25:31,840
So actually, let's
look at it here.

1184
01:25:31,840 --> 01:25:33,910
Here, the main
thing of an ellipse,

1185
01:25:33,910 --> 01:25:36,670
if you have the focus here,
is that this line is not--

1186
01:25:36,670 --> 01:25:39,420
the tangent is not horizontal.

1187
01:25:39,420 --> 01:25:42,480
So the momentum is here.

1188
01:25:42,480 --> 01:25:48,120
L is out of the blackboard,
but p cross L now is like that.

1189
01:25:53,930 --> 01:25:57,500
And the radial vector is here.

1190
01:25:57,500 --> 01:26:00,120
And they don't cancel.

1191
01:26:00,120 --> 01:26:02,870
So the only thing
that can happen--

1192
01:26:02,870 --> 01:26:04,750
since this is vertical,
this is vertical.

1193
01:26:04,750 --> 01:26:09,190
It's a little bit to the left--
is that the r vector must

1194
01:26:09,190 --> 01:26:14,260
be a little vector
horizontal here.

1195
01:26:14,260 --> 01:26:19,380
Because the sum of this
vector and this vector--

1196
01:26:19,380 --> 01:26:20,900
it has to be horizontal.

1197
01:26:20,900 --> 01:26:23,100
Here we don't know
if they can cancel.

1198
01:26:23,100 --> 01:26:26,410
But if they don't cancel,
it's definitely horizontal.

1199
01:26:26,410 --> 01:26:30,220
We know it's conserved, so
it must be horizontal here.

1200
01:26:30,220 --> 01:26:32,010
So it points in.

1201
01:26:32,010 --> 01:26:36,950
So the Runge-Lenz
vector r points in.

1202
01:26:36,950 --> 01:26:40,820
And it's, in fact, that.

1203
01:26:40,820 --> 01:26:46,520
So here you go, this is a
vector that is conserved.

1204
01:26:46,520 --> 01:26:49,010
And its properties
that is really

1205
01:26:49,010 --> 01:26:51,720
about the axis of the ellipse.

1206
01:26:51,720 --> 01:26:54,500
It tells you where the axis is.

1207
01:26:54,500 --> 01:26:59,470
In Einstein's theory of gravity,
the potential is not 1/r

1208
01:26:59,470 --> 01:27:02,590
and the ellipsis [? precess ?]
and the Runge vector

1209
01:27:02,590 --> 01:27:03,920
is not conserved.

1210
01:27:03,920 --> 01:27:07,810
But in 1/r potentials,
it is conserved.

1211
01:27:07,810 --> 01:27:11,540
The final thing-- sorry
for taking so long--

1212
01:27:11,540 --> 01:27:14,940
is that the magnitude
of r is precisely

1213
01:27:14,940 --> 01:27:17,730
the eccentricity of the orbit.

1214
01:27:17,730 --> 01:27:21,110
So it's a really nice way
of characterizing the orbits

1215
01:27:21,110 --> 01:27:24,930
and we'll be using it
in the next lecture.

1216
01:27:24,930 --> 01:27:26,907
See you on Wednesday.