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PROFESSOR: So it's a
beautiful recording.

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00:00:26,780 --> 00:00:30,180
OK, so to get
started, questions?

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00:00:33,090 --> 00:00:34,040
From last time?

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00:00:34,040 --> 00:00:35,530
Barton covered for me last time.

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00:00:35,530 --> 00:00:36,610
I fled.

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I was out of town.

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00:00:38,350 --> 00:00:40,654
I was at a math conference.

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It was pretty surreal.

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00:00:41,570 --> 00:00:42,830
Questions, yes.

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00:00:42,830 --> 00:00:44,590
AUDIENCE: [INAUDIBLE]
about the exam?

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00:00:44,590 --> 00:00:45,000
PROFESSOR: About--?

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00:00:45,000 --> 00:00:45,520
AUDIENCE: The exam.

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00:00:45,520 --> 00:00:46,020
PROFESSOR: The exam.

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00:00:46,020 --> 00:00:46,720
Yes, absolutely.

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00:00:46,720 --> 00:00:52,340
So the exam is, as you all
know, on Thursday, a week hence.

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00:00:52,340 --> 00:00:54,380
So on Tuesday we
will have a lecture.

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00:00:54,380 --> 00:00:58,340
The material Tuesday will
not be covered on the exam.

25
00:00:58,340 --> 00:01:00,580
The exam will be a
review of everything

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00:01:00,580 --> 00:01:03,010
through today's lecture,
including the problems that,

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00:01:03,010 --> 00:01:04,486
which for some
technical reason I

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00:01:04,486 --> 00:01:05,860
don't know why
didn't get posted.

29
00:01:05,860 --> 00:01:09,440
But it should be up
after lecture today.

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00:01:09,440 --> 00:01:13,120
The exam will be a
combination of short questions

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00:01:13,120 --> 00:01:15,012
and computations.

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00:01:15,012 --> 00:01:17,345
It will not focus on an
enormous number of computations.

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00:01:17,345 --> 00:01:19,030
It will focus more
on conceptual things.

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00:01:19,030 --> 00:01:22,440
But there will be a few
calculations on the exam.

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00:01:22,440 --> 00:01:25,230
And I will post some
practice problems

36
00:01:25,230 --> 00:01:27,092
over the next couple of days.

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00:01:27,092 --> 00:01:29,050
AUDIENCE: Do we have a
problem set [INAUDIBLE]?

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00:01:29,050 --> 00:01:31,230
PROFESSOR: You do have a
problem set due Tuesday.

39
00:01:31,230 --> 00:01:33,790
And that is part of your
preparation for the exam.

40
00:01:33,790 --> 00:01:36,560
Here's a basic strategy
for exams for this class.

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00:01:36,560 --> 00:01:39,170
Anything that's on a
problem set is fair game.

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00:01:39,170 --> 00:01:41,330
Anything that's not
covered on a problem set

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00:01:41,330 --> 00:01:42,694
is not going to be fair game.

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00:01:42,694 --> 00:01:44,360
If you haven't seen
a new problem on it,

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00:01:44,360 --> 00:01:47,800
broadly construed, then you
won't-- I won't test you

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00:01:47,800 --> 00:01:50,800
on a topic you haven't
done problems on before.

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00:01:50,800 --> 00:01:53,180
But I will take
problems and ideas

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00:01:53,180 --> 00:01:55,920
that you've studied before and
spin them slightly differently

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00:01:55,920 --> 00:01:58,360
to make you think through
them in real time on the exam.

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00:01:58,360 --> 00:01:58,860
OK?

51
00:02:01,210 --> 00:02:03,520
From my point of view,
the purpose of these exams

52
00:02:03,520 --> 00:02:04,682
is not to give you a grade.

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00:02:04,682 --> 00:02:05,890
I don't care about the grade.

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00:02:05,890 --> 00:02:08,830
The purpose of these
exams is to give you

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00:02:08,830 --> 00:02:11,402
feedback on your understanding.

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00:02:11,402 --> 00:02:13,860
It's very easy to slip through
quantum mechanics and think,

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00:02:13,860 --> 00:02:15,345
oh yeah, I totally-- I got this.

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00:02:15,345 --> 00:02:16,530
This is fine.

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00:02:16,530 --> 00:02:20,890
But it's not always
an accurate read.

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So that's the point.

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00:02:22,662 --> 00:02:25,010
Did that answer your question?

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00:02:25,010 --> 00:02:26,060
Other questions?

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00:02:26,060 --> 00:02:27,039
Exam or-- yeah.

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00:02:27,039 --> 00:02:29,080
AUDIENCE: About the harmonic
oscillator actually.

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PROFESSOR: Excellent.

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AUDIENCE: So when
we solved it Tuesday

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using the series method,
so there are two solutions

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00:02:34,470 --> 00:02:37,294
technically, the even solution
and the odd term solution.

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00:02:37,294 --> 00:02:39,210
So did boundary conditions
force the other one

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to be completely zero, like
the coefficient in front of it?

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So there's like an A0 term which
determines all the other ones.

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00:02:47,050 --> 00:02:50,104
But there's an A0 term and an
A1 term for the evens and odds.

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00:02:50,104 --> 00:02:51,770
So did the other ones
just have to be 0?

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00:02:51,770 --> 00:02:53,460
PROFESSOR: This is a
really good question.

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00:02:53,460 --> 00:02:54,670
This is an excellent question.

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Let me ask the question
slightly differently.

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00:02:56,544 --> 00:02:58,650
And tell me if this
is the same question.

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00:02:58,650 --> 00:03:00,130
When we wrote down our
differential equation--

79
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so last time we did the
harmonic oscillator.

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00:03:01,963 --> 00:03:04,840
And Barton did give you
the brute force strategy

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00:03:04,840 --> 00:03:06,150
for the harmonic oscillator.

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00:03:06,150 --> 00:03:07,775
We want to find the
energy eigenstates,

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00:03:07,775 --> 00:03:10,780
because that's what we do to
solve the Schrodinger equation.

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00:03:10,780 --> 00:03:13,000
And we turn that into a
differential equation.

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00:03:13,000 --> 00:03:14,625
And we solve this
differential equation

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00:03:14,625 --> 00:03:18,190
by doing an asymptotic analysis
and then a series expansion.

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Now, this is a second order
differential equation.

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00:03:21,590 --> 00:03:22,980
Everyone agree with that?

89
00:03:22,980 --> 00:03:24,730
It's a second order
differential equation.

90
00:03:24,730 --> 00:03:27,420
However, in our
series expansion we

91
00:03:27,420 --> 00:03:33,010
ended up with one integration
constant, not two.

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00:03:33,010 --> 00:03:34,494
How does that work?

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00:03:34,494 --> 00:03:36,910
How can it be that there was
only one integration constant

94
00:03:36,910 --> 00:03:37,410
and not two?

95
00:03:37,410 --> 00:03:39,407
It's a second order
differential equation.

96
00:03:39,407 --> 00:03:40,240
Is this he question?

97
00:03:40,240 --> 00:03:40,540
AUDIENCE: Yeah.

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00:03:40,540 --> 00:03:42,020
PROFESSOR: OK, and this
is an excellent question.

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00:03:42,020 --> 00:03:44,830
Because it must be true,
that there are two solutions.

100
00:03:44,830 --> 00:03:46,930
It cannot be that there
is just one solution.

101
00:03:46,930 --> 00:03:48,750
It's a second order
differential equation.

102
00:03:48,750 --> 00:03:51,945
Their existence in uniqueness
theorems, which tell us there

103
00:03:51,945 --> 00:03:54,027
are two integration constants.

104
00:03:54,027 --> 00:03:56,110
So how can it possibly be
that there was only one?

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00:03:56,110 --> 00:03:57,930
Well, we did something
rather subtle

106
00:03:57,930 --> 00:04:00,350
in that series expansion.

107
00:04:00,350 --> 00:04:02,950
For that series expansion
there was a critical moment,

108
00:04:02,950 --> 00:04:03,880
which I'm not going
to go through but you

109
00:04:03,880 --> 00:04:05,520
can come to my office
hours again, but just

110
00:04:05,520 --> 00:04:06,180
look through the notes.

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00:04:06,180 --> 00:04:08,055
There's an important
moment in the notes when

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00:04:08,055 --> 00:04:10,500
we say, aha, these terms matter.

113
00:04:10,500 --> 00:04:15,980
But what we did is we
suppressed a singular solution.

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00:04:15,980 --> 00:04:18,500
There's a solution of
that differential equation

115
00:04:18,500 --> 00:04:20,640
which is not well-behaved,
which is not smooth,

116
00:04:20,640 --> 00:04:23,070
and in particular
which diverges.

117
00:04:23,070 --> 00:04:25,400
And we already did, from
the asymptotic analysis,

118
00:04:25,400 --> 00:04:26,930
we already fixed that the
asymptotic behavior was

119
00:04:26,930 --> 00:04:27,850
exponentially falling.

120
00:04:27,850 --> 00:04:30,620
But there's a second solution
which is exponentially growing.

121
00:04:30,620 --> 00:04:33,427
So what we did, remember
how we did this story?

122
00:04:33,427 --> 00:04:35,010
We took our wave
function and we said,

123
00:04:35,010 --> 00:04:36,130
OK, look, we're
going to pull off--

124
00:04:36,130 --> 00:04:37,838
we're going to first
asymptotic analysis.

125
00:04:37,838 --> 00:04:40,182
And asymptotic analysis
tells us that either we

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00:04:40,182 --> 00:04:42,390
have exponentially growing
or exponentially shrinking

127
00:04:42,390 --> 00:04:43,070
solutions.

128
00:04:43,070 --> 00:04:45,410
Let's pick the exponentially
shrinking solutions.

129
00:04:45,410 --> 00:04:48,370
So phi e is equal
to e to the minus

130
00:04:48,370 --> 00:04:51,545
x over 2a squared
squared, times some--

131
00:04:51,545 --> 00:04:53,170
I don't remember what
Barton called it.

132
00:04:53,170 --> 00:04:55,300
I'll call it u of x.

133
00:04:55,300 --> 00:04:58,910
So we've extracted, because
we know that asymptotically it

134
00:04:58,910 --> 00:04:59,840
takes this form.

135
00:04:59,840 --> 00:05:01,060
Well, it could also
take the other form.

136
00:05:01,060 --> 00:05:02,250
It could be e to the
plus, which would

137
00:05:02,250 --> 00:05:03,417
be bad and not normalizable.

138
00:05:03,417 --> 00:05:05,249
We've extracted that,
and then we write down

139
00:05:05,249 --> 00:05:06,720
the differential equation for u.

140
00:05:06,720 --> 00:05:08,553
And then we solve that
differential equation

141
00:05:08,553 --> 00:05:10,570
by series analysis, yeah?

142
00:05:10,570 --> 00:05:17,151
However, if I have a secondary
differential equation for phi,

143
00:05:17,151 --> 00:05:19,150
this change of variables
doesn't change the fact

144
00:05:19,150 --> 00:05:20,840
that it's a secondary
differential equation for u,

145
00:05:20,840 --> 00:05:21,610
right?

146
00:05:21,610 --> 00:05:23,920
There's still two
solutions for u.

147
00:05:23,920 --> 00:05:25,984
One of those solutions
will be the solution

148
00:05:25,984 --> 00:05:27,900
of the equation that has
this asymptotic form.

149
00:05:27,900 --> 00:05:30,683
But the other solution
will be one that has an e

150
00:05:30,683 --> 00:05:33,320
to the plus x squared
over a squared

151
00:05:33,320 --> 00:05:35,070
so that it cancels off
this leading factor

152
00:05:35,070 --> 00:05:37,070
and gives me the exponentially
growing solution.

153
00:05:40,490 --> 00:05:42,650
Everyone cool with that?

154
00:05:42,650 --> 00:05:46,330
So in that series
analysis there's

155
00:05:46,330 --> 00:05:47,880
sort of a subtle
moment where you

156
00:05:47,880 --> 00:05:51,527
impose that you have
the convergent solution.

157
00:05:51,527 --> 00:05:53,860
So the answer of, why did we
get a first order relation,

158
00:05:53,860 --> 00:05:57,820
is that we very carefully,
although it may not

159
00:05:57,820 --> 00:06:00,740
have been totally obvious,
when doing this calculation

160
00:06:00,740 --> 00:06:03,360
one carefully chooses the
convergent solution that

161
00:06:03,360 --> 00:06:06,380
doesn't have this
function blowing up so

162
00:06:06,380 --> 00:06:08,957
as to overwhelm the envelope.

163
00:06:08,957 --> 00:06:10,040
That answer your question?

164
00:06:10,040 --> 00:06:10,530
AUDIENCE: Yep.

165
00:06:10,530 --> 00:06:11,260
PROFESSOR: Great.

166
00:06:11,260 --> 00:06:12,530
It's a very good question.

167
00:06:12,530 --> 00:06:13,500
This is an important
subtlety that

168
00:06:13,500 --> 00:06:16,190
comes up all over the place
when you do asymptotic analysis.

169
00:06:16,190 --> 00:06:17,210
I speak from my heart.

170
00:06:17,210 --> 00:06:18,420
It's an important
thing in the research

171
00:06:18,420 --> 00:06:19,795
that I'm doing
right now, getting

172
00:06:19,795 --> 00:06:22,490
these sorts of subtleties right.

173
00:06:22,490 --> 00:06:23,532
It can be very confusing.

174
00:06:23,532 --> 00:06:25,490
It's important to think
carefully through them.

175
00:06:25,490 --> 00:06:26,700
So it's a very good question.

176
00:06:26,700 --> 00:06:29,940
Other questions
before we move on?

177
00:06:29,940 --> 00:06:31,830
OK.

178
00:06:31,830 --> 00:06:34,386
So I'm going to erase
this, because it's not

179
00:06:34,386 --> 00:06:36,430
directly germane,
but it is great.

180
00:06:36,430 --> 00:06:40,690
OK, so one of the lessons
of this brute force analysis

181
00:06:40,690 --> 00:06:43,120
was that we constructed
the spectrum, i.e., the set

182
00:06:43,120 --> 00:06:48,190
of energy eigenvalues allowed
for the quantum harmonic

183
00:06:48,190 --> 00:06:51,100
oscillator, and we constructed
the wave functions.

184
00:06:51,100 --> 00:06:52,490
We constructed
the wave functions

185
00:06:52,490 --> 00:06:53,990
by solving the
differential equation

186
00:06:53,990 --> 00:06:57,180
through asymptotic analysis,
which give us the Gaussian

187
00:06:57,180 --> 00:06:59,050
envelope, and series
expansion, which

188
00:06:59,050 --> 00:07:01,490
give us the Hermite polynomials.

189
00:07:01,490 --> 00:07:04,059
And then there's some
normalization coefficient.

190
00:07:04,059 --> 00:07:06,100
And then we got the energy
eigenvalues by asking,

191
00:07:06,100 --> 00:07:09,290
when does this series
expansion converge?

192
00:07:09,290 --> 00:07:11,170
When does it, in fact
truncate, terminate,

193
00:07:11,170 --> 00:07:13,570
so that we can write
down an answer?

194
00:07:13,570 --> 00:07:15,700
And that was what gave
us these discrete values.

195
00:07:15,700 --> 00:07:17,950
But fine, we can see that
it would be discrete values.

196
00:07:17,950 --> 00:07:18,480
We're cool with that.

197
00:07:18,480 --> 00:07:21,104
In fact, Barton went through the
discussion of the node theorem

198
00:07:21,104 --> 00:07:26,570
and the lack of degeneracy
in one dimensional quantum

199
00:07:26,570 --> 00:07:27,590
mechanics.

200
00:07:27,590 --> 00:07:30,250
So it's reasonable that we
get a bunch of discrete energy

201
00:07:30,250 --> 00:07:32,800
eigenvalues, as we've talked
about now for two lectures.

202
00:07:32,800 --> 00:07:34,880
However, there's
a surprise here,

203
00:07:34,880 --> 00:07:36,680
which is that these
aren't just discrete,

204
00:07:36,680 --> 00:07:38,020
they're evenly spaced.

205
00:07:38,020 --> 00:07:41,040
We get a tower, starting with
the lowest possible energy

206
00:07:41,040 --> 00:07:44,170
corresponding to a--
sorry, E0-- starting

207
00:07:44,170 --> 00:07:48,190
with the lowest possible
energy, which is greater than 0,

208
00:07:48,190 --> 00:07:52,900
and a corresponding ground
state wave function.

209
00:07:52,900 --> 00:07:55,816
And then we have a whole bunch
of other states, phi 1, phi 2,

210
00:07:55,816 --> 00:08:00,009
phi 3, phi 4, labeled
by their energies

211
00:08:00,009 --> 00:08:01,550
where the energies
are evenly spaced.

212
00:08:04,890 --> 00:08:09,410
They needed to be discrete,
because these are bound states.

213
00:08:09,410 --> 00:08:11,700
But evenly spaced is a surprise.

214
00:08:11,700 --> 00:08:14,430
So why are they evenly spaced?

215
00:08:14,430 --> 00:08:18,904
Anyone, based on the
last lecture's analysis?

216
00:08:18,904 --> 00:08:20,320
Yeah, you don't
have a good answer

217
00:08:20,320 --> 00:08:21,540
to that from last
lecture's analysis.

218
00:08:21,540 --> 00:08:22,870
It's one of the
mysteries that comes out

219
00:08:22,870 --> 00:08:23,780
of the first analysis.

220
00:08:23,780 --> 00:08:25,363
When you take a
differential equation,

221
00:08:25,363 --> 00:08:28,440
you just beat the crap out of
it with a stick by solving it.

222
00:08:28,440 --> 00:08:31,500
With differential equations
strategies like this

223
00:08:31,500 --> 00:08:34,772
you don't necessarily get some
of the more subtle structure.

224
00:08:34,772 --> 00:08:36,230
One of the goals
of today's lecture

225
00:08:36,230 --> 00:08:39,909
is going to be to explain
why we get this structure.

226
00:08:39,909 --> 00:08:41,706
Why just from the
physics of the problem,

227
00:08:41,706 --> 00:08:43,122
the underlying
physics, should you

228
00:08:43,122 --> 00:08:46,300
know that the system is going to
have evenly spaced eigenvalues?

229
00:08:46,300 --> 00:08:47,600
What's the structure?

230
00:08:47,600 --> 00:08:50,121
And secondly, I want to
show you a way of repeating

231
00:08:50,121 --> 00:08:52,620
this calculation without doing
the brute force analysis that

232
00:08:52,620 --> 00:08:56,200
reveals some of that more fine
grain structure of the problem.

233
00:08:56,200 --> 00:08:57,820
And this is going
to turn out to be

234
00:08:57,820 --> 00:09:00,200
one of the canonical
moves in the analysis

235
00:09:00,200 --> 00:09:01,799
of quantum mechanical systems.

236
00:09:01,799 --> 00:09:03,840
So from quantum mechanics
to quantum field theory

237
00:09:03,840 --> 00:09:07,150
this is a basic
series of logic moves.

238
00:09:07,150 --> 00:09:10,120
What I'm going to do today
also has an independent life

239
00:09:10,120 --> 00:09:13,765
in mathematics, in algebra.

240
00:09:13,765 --> 00:09:15,140
And that will be
something you'll

241
00:09:15,140 --> 00:09:17,390
studying in more detail in
8.05, but I would encourage

242
00:09:17,390 --> 00:09:19,600
you to ask your recitation
instructors about it,

243
00:09:19,600 --> 00:09:21,386
or me in office hours.

244
00:09:25,520 --> 00:09:30,440
So our goal is to understand
that even spacing and also

245
00:09:30,440 --> 00:09:32,300
to re-derive these
results without the sort

246
00:09:32,300 --> 00:09:37,692
of brutal direct assault
methods we used last time.

247
00:09:37,692 --> 00:09:39,400
So what I'm going to
tell you about today

248
00:09:39,400 --> 00:09:40,950
is something called
the operator method.

249
00:09:40,950 --> 00:09:43,200
It usually goes under the
name of the operator method.

250
00:09:57,340 --> 00:10:01,780
To get us started let's go back
to look at the energy operator

251
00:10:01,780 --> 00:10:04,310
for the harmonic
oscillator, which

252
00:10:04,310 --> 00:10:08,310
is what, at the end of
the day, we want to solve.

253
00:10:08,310 --> 00:10:15,230
p squared over 2m plus m omega
squared upon 2, x squared.

254
00:10:15,230 --> 00:10:18,615
And this is the operator that we
want to-- whose eigenvalues we

255
00:10:18,615 --> 00:10:20,240
want to construct,
whose eigenfunctions

256
00:10:20,240 --> 00:10:21,600
we want to construct.

257
00:10:21,600 --> 00:10:25,720
Before we do anything else, we
should do dimensional analysis.

258
00:10:25,720 --> 00:10:27,470
First thing you when
you look at a problem

259
00:10:27,470 --> 00:10:28,600
is do some dimensional analysis.

260
00:10:28,600 --> 00:10:30,433
Identify the salient
scales and make things,

261
00:10:30,433 --> 00:10:32,210
to the degree possible,
dimensionless.

262
00:10:32,210 --> 00:10:33,470
Your life will be better.

263
00:10:33,470 --> 00:10:35,780
So what are the parameters
we have available to us?

264
00:10:35,780 --> 00:10:38,529
We have h bar, because
it's quantum mechanics.

265
00:10:38,529 --> 00:10:40,570
We have m, because we have
a particle of mass, m.

266
00:10:40,570 --> 00:10:42,850
We have omega,
because this potential

267
00:10:42,850 --> 00:10:44,740
has a characteristic
frequency of omega.

268
00:10:44,740 --> 00:10:46,940
What other parameters do
we have available to us?

269
00:10:46,940 --> 00:10:47,930
Well, we have c.

270
00:10:47,930 --> 00:10:49,230
That's available to us.

271
00:10:49,230 --> 00:10:51,360
But is it relevant?

272
00:10:51,360 --> 00:10:52,087
No.

273
00:10:52,087 --> 00:10:54,420
If you get an answer that
depends on the speed of light,

274
00:10:54,420 --> 00:10:56,510
you made some horrible mistake.

275
00:10:56,510 --> 00:10:57,260
So not there.

276
00:10:57,260 --> 00:11:00,230
What about the number
of students in 8.04?

277
00:11:00,230 --> 00:11:00,730
No.

278
00:11:00,730 --> 00:11:01,990
There are an infinite
number of parameters

279
00:11:01,990 --> 00:11:03,430
that don't matter
to this problem.

280
00:11:03,430 --> 00:11:05,270
What you want to know is, when
you do dimensional analysis,

281
00:11:05,270 --> 00:11:07,110
what parameters matter
for the problem.

282
00:11:07,110 --> 00:11:09,600
What parameters could possibly
appear during the answer?

283
00:11:09,600 --> 00:11:10,510
And that's it.

284
00:11:10,510 --> 00:11:14,120
There are no other
parameters in this problem.

285
00:11:14,120 --> 00:11:16,730
So that's a full set of
parameters available to us.

286
00:11:16,730 --> 00:11:19,470
This has dimensions of
momentum times length.

287
00:11:19,470 --> 00:11:22,850
This has dimensions of mass,
and this has dimensions of one

288
00:11:22,850 --> 00:11:23,840
upon the time.

289
00:11:23,840 --> 00:11:25,890
And so what
characteristic scales can

290
00:11:25,890 --> 00:11:29,050
we build using these
three parameters?

291
00:11:29,050 --> 00:11:33,310
Well, this is a
moment times a length.

292
00:11:33,310 --> 00:11:38,340
If we multiply by a mass,
that's momentum times mass

293
00:11:38,340 --> 00:11:41,020
times x, which is
almost momentum again.

294
00:11:41,020 --> 00:11:42,700
We need a velocity
and not a position,

295
00:11:42,700 --> 00:11:44,020
but we have 1 over time.

296
00:11:44,020 --> 00:11:48,570
So if we take h bar
times and omega,

297
00:11:48,570 --> 00:11:51,324
so that's px times
m over t, that

298
00:11:51,324 --> 00:11:52,573
has units of momentum squared.

299
00:11:59,330 --> 00:12:00,990
And similarly, this
is momentum which

300
00:12:00,990 --> 00:12:03,960
is x, which is length
mass over time.

301
00:12:03,960 --> 00:12:05,870
I can divide by mass
and divide by frequency

302
00:12:05,870 --> 00:12:11,200
or multiply by time,
so h bar upon m omega.

303
00:12:11,200 --> 00:12:15,411
And this is going to have
units of length squared.

304
00:12:15,411 --> 00:12:17,660
And with a little bit of
foresight from factors of two

305
00:12:17,660 --> 00:12:20,860
I'm going to use these to
define two link scales.

306
00:12:20,860 --> 00:12:26,390
x0 is equal to h
bar-- I want to be

307
00:12:26,390 --> 00:12:28,676
careful to get my coefficients.

308
00:12:28,676 --> 00:12:30,390
I always put the two
in the wrong place.

309
00:12:30,390 --> 00:12:33,100
So 2 h bar upon m omega.

310
00:12:33,100 --> 00:12:34,890
Square root.

311
00:12:34,890 --> 00:12:39,660
And I'm going to define p0 as
equal to square root of 2 h bar

312
00:12:39,660 --> 00:12:40,440
times m omega.

313
00:12:45,900 --> 00:12:46,860
So here's my claim.

314
00:12:46,860 --> 00:12:49,650
My claim is at the end of
the day the salient link

315
00:12:49,650 --> 00:12:51,250
scales for this
problem should be

316
00:12:51,250 --> 00:12:54,780
integers or dimensionless
numbers times this link scale.

317
00:12:54,780 --> 00:12:58,480
And salient momentum scales
should be this scale.

318
00:12:58,480 --> 00:12:59,930
Just from dimensional analysis.

319
00:12:59,930 --> 00:13:01,420
So if someone at
this point says,

320
00:13:01,420 --> 00:13:03,840
what do you think is
the typical scale, what

321
00:13:03,840 --> 00:13:06,670
is the typical size of
the ground state wave

322
00:13:06,670 --> 00:13:10,530
function, the typical link scale
over which the wave function is

323
00:13:10,530 --> 00:13:12,670
not 0?

324
00:13:12,670 --> 00:13:15,220
Well, that can't possibly
be the size of Manhattan.

325
00:13:15,220 --> 00:13:16,609
It's not the size of a proton.

326
00:13:16,609 --> 00:13:18,900
There's only one link scale
associated with the system.

327
00:13:18,900 --> 00:13:20,730
It should be of order x0.

328
00:13:20,730 --> 00:13:22,510
Always start with
dimensional analysis.

329
00:13:22,510 --> 00:13:23,180
Always.

330
00:13:23,180 --> 00:13:26,432
OK, so with that we can
rewrite this energy.

331
00:13:26,432 --> 00:13:29,050
Sorry, and there's
one last energy.

332
00:13:29,050 --> 00:13:32,780
We can write an energy,
the thing with energy,

333
00:13:32,780 --> 00:13:35,880
which is equal to h bar omega.

334
00:13:35,880 --> 00:13:43,390
And this times a frequency
gives us an energy.

335
00:13:43,390 --> 00:13:47,360
So we can rewrite
this energy operator

336
00:13:47,360 --> 00:13:52,784
as h bar omega times
p squared over p0.

337
00:13:52,784 --> 00:13:53,950
So this has units of energy.

338
00:13:53,950 --> 00:13:55,820
So everything here
must be dimensionless.

339
00:13:55,820 --> 00:13:58,340
And it turns out to be p
squared over p0 squared

340
00:13:58,340 --> 00:14:02,760
plus x operator squared
over x0 squared.

341
00:14:02,760 --> 00:14:05,515
So that's convenient.

342
00:14:05,515 --> 00:14:07,640
So this has nothing to do
with the operator method.

343
00:14:07,640 --> 00:14:08,889
This is just being reasonable.

344
00:14:15,190 --> 00:14:25,380
Quick thing to note, x0 times p0
is just-- the m omegas cancel,

345
00:14:25,380 --> 00:14:28,070
so we get root 2h
bar squared, 2 h bar.

346
00:14:31,410 --> 00:14:33,160
Little tricks like
that are useful to keep

347
00:14:33,160 --> 00:14:34,180
track of as you go.

348
00:14:36,760 --> 00:14:40,930
So we're interested in
this energy operator.

349
00:14:40,930 --> 00:14:42,030
And it has a nice form.

350
00:14:42,030 --> 00:14:43,140
It's a sum of squares.

351
00:14:43,140 --> 00:14:46,320
And we see the sum of squares,
a very tempting thing to do

352
00:14:46,320 --> 00:14:47,820
is to factor it.

353
00:14:47,820 --> 00:14:51,190
So for example, if I have
two classical numbers,

354
00:14:51,190 --> 00:14:55,030
c squared plus d squared,
the mathematician in me

355
00:14:55,030 --> 00:14:59,940
screams out to write c
minus id times c plus id.

356
00:15:02,640 --> 00:15:03,630
I have factored this.

357
00:15:03,630 --> 00:15:05,500
And that's usually a step
in the right direction.

358
00:15:05,500 --> 00:15:06,208
And is this true?

359
00:15:06,208 --> 00:15:08,910
Well yes, it's true. c
squared plus d squared

360
00:15:08,910 --> 00:15:10,312
and the cross terms cancel.

361
00:15:10,312 --> 00:15:11,020
OK, that's great.

362
00:15:11,020 --> 00:15:12,840
Four complex numbers,
or four C numbers.

363
00:15:15,819 --> 00:15:17,110
Now is this true for operators?

364
00:15:17,110 --> 00:15:17,920
Can I do this for operators?

365
00:15:17,920 --> 00:15:20,340
Here we have the energy
operator as a sum of squares.

366
00:15:22,910 --> 00:15:26,150
Well, let's try it.

367
00:15:26,150 --> 00:15:28,290
I'd like to write that
in terms of x and p.

368
00:15:28,290 --> 00:15:38,070
So what about writing the
quantity x minus ip over x0

369
00:15:38,070 --> 00:15:46,252
over p0, operator, times
x over x0 plus ip over p0.

370
00:15:49,647 --> 00:15:50,480
We can compute this.

371
00:15:50,480 --> 00:15:51,270
This is easy.

372
00:15:51,270 --> 00:15:56,700
So the first term gives us
the x squared over x0 squared.

373
00:15:56,700 --> 00:15:58,650
That last term gives
us-- the i's cancel,

374
00:15:58,650 --> 00:16:03,579
so we get p squared
over p0, squared.

375
00:16:03,579 --> 00:16:04,870
But then there are cross terms.

376
00:16:04,870 --> 00:16:09,950
We have an xp and a minus
px, with an overall i.

377
00:16:09,950 --> 00:16:17,320
So plus i times xp
over x0 times p0.

378
00:16:17,320 --> 00:16:19,350
x0 times p0,
however, is 2 h bar.

379
00:16:19,350 --> 00:16:20,570
So that's over 2 h bar.

380
00:16:23,300 --> 00:16:25,150
And then we have the
other term, minus px.

381
00:16:25,150 --> 00:16:25,650
Same thing.

382
00:16:25,650 --> 00:16:30,676
So I could write that as
a commutator, xp minus px.

383
00:16:30,676 --> 00:16:31,675
Everyone cool with what?

384
00:16:35,152 --> 00:16:36,860
Unfortunately this is
not what we wanted.

385
00:16:36,860 --> 00:16:38,740
We wanted just p
squared plus x squared.

386
00:16:38,740 --> 00:16:40,420
And what we got
instead was p squared

387
00:16:40,420 --> 00:16:42,200
plus x squared close
plus a commutator.

388
00:16:42,200 --> 00:16:43,616
Happily this
commutator is simple.

389
00:16:43,616 --> 00:16:45,054
What's the commutator
of x with p?

390
00:16:45,054 --> 00:16:45,970
AUDIENCE: [INAUDIBLE].

391
00:16:45,970 --> 00:16:47,011
PROFESSOR: Yeah, exactly.

392
00:16:47,011 --> 00:16:49,850
Commit this to memory.

393
00:16:49,850 --> 00:16:53,070
This is your friend.

394
00:16:53,070 --> 00:16:54,900
So this is just i
h bar, so this is

395
00:16:54,900 --> 00:16:58,780
equal to ditto plus
ditto plus i h bar.

396
00:17:01,390 --> 00:17:02,690
Somewhere I got a minus sign.

397
00:17:02,690 --> 00:17:06,255
Where did I get my minus
sign wrong? x with ip.

398
00:17:06,255 --> 00:17:06,900
Oh now, good.

399
00:17:06,900 --> 00:17:07,560
This is good.

400
00:17:07,560 --> 00:17:10,901
So x with p is i--
so we get an i h bar.

401
00:17:10,901 --> 00:17:12,359
No, I really did
screw up the sign.

402
00:17:12,359 --> 00:17:13,530
How did I screw up the sign?

403
00:17:16,160 --> 00:17:17,770
No I didn't.

404
00:17:17,770 --> 00:17:18,440
Wait.

405
00:17:18,440 --> 00:17:18,720
Oh!

406
00:17:18,720 --> 00:17:19,220
Of course.

407
00:17:19,220 --> 00:17:20,099
No, good.

408
00:17:20,099 --> 00:17:21,630
Sorry, sorry.

409
00:17:21,630 --> 00:17:24,030
Trust your calculation,
not your memory.

410
00:17:24,030 --> 00:17:25,407
So the calculation gave us this.

411
00:17:25,407 --> 00:17:26,490
So what does this give us?

412
00:17:26,490 --> 00:17:28,740
It gives us i h bar.

413
00:17:28,740 --> 00:17:29,360
So plus.

414
00:17:29,360 --> 00:17:31,675
But the i h bar times i is
going to give me a minus.

415
00:17:31,675 --> 00:17:33,050
And the h bar is
going to cancel,

416
00:17:33,050 --> 00:17:35,050
because I've got an h bar
from here and an h bar

417
00:17:35,050 --> 00:17:37,460
at the denominator minus 1/2.

418
00:17:37,460 --> 00:17:43,222
So this quantity is equal to the
quantity we wanted minus 1/2.

419
00:17:43,222 --> 00:17:44,680
And what is the
quantity we wanted,

420
00:17:44,680 --> 00:17:48,260
x0 squared plus p0 squared?

421
00:17:48,260 --> 00:17:50,902
This guy.

422
00:17:50,902 --> 00:17:52,360
So putting that
all together we can

423
00:17:52,360 --> 00:17:55,680
write that the energy
operator, which

424
00:17:55,680 --> 00:18:05,841
was equal to h bar omega
times the quantity we wanted,

425
00:18:05,841 --> 00:18:07,590
is equal to-- well,
the quantity we wanted

426
00:18:07,590 --> 00:18:10,290
is this quantity plus 1/2.

427
00:18:10,290 --> 00:18:14,290
h bar omega times--
I'll write this

428
00:18:14,290 --> 00:18:31,450
as x over x0 plus ip over p0,
x over x0 plus ip over p0, hat,

429
00:18:31,450 --> 00:18:33,700
hat, hat, plus 1/2.

430
00:18:37,860 --> 00:18:39,370
Everyone cool with that?

431
00:18:39,370 --> 00:18:40,230
So it almost worked.

432
00:18:40,230 --> 00:18:41,150
We can almost factor.

433
00:18:44,709 --> 00:18:46,500
So at this point it's
tempting to say, well

434
00:18:46,500 --> 00:18:48,800
that isn't really
much an improvement.

435
00:18:48,800 --> 00:18:50,590
You've just made it uglier.

436
00:18:50,590 --> 00:18:54,117
But consider the following.

437
00:18:54,117 --> 00:18:56,700
And just trust me on this one,
that this is not a stupid thing

438
00:18:56,700 --> 00:18:58,694
to do.

439
00:18:58,694 --> 00:19:00,360
That's a stupid symbol
to write, though.

440
00:19:00,360 --> 00:19:04,780
So let's define
an operator called

441
00:19:04,780 --> 00:19:13,740
a, which is equal to x
over x0 plus ip over p0,

442
00:19:13,740 --> 00:19:17,062
and an operator, which
I will call a dagger.

443
00:19:17,062 --> 00:19:19,550
"Is that a dagger
I see before me?"

444
00:19:19,550 --> 00:19:23,950
Sorry. x over x0
minus ip over p0.

445
00:19:27,420 --> 00:19:29,360
Hamlet quotes are harder.

446
00:19:29,360 --> 00:19:31,500
So this is a dagger.

447
00:19:31,500 --> 00:19:33,960
And we can now write
the energy operator

448
00:19:33,960 --> 00:19:37,930
for the harmonic oscillator
is equal to h bar omega times

449
00:19:37,930 --> 00:19:41,370
a dagger a plus 1/2.

450
00:19:45,740 --> 00:19:46,740
Everyone cool with that?

451
00:19:49,175 --> 00:19:50,550
Now, this should
look suggestive.

452
00:19:50,550 --> 00:19:52,966
You should say, aha, this looks
like h bar omega something

453
00:19:52,966 --> 00:19:53,620
plus 1/2.

454
00:19:53,620 --> 00:19:57,080
That sure looks familiar from
our brute force calculation.

455
00:19:57,080 --> 00:20:02,325
But, OK, that familiarity is
not an answer to the question.

456
00:20:02,325 --> 00:20:04,200
Meanwhile you should
say something like this.

457
00:20:04,200 --> 00:20:06,330
Look, this looks kind of
like the complex conjugate

458
00:20:06,330 --> 00:20:08,140
of this guy.

459
00:20:08,140 --> 00:20:10,859
Because there's an i and you
change the sign of the i.

460
00:20:10,859 --> 00:20:12,900
But what is the complex
conjugate of an operator?

461
00:20:12,900 --> 00:20:13,733
What does that mean?

462
00:20:15,885 --> 00:20:17,760
An operator is like take
a vector and rotate.

463
00:20:17,760 --> 00:20:19,920
What is the complex
conjugation of that?

464
00:20:19,920 --> 00:20:20,477
I don't know.

465
00:20:20,477 --> 00:20:21,560
So we have to define that.

466
00:20:24,740 --> 00:20:27,430
So I'm now going to start
with this quick math aside.

467
00:20:30,280 --> 00:20:32,249
And morally, this
is about what is

468
00:20:32,249 --> 00:20:33,790
the complex conjugate
of an operator.

469
00:20:33,790 --> 00:20:35,123
But before I move on, questions?

470
00:20:40,390 --> 00:20:42,630
OK.

471
00:20:42,630 --> 00:20:45,830
So here's a mathematical
series of a facts and claims.

472
00:20:48,630 --> 00:20:49,680
I claim the following.

473
00:20:49,680 --> 00:21:04,327
Given any linear operator we can
build-- there's a natural way

474
00:21:04,327 --> 00:21:06,410
to build without making
any additional assumptions

475
00:21:06,410 --> 00:21:07,660
or any additional ingredients.

476
00:21:07,660 --> 00:21:14,960
We can build another
operator, o dagger, hat, hat,

477
00:21:14,960 --> 00:21:16,860
in the following way.

478
00:21:16,860 --> 00:21:20,600
Consider the inner
product of f with g,

479
00:21:20,600 --> 00:21:22,280
or the bracket of f with g.

480
00:21:22,280 --> 00:21:27,130
So integral dx of f
complex conjugate g.

481
00:21:27,130 --> 00:21:29,830
Consider the function
we're taking here

482
00:21:29,830 --> 00:21:31,910
is actually the
operator we have on g.

483
00:21:35,102 --> 00:21:37,560
I'm going to define my-- so
this is a perfectly good thing.

484
00:21:37,560 --> 00:21:39,790
What this expression says
is, take your function g.

485
00:21:39,790 --> 00:21:41,744
Act on it with the operator o.

486
00:21:41,744 --> 00:21:43,160
Multiply by the
complex conjugate.

487
00:21:43,160 --> 00:21:43,860
Take the integral.

488
00:21:43,860 --> 00:21:45,401
This is what we
would have done if we

489
00:21:45,401 --> 00:21:50,430
had taken the inner product
of f with the function we get

490
00:21:50,430 --> 00:21:52,320
by taking o and acting
on the function g.

491
00:21:56,347 --> 00:21:57,180
So here's the thing.

492
00:21:57,180 --> 00:22:01,379
What we want-- just an
aside-- what we want to do

493
00:22:01,379 --> 00:22:02,420
is define a new operator.

494
00:22:02,420 --> 00:22:03,410
And here's how I'm
going to define it.

495
00:22:03,410 --> 00:22:04,932
We can define it by
choosing how it acts.

496
00:22:04,932 --> 00:22:06,790
I'm going to tell you
exactly how it acts,

497
00:22:06,790 --> 00:22:08,330
and then we'll
define the operator.

498
00:22:08,330 --> 00:22:12,837
So this operator, o with a
dagger, called the adjoint,

499
00:22:12,837 --> 00:22:14,170
is defined in the following way.

500
00:22:14,170 --> 00:22:17,030
This is whatever
operator you need, such

501
00:22:17,030 --> 00:22:21,035
that the integral
gives you-- such

502
00:22:21,035 --> 00:22:22,160
that the following is true.

503
00:22:22,160 --> 00:22:29,050
Integral dx o on f,
complex conjugate, g.

504
00:22:29,050 --> 00:22:31,960
So this is the
definition of this dagger

505
00:22:31,960 --> 00:22:35,530
action, the adjoint action.

506
00:22:35,530 --> 00:22:38,870
OK so o dagger is the adjoint.

507
00:22:41,670 --> 00:22:44,012
And sometimes it's called
the Hermetian adjoint.

508
00:22:44,012 --> 00:22:46,220
I'll occasionally say
Hermetian and occasionally not,

509
00:22:46,220 --> 00:22:49,580
with no particular order to it.

510
00:22:49,580 --> 00:22:51,970
So what does this mean?

511
00:22:51,970 --> 00:22:54,480
This means that
whatever o dagger is,

512
00:22:54,480 --> 00:22:57,760
it's that operator
that when acting on g

513
00:22:57,760 --> 00:22:59,440
and then taking the
inner product with f

514
00:22:59,440 --> 00:23:01,565
gives me same answer as
taking my original operator

515
00:23:01,565 --> 00:23:05,805
and acting on f and taking
the inner product with g.

516
00:23:05,805 --> 00:23:06,305
Cool?

517
00:23:09,210 --> 00:23:12,655
So we know how-- if we know
what our operator o is,

518
00:23:12,655 --> 00:23:15,280
the challenge now is going to be
to figure out what must this o

519
00:23:15,280 --> 00:23:18,220
dagger operator be such that
this expression is true.

520
00:23:18,220 --> 00:23:21,850
That's going to be my
definition of the adjoint.

521
00:23:21,850 --> 00:23:22,625
Cool?

522
00:23:22,625 --> 00:23:24,250
So I'm going to do
a bunch of examples.

523
00:23:24,250 --> 00:23:25,541
I'm going to walk through this.

524
00:23:30,100 --> 00:23:31,620
So the mathematical
definition is

525
00:23:31,620 --> 00:23:34,110
that an operator o
defined in this fashion

526
00:23:34,110 --> 00:23:39,920
is the Hermetian adjoint of o.

527
00:23:44,050 --> 00:23:47,330
So that's the
mathematical definition.

528
00:23:47,330 --> 00:23:51,784
Well, that's our version of
the mathematical definition.

529
00:23:51,784 --> 00:23:53,450
I just came back from
a math conference,

530
00:23:53,450 --> 00:23:55,866
so I'm particularly chastened
at the moment to be careful.

531
00:23:58,580 --> 00:24:00,660
So let's do some quick examples.

532
00:24:00,660 --> 00:24:02,780
Example one.

533
00:24:02,780 --> 00:24:05,010
Suppose c is a complex number.

534
00:24:09,554 --> 00:24:11,095
I claim a number is
also an operator.

535
00:24:11,095 --> 00:24:12,592
It acts by multiplication.

536
00:24:12,592 --> 00:24:14,800
The number 7 is an operator
because it takes a vector

537
00:24:14,800 --> 00:24:17,810
and it gives you 7
times that vector.

538
00:24:17,810 --> 00:24:21,980
So this number is a particularly
simple kind of operator.

539
00:24:21,980 --> 00:24:22,960
And what's the adjoint?

540
00:24:26,816 --> 00:24:27,440
We can do that.

541
00:24:27,440 --> 00:24:28,026
That's easy.

542
00:24:28,026 --> 00:24:30,400
So c adjoint is going to be
defined in the following way.

543
00:24:30,400 --> 00:24:35,740
It's integral dx f
star of c adjoint g

544
00:24:35,740 --> 00:24:43,430
is equal to the integral dx of c
on f complex conjugate times g.

545
00:24:43,430 --> 00:24:46,060
But what is this?

546
00:24:46,060 --> 00:24:47,399
Well, c is just a number.

547
00:24:47,399 --> 00:24:48,940
So when we take its
complex conjugate

548
00:24:48,940 --> 00:24:50,190
we can just pull it out.

549
00:24:50,190 --> 00:24:54,810
So this is equal to the
integral dx c complex conjugate,

550
00:24:54,810 --> 00:24:58,174
f star g.

551
00:24:58,174 --> 00:24:59,590
But I'm now going
to rewrite this,

552
00:24:59,590 --> 00:25:02,410
using the awesome power of
reordering multiplication,

553
00:25:02,410 --> 00:25:04,187
as c star.

554
00:25:04,187 --> 00:25:06,020
And I'm going to put
parentheses around this

555
00:25:06,020 --> 00:25:07,820
because it seems like fun.

556
00:25:07,820 --> 00:25:09,320
So now we have this
nice expression.

557
00:25:09,320 --> 00:25:12,540
The integral dx of
f c adjoint g is

558
00:25:12,540 --> 00:25:18,880
equal to the integral dx of f c
star g, c complex conjugate g.

559
00:25:18,880 --> 00:25:21,350
But notice that this
must be true for all f.

560
00:25:24,691 --> 00:25:25,440
It's true for all.

561
00:25:25,440 --> 00:25:27,064
Because I made no
assumption about what

562
00:25:27,064 --> 00:25:31,340
f and g are, true
for all f and g.

563
00:25:31,340 --> 00:25:34,269
And therefore the adjoint
of a complex number

564
00:25:34,269 --> 00:25:35,310
is its complex conjugate.

565
00:25:38,500 --> 00:25:41,470
And this is the basic
strategy for determining

566
00:25:41,470 --> 00:25:43,660
the adjoint of any operator.

567
00:25:43,660 --> 00:25:45,790
We're going to play
exactly this sort of game.

568
00:25:45,790 --> 00:25:47,460
We'll put the adjoint in here.

569
00:25:47,460 --> 00:25:49,220
We'll use the definition
of the adjoint.

570
00:25:49,220 --> 00:25:50,870
And then we'll do
whatever machinations

571
00:25:50,870 --> 00:25:54,280
are necessary to rewrite
this as some operator acting

572
00:25:54,280 --> 00:25:55,810
on the first factor.

573
00:25:55,810 --> 00:25:56,310
Cool?

574
00:25:59,050 --> 00:26:00,820
Questions?

575
00:26:00,820 --> 00:26:03,790
OK, let's do a more
interesting operator.

576
00:26:03,790 --> 00:26:05,330
By the way, to
check at home, and I

577
00:26:05,330 --> 00:26:07,830
think this might be
on your problem set--

578
00:26:07,830 --> 00:26:09,484
but I don't remember
if it's on or not.

579
00:26:09,484 --> 00:26:11,150
So if it's not, check
this for yourself.

580
00:26:11,150 --> 00:26:13,827
Check that the
adjoin of the adjoint

581
00:26:13,827 --> 00:26:15,160
is equal to the operator itself.

582
00:26:18,640 --> 00:26:20,380
It's an easy thing to check.

583
00:26:20,380 --> 00:26:21,650
So next example.

584
00:26:27,550 --> 00:26:32,550
What is the adjoint of
the operator derivative

585
00:26:32,550 --> 00:26:33,540
with respect to x?

586
00:26:35,421 --> 00:26:37,920
Consider the operator, which
is just derivative with respect

587
00:26:37,920 --> 00:26:38,750
to x.

588
00:26:38,750 --> 00:26:43,280
And I want to know what is
the adjoint of this beast.

589
00:26:43,280 --> 00:26:45,700
So how do we do this?

590
00:26:45,700 --> 00:26:46,820
Same logic as before.

591
00:26:46,820 --> 00:26:49,520
Whatever the operator is, it's
defined in the following way.

592
00:26:49,520 --> 00:26:59,020
Integral dx, f complex
conjugate on dx dagger on g.

593
00:26:59,020 --> 00:27:02,920
This is equal to the integral
dx of-- how we doing on time?

594
00:27:02,920 --> 00:27:12,961
Good-- integral dx of dx,
f complex conjugate on g.

595
00:27:12,961 --> 00:27:15,460
Now, what we want is we want
to turn this into an expression

596
00:27:15,460 --> 00:27:17,085
where the operator
is acting on g, just

597
00:27:17,085 --> 00:27:19,680
as our familiar operator ddx.

598
00:27:19,680 --> 00:27:23,370
So how do I get
the ddx over here?

599
00:27:26,160 --> 00:27:27,170
I need to do two things.

600
00:27:27,170 --> 00:27:28,380
First, what's the
complex conjugate

601
00:27:28,380 --> 00:27:30,796
of the derivative with respect
to x of a complex function?

602
00:27:35,210 --> 00:27:36,940
MIT has indigestion.

603
00:27:36,940 --> 00:27:39,460
So this is integral
dx, derivative

604
00:27:39,460 --> 00:27:42,200
with respect to x of f
complex conjugate, g.

605
00:27:44,870 --> 00:27:46,790
And now I want this operator.

606
00:27:46,790 --> 00:27:48,229
I want derivative acting on g.

607
00:27:48,229 --> 00:27:49,145
That's the definition.

608
00:27:57,630 --> 00:28:00,650
Because I want to know
what is this operator.

609
00:28:00,650 --> 00:28:04,300
And so I'm going to do
integration by parts.

610
00:28:04,300 --> 00:28:07,920
So this is equal to
the integral, dx.

611
00:28:07,920 --> 00:28:11,460
When I integrate by
parts I get an F complex

612
00:28:11,460 --> 00:28:16,610
conjugate, and then an overall
minus sign from the integration

613
00:28:16,610 --> 00:28:19,684
by parts minus f
complex conjugate dx g.

614
00:28:22,851 --> 00:28:24,350
Was I telling you
the truth earlier?

615
00:28:24,350 --> 00:28:25,183
Or did I lie to you?

616
00:28:30,870 --> 00:28:34,430
OK, keep thinking about that.

617
00:28:34,430 --> 00:28:37,670
And this is equal
to, well the integral

618
00:28:37,670 --> 00:28:42,420
of dx, f complex conjugate
if minus the derivative

619
00:28:42,420 --> 00:28:45,450
with respect to x acting on g.

620
00:28:45,450 --> 00:28:47,120
Everyone cool with that?

621
00:28:47,120 --> 00:28:50,110
But if you look at
these equalities,

622
00:28:50,110 --> 00:28:55,620
dx adjoint acting on g is the
same as minus dx acting on g.

623
00:28:55,620 --> 00:28:59,060
So this tells me that
the adjoint of dx

624
00:28:59,060 --> 00:29:00,973
is equal to minus dx.

625
00:29:05,403 --> 00:29:05,903
Yeah?

626
00:29:05,903 --> 00:29:07,880
AUDIENCE: Are you assuming
that your surface terms vanish?

627
00:29:07,880 --> 00:29:08,755
PROFESSOR: Thank you!

628
00:29:08,755 --> 00:29:09,360
I lied to you.

629
00:29:09,360 --> 00:29:12,445
So I assumed in this that
my surface terms vanished.

630
00:29:12,445 --> 00:29:13,570
I did a variation by parts.

631
00:29:13,570 --> 00:29:15,995
And that leaves me with
a total derivative.

632
00:29:15,995 --> 00:29:18,120
And that total derivative
gives me a boundary term.

633
00:29:18,120 --> 00:29:20,010
Remember how integration
by parts works.

634
00:29:20,010 --> 00:29:26,100
Integration by parts
says the integral of AdxB

635
00:29:26,100 --> 00:29:28,170
is equal to the
integral of-- well,

636
00:29:28,170 --> 00:29:31,530
AdxB can be written as
derivative with respect

637
00:29:31,530 --> 00:29:43,500
to x of AB minus B derivative
with respect to x of A.

638
00:29:43,500 --> 00:29:47,956
Because this is A prime B plus B
prime A. Here we have AB prime.

639
00:29:47,956 --> 00:29:49,830
So we just subtract off
the appropriate term.

640
00:29:49,830 --> 00:29:51,500
But this is a total derivative.

641
00:29:51,500 --> 00:29:53,740
So it only gives
us a boundary term.

642
00:29:53,740 --> 00:29:57,410
So this integral is equal
to-- can move the integral

643
00:29:57,410 --> 00:30:01,690
over here-- the integral
and the derivative,

644
00:30:01,690 --> 00:30:04,386
because an integral is
nothing but an antiderivative.

645
00:30:04,386 --> 00:30:06,010
The integral and the
derivative cancel,

646
00:30:06,010 --> 00:30:07,570
leaving us with
the boundary terms.

647
00:30:07,570 --> 00:30:09,320
And in this case, it's
from our boundaries

648
00:30:09,320 --> 00:30:13,880
which are minus infinity
plus infinity, minus infinity

649
00:30:13,880 --> 00:30:16,420
and plus infinity.

650
00:30:16,420 --> 00:30:18,480
Now, this tells us
something very important.

651
00:30:18,480 --> 00:30:19,990
And I'm not going to speak
about this in detail,

652
00:30:19,990 --> 00:30:21,360
but I encourage the
recitation instructors

653
00:30:21,360 --> 00:30:23,020
who might happen
to be here to think

654
00:30:23,020 --> 00:30:25,490
to mention this in recitation.

655
00:30:25,490 --> 00:30:29,170
And I encourage you
all to think about it.

656
00:30:29,170 --> 00:30:30,855
If I ask you, what
is the adjoint

657
00:30:30,855 --> 00:30:32,680
of the derivative
operator acting

658
00:30:32,680 --> 00:30:37,080
on the space of functions
which are normalizable,

659
00:30:37,080 --> 00:30:40,180
so that they vanish
at infinity, what

660
00:30:40,180 --> 00:30:43,490
is the adjoint of the
derivative operator acting

661
00:30:43,490 --> 00:30:46,060
on the space of functions which
is normalizable at infinity?

662
00:30:46,060 --> 00:30:47,143
We just derive the answer.

663
00:30:47,143 --> 00:30:51,300
Because we assume that
these surface terms vanish.

664
00:30:51,300 --> 00:30:55,110
Because our wave functions,
f and g, vanish at infinity.

665
00:30:55,110 --> 00:30:56,674
They're normalizable.

666
00:30:56,674 --> 00:30:59,090
However, if I had asked you a
slightly different question,

667
00:30:59,090 --> 00:31:01,410
if I had asked you,
what's the adjoint

668
00:31:01,410 --> 00:31:03,784
of the derivative
operator acting

669
00:31:03,784 --> 00:31:05,450
on a different set
of functions, the set

670
00:31:05,450 --> 00:31:06,900
of functions that
don't necessarily

671
00:31:06,900 --> 00:31:09,025
vanish at infinity, including
sinusoids that go off

672
00:31:09,025 --> 00:31:10,860
to infinity and don't vanish.

673
00:31:10,860 --> 00:31:13,400
Is this the correct answer?

674
00:31:13,400 --> 00:31:13,900
No.

675
00:31:13,900 --> 00:31:14,860
This would not be
the correct answer,

676
00:31:14,860 --> 00:31:16,630
because there are
boundary terms.

677
00:31:16,630 --> 00:31:20,292
So the point I'm making
here, first off, in physics

678
00:31:20,292 --> 00:31:22,750
we're always going to be talking
about normalizable beasts.

679
00:31:22,750 --> 00:31:24,380
At the end of the day, the
physical objects we care about

680
00:31:24,380 --> 00:31:25,230
are in a room.

681
00:31:25,230 --> 00:31:26,651
They're not off infinity.

682
00:31:26,651 --> 00:31:28,400
So everything is going
to be normalizable.

683
00:31:28,400 --> 00:31:30,610
That is just how
the world works.

684
00:31:30,610 --> 00:31:33,760
However, you've got to
be careful in making

685
00:31:33,760 --> 00:31:35,260
these sorts of
arguments and realize

686
00:31:35,260 --> 00:31:38,710
that when I ask you, what is
the adjoint of this operator,

687
00:31:38,710 --> 00:31:40,976
I need to tell you
something more precise.

688
00:31:40,976 --> 00:31:42,350
I need to say,
what's the adjoint

689
00:31:42,350 --> 00:31:45,640
of the derivative acting when
this operator's understood

690
00:31:45,640 --> 00:31:48,490
as acting on some
particular set of functions,

691
00:31:48,490 --> 00:31:50,225
acting on normalizable
functions?

692
00:31:58,390 --> 00:31:59,840
Good.

693
00:31:59,840 --> 00:32:02,920
So anyway, I'll leave that
aside as something to ponder.

694
00:32:06,430 --> 00:32:08,342
But with that
technical detail aside,

695
00:32:08,342 --> 00:32:10,550
as long as we're talking
about normalizable functions

696
00:32:10,550 --> 00:32:13,220
so these boundary terms from
the integration by parts cancel,

697
00:32:13,220 --> 00:32:15,010
the adjoint of the
derivative operator

698
00:32:15,010 --> 00:32:17,650
is minus the
derivative operator.

699
00:32:17,650 --> 00:32:20,300
Cool?

700
00:32:20,300 --> 00:32:23,760
OK, let's do another example.

701
00:32:23,760 --> 00:32:26,008
And where do I want to do this?

702
00:32:26,008 --> 00:32:27,490
I'll do it here.

703
00:32:27,490 --> 00:32:30,300
So another example.

704
00:32:30,300 --> 00:32:30,970
Actually, no.

705
00:32:34,300 --> 00:32:35,210
I will do it here.

706
00:32:41,910 --> 00:32:50,050
So we have another
example, which is three.

707
00:32:50,050 --> 00:32:52,650
What's the adjoint of
the position operator?

708
00:32:59,540 --> 00:33:00,480
OK, take two minutes.

709
00:33:00,480 --> 00:33:02,757
Do this on a piece of
paper in front of you.

710
00:33:02,757 --> 00:33:03,965
I'm not going to call on you.

711
00:33:03,965 --> 00:33:05,960
So you can raise you hand if
you-- OK, chat with the person

712
00:33:05,960 --> 00:33:06,610
next to you.

713
00:33:10,046 --> 00:33:11,420
I mean chat about
physics, right?

714
00:33:11,420 --> 00:33:13,380
Just not-- [LAUGHS].

715
00:33:13,380 --> 00:33:16,810
AUDIENCE: [CHATTING]

716
00:33:42,931 --> 00:33:44,930
PROFESSOR: OK, so how do
we go about doing this?

717
00:33:44,930 --> 00:33:49,750
We go about solving
this problem by using

718
00:33:49,750 --> 00:33:51,580
the definition of the adjoint.

719
00:33:51,580 --> 00:33:52,940
So what is x adjoint?

720
00:33:52,940 --> 00:33:54,690
It's that operator
such that the following

721
00:33:54,690 --> 00:34:01,210
is true, such that the integral
dx of f complex conjugate

722
00:34:01,210 --> 00:34:06,598
with x dagger acting on g
is equal to the integral dx

723
00:34:06,598 --> 00:34:08,889
of what I get by taking the
complex conjugate of taking

724
00:34:08,889 --> 00:34:12,850
x and acting on f and then
integrating this against g.

725
00:34:12,850 --> 00:34:15,179
But now we can use
the action of x

726
00:34:15,179 --> 00:34:19,510
and say that this is equal
to the integral dx of x f

727
00:34:19,510 --> 00:34:22,096
complex conjugate g.

728
00:34:22,096 --> 00:34:23,179
But here's the nice thing.

729
00:34:23,179 --> 00:34:24,730
What is the complex
conjugate of f

730
00:34:24,730 --> 00:34:26,890
times the complex function of f?

731
00:34:26,890 --> 00:34:28,719
x is real.

732
00:34:28,719 --> 00:34:29,610
Positions are real.

733
00:34:29,610 --> 00:34:32,989
So that's just x times the
complex conjugate of f.

734
00:34:32,989 --> 00:34:34,949
So that was
essential move there.

735
00:34:34,949 --> 00:34:37,790
And now we can
rewrite this as equal

736
00:34:37,790 --> 00:34:41,860
the integral of f
complex conjugate xg.

737
00:34:41,860 --> 00:34:44,359
And now, eyeballing
this, x dagger

738
00:34:44,359 --> 00:34:46,590
is that operator
which acts by acting

739
00:34:46,590 --> 00:34:49,630
by multiplying with little x.

740
00:34:49,630 --> 00:34:52,280
Therefore, the adjoint
of the operator x

741
00:34:52,280 --> 00:34:55,690
is equal to the same operator.

742
00:34:55,690 --> 00:34:59,330
x is equal to its own adjoint.

743
00:34:59,330 --> 00:35:00,450
OK?

744
00:35:00,450 --> 00:35:01,930
Cool?

745
00:35:01,930 --> 00:35:04,160
So we've just learned a
couple of really nice things.

746
00:35:04,160 --> 00:35:06,380
So the first is-- where
we I want to do this?

747
00:35:06,380 --> 00:35:07,690
Yeah, good.

748
00:35:07,690 --> 00:35:11,051
So we've learned a
couple of nice things.

749
00:35:11,051 --> 00:35:13,300
And I want to encode them
in the following definition.

750
00:35:15,830 --> 00:35:19,090
Definition-- an
operator, which I

751
00:35:19,090 --> 00:35:25,210
will call o, whose
adjoint is equal to o,

752
00:35:25,210 --> 00:35:28,470
so an operator whose
adjoint is equal to itself

753
00:35:28,470 --> 00:35:31,685
is called Hermetian.

754
00:35:36,500 --> 00:35:38,500
So an operator which is
equal to its own adjoint

755
00:35:38,500 --> 00:35:39,333
is called Hermetian.

756
00:35:42,760 --> 00:35:46,080
And so I want to note a couple
of nice examples of that.

757
00:35:46,080 --> 00:35:58,031
So note a number which
is Hermetian is what?

758
00:35:58,031 --> 00:35:58,530
Real.

759
00:36:01,110 --> 00:36:04,880
An operator-- we found
an operator which

760
00:36:04,880 --> 00:36:08,820
is equal to its own adjoint.
x dagger is equal to x.

761
00:36:08,820 --> 00:36:11,829
And what can you say about the
eigenvalues of this operator?

762
00:36:11,829 --> 00:36:12,370
They're real.

763
00:36:12,370 --> 00:36:13,920
We use that in the
proof, actually.

764
00:36:13,920 --> 00:36:15,200
So this is real.

765
00:36:15,200 --> 00:36:18,260
I will call an operator
real if it's Hermetian.

766
00:36:18,260 --> 00:36:21,190
And here's a
mathematical fact, which

767
00:36:21,190 --> 00:36:23,540
is that any operator
which is Hermetian

768
00:36:23,540 --> 00:36:25,640
has all real eigenvalues.

769
00:36:25,640 --> 00:36:27,820
So this is really-- I'll
state it as a theorem,

770
00:36:27,820 --> 00:36:29,620
but it's just a fact for us.

771
00:36:32,360 --> 00:36:38,045
o has all real eigenvalues.

772
00:36:48,338 --> 00:36:49,254
AUDIENCE: [INAUDIBLE].

773
00:36:49,254 --> 00:36:49,920
PROFESSOR: Yeah?

774
00:36:49,920 --> 00:36:52,777
AUDIENCE: Is it if
and only [INAUDIBLE]?

775
00:36:52,777 --> 00:36:53,360
PROFESSOR: No.

776
00:37:00,280 --> 00:37:01,410
Let's see.

777
00:37:01,410 --> 00:37:04,420
If you have all
real eigenvalues,

778
00:37:04,420 --> 00:37:06,190
it does not imply
that you're Hermetian.

779
00:37:06,190 --> 00:37:08,790
However, if you have
all real eigenvalues

780
00:37:08,790 --> 00:37:11,390
and you can be
diagonalized, it does imply.

781
00:37:11,390 --> 00:37:12,640
So let me give you an example.

782
00:37:12,640 --> 00:37:16,830
So consider the
following operator.

783
00:37:16,830 --> 00:37:18,330
We've done this
many times, rotation

784
00:37:18,330 --> 00:37:20,121
in real three-dimensional
space of a vector

785
00:37:20,121 --> 00:37:21,980
around the vertical axis.

786
00:37:21,980 --> 00:37:26,690
It has one eigenvector,
which is the vertical vector.

787
00:37:26,690 --> 00:37:30,859
And the eigenvalue
is 1, so it's real.

788
00:37:30,859 --> 00:37:32,650
But that's not enough
to make it Hermetian.

789
00:37:32,650 --> 00:37:33,920
Because there's another
fact that we haven't

790
00:37:33,920 --> 00:37:35,420
got to yet with
Hermetian operators,

791
00:37:35,420 --> 00:37:38,080
which is going to tell us
that a Hermetian operator has

792
00:37:38,080 --> 00:37:40,990
as many eigenvectors
as there are dimensions

793
00:37:40,990 --> 00:37:46,124
in the space, i.e., that the
eigenvectors form a basis.

794
00:37:46,124 --> 00:37:48,040
But there's only one
eigenvector for this guy,

795
00:37:48,040 --> 00:37:50,290
even though we're in a
three-dimensional vector space.

796
00:37:50,290 --> 00:37:52,520
So this operator, rotation
by an angle theta,

797
00:37:52,520 --> 00:37:55,540
is not Hermetian, even
though its only eigenvalue

798
00:37:55,540 --> 00:37:57,120
isn't in fact real.

799
00:37:57,120 --> 00:37:58,500
So it's not an only if.

800
00:37:58,500 --> 00:38:01,000
If you are Hermetian, your
eigenvalues are all real.

801
00:38:01,000 --> 00:38:03,560
And you'll prove this
on a problem set.

802
00:38:03,560 --> 00:38:04,060
Yeah?

803
00:38:04,060 --> 00:38:08,420
AUDIENCE: If you're Hermetian,
are your eigenfunctions normal?

804
00:38:10,565 --> 00:38:11,690
PROFESSOR: Not necessarily.

805
00:38:11,690 --> 00:38:13,400
But they can be made normal.

806
00:38:13,400 --> 00:38:15,970
We'll talk about this
in more detail later.

807
00:38:15,970 --> 00:38:16,470
OK.

808
00:38:19,730 --> 00:38:23,647
Let's do a quick
check, last example.

809
00:38:23,647 --> 00:38:25,980
And I'm not actually going
to go through this in detail,

810
00:38:25,980 --> 00:38:27,000
but what about p?

811
00:38:27,000 --> 00:38:28,750
What about the
momentum operator?

812
00:38:28,750 --> 00:38:31,800
First off, do you think
the momentum is real?

813
00:38:31,800 --> 00:38:33,480
It sure would be nice.

814
00:38:33,480 --> 00:38:37,070
Because its eigenvalues are the
observable values of momentum.

815
00:38:37,070 --> 00:38:39,070
And so its eigenvalues
should all be real.

816
00:38:39,070 --> 00:38:40,236
Does that make it Hermetian?

817
00:38:40,236 --> 00:38:41,910
Not necessarily,
but let's check.

818
00:38:41,910 --> 00:38:44,980
So what is the adjoint of p?

819
00:38:44,980 --> 00:38:47,140
Well, this actually
we can do very easily.

820
00:38:47,140 --> 00:38:49,390
And I'm not going to go
through an elaborate argument.

821
00:38:49,390 --> 00:38:50,931
I'm just going to
know the following.

822
00:38:50,931 --> 00:38:58,150
p is equal to h bar upon i ddx.

823
00:39:01,420 --> 00:39:02,420
And this is an operator.

824
00:39:02,420 --> 00:39:03,410
This is an operator.

825
00:39:03,410 --> 00:39:06,400
So what's the adjoint
of this operator?

826
00:39:06,400 --> 00:39:10,280
Well, this under an adjoint
gets a minus sign, right?

827
00:39:10,280 --> 00:39:11,860
It's itself up to a minus sign.

828
00:39:11,860 --> 00:39:14,414
So is the derivative Hermetian?

829
00:39:14,414 --> 00:39:16,080
No, it's in fact what
we anti-Hermetian.

830
00:39:16,080 --> 00:39:18,530
Its adjoint is minus itself.

831
00:39:18,530 --> 00:39:20,530
What about i?

832
00:39:20,530 --> 00:39:22,680
What's its adjoint?

833
00:39:22,680 --> 00:39:24,070
Minus i.

834
00:39:24,070 --> 00:39:24,602
Sweet.

835
00:39:24,602 --> 00:39:26,310
So this has an adjoint,
picks up a minus.

836
00:39:26,310 --> 00:39:28,460
This has an adjoint,
picks up a minus.

837
00:39:28,460 --> 00:39:29,800
The minuses cancel.

838
00:39:29,800 --> 00:39:30,890
p adjoint is p.

839
00:39:33,987 --> 00:39:35,070
So p is in fact Hermetian.

840
00:39:38,410 --> 00:39:40,150
And here's a stronger
physical fact.

841
00:39:40,150 --> 00:39:43,370
So now we've seen that each
of the operators we built

842
00:39:43,370 --> 00:39:45,930
is x and p, true
of the operators

843
00:39:45,930 --> 00:39:47,840
we've looked at so
far is Hermetian,

844
00:39:47,840 --> 00:39:49,810
those that correspond
to physical observables.

845
00:39:49,810 --> 00:39:50,990
Here's a physical fact.

846
00:39:57,310 --> 00:40:03,990
All the observables you
measure with sticks are real.

847
00:40:07,220 --> 00:40:09,740
And the corresponding
statement is

848
00:40:09,740 --> 00:40:21,190
that all operators
corresponding to observables,

849
00:40:21,190 --> 00:40:22,640
all operators must be Hermetian.

850
00:40:32,480 --> 00:40:34,749
To the postulate that
says, "Observables

851
00:40:34,749 --> 00:40:36,290
are represented by
operators," should

852
00:40:36,290 --> 00:40:37,930
be adjoined the
word "Hermetian."

853
00:40:37,930 --> 00:40:40,160
Observables are represented
in quantum mechanics

854
00:40:40,160 --> 00:40:42,787
by Hermetian operators,
which are operators

855
00:40:42,787 --> 00:40:44,370
that have a number
of nice properties,

856
00:40:44,370 --> 00:40:47,250
including they have
all real eigenvalues.

857
00:40:47,250 --> 00:40:47,750
Cool?

858
00:40:50,850 --> 00:40:53,250
OK.

859
00:40:53,250 --> 00:40:54,460
Questions?

860
00:40:54,460 --> 00:40:55,440
Yeah.

861
00:40:55,440 --> 00:40:57,739
AUDIENCE: If it has to be
Hermetian and not just have

862
00:40:57,739 --> 00:41:00,030
real eigenvalues, does that
mean the eigenvalues always

863
00:41:00,030 --> 00:41:01,650
need to form some kind of basis?

864
00:41:01,650 --> 00:41:04,539
PROFESSOR: Yeah, the
eigenvectors will.

865
00:41:04,539 --> 00:41:06,580
This is connected to the
fact we've already seen.

866
00:41:06,580 --> 00:41:07,850
If you take an
arbitrary wave function

867
00:41:07,850 --> 00:41:09,933
you can expand it in states
with definite momentum

868
00:41:09,933 --> 00:41:10,840
as a superposition.

869
00:41:10,840 --> 00:41:14,720
You can also expand it in a set
of states of definite energy

870
00:41:14,720 --> 00:41:16,210
or of definite position.

871
00:41:16,210 --> 00:41:18,510
Anytime you have a
Hermetian operator,

872
00:41:18,510 --> 00:41:24,880
its eigenvectors suffice
to expand any function.

873
00:41:24,880 --> 00:41:27,270
They provide a basis for
representing any function.

874
00:41:29,840 --> 00:41:31,750
So that's the end of
the mathematical side.

875
00:41:31,750 --> 00:41:36,350
Let's get back to
this physical point.

876
00:41:36,350 --> 00:41:39,800
So we've defined this operator
a and this other operator

877
00:41:39,800 --> 00:41:41,140
a dagger.

878
00:41:41,140 --> 00:41:42,800
And here's my question first.

879
00:41:42,800 --> 00:41:45,560
Is a Hermetian?

880
00:41:45,560 --> 00:41:46,060
No.

881
00:41:46,060 --> 00:41:47,000
That's Hermetian.

882
00:41:47,000 --> 00:41:47,708
That's Hermetian.

883
00:41:47,708 --> 00:41:48,620
But there's an i.

884
00:41:48,620 --> 00:41:50,080
That i will pick
up the minus sign

885
00:41:50,080 --> 00:41:52,490
when we do the
complex conjugation.

886
00:41:52,490 --> 00:41:54,510
Oh, look.

887
00:41:54,510 --> 00:41:57,180
Sure was fortuitous that
I called this a dagger,

888
00:41:57,180 --> 00:42:00,690
since this is equal to a dagger.

889
00:42:03,700 --> 00:42:05,175
So this is the adjoint of a.

890
00:42:05,175 --> 00:42:07,425
So this immediately tells
you something interesting. x

891
00:42:07,425 --> 00:42:08,440
and p are both observables.

892
00:42:08,440 --> 00:42:09,898
Does a correspond
to an observable?

893
00:42:13,750 --> 00:42:14,520
Is it Hermetian?

894
00:42:17,110 --> 00:42:21,280
Every intervals is associated
to a Hermetian operator.

895
00:42:21,280 --> 00:42:23,090
This is not Hermetian.

896
00:42:23,090 --> 00:42:27,210
So a does not represent
an observable operator.

897
00:42:32,200 --> 00:42:35,239
And I will post notes on
the web page, which give us

898
00:42:35,239 --> 00:42:36,780
a somewhat lengthy
discussion-- or it

899
00:42:36,780 --> 00:42:37,830
might be in one
of the solutions--

900
00:42:37,830 --> 00:42:39,371
a somewhat lengthy
discussion of what

901
00:42:39,371 --> 00:42:42,550
it means for a and a dagger
to not be observable.

902
00:42:42,550 --> 00:42:46,390
You'll get more
discussion of that there.

903
00:42:46,390 --> 00:42:49,155
Meanwhile, if a is not
observable, it's not Hermetian,

904
00:42:49,155 --> 00:42:50,405
does it have real eigenvalues?

905
00:42:53,030 --> 00:42:54,400
Well, here's an important thing.

906
00:42:54,400 --> 00:42:57,344
I said if you're Hermetian,
all the eigenvalues are real.

907
00:42:57,344 --> 00:42:59,260
If you're not Hermetian,
that doesn't tell you

908
00:42:59,260 --> 00:43:01,030
you can't have any
real eigenvalues.

909
00:43:01,030 --> 00:43:02,990
It just says that I
haven't guaranteed for you

910
00:43:02,990 --> 00:43:05,170
that all the
eigenvalues are real.

911
00:43:05,170 --> 00:43:07,337
So what we'll discover
towards the end of the course

912
00:43:07,337 --> 00:43:09,795
when we talk about something
called coherent states is that

913
00:43:09,795 --> 00:43:11,890
in fact, a does have a
nice set of eigenvectors.

914
00:43:11,890 --> 00:43:12,750
They're very nice.

915
00:43:12,750 --> 00:43:13,370
They're great.

916
00:43:13,370 --> 00:43:15,520
We use them for lasers.

917
00:43:15,520 --> 00:43:16,510
They're very useful.

918
00:43:16,510 --> 00:43:18,030
And they're called
coherent states.

919
00:43:18,030 --> 00:43:20,820
But their eigenvalues
are not in general real.

920
00:43:20,820 --> 00:43:23,320
They're generically
complex numbers.

921
00:43:23,320 --> 00:43:25,320
Are they things you can measure?

922
00:43:25,320 --> 00:43:26,447
Not directly.

923
00:43:26,447 --> 00:43:28,530
They're related to things
you can measure, though,

924
00:43:28,530 --> 00:43:30,030
in some pretty nice ways.

925
00:43:34,560 --> 00:43:36,240
So why are we bothering
with these guys

926
00:43:36,240 --> 00:43:38,800
if they're not observable?

927
00:43:38,800 --> 00:43:39,891
Yeah.

928
00:43:39,891 --> 00:43:40,890
AUDIENCE: E [INAUDIBLE].

929
00:43:40,890 --> 00:43:41,806
PROFESSOR: Yeah, good.

930
00:43:41,806 --> 00:43:42,307
Excellent.

931
00:43:42,307 --> 00:43:43,097
That's really good.

932
00:43:43,097 --> 00:43:44,150
So two things about it.

933
00:43:44,150 --> 00:43:46,900
So one thing is this form
for the energy operator

934
00:43:46,900 --> 00:43:49,150
is particularly simple.

935
00:43:49,150 --> 00:43:49,850
We see the 1/2.

936
00:43:49,850 --> 00:43:51,770
This looks suggestive
from before.

937
00:43:51,770 --> 00:43:55,150
But it makes it obvious
that E is Hermetian.

938
00:43:55,150 --> 00:43:56,830
And that may not be
obvious to you guys.

939
00:43:56,830 --> 00:43:58,360
So let's just check.

940
00:43:58,360 --> 00:44:00,740
Here's something that you'll
show on the problem set.

941
00:44:00,740 --> 00:44:05,310
AB adjoint is equal to
B adjoint A adjoint.

942
00:44:05,310 --> 00:44:06,920
The order matters.

943
00:44:06,920 --> 00:44:08,070
These are operators.

944
00:44:08,070 --> 00:44:09,760
And so if we take the
adjective of this,

945
00:44:09,760 --> 00:44:10,968
what's this going to give us?

946
00:44:10,968 --> 00:44:12,270
Well we change the order.

947
00:44:12,270 --> 00:44:13,520
So it's going to be
a dagger, and then we

948
00:44:13,520 --> 00:44:15,300
take the dagger of
both of the a dagger.

949
00:44:15,300 --> 00:44:17,730
So this is self-adjoint,
or Hermetian.

950
00:44:17,730 --> 00:44:18,590
So that's good.

951
00:44:18,590 --> 00:44:19,760
Of course, we already
knew that, because we

952
00:44:19,760 --> 00:44:21,880
could have written it
in terms of x and p.

953
00:44:21,880 --> 00:44:23,450
But this is somehow simpler.

954
00:44:23,450 --> 00:44:25,990
And it in particular
emphasizes the form,

955
00:44:25,990 --> 00:44:28,565
or recapitulates the form
of the energy eigenvalues.

956
00:44:31,170 --> 00:44:33,270
Why else would we care
about a and a dagger?

957
00:44:36,721 --> 00:44:39,179
OK, now this is a good moment.

958
00:44:39,179 --> 00:44:40,220
Here's the second reason.

959
00:44:40,220 --> 00:44:41,928
So the first reason
you care is this sort

960
00:44:41,928 --> 00:44:43,590
of structural
similarity and the fact

961
00:44:43,590 --> 00:44:46,270
that it's nicely Hermetian
in a different way.

962
00:44:46,270 --> 00:44:48,060
Here's the key thing.

963
00:44:48,060 --> 00:44:50,480
Key.

964
00:44:50,480 --> 00:44:55,330
a and a dagger satisfy the
simplest commutation relation

965
00:44:55,330 --> 00:44:57,160
in the world.

966
00:44:57,160 --> 00:44:58,280
Well, the second simplest.

967
00:44:58,280 --> 00:45:00,405
The simplest is that it's
0 on the right-hand side.

968
00:45:00,405 --> 00:45:02,830
But the simplest not trivial
commutation relationship.

969
00:45:02,830 --> 00:45:09,324
a with a dagger is equal to--
so what is a dagger equal to?

970
00:45:09,324 --> 00:45:10,490
We just take the definition.

971
00:45:10,490 --> 00:45:11,240
Let's put this in.

972
00:45:11,240 --> 00:45:21,050
So this is x over x0
plus ip over p0, comma,

973
00:45:21,050 --> 00:45:27,140
x over x0 minus i, p over p0,
hat, hat, hat, hat, bracket,

974
00:45:27,140 --> 00:45:28,080
bracket.

975
00:45:28,080 --> 00:45:28,690
Good.

976
00:45:28,690 --> 00:45:32,030
So here there are
going to be four terms.

977
00:45:32,030 --> 00:45:33,230
There's x commutator x.

978
00:45:33,230 --> 00:45:36,040
What is that?

979
00:45:36,040 --> 00:45:39,551
What is the commutator of
an operator with itself?

980
00:45:39,551 --> 00:45:40,050
0.

981
00:45:40,050 --> 00:45:43,145
Because remember the
definition of the commutator A,

982
00:45:43,145 --> 00:45:47,410
B is AB minus BA.

983
00:45:47,410 --> 00:45:52,270
So A with A is equal
to AA minus AA.

984
00:45:52,270 --> 00:45:55,680
And you have no options there.

985
00:45:55,680 --> 00:45:56,480
That's 0.

986
00:45:56,480 --> 00:45:59,667
So x with x is 0. p with p is 0.

987
00:45:59,667 --> 00:46:01,750
So the only terms that
matter are the cross terms.

988
00:46:01,750 --> 00:46:04,500
We have an x with p.

989
00:46:04,500 --> 00:46:08,192
And notice that's going to be
times a minus i with p0 and x0.

990
00:46:08,192 --> 00:46:09,650
And then we have
another term which

991
00:46:09,650 --> 00:46:14,650
is p with x, which
is i, p0 over x0.

992
00:46:14,650 --> 00:46:16,650
So you change the order
and you change the sign.

993
00:46:16,650 --> 00:46:18,483
But if you change the
order of a commutator,

994
00:46:18,483 --> 00:46:19,500
you change the side.

995
00:46:19,500 --> 00:46:21,845
So we can put them
both in the same order.

996
00:46:21,845 --> 00:46:22,970
Let me just write this out.

997
00:46:22,970 --> 00:46:24,950
So this is i over x0 p0.

998
00:46:24,950 --> 00:46:27,590
So this guy, minus i over x0p0.

999
00:46:27,590 --> 00:46:31,990
But x0p0 is equal to 2h
bar, as we checked before.

1000
00:46:31,990 --> 00:46:34,520
This was x with p.

1001
00:46:34,520 --> 00:46:40,190
And then the second
term was plus i, again

1002
00:46:40,190 --> 00:46:43,400
over x0p0, which is
2h bar, p with x.

1003
00:46:46,810 --> 00:46:50,610
This x with p is equal to?

1004
00:46:50,610 --> 00:46:51,330
i h bar.

1005
00:46:51,330 --> 00:46:52,280
So the h bar cancels.

1006
00:46:52,280 --> 00:46:56,800
The i gives me a plus 1.

1007
00:46:56,800 --> 00:46:59,370
And p with x gives
me minus i h bar.

1008
00:46:59,370 --> 00:47:04,422
So the h bar and the
minus i gives me plus 1.

1009
00:47:04,422 --> 00:47:05,130
Well that's nice.

1010
00:47:05,130 --> 00:47:07,370
This is equal to 1.

1011
00:47:07,370 --> 00:47:16,875
So plus 1/2, therefore a
with a dagger is equal to 1.

1012
00:47:20,800 --> 00:47:25,110
As advertised, that is
about as simple as it gets.

1013
00:47:25,110 --> 00:47:26,610
Notice a couple of
other commutators

1014
00:47:26,610 --> 00:47:29,230
that follow from this.

1015
00:47:29,230 --> 00:47:35,637
a dagger with a is
equal to minus 1.

1016
00:47:35,637 --> 00:47:36,720
We just changed the order.

1017
00:47:36,720 --> 00:47:38,520
And that's just an
overall minus sign.

1018
00:47:38,520 --> 00:47:40,430
And a with a is what?

1019
00:47:40,430 --> 00:47:41,390
0.

1020
00:47:41,390 --> 00:47:43,310
a dagger with a dagger?

1021
00:47:43,310 --> 00:47:43,940
Good.

1022
00:47:43,940 --> 00:47:46,060
OK.

1023
00:47:46,060 --> 00:47:53,930
So we are now going to use
this commutation relation

1024
00:47:53,930 --> 00:47:56,960
to totally crush the
problem into submission.

1025
00:47:56,960 --> 00:48:00,700
It's going to be weeping before
us like the Romans in front

1026
00:48:00,700 --> 00:48:01,770
of the Visigoths.

1027
00:48:01,770 --> 00:48:03,570
It's going to be dramatic.

1028
00:48:03,570 --> 00:48:06,900
OK, so let's check.

1029
00:48:06,900 --> 00:48:10,080
So let's combine the two things.

1030
00:48:10,080 --> 00:48:12,360
So we had the first thing
is that this form is simple.

1031
00:48:12,360 --> 00:48:14,010
The second is that the
commutator is simple.

1032
00:48:14,010 --> 00:48:15,930
Let's combine these
together and really milk

1033
00:48:15,930 --> 00:48:16,880
the system for what it's got.

1034
00:48:16,880 --> 00:48:18,786
And to do that, I need
two more commutators.

1035
00:48:21,880 --> 00:48:24,124
And the lesson of this
series of machinations,

1036
00:48:24,124 --> 00:48:25,540
it's very tempting
to look at this

1037
00:48:25,540 --> 00:48:26,650
and be like, why
are you doing this?

1038
00:48:26,650 --> 00:48:28,233
And the reason is,
I want to encourage

1039
00:48:28,233 --> 00:48:32,270
you to see the power of
these commutation relations.

1040
00:48:32,270 --> 00:48:34,950
They're telling you a tremendous
amount about the system.

1041
00:48:34,950 --> 00:48:36,440
So we're going through and
doing some relatively simple

1042
00:48:36,440 --> 00:48:36,950
calculations.

1043
00:48:36,950 --> 00:48:38,324
We're just computing
commutators.

1044
00:48:38,324 --> 00:48:39,740
We're following our nose.

1045
00:48:39,740 --> 00:48:43,030
And we're going to
derive something awesome.

1046
00:48:43,030 --> 00:48:46,310
So don't just bear with it.

1047
00:48:46,310 --> 00:48:48,830
Learn from this, that
there's something very useful

1048
00:48:48,830 --> 00:48:50,740
and powerful about
commutation relations.

1049
00:48:50,740 --> 00:48:51,510
You'll see that at the end.

1050
00:48:51,510 --> 00:48:53,426
But I want you to on to
the slight awkwardness

1051
00:48:53,426 --> 00:48:56,520
right now, that it's not
totally obvious beforehand where

1052
00:48:56,520 --> 00:48:58,700
this is going.

1053
00:48:58,700 --> 00:49:00,786
So what is E with a?

1054
00:49:00,786 --> 00:49:01,660
That's easy.

1055
00:49:01,660 --> 00:49:06,210
It's the h bar omega
a dagger a plus 1/2.

1056
00:49:06,210 --> 00:49:09,385
So the 1/2, what's 1/2
commutator with an operator?

1057
00:49:12,120 --> 00:49:12,620
0.

1058
00:49:12,620 --> 00:49:14,494
Because any number
commutes with an operator.

1059
00:49:14,494 --> 00:49:16,380
1/2 operator is operator 1/2.

1060
00:49:16,380 --> 00:49:18,270
It's just a constant.

1061
00:49:18,270 --> 00:49:19,020
That term is gone.

1062
00:49:19,020 --> 00:49:20,436
So the only thing
that's left over

1063
00:49:20,436 --> 00:49:25,764
is h bar omega, a
dagger a with a.

1064
00:49:25,764 --> 00:49:27,180
The h bar omega's
just a constant.

1065
00:49:27,180 --> 00:49:29,721
It's going to pull out no matter
which term we're looking at.

1066
00:49:29,721 --> 00:49:33,450
So I could just pull
that factor out.

1067
00:49:33,450 --> 00:49:40,840
So this is equal to h
bar omega times a dagger

1068
00:49:40,840 --> 00:49:46,420
a minus a a dagger a.

1069
00:49:46,420 --> 00:49:49,360
But this is equal to
h bar omega-- well,

1070
00:49:49,360 --> 00:49:51,420
that's a dagger a
a, a a dagger a.

1071
00:49:51,420 --> 00:49:53,690
You can just pull out
the a on the right.

1072
00:49:53,690 --> 00:49:58,230
a dagger a minus a a dagger a.

1073
00:49:58,230 --> 00:50:00,130
That's equal to h bar omega.

1074
00:50:00,130 --> 00:50:08,360
Well, a dagger with a is equal
to a dagger with a minus 1

1075
00:50:08,360 --> 00:50:10,150
is equal to minus h bar omega.

1076
00:50:10,150 --> 00:50:12,070
And we have this a leftover, a.

1077
00:50:12,070 --> 00:50:17,960
So E with a is equal to minus a.

1078
00:50:17,960 --> 00:50:20,500
Well, that's interesting.

1079
00:50:20,500 --> 00:50:22,000
Now, the second
commutator-- I'm not

1080
00:50:22,000 --> 00:50:25,092
going to do it--
E with a dagger is

1081
00:50:25,092 --> 00:50:26,800
going to be equal to--
let's just eyeball

1082
00:50:26,800 --> 00:50:27,758
what's going to happen.

1083
00:50:27,758 --> 00:50:30,600
They can be a dagger.

1084
00:50:30,600 --> 00:50:32,870
So we're going to
have a dagger a dagger

1085
00:50:32,870 --> 00:50:34,289
minus a dagger a dagger a.

1086
00:50:34,289 --> 00:50:36,080
So we're going to have
an a dagger in front

1087
00:50:36,080 --> 00:50:38,661
and then a dagger.

1088
00:50:38,661 --> 00:50:40,160
So all we're going
to get is a sign.

1089
00:50:40,160 --> 00:50:43,394
And it's going to be a
dagger plus a dagger.

1090
00:50:46,355 --> 00:50:47,980
I shouldn't written
that in the center.

1091
00:50:52,420 --> 00:50:54,195
Everyone cool with that?

1092
00:50:54,195 --> 00:50:54,695
Yeah.

1093
00:50:54,695 --> 00:50:57,360
AUDIENCE: The h bar where?

1094
00:50:57,360 --> 00:50:59,740
PROFESSOR: Oh shoot, thank you!

1095
00:50:59,740 --> 00:51:00,530
h bar here.

1096
00:51:00,530 --> 00:51:01,105
Thank you.

1097
00:51:03,860 --> 00:51:07,760
We would have
misruled the galaxy.

1098
00:51:07,760 --> 00:51:08,859
OK, good.

1099
00:51:08,859 --> 00:51:09,525
Other questions?

1100
00:51:12,410 --> 00:51:16,102
You don't notice-- you haven't
noticed yet, but we just won.

1101
00:51:16,102 --> 00:51:17,560
We just totally
solved the problem.

1102
00:51:17,560 --> 00:51:18,590
And here's why.

1103
00:51:18,590 --> 00:51:24,050
Once you see this, any
time you see this, anytime

1104
00:51:24,050 --> 00:51:27,325
you see this commutator,
an operator with an a

1105
00:51:27,325 --> 00:51:31,050
is equal to plus a times some
constant, anytime you see this,

1106
00:51:31,050 --> 00:51:32,460
cheer.

1107
00:51:32,460 --> 00:51:33,970
And here's why.

1108
00:51:33,970 --> 00:51:34,500
Yeah, right.

1109
00:51:34,500 --> 00:51:35,000
Exactly.

1110
00:51:35,000 --> 00:51:35,680
Now.

1111
00:51:35,680 --> 00:51:37,920
Whoo!

1112
00:51:37,920 --> 00:51:39,064
Here's why.

1113
00:51:39,064 --> 00:51:40,230
Here's why you should cheer.

1114
00:51:40,230 --> 00:51:42,430
Because you no longer have
to solve any problems.

1115
00:51:42,430 --> 00:51:44,800
You no longer have to solve
any differential equations.

1116
00:51:44,800 --> 00:51:46,600
You can simply write
down the problem.

1117
00:51:46,600 --> 00:51:48,850
And let's see that you can
just write down the answer.

1118
00:51:51,740 --> 00:51:54,410
Suppose that we already
happened to have access--

1119
00:51:54,410 --> 00:51:57,710
here in my sleeve I have
access to an eigenfunction

1120
00:51:57,710 --> 00:51:58,810
of the energy operator.

1121
00:51:58,810 --> 00:52:05,780
E on phi E is equal to E phi
E. Suppose I have this guy.

1122
00:52:05,780 --> 00:52:08,500
Cool?

1123
00:52:08,500 --> 00:52:10,260
Check this out.

1124
00:52:10,260 --> 00:52:15,650
Consider a new
state, psi, which is

1125
00:52:15,650 --> 00:52:18,810
equal to a-- which do
I want to do first?

1126
00:52:18,810 --> 00:52:20,740
Doesn't really matter,
but let's do a.

1127
00:52:20,740 --> 00:52:26,910
Consider psi is
equal to a on phi E.

1128
00:52:26,910 --> 00:52:28,625
What can you say
about this state?

1129
00:52:28,625 --> 00:52:31,000
Well, it's the state you get
by taking this wave function

1130
00:52:31,000 --> 00:52:32,300
and acting with a.

1131
00:52:32,300 --> 00:52:33,550
Not terribly illuminating.

1132
00:52:33,550 --> 00:52:39,670
However, E on psi
is equal to what?

1133
00:52:39,670 --> 00:52:41,730
Maybe this has some nice
property under acting

1134
00:52:41,730 --> 00:52:47,390
with E. This is equal
to E on a with pfi E.

1135
00:52:47,390 --> 00:52:48,660
Now, this is tantalizing.

1136
00:52:48,660 --> 00:52:50,700
Because at this point
it's very-- look, that E,

1137
00:52:50,700 --> 00:52:52,840
it really wants to hit this phi.

1138
00:52:52,840 --> 00:52:53,840
It just really wants to.

1139
00:52:53,840 --> 00:52:55,256
There's an E it
wants to pull out.

1140
00:52:55,256 --> 00:52:56,130
It'll be great.

1141
00:52:56,130 --> 00:52:57,920
The problem is it's not there.

1142
00:52:57,920 --> 00:52:59,030
There's an a in the way.

1143
00:52:59,030 --> 00:53:01,880
And so at this point we add 0.

1144
00:53:01,880 --> 00:53:03,800
And this is a very
powerful technique.

1145
00:53:03,800 --> 00:53:17,110
This is equal to Ea
minus aE plus aE, phi E.

1146
00:53:17,110 --> 00:53:18,770
But that has a nice expression.

1147
00:53:18,770 --> 00:53:21,610
This is equal to Ea minus aE.

1148
00:53:21,610 --> 00:53:25,960
That's the commutator
of E with a.

1149
00:53:25,960 --> 00:53:27,110
Plus a.

1150
00:53:27,110 --> 00:53:29,960
What's E acting on phi E?

1151
00:53:29,960 --> 00:53:31,960
Actually, let me just
leave this as aE.

1152
00:53:31,960 --> 00:53:34,932
So what have we done here
before we actually act?

1153
00:53:34,932 --> 00:53:36,390
What we've done is
something called

1154
00:53:36,390 --> 00:53:37,905
commuting an operator through.

1155
00:53:37,905 --> 00:53:40,030
So what do I mean by
commuting an operator through?

1156
00:53:40,030 --> 00:53:44,016
If we have an operator A and
an operator B and a state f,

1157
00:53:44,016 --> 00:53:46,800
and I want A to act
on f, I can always

1158
00:53:46,800 --> 00:53:51,070
write this as-- this is
equal to the commutator of A

1159
00:53:51,070 --> 00:53:55,470
would be plus BA acting on f.

1160
00:53:55,470 --> 00:53:58,266
So this lets me act A
on f directly without B.

1161
00:53:58,266 --> 00:54:00,390
But I have to know what
the commutator of these two

1162
00:54:00,390 --> 00:54:01,365
operators is.

1163
00:54:01,365 --> 00:54:03,490
So if I know what the
commutator is, I can do this.

1164
00:54:03,490 --> 00:54:04,962
I can simplify.

1165
00:54:04,962 --> 00:54:07,170
When one does this, when
one takes AB and replaces it

1166
00:54:07,170 --> 00:54:10,310
by the commutator of A with B,
plus BA, changing the order,

1167
00:54:10,310 --> 00:54:14,470
the phrase that one uses is
I have commuted A through B.

1168
00:54:14,470 --> 00:54:16,520
And commuting
operators through other

1169
00:54:16,520 --> 00:54:19,940
is an extraordinarily useful
tool, useful technique.

1170
00:54:19,940 --> 00:54:20,860
Now let's do y.

1171
00:54:20,860 --> 00:54:25,230
So here what's the
commutator of E with a?

1172
00:54:25,230 --> 00:54:26,850
We just did that.

1173
00:54:26,850 --> 00:54:30,360
It's minus h bar omega a.

1174
00:54:30,360 --> 00:54:33,230
And what's aE on phi E?

1175
00:54:33,230 --> 00:54:35,740
What's E on phi E?

1176
00:54:35,740 --> 00:54:37,820
E. Exactly.

1177
00:54:37,820 --> 00:54:42,290
Plus Ea on phi.

1178
00:54:46,240 --> 00:54:48,180
And now we're cooking with gas.

1179
00:54:48,180 --> 00:54:54,750
Because this is equal to minus
h bar omega a plus Ea, hat.

1180
00:54:54,750 --> 00:54:57,450
I'm going to pull out
this common factor of a.

1181
00:54:57,450 --> 00:55:04,904
So if I pull out that common
factor of a, plus E, a phi E,

1182
00:55:04,904 --> 00:55:06,570
and now I'm going to
just slightly write

1183
00:55:06,570 --> 00:55:08,752
this instead of minus
h bar omega plus E,

1184
00:55:08,752 --> 00:55:10,710
I'm going to write this
as E minus h bar omega.

1185
00:55:10,710 --> 00:55:12,918
I'm just literally changing
the order of the algebra.

1186
00:55:12,918 --> 00:55:15,340
E minus h bar omega.

1187
00:55:15,340 --> 00:55:18,330
And what is aE?

1188
00:55:18,330 --> 00:55:18,830
Psi.

1189
00:55:18,830 --> 00:55:22,820
That was the original
state we started with, psi.

1190
00:55:22,820 --> 00:55:24,230
Well, that's cool.

1191
00:55:24,230 --> 00:55:27,460
If I have a state with
energy E and I act on it

1192
00:55:27,460 --> 00:55:32,420
with the operator a, I
get a new state, psi,

1193
00:55:32,420 --> 00:55:36,900
which is also an eigenstate
of the energy operator,

1194
00:55:36,900 --> 00:55:39,940
but with a slightly
different energy eigenvalue.

1195
00:55:39,940 --> 00:55:44,115
The eigenvalue is now
decreased by h bar omega.

1196
00:55:44,115 --> 00:55:44,615
Cool?

1197
00:55:50,090 --> 00:55:54,350
And that is what we wanted.

1198
00:55:54,350 --> 00:55:58,560
Let's explore the
consequences of this.

1199
00:55:58,560 --> 00:56:03,160
So if we have a state
with eigenvalue E,

1200
00:56:03,160 --> 00:56:08,440
we have phi E such that E on
phi E is equal to E phi E.

1201
00:56:08,440 --> 00:56:17,780
Then the state a
phi E has eigenvalue

1202
00:56:17,780 --> 00:56:26,100
as energy, eigenvalue
E minus h bar omega.

1203
00:56:30,380 --> 00:56:39,230
So I could call this phi
sub E minus h bar omega.

1204
00:56:39,230 --> 00:56:41,940
It's an eigenfunction
of the energy operator,

1205
00:56:41,940 --> 00:56:44,530
the eigenvalue, E
minus h bar omega.

1206
00:56:44,530 --> 00:56:45,740
Agreed?

1207
00:56:45,740 --> 00:56:50,250
Do I know that this is in
fact properly normalized?

1208
00:56:50,250 --> 00:56:51,830
No, because 12
times it would also

1209
00:56:51,830 --> 00:56:53,204
be a perfectly
good eigenfunction

1210
00:56:53,204 --> 00:56:54,210
of the energy operator.

1211
00:56:54,210 --> 00:56:57,880
So this is proportional to
the properly normalized guy,

1212
00:56:57,880 --> 00:57:00,937
with some, at the moment,
unknown constant coefficient

1213
00:57:00,937 --> 00:57:01,520
normalization.

1214
00:57:01,520 --> 00:57:03,590
Everyone cool with that?

1215
00:57:03,590 --> 00:57:05,820
So now let's think about
what this tells us.

1216
00:57:05,820 --> 00:57:07,335
This tells us if
we have a state phi

1217
00:57:07,335 --> 00:57:10,330
E, which I will denote
its energy by this level,

1218
00:57:10,330 --> 00:57:13,460
then if I act on
it with a phi E I

1219
00:57:13,460 --> 00:57:16,790
get another state where the
energy, instead of being E,

1220
00:57:16,790 --> 00:57:19,270
is equal E minus h bar omega.

1221
00:57:19,270 --> 00:57:22,660
So this distance in
energy is h bar omega.

1222
00:57:22,660 --> 00:57:24,180
Cool?

1223
00:57:24,180 --> 00:57:26,110
Let me do it again.

1224
00:57:26,110 --> 00:57:29,070
We'll tack a on phi E. By
exactly the same argument, if I

1225
00:57:29,070 --> 00:57:34,660
make psi as equal to a on
a phi E, a squared phi E,

1226
00:57:34,660 --> 00:57:38,819
I get another state, again
separated by h bar omega, E

1227
00:57:38,819 --> 00:57:39,610
minus 2h bar omega.

1228
00:57:44,699 --> 00:57:45,740
Turtles all the way down.

1229
00:57:50,911 --> 00:57:51,910
Everyone cool with that?

1230
00:57:54,485 --> 00:57:56,235
Let's do a slightly
different calculation.

1231
00:58:00,590 --> 00:58:05,052
But before we do that,
I want to give a a name.

1232
00:58:05,052 --> 00:58:06,260
a does something really cool.

1233
00:58:06,260 --> 00:58:08,820
When you take the state phi
E that has definite energy E,

1234
00:58:08,820 --> 00:58:10,860
it's an energy eigenfunction,
and you act on it

1235
00:58:10,860 --> 00:58:13,880
with a, what happens?

1236
00:58:13,880 --> 00:58:16,180
It lowers the energy
by h bar omega.

1237
00:58:16,180 --> 00:58:18,470
So I'm going to call a
the lowering operator.

1238
00:58:21,270 --> 00:58:24,890
Because what it does is it
takes a state with phi E,

1239
00:58:24,890 --> 00:58:26,530
with energy
eigenvalue E to state

1240
00:58:26,530 --> 00:58:28,730
with energy E minus h bar omega.

1241
00:58:28,730 --> 00:58:30,980
And I can just keep doing
this as many times as I like

1242
00:58:30,980 --> 00:58:31,813
and I build a tower.

1243
00:58:31,813 --> 00:58:32,600
Yes?

1244
00:58:32,600 --> 00:58:34,330
AUDIENCE: [INAUDIBLE]

1245
00:58:34,330 --> 00:58:35,580
PROFESSOR: Very good question.

1246
00:58:35,580 --> 00:58:37,832
Hold on to that for a second.

1247
00:58:37,832 --> 00:58:39,540
We'll come back to
that in just a second.

1248
00:58:42,920 --> 00:58:46,500
So this seems to build
for me a ladder downwards.

1249
00:58:46,500 --> 00:58:47,684
Everyone cool with that?

1250
00:58:47,684 --> 00:58:49,850
But we could have done the
same thing with a dagger.

1251
00:58:49,850 --> 00:58:51,141
And how does this story change?

1252
00:58:51,141 --> 00:58:53,240
What happens if we take
a dagger instead of a?

1253
00:58:53,240 --> 00:58:54,865
Well, let's go through
every step here.

1254
00:58:54,865 --> 00:58:57,290
So this is going to
be E on a dagger.

1255
00:58:57,290 --> 00:59:01,610
And now we have E a dagger,
a dagger, E, a dagger.

1256
00:59:01,610 --> 00:59:02,910
What's E with a dagger?

1257
00:59:08,780 --> 00:59:11,990
E with a dagger is equal to
same thing but with a plus.

1258
00:59:14,810 --> 00:59:15,840
And again, psi.

1259
00:59:15,840 --> 00:59:20,020
Same thing, because the
a dagger factors out.

1260
00:59:20,020 --> 00:59:22,050
Yeah?

1261
00:59:22,050 --> 00:59:28,600
So we go down by acting with a.

1262
00:59:28,600 --> 00:59:30,500
We go up by acting
with a dagger.

1263
00:59:33,760 --> 00:59:36,510
And again, the spacing
is h bar omega.

1264
00:59:36,510 --> 00:59:38,520
And we go up by acting
with a dagger again.

1265
00:59:46,560 --> 00:59:53,560
So a and a dagger are called the
raising and lowering operators.

1266
00:59:58,770 --> 01:00:01,650
a dagger, the raising operator.

1267
01:00:12,670 --> 01:00:17,563
a dagger phi E plus h bar omega.

1268
01:00:24,660 --> 01:00:31,990
So what that lets us do is
build a tower of states,

1269
01:00:31,990 --> 01:00:34,360
an infinite number of
states where, given a state,

1270
01:00:34,360 --> 01:00:37,400
we can walk up this ladder
with the raising operator,

1271
01:00:37,400 --> 01:00:41,450
and we can walk down it
by the lowering operator.

1272
01:00:41,450 --> 01:00:43,780
So now I ask you
the question, why

1273
01:00:43,780 --> 01:00:45,865
is this ladder evenly spaced?

1274
01:00:49,630 --> 01:00:52,089
There's one equation on the
board that you can point to-- I

1275
01:00:52,089 --> 01:00:54,088
guess two, technically--
there are two equations

1276
01:00:54,088 --> 01:00:55,860
on the board that you
could point to that

1277
01:00:55,860 --> 01:00:58,100
suffice to immediately
answer the question,

1278
01:00:58,100 --> 01:01:03,944
why is the tower of energy
eigenstates evenly spaced.

1279
01:01:03,944 --> 01:01:04,860
What is that equation?

1280
01:01:07,005 --> 01:01:07,880
AUDIENCE: [INAUDIBLE]

1281
01:01:07,880 --> 01:01:10,260
PROFESSOR: Yeah,
those commutators.

1282
01:01:10,260 --> 01:01:11,969
These commutators
are all we needed.

1283
01:01:11,969 --> 01:01:13,510
We didn't need to
know anything else.

1284
01:01:13,510 --> 01:01:17,120
We didn't even need to know
what the potential was.

1285
01:01:17,120 --> 01:01:19,200
If I just told you there's
an energy operator E

1286
01:01:19,200 --> 01:01:21,200
and there's an operator
a that you can build out

1287
01:01:21,200 --> 01:01:22,700
of the observables
of the system, such

1288
01:01:22,700 --> 01:01:24,366
that you have this
commutation relation,

1289
01:01:24,366 --> 01:01:26,610
what do you immediately know?

1290
01:01:26,610 --> 01:01:29,360
You immediately know that
you get a tower of operators.

1291
01:01:29,360 --> 01:01:31,630
Because you can act with
a and raise the energy

1292
01:01:31,630 --> 01:01:33,494
by a finite amount,
which is the coefficient

1293
01:01:33,494 --> 01:01:34,660
of that a in the commutator.

1294
01:01:38,010 --> 01:01:40,350
This didn't have to be
the quantum mechanics

1295
01:01:40,350 --> 01:01:42,220
of the harmonic
oscillator at this point.

1296
01:01:42,220 --> 01:01:45,390
We just needed this commutator
relation, E with a, E

1297
01:01:45,390 --> 01:01:47,539
with a dagger.

1298
01:01:47,539 --> 01:01:49,080
And one of the
totally awesome things

1299
01:01:49,080 --> 01:01:50,149
is how often it shows up.

1300
01:01:50,149 --> 01:01:52,190
If you take a bunch of
electrons and you put them

1301
01:01:52,190 --> 01:01:55,290
in a magnetic field,
bunch of electrons,

1302
01:01:55,290 --> 01:01:57,767
very strong magnetic
field, what you discover

1303
01:01:57,767 --> 01:01:59,350
is the quantum
mechanics of those guys

1304
01:01:59,350 --> 01:02:01,083
has nothing to do with the
harmonic oscillator on the face

1305
01:02:01,083 --> 01:02:04,160
if it's magnetic fields, Lorentz
force law, the whole thing.

1306
01:02:04,160 --> 01:02:07,275
What you discover is
there's an operator, which

1307
01:02:07,275 --> 01:02:09,455
isn't usually called a, but
it depends on which book

1308
01:02:09,455 --> 01:02:11,440
you use-- it's n
or m or l-- there's

1309
01:02:11,440 --> 01:02:13,180
an operator that
commutes with the energy

1310
01:02:13,180 --> 01:02:16,180
operator in precisely this
fashion, which tells you

1311
01:02:16,180 --> 01:02:20,070
that the energy eigenstates
live in a ladder.

1312
01:02:20,070 --> 01:02:21,926
They're called Landau levels.

1313
01:02:21,926 --> 01:02:23,300
This turns out to
be very useful.

1314
01:02:23,300 --> 01:02:24,883
Any of you who are
doing a [INAUDIBLE]

1315
01:02:24,883 --> 01:02:27,970
in the lab that has
graphene or any material,

1316
01:02:27,970 --> 01:02:30,620
really, with a magnetic
field, then this matters.

1317
01:02:30,620 --> 01:02:33,800
So this commutator
encodes an enormous amount

1318
01:02:33,800 --> 01:02:36,420
of the structure of
the energy eigenvalues.

1319
01:02:36,420 --> 01:02:38,460
And the trick for us
was showing that we

1320
01:02:38,460 --> 01:02:40,210
could write the harmonic
oscillator energy

1321
01:02:40,210 --> 01:02:43,200
operator in terms of operators
that commute in this fashion.

1322
01:02:46,040 --> 01:02:49,550
So we're going to run into this
structure over and over again.

1323
01:02:49,550 --> 01:02:51,070
This operator
commutes with this one

1324
01:02:51,070 --> 01:02:53,120
to the same operator times
a constant that tells you

1325
01:02:53,120 --> 01:02:53,770
have a ladder.

1326
01:02:53,770 --> 01:02:55,020
We're going to run
into that over and over

1327
01:02:55,020 --> 01:02:57,030
again when we talk about
Landau levels, if we get there.

1328
01:02:57,030 --> 01:02:58,260
When we talk about
angular momentum

1329
01:02:58,260 --> 01:02:59,390
we'll get the same thing.

1330
01:02:59,390 --> 01:03:01,139
When we talk about the
harmonic oscillator

1331
01:03:01,139 --> 01:03:02,700
we'll get the same thing.

1332
01:03:02,700 --> 01:03:04,960
Sorry, the hydrogen system.

1333
01:03:04,960 --> 01:03:07,680
We'll get the same thing.

1334
01:03:07,680 --> 01:03:15,285
So second question, does this
ladder extend infinitely up?

1335
01:03:15,285 --> 01:03:16,430
Yeah, why not?

1336
01:03:16,430 --> 01:03:17,785
Can it extend infinitely down?

1337
01:03:17,785 --> 01:03:18,410
AUDIENCE: Nope.

1338
01:03:18,410 --> 01:03:20,492
PROFESSOR: Why?

1339
01:03:20,492 --> 01:03:21,450
AUDIENCE: Ground state.

1340
01:03:21,450 --> 01:03:22,850
PROFESSOR: Well, people
are saying ground state.

1341
01:03:22,850 --> 01:03:24,370
Well, we know that from the
brute force calculation.

1342
01:03:24,370 --> 01:03:26,036
But without the brute
force calculation,

1343
01:03:26,036 --> 01:03:27,953
can this ladder extend
infinitely down?

1344
01:03:27,953 --> 01:03:30,120
AUDIENCE: [INAUDIBLE]
you can't go [INAUDIBLE].

1345
01:03:30,120 --> 01:03:30,480
PROFESSOR: Brilliant.

1346
01:03:30,480 --> 01:03:30,980
OK, good.

1347
01:03:30,980 --> 01:03:32,140
And as you'll prove
on the problems,

1348
01:03:32,140 --> 01:03:34,306
that you can't make the
energy arbitrarily negative.

1349
01:03:34,306 --> 01:03:36,990
But let me make that sharp.

1350
01:03:36,990 --> 01:03:40,930
I don't want to appeal to
something we haven't proven.

1351
01:03:40,930 --> 01:03:42,695
Let me show you that concretely.

1352
01:03:48,910 --> 01:03:52,370
In some state, in any state,
the energy expectation value

1353
01:03:52,370 --> 01:03:54,492
can be written as the
integral of phi complex

1354
01:03:54,492 --> 01:03:56,200
conjugate-- we'll say
in this state phi--

1355
01:03:56,200 --> 01:03:58,110
phi complex conjugate E phi.

1356
01:04:01,570 --> 01:04:03,510
But I can write this
as the integral,

1357
01:04:03,510 --> 01:04:07,460
and let's say dx, integral dx.

1358
01:04:07,460 --> 01:04:10,160
Let's just put in what the
energy operator looks like.

1359
01:04:10,160 --> 01:04:12,235
So psi tilda, we can
take the 4a transfer

1360
01:04:12,235 --> 01:04:16,145
and write the psi tilda p,
p squared upon 2m-- whoops,

1361
01:04:16,145 --> 01:04:19,939
dp-- for the kinetic energy
term, plus the integral--

1362
01:04:19,939 --> 01:04:21,730
and now I'm using the
harmonic oscillator--

1363
01:04:21,730 --> 01:04:29,640
plus the integral dx of psi of
x norm squared, norm squared,

1364
01:04:29,640 --> 01:04:32,030
m omega squared upon 2x squared.

1365
01:04:34,960 --> 01:04:36,409
Little bit of a
quick move there,

1366
01:04:36,409 --> 01:04:38,200
doing the 4a transfer
for the momentum term

1367
01:04:38,200 --> 01:04:39,610
and not doing the 4a
[INAUDIBLE] but it's OK.

1368
01:04:39,610 --> 01:04:40,735
They're separate integrals.

1369
01:04:40,735 --> 01:04:41,450
I can do this.

1370
01:04:41,450 --> 01:04:43,870
And the crucial thing here
is, this is positive definite.

1371
01:04:43,870 --> 01:04:45,850
This is positive definite,
positive definite,

1372
01:04:45,850 --> 01:04:46,800
positive definite.

1373
01:04:46,800 --> 01:04:48,410
All these terms are
strictly positive.

1374
01:04:48,410 --> 01:04:50,076
This must be greater
than or equal to 0.

1375
01:04:50,076 --> 01:04:52,240
It can never be negative.

1376
01:04:52,240 --> 01:04:54,890
Yeah?

1377
01:04:54,890 --> 01:04:56,670
So what that tells
us is there must

1378
01:04:56,670 --> 01:05:02,950
be a minimum E. There
must be a minimum energy.

1379
01:05:02,950 --> 01:05:04,360
And I will call it minimum E0.

1380
01:05:07,026 --> 01:05:08,400
We can't lower
the tower forever.

1381
01:05:15,030 --> 01:05:16,250
So how is this possible?

1382
01:05:16,250 --> 01:05:18,412
How is it possible that,
look, on the one hand,

1383
01:05:18,412 --> 01:05:20,870
if we want, if we have a state,
we can always build a lower

1384
01:05:20,870 --> 01:05:24,450
energy state by acting
with lowering operator a.

1385
01:05:24,450 --> 01:05:26,140
And yet this is telling
me that I can't.

1386
01:05:26,140 --> 01:05:29,030
There must be a last one where
I can't lower it anymore.

1387
01:05:29,030 --> 01:05:30,870
So what reaches out
of the chalkboard

1388
01:05:30,870 --> 01:05:34,170
and stops me from
acting with a again?

1389
01:05:34,170 --> 01:05:36,410
How can it possibly be
true that a always lowers

1390
01:05:36,410 --> 01:05:38,670
the eigenfunction but
there's at least one

1391
01:05:38,670 --> 01:05:41,580
that can't be
lowered any further.

1392
01:05:41,580 --> 01:05:43,070
Normalizable's a good guess.

1393
01:05:43,070 --> 01:05:43,860
Very good guess.

1394
01:05:43,860 --> 01:05:45,840
Not the case.

1395
01:05:45,840 --> 01:05:49,615
Because from this argument we
don't even use wave functions.

1396
01:05:49,615 --> 01:05:50,490
AUDIENCE: [INAUDIBLE]

1397
01:05:55,720 --> 01:05:56,930
PROFESSOR: That would be bad.

1398
01:05:56,930 --> 01:05:57,670
Yes, exactly.

1399
01:05:57,670 --> 01:05:59,570
So that would be
bad, but that's just

1400
01:05:59,570 --> 01:06:01,502
saying that there's
an inconsistency here.

1401
01:06:01,502 --> 01:06:03,210
So I'm going to come
back to your answer,

1402
01:06:03,210 --> 01:06:04,250
a non-normalizable.

1403
01:06:04,250 --> 01:06:06,620
It's correct, but
in a sneaky way.

1404
01:06:06,620 --> 01:06:09,950
Here's the way it's sneaky.

1405
01:06:09,950 --> 01:06:13,420
Consider a state
a on phi-- let's

1406
01:06:13,420 --> 01:06:16,530
say this is the lowest state,
the lowest possible state.

1407
01:06:16,530 --> 01:06:18,930
It must be true that the
resulting statement is not

1408
01:06:18,930 --> 01:06:20,450
phi minus h bar omega.

1409
01:06:20,450 --> 01:06:22,947
There can't be any such state.

1410
01:06:22,947 --> 01:06:23,780
And how can that be?

1411
01:06:23,780 --> 01:06:26,370
That can be true if it's 0.

1412
01:06:26,370 --> 01:06:30,660
So if the lowering operator acts
on some state and gives me 0,

1413
01:06:30,660 --> 01:06:33,700
well, OK, that's an eigenstate.

1414
01:06:33,700 --> 01:06:35,070
But it's a stupid eigenstate.

1415
01:06:35,070 --> 01:06:36,520
It's not normalizable.

1416
01:06:36,520 --> 01:06:39,287
It can't be used to describe
any real physical object.

1417
01:06:39,287 --> 01:06:40,120
Because where is it?

1418
01:06:40,120 --> 01:06:40,660
Well, it's nowhere.

1419
01:06:40,660 --> 01:06:42,659
The probability density,
you'd find it anywhere.

1420
01:06:42,659 --> 01:06:45,400
It's nowhere, nothing, zero.

1421
01:06:45,400 --> 01:06:48,540
So the way that this
tower terminates

1422
01:06:48,540 --> 01:06:50,850
is by having a last
state, which we'll

1423
01:06:50,850 --> 01:06:55,135
call phi 0, such that
lowering it gives me 0.

1424
01:06:58,110 --> 01:07:00,127
Not the state called 0,
which I would call this,

1425
01:07:00,127 --> 01:07:02,710
but actually the function called
0, which is not normalizable,

1426
01:07:02,710 --> 01:07:05,390
which is not a good state.

1427
01:07:05,390 --> 01:07:06,590
So there's a minimum E0.

1428
01:07:06,590 --> 01:07:11,470
Associated with that is a
lowest energy eigenstate

1429
01:07:11,470 --> 01:07:14,670
called the ground state.

1430
01:07:14,670 --> 01:07:18,930
Now, can the energy
get arbitrarily large?

1431
01:07:18,930 --> 01:07:19,430
Sure.

1432
01:07:19,430 --> 01:07:20,660
That's a positive
definite thing,

1433
01:07:20,660 --> 01:07:22,220
and this could get
as large as you like.

1434
01:07:22,220 --> 01:07:23,910
There's no problem with the
energy eigenvalues getting

1435
01:07:23,910 --> 01:07:24,500
arbitrarily large.

1436
01:07:24,500 --> 01:07:26,541
We can just keep raising
and raising and raising.

1437
01:07:26,541 --> 01:07:28,240
I mention that
because later on in

1438
01:07:28,240 --> 01:07:30,090
the semester we
will find a system

1439
01:07:30,090 --> 01:07:32,617
with exactly that commutation
relation, precisely

1440
01:07:32,617 --> 01:07:34,200
that commutation
relation, where there

1441
01:07:34,200 --> 01:07:38,044
will be a minimum and a maximum.

1442
01:07:38,044 --> 01:07:39,960
So the communication
relation is a good start,

1443
01:07:39,960 --> 01:07:40,860
but it doesn't
tell you anything.

1444
01:07:40,860 --> 01:07:42,990
We have to add in some
physics like the energy

1445
01:07:42,990 --> 01:07:46,100
operators bounded below for
the harmonic oscillator.

1446
01:07:49,190 --> 01:07:50,572
Questions at this point?

1447
01:07:50,572 --> 01:07:51,880
Yeah?

1448
01:07:51,880 --> 01:07:54,700
AUDIENCE: So you basically
[INAUDIBLE] this ladder

1449
01:07:54,700 --> 01:07:57,206
has to [INAUDIBLE] my
particular energy eigenstate

1450
01:07:57,206 --> 01:07:58,747
and I can kind of
construct a ladder.

1451
01:07:58,747 --> 01:08:00,246
How do I know that
I can't construct

1452
01:08:00,246 --> 01:08:01,693
other, intersecting ladders?

1453
01:08:01,693 --> 01:08:04,150
PROFESSOR: Yeah, that's
an excellent question.

1454
01:08:04,150 --> 01:08:08,790
I remember vividly when I
saw this lecture in 143A,

1455
01:08:08,790 --> 01:08:10,290
and that question plagued me.

1456
01:08:10,290 --> 01:08:11,992
And foolishly I didn't ask it.

1457
01:08:11,992 --> 01:08:12,950
So here's the question.

1458
01:08:12,950 --> 01:08:15,075
The question is, look, you
found a bunch of states.

1459
01:08:15,075 --> 01:08:16,622
How do you know
that's all of them?

1460
01:08:16,622 --> 01:08:18,080
How do you know
that's all of them?

1461
01:08:18,080 --> 01:08:19,140
So let's think through that.

1462
01:08:19,140 --> 01:08:20,040
That's a very good question.

1463
01:08:20,040 --> 01:08:21,831
I'm not going to worry
about normalization.

1464
01:08:21,831 --> 01:08:25,054
There's a discussion of
normalization in the notes.

1465
01:08:25,054 --> 01:08:26,470
How do we know
that's all of them?

1466
01:08:29,344 --> 01:08:30,540
That's a little bit tricky.

1467
01:08:30,540 --> 01:08:32,960
So let's think through it.

1468
01:08:32,960 --> 01:08:35,350
Imagine it's not all of them.

1469
01:08:35,350 --> 01:08:37,120
In particular, what
would that mean?

1470
01:08:37,120 --> 01:08:39,359
In order for there to be
more states than the ones

1471
01:08:39,359 --> 01:08:41,170
that we've written
down, there must

1472
01:08:41,170 --> 01:08:43,653
be states that are
not on that tower.

1473
01:08:43,653 --> 01:08:46,319
And how can we possi-- wow, this
thing is totally falling apart.

1474
01:08:46,319 --> 01:08:47,160
How do we do that?

1475
01:08:47,160 --> 01:08:49,479
How is that possible?

1476
01:08:49,479 --> 01:08:52,040
There are two ways to do it.

1477
01:08:52,040 --> 01:08:53,215
Here's my tower of states.

1478
01:08:56,109 --> 01:08:59,529
I'll call this one phi 0.

1479
01:08:59,529 --> 01:09:05,576
so I raise with a dagger
and I lower with a.

1480
01:09:05,576 --> 01:09:07,450
So how could it be that
I missed some states?

1481
01:09:07,450 --> 01:09:08,700
Well, there are ways to do it.

1482
01:09:08,700 --> 01:09:11,910
One is there could be extra
states that are in between.

1483
01:09:11,910 --> 01:09:13,910
So let's say that there's
one extra state that's

1484
01:09:13,910 --> 01:09:15,779
in between these two.

1485
01:09:15,779 --> 01:09:17,430
Just imagine that's true.

1486
01:09:17,430 --> 01:09:21,540
If there is such a state,
by that commutation relation

1487
01:09:21,540 --> 01:09:24,124
there must be another tower.

1488
01:09:24,124 --> 01:09:26,540
So there must be this state,
and there must be this state,

1489
01:09:26,540 --> 01:09:29,319
and there must be this state,
and there must be this state.

1490
01:09:29,319 --> 01:09:30,840
Yeah?

1491
01:09:30,840 --> 01:09:32,260
OK, so that's good so far.

1492
01:09:32,260 --> 01:09:33,240
But what happens?

1493
01:09:33,240 --> 01:09:38,000
Well, A on this guy gave me 0.

1494
01:09:38,000 --> 01:09:41,649
And this is going to
be some phi tilde 0.

1495
01:09:41,649 --> 01:09:44,590
Suppose that this tower ends.

1496
01:09:44,590 --> 01:09:46,426
And now you have to
ask the question,

1497
01:09:46,426 --> 01:09:47,800
can there be two
different states

1498
01:09:47,800 --> 01:09:50,734
with two different
energies with a0?

1499
01:09:50,734 --> 01:09:52,109
Can there be two
different states

1500
01:09:52,109 --> 01:09:54,610
that are annihilated by a0?

1501
01:09:54,610 --> 01:09:56,290
Well, let's check.

1502
01:09:56,290 --> 01:09:59,130
What must be true of any
state annihilated by a0?

1503
01:09:59,130 --> 01:10:02,420
Well, let's write the energy
operator acting on that state.

1504
01:10:02,420 --> 01:10:04,040
What's the energy of that state?

1505
01:10:04,040 --> 01:10:06,268
Energy on phi 0 is
equal to h bar omega.

1506
01:10:06,268 --> 01:10:08,476
This is a very good question,
so let's go through it.

1507
01:10:08,476 --> 01:10:10,920
So it's equal to h bar
omega times a dagger

1508
01:10:10,920 --> 01:10:15,967
a plus 1/2 on phi 0.

1509
01:10:15,967 --> 01:10:17,300
But what can you say about this?

1510
01:10:17,300 --> 01:10:20,220
Well, a annihilates phi 0.

1511
01:10:20,220 --> 01:10:21,814
It gives us 0.

1512
01:10:21,814 --> 01:10:24,480
So in addition to a being called
the lowering operator it's also

1513
01:10:24,480 --> 01:10:26,730
called the
annihilation operator,

1514
01:10:26,730 --> 01:10:30,320
because, I don't know, we're
a brutal and warlike species.

1515
01:10:30,320 --> 01:10:32,950
So this is equal to h bar
omega-- this term kills phi

1516
01:10:32,950 --> 01:10:36,440
0-- again with the kills-- and
gives me a 1/2 half leftover.

1517
01:10:36,440 --> 01:10:39,650
1/2 h bar omega phi 0.

1518
01:10:39,650 --> 01:10:46,080
So the ground state, any state--
any state annihilated by a

1519
01:10:46,080 --> 01:10:49,640
must have the same energy.

1520
01:10:49,640 --> 01:10:51,633
The only way you can
be annihilated by a

1521
01:10:51,633 --> 01:10:53,750
is if your energy is this.

1522
01:10:53,750 --> 01:10:54,590
Cool?

1523
01:10:54,590 --> 01:10:56,673
So what does that tell you
about the second ladder

1524
01:10:56,673 --> 01:10:58,290
of hidden seats that we missed?

1525
01:10:58,290 --> 01:10:59,642
It's got to be degenerate.

1526
01:10:59,642 --> 01:11:01,100
It's got to have
the same energies.

1527
01:11:08,330 --> 01:11:09,850
I drew that really
badly, didn't I?

1528
01:11:09,850 --> 01:11:12,402
Those are evenly spaced.

1529
01:11:12,402 --> 01:11:13,610
So it's got to be degenerate.

1530
01:11:13,610 --> 01:11:16,450
However, Barton proved for you
the node theorem last time,

1531
01:11:16,450 --> 01:11:16,950
right?

1532
01:11:16,950 --> 01:11:20,554
He gave you my spread
argument for the node theorem?

1533
01:11:20,554 --> 01:11:22,470
In particular, one of
the consequences of that

1534
01:11:22,470 --> 01:11:25,125
is it in a system
with bound states,

1535
01:11:25,125 --> 01:11:28,790
in a system with
potential that goes up,

1536
01:11:28,790 --> 01:11:32,420
you can never have
degeneracies in one dimension.

1537
01:11:32,420 --> 01:11:34,520
We're not going to prove
that carefully in here.

1538
01:11:34,520 --> 01:11:35,945
But it's relatively
easy to prove.

1539
01:11:35,945 --> 01:11:37,570
In fact, if you come
to my office hours

1540
01:11:37,570 --> 01:11:37,935
I'll prove it for you.

1541
01:11:37,935 --> 01:11:38,680
It takes three minutes.

1542
01:11:38,680 --> 01:11:40,150
But I don't want to set
up the math right now.

1543
01:11:40,150 --> 01:11:41,990
So how many people know
about the Wronskian?

1544
01:11:44,870 --> 01:11:45,630
That's awesome.

1545
01:11:45,630 --> 01:11:48,640
OK, so I leave it to you as an
exercise to use the Wronskian

1546
01:11:48,640 --> 01:11:52,070
to show that there cannot be
degeneracies in one dimension,

1547
01:11:52,070 --> 01:11:52,960
which is cool.

1548
01:11:52,960 --> 01:11:55,340
Anyway, so the Wronskian for
the differential equation,

1549
01:11:55,340 --> 01:11:58,100
which is the energy
eigenvalue equation.

1550
01:11:58,100 --> 01:12:01,610
There can be no degeneracies
in one dimensional potentials

1551
01:12:01,610 --> 01:12:03,297
with bound states.

1552
01:12:03,297 --> 01:12:05,630
So what we've just shown is
that the only way that there

1553
01:12:05,630 --> 01:12:07,046
can be extra states
that we missed

1554
01:12:07,046 --> 01:12:10,230
is if there's a tower with
exactly identical energies all

1555
01:12:10,230 --> 01:12:11,040
the way up.

1556
01:12:11,040 --> 01:12:13,527
But if they have exactly
identical energies,

1557
01:12:13,527 --> 01:12:14,860
that means there's a degenerate.

1558
01:12:14,860 --> 01:12:18,350
But we can prove that there
can't be degeneracies in 1D.

1559
01:12:18,350 --> 01:12:21,130
So can there be an extra
tower of states we missed?

1560
01:12:21,130 --> 01:12:21,630
No.

1561
01:12:21,630 --> 01:12:24,390
Can we have missed any states?

1562
01:12:24,390 --> 01:12:25,180
No.

1563
01:12:25,180 --> 01:12:26,850
Those are all the
states there are.

1564
01:12:26,850 --> 01:12:29,210
And we've done it without
ever solving a differential

1565
01:12:29,210 --> 01:12:32,297
equation, just by using
that commutation relation.

1566
01:12:32,297 --> 01:12:34,130
Now at this point it's
very tempting to say,

1567
01:12:34,130 --> 01:12:36,140
that was just sort of
magical mystery stuff.

1568
01:12:36,140 --> 01:12:38,610
But what we really did
last time was very honest.

1569
01:12:38,610 --> 01:12:39,540
We wrote down a
differential equation.

1570
01:12:39,540 --> 01:12:40,510
We found the solution.

1571
01:12:40,510 --> 01:12:42,330
And we got the wave functions.

1572
01:12:42,330 --> 01:12:44,342
So, Professor Adams,
you just monkeyed around

1573
01:12:44,342 --> 01:12:46,300
at the chalkboard with
commutators for a while,

1574
01:12:46,300 --> 01:12:47,930
but what are the
damn wave functions?

1575
01:12:47,930 --> 01:12:48,430
Right?

1576
01:12:51,190 --> 01:12:52,770
We already have the answer.

1577
01:12:52,770 --> 01:12:54,870
This is really quite nice.

1578
01:12:54,870 --> 01:12:57,470
Last time we solved that
differential equation.

1579
01:12:57,470 --> 01:12:59,386
And we had to solve that
differential equation

1580
01:12:59,386 --> 01:13:02,470
many, many times,
different levels.

1581
01:13:02,470 --> 01:13:04,550
But now we have a very
nice thing we can do.

1582
01:13:04,550 --> 01:13:06,849
What's true of the ground state?

1583
01:13:06,849 --> 01:13:08,390
Well, the ground
state is annihilated

1584
01:13:08,390 --> 01:13:10,200
by the lowering operator.

1585
01:13:10,200 --> 01:13:15,400
So that means that a acting
on phi 0 of x is equal to 0.

1586
01:13:15,400 --> 01:13:19,170
But a has a nice expression,
which unfortunately I erased.

1587
01:13:19,170 --> 01:13:19,920
Sorry about that.

1588
01:13:19,920 --> 01:13:21,750
So a has a nice expression.

1589
01:13:21,750 --> 01:13:27,670
a is equal to x over
x0 plus ip over p0.

1590
01:13:27,670 --> 01:13:30,510
And so if you write
that out and multiply

1591
01:13:30,510 --> 01:13:33,640
from appropriate
constants, this becomes

1592
01:13:33,640 --> 01:13:35,140
the following
differential equation.

1593
01:13:35,140 --> 01:13:36,589
The x is just multiplied by x.

1594
01:13:36,589 --> 01:13:38,380
And the p is take a
derivative with respect

1595
01:13:38,380 --> 01:13:40,837
to x, multiply by h bar upon i.

1596
01:13:40,837 --> 01:13:42,420
And multiplying by
i over h bar to get

1597
01:13:42,420 --> 01:13:52,612
that equation, this gives us dx
plus p over h bar x0-- sorry,

1598
01:13:52,612 --> 01:13:53,570
that shouldn't be an i.

1599
01:13:53,570 --> 01:13:57,710
That should be p0.

1600
01:13:57,710 --> 01:14:00,550
x on phi 0 is equal to 0.

1601
01:14:03,780 --> 01:14:05,710
And you solved
this last time when

1602
01:14:05,710 --> 01:14:08,420
you did the asymptotic analysis.

1603
01:14:08,420 --> 01:14:10,770
This is actually a
ridiculously easy equation.

1604
01:14:10,770 --> 01:14:13,820
It's a first order
differential equation.

1605
01:14:13,820 --> 01:14:15,385
There's one
integration constant.

1606
01:14:15,385 --> 01:14:17,260
That's going to be the
overall normalization.

1607
01:14:17,260 --> 01:14:19,180
And so the form is
completely fixed.

1608
01:14:19,180 --> 01:14:20,596
First order
differential equation.

1609
01:14:20,596 --> 01:14:22,260
So what's the
solution of this guy?

1610
01:14:22,260 --> 01:14:23,030
It's a Gaussian.

1611
01:14:23,030 --> 01:14:24,613
And what's the width
of that Gaussian?

1612
01:14:24,613 --> 01:14:28,000
Well, look at p0 over h bar x0.

1613
01:14:28,000 --> 01:14:31,660
We know that p0 times
x0 is twice h bar.

1614
01:14:31,660 --> 01:14:33,960
So if I multiply by x amount
on the top and bottom,

1615
01:14:33,960 --> 01:14:35,520
you get 2 h bar.

1616
01:14:35,520 --> 01:14:37,060
The h bars cancel.

1617
01:14:37,060 --> 01:14:40,904
So this gives me
two upon x0 squared.

1618
01:14:40,904 --> 01:14:42,820
Remember I said it would
be useful to remember

1619
01:14:42,820 --> 01:14:45,310
that p0 times x0 is 2h bar?

1620
01:14:45,310 --> 01:14:46,870
It's useful.

1621
01:14:46,870 --> 01:14:48,990
So it gives us this.

1622
01:14:48,990 --> 01:14:51,180
And so the result is
that phi 0 is equal to,

1623
01:14:51,180 --> 01:14:55,280
up to an overall normalization
coefficient, e to the minus

1624
01:14:55,280 --> 01:14:58,980
x squared over x0 squared.

1625
01:14:58,980 --> 01:15:01,081
Solid.

1626
01:15:01,081 --> 01:15:01,580
So there.

1627
01:15:01,580 --> 01:15:02,640
We've solved that
differential equation.

1628
01:15:02,640 --> 01:15:04,780
Is the easiest, second
easiest differential equation.

1629
01:15:04,780 --> 01:15:05,780
It's our first order
differential equation

1630
01:15:05,780 --> 01:15:07,820
with a linear term
rather than a constant.

1631
01:15:07,820 --> 01:15:08,720
We get a Gaussian.

1632
01:15:08,720 --> 01:15:10,950
And now that we've
got this guy-- look,

1633
01:15:10,950 --> 01:15:12,990
do you remember the
third Hermite polynomial?

1634
01:15:12,990 --> 01:15:15,520
Because we know the
third excited state

1635
01:15:15,520 --> 01:15:19,090
is given by h3
times this Gaussian.

1636
01:15:19,090 --> 01:15:21,290
Do you remember it off
the top of your head?

1637
01:15:21,290 --> 01:15:23,110
How do you solve what it is?

1638
01:15:23,110 --> 01:15:24,940
How do we get phi 3?

1639
01:15:24,940 --> 01:15:26,539
First off, how do we get phi 1?

1640
01:15:26,539 --> 01:15:28,330
How do we get the next
state in the ladder?

1641
01:15:28,330 --> 01:15:30,042
How do we get the wave function?

1642
01:15:30,042 --> 01:15:30,750
Raising operator.

1643
01:15:30,750 --> 01:15:32,240
But what is the
raising operator?

1644
01:15:32,240 --> 01:15:35,120
Oh, it's the differential
operator I take with-- OK,

1645
01:15:35,120 --> 01:15:38,230
but if I had a dagger, it's just
going to change the sign here.

1646
01:15:38,230 --> 01:15:40,590
So how do I get phi 1?

1647
01:15:40,590 --> 01:15:43,380
Phi 1 is equal to up
to some normalization.

1648
01:15:43,380 --> 01:15:50,180
dx minus 2 over x0
squared, x0, phi 0.

1649
01:15:52,365 --> 01:15:54,490
So now do I have to solve
the differential equation

1650
01:15:54,490 --> 01:15:56,601
to get the higher states?

1651
01:15:56,601 --> 01:15:57,100
No.

1652
01:15:57,100 --> 01:16:00,680
I take derivatives and
multiply by constants.

1653
01:16:00,680 --> 01:16:03,610
So to get the third Hermite
polynomial what do you do?

1654
01:16:03,610 --> 01:16:06,740
You do this three times.

1655
01:16:06,740 --> 01:16:08,640
This is actually an
extremely efficient way--

1656
01:16:08,640 --> 01:16:10,100
it's related to something
called the generating function,

1657
01:16:10,100 --> 01:16:11,400
and an extremely efficient
way to write down

1658
01:16:11,400 --> 01:16:12,752
the Hermite polynomials.

1659
01:16:12,752 --> 01:16:14,460
They're the things
that you get by acting

1660
01:16:14,460 --> 01:16:18,920
on this with this operator
as many times as you want.

1661
01:16:18,920 --> 01:16:21,880
That is a nice formal definition
of the Hermite polynomials.

1662
01:16:21,880 --> 01:16:24,610
The upshot of all of
this is the following.

1663
01:16:24,610 --> 01:16:32,160
The upshot of all this is that
we've derived that without ever

1664
01:16:32,160 --> 01:16:34,860
solving the differential
equation the spectrum just from

1665
01:16:34,860 --> 01:16:37,190
that commutation relation,
just from that commutation

1666
01:16:37,190 --> 01:16:39,030
relation-- I cannot emphasize
this strongly enough--

1667
01:16:39,030 --> 01:16:40,488
just from the
commutation relation,

1668
01:16:40,488 --> 01:16:42,270
Ea is minus a
times the constant,

1669
01:16:42,270 --> 01:16:44,610
and Ea dagger is a dagger
times the constant.

1670
01:16:44,610 --> 01:16:48,675
We derive that the energy
eigenstates come in a tower.

1671
01:16:48,675 --> 01:16:51,050
You can move along this tower
by raising with the raising

1672
01:16:51,050 --> 01:16:52,966
operator, lowering with
the lowering operator.

1673
01:16:52,966 --> 01:16:54,610
You can construct
the ground state

1674
01:16:54,610 --> 01:16:57,999
by building that simple
wave function, which

1675
01:16:57,999 --> 01:16:59,665
is annihilated by the
lowering operator.

1676
01:16:59,665 --> 01:17:01,210
You can build all
the other states

1677
01:17:01,210 --> 01:17:03,860
by raising them, which is
just taking derivatives

1678
01:17:03,860 --> 01:17:07,540
instead of solving differential
equations, which is hard.

1679
01:17:07,540 --> 01:17:10,340
And all of this came from
this commutation relation.

1680
01:17:10,340 --> 01:17:13,030
And since we are going to see
this over and over again--

1681
01:17:13,030 --> 01:17:15,787
and depending on how
far you take physics,

1682
01:17:15,787 --> 01:17:16,870
you will see this in 8.05.

1683
01:17:16,870 --> 01:17:17,780
You will see this in 8.06.

1684
01:17:17,780 --> 01:17:19,639
You will see this in
quantum field theory.

1685
01:17:19,639 --> 01:17:20,680
This shows up everywhere.

1686
01:17:20,680 --> 01:17:22,340
It's absolutely
at the core of how

1687
01:17:22,340 --> 01:17:25,130
we organize the
degrees of freedom.

1688
01:17:25,130 --> 01:17:29,220
This structure is something you
should see and declare victory

1689
01:17:29,220 --> 01:17:30,020
upon seeing.

1690
01:17:30,020 --> 01:17:32,420
Should see this and immediately
say, I know the answer,

1691
01:17:32,420 --> 01:17:34,520
and I can write it down.

1692
01:17:34,520 --> 01:17:36,310
OK?

1693
01:17:36,310 --> 01:17:39,604
In the next lecture we're
going to do a review which

1694
01:17:39,604 --> 01:17:42,270
is going to introduce a slightly
more formal presentation of all

1695
01:17:42,270 --> 01:17:43,214
these ideas.

1696
01:17:43,214 --> 01:17:45,380
That's not going to be
material covered on the exam,

1697
01:17:45,380 --> 01:17:47,338
but it's going to help
you with the exam, which

1698
01:17:47,338 --> 01:17:48,390
will be on Thursday.

1699
01:17:48,390 --> 01:17:50,210
See you Tuesday.