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PROFESSOR: All right.

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00:00:27,000 --> 00:00:28,664
Hi, everyone.

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00:00:28,664 --> 00:00:30,276
AUDIENCE: Hi.

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00:00:30,276 --> 00:00:32,650
PROFESSOR: We're getting
towards the end of the semester.

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00:00:32,650 --> 00:00:37,000
Things are starting to
cohere and come together.

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00:00:37,000 --> 00:00:40,540
We have one more midterm exam.

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00:00:40,540 --> 00:00:44,880
So there is an exam
next Thursday, the 18th.

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00:00:44,880 --> 00:00:45,380
OK?

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00:00:45,380 --> 00:00:47,730
There will be a problem set due.

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00:00:47,730 --> 00:00:50,310
It'll be posted later today,
and it will be due next week

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00:00:50,310 --> 00:00:51,920
on Tuesday as usual.

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00:00:51,920 --> 00:00:55,840
Of course, next week on Tuesday
is a holiday technically,

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00:00:55,840 --> 00:00:58,530
so we'll actually make the
due date be on Wednesday.

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00:00:58,530 --> 00:01:01,280
So on Wednesday at 10 o'clock.

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00:01:01,280 --> 00:01:02,870
You should think
of this problem set

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00:01:02,870 --> 00:01:04,930
as part of the
review for the exam.

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00:01:04,930 --> 00:01:09,820
Material that is covered
today and on Thursday

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00:01:09,820 --> 00:01:11,900
will be fair game for the exam.

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00:01:14,560 --> 00:01:16,070
So the format of
the exam is going

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00:01:16,070 --> 00:01:17,722
to be much more canonical.

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00:01:17,722 --> 00:01:19,430
It's going to be a
series of short answer

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00:01:19,430 --> 00:01:21,450
plus a series of computations.

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00:01:21,450 --> 00:01:25,070
They'll be, roughly speaking,
at the level of the problem

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00:01:25,070 --> 00:01:26,990
sets, and a practice
exam will be

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00:01:26,990 --> 00:01:28,650
posted in the next
couple of days.

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00:01:32,130 --> 00:01:35,350
The practice exam is going
to be of the same general

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00:01:35,350 --> 00:01:36,920
intellectual
difficulty, but it's

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00:01:36,920 --> 00:01:40,140
going to be considerably
longer than the actual exam.

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00:01:40,140 --> 00:01:42,560
So as a check for yourself
here's what I would recommend.

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00:01:42,560 --> 00:01:44,820
I would recommend sitting
down and giving yourself

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00:01:44,820 --> 00:01:48,660
an hour and a half or two
hours with the practice exam

39
00:01:48,660 --> 00:01:51,510
and see how that goes.

40
00:01:51,510 --> 00:01:53,380
OK?

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00:01:53,380 --> 00:01:55,290
Try to give yourself
the time constraint.

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00:01:55,290 --> 00:01:59,240
And here's one of the
things that-- there's

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00:01:59,240 --> 00:02:01,810
a very strong correlation
between having

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00:02:01,810 --> 00:02:05,540
nailed the problem sets, and
worked through all the practice

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00:02:05,540 --> 00:02:07,780
exams, and just done
a lot of problems,

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00:02:07,780 --> 00:02:09,130
and doing well on exams.

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00:02:09,130 --> 00:02:10,694
You'll be more confident.

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00:02:10,694 --> 00:02:12,610
The only way to study
for these things is just

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00:02:12,610 --> 00:02:13,660
do a lot of problems.

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00:02:13,660 --> 00:02:17,380
So on Stellar, for example,
are a bunch of previous problem

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00:02:17,380 --> 00:02:19,780
sets, and previous
exams and practice

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00:02:19,780 --> 00:02:22,080
exams from previous years.

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00:02:22,080 --> 00:02:25,700
I would encourage you to
look at those also on OCW

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00:02:25,700 --> 00:02:27,240
and use those as practice.

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00:02:27,240 --> 00:02:28,794
OK, any practical
questions before we

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00:02:28,794 --> 00:02:29,710
get on to the physics.

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00:02:29,710 --> 00:02:30,417
Yeah?

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00:02:30,417 --> 00:02:32,802
AUDIENCE: Will the exam
cover all the material,

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00:02:32,802 --> 00:02:34,427
or all the material [INAUDIBLE]?

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00:02:34,427 --> 00:02:36,010
PROFESSOR: This is
a cumulative topic.

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00:02:36,010 --> 00:02:37,570
So the question is does it
cover everything, or just

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00:02:37,570 --> 00:02:38,560
last couple of weeks.

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00:02:38,560 --> 00:02:40,800
And it's a cumulative topic,
so the entire semester

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00:02:40,800 --> 00:02:44,820
is necessary in order
to answer the problems.

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00:02:44,820 --> 00:02:47,620
Other questions?

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00:02:47,620 --> 00:02:48,621
OK.

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00:02:48,621 --> 00:02:50,980
Anything else?

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00:02:50,980 --> 00:02:57,350
So quick couple of
review questions

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00:02:57,350 --> 00:03:00,234
before we launch into the
main material of today.

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00:03:00,234 --> 00:03:01,900
These are going to
turn out to be useful

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00:03:01,900 --> 00:03:04,970
reminders later on
in today's lecture.

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00:03:04,970 --> 00:03:06,890
So first, suppose
I tell you I have

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00:03:06,890 --> 00:03:11,700
a system with energy operator
E, which has an operator A such

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00:03:11,700 --> 00:03:18,380
that the commutator of E with
A would say plus h bar A. OK?

75
00:03:18,380 --> 00:03:21,120
What does that tell you about
the spectrum of the energy

76
00:03:21,120 --> 00:03:21,620
operator?

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00:03:24,876 --> 00:03:26,064
AUDIENCE: It's a ladder.

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00:03:26,064 --> 00:03:27,480
PROFESSOR: It's a
ladder, exactly.

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00:03:27,480 --> 00:03:30,400
So this tells you that the
spectrum of E, or the energy

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00:03:30,400 --> 00:03:34,060
eigenvalues En,
are evenly spaced--

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00:03:34,060 --> 00:03:40,030
and we need a dimensional
constant here,

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00:03:40,030 --> 00:03:49,050
omega-- evenly spaced
by h bar omega.

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00:03:49,050 --> 00:03:54,530
And more precisely, that
given a state phi E,

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00:03:54,530 --> 00:03:58,030
we can act on it
with the operator A

85
00:03:58,030 --> 00:04:02,310
to give us a new state, which
is also an energy eigenstate,

86
00:04:02,310 --> 00:04:06,250
with energy E plus h bar omega.

87
00:04:06,250 --> 00:04:07,900
right?

88
00:04:07,900 --> 00:04:10,460
So any time you see that
commutation relation,

89
00:04:10,460 --> 00:04:13,610
you know this fact to be true.

90
00:04:13,610 --> 00:04:14,540
Second statement.

91
00:04:14,540 --> 00:04:17,220
Suppose I have an
operator B which

92
00:04:17,220 --> 00:04:19,860
commutes with the
energy operator.

93
00:04:19,860 --> 00:04:20,360
OK?

94
00:04:20,360 --> 00:04:22,547
So that the commutator vanishes.

95
00:04:22,547 --> 00:04:24,255
What does that tell
you about the system?

96
00:04:27,814 --> 00:04:30,130
AUDIENCE: Simultaneous
eigenfunctions [INAUDIBLE].

97
00:04:30,130 --> 00:04:31,005
PROFESSOR: Excellent.

98
00:04:31,005 --> 00:04:32,970
So one one consequence
is that there

99
00:04:32,970 --> 00:04:37,100
exists simultaneous
eigenfunctions phi sub E, B,

100
00:04:37,100 --> 00:04:46,576
which are simultaneous
eigenfunctions of both E and B.

101
00:04:46,576 --> 00:04:47,700
What else does it tell you?

102
00:04:52,090 --> 00:04:56,557
Well, notice the following.

103
00:04:56,557 --> 00:04:58,890
Notice that if we took this
computation relation and set

104
00:04:58,890 --> 00:05:01,670
omega to 0, we get this
commutation relation.

105
00:05:01,670 --> 00:05:04,440
So this commutation relation is
of the form of this commutation

106
00:05:04,440 --> 00:05:06,770
relation with omega equals 0.

107
00:05:06,770 --> 00:05:09,890
So what does that tell
you about the states?

108
00:05:09,890 --> 00:05:11,210
About the energy eigenvalues?

109
00:05:14,490 --> 00:05:17,740
What happens if I
take a state phi sub E

110
00:05:17,740 --> 00:05:20,060
and I act on it with B?

111
00:05:20,060 --> 00:05:20,780
What do I get?

112
00:05:25,819 --> 00:05:27,360
What can you say
about this function?

113
00:05:31,300 --> 00:05:34,287
Well, what is its
eigenvalue under E?

114
00:05:34,287 --> 00:05:36,370
It's E. It's the same
thing, because they commute.

115
00:05:36,370 --> 00:05:38,300
You can pull the E
through, multiply it,

116
00:05:38,300 --> 00:05:40,760
you get the constant E,
you pull it back out.

117
00:05:40,760 --> 00:05:44,770
This is still an
eigenfunction of phi

118
00:05:44,770 --> 00:05:47,345
of the energy operator
with the same eigenvalue.

119
00:05:47,345 --> 00:05:49,220
But is it necessarily
the same eigenfunction?

120
00:05:52,610 --> 00:05:54,590
No, because B may act
on phi and just give you

121
00:05:54,590 --> 00:05:55,340
a different state.

122
00:05:55,340 --> 00:06:00,520
So this is some eigenfunction
with the same energy,

123
00:06:00,520 --> 00:06:03,994
but it may be a
different eigenfunction.

124
00:06:03,994 --> 00:06:04,669
OK?

125
00:06:04,669 --> 00:06:06,210
So let's think of
an example of this.

126
00:06:06,210 --> 00:06:08,300
An example of this is
consider a free particle.

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00:06:11,150 --> 00:06:13,210
In the case of a
free particle, we

128
00:06:13,210 --> 00:06:18,590
have E is equal to
P squared upon 2m.

129
00:06:18,590 --> 00:06:21,500
And as a consequence
E and P commute.

130
00:06:26,510 --> 00:06:29,160
Everyone agree with that?

131
00:06:29,160 --> 00:06:30,630
Everyone happy with
that statement?

132
00:06:30,630 --> 00:06:31,420
They commute.

133
00:06:31,420 --> 00:06:32,950
So plus 0 if you will.

134
00:06:35,750 --> 00:06:38,600
And here what I wanted to say
is when you have an operator

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00:06:38,600 --> 00:06:41,990
that commutes with
the energy, then

136
00:06:41,990 --> 00:06:45,860
there can be multiple states
with the same energy which

137
00:06:45,860 --> 00:06:48,200
are different states, right?

138
00:06:48,200 --> 00:06:49,430
Different states entirely.

139
00:06:49,430 --> 00:06:51,230
So for example,
in this case, what

140
00:06:51,230 --> 00:06:51,880
are the energy eigenfunctions?

141
00:06:51,880 --> 00:06:52,850
Well, e to the ikx.

142
00:06:55,718 --> 00:07:00,391
And this has energy h bar
squared k squared upon 2m.

143
00:07:00,391 --> 00:07:02,890
But there's another state, which
is a different state, which

144
00:07:02,890 --> 00:07:05,325
has the same energy.

145
00:07:05,325 --> 00:07:08,340
e to the minus ikx.

146
00:07:08,340 --> 00:07:09,610
OK?

147
00:07:09,610 --> 00:07:13,320
So when you have an operator
that commutes with the energy

148
00:07:13,320 --> 00:07:19,770
operator, you can have
simultaneous eigenfunctions.

149
00:07:19,770 --> 00:07:24,950
And you can also have
multiple eigenfunctions

150
00:07:24,950 --> 00:07:26,900
that have the same
energy eigenvalue,

151
00:07:26,900 --> 00:07:28,080
but are different functions.

152
00:07:28,080 --> 00:07:29,920
For example, this.

153
00:07:29,920 --> 00:07:31,390
Everyone cool with that?

154
00:07:31,390 --> 00:07:34,040
Now, just to make clear
that it's actually

155
00:07:34,040 --> 00:07:36,320
the commuting that
matters, imagine

156
00:07:36,320 --> 00:07:39,020
we took not the free particle,
but the harmonic oscillator.

157
00:07:42,940 --> 00:07:51,700
E is p squared upon 2m plus m
omega squared upon 2 x squared.

158
00:07:51,700 --> 00:07:57,170
Is it true that E-- I'll call
this harmonic oscillator--

159
00:07:57,170 --> 00:08:00,869
does E harmonic
oscillator commute with P?

160
00:08:00,869 --> 00:08:02,410
No, you're going to
get a term that's

161
00:08:02,410 --> 00:08:06,282
m omega squared x
from the commutator.

162
00:08:06,282 --> 00:08:08,240
The potential is not
translationally invariant.

163
00:08:08,240 --> 00:08:09,780
It does not commute
with momentum.

164
00:08:09,780 --> 00:08:11,860
So this is not equal to 0.

165
00:08:11,860 --> 00:08:14,865
So that suggests that this
degeneracy, two states having

166
00:08:14,865 --> 00:08:16,490
the same energy,
should not be present.

167
00:08:16,490 --> 00:08:18,490
And indeed, are the
states degenerate

168
00:08:18,490 --> 00:08:20,530
for the harmonic oscillator.

169
00:08:20,530 --> 00:08:22,100
No.

170
00:08:22,100 --> 00:08:22,800
No degeneracy.

171
00:08:28,310 --> 00:08:28,930
Yeah?

172
00:08:28,930 --> 00:08:31,798
AUDIENCE: We did
something [INAUDIBLE]

173
00:08:31,798 --> 00:08:34,409
where I think it said that the
eigenfunctions were complete.

174
00:08:34,409 --> 00:08:35,034
PROFESSOR: Yes.

175
00:08:35,034 --> 00:08:36,020
AUDIENCE: What does that mean?

176
00:08:36,020 --> 00:08:37,340
PROFESSOR: What does it
mean for the eigenfunctions

177
00:08:37,340 --> 00:08:38,110
to be complete?

178
00:08:38,110 --> 00:08:39,860
What that means is
that they form a basis.

179
00:08:42,369 --> 00:08:44,119
AUDIENCE: So the basis
doesn't necessarily

180
00:08:44,119 --> 00:08:46,674
mean not [INAUDIBLE].

181
00:08:46,674 --> 00:08:49,340
PROFESSOR: Yeah, no one told you
the basis had to be degenerate,

182
00:08:49,340 --> 00:08:50,700
and in particular,
that's a excellent--

183
00:08:50,700 --> 00:08:52,324
so the question here
is, wait a minute,

184
00:08:52,324 --> 00:08:54,726
I thought a basis had to be
a complete set-- if you had

185
00:08:54,726 --> 00:08:57,100
an energy operator and you
constructed the energy eigen--

186
00:08:57,100 --> 00:08:57,970
this is a very good question.

187
00:08:57,970 --> 00:08:58,620
Thank you.

188
00:08:58,620 --> 00:09:01,220
If I have the energy operator
and I construct it's energy

189
00:09:01,220 --> 00:09:03,220
eigenfunctions, then those
energy eigenfunctions

190
00:09:03,220 --> 00:09:06,060
form a complete basis for
any arbitrary function.

191
00:09:06,060 --> 00:09:07,927
Any function can be
expanded in it, right?

192
00:09:07,927 --> 00:09:09,510
So for example, for
the free particle,

193
00:09:09,510 --> 00:09:11,301
the energy eigenfunctions
are e to the ikx,

194
00:09:11,301 --> 00:09:13,890
the momentum eigenfunctions
for any value of k.

195
00:09:13,890 --> 00:09:15,800
But wait, how can there
be multiple states

196
00:09:15,800 --> 00:09:17,804
with the same energy?

197
00:09:17,804 --> 00:09:19,470
Isn't that double
counting or something?

198
00:09:19,470 --> 00:09:21,680
And the important thing is
those guys have the same energy,

199
00:09:21,680 --> 00:09:23,150
but they have different momenta.

200
00:09:23,150 --> 00:09:24,630
They're different states.

201
00:09:24,630 --> 00:09:26,390
One has momentum h bar k.

202
00:09:26,390 --> 00:09:28,037
One has momentum minus h bar k.

203
00:09:28,037 --> 00:09:29,620
And you know that
that has to be true,

204
00:09:29,620 --> 00:09:31,400
because the Fourier
theorem tells you,

205
00:09:31,400 --> 00:09:33,830
in order to form a complete
basis, you need all of them,

206
00:09:33,830 --> 00:09:36,080
all possible values of k.

207
00:09:36,080 --> 00:09:38,340
So there's no problem with
being a complete basis

208
00:09:38,340 --> 00:09:43,120
and having states have the
same energy eigenvalue, OK?

209
00:09:43,120 --> 00:09:44,720
It's a good question.

210
00:09:44,720 --> 00:09:45,920
Other questions?

211
00:09:45,920 --> 00:09:46,420
Yeah?

212
00:09:46,420 --> 00:09:49,300
AUDIENCE: So if we have a
potential that only admits

213
00:09:49,300 --> 00:09:51,940
bound states, we'll never
have this commutation

214
00:09:51,940 --> 00:09:52,670
happen basically?

215
00:09:52,670 --> 00:09:53,550
PROFESSOR: Yeah, exactly.

216
00:09:53,550 --> 00:09:54,050
Excellent.

217
00:09:54,050 --> 00:09:55,360
So the observation is this.

218
00:09:55,360 --> 00:09:58,810
Look, imagine I have a
potential that's not trivial.

219
00:09:58,810 --> 00:10:02,200
It's not 0, OK?

220
00:10:02,200 --> 00:10:05,860
Will the momentum commute
with the energy operator.

221
00:10:05,860 --> 00:10:08,029
No, because it's got
a potential that's

222
00:10:08,029 --> 00:10:10,570
going to be acted upon by P, so
you'll get a derivative term.

223
00:10:10,570 --> 00:10:14,940
But more precisely, if I have
a system with bound states,

224
00:10:14,940 --> 00:10:17,980
I have to have a
potential, right?

225
00:10:17,980 --> 00:10:21,360
And then I can't have P
commuting with the energy

226
00:10:21,360 --> 00:10:24,160
operator, which means I
can't have degeneracies.

227
00:10:24,160 --> 00:10:25,770
So indeed, if you
have bound states,

228
00:10:25,770 --> 00:10:27,626
you cannot have degeneracies.

229
00:10:27,626 --> 00:10:28,500
That's exactly right.

230
00:10:28,500 --> 00:10:29,102
Yeah?

231
00:10:29,102 --> 00:10:31,560
AUDIENCE: But doesn't this
break down in higher dimensions?

232
00:10:31,560 --> 00:10:33,060
PROFESSOR: Excellent,
so we're going

233
00:10:33,060 --> 00:10:35,150
to come back to higher
dimensions later.

234
00:10:35,150 --> 00:10:37,069
So the question
predicts what's going

235
00:10:37,069 --> 00:10:38,610
to happen in the
rest of the lecture.

236
00:10:38,610 --> 00:10:39,810
What we're going to
do in just a minute

237
00:10:39,810 --> 00:10:41,060
is we're going to
start working in three

238
00:10:41,060 --> 00:10:42,330
dimensions for the first time.

239
00:10:42,330 --> 00:10:43,840
We're going to leave 1D behind.

240
00:10:43,840 --> 00:10:45,690
We're going to take our
tripped out tricycle

241
00:10:45,690 --> 00:10:49,020
and replace it with a Yamaha.

242
00:10:49,020 --> 00:10:52,170
As you'll see, it has the
same basic physics driving

243
00:10:52,170 --> 00:10:54,800
its, well, self.

244
00:10:54,800 --> 00:10:56,312
It's the same dynamics.

245
00:10:56,312 --> 00:10:58,145
But I want to emphasize
a couple things that

246
00:10:58,145 --> 00:10:59,360
are going to show up.

247
00:10:59,360 --> 00:11:01,130
So the question is,
isn't this story

248
00:11:01,130 --> 00:11:02,590
different in three dimensions?

249
00:11:02,590 --> 00:11:06,192
And we shall see exactly what
happens in higher dimensions.

250
00:11:06,192 --> 00:11:07,400
We'll work in two dimensions.

251
00:11:07,400 --> 00:11:08,691
We'll work in three dimensions.

252
00:11:08,691 --> 00:11:10,882
We'll work in more.

253
00:11:10,882 --> 00:11:13,090
Doesn't really matter how
many dimensions we work in.

254
00:11:13,090 --> 00:11:14,900
You'll see it.

255
00:11:14,900 --> 00:11:16,394
OK, third thing.

256
00:11:16,394 --> 00:11:18,560
You studied this in some
detail in your problem set.

257
00:11:18,560 --> 00:11:20,040
Suppose I have an
energy operator

258
00:11:20,040 --> 00:11:23,240
that commutes with a
unitary operator, U, OK?

259
00:11:23,240 --> 00:11:25,800
So it commutes to 0.

260
00:11:25,800 --> 00:11:30,540
And U is unitary, so U dagger
U is one, is the identity.

261
00:11:30,540 --> 00:11:34,100
So what does this tell you?

262
00:11:34,100 --> 00:11:38,420
Well, first off
from these guys it

263
00:11:38,420 --> 00:11:48,510
tells us that we can have
simultaneous eigenfunctions.

264
00:11:55,400 --> 00:12:00,190
It also tells us too that
if we take our state phi

265
00:12:00,190 --> 00:12:02,860
and we act on it with
U, this could give us

266
00:12:02,860 --> 00:12:09,310
a new state, phi tilde sub
E, which will necessarily

267
00:12:09,310 --> 00:12:12,210
have the same energy eigenvalue,
because U and E commute.

268
00:12:12,210 --> 00:12:13,930
But it may be a different state.

269
00:12:16,510 --> 00:12:18,472
We'll come to this
in more detail later.

270
00:12:18,472 --> 00:12:20,555
But the third thing, and
I want to emphasize this,

271
00:12:20,555 --> 00:12:23,116
is this tells us, look, we
have a unitary operator.

272
00:12:23,116 --> 00:12:24,990
We can always write the
unitary operator as e

273
00:12:24,990 --> 00:12:26,720
to the i of a
Hermitian operator.

274
00:12:31,490 --> 00:12:34,410
So what is the meaning of
the Hermitian operator?

275
00:12:34,410 --> 00:12:35,530
What is this guy?

276
00:12:35,530 --> 00:12:37,180
So in your problem
set, you looked

277
00:12:37,180 --> 00:12:40,660
at what unitary operators are.

278
00:12:40,660 --> 00:12:43,365
And in the problem set, it's
discussed in some detail

279
00:12:43,365 --> 00:12:45,230
that there's a
relationship between

280
00:12:45,230 --> 00:12:47,630
a unitary transformation,
or unitary operator,

281
00:12:47,630 --> 00:12:49,455
and the symmetry.

282
00:12:49,455 --> 00:12:51,080
A symmetry is when
you take your system

283
00:12:51,080 --> 00:12:55,207
and you do something to it,
like a rotation or translation,

284
00:12:55,207 --> 00:12:57,290
and it's a symmetry if it
doesn't change anything,

285
00:12:57,290 --> 00:12:59,034
if the energy remains invariant.

286
00:12:59,034 --> 00:13:01,450
So if the energy doesn't change
under this transformation,

287
00:13:01,450 --> 00:13:03,880
we call that a symmetry.

288
00:13:03,880 --> 00:13:07,080
And we also showed that
symmetries, or translations,

289
00:13:07,080 --> 00:13:08,850
are generated by
unitary operators.

290
00:13:08,850 --> 00:13:12,350
For example, my favorite
examples are the translate by L

291
00:13:12,350 --> 00:13:14,350
operator, which is unitary.

292
00:13:14,350 --> 00:13:18,060
And also e to the minus dxl.

293
00:13:20,980 --> 00:13:25,890
And the boost by
q operator, which

294
00:13:25,890 --> 00:13:28,460
similarly is e to the minus qdp.

295
00:13:37,760 --> 00:13:40,610
And the time translation
operator, U sub t,

296
00:13:40,610 --> 00:13:46,690
which is equal to e to the minus
i t over h bar energy operator.

297
00:13:46,690 --> 00:13:47,310
OK?

298
00:13:47,310 --> 00:13:50,470
So these are
transformation operators.

299
00:13:50,470 --> 00:13:53,420
These are symmetry operators,
which translate you by L,

300
00:13:53,420 --> 00:13:55,900
boost or speed you
up by momentum q,

301
00:13:55,900 --> 00:13:57,510
evolve you forward in time t.

302
00:13:57,510 --> 00:14:00,920
And they can all be expressed
as e to the unitary operator.

303
00:14:00,920 --> 00:14:11,250
So this in particular
is l i over h bar p.

304
00:14:11,250 --> 00:14:14,530
And this is similarly x.

305
00:14:14,530 --> 00:14:19,402
And earlier we understood
the role of momentum

306
00:14:19,402 --> 00:14:21,360
having to do translations
in the following way.

307
00:14:21,360 --> 00:14:23,060
There's a beautiful
theorem about this.

308
00:14:23,060 --> 00:14:25,900
If you take a system and
its translation invariant,

309
00:14:25,900 --> 00:14:27,890
the classical statement
of Noether's theorem

310
00:14:27,890 --> 00:14:30,670
is that there's a conserved
quantity associated

311
00:14:30,670 --> 00:14:31,780
with that translation.

312
00:14:31,780 --> 00:14:34,260
That conserved quantity
is the momentum.

313
00:14:34,260 --> 00:14:36,100
And quantum mechanically,
the generator

314
00:14:36,100 --> 00:14:38,770
of that transformation
the Hermitian operator

315
00:14:38,770 --> 00:14:40,770
that goes upstairs
in the unitary

316
00:14:40,770 --> 00:14:43,260
is the operator associated
to that conserved quantity,

317
00:14:43,260 --> 00:14:45,280
associated to that observable.

318
00:14:45,280 --> 00:14:47,010
You have translations.

319
00:14:47,010 --> 00:14:49,011
There's a conserved
quantity, which is momentum.

320
00:14:49,011 --> 00:14:51,510
And the thing that generates
translations, the operator that

321
00:14:51,510 --> 00:14:54,300
generates translations, is the
operator representing momentum.

322
00:15:00,637 --> 00:15:02,845
So each of these are going
to come up later in today,

323
00:15:02,845 --> 00:15:06,250
and I just wanted to flag
them down before the moment.

324
00:15:06,250 --> 00:15:10,345
OK, questions before we move on?

325
00:15:10,345 --> 00:15:11,315
Yeah?

326
00:15:11,315 --> 00:15:14,952
AUDIENCE: So you made the claim
that every unitary operator can

327
00:15:14,952 --> 00:15:17,195
be expressed as p to
the eigenfunction.

328
00:15:17,195 --> 00:15:19,570
PROFESSOR: OK, I should be a
little bit careful, but yes.

329
00:15:19,570 --> 00:15:21,910
That's right.

330
00:15:21,910 --> 00:15:25,134
AUDIENCE: But if I
take the [INAUDIBLE] I

331
00:15:25,134 --> 00:15:26,800
should be able to
figure out what it is,

332
00:15:26,800 --> 00:15:28,230
but you can't take
the [INAUDIBLE]

333
00:15:28,230 --> 00:15:29,300
PROFESSOR: The more
precise statement

334
00:15:29,300 --> 00:15:31,140
is that any unitary--
any one parameter

335
00:15:31,140 --> 00:15:34,142
family of unitary operators
can be expressed in that form.

336
00:15:34,142 --> 00:15:35,600
And then you can
take a derivative.

337
00:15:35,600 --> 00:15:38,060
And that's the theory
of [INAUDIBLE],

338
00:15:38,060 --> 00:15:40,350
which is beyond the scope.

339
00:15:43,080 --> 00:15:45,350
Let me make a very
specific statement, which

340
00:15:45,350 --> 00:15:47,190
is that one parameter
of [INAUDIBLE]

341
00:15:47,190 --> 00:15:48,148
unitary transformation.

342
00:15:48,148 --> 00:15:50,240
So translations by l,
where you can vary l,

343
00:15:50,240 --> 00:15:52,150
can be expressed in that form.

344
00:15:52,150 --> 00:15:53,900
And that's a very
general statement.

345
00:15:53,900 --> 00:15:58,535
OK, so with all that as
prelude, let's go back to 3D.

346
00:16:03,360 --> 00:16:07,540
So in 3D, the energy operator--
so what's going to change?

347
00:16:07,540 --> 00:16:13,180
Now instead of just having
position and its momentum,

348
00:16:13,180 --> 00:16:16,090
we now also have-- I'll call
this P sub x-- we can also

349
00:16:16,090 --> 00:16:19,090
have a y-coordinate and
we have a z-coordinate.

350
00:16:19,090 --> 00:16:20,820
And each of them
has its momentum.

351
00:16:20,820 --> 00:16:23,610
P sub z and P sub y.

352
00:16:26,290 --> 00:16:28,740
And here's just a quick
practical question.

353
00:16:28,740 --> 00:16:36,020
We know that x with Px
is equal to i h bar.

354
00:16:44,170 --> 00:16:46,750
So what do you expect
to be true of x with y?

355
00:16:50,089 --> 00:16:51,050
AUDIENCE: 0.

356
00:16:51,050 --> 00:16:52,729
PROFESSOR: Why?

357
00:16:52,729 --> 00:16:53,645
AUDIENCE: [INAUDIBLE].

358
00:16:56,967 --> 00:16:58,800
PROFESSOR: What does
this equation tell you?

359
00:16:58,800 --> 00:17:00,008
What is its physical content?

360
00:17:02,170 --> 00:17:03,670
Well, that they
don't commute, good.

361
00:17:03,670 --> 00:17:05,140
What does that tell
you physically?

362
00:17:05,140 --> 00:17:05,479
Yes?

363
00:17:05,479 --> 00:17:07,437
AUDIENCE: That there's
an uncertainty principle

364
00:17:07,437 --> 00:17:08,420
connecting the two.

365
00:17:08,420 --> 00:17:08,720
PROFESSOR: Excellent.

366
00:17:08,720 --> 00:17:09,550
So that's one statement.

367
00:17:09,550 --> 00:17:11,000
So the consequence of
this is that there's

368
00:17:11,000 --> 00:17:12,041
an uncertainty principle.

369
00:17:12,041 --> 00:17:16,130
Delta x delta Px must be greater
than or equal to h bar upon 2.

370
00:17:16,130 --> 00:17:17,550
What's another way
of saying this?

371
00:17:22,349 --> 00:17:24,999
Do there exist simultaneous
eigenfunctions of x and P?

372
00:17:24,999 --> 00:17:25,540
AUDIENCE: No.

373
00:17:25,540 --> 00:17:26,123
PROFESSOR: No.

374
00:17:26,123 --> 00:17:28,400
No simultaneous eigenfunctions.

375
00:17:28,400 --> 00:17:32,280
OK, so you can't
have a definite value

376
00:17:32,280 --> 00:17:34,234
of x and a definite value
of P simultaneously.

377
00:17:34,234 --> 00:17:35,150
There's no such state.

378
00:17:35,150 --> 00:17:36,358
It's not that you can't know.

379
00:17:36,358 --> 00:17:39,180
It's that there's no such state.

380
00:17:39,180 --> 00:17:42,416
Do you expect to be able
to know the position in x

381
00:17:42,416 --> 00:17:45,710
and the position in
y simultaneously?

382
00:17:45,710 --> 00:17:46,240
Sure.

383
00:17:46,240 --> 00:17:48,551
OK, so this turns out to be 0.

384
00:17:48,551 --> 00:17:50,050
And in some sense,
you can take that

385
00:17:50,050 --> 00:17:51,850
as a definition of
quantum mechanics.

386
00:17:51,850 --> 00:17:54,500
x and y need to be 0.

387
00:17:54,500 --> 00:18:00,780
And similarly, Px
and Py commute.

388
00:18:00,780 --> 00:18:02,340
The momenta are independent.

389
00:18:02,340 --> 00:18:10,890
However, Py and y should
be equal to minus i h bar.

390
00:18:10,890 --> 00:18:11,390
Good.

391
00:18:11,390 --> 00:18:11,740
Exactly.

392
00:18:11,740 --> 00:18:13,290
So the commutators
work out exactly

393
00:18:13,290 --> 00:18:15,600
as you'd naively expect.

394
00:18:15,600 --> 00:18:20,450
Every pair of position and its
momenta commute canonically

395
00:18:20,450 --> 00:18:21,920
to i h bar.

396
00:18:21,920 --> 00:18:25,480
And every pair of
coordinates commute to 0.

397
00:18:25,480 --> 00:18:27,670
Every pair of
momenta commute to 0.

398
00:18:27,670 --> 00:18:28,170
Cool?

399
00:18:33,240 --> 00:18:36,835
So what kind of systems are
we going to interested in?

400
00:18:36,835 --> 00:18:38,710
Well, we're going to be
interested in systems

401
00:18:38,710 --> 00:18:41,430
where the energy
operator is equal to P

402
00:18:41,430 --> 00:18:49,790
vector hat squared upon 2m plus
U of x and y and z, hat, hat.

403
00:18:49,790 --> 00:18:57,770
You can see why dropping
the hats becomes almost /

404
00:18:57,770 --> 00:19:03,710
So in this language, we can
write the Schrodinger equation.

405
00:19:03,710 --> 00:19:07,220
This is just a direct extension
of the 1D Schrodinger equation.

406
00:19:07,220 --> 00:19:09,040
i h bar dt of psi.

407
00:19:09,040 --> 00:19:12,770
Now our wave function is a
function of x and y and z.

408
00:19:12,770 --> 00:19:14,350
There's some finite
probability then

409
00:19:14,350 --> 00:19:15,850
to find a particle
at some position.

410
00:19:15,850 --> 00:19:18,850
That position is labeled
by the three coordinates.

411
00:19:18,850 --> 00:19:21,630
Is equal to-- and of t.

412
00:19:24,890 --> 00:19:26,767
Is equal to-- well,
I'm actually write

413
00:19:26,767 --> 00:19:28,100
this in slightly different form.

414
00:19:28,100 --> 00:19:30,510
This is going to be easier
if I use vector notation.

415
00:19:30,510 --> 00:19:33,750
So I'm going to write
this as psi of r and t,

416
00:19:33,750 --> 00:19:36,260
where r denotes the
position vector,

417
00:19:36,260 --> 00:19:39,460
is equal to the energy
operator acting on it.

418
00:19:39,460 --> 00:19:46,790
And P is just equal to
minus i h bar the gradient.

419
00:19:49,830 --> 00:19:53,780
So this is minus i h bar
squared, or minus h bar squared

420
00:19:53,780 --> 00:20:04,960
upon 2m gradient squared plus
u of x or now u of r psi of r

421
00:20:04,960 --> 00:20:05,950
and t.

422
00:20:11,770 --> 00:20:13,520
Quick question, what
are the units or what

423
00:20:13,520 --> 00:20:18,940
are the dimensions
of psi of r in 3D?

424
00:20:22,692 --> 00:20:24,570
AUDIENCE: [INAUDIBLE].

425
00:20:24,570 --> 00:20:29,170
PROFESSOR: Yeah, one over
length to the root three halves.

426
00:20:29,170 --> 00:20:32,240
And the reason is this norm
squared gives us a probability

427
00:20:32,240 --> 00:20:34,500
density, something
that when we integrated

428
00:20:34,500 --> 00:20:38,660
over all positions in a
region integral d 3x is going

429
00:20:38,660 --> 00:20:40,620
to give us a number,
a probability.

430
00:20:40,620 --> 00:20:42,290
So its actual
magnitude must be--

431
00:20:42,290 --> 00:20:45,670
or its dimension must
be 1 over L to the 3/2.

432
00:20:45,670 --> 00:20:47,580
Just the cube of
what it was in 1D.

433
00:20:50,230 --> 00:20:54,415
And as you'll see
on the problems set

434
00:20:54,415 --> 00:20:57,580
and as we'll do in a couple
lectures down the road,

435
00:20:57,580 --> 00:21:02,430
it's convenient sometimes
to work in Cartesian,

436
00:21:02,430 --> 00:21:04,395
but it's also
sometimes convenient

437
00:21:04,395 --> 00:21:05,770
to work in spherical
coordinates.

438
00:21:09,632 --> 00:21:10,590
And it does not matter.

439
00:21:10,590 --> 00:21:12,006
And here's a really
deep statement

440
00:21:12,006 --> 00:21:13,750
that goes way beyond
quantum mechanics.

441
00:21:13,750 --> 00:21:15,170
It does not matter which
coordinates you work in.

442
00:21:15,170 --> 00:21:17,020
You cannot possibly get a
different answer by using

443
00:21:17,020 --> 00:21:18,140
different coordinates.

444
00:21:18,140 --> 00:21:20,760
So we're going to be ruthless
in exploiting coordinates that

445
00:21:20,760 --> 00:21:22,870
will simplify our problem
throughout the rest

446
00:21:22,870 --> 00:21:24,600
of this course.

447
00:21:24,600 --> 00:21:27,800
In the notes is it a short
discussion of the form

448
00:21:27,800 --> 00:21:30,540
of the Laplacian,
or the gradient

449
00:21:30,540 --> 00:21:34,532
squared in Cartesian spherical
and cylindrical coordinates.

450
00:21:34,532 --> 00:21:36,740
You should feel free to use
any coordinate system you

451
00:21:36,740 --> 00:21:37,630
want at any point.

452
00:21:37,630 --> 00:21:39,690
You just have to be
consistent about it.

453
00:21:42,630 --> 00:21:44,345
So let's work out a
couple of examples.

454
00:21:49,160 --> 00:21:51,010
And here are all
we're going to do

455
00:21:51,010 --> 00:21:53,120
is apply exactly the same
logic that we see over

456
00:21:53,120 --> 00:21:56,410
and over in 1D to
our 3D problems.

457
00:21:56,410 --> 00:22:07,070
So the first example is
a free particle in 3D.

458
00:22:07,070 --> 00:22:09,192
So before I get started
on this, any questions?

459
00:22:09,192 --> 00:22:10,400
Just in general 3D questions?

460
00:22:13,150 --> 00:22:14,150
OK.

461
00:22:14,150 --> 00:22:15,447
So this stuff starts off easy.

462
00:22:15,447 --> 00:22:17,405
And I'm going to work in
Cartesian coordinates.

463
00:22:21,220 --> 00:22:24,990
And a fun problem is
to repeat this analysis

464
00:22:24,990 --> 00:22:28,740
in spherical coordinates,
and we'll do that later on.

465
00:22:28,740 --> 00:22:30,420
OK, so free particle
in 3D, so what

466
00:22:30,420 --> 00:22:32,378
is the energy eigenfunction
equation look like?

467
00:22:32,378 --> 00:22:34,480
We want to find-- the
Schrodinger equation has

468
00:22:34,480 --> 00:22:36,790
exactly the same
structure as before.

469
00:22:36,790 --> 00:22:38,840
It's a linear
differential equation.

470
00:22:38,840 --> 00:22:41,800
So if we find the eigenfunctions
of the energy operator,

471
00:22:41,800 --> 00:22:45,030
we can use superposition to
construct the general solution,

472
00:22:45,030 --> 00:22:46,160
right?

473
00:22:46,160 --> 00:22:49,980
So exactly as in 1D, I'm going
to construct first the energy

474
00:22:49,980 --> 00:22:52,700
eigenfunctions , and then use
them in superposition to find

475
00:22:52,700 --> 00:22:54,960
a general solution to
the Schrodinger equation.

476
00:22:54,960 --> 00:22:55,640
OK?

477
00:22:55,640 --> 00:22:57,910
So let's construct the
energy eigenfunctions.

478
00:22:57,910 --> 00:23:00,540
So what is the energy
eigenvalue equation look like?

479
00:23:00,540 --> 00:23:06,000
Well, E on psi is equal to
minus h bar squared upon 2m.

480
00:23:06,000 --> 00:23:08,860
And in Cartesian, the
Laplacian is derivative respect

481
00:23:08,860 --> 00:23:10,700
to x squared plus
derivative with respect

482
00:23:10,700 --> 00:23:15,280
to y squared plus derivative
with respect to z squared.

483
00:23:15,280 --> 00:23:17,904
And we have no potential,
so this is just psi.

484
00:23:17,904 --> 00:23:19,820
So that would be energy
operator acting on it.

485
00:23:19,820 --> 00:23:21,694
And the eigenvalue
equation is at a constant,

486
00:23:21,694 --> 00:23:24,886
the energy E on psi
satisfies this equation.

487
00:23:24,886 --> 00:23:26,260
I'm going to write
this phi sub e

488
00:23:26,260 --> 00:23:28,560
to continue with
our notation of phi

489
00:23:28,560 --> 00:23:30,070
being the energy eigenfunctions.

490
00:23:30,070 --> 00:23:31,903
It of course, doesn't
matter what I call it,

491
00:23:31,903 --> 00:23:36,270
but just for consistency I'm
going to use the letter phi.

492
00:23:36,270 --> 00:23:39,240
So this is a very easy
equation to solve.

493
00:23:41,770 --> 00:23:45,240
In particular, it has
a lovely property,

494
00:23:45,240 --> 00:23:46,560
which is that it's separable.

495
00:23:50,600 --> 00:23:51,150
OK?

496
00:23:51,150 --> 00:23:52,525
So separable means
the following.

497
00:23:52,525 --> 00:23:58,350
It means, look, I note, I just
observed that this differential

498
00:23:58,350 --> 00:24:00,749
equation can be written
as a sum of terms

499
00:24:00,749 --> 00:24:03,290
where there's a derivative with
respect to only one variable.

500
00:24:03,290 --> 00:24:04,390
There's a differential
operator with respect

501
00:24:04,390 --> 00:24:06,640
to only one variable and
another differential operator

502
00:24:06,640 --> 00:24:08,930
with respect to only one
variable added together.

503
00:24:08,930 --> 00:24:12,230
And when you see that,
you can separate.

504
00:24:12,230 --> 00:24:14,140
And here's what I
mean by separate.

505
00:24:14,140 --> 00:24:15,550
I'm going to just construct.

506
00:24:15,550 --> 00:24:17,250
I'm not going to say that
this is a general solution.

507
00:24:17,250 --> 00:24:18,666
I'm just going to
try to construct

508
00:24:18,666 --> 00:24:20,470
a set of solutions of
the following form.

509
00:24:20,470 --> 00:24:24,610
Psi E of x, y, and
z is equal to psi

510
00:24:24,610 --> 00:24:33,800
E-- I will call this psi
sub x of x times phi sub

511
00:24:33,800 --> 00:24:38,678
y of y times phi sub z of z.

512
00:24:41,460 --> 00:24:44,250
And if we take this and we plug
it in, let's see what we get.

513
00:24:44,250 --> 00:24:48,940
This gives us that E on phi E
is equal to minus h bar squared

514
00:24:48,940 --> 00:24:49,440
upon 2m.

515
00:24:52,210 --> 00:24:55,240
Well the dx squared
acting on phi sub e

516
00:24:55,240 --> 00:24:56,610
is only going to hit this guy.

517
00:24:56,610 --> 00:25:03,580
So I'm going to get phi x
prime prime phi y phi z.

518
00:25:03,580 --> 00:25:08,870
Plus from the next term phi
y prime prime phi x phi z.

519
00:25:08,870 --> 00:25:13,301
And from the next term phi
z prime prime phi x phi y.

520
00:25:15,817 --> 00:25:18,400
But now I can do a sneaky thing
and divide the entire equation

521
00:25:18,400 --> 00:25:19,470
by phi sub e.

522
00:25:19,470 --> 00:25:21,800
Phi sub e is phi x phi y phi z.

523
00:25:21,800 --> 00:25:28,420
So if I do so, I lose the phi
y phi z, and I divide by phi x.

524
00:25:28,420 --> 00:25:30,740
And in this term, when I
divide by phi x phi y phi z,

525
00:25:30,740 --> 00:25:36,100
I lose the x and z, and I have
a left over phi sub y, or phi y.

526
00:25:36,100 --> 00:25:38,240
And similarly here,
phi sub z upon phi z.

527
00:25:42,994 --> 00:25:44,780
Everyone cool with that?

528
00:25:44,780 --> 00:25:46,036
Yeah?

529
00:25:46,036 --> 00:25:50,350
AUDIENCE: Can we also
lose the [INAUDIBLE]?

530
00:25:50,350 --> 00:25:51,790
PROFESSOR: I don't think so.

531
00:25:51,790 --> 00:25:54,980
Minus h bar squared over 2m
just hangs out for the ride.

532
00:25:58,870 --> 00:26:01,390
So when I take the
derivatives, I get these guys.

533
00:26:01,390 --> 00:26:04,080
And I have E times the function.

534
00:26:04,080 --> 00:26:06,836
We could certainly write
this as 2m over h bar squared

535
00:26:06,836 --> 00:26:07,710
and put it over here.

536
00:26:07,710 --> 00:26:10,630
That's fine.

537
00:26:10,630 --> 00:26:13,460
OK, so this is the form
of the equation we have,

538
00:26:13,460 --> 00:26:15,680
and what does this give us?

539
00:26:15,680 --> 00:26:17,580
What content does this give us?

540
00:26:17,580 --> 00:26:19,120
Well, note the following.

541
00:26:19,120 --> 00:26:20,480
This is a funny system.

542
00:26:20,480 --> 00:26:24,020
This is a function of x.

543
00:26:24,020 --> 00:26:28,080
This is a function of y, g
of y only, and not of x or z.

544
00:26:28,080 --> 00:26:33,280
And this is a function only
of z, and not of x or y.

545
00:26:33,280 --> 00:26:35,750
Yeah?

546
00:26:35,750 --> 00:26:38,440
So we have that E,
and let's put this

547
00:26:38,440 --> 00:26:42,160
as minus 2m over
h bar squared is

548
00:26:42,160 --> 00:26:46,300
equal to a function of
x plus a function of y

549
00:26:46,300 --> 00:26:47,310
plus a function of h.

550
00:26:51,100 --> 00:26:52,120
What does this tell you?

551
00:26:52,120 --> 00:26:53,570
AUDIENCE: They're all constant.

552
00:26:53,570 --> 00:26:54,930
PROFESSOR: They're
all constant, right.

553
00:26:54,930 --> 00:26:56,320
So the important
thing is this equation

554
00:26:56,320 --> 00:26:58,260
has to be true for every
value of x, y, and z.

555
00:26:58,260 --> 00:26:59,130
It's a differential equation.

556
00:26:59,130 --> 00:27:00,005
It's true everywhere.

557
00:27:00,005 --> 00:27:00,650
It's true here.

558
00:27:00,650 --> 00:27:01,316
It's true there.

559
00:27:01,316 --> 00:27:03,288
It's true at every point.

560
00:27:03,288 --> 00:27:04,100
Yeah?

561
00:27:04,100 --> 00:27:07,450
So for any value of x, y, and
z, this equation must be true.

562
00:27:07,450 --> 00:27:10,117
So now imagine I have a
particular solution g at h.

563
00:27:10,117 --> 00:27:12,200
I'm going to fix y and z
to some particular point.

564
00:27:12,200 --> 00:27:13,630
I'm going to look right here.

565
00:27:13,630 --> 00:27:15,570
And here that fixes y and z.

566
00:27:15,570 --> 00:27:17,510
So these are just some numbers.

567
00:27:17,510 --> 00:27:19,970
And suppose we
satisfy this equation.

568
00:27:19,970 --> 00:27:23,280
Then there's a very x
leaving y and z fixed.

569
00:27:23,280 --> 00:27:27,054
What must be true of f of x?

570
00:27:27,054 --> 00:27:27,930
AUDIENCE: Constant.

571
00:27:27,930 --> 00:27:29,763
PROFESSOR: It's got to
be constant, exactly.

572
00:27:29,763 --> 00:27:31,870
So this tells us that
f of x is a constant.

573
00:27:31,870 --> 00:27:32,620
I.e.

574
00:27:32,620 --> 00:27:36,920
phi x prime prime of
over phi x is a constant.

575
00:27:36,920 --> 00:27:38,100
I'll call it epsilon x.

576
00:27:40,910 --> 00:27:46,560
And phi y prime prime
of over phi y and this

577
00:27:46,560 --> 00:27:51,950
is a function of y of x
is equal to epsilon y.

578
00:27:51,950 --> 00:27:56,840
And actually for--
yeah, that's fine.

579
00:27:56,840 --> 00:27:58,340
For fun I'm going
to put in a minus.

580
00:27:58,340 --> 00:28:00,298
It doesn't matter what
I call this coefficient.

581
00:28:00,298 --> 00:28:04,674
And similarly for phi z
prime prime z over phi z

582
00:28:04,674 --> 00:28:05,840
is equal to minus epsilon z.

583
00:28:08,410 --> 00:28:13,910
So this tells us that
minus 2m upon h bar squared

584
00:28:13,910 --> 00:28:19,400
e is equal to minus epsilon x
minus epsilon y minus epsilon

585
00:28:19,400 --> 00:28:21,080
z.

586
00:28:21,080 --> 00:28:23,160
And any solutions
of these equations

587
00:28:23,160 --> 00:28:25,316
with some constant value
of epsilon x, epsilon

588
00:28:25,316 --> 00:28:26,880
y, and epsilon z
is going to give me

589
00:28:26,880 --> 00:28:29,880
a solution of my original energy
eigenvalue equation, where

590
00:28:29,880 --> 00:28:33,000
the value of capital
E is equal to the sum.

591
00:28:33,000 --> 00:28:37,700
And I can take the minus
signs make this plus plus.

592
00:28:37,700 --> 00:28:38,200
Yeah?

593
00:28:41,539 --> 00:28:46,360
AUDIENCE: Can
epsilon x, epsilon y,

594
00:28:46,360 --> 00:28:50,495
and epsilon z-- can
one of them be negative

595
00:28:50,495 --> 00:28:52,942
if the other's are sufficiently
positive or vice versa?

596
00:28:52,942 --> 00:28:53,900
Or is that [INAUDIBLE]?

597
00:28:53,900 --> 00:28:54,890
PROFESSOR: Let's check.

598
00:28:54,890 --> 00:28:55,570
Let's check.

599
00:28:55,570 --> 00:28:57,640
So what are the solutions
of this equation?

600
00:29:03,170 --> 00:29:03,670
Yeah.

601
00:29:03,670 --> 00:29:05,357
So solutions to
this equation phi

602
00:29:05,357 --> 00:29:07,940
double-- so let's write this in
a slightly more familiar form.

603
00:29:07,940 --> 00:29:12,140
This says that phi prime
prime plus epsilon x

604
00:29:12,140 --> 00:29:15,800
phi is equal to 0, OK?

605
00:29:15,800 --> 00:29:18,730
But this just tells you
that phi is exponential.

606
00:29:18,730 --> 00:29:25,670
Phi is equal to a e
to the Ikx kxx plus B

607
00:29:25,670 --> 00:29:34,580
e to the minus ikxx,
where k squared is

608
00:29:34,580 --> 00:29:37,609
equal to kx squared
is equal to epsilon x.

609
00:29:37,609 --> 00:29:38,108
OK?

610
00:29:41,840 --> 00:29:47,340
So this becomes-- and similarly
for epsilon y and epsilon z,

611
00:29:47,340 --> 00:29:50,270
each with their own
value of ky and kz

612
00:29:50,270 --> 00:29:54,260
who squares the
epsilon accordingly.

613
00:29:54,260 --> 00:29:57,900
So what can you say
about these epsilons?

614
00:29:57,900 --> 00:30:01,945
Well, the epsilons are
strictly positive numbers.

615
00:30:01,945 --> 00:30:03,070
So to answer your question.

616
00:30:03,070 --> 00:30:05,710
So the epsilons
have to be positive.

617
00:30:05,710 --> 00:30:08,325
OK, so this equation becomes,
though, E is equal to-- I'm

618
00:30:08,325 --> 00:30:10,908
going to put the h bar squared
over 2m back on the other side.

619
00:30:10,908 --> 00:30:13,150
h bar squared upon 2m.

620
00:30:13,150 --> 00:30:15,150
Epsilon x, epsilon y, epsilon z.

621
00:30:15,150 --> 00:30:17,000
But those are just
Kx squared plus Ky

622
00:30:17,000 --> 00:30:18,420
squared plus Kz squared.

623
00:30:18,420 --> 00:30:21,650
Kx squared plus
Ky squared plus Kz

624
00:30:21,650 --> 00:30:32,680
squared, also known as
h bar squared upon 2m k

625
00:30:32,680 --> 00:30:36,890
vector squared, where the
wave function, phi sub

626
00:30:36,890 --> 00:30:41,550
E is equal to some overall
normalization constant times,

627
00:30:41,550 --> 00:30:46,320
for the first function--
where did the definition go?

628
00:30:46,320 --> 00:30:46,970
Right here.

629
00:30:46,970 --> 00:30:49,430
So from here, phi x is
the exponential with Kx.

630
00:30:49,430 --> 00:30:50,920
This is an
exponential y with Ky.

631
00:30:50,920 --> 00:30:53,060
And then the exponential
in z with Kz.

632
00:30:53,060 --> 00:31:00,700
e to the ikx times x plus
Ky times y plus Kz times z.

633
00:31:04,510 --> 00:31:06,850
Let me write this as e to the i.

634
00:31:06,850 --> 00:31:09,070
e to the i.

635
00:31:09,070 --> 00:31:13,970
Also known as some normalization
constant e to the i k vector

636
00:31:13,970 --> 00:31:17,772
dot r vector.

637
00:31:17,772 --> 00:31:18,272
OK?

638
00:31:21,580 --> 00:31:25,090
So the energy, if we have
a free particle, the energy

639
00:31:25,090 --> 00:31:28,066
eigenfunctions can
be put in the form,

640
00:31:28,066 --> 00:31:30,770
or at least we can build energy
eigenfunctions of the form,

641
00:31:30,770 --> 00:31:36,520
plane waves with some 3D
momentum with energy E

642
00:31:36,520 --> 00:31:41,370
is equal to h bar squared
k squared upon 2m,

643
00:31:41,370 --> 00:31:44,286
just as we saw for the free
particle in one dimension.

644
00:31:44,286 --> 00:31:45,910
And the actual wave
function is nothing

645
00:31:45,910 --> 00:31:48,580
but a product of
wave functions in 1D.

646
00:31:48,580 --> 00:31:49,530
Yeah?

647
00:31:49,530 --> 00:31:53,751
AUDIENCE: What happened
to the minus ikx minus iky

648
00:31:53,751 --> 00:31:55,280
minus ikz terms?

649
00:31:55,280 --> 00:31:57,710
PROFESSOR: Good, so here
I just dropped these guys.

650
00:31:57,710 --> 00:32:02,682
So I just picked examples where
we just picked e to the ikz.

651
00:32:02,682 --> 00:32:03,890
That's an excellent question.

652
00:32:03,890 --> 00:32:05,750
So I've done something here.

653
00:32:05,750 --> 00:32:08,654
In particular, I looked
at a special case.

654
00:32:08,654 --> 00:32:10,570
And here's an important
lesson from the theory

655
00:32:10,570 --> 00:32:13,010
of the separable
equations, which

656
00:32:13,010 --> 00:32:22,750
is that once I separate-- so
if I have a separable equation

657
00:32:22,750 --> 00:32:25,010
and I find it
separated solution,

658
00:32:25,010 --> 00:32:33,740
phi E is equal to phi x of
x, phi y of y phi z of z,

659
00:32:33,740 --> 00:32:36,310
not all functions-- not all
solutions of the equation

660
00:32:36,310 --> 00:32:37,520
are of this form.

661
00:32:37,520 --> 00:32:38,020
They're not.

662
00:32:38,020 --> 00:32:39,269
I had to make that assumption.

663
00:32:39,269 --> 00:32:42,070
I said suppose it's of
this form right here.

664
00:32:42,070 --> 00:32:43,150
So this is an assumption.

665
00:32:46,840 --> 00:32:47,340
OK?

666
00:32:47,340 --> 00:32:51,520
However, these
form a good basis.

667
00:32:51,520 --> 00:32:54,390
By taking suitable
linear combinations

668
00:32:54,390 --> 00:32:56,140
of them, suitable
super positions,

669
00:32:56,140 --> 00:32:58,320
I can build a completely
general solution.

670
00:32:58,320 --> 00:33:02,360
For example, as was noted, the
true solution of this equation,

671
00:33:02,360 --> 00:33:04,690
even just focusing on
the x, is a superposition

672
00:33:04,690 --> 00:33:08,030
of plus and minus
waves, waves with

673
00:33:08,030 --> 00:33:09,650
plus positive negative momentum.

674
00:33:09,650 --> 00:33:10,920
So how do we get that?

675
00:33:10,920 --> 00:33:12,810
Well, we could
write that as phi is

676
00:33:12,810 --> 00:33:18,940
equal to e to the ikx
phi y phi z plus e

677
00:33:18,940 --> 00:33:29,550
to the minus ikx phi y phi z
with the same phi y and phi z.

678
00:33:29,550 --> 00:33:30,930
So these are actually in there.

679
00:33:34,390 --> 00:33:34,950
Yeah?

680
00:33:34,950 --> 00:33:38,280
AUDIENCE: My other
question is so I still

681
00:33:38,280 --> 00:33:39,828
don't see why any
of the epsilons

682
00:33:39,828 --> 00:33:40,952
can't have a negative sign.

683
00:33:40,952 --> 00:33:44,482
You have an exponential,
a real exponential as one

684
00:33:44,482 --> 00:33:45,580
of your products.

685
00:33:45,580 --> 00:33:51,230
PROFESSOR: OK, so if we
had a negative epsilon,

686
00:33:51,230 --> 00:33:55,130
is that wave function
going to be normalizable?

687
00:33:55,130 --> 00:33:57,100
AUDIENCE: Oh, as r
goes to-- but can you

688
00:33:57,100 --> 00:33:59,440
just keep the minus term?

689
00:33:59,440 --> 00:34:01,417
PROFESSOR: In which direction?

690
00:34:01,417 --> 00:34:02,250
AUDIENCE: Oh, right.

691
00:34:02,250 --> 00:34:03,950
PROFESSOR: So if it's
converging in this direction,

692
00:34:03,950 --> 00:34:05,790
it's got to be growing
in this direction.

693
00:34:05,790 --> 00:34:07,590
And that's not going
to be normalizable.

694
00:34:07,590 --> 00:34:14,000
And so as usual
with the plane wave,

695
00:34:14,000 --> 00:34:17,830
we can pick the oscillating
solutions that are also

696
00:34:17,830 --> 00:34:19,760
not normalizable
to one, but they're

697
00:34:19,760 --> 00:34:20,929
delta function normalizable.

698
00:34:20,929 --> 00:34:21,902
And so that's what
we've done here.

699
00:34:21,902 --> 00:34:23,280
It's exactly the
same thing as in 1D.

700
00:34:23,280 --> 00:34:23,780
Yeah?

701
00:34:23,780 --> 00:34:27,560
AUDIENCE: So does this mean
that any superposition of plane

702
00:34:27,560 --> 00:34:30,913
waves with wave vector
equal in magnitude

703
00:34:30,913 --> 00:34:33,620
will also be an eigenfunction
of the same energy?

704
00:34:33,620 --> 00:34:34,850
PROFESSOR: Absolutely.

705
00:34:34,850 --> 00:34:35,380
Awesome.

706
00:34:35,380 --> 00:34:36,130
Great observation.

707
00:34:36,130 --> 00:34:37,530
So the observation is this.

708
00:34:37,530 --> 00:34:41,199
Suppose I take K,
I'll call it K1.

709
00:34:41,199 --> 00:34:48,530
So this is a vector
such that K1 squared

710
00:34:48,530 --> 00:34:55,320
times h bar squared
over 2m is equal to E.

711
00:34:55,320 --> 00:34:59,890
Now there are many vectors k
which have the same magnitude,

712
00:34:59,890 --> 00:35:01,540
but not the same direction.

713
00:35:01,540 --> 00:35:05,010
So we could also make this equal
to h bar squared K2 squared

714
00:35:05,010 --> 00:35:12,320
vector squared over 2m where K2
is not equal to K1 as a vector,

715
00:35:12,320 --> 00:35:14,125
although they share
the same magnitude.

716
00:35:16,660 --> 00:35:17,940
So that's interesting.

717
00:35:17,940 --> 00:35:19,400
So that looks a lot like before.

718
00:35:19,400 --> 00:35:22,555
In 1D, we saw that if
we have k or minus k,

719
00:35:22,555 --> 00:35:23,680
these have the same energy.

720
00:35:26,750 --> 00:35:28,000
All right?

721
00:35:28,000 --> 00:35:32,800
Now if we have any K,
K1-- so this is 1D.

722
00:35:32,800 --> 00:35:44,920
In 3D, if we have K1 and
K2 with the same magnitude

723
00:35:44,920 --> 00:35:51,730
and the same energy,
they're degenerate.

724
00:35:57,010 --> 00:35:58,730
That's interesting.

725
00:35:58,730 --> 00:35:59,656
Why?

726
00:35:59,656 --> 00:36:01,962
Why do we have this
gigantic degeneracy

727
00:36:01,962 --> 00:36:04,420
of the energy eigenfunctions
for the free particle in three

728
00:36:04,420 --> 00:36:04,919
dimensions?

729
00:36:07,450 --> 00:36:07,950
Yeah?

730
00:36:07,950 --> 00:36:09,600
AUDIENCE: Well, there
are an infinite number

731
00:36:09,600 --> 00:36:11,910
of directions it could be going
in with the same momentum.

732
00:36:11,910 --> 00:36:12,370
PROFESSOR: Awesome.

733
00:36:12,370 --> 00:36:14,744
So this is clearly true that
there are an infinite number

734
00:36:14,744 --> 00:36:17,330
of momenta with
the same magnitude.

735
00:36:17,330 --> 00:36:19,935
So there are many,
many, but why?

736
00:36:19,935 --> 00:36:22,480
Why do they have
the same energy?

737
00:36:22,480 --> 00:36:24,390
Couldn't they have
different energy?

738
00:36:24,390 --> 00:36:26,784
Couldn't this one have
a different energy?

739
00:36:26,784 --> 00:36:27,700
AUDIENCE: [INAUDIBLE].

740
00:36:27,700 --> 00:36:28,740
PROFESSOR: Excellent,
it's symmetric.

741
00:36:28,740 --> 00:36:30,360
The system is invariant
under rotation.

742
00:36:30,360 --> 00:36:32,151
Who are you to say this
is the x direction?

743
00:36:32,151 --> 00:36:33,470
I call it y.

744
00:36:33,470 --> 00:36:34,910
Right?

745
00:36:34,910 --> 00:36:38,907
So the system has a symmetry.

746
00:36:38,907 --> 00:36:39,990
The symmetry is rotations.

747
00:36:42,560 --> 00:36:44,710
And when we have
a symmetry, that

748
00:36:44,710 --> 00:36:47,070
means there's an operator,
a unitary operator, which

749
00:36:47,070 --> 00:36:50,119
affects that rotation,
the rotation operator.

750
00:36:50,119 --> 00:36:51,660
It's the operator
that takes a vector

751
00:36:51,660 --> 00:36:54,250
and rotates in some
particular way.

752
00:36:54,250 --> 00:36:56,040
We have a unitary
operator that's

753
00:36:56,040 --> 00:36:59,070
a symmetry that means it
commutes with the energy

754
00:36:59,070 --> 00:36:59,807
operator.

755
00:36:59,807 --> 00:37:01,640
But if it commutes with
the energy operator,

756
00:37:01,640 --> 00:37:03,040
we get can degeneracies.

757
00:37:03,040 --> 00:37:06,004
We can get states that are
different states mapped

758
00:37:06,004 --> 00:37:08,545
to each other under our unitary
operator, under our rotation.

759
00:37:08,545 --> 00:37:11,749
We get states which are
different states manifestly.

760
00:37:11,749 --> 00:37:13,290
But which have the
same energy, which

761
00:37:13,290 --> 00:37:16,060
are shared energy eigenvalues.

762
00:37:16,060 --> 00:37:17,160
Cool?

763
00:37:17,160 --> 00:37:21,210
And this is a really lovely
example, both in 1D and 3D,

764
00:37:21,210 --> 00:37:25,230
that when you have a symmetry,
you get degeneracies.

765
00:37:29,570 --> 00:37:32,107
And when you have
a degeneracy, you

766
00:37:32,107 --> 00:37:33,690
should be very
suspicious that there's

767
00:37:33,690 --> 00:37:39,110
a symmetry hanging around,
lurking around ensuring it, OK?

768
00:37:39,110 --> 00:37:40,770
And this is an
important general lesson

769
00:37:40,770 --> 00:37:45,300
that goes way beyond the
specifics of the free particle.

770
00:37:45,300 --> 00:37:46,210
Yeah?

771
00:37:46,210 --> 00:37:48,370
AUDIENCE: So that
occurs in systems

772
00:37:48,370 --> 00:37:50,439
with bound states [INAUDIBLE]?

773
00:37:50,439 --> 00:37:52,730
PROFESSOR: Yeah, it occurs
in systems with bound states

774
00:37:52,730 --> 00:37:54,146
and systems with
non bound states.

775
00:37:54,146 --> 00:37:56,460
So here we're talking
about a free particle.

776
00:37:56,460 --> 00:37:57,600
Certainly not bound.

777
00:37:57,600 --> 00:37:59,430
And its true.

778
00:37:59,430 --> 00:38:01,320
For bound states, we'll
also see that there

779
00:38:01,320 --> 00:38:03,612
will be a degeneracy
associated with symmetry.

780
00:38:03,612 --> 00:38:05,570
Now your question is a
really, really good one,

781
00:38:05,570 --> 00:38:07,778
because what we found-- let
me rephrase the question.

782
00:38:07,778 --> 00:38:10,790
The question is, look, in
1D when we had bound states,

783
00:38:10,790 --> 00:38:12,150
there was no degeneracy.

784
00:38:12,150 --> 00:38:13,858
Didn't matter what
you did to the system.

785
00:38:13,858 --> 00:38:16,980
When you had bound states, bound
states were non degenerate.

786
00:38:16,980 --> 00:38:19,339
In 3D, we see that when
you have a free particle,

787
00:38:19,339 --> 00:38:20,380
you again get degeneracy.

788
00:38:20,380 --> 00:38:22,260
In fact, you get a heck
of a lot more degeneracy.

789
00:38:22,260 --> 00:38:23,140
You get a sphere's worth.

790
00:38:23,140 --> 00:38:25,290
Although actually, that's a
sphere in 0 dimension, right?

791
00:38:25,290 --> 00:38:26,962
It's a 0 dimensional
sphere, two points.

792
00:38:26,962 --> 00:38:29,270
So you get a sphere's
worth of degenerate states

793
00:38:29,270 --> 00:38:29,850
for the free particle.

794
00:38:29,850 --> 00:38:31,100
Well, what about bound states?

795
00:38:31,100 --> 00:38:34,110
Are bound states non
degenerate still?

796
00:38:34,110 --> 00:38:35,840
Fantastic question.

797
00:38:35,840 --> 00:38:36,910
Let's find out.

798
00:38:36,910 --> 00:38:38,410
So let's do the
harmonic oscillator.

799
00:38:38,410 --> 00:38:40,310
Let's do the 3D harmonic
oscillator to check.

800
00:38:44,990 --> 00:38:48,240
So the 3D harmonic oscillator,
the potential is h bar squared.

801
00:38:48,240 --> 00:38:52,020
And let's pick for fun the
rotationally symmetric 3D

802
00:38:52,020 --> 00:38:53,575
harmonic oscillator.

803
00:38:53,575 --> 00:39:01,050
m omega squared upon 2 x squared
plus y squared plus z squared.

804
00:39:01,050 --> 00:39:08,845
This could also be written m
omega squared upon 2 r squared.

805
00:39:11,287 --> 00:39:13,120
So I could write it in
spherical coordinates

806
00:39:13,120 --> 00:39:16,310
or in Cartesian coordinates.

807
00:39:16,310 --> 00:39:18,256
This is really in
vector notation.

808
00:39:18,256 --> 00:39:18,880
Doesn't matter.

809
00:39:18,880 --> 00:39:19,713
It's the same thing.

810
00:39:19,713 --> 00:39:22,780
3D harmonic oscillator.

811
00:39:22,780 --> 00:39:25,250
So what you immediately deduce
about the form of the energy

812
00:39:25,250 --> 00:39:25,875
eigenfunctions?

813
00:39:28,590 --> 00:39:33,400
Well, we have that E phi E
is equal to P squared over

814
00:39:33,400 --> 00:39:35,830
minus h bar squared
over 2m-- so here's

815
00:39:35,830 --> 00:39:42,430
the energy eigenvalue
equation-- times

816
00:39:42,430 --> 00:39:50,680
dx squared plus dy squared plus
dz squared plus m omega squared

817
00:39:50,680 --> 00:40:03,440
upon 2 times x squared plus y
squared plus z squared phi E.

818
00:40:03,440 --> 00:40:05,610
But I can put this
together in a nice way.

819
00:40:05,610 --> 00:40:11,330
This is minus h bar
squared upon 2m dx squared

820
00:40:11,330 --> 00:40:18,620
plus m omega squared upon 2
x squared plus ditto for y

821
00:40:18,620 --> 00:40:26,940
plus ditto for z phi E. Yeah?

822
00:40:26,940 --> 00:40:28,241
Everyone agree?

823
00:40:28,241 --> 00:40:30,490
So this is differential
operator that only involves x.

824
00:40:30,490 --> 00:40:31,930
Doesn't involve y or z.

825
00:40:31,930 --> 00:40:33,480
Ditto y, but no x or z.

826
00:40:33,480 --> 00:40:35,710
And ditto z, but no x or y.

827
00:40:35,710 --> 00:40:39,597
Aha, this is separable
just as before.

828
00:40:39,597 --> 00:40:41,180
So now we have a
nice separable system

829
00:40:41,180 --> 00:40:44,180
where I want to solve the
equations 3 times, once

830
00:40:44,180 --> 00:40:44,960
for x, y, and z.

831
00:40:44,960 --> 00:40:46,626
And I'm just going
to write it for x, y,

832
00:40:46,626 --> 00:40:51,530
and z epsilon sub x phi x is
equal to minus h bar squared

833
00:40:51,530 --> 00:40:57,680
over 2m dx squared plus
m omega squared upon 2 x

834
00:40:57,680 --> 00:41:02,000
squared phi sub x.

835
00:41:02,000 --> 00:41:04,180
And ditto for x, y, z.

836
00:41:04,180 --> 00:41:08,400
And then phi E is going
to be equal to phi

837
00:41:08,400 --> 00:41:15,600
x of x, phi y of y,
and phi z of z, OK?

838
00:41:15,600 --> 00:41:21,797
Where E is equal to epsilon x
plus epsilon y plus epsilon z.

839
00:41:21,797 --> 00:41:24,380
Note that I've used a slightly
different definition of epsilon

840
00:41:24,380 --> 00:41:25,230
here as before.

841
00:41:25,230 --> 00:41:28,715
Here it's explicitly
the energy eigenvalue.

842
00:41:28,715 --> 00:41:30,090
So what this is
telling is, look,

843
00:41:30,090 --> 00:41:31,130
we know what this equation is.

844
00:41:31,130 --> 00:41:32,530
This equation is
the same equation

845
00:41:32,530 --> 00:41:34,280
we ran into in the 1D
harmonic oscillator.

846
00:41:34,280 --> 00:41:38,590
It's exactly the 1D
harmonic oscillator problem.

847
00:41:38,590 --> 00:41:41,140
So the solution to the 3D
harmonic oscillator problem

848
00:41:41,140 --> 00:41:43,010
can be written for
energy eigenfunctions,

849
00:41:43,010 --> 00:41:45,850
can be constructed
by taking a harmonic

850
00:41:45,850 --> 00:41:48,750
oscillator in the x direction--
and we know what those are.

851
00:41:48,750 --> 00:41:49,750
There's a tower of them.

852
00:41:49,750 --> 00:41:52,335
There's a ladder of them
created by the raising operator

853
00:41:52,335 --> 00:41:56,160
and lowered by the
lowering operator.

854
00:41:56,160 --> 00:41:59,862
And similarly for y,
and similarly for z.

855
00:41:59,862 --> 00:42:01,820
So I'm going to write
this slightly differently

856
00:42:01,820 --> 00:42:10,540
in the same place as phi sub
E is equal to phi sub nx of x.

857
00:42:10,540 --> 00:42:13,560
The state with energy E sub nx.

858
00:42:13,560 --> 00:42:16,530
Phi sub ny of y.

859
00:42:16,530 --> 00:42:18,820
Phi sub nz of z.

860
00:42:18,820 --> 00:42:21,030
Where these are all
the single dimension,

861
00:42:21,030 --> 00:42:23,970
one dimensional harmonic
oscillator eigenfunctions.

862
00:42:23,970 --> 00:42:34,658
And E is equal to Ex E sub nx
plus E sub ny plus E sub nz.

863
00:42:34,658 --> 00:42:35,158
OK?

864
00:42:43,380 --> 00:42:46,845
So let's look at the
consequences of this.

865
00:42:46,845 --> 00:42:48,720
So first off, does anyone
have any questions?

866
00:42:48,720 --> 00:42:50,870
I went through
the kind of quick.

867
00:42:50,870 --> 00:42:52,485
Any questions about that?

868
00:42:52,485 --> 00:42:54,610
If you're not comfortable
with separable equations,

869
00:42:54,610 --> 00:42:56,970
you need to become super
comfortable with separable

870
00:42:56,970 --> 00:42:58,084
equations.

871
00:42:58,084 --> 00:42:59,250
It's an important technique.

872
00:42:59,250 --> 00:43:01,450
We're going to use it a lot.

873
00:43:01,450 --> 00:43:04,099
So the upshot is that
if I write-- in fact,

874
00:43:04,099 --> 00:43:06,390
I'm going to write that in
slightly different notation.

875
00:43:06,390 --> 00:43:12,410
If I write phi E
is equal to phi n

876
00:43:12,410 --> 00:43:22,474
of x, where it's the-- and phi
l of y and phi m of z-- actually

877
00:43:22,474 --> 00:43:23,515
that's a stupid ordering.

878
00:43:23,515 --> 00:43:24,470
Let's try that again.

879
00:43:28,830 --> 00:43:30,670
l, m, n.

880
00:43:30,670 --> 00:43:32,350
That is the
alphabetical ordering.

881
00:43:32,350 --> 00:43:34,500
With the energy is
now equal to-- we

882
00:43:34,500 --> 00:43:37,610
know the energy is of a
state with of the harmonic

883
00:43:37,610 --> 00:43:39,970
oscillator with
excitation number l.

884
00:43:39,970 --> 00:43:47,480
It's h bar omega, the overall
omega, times l plus 1/2.

885
00:43:47,480 --> 00:43:50,840
But from this guy it's
got excitation number m,

886
00:43:50,840 --> 00:43:56,140
so energy of that is h bar
omega m plus 1/2 plus m.

887
00:43:56,140 --> 00:43:59,010
And so now that's plus 1.

888
00:43:59,010 --> 00:44:00,850
And for this guy
similarly, h bar omega

889
00:44:00,850 --> 00:44:04,475
n plus n and now plus
1/2 again plus 3/2.

890
00:44:10,380 --> 00:44:12,781
This is a basis of solutions
of the energy eigenfunction

891
00:44:12,781 --> 00:44:13,280
equations.

892
00:44:13,280 --> 00:44:20,520
These are the solutions of
the energy eigenfunctions

893
00:44:20,520 --> 00:44:24,104
for the 3D harmonic oscillator.

894
00:44:24,104 --> 00:44:25,270
And now here's the question.

895
00:44:25,270 --> 00:44:27,460
The question that
was asked is, look,

896
00:44:27,460 --> 00:44:31,180
there are no degeneracies
in bound states in 1D.

897
00:44:31,180 --> 00:44:33,970
Here we have manifestly
a 3D bound state system.

898
00:44:33,970 --> 00:44:35,299
Are there degeneracies?

899
00:44:35,299 --> 00:44:35,882
AUDIENCE: Yes.

900
00:44:35,882 --> 00:44:37,610
PROFESSOR: Yeah,
obviously, right?

901
00:44:37,610 --> 00:44:40,205
So for example, if I call
l1 this 0 and this 0.

902
00:44:40,205 --> 00:44:44,780
Or if I call this 010 or 001,
those all have the same energy.

903
00:44:44,780 --> 00:44:47,610
They have the energy h bar
omega 0 times 1 plus 3/2 or 5/2.

904
00:44:50,320 --> 00:44:53,970
So let's look at this
in a little more detail.

905
00:44:53,970 --> 00:44:57,394
Let's write a list
of the degeneracies

906
00:44:57,394 --> 00:44:58,560
as a function of the energy.

907
00:45:03,120 --> 00:45:06,660
So at energy what's
the ground state energy

908
00:45:06,660 --> 00:45:09,520
for the 3D harmonic oscillator?

909
00:45:09,520 --> 00:45:10,466
3 halves h bar omega.

910
00:45:10,466 --> 00:45:11,840
It's three times
the ground state

911
00:45:11,840 --> 00:45:13,830
energy for the single
1D harmonic oscillator.

912
00:45:13,830 --> 00:45:15,990
So 3/2 h bar omega.

913
00:45:19,420 --> 00:45:19,974
Yeah?

914
00:45:19,974 --> 00:45:20,890
AUDIENCE: [INAUDIBLE].

915
00:45:23,830 --> 00:45:26,150
PROFESSOR: Good, the way
we arrived that this was we

916
00:45:26,150 --> 00:45:31,300
found that the energy-- so
the energy operator acting

917
00:45:31,300 --> 00:45:34,040
on the 3D wave
function is what I

918
00:45:34,040 --> 00:45:36,870
get by taking the
energy operator in 1D

919
00:45:36,870 --> 00:45:41,186
and acting on the wave function,
and the energy operator in y

920
00:45:41,186 --> 00:45:43,685
acting on the wave function,
the energy operator in z acting

921
00:45:43,685 --> 00:45:45,000
on the wave function, where
the energy operator for each

922
00:45:45,000 --> 00:45:46,660
of those is the 1D
harmonic oscillator

923
00:45:46,660 --> 00:45:48,560
with the same frequency.

924
00:45:48,560 --> 00:45:49,150
OK?

925
00:45:49,150 --> 00:45:50,120
And then I separated.

926
00:45:50,120 --> 00:45:51,600
I said look, let
the wave function,

927
00:45:51,600 --> 00:45:55,400
the 3D wave function, since
I know this is separable

928
00:45:55,400 --> 00:45:58,560
and each separated part
of the wave function

929
00:45:58,560 --> 00:46:00,990
satisfies the 1D harmonic
oscillator equation,

930
00:46:00,990 --> 00:46:04,380
I know what the eigenfunctions
of the 1D harmonic oscillator

931
00:46:04,380 --> 00:46:05,790
energy eigenvalue problem are.

932
00:46:05,790 --> 00:46:08,780
They are the phi n's.

933
00:46:08,780 --> 00:46:10,840
And so I can just take
my 3D wave function.

934
00:46:10,840 --> 00:46:13,510
I can say the 3D wave function
is going to be the separated

935
00:46:13,510 --> 00:46:15,180
form, the product
of 1D wave function

936
00:46:15,180 --> 00:46:19,610
in x, a 1D wave function in
y, and 1D wave function in z.

937
00:46:19,610 --> 00:46:20,240
Cool?

938
00:46:20,240 --> 00:46:22,220
So now let's look back
at what let's think

939
00:46:22,220 --> 00:46:23,740
about what happens when I
take the energy operator

940
00:46:23,740 --> 00:46:25,270
and I act on that [INAUDIBLE].

941
00:46:25,270 --> 00:46:27,160
When I take the 3D
energy operator, which

942
00:46:27,160 --> 00:46:29,209
is the sum of the three
1D energy operators,

943
00:46:29,209 --> 00:46:30,750
harmonic oscillator
energy operators.

944
00:46:30,750 --> 00:46:32,541
When thinking I'm going
to act on this guy,

945
00:46:32,541 --> 00:46:35,150
the first one, which only
knows about the x direction,

946
00:46:35,150 --> 00:46:38,060
sees the y and z
parts as constants.

947
00:46:38,060 --> 00:46:40,210
And it's a phi x, and
what does it give us back?

948
00:46:40,210 --> 00:46:44,440
E on phi x is just h bar
omega n plus one half.

949
00:46:44,440 --> 00:46:45,654
Ditto for this guy.

950
00:46:45,654 --> 00:46:47,070
And then the energy
operator in 3D

951
00:46:47,070 --> 00:46:49,640
is the sum of the three
1D energy operators.

952
00:46:49,640 --> 00:46:52,240
So that tells us the energy is
the sum of the three energies.

953
00:46:52,240 --> 00:46:53,840
Is that cool?

954
00:46:53,840 --> 00:46:54,460
OK good.

955
00:46:54,460 --> 00:46:55,370
Other questions?

956
00:46:55,370 --> 00:46:56,193
Yeah?

957
00:46:56,193 --> 00:46:58,508
AUDIENCE: Is the number of
degeneracies essentially

958
00:46:58,508 --> 00:47:01,760
[INAUDIBLE] of number theory.

959
00:47:01,760 --> 00:47:03,230
PROFESSOR: Ask me
that after class.

960
00:47:07,120 --> 00:47:08,810
So let's look at
the degeneracies

961
00:47:08,810 --> 00:47:11,530
as a function of the energy.

962
00:47:11,530 --> 00:47:15,160
So at the lowest possible
energy, 3/2 omega, what states

963
00:47:15,160 --> 00:47:17,584
can I possibly have?

964
00:47:17,584 --> 00:47:20,250
I'm going to label the states by
the three numbers, l, m, and n.

965
00:47:20,250 --> 00:47:22,670
So this is just the
ground state 0 0 0.

966
00:47:22,670 --> 00:47:24,740
So is that degenerate?

967
00:47:24,740 --> 00:47:27,720
No, because there's
just the one state.

968
00:47:27,720 --> 00:47:29,040
What about at the next level?

969
00:47:29,040 --> 00:47:30,727
What's the next allowed energy?

970
00:47:30,727 --> 00:47:31,310
AUDIENCE: 5/2.

971
00:47:31,310 --> 00:47:32,340
PROFESSOR: 5/2.

972
00:47:32,340 --> 00:47:33,660
OK, 5/2.

973
00:47:33,660 --> 00:47:35,660
So at 5/2 what
states do we have?

974
00:47:35,660 --> 00:47:36,620
Well, we have 1 0 0.

975
00:47:36,620 --> 00:47:38,520
But we also have 0, 1, 0.

976
00:47:38,520 --> 00:47:41,410
And we also 0 0 1.

977
00:47:41,410 --> 00:47:44,044
Aha, this is looking good.

978
00:47:44,044 --> 00:47:45,710
What does this
correspond to physically?

979
00:47:45,710 --> 00:47:48,860
This says you have
excitation, so you've

980
00:47:48,860 --> 00:47:51,110
got a node in the x direction.

981
00:47:51,110 --> 00:47:55,220
But your Gaussian in
the y and z directions.

982
00:47:55,220 --> 00:48:00,340
This one says your Gaussian
in the x direction.

983
00:48:00,340 --> 00:48:02,590
You have a node in the y
direction as a function of y,

984
00:48:02,590 --> 00:48:05,280
because it's phi 1 of y.

985
00:48:05,280 --> 00:48:07,440
And you're Gaussian
in the z directions.

986
00:48:07,440 --> 00:48:11,310
And this one says you have 1
excitation in the z direction.

987
00:48:11,310 --> 00:48:15,290
So they sound sort of
rotated from each other.

988
00:48:15,290 --> 00:48:18,500
That sounds promising.

989
00:48:18,500 --> 00:48:20,740
But in particular, what
we just discovered sort of

990
00:48:20,740 --> 00:48:22,370
by construction
is that there can

991
00:48:22,370 --> 00:48:35,580
be degeneracies among
bound states in 3D.

992
00:48:35,580 --> 00:48:38,310
This was not possible
in 1D, but it

993
00:48:38,310 --> 00:48:40,950
is possible in
3D, which is cool.

994
00:48:40,950 --> 00:48:42,360
But we've actually learned more.

995
00:48:42,360 --> 00:48:43,570
What's the form of
the degeneracies?

996
00:48:43,570 --> 00:48:46,028
So here it looks like they're
just rotations of each other.

997
00:48:46,028 --> 00:48:48,460
You call this x, I call it
y, someone else calls it z.

998
00:48:48,460 --> 00:48:50,510
These can't possibly
look different functions

999
00:48:50,510 --> 00:48:53,330
because they're just
rotations of each other.

1000
00:48:53,330 --> 00:48:55,017
However, things get
a little more messy

1001
00:48:55,017 --> 00:48:56,850
when you write, well,
what's the next level?

1002
00:48:56,850 --> 00:48:59,100
What's the next energy
after 5/2 h bar omega?

1003
00:48:59,100 --> 00:49:00,750
7/2 h bar omega, exactly.

1004
00:49:00,750 --> 00:49:01,354
7/2.

1005
00:49:01,354 --> 00:49:02,270
And that's not so bad.

1006
00:49:02,270 --> 00:49:08,790
So for that one, we get
2 0 0, 0 2 0, 0 0 2.

1007
00:49:08,790 --> 00:49:09,610
But is that it?

1008
00:49:09,610 --> 00:49:10,370
AUDIENCE: No.

1009
00:49:10,370 --> 00:49:12,410
PROFESSOR: What else do we get?

1010
00:49:12,410 --> 00:49:17,560
1 1 0, 0 1 1, 1 0 1.

1011
00:49:17,560 --> 00:49:19,200
So first off, let's
look at the number.

1012
00:49:19,200 --> 00:49:21,270
The degeneracy number
here-- I'll call this d sub

1013
00:49:21,270 --> 00:49:26,250
0-- the degeneracy of the
ground state is 1, OK?

1014
00:49:26,250 --> 00:49:27,790
The degeneracy--
and in fact, I'm

1015
00:49:27,790 --> 00:49:31,530
going to write this as a table--
the degeneracy as level n.

1016
00:49:31,530 --> 00:49:35,150
So for d0 is equal to 1.

1017
00:49:35,150 --> 00:49:37,990
d1 is equal to 3.

1018
00:49:37,990 --> 00:49:44,150
And d2 is equal to 6.

1019
00:49:44,150 --> 00:49:46,590
Now it's less clear
here what's going on,

1020
00:49:46,590 --> 00:49:50,650
because is this just
this guy relabelled?

1021
00:49:50,650 --> 00:49:52,100
No.

1022
00:49:52,100 --> 00:49:53,870
So this is weird,
because we already

1023
00:49:53,870 --> 00:50:00,280
said that the reason we expect
that there might be degeneracy,

1024
00:50:00,280 --> 00:50:02,369
is because of
rotational symmetry.

1025
00:50:02,369 --> 00:50:03,910
The system is
rotationally invariant.

1026
00:50:03,910 --> 00:50:06,326
The potential, which is the
harmonic oscillator potential,

1027
00:50:06,326 --> 00:50:15,600
doesn't care in what direction
the radial displacement

1028
00:50:15,600 --> 00:50:17,380
vector is pointing.

1029
00:50:17,380 --> 00:50:18,630
It's rotationally symmetrical.

1030
00:50:18,630 --> 00:50:21,180
When we have symmetry--
on general grounds, when

1031
00:50:21,180 --> 00:50:23,350
we have a symmetry, we
expect to have degeneracies.

1032
00:50:26,422 --> 00:50:28,130
But this are kind of
weird, because these

1033
00:50:28,130 --> 00:50:30,129
don't seem to be simple
rotations of each other,

1034
00:50:30,129 --> 00:50:32,410
and yet they're degenerate.

1035
00:50:32,410 --> 00:50:35,200
So what's up with that?

1036
00:50:35,200 --> 00:50:35,711
Question?

1037
00:50:35,711 --> 00:50:36,210
Yeah?

1038
00:50:36,210 --> 00:50:40,230
AUDIENCE: [INAUDIBLE] Gaussian
in certain directions?

1039
00:50:40,230 --> 00:50:41,350
PROFESSOR: Yeah, sure OK.

1040
00:50:41,350 --> 00:50:44,420
So let me just explain what
this notation means again.

1041
00:50:44,420 --> 00:50:50,260
So by 1 0 0, what I mean is
that the number l is equal to 1,

1042
00:50:50,260 --> 00:50:52,095
the number m is equal
to 0, n is equal to 0.

1043
00:50:52,095 --> 00:50:53,470
That means that
the wave function

1044
00:50:53,470 --> 00:51:03,280
phi 3D is equal to phi 1 of
x, phi 0 of y, phi 0 of z.

1045
00:51:03,280 --> 00:51:05,230
But what's phi 0 of z?

1046
00:51:05,230 --> 00:51:07,350
What's a Gaussian
in the z direction?

1047
00:51:07,350 --> 00:51:08,790
Phi 0 of y.

1048
00:51:08,790 --> 00:51:10,550
That's the ground state
in the y direction

1049
00:51:10,550 --> 00:51:11,260
of the harmonic oscillator.

1050
00:51:11,260 --> 00:51:12,710
It's Gaussian in
the y direction.

1051
00:51:12,710 --> 00:51:15,255
If I wanted x, that's
not the ground state.

1052
00:51:15,255 --> 00:51:16,350
That's the excited state.

1053
00:51:16,350 --> 00:51:19,450
And in particular, sort of being
a Gaussian it goes through 0.

1054
00:51:19,450 --> 00:51:20,900
It has a node.

1055
00:51:20,900 --> 00:51:24,250
So this wave function is
not rotationally invariant.

1056
00:51:24,250 --> 00:51:26,890
It as a node in the x direction,
but no nodes and y and z

1057
00:51:26,890 --> 00:51:29,000
direction.

1058
00:51:29,000 --> 00:51:30,750
And similarly for these guys.

1059
00:51:30,750 --> 00:51:32,280
Did that answer your questions?

1060
00:51:32,280 --> 00:51:33,510
Great.

1061
00:51:33,510 --> 00:51:38,596
OK, so we have
these degeneracies,

1062
00:51:38,596 --> 00:51:40,160
and they beg an explanation.

1063
00:51:40,160 --> 00:51:41,650
And if you look
at the next level,

1064
00:51:41,650 --> 00:51:43,930
it turns out that d3-- and
you can do this quickly

1065
00:51:43,930 --> 00:51:48,300
on a scrap of
paper-- d3 is 10, OK?

1066
00:51:48,300 --> 00:51:49,440
And they go on.

1067
00:51:49,440 --> 00:51:52,802
And if you keep
writing this list out,

1068
00:51:52,802 --> 00:51:54,510
I guess it goes up--
what's the next one?

1069
00:51:54,510 --> 00:51:57,260
15.

1070
00:51:57,260 --> 00:51:58,870
21, yeah.

1071
00:51:58,870 --> 00:52:01,260
So this has a simple
mathematical structure,

1072
00:52:01,260 --> 00:52:03,150
and you can very quickly
convince yourself

1073
00:52:03,150 --> 00:52:05,740
of the form of this degeneracy.

1074
00:52:05,740 --> 00:52:09,680
dn is n n plus 1 over 2.

1075
00:52:09,680 --> 00:52:12,090
So let's just make
sure that works.

1076
00:52:12,090 --> 00:52:15,940
1 1 plus 1 over 2.

1077
00:52:15,940 --> 00:52:17,606
Sorry I should
really call this 1.

1078
00:52:20,868 --> 00:52:25,740
n plus 1, n plus 2
if I count from 0.

1079
00:52:25,740 --> 00:52:30,620
So for 0, this is going to
give us 1 times 2 over 2.

1080
00:52:30,620 --> 00:52:31,190
That's 1.

1081
00:52:31,190 --> 00:52:31,940
That works.

1082
00:52:31,940 --> 00:52:35,620
So for 1 that gives
2 times 3 over 2,

1083
00:52:35,620 --> 00:52:38,594
which is 3, and so
on and so forth.

1084
00:52:38,594 --> 00:52:39,760
So where did this come from?

1085
00:52:39,760 --> 00:52:42,140
This is something we're
going to have to answer.

1086
00:52:42,140 --> 00:52:43,950
Why that degeneracy?

1087
00:52:43,950 --> 00:52:45,420
That seems important.

1088
00:52:45,420 --> 00:52:46,630
Why is it that number?

1089
00:52:46,630 --> 00:52:48,210
Why do we have that
much degeneracy?

1090
00:52:48,210 --> 00:52:50,460
But the thing I really want
to emphasize at this point

1091
00:52:50,460 --> 00:52:55,050
is that there's an absolutely
essential deep connection

1092
00:52:55,050 --> 00:52:57,770
between symmetries
and degeneracies.

1093
00:52:57,770 --> 00:53:00,140
If we didn't have symmetry,
we wouldn't have degeneracy,

1094
00:53:00,140 --> 00:53:03,870
and we can see that
very easily here.

1095
00:53:03,870 --> 00:53:07,640
Imagine that this potential
was not exactly symmetric.

1096
00:53:07,640 --> 00:53:10,530
Imagine we made it slightly
different by adding

1097
00:53:10,530 --> 00:53:17,200
a little bit of extra
frequency to z direction.

1098
00:53:17,200 --> 00:53:19,300
Make the z frequency
slightly different.

1099
00:53:19,300 --> 00:53:24,210
Plus m omega tilde
squared upon 2 z

1100
00:53:24,210 --> 00:53:29,310
squared, where omega tilde
is not equal to omega 0.

1101
00:53:29,310 --> 00:53:30,540
OK?

1102
00:53:30,540 --> 00:53:33,230
The system is still
separable, but this guy

1103
00:53:33,230 --> 00:53:34,982
has frequency omega 0.

1104
00:53:34,982 --> 00:53:36,970
The x part has omega 0.

1105
00:53:36,970 --> 00:53:38,040
This has omega 0.

1106
00:53:38,040 --> 00:53:41,205
But this has omega tilde.

1107
00:53:41,205 --> 00:53:42,690
OK?

1108
00:53:42,690 --> 00:53:45,390
And so exactly the same
argument is going to go through,

1109
00:53:45,390 --> 00:53:48,530
but the energy now is going
to have a different form.

1110
00:53:48,530 --> 00:53:53,500
The energy is going to have h
bar omega-- h bar omega 0 times

1111
00:53:53,500 --> 00:53:57,060
l plus m plus 1.

1112
00:53:57,060 --> 00:53:59,890
But from the z part
it's going to have

1113
00:53:59,890 --> 00:54:05,010
plus h bar omega tilde
times n plus 1/2.

1114
00:54:09,861 --> 00:54:11,360
And now these
degeneracies are going

1115
00:54:11,360 --> 00:54:13,420
to be broken, because
this state will not

1116
00:54:13,420 --> 00:54:16,430
have the same
energy as these two.

1117
00:54:16,430 --> 00:54:18,540
Everyone see that?

1118
00:54:18,540 --> 00:54:20,810
When you have symmetry,
you get to degeneracy.

1119
00:54:20,810 --> 00:54:23,740
When you don't have a symmetry,
you do not get degeneracy.

1120
00:54:23,740 --> 00:54:25,770
This connection is
extremely important,

1121
00:54:25,770 --> 00:54:27,550
because it allows
you to do two things.

1122
00:54:27,550 --> 00:54:29,504
It allows you to
first not solve things

1123
00:54:29,504 --> 00:54:30,670
you don't need to solve for.

1124
00:54:30,670 --> 00:54:33,253
If you know there's a symmetry,
solve it once and then compute

1125
00:54:33,253 --> 00:54:34,820
the degeneracy and you're done.

1126
00:54:34,820 --> 00:54:36,620
On the other hand,
if you have a system

1127
00:54:36,620 --> 00:54:40,040
and you see just manifestly
you measure the energies,

1128
00:54:40,040 --> 00:54:42,990
and you measure that the
energies are degenerate,

1129
00:54:42,990 --> 00:54:46,279
you know there's a symmetry
protecting those degeneracies.

1130
00:54:46,279 --> 00:54:47,820
You actually can't
be 100% confident,

1131
00:54:47,820 --> 00:54:49,695
because I didn't prove
that these are related

1132
00:54:49,695 --> 00:54:52,760
to each other, but you
should be highly suspicious.

1133
00:54:52,760 --> 00:54:55,130
And in fact, this is an
incredibly powerful tool

1134
00:54:55,130 --> 00:54:57,140
in building models
of physical systems.

1135
00:54:57,140 --> 00:54:59,696
If you see a degeneracy or
an approximate degeneracy,

1136
00:54:59,696 --> 00:55:02,070
you can exploit that to learn
things about the underlying

1137
00:55:02,070 --> 00:55:02,680
system.

1138
00:55:02,680 --> 00:55:03,263
Yeah?

1139
00:55:03,263 --> 00:55:05,096
AUDIENCE: So we just
add the different omega

1140
00:55:05,096 --> 00:55:08,180
to each omega
[INAUDIBLE] number there

1141
00:55:08,180 --> 00:55:10,430
is still a possibility
to get a degeneracy.

1142
00:55:10,430 --> 00:55:12,320
PROFESSOR: Exactly.

1143
00:55:12,320 --> 00:55:20,120
So it's possible for these
omegas to be specially tuned

1144
00:55:20,120 --> 00:55:24,010
so that rational combinations
of them give you a degeneracy.

1145
00:55:24,010 --> 00:55:26,330
But it's extraordinarily
unlikely for that

1146
00:55:26,330 --> 00:55:28,470
to happen accidentally,
because they

1147
00:55:28,470 --> 00:55:30,700
have to be rationally
related to each other,

1148
00:55:30,700 --> 00:55:34,220
and the rationals are a set
of measures 0 in the reels.

1149
00:55:34,220 --> 00:55:36,560
So if you just randomly
pick some frequencies,

1150
00:55:36,560 --> 00:55:37,380
they'll be totally
incommensurate,

1151
00:55:37,380 --> 00:55:38,900
and you'll never
get a degeneracy.

1152
00:55:38,900 --> 00:55:41,750
So it is possible to have
an accidental degeneracy.

1153
00:55:41,750 --> 00:55:44,240
Whoops, just pure coincidence.

1154
00:55:44,240 --> 00:55:45,960
But it's extraordinarily
unlikely.

1155
00:55:45,960 --> 00:55:48,210
And as you'll see when you
get to perturbation theory,

1156
00:55:48,210 --> 00:55:50,110
it's more than unlikely.

1157
00:55:50,110 --> 00:55:51,682
It's almost impossible.

1158
00:55:51,682 --> 00:55:53,890
So it's very rare that you
get accidental degeneracy.

1159
00:55:53,890 --> 00:55:57,000
It happens, but it's rare.

1160
00:55:57,000 --> 00:55:58,960
Other questions?

1161
00:55:58,960 --> 00:56:00,950
OK, nothing?

1162
00:56:00,950 --> 00:56:03,170
OK so here we're now
going to launch into--

1163
00:56:03,170 --> 00:56:05,740
so this leads us into
a very simple question.

1164
00:56:05,740 --> 00:56:07,620
At the end of the
day, the degeneracies

1165
00:56:07,620 --> 00:56:11,420
that we see for the 3D
free particle, which

1166
00:56:11,420 --> 00:56:13,610
is a whole sphere's
worth of degeneracy,

1167
00:56:13,610 --> 00:56:16,130
and the degeneracy we see for
the 3D harmonic oscillator,

1168
00:56:16,130 --> 00:56:18,496
the bound states,
which is discrete,

1169
00:56:18,496 --> 00:56:19,870
but with more and
more degeneracy

1170
00:56:19,870 --> 00:56:22,610
the higher and
higher energy you go.

1171
00:56:22,610 --> 00:56:24,640
Those we're blaming,
at the moment,

1172
00:56:24,640 --> 00:56:26,540
on a symmetry, on
rotational symmetry,

1173
00:56:26,540 --> 00:56:29,050
rotational invariance.

1174
00:56:29,050 --> 00:56:32,240
So it seems wise
to study rotations,

1175
00:56:32,240 --> 00:56:35,220
to study rotational invariance
and rotational transformations

1176
00:56:35,220 --> 00:56:36,110
in the first place.

1177
00:56:36,110 --> 00:56:38,890
In the first part of the
course, in 1D quantum mechanics,

1178
00:56:38,890 --> 00:56:42,000
we got an awful lot of juice
out of studying translations.

1179
00:56:42,000 --> 00:56:44,354
And the generator of
translations was momentum.

1180
00:56:44,354 --> 00:56:46,020
So we're going to do
the same thing now.

1181
00:56:46,020 --> 00:56:48,370
We're going to study
rotations and the generators

1182
00:56:48,370 --> 00:56:50,700
of rotations, which are the
angular momentum operators,

1183
00:56:50,700 --> 00:56:52,824
and that's going to occupy
us for the rest of today

1184
00:56:52,824 --> 00:56:53,430
and Thursday.

1185
00:56:53,430 --> 00:56:53,930
Yeah?

1186
00:56:53,930 --> 00:56:56,720
AUDIENCE: So your
rotational symmetry

1187
00:56:56,720 --> 00:57:03,690
will explain a factor of three
in your degeneracy, right?

1188
00:57:03,690 --> 00:57:07,500
But what's the symmetry that
explains the way this grows.

1189
00:57:07,500 --> 00:57:10,150
Because this very
clearly appears

1190
00:57:10,150 --> 00:57:12,835
there's 1 up to a
factor of three.

1191
00:57:12,835 --> 00:57:14,626
And then there's 2 up
to a factor of three.

1192
00:57:14,626 --> 00:57:18,300
And there's even more that's
not even a multiple of three.

1193
00:57:18,300 --> 00:57:19,895
PROFESSOR: Right,
actually so here's

1194
00:57:19,895 --> 00:57:21,270
a very tempting
bit of intuition.

1195
00:57:21,270 --> 00:57:23,811
Very tempting bit of intuition
is going to say the following.

1196
00:57:23,811 --> 00:57:26,010
Look, rotational invariance,
there's x, there's y,

1197
00:57:26,010 --> 00:57:26,220
and there's z.

1198
00:57:26,220 --> 00:57:28,386
It's going to explain
rotations amongst those three.

1199
00:57:28,386 --> 00:57:31,130
So that could only possibly
give you a factor of three.

1200
00:57:31,130 --> 00:57:33,415
But it's important
to keep in mind

1201
00:57:33,415 --> 00:57:35,100
that that's not
correct intuition.

1202
00:57:35,100 --> 00:57:36,580
It's tempting intuition,
but it's not correct.

1203
00:57:36,580 --> 00:57:38,371
And an easy way to see
that its not correct

1204
00:57:38,371 --> 00:57:41,540
is that for the
free particle, there

1205
00:57:41,540 --> 00:57:44,340
is a continuum, a
whole sphere's worth

1206
00:57:44,340 --> 00:57:46,600
of degenerate states
in any energy.

1207
00:57:46,600 --> 00:57:49,550
And all of those are related to
each other by simple rotation

1208
00:57:49,550 --> 00:57:52,430
of the k vector, of
the wave vector, right?

1209
00:57:52,430 --> 00:57:54,850
So the rotational
symmetry is giving us

1210
00:57:54,850 --> 00:57:56,400
a lot more than a
factor of three.

1211
00:57:56,400 --> 00:57:58,150
And in fact, as
we'll see, it's going

1212
00:57:58,150 --> 00:58:00,360
to explain exactly the n
plus 1 n plus 2 over 2.

1213
00:58:02,890 --> 00:58:05,450
OK, so with that
motivation let's

1214
00:58:05,450 --> 00:58:07,335
start talking about
angular momentum.

1215
00:58:11,900 --> 00:58:19,790
So I found this topic
to be not obviously

1216
00:58:19,790 --> 00:58:23,500
the most powerful or interesting
thing in the world when I first

1217
00:58:23,500 --> 00:58:24,150
studied it.

1218
00:58:24,150 --> 00:58:25,697
And my professor
was like no, no, no.

1219
00:58:25,697 --> 00:58:26,780
This is the deepest thing.

1220
00:58:26,780 --> 00:58:28,412
And recently I had a fun
conversation with one

1221
00:58:28,412 --> 00:58:29,786
of my colleagues,
Frank Wilczeck,

1222
00:58:29,786 --> 00:58:32,530
who said yeah, in intro
to quantum mechanics

1223
00:58:32,530 --> 00:58:35,310
the single most interesting
thing is the angular momentum

1224
00:58:35,310 --> 00:58:36,930
and the addition of
angular momentum.

1225
00:58:36,930 --> 00:58:39,890
And something has happened to
me in the intervening 20 years

1226
00:58:39,890 --> 00:58:41,470
that I totally agree with him.

1227
00:58:41,470 --> 00:58:46,020
So I will attempt to convey to
you the awesomeness of this.

1228
00:58:46,020 --> 00:58:48,050
But you have to buy in a little.

1229
00:58:48,050 --> 00:58:50,740
So work with me in the math
at the beginning of this,

1230
00:58:50,740 --> 00:58:53,380
and it has a great payoff.

1231
00:58:53,380 --> 00:58:55,927
OK so the question is,
what is the operator.

1232
00:58:55,927 --> 00:58:58,010
So we're going to talk
about angular momentum now.

1233
00:59:01,770 --> 00:59:05,510
And I want to start with
the following question.

1234
00:59:05,510 --> 00:59:07,170
In the same sense
as we started out

1235
00:59:07,170 --> 00:59:08,640
by asking what
represents position

1236
00:59:08,640 --> 00:59:11,970
and momentum, linear momentum,
in quantum mechanics, what

1237
00:59:11,970 --> 00:59:17,020
represents what operator by
our first, second, or third

1238
00:59:17,020 --> 00:59:19,020
postulate-- I don't even
remember the order now.

1239
00:59:19,020 --> 00:59:26,060
What operator represents angular
momentum in quantum mechanics?

1240
00:59:30,980 --> 00:59:33,370
And let's start by remembering
what angular momentum is

1241
00:59:33,370 --> 00:59:35,550
in classical mechanics.

1242
00:59:35,550 --> 00:59:41,070
So L in classical
mechanics is r cross p.

1243
00:59:41,070 --> 00:59:43,702
In classical mechanics.

1244
00:59:43,702 --> 00:59:44,660
So let's just try this.

1245
00:59:44,660 --> 00:59:45,950
Let's construct that operator.

1246
00:59:45,950 --> 00:59:48,450
This is not the world's most
beautiful way of deriving this,

1247
00:59:48,450 --> 00:59:50,880
but let's just write
down natural guess.

1248
00:59:50,880 --> 00:59:53,320
For in quantum mechanics
what's the operator we want?

1249
00:59:53,320 --> 00:59:56,430
Well, we want a vectors
worth of operators,

1250
00:59:56,430 --> 00:59:58,267
because angular
momentum is a vector.

1251
00:59:58,267 --> 01:00:00,100
It's a vector of
operators, three operators.

1252
01:00:00,100 --> 01:00:06,280
And I'm going to write these
as r vector the operators x, y,

1253
01:00:06,280 --> 01:00:12,920
z cross p the vector
of momentum operators.

1254
01:00:12,920 --> 01:00:14,670
So at this point, you
should really worry,

1255
01:00:14,670 --> 01:00:18,050
because do r and p commute?

1256
01:00:18,050 --> 01:00:19,770
Not so much, right?

1257
01:00:19,770 --> 01:00:22,150
However, the situation is
better than it first appears.

1258
01:00:22,150 --> 01:00:24,840
Let's write this out
in terms of components.

1259
01:00:24,840 --> 01:00:27,387
So this is in components.

1260
01:00:27,387 --> 01:00:28,970
And I'm going to
work, for the moment,

1261
01:00:28,970 --> 01:00:30,560
in Cartesian coordinates.

1262
01:00:30,560 --> 01:00:34,120
So Lx is equal to?

1263
01:00:34,120 --> 01:00:37,400
Lx is equal to?

1264
01:00:37,400 --> 01:00:38,840
You all took mechanics.

1265
01:00:38,840 --> 01:00:41,332
Lx is equal to?

1266
01:00:41,332 --> 01:00:43,575
AUDIENCE: [INAUDIBLE].

1267
01:00:43,575 --> 01:00:44,450
PROFESSOR: Thank you.

1268
01:00:44,450 --> 01:00:47,920
YPz minus ZPy.

1269
01:00:47,920 --> 01:00:50,960
And that's the curl, the
x component of the curl.

1270
01:00:50,960 --> 01:00:55,650
And similarly, the x component--
so the way to remember this

1271
01:00:55,650 --> 01:00:56,740
is that its cyclic.

1272
01:00:56,740 --> 01:00:58,644
x, y, z.

1273
01:00:58,644 --> 01:01:00,470
y, z, x.

1274
01:01:00,470 --> 01:01:02,400
So z, p, y.

1275
01:01:05,450 --> 01:01:10,310
PX minus XPz.

1276
01:01:10,310 --> 01:01:14,030
And then we have
z XPy minus YPx.

1277
01:01:18,210 --> 01:01:21,200
So we were worried here about
maybe an ordering problem.

1278
01:01:21,200 --> 01:01:24,930
Is there an ordering
problem here?

1279
01:01:24,930 --> 01:01:27,923
Does it matter if
I write YPz or PZy?

1280
01:01:27,923 --> 01:01:28,465
AUDIENCE: No.

1281
01:01:28,465 --> 01:01:30,631
PROFESSOR: No, because they
commute with each other.

1282
01:01:30,631 --> 01:01:32,570
PZ is momentum for
the z-coordinate, not

1283
01:01:32,570 --> 01:01:34,700
the y-coordinate, and they
commute with each other.

1284
01:01:34,700 --> 01:01:36,200
So there's no ambiguity.

1285
01:01:36,200 --> 01:01:37,480
It's perfectly well defined.

1286
01:01:37,480 --> 01:01:39,063
So we're just going
to take this to be

1287
01:01:39,063 --> 01:01:41,430
the definition of the components
of the angular momentum

1288
01:01:41,430 --> 01:01:44,710
operator Lx, Ly, and Lz.

1289
01:01:50,460 --> 01:01:53,180
And just for fun, I
want to write this out.

1290
01:01:53,180 --> 01:01:56,870
So because we know that Px,
Py, and Pz can be expressed

1291
01:01:56,870 --> 01:01:59,600
in terms of derivatives
or differential operators,

1292
01:01:59,600 --> 01:02:01,970
we can write the same operator
in Cartesian coordinates

1293
01:02:01,970 --> 01:02:02,960
in the following way.

1294
01:02:02,960 --> 01:02:06,880
So clearly we could
write this as Y d dx i

1295
01:02:06,880 --> 01:02:09,120
upon h bar-- or
sorry, h bar upon i.

1296
01:02:09,120 --> 01:02:12,232
And z h bar upon i d dy.

1297
01:02:12,232 --> 01:02:14,190
So we could write that
in Cartesian coordinates

1298
01:02:14,190 --> 01:02:16,320
as a differential operator.

1299
01:02:16,320 --> 01:02:19,350
But we can also write this
in spherical coordinates.

1300
01:02:19,350 --> 01:02:21,300
I'm just going to take
a quick side note just

1301
01:02:21,300 --> 01:02:22,350
to write down what it is.

1302
01:02:22,350 --> 01:02:24,391
If we did this
spherical coordinates,

1303
01:02:24,391 --> 01:02:26,016
it's particularly
convenient to write--

1304
01:02:26,016 --> 01:02:27,980
let me just write
Lz for the moment.

1305
01:02:27,980 --> 01:02:31,890
This is equal to minus i h bar
derivative with respect to phi,

1306
01:02:31,890 --> 01:02:33,430
where the coordinates,
[INAUDIBLE]

1307
01:02:33,430 --> 01:02:36,010
coordinates and spherical
coordinates in this class

1308
01:02:36,010 --> 01:02:38,670
is theta is going
to be the angle down

1309
01:02:38,670 --> 01:02:40,550
from the vertical
axis, from the z-axis.

1310
01:02:40,550 --> 01:02:43,370
And phi is going to be
the angle of a period 2pi

1311
01:02:43,370 --> 01:02:44,860
that goes around
the equator, OK?

1312
01:02:44,860 --> 01:02:45,860
Just to give you a name.

1313
01:02:45,860 --> 01:02:47,510
This is typically what
physicists call them.

1314
01:02:47,510 --> 01:02:49,676
This is typically not what
mathematicians call them.

1315
01:02:49,676 --> 01:02:51,180
This leads to
enormous confusion.

1316
01:02:51,180 --> 01:02:52,520
I apologize for my field.

1317
01:02:52,520 --> 01:02:55,420
So here it is.

1318
01:02:55,420 --> 01:02:57,450
I can also construct
the operator associated

1319
01:02:57,450 --> 01:02:59,150
with the square of the momentum.

1320
01:02:59,150 --> 01:03:01,640
And why would we care about
the square of the momentum?

1321
01:03:01,640 --> 01:03:03,848
Well, that's what shows up
in the Hamiltonian, that's

1322
01:03:03,848 --> 01:03:05,020
what shows up in the energy.

1323
01:03:05,020 --> 01:03:07,603
So I can construct the operator
for the square of the momentum

1324
01:03:07,603 --> 01:03:10,280
and write it, and it takes
a surprisingly simple form.

1325
01:03:10,280 --> 01:03:12,780
When I see surprisingly simple,
you might disagree with me,

1326
01:03:12,780 --> 01:03:14,750
but if you actually do
the derivation of this,

1327
01:03:14,750 --> 01:03:16,190
it's much worse in between.

1328
01:03:16,190 --> 01:03:21,340
1 over sine theta
d theta sine theta

1329
01:03:21,340 --> 01:03:28,020
d theta plus 1 over sine
squared theta d phi.

1330
01:03:30,620 --> 01:03:31,770
Couple quick things.

1331
01:03:31,770 --> 01:03:33,615
What are the dimensions
of angular momentum?

1332
01:03:36,890 --> 01:03:38,170
Length and momentum.

1333
01:03:38,170 --> 01:03:40,880
What else has units
of angular momentum?

1334
01:03:40,880 --> 01:03:41,960
AUDIENCE: h bar.

1335
01:03:41,960 --> 01:03:44,226
PROFESSOR: Solid. h
bar, dimensionless.

1336
01:03:44,226 --> 01:03:46,350
Angular momentum squared,
angular momentum squared.

1337
01:03:46,350 --> 01:03:46,880
OK, great.

1338
01:03:46,880 --> 01:03:48,463
So that's going to
be very convenient.

1339
01:03:48,463 --> 01:03:52,540
h bars are just going to
float around willy nilly.

1340
01:03:52,540 --> 01:03:57,800
OK, so suppose I ask you
the following-- bless you.

1341
01:03:57,800 --> 01:03:59,680
Suppose I ask you the
following questions.

1342
01:03:59,680 --> 01:04:02,950
I say look, here are the
operators of angular momentum.

1343
01:04:02,950 --> 01:04:03,550
This is Lz.

1344
01:04:03,550 --> 01:04:05,040
We could have written down
the same expression for Lx,

1345
01:04:05,040 --> 01:04:06,490
and a Ly, and L squared.

1346
01:04:06,490 --> 01:04:08,650
What are the eigenfunctions
of these operators?

1347
01:04:08,650 --> 01:04:10,070
Suppose I ask you this question.

1348
01:04:10,070 --> 01:04:12,270
You all know how to
answer this question.

1349
01:04:12,270 --> 01:04:14,020
You take these operators--
so for example,

1350
01:04:14,020 --> 01:04:17,360
if I ask you what are
the eigenfunctions of Lz?

1351
01:04:17,360 --> 01:04:18,910
Well, that's not so bad, right?

1352
01:04:18,910 --> 01:04:20,409
The eigenfunction
of Lz is something

1353
01:04:20,409 --> 01:04:25,010
where Lz on phi--
I'll call little m--

1354
01:04:25,010 --> 01:04:29,680
is equal to minus I h
bar d d theta phi sub m.

1355
01:04:29,680 --> 01:04:31,550
But I want the
eigenvalue, so I'll

1356
01:04:31,550 --> 01:04:35,130
call this h bar
times some number.

1357
01:04:35,130 --> 01:04:37,920
Let's call it m, because
Lz is an angular momentum.

1358
01:04:37,920 --> 01:04:40,230
It carries units of
h bar, and its h bar

1359
01:04:40,230 --> 01:04:41,800
times some number which is
dimensional, so we'll call it

1360
01:04:41,800 --> 01:04:42,350
m.

1361
01:04:42,350 --> 01:04:44,510
And we all know the
solution of this equation.

1362
01:04:44,510 --> 01:04:47,290
The derivative is equal
to a constant times--

1363
01:04:47,290 --> 01:04:49,340
we can lose the h bar.

1364
01:04:49,340 --> 01:04:51,335
We get a minus i,
so we pick up an i.

1365
01:04:51,335 --> 01:04:55,640
So therefore phi sub m is
equal to some constant times e

1366
01:04:55,640 --> 01:04:58,200
to the im phi.

1367
01:04:58,200 --> 01:05:00,590
What can we say about m?

1368
01:05:00,590 --> 01:05:02,570
Well, heres an important
thing-- oh shoot.

1369
01:05:02,570 --> 01:05:05,139
I'm using phi in so many
different ways here.

1370
01:05:05,139 --> 01:05:06,680
Let's call this not
phi, because it's

1371
01:05:06,680 --> 01:05:08,180
going to confuse
the heck out of us.

1372
01:05:08,180 --> 01:05:12,600
Let's call it Y. So
we'll call it Y sub m.

1373
01:05:12,600 --> 01:05:13,200
Why not?

1374
01:05:16,650 --> 01:05:17,970
It's not my joke.

1375
01:05:17,970 --> 01:05:22,070
This goes back to a bunch of--
yeah well, it goes way back.

1376
01:05:22,070 --> 01:05:23,380
So phi is the variable.

1377
01:05:23,380 --> 01:05:26,670
Y is the deciding eigenfunction
of Lz, and that's great.

1378
01:05:26,670 --> 01:05:30,170
But what can you say about m?

1379
01:05:30,170 --> 01:05:32,970
Well phi is the variable
around the equator

1380
01:05:32,970 --> 01:05:34,957
is periodic with period 2pi.

1381
01:05:34,957 --> 01:05:37,040
And our wave function had
better be single valued.

1382
01:05:37,040 --> 01:05:40,040
So what does that
tell you about m?

1383
01:05:40,040 --> 01:05:41,850
Well, under phi goes
to phi plus 2pi.

1384
01:05:41,850 --> 01:05:44,859
This shifts by im 2pi.

1385
01:05:44,859 --> 01:05:46,650
And that's only one to
make a single valued

1386
01:05:46,650 --> 01:05:48,707
if m is an integer.

1387
01:05:48,707 --> 01:05:49,790
So m has to be an integer.

1388
01:05:52,654 --> 01:05:53,320
m is an integer.

1389
01:05:53,320 --> 01:05:55,050
Now we did that for Lz.

1390
01:05:55,050 --> 01:05:56,500
We found the
eigenfunctions of Lz.

1391
01:05:56,500 --> 01:05:59,310
What about finding the
eigenfunctions L squared?

1392
01:05:59,310 --> 01:06:00,360
Exactly the same thing.

1393
01:06:00,360 --> 01:06:02,234
We're going to solve
the eigenvalue equation,

1394
01:06:02,234 --> 01:06:04,970
but it's going to be
horrible, horrible to find

1395
01:06:04,970 --> 01:06:05,965
these functions, right?

1396
01:06:05,965 --> 01:06:06,840
Because look at this.

1397
01:06:06,840 --> 01:06:08,370
1 over sine squared d d phi.

1398
01:06:08,370 --> 01:06:11,230
And then 1 over sine
d theta sine d theta.

1399
01:06:11,230 --> 01:06:13,590
This is not going to
be a fun thing to do.

1400
01:06:13,590 --> 01:06:17,020
So we could just brute
force this, but let's not.

1401
01:06:17,020 --> 01:06:20,730
Let's all agree that
that's probably a bad idea.

1402
01:06:20,730 --> 01:06:23,340
Let's find a better way to
construct the eigenfunctions

1403
01:06:23,340 --> 01:06:25,720
of the angular
momentum operators.

1404
01:06:25,720 --> 01:06:26,840
So let's do it.

1405
01:06:26,840 --> 01:06:28,590
So we ran into a
situation like this

1406
01:06:28,590 --> 01:06:31,625
before when we dealt with
the harmonic oscillator.

1407
01:06:31,625 --> 01:06:33,000
There was a
differential equation

1408
01:06:33,000 --> 01:06:34,290
that we wanted to solve.

1409
01:06:34,290 --> 01:06:37,760
And OK, this one isn't nearly
as bad, not nearly as bad

1410
01:06:37,760 --> 01:06:40,950
as that one would have been.

1411
01:06:40,950 --> 01:06:44,740
But still it was more useful
to work with operator methods.

1412
01:06:44,740 --> 01:06:48,270
So let's take a hint from that
and work with operator methods.

1413
01:06:48,270 --> 01:06:53,090
So now we need to study the
operators of angular momentum.

1414
01:06:53,090 --> 01:06:54,924
So let's study them in
a little more detail.

1415
01:06:54,924 --> 01:06:57,131
So something you're going
to show in your problem set

1416
01:06:57,131 --> 01:06:58,010
is the following.

1417
01:06:58,010 --> 01:07:03,280
The commutator of Lx with Ly
takes a really simple form.

1418
01:07:03,280 --> 01:07:07,409
This is equal to i h bar--
let's just do this out.

1419
01:07:07,409 --> 01:07:08,450
Let's do this commutator.

1420
01:07:08,450 --> 01:07:09,510
We're OK.

1421
01:07:09,510 --> 01:07:12,210
So Lx with Ly, this is
equal to the commutator

1422
01:07:12,210 --> 01:07:21,400
of YPz minus ZPy with
Ly ZPx minus XPz.

1423
01:07:24,830 --> 01:07:26,630
So let's look at
these term by term.

1424
01:07:26,630 --> 01:07:29,790
So the first one is YPz ZPx.

1425
01:07:29,790 --> 01:07:33,130
YPz ZPx.

1426
01:07:33,130 --> 01:07:34,310
That's a Px.

1427
01:07:34,310 --> 01:07:34,810
Sorry, ZPy.

1428
01:07:37,540 --> 01:07:40,230
ZPx, this term.

1429
01:07:40,230 --> 01:07:41,900
X, good.

1430
01:07:41,900 --> 01:07:43,360
That's the first commutator.

1431
01:07:43,360 --> 01:07:51,310
The second commutator, it can
be YPz an XPz minus YPz XPz.

1432
01:07:54,470 --> 01:07:56,665
And then these
commutators, the next two,

1433
01:07:56,665 --> 01:08:03,400
are going to be ZPy with ZPx.

1434
01:08:06,060 --> 01:08:08,390
And finally ZPy.

1435
01:08:08,390 --> 01:08:10,260
And that's minus
and this is a plus.

1436
01:08:10,260 --> 01:08:12,030
ZPy and XPz.

1437
01:08:15,810 --> 01:08:19,240
OK, so let's look at these.

1438
01:08:19,240 --> 01:08:21,180
These look kind
of scary at first.

1439
01:08:21,180 --> 01:08:22,760
But in this one
notice the following.

1440
01:08:22,760 --> 01:08:28,630
This is XPz ZPx minus ZPx YPz.

1441
01:08:28,630 --> 01:08:33,779
But what you say about Y with
all these other operators?

1442
01:08:33,779 --> 01:08:35,210
Y commutes with all of them.

1443
01:08:35,210 --> 01:08:36,740
So in each term I
could just pull Y

1444
01:08:36,740 --> 01:08:38,689
all the way out to one side.

1445
01:08:38,689 --> 01:08:39,505
Yeah?

1446
01:08:39,505 --> 01:08:43,460
So I could just pull out this
Y. So for the first term,

1447
01:08:43,460 --> 01:08:48,582
I'm going to write this as
Y commutator PZ with ZPx.

1448
01:08:48,582 --> 01:08:50,040
And let me just do
that explicitly.

1449
01:08:50,040 --> 01:08:51,200
There's no reason to.

1450
01:08:51,200 --> 01:09:02,580
So this is YPz
ZPx minus ZPx YPz.

1451
01:09:02,580 --> 01:09:05,560
And I can pull the Y out front,
because this commutes with Px

1452
01:09:05,560 --> 01:09:12,979
and with Z. So I can make this
Y times PZ ZPx minus ZPx Pz.

1453
01:09:12,979 --> 01:09:15,260
But now note that I can
do exactly the same thing

1454
01:09:15,260 --> 01:09:16,179
with the Px.

1455
01:09:16,179 --> 01:09:16,970
Px commutes with z.

1456
01:09:16,970 --> 01:09:18,240
Px commutes with Pz.

1457
01:09:18,240 --> 01:09:19,529
And it commutes with y.

1458
01:09:19,529 --> 01:09:22,310
So I can pull the px
from each term out.

1459
01:09:22,310 --> 01:09:27,415
Px Y. And now I lose the Px.

1460
01:09:27,415 --> 01:09:28,620
I lose the Px.

1461
01:09:28,620 --> 01:09:30,569
But now this is looking good.

1462
01:09:30,569 --> 01:09:38,210
This is Px times Y. And Pz
minus ZPz PZz minus ZPz.

1463
01:09:38,210 --> 01:09:47,047
This is also known as PXy
times commutator of PZ with Z.

1464
01:09:47,047 --> 01:09:48,130
And what is this equal to?

1465
01:09:51,497 --> 01:09:52,580
Let's get our signs right.

1466
01:09:52,580 --> 01:09:55,930
This is Px with Y
time-- OK, we all

1467
01:09:55,930 --> 01:09:57,930
agree that this is going
to have an h bar in it.

1468
01:09:57,930 --> 01:09:59,290
Let's write units.

1469
01:09:59,290 --> 01:10:00,950
There's going to be an i.

1470
01:10:00,950 --> 01:10:04,030
And is it a plus or minus?

1471
01:10:04,030 --> 01:10:05,020
Minus.

1472
01:10:05,020 --> 01:10:05,770
OK, good.

1473
01:10:05,770 --> 01:10:08,730
So we get minus h bar XPy.

1474
01:10:08,730 --> 01:10:17,060
So this term is going to give
us minus i h bar Py XPx times Y.

1475
01:10:17,060 --> 01:10:18,480
And let's look at this term.

1476
01:10:18,480 --> 01:10:24,440
OK, this is Y, and Y commutes
with PZx and PZ, right?

1477
01:10:24,440 --> 01:10:26,080
So I could just pull out the y.

1478
01:10:26,080 --> 01:10:28,700
And x commutes with everything,
so I can pull out the x.

1479
01:10:28,700 --> 01:10:31,410
And I'm left with the
commutator of Pz with Pz.

1480
01:10:31,410 --> 01:10:32,910
What's the commutator
of Pz with Pz?

1481
01:10:32,910 --> 01:10:33,410
AUDIENCE: 0.

1482
01:10:33,410 --> 01:10:33,951
PROFESSOR: 0.

1483
01:10:33,951 --> 01:10:35,170
This term gives me a 0.

1484
01:10:35,170 --> 01:10:37,150
Similarly here, Py
commutes with everything.

1485
01:10:37,150 --> 01:10:38,696
Z ZPx.

1486
01:10:38,696 --> 01:10:40,100
Px commutes with everything.

1487
01:10:40,100 --> 01:10:41,590
Z ZPy.

1488
01:10:41,590 --> 01:10:44,870
So I can pull out the Px Py,
and I get Z commutator Z,

1489
01:10:44,870 --> 01:10:45,610
and what's that?

1490
01:10:45,610 --> 01:10:46,159
AUDIENCE: 0.

1491
01:10:46,159 --> 01:10:46,700
PROFESSOR: 0.

1492
01:10:46,700 --> 01:10:47,710
So this gives me 0.

1493
01:10:47,710 --> 01:10:51,480
And now this term, ZPy
XPz, the only two things

1494
01:10:51,480 --> 01:10:54,600
that don't commute with each
other are the Z and the Pz.

1495
01:10:54,600 --> 01:10:56,580
The Py and the X
I pull out, so I

1496
01:10:56,580 --> 01:11:00,640
get it a term that's XPy and
the commutator of Z with Pz.

1497
01:11:00,640 --> 01:11:03,190
And what is that
going to give me?

1498
01:11:03,190 --> 01:11:05,250
PyX i h bar.

1499
01:11:11,685 --> 01:11:14,870
Aha, look at what we got.

1500
01:11:14,870 --> 01:11:22,050
This is equal to i h bar times--
did I screw up the signs?

1501
01:11:22,050 --> 01:11:24,765
XPy minus YPz.

1502
01:11:30,060 --> 01:11:33,130
Oh sorry, Px.

1503
01:11:33,130 --> 01:11:35,036
And what is this equal to?

1504
01:11:35,036 --> 01:11:36,380
AUDIENCE: [INAUDIBLE].

1505
01:11:36,380 --> 01:11:37,801
PROFESSOR: Yeah, i h bar Lz.

1506
01:11:43,580 --> 01:11:46,964
And more generally, as you'll
show on the problem set,

1507
01:11:46,964 --> 01:11:48,380
you get the following
commutators.

1508
01:11:48,380 --> 01:11:51,510
Once you've done this once, you
can do the rest very easily.

1509
01:11:51,510 --> 01:11:57,980
Lx Ly is-- so Lx with
Ly is i h bar Lz.

1510
01:12:03,010 --> 01:12:06,000
And then the rest can be
got from cyclic rotations

1511
01:12:06,000 --> 01:12:16,970
Ly with Lz is i h bar Lx and
Lz with Lx is i h bar Ly.

1512
01:12:21,460 --> 01:12:22,930
And now here's a fancier one.

1513
01:12:22,930 --> 01:12:25,536
This is less obvious, but
exactly the same machinations

1514
01:12:25,536 --> 01:12:27,660
will give you this result,
and you'll do this again

1515
01:12:27,660 --> 01:12:28,580
on the problems set.

1516
01:12:28,580 --> 01:12:31,840
If I take L squared,
k and L squared

1517
01:12:31,840 --> 01:12:36,150
here is going to be
L squared, I just

1518
01:12:36,150 --> 01:12:42,520
mean Lx squared plus Ly
squared plus Lz squared.

1519
01:12:42,520 --> 01:12:46,130
This is the norm squared of
the vector, the operator form.

1520
01:12:46,130 --> 01:12:49,470
L squared with Lz--
or sorry, with Lx.

1521
01:12:49,470 --> 01:12:50,350
Yeah, fine.

1522
01:12:50,350 --> 01:12:52,570
Lx is equal to 0.

1523
01:12:52,570 --> 01:12:55,910
So Lx commutes with the
magnitude L squared.

1524
01:12:55,910 --> 01:12:58,910
Similarly L squared, now just
by rotational invariance,

1525
01:12:58,910 --> 01:13:00,814
if Lx commutes
with it, Ly and Lz

1526
01:13:00,814 --> 01:13:02,480
had better also commute
with it, because

1527
01:13:02,480 --> 01:13:06,180
of who you to see what's Lx.

1528
01:13:06,180 --> 01:13:09,290
And L squared with Lz
must be equal to 0.

1529
01:13:15,340 --> 01:13:15,840
Yeah?

1530
01:13:15,840 --> 01:13:16,860
Everyone cool with that?

1531
01:13:21,160 --> 01:13:23,320
And the thing is
we've just-- and I'm

1532
01:13:23,320 --> 01:13:24,950
going to put big
flags around this.

1533
01:13:24,950 --> 01:13:26,620
We've just learned
a tremendous amount

1534
01:13:26,620 --> 01:13:29,520
about the eigenfunctions of
the angular momentum operators.

1535
01:13:29,520 --> 01:13:31,411
Why?

1536
01:13:31,411 --> 01:13:33,900
Let's leave that up.

1537
01:13:33,900 --> 01:13:36,590
So what have we just learned
about the eigenfunctions

1538
01:13:36,590 --> 01:13:39,221
of the angular momentum
operators Lx, Ly, Lz, and L

1539
01:13:39,221 --> 01:13:39,720
squared?

1540
01:13:44,890 --> 01:13:45,562
Anyone?

1541
01:13:45,562 --> 01:13:47,770
We just learned something
totally awesome about them.

1542
01:13:57,620 --> 01:13:58,750
Anyone?

1543
01:13:58,750 --> 01:14:03,174
So can you find simultaneous
eigenfunctions of Lx and Ly?

1544
01:14:03,174 --> 01:14:03,912
AUDIENCE: No.

1545
01:14:03,912 --> 01:14:04,870
PROFESSOR: Not so much.

1546
01:14:04,870 --> 01:14:06,000
They don't commute to 0.

1547
01:14:06,000 --> 01:14:07,810
What about Ly and Lz?

1548
01:14:07,810 --> 01:14:08,800
Nope.

1549
01:14:08,800 --> 01:14:11,080
Lz Lx, nope.

1550
01:14:11,080 --> 01:14:14,060
So you cannot find simultaneous
eigenfunctions of Lx and Ly.

1551
01:14:17,010 --> 01:14:20,680
What about Lx and L squared?

1552
01:14:20,680 --> 01:14:22,120
Yes.

1553
01:14:22,120 --> 01:14:24,480
So we can find
simultaneously eigenfunctions

1554
01:14:24,480 --> 01:14:29,740
of L squared and Lx.

1555
01:14:29,740 --> 01:14:32,850
OK, what about can we find
simultaneous eigenfunctions

1556
01:14:32,850 --> 01:14:35,370
of L squared and Ly?

1557
01:14:35,370 --> 01:14:35,990
Yep.

1558
01:14:35,990 --> 01:14:36,489
Exactly.

1559
01:14:36,489 --> 01:14:39,840
What about L
squared, Ly, and Lz?

1560
01:14:39,840 --> 01:14:40,580
Nope.

1561
01:14:40,580 --> 01:14:42,530
No such luck.

1562
01:14:42,530 --> 01:14:44,720
And this leads us to
the following idea.

1563
01:14:44,720 --> 01:14:51,130
The idea is a complete set
of commuting observables.

1564
01:14:54,790 --> 01:14:58,180
And here's what this
idea is meant to contain.

1565
01:14:58,180 --> 01:15:00,350
You can always write
down a lot of operators.

1566
01:15:00,350 --> 01:15:02,710
So let me step back and
ask a classical question.

1567
01:15:02,710 --> 01:15:04,730
Classically, suppose I have a
particle in three dimensions,

1568
01:15:04,730 --> 01:15:07,063
a particle moving around in
this room, non relativistic,

1569
01:15:07,063 --> 01:15:08,380
familiar 801.

1570
01:15:08,380 --> 01:15:10,720
I have a particle moving
around in this room.

1571
01:15:10,720 --> 01:15:12,835
How much data must
I specify to specify

1572
01:15:12,835 --> 01:15:14,210
the configuration
of this system?

1573
01:15:14,210 --> 01:15:16,290
Well, I have to tell you
where the particle is,

1574
01:15:16,290 --> 01:15:18,530
and I have to tell you what
its momentum is, right?

1575
01:15:18,530 --> 01:15:21,130
So I have to tell you the
three coordinates and the three

1576
01:15:21,130 --> 01:15:22,230
momenta.

1577
01:15:22,230 --> 01:15:26,480
If I give you five numbers,
that's not enough, right?

1578
01:15:26,480 --> 01:15:29,902
I need to give you
six bits of data.

1579
01:15:29,902 --> 01:15:31,860
On the other hand, if I
give you seven numbers,

1580
01:15:31,860 --> 01:15:37,750
like I give x, y, z, Px, Py,
Pz, and e, that's over complete.

1581
01:15:37,750 --> 01:15:38,250
Right?

1582
01:15:38,250 --> 01:15:41,214
That was unnecessary.

1583
01:15:41,214 --> 01:15:42,630
So in classical
mechanics, you can

1584
01:15:42,630 --> 01:15:45,570
ask what data must you
specify to completely specify

1585
01:15:45,570 --> 01:15:46,570
the state of the system.

1586
01:15:46,570 --> 01:15:47,600
And that's usually pretty easy.

1587
01:15:47,600 --> 01:15:48,620
You specify the
number coordinates

1588
01:15:48,620 --> 01:15:49,802
and the number of momenta.

1589
01:15:49,802 --> 01:15:52,260
In quantum mechanics, we ask
a slightly different question.

1590
01:15:52,260 --> 01:15:54,970
We ask, in order to specify
the state of a system,

1591
01:15:54,970 --> 01:15:57,860
we say we want to specify
which state it is.

1592
01:15:57,860 --> 01:15:59,930
You can specify
that by saying which

1593
01:15:59,930 --> 01:16:01,830
superposition in a
particular basis.

1594
01:16:01,830 --> 01:16:04,440
So you pick a basis,
and you specify

1595
01:16:04,440 --> 01:16:05,980
which particular
superposition, what

1596
01:16:05,980 --> 01:16:09,970
are the eigenvalues of
the operators you've

1597
01:16:09,970 --> 01:16:12,410
diagonalized in that basis?

1598
01:16:12,410 --> 01:16:16,619
So another way to phrase
this is for a 1D problem,

1599
01:16:16,619 --> 01:16:18,160
say we have just a
simple 1D problem.

1600
01:16:18,160 --> 01:16:23,140
We have X and we have P. What
is a complete set of commuting

1601
01:16:23,140 --> 01:16:23,780
operators?

1602
01:16:23,780 --> 01:16:25,405
Well, you have to
have enough operators

1603
01:16:25,405 --> 01:16:27,540
so that the eigenvalue
specifies a state.

1604
01:16:27,540 --> 01:16:30,130
So X would be-- is that enough?

1605
01:16:32,650 --> 01:16:35,520
So if I take the operator
X and I say look,

1606
01:16:35,520 --> 01:16:37,881
my system is in the state
with an eigenvalue X not of X,

1607
01:16:37,881 --> 01:16:39,630
does that specify the
state of the system?

1608
01:16:42,817 --> 01:16:45,150
It tells you the particles
are in a delta function state

1609
01:16:45,150 --> 01:16:46,590
right here.

1610
01:16:46,590 --> 01:16:47,820
Does that specify the state?

1611
01:16:47,820 --> 01:16:48,130
AUDIENCE: Yes.

1612
01:16:48,130 --> 01:16:49,380
PROFESSOR: Fantastic, it does.

1613
01:16:49,380 --> 01:16:51,030
Now what if I tell
you instead, oh it's

1614
01:16:51,030 --> 01:16:55,460
in a state of definite P.
Does that specify the state?

1615
01:16:55,460 --> 01:16:56,570
Absolutely.

1616
01:16:56,570 --> 01:16:59,910
Can I say it's in a state with
definite X and definite P?

1617
01:16:59,910 --> 01:17:00,730
AUDIENCE: No.

1618
01:17:00,730 --> 01:17:01,480
PROFESSOR: No.

1619
01:17:01,480 --> 01:17:03,839
These don't commute.

1620
01:17:03,839 --> 01:17:06,130
So a complete set of commuting
observables in this case

1621
01:17:06,130 --> 01:17:09,300
would be either X
or P, but not both.

1622
01:17:09,300 --> 01:17:10,410
Yeah?

1623
01:17:10,410 --> 01:17:15,090
Now if we're in three
dimensions, is x a complete set

1624
01:17:15,090 --> 01:17:16,466
of commuting observables?

1625
01:17:16,466 --> 01:17:17,205
AUDIENCE: No.

1626
01:17:17,205 --> 01:17:18,830
PROFESSOR: No, because
it's not enough.

1627
01:17:18,830 --> 01:17:20,880
You tell me that it's
X, that doesn't tell me

1628
01:17:20,880 --> 01:17:22,255
what state it is
because it could

1629
01:17:22,255 --> 01:17:24,420
have y dependence
or z dependence.

1630
01:17:24,420 --> 01:17:30,760
So in 3D, we take, for
example, x, and y, and z.

1631
01:17:30,760 --> 01:17:37,010
Or Px, and Py, and Pz.

1632
01:17:37,010 --> 01:17:42,940
We could also pick
z, and Px, and y.

1633
01:17:42,940 --> 01:17:45,710
Are these complete?

1634
01:17:45,710 --> 01:17:47,690
If I tell you I have
definite position in z,

1635
01:17:47,690 --> 01:17:50,050
definite position in y,
and definite momentum in x?

1636
01:17:50,050 --> 01:17:51,380
Does that completely
specify my state?

1637
01:17:51,380 --> 01:17:51,855
AUDIENCE: Yes.

1638
01:17:51,855 --> 01:17:53,479
PROFESSOR: Yeah,
totally unambiguously.

1639
01:17:53,479 --> 01:18:00,030
e to the ikx delta of y delta
of z totally fixes my function,

1640
01:18:00,030 --> 01:18:01,030
completely specifies it.

1641
01:18:01,030 --> 01:18:03,190
If I'd only picked
two of these, it

1642
01:18:03,190 --> 01:18:04,712
would not have been complete.

1643
01:18:04,712 --> 01:18:06,170
It would have told
me, for example,

1644
01:18:06,170 --> 01:18:09,004
e to the ikx delta of y,
but it doesn't tell me

1645
01:18:09,004 --> 01:18:11,850
how it depends on z, how
the wave functions on z.

1646
01:18:11,850 --> 01:18:16,154
And if I added another
operator, for example, Py,

1647
01:18:16,154 --> 01:18:17,320
this is no longer commuting.

1648
01:18:17,320 --> 01:18:18,751
These two operators
don't commute.

1649
01:18:18,751 --> 01:18:20,500
So a complete set of
commuting observables

1650
01:18:20,500 --> 01:18:22,270
can be thought of as
the most operators

1651
01:18:22,270 --> 01:18:25,020
you can write down that
all commute with each other

1652
01:18:25,020 --> 01:18:28,060
and the minimum number
whose eigenvalues completely

1653
01:18:28,060 --> 01:18:30,008
specify the state of the system.

1654
01:18:30,008 --> 01:18:30,508
Cool?

1655
01:18:30,508 --> 01:18:31,340
OK.

1656
01:18:31,340 --> 01:18:33,750
So with all that said,
what is a complete set

1657
01:18:33,750 --> 01:18:36,400
of commuting observables for
the angular momentum system?

1658
01:18:40,730 --> 01:18:44,552
Well, it can't be any
two of Lx, Ly, and Lz.

1659
01:18:44,552 --> 01:18:45,510
So let's just pick one.

1660
01:18:45,510 --> 01:18:47,210
I'll call it Lz.

1661
01:18:47,210 --> 01:18:48,220
I could've called it Lx.

1662
01:18:48,220 --> 01:18:48,650
It doesn't matter.

1663
01:18:48,650 --> 01:18:50,210
It's up to you
what axis is what.

1664
01:18:50,210 --> 01:18:53,950
I'll just call it
Lz conventionally.

1665
01:18:53,950 --> 01:18:56,280
And then L squared
also commutes,

1666
01:18:56,280 --> 01:18:57,780
because L squared
commutes with Lx.

1667
01:18:57,780 --> 01:18:59,420
It commutes Ly and with Lz.

1668
01:18:59,420 --> 01:19:01,100
So this actually
forms a complete set

1669
01:19:01,100 --> 01:19:06,010
of commuting observables for
the angular momentum system.

1670
01:19:06,010 --> 01:19:08,290
Complete set of
commuting observables

1671
01:19:08,290 --> 01:19:09,840
for angular momentum.

1672
01:19:15,670 --> 01:19:21,000
So this idea will come
up more in the future.

1673
01:19:21,000 --> 01:19:28,860
And here is going to
be the key [INAUDIBLE].

1674
01:19:28,860 --> 01:19:31,320
So we'd like to use
the following fact.

1675
01:19:31,320 --> 01:19:33,010
We want to construct
the eigenfunctions

1676
01:19:33,010 --> 01:19:34,731
of our complete set
of-- yeah, question?

1677
01:19:34,731 --> 01:19:36,314
AUDIENCE: Really
quick can you explain

1678
01:19:36,314 --> 01:19:38,470
how you got that L
squared and Lx commute?

1679
01:19:38,470 --> 01:19:39,970
PROFESSOR: Yeah, I
got it by knowing

1680
01:19:39,970 --> 01:19:42,010
what you're going to write
on your solution set.

1681
01:19:42,010 --> 01:19:43,530
So this is on your problem set.

1682
01:19:43,530 --> 01:19:46,230
So the way it goes-- so there
are fancy ways of doing it,

1683
01:19:46,230 --> 01:19:48,170
but the just direct
way of doing,

1684
01:19:48,170 --> 01:19:49,930
how do you construct
these commutators?

1685
01:19:49,930 --> 01:19:51,388
Is you know what
the operators are.

1686
01:19:51,388 --> 01:19:52,802
You know what L squared is.

1687
01:19:52,802 --> 01:19:54,510
And you know that L
squared is Lx squared

1688
01:19:54,510 --> 01:19:55,950
plus Ly squared plus Lz squared.

1689
01:19:55,950 --> 01:19:59,310
It's built in that
fashion out of x and Py.

1690
01:19:59,310 --> 01:20:02,460
And then I literally just put
in the definitions of Lx, Ly,

1691
01:20:02,460 --> 01:20:04,910
and Lz into that
expression for L squared

1692
01:20:04,910 --> 01:20:06,910
and compute the
commutator with Lx, again,

1693
01:20:06,910 --> 01:20:10,880
using the definition
in terms of Py and z.

1694
01:20:10,880 --> 01:20:13,390
And then you just chug
through the commutators.

1695
01:20:13,390 --> 01:20:14,390
Yeah, it just works out.

1696
01:20:14,390 --> 01:20:17,700
So it's not obvious from
the way I just phrased it

1697
01:20:17,700 --> 01:20:20,022
that it works out like that.

1698
01:20:20,022 --> 01:20:21,980
Later on you'll probably
develop some intuition

1699
01:20:21,980 --> 01:20:23,370
that it should be obvious.

1700
01:20:23,370 --> 01:20:24,786
But for the moment,
I'm just going

1701
01:20:24,786 --> 01:20:26,636
to call it a brute
force computation.

1702
01:20:26,636 --> 01:20:29,010
And that's how you're going
to do it on your problem set.

1703
01:20:29,010 --> 01:20:32,480
So let me tell you
where we're going next.

1704
01:20:32,480 --> 01:20:35,310
So the question I
really want to deal with

1705
01:20:35,310 --> 01:20:37,780
is what are the eigenfunctions?

1706
01:20:37,780 --> 01:20:39,730
So this is where
we're leaving off.

1707
01:20:42,270 --> 01:20:47,530
What are the eigenfunctions
of our complete set

1708
01:20:47,530 --> 01:20:50,950
of commuting variable
L squared and Lz.

1709
01:20:55,270 --> 01:20:56,120
What are these guys?

1710
01:20:56,120 --> 01:20:57,190
And we know that
we could solve them

1711
01:20:57,190 --> 01:20:58,960
by solving the differential
equations using

1712
01:20:58,960 --> 01:21:00,520
those operators, but
that would be horrible.

1713
01:21:00,520 --> 01:21:01,980
We'd like to do something
a little smarter.

1714
01:21:01,980 --> 01:21:04,438
We'd like to use the commutation
relations and the algebra.

1715
01:21:06,650 --> 01:21:11,690
And here a really beautiful
thing is going to happen.

1716
01:21:11,690 --> 01:21:13,601
When you look at these
commutation relations,

1717
01:21:13,601 --> 01:21:16,100
one thing they're telling us
is that we can't simultaneously

1718
01:21:16,100 --> 01:21:20,030
have eigenfunctions
of Lx and Ly.

1719
01:21:20,030 --> 01:21:24,919
However, the way that Lx and Ly
commute together is to form Lz.

1720
01:21:24,919 --> 01:21:26,210
That gives us some information.

1721
01:21:26,210 --> 01:21:29,910
That gives us some
magic, some power.

1722
01:21:29,910 --> 01:21:32,339
And in particular, much like
that moment in the harmonic

1723
01:21:32,339 --> 01:21:34,380
oscillator I said, well
look, we could write down

1724
01:21:34,380 --> 01:21:37,040
these operators as a.

1725
01:21:37,040 --> 01:21:40,170
Well look, we can write
down these operators,

1726
01:21:40,170 --> 01:21:43,320
which I'm going to call
L plus and L minus.

1727
01:21:43,320 --> 01:21:48,200
L plus is going to be
equal to Lx plus i Ly.

1728
01:21:48,200 --> 01:21:53,390
And L minus is going to
be equal to Lx minus I Ly.

1729
01:21:53,390 --> 01:21:55,880
Now Lx, Ly, those
are observables?

1730
01:21:55,880 --> 01:21:59,464
What can you say about
them as operators?

1731
01:21:59,464 --> 01:22:01,880
What kind of operators are
they since they're observables?

1732
01:22:01,880 --> 01:22:03,180
Hermitian, exactly.

1733
01:22:03,180 --> 01:22:07,320
So what's the Hermitian adjoint
of Hermitian plus i Hermitian?

1734
01:22:07,320 --> 01:22:08,571
Hermitian minus i Hermitian.

1735
01:22:08,571 --> 01:22:09,070
OK, good.

1736
01:22:09,070 --> 01:22:11,120
So L minus is the
adjoint of L plus.

1737
01:22:15,392 --> 01:22:17,100
So this is just going
to be a definition.

1738
01:22:17,100 --> 01:22:19,308
Let's take these to be the
definitions of these guys.

1739
01:22:22,902 --> 01:22:25,110
If we take their commutator,
something totally lovely

1740
01:22:25,110 --> 01:22:26,080
happens.

1741
01:22:26,080 --> 01:22:26,930
I'm not going to write
all the commutators.

1742
01:22:26,930 --> 01:22:28,305
I'm just going to
write a couple.

1743
01:22:28,305 --> 01:22:33,560
The first is if I take L squared
and I commute with L plus,

1744
01:22:33,560 --> 01:22:36,010
well, L plus is Lx
plus Ly, and we already

1745
01:22:36,010 --> 01:22:37,510
know that L squared
commutes with Lx

1746
01:22:37,510 --> 01:22:38,510
and it commutes with Ly.

1747
01:22:38,510 --> 01:22:42,600
So L squared
commutes with L plus.

1748
01:22:42,600 --> 01:22:46,140
And similarly for L minus,
it commutes with each term.

1749
01:22:46,140 --> 01:22:48,737
Question?

1750
01:22:48,737 --> 01:22:49,320
Oh, I'm sorry.

1751
01:22:49,320 --> 01:22:50,360
Lx.

1752
01:22:50,360 --> 01:22:51,320
Shoot, that was an x.

1753
01:22:51,320 --> 01:22:52,630
It just didn't look like it.

1754
01:22:52,630 --> 01:22:54,000
Lx minus i Ly.

1755
01:22:56,760 --> 01:22:59,796
So it commutes both
L plus and L minus.

1756
01:22:59,796 --> 01:23:02,350
But here's the
real beauty of it.

1757
01:23:02,350 --> 01:23:07,760
Lz with L plus is
equal to-- well,

1758
01:23:07,760 --> 01:23:10,110
let's just do
dimensional analysis.

1759
01:23:10,110 --> 01:23:12,070
L plus is Lx plus i Ly.

1760
01:23:12,070 --> 01:23:14,930
We know that Lz with Lx
is something like Ly.

1761
01:23:14,930 --> 01:23:17,930
And Lz with Ly is
something like Lx.

1762
01:23:17,930 --> 01:23:19,560
With factors of i's an h bars.

1763
01:23:22,340 --> 01:23:26,930
And when you work out the
commutator, which should only

1764
01:23:26,930 --> 01:23:32,040
take you second, you
get i h bar L plus.

1765
01:23:32,040 --> 01:23:37,020
And similarly, when we
construct Lz and L minus,

1766
01:23:37,020 --> 01:23:42,630
we get minus h bar L minus.

1767
01:23:42,630 --> 01:23:46,160
Are L plus an L minus Hermitian?

1768
01:23:46,160 --> 01:23:49,310
No, they're each
other's adjoints.

1769
01:23:49,310 --> 01:23:50,620
Lz is Hermitian.

1770
01:23:53,360 --> 01:23:56,111
And look at this
commutation relation.

1771
01:23:56,111 --> 01:23:57,110
What does that tell you?

1772
01:24:00,520 --> 01:24:02,840
From the first observation,
if we have an energy

1773
01:24:02,840 --> 01:24:05,070
or if we have an operator
e and an operator a

1774
01:24:05,070 --> 01:24:06,850
that commute in this
fashion, then this

1775
01:24:06,850 --> 01:24:09,010
tells you that the
eigenfunctions of this operator

1776
01:24:09,010 --> 01:24:11,660
are staggered in a the
ladder spaced by h bar.

1777
01:24:11,660 --> 01:24:16,680
The eigenvalues of Lz come
in a ladder spaced by h bar.

1778
01:24:16,680 --> 01:24:20,030
We can raise with
L plus, and we can

1779
01:24:20,030 --> 01:24:22,670
lower with L minus just like
in the harmonic oscillator

1780
01:24:22,670 --> 01:24:25,440
problem.

1781
01:24:25,440 --> 01:24:28,187
And we'll exploit the
rest of the-- we'll

1782
01:24:28,187 --> 01:24:30,520
deduce the rest of the structure
of the angular momentum

1783
01:24:30,520 --> 01:24:33,920
operator eigenfunctions next
time using this computation

1784
01:24:33,920 --> 01:24:34,826
relation.

1785
01:24:34,826 --> 01:24:37,580
See you next time.