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PROFESSOR: Hi everyone.

9
00:00:29,146 --> 00:00:30,020
Spring has regressed.

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00:00:32,990 --> 00:00:35,920
So, we have--
we're going to have

11
00:00:35,920 --> 00:00:38,030
a guest at the end of
lecture today, which should

12
00:00:38,030 --> 00:00:39,210
kind of entertaining.

13
00:00:39,210 --> 00:00:42,960
Just as a warning, if
you see someone come in.

14
00:00:42,960 --> 00:00:44,460
So questions, before
we get started?

15
00:00:49,340 --> 00:00:51,870
No questions about anything?

16
00:00:51,870 --> 00:00:54,386
At all?

17
00:00:54,386 --> 00:00:56,751
Math?

18
00:00:56,751 --> 00:00:57,697
Nothing?

19
00:00:57,697 --> 00:00:58,197
Yeah?

20
00:00:58,197 --> 00:01:00,322
AUDIENCE: Can you explain
the physical significance

21
00:01:00,322 --> 00:01:01,481
of the crystal momentum?

22
00:01:01,481 --> 00:01:02,330
PROFESSOR: Yeah.

23
00:01:02,330 --> 00:01:02,510
OK.

24
00:01:02,510 --> 00:01:03,480
Let me go over that.

25
00:01:03,480 --> 00:01:03,920
That's a good question.

26
00:01:03,920 --> 00:01:06,830
So the question is what again is
the significance of the crystal

27
00:01:06,830 --> 00:01:07,330
momentum?

28
00:01:13,120 --> 00:01:16,651
So let me answer that in
a slightly backward way.

29
00:01:16,651 --> 00:01:18,150
So this is a form
of the explanation

30
00:01:18,150 --> 00:01:19,101
I haven't given you.

31
00:01:19,101 --> 00:01:20,850
It's going to be a
slightly different one.

32
00:01:20,850 --> 00:01:22,725
Let's step back and
think about the momentum,

33
00:01:22,725 --> 00:01:24,110
and ask what the momentum is.

34
00:01:24,110 --> 00:01:28,280
Now you guys showed on a
problem set, the following fact.

35
00:01:28,280 --> 00:01:32,870
That if you have a wave
function, sine of x, such that,

36
00:01:32,870 --> 00:01:42,180
the expectation value in the
state SI of x in the state SI

37
00:01:42,180 --> 00:01:45,800
is equal to x naught,
and the expectation value

38
00:01:45,800 --> 00:01:50,210
of p in the state
SI is p naught.

39
00:01:50,210 --> 00:01:52,930
Hat, hat.

40
00:01:52,930 --> 00:01:57,040
Then if you want to change the
momentum, increase momentum

41
00:01:57,040 --> 00:02:02,170
by h bar k, the way to do
that is to take SI and build

42
00:02:02,170 --> 00:02:04,370
a new wave function,
SI tilda, is

43
00:02:04,370 --> 00:02:11,140
equal to e to the
i, k x, SI of x.

44
00:02:11,140 --> 00:02:13,300
And then the
expectation value of x

45
00:02:13,300 --> 00:02:16,580
is the same, SI tilda,
still equal to x naught,

46
00:02:16,580 --> 00:02:20,610
because this phase goes
away from the two complex,

47
00:02:20,610 --> 00:02:22,350
from the wave function
is complex content

48
00:02:22,350 --> 00:02:23,740
we give the inner product.

49
00:02:23,740 --> 00:02:25,920
But the expectation
value, the momentum

50
00:02:25,920 --> 00:02:30,195
is shifted in state SI tilda,
is shifted by each h bar

51
00:02:30,195 --> 00:02:32,930
k, p naught plus h bar k.

52
00:02:32,930 --> 00:02:37,680
So all the intuition
you have about momentum,

53
00:02:37,680 --> 00:02:41,720
you can translate into intuition
about the spatial variation

54
00:02:41,720 --> 00:02:44,830
of the phase of
the wave function.

55
00:02:44,830 --> 00:02:46,509
Yeah?

56
00:02:46,509 --> 00:02:47,455
AUDIENCE: [INAUDIBLE]

57
00:02:47,455 --> 00:02:49,050
PROFESSOR: OK, good.

58
00:02:49,050 --> 00:02:52,310
OK, good, So we have
a sneaky [INAUDIBLE].

59
00:02:52,310 --> 00:02:54,940
So, the information
about the momentum

60
00:02:54,940 --> 00:02:57,600
can be encoded in
these spatial variation

61
00:02:57,600 --> 00:03:00,830
of the phase of
the wave function.

62
00:03:00,830 --> 00:03:04,650
So another way to answer the
question of what is momentum,

63
00:03:04,650 --> 00:03:06,410
apart from it's the
thing that-- so what

64
00:03:06,410 --> 00:03:08,520
are ways to answer the
question, what is momentum,

65
00:03:08,520 --> 00:03:10,260
you could ask well
what is momentum?

66
00:03:10,260 --> 00:03:14,940
It's the thing that commutes
with p or with x by i h bar.

67
00:03:14,940 --> 00:03:16,180
That's one way to answer it.

68
00:03:16,180 --> 00:03:19,170
Another way to answer is to
say that translations by l

69
00:03:19,170 --> 00:03:22,960
can be expressed in terms of
momentum as e to the minus i

70
00:03:22,960 --> 00:03:28,380
upon h bar p l.

71
00:03:31,080 --> 00:03:32,580
So these are both
ways of describing

72
00:03:32,580 --> 00:03:33,500
what the momentum is.

73
00:03:33,500 --> 00:03:35,374
But another way of
talking about the momentum

74
00:03:35,374 --> 00:03:42,435
is the momentum p governance
the spatial variation,

75
00:03:42,435 --> 00:03:45,630
the x dependence of the
phase of the wave function.

76
00:03:53,119 --> 00:03:55,410
So these are always talking
about what the momentum is.

77
00:03:55,410 --> 00:03:56,660
So now let's turn
this around, and let's

78
00:03:56,660 --> 00:03:58,320
ask about the crystal momentum.

79
00:03:58,320 --> 00:04:01,210
Oh, and one last thing,
a last defining property

80
00:04:01,210 --> 00:04:07,900
of the momentum,
a central property

81
00:04:07,900 --> 00:04:13,890
from the Schrodinger equation is
at the time variation d dt of p

82
00:04:13,890 --> 00:04:19,290
is equal to the expectation
value of minus d the potential

83
00:04:19,290 --> 00:04:21,750
of x d x.

84
00:04:21,750 --> 00:04:23,740
Also known as the force.

85
00:04:23,740 --> 00:04:27,810
So this is the Ehrenfest Theorem
Statement that the classical

86
00:04:27,810 --> 00:04:32,050
equation of motion, p dot, is
equal to the minus d v d x is

87
00:04:32,050 --> 00:04:34,360
equal to the force, Ehrenfest's

88
00:04:34,360 --> 00:04:37,370
Theorem tells us that the
classical equations of motion

89
00:04:37,370 --> 00:04:39,430
are realized as
expectation values.

90
00:04:39,430 --> 00:04:41,865
And equivantly, if
there's no potential,

91
00:04:41,865 --> 00:04:43,700
the potential is
constant, this tells us

92
00:04:43,700 --> 00:04:46,640
that the momentum expectation
value is time independent.

93
00:04:46,640 --> 00:04:47,380
Right?

94
00:04:47,380 --> 00:04:49,700
A familiar fact.

95
00:04:49,700 --> 00:04:54,510
So these are all true lovely
and things about the momentum.

96
00:04:54,510 --> 00:04:56,060
So let's turn all
these facts around

97
00:04:56,060 --> 00:04:57,143
into the crystal momentum.

98
00:04:57,143 --> 00:05:02,110
So let's talk about
crystal momentum.

99
00:05:02,110 --> 00:05:06,430
Which was the question, what
is the crystal momentum?

100
00:05:06,430 --> 00:05:08,644
So the crystal momentum
is defined from beginning,

101
00:05:08,644 --> 00:05:09,810
from the following property.

102
00:05:09,810 --> 00:05:11,770
If we have a potential
v of x, which

103
00:05:11,770 --> 00:05:16,225
is invariant under shifting, by
one lattice spacing, by some l,

104
00:05:16,225 --> 00:05:22,940
v of x, then this tells us
that the energy operator

105
00:05:22,940 --> 00:05:25,390
is invariant if we shift by l.

106
00:05:25,390 --> 00:05:28,500
If we translate
by l equals zero.

107
00:05:28,500 --> 00:05:32,966
And from this fact, we deduced
via block or a la block,

108
00:05:32,966 --> 00:05:34,840
that the wave functions
are really the energy

109
00:05:34,840 --> 00:05:38,850
eigenfunctions, can
be written in the form

110
00:05:38,850 --> 00:05:46,010
e cubed is equal to e to the
i q x, u of x, where u, we're

111
00:05:46,010 --> 00:05:47,720
going to take to be
a periodic function.

112
00:05:51,460 --> 00:05:55,260
So what is this
parameter q doing?

113
00:05:55,260 --> 00:05:58,160
Q is governing the
spatial variation

114
00:05:58,160 --> 00:06:00,950
of the phase of
the wave function.

115
00:06:00,950 --> 00:06:02,300
Cool?

116
00:06:02,300 --> 00:06:08,770
So in precisely this sense,
the momentum difference

117
00:06:08,770 --> 00:06:10,020
is space of the wave function.

118
00:06:10,020 --> 00:06:12,180
Here, in the case of
a periodic potential,

119
00:06:12,180 --> 00:06:15,940
the crystal momentum q is
governing the spatial variation

120
00:06:15,940 --> 00:06:19,270
of the phase of
the wave function.

121
00:06:19,270 --> 00:06:25,240
So q is the thing the governs
the phase as a function of x.

122
00:06:27,760 --> 00:06:33,560
Well what about-- another fact
about the crystal momentum

123
00:06:33,560 --> 00:06:35,560
which you show in
your problems set,

124
00:06:35,560 --> 00:06:42,470
is that if you impose an
external force d q d t,

125
00:06:42,470 --> 00:06:51,078
and really d h bar q.
d t is equal to-- d

126
00:06:51,078 --> 00:06:57,920
dt of the expectation
value of h bar q,

127
00:06:57,920 --> 00:07:03,300
is equal to the expectation
value of the force.

128
00:07:03,300 --> 00:07:05,870
I'll just write-- OK?

129
00:07:05,870 --> 00:07:10,350
So again, this is a
quantity, and this

130
00:07:10,350 --> 00:07:14,160
was assuming that we had a
sharply peaked wave packet.

131
00:07:14,160 --> 00:07:27,800
So this is for a wave packet
sharply peaked at q naught.

132
00:07:27,800 --> 00:07:32,320
And so let me just write
this as h bar q naught.

133
00:07:32,320 --> 00:07:34,120
So the central value
of your wave packet--

134
00:07:34,120 --> 00:07:35,250
so this is what you've
shown on the problem

135
00:07:35,250 --> 00:07:37,370
set that the central
value of your wave

136
00:07:37,370 --> 00:07:39,650
packet, the peak
of your wave packet

137
00:07:39,650 --> 00:07:41,889
varies in time according
to the external force.

138
00:07:41,889 --> 00:07:43,680
And so in particular,
if the force is zero,

139
00:07:43,680 --> 00:07:45,380
we turn no external
driving force,

140
00:07:45,380 --> 00:07:47,664
your wave packet maintains
its crystal momentum.

141
00:07:47,664 --> 00:07:48,580
It's time independent.

142
00:07:48,580 --> 00:07:49,670
So the crystal
momentum is something

143
00:07:49,670 --> 00:07:52,128
that time independent, unless
an external force is applied,

144
00:07:52,128 --> 00:07:53,352
just like the momentum.

145
00:07:53,352 --> 00:07:55,560
And it's something that
governs the phase of the wave

146
00:07:55,560 --> 00:07:58,040
function just like the momentum.

147
00:07:58,040 --> 00:08:00,030
However, it's different
in a crucial way.

148
00:08:00,030 --> 00:08:05,350
It is not the eigenvalue
p on five sub e q

149
00:08:05,350 --> 00:08:13,330
is not equal to a constant
p naught times 5 sub e q.

150
00:08:13,330 --> 00:08:15,302
Because when we take--
when we active p

151
00:08:15,302 --> 00:08:17,260
or we active the derivative,
you pick up a term

152
00:08:17,260 --> 00:08:18,801
from here, which
gives us a constant,

153
00:08:18,801 --> 00:08:21,310
but we also have this
overall periodic piece.

154
00:08:21,310 --> 00:08:24,872
And its spatial variation
is generically non-zero.

155
00:08:24,872 --> 00:08:26,330
And if the potential
is nontrivial,

156
00:08:26,330 --> 00:08:28,260
it's always non constant.

157
00:08:28,260 --> 00:08:31,330
So when the momentum
operator hits this guy,

158
00:08:31,330 --> 00:08:33,070
it will generically
not give us zero.

159
00:08:33,070 --> 00:08:35,700
It'll get two terms and we will
not get an eigenvalue equation.

160
00:08:35,700 --> 00:08:45,340
So q is not the eigenvalue h bar
q is not the eigenvalue of p.

161
00:08:45,340 --> 00:08:47,430
And what's the last
important property

162
00:08:47,430 --> 00:08:49,240
of q that's different
from the momentum?

163
00:08:52,720 --> 00:08:56,090
It comes from the
commutator, which tells us

164
00:08:56,090 --> 00:09:01,557
that the thing that's conserved
is the expectation value of p l

165
00:09:01,557 --> 00:09:02,890
is really the precise statement.

166
00:09:06,950 --> 00:09:15,230
And in particular,
what this tells

167
00:09:15,230 --> 00:09:18,940
us is that the eigenfunction,
or the eigenvalue of our wave

168
00:09:18,940 --> 00:09:21,400
function, under
translations by l,

169
00:09:21,400 --> 00:09:23,810
is a quantity that can be
determined simultaneously

170
00:09:23,810 --> 00:09:24,890
with knowing the energy.

171
00:09:24,890 --> 00:09:30,580
However, the eigenvalue
of t sub l, on this state,

172
00:09:30,580 --> 00:09:32,200
is equal to e to the i q l.

173
00:09:35,580 --> 00:09:39,430
Which means that
q is only defined

174
00:09:39,430 --> 00:09:42,160
for determining the
eigenvalue up to 2 pi over l.

175
00:09:42,160 --> 00:09:46,450
If you have q, which is 0, and
you increase it to pi over l,

176
00:09:46,450 --> 00:09:48,030
that value, pi over
l, is effectively

177
00:09:48,030 --> 00:09:50,680
the same as the value
minus pi over l.

178
00:09:50,680 --> 00:09:52,555
Because at least they're
the same eigenvalue.

179
00:09:55,177 --> 00:09:56,760
But that's really
strange because that

180
00:09:56,760 --> 00:10:00,450
means that q itself, it's
not strictly conserved.

181
00:10:00,450 --> 00:10:03,240
It's conserved mod 2 pi over l.

182
00:10:03,240 --> 00:10:05,500
When you have
momentum conservation,

183
00:10:05,500 --> 00:10:07,805
momentum is strictly
conserved if there's no force.

184
00:10:10,950 --> 00:10:13,337
And even if there is a force,
it's increasing control

185
00:10:13,337 --> 00:10:14,920
by the force as you
turn on the force,

186
00:10:14,920 --> 00:10:16,170
it just constantly increases.

187
00:10:16,170 --> 00:10:18,086
For the crystal momentum,
that's not the case.

188
00:10:18,086 --> 00:10:19,480
You turn on a
force, it increases

189
00:10:19,480 --> 00:10:21,350
according to the
conservation law.

190
00:10:21,350 --> 00:10:24,137
But it's not
increasing constantly.

191
00:10:24,137 --> 00:10:24,720
It's periodic.

192
00:10:24,720 --> 00:10:25,890
It's periodically defined.

193
00:10:25,890 --> 00:10:28,015
So it increases then it
ends up at a smaller value.

194
00:10:28,015 --> 00:10:30,240
It increases and ends
up at a smaller value.

195
00:10:30,240 --> 00:10:30,740
OK?

196
00:10:33,290 --> 00:10:35,040
So it carries many of
the same properties.

197
00:10:35,040 --> 00:10:36,030
It governs the phase.

198
00:10:36,030 --> 00:10:38,570
It's time independent
unless there's

199
00:10:38,570 --> 00:10:39,790
an external force applied.

200
00:10:39,790 --> 00:10:40,640
It's the eigenvalue.

201
00:10:40,640 --> 00:10:42,580
Controls the eigenvalue
of an operator that

202
00:10:42,580 --> 00:10:46,750
commutes with the energy when
you have a periodic potential,

203
00:10:46,750 --> 00:10:48,730
in the same way that
the momentum commutes

204
00:10:48,730 --> 00:10:50,792
with the energy when you
have no external force,

205
00:10:50,792 --> 00:10:52,250
when you have a
constant potential.

206
00:10:54,840 --> 00:10:57,161
Does that help?

207
00:10:57,161 --> 00:10:57,660
Good.

208
00:10:57,660 --> 00:10:58,210
OK.

209
00:10:58,210 --> 00:11:01,350
So developing an intuition for
the crystal momentum, I think,

210
00:11:01,350 --> 00:11:03,839
is best done by just
playing with examples.

211
00:11:03,839 --> 00:11:05,380
And you'll do that
more in the course

212
00:11:05,380 --> 00:11:07,255
on solids, which I
encourage you all to take.

213
00:11:07,255 --> 00:11:09,680
Because it's really
beautiful stuff.

214
00:11:09,680 --> 00:11:11,440
But for our purposes,
this is going

215
00:11:11,440 --> 00:11:15,780
to be the full set of
ideas we'll need for 8.04.

216
00:11:15,780 --> 00:11:16,280
Yeah?

217
00:11:16,280 --> 00:11:17,280
AUDIENCE: [INAUDIBLE]

218
00:11:20,280 --> 00:11:21,280
PROFESSOR: Ah.

219
00:11:24,280 --> 00:11:25,320
So good.

220
00:11:25,320 --> 00:11:28,880
So thank you.

221
00:11:28,880 --> 00:11:31,890
So this involves
a slight subtlety,

222
00:11:31,890 --> 00:11:35,452
which I've been glossing over
in the entire story here.

223
00:11:35,452 --> 00:11:36,410
Which of the following.

224
00:11:36,410 --> 00:11:38,385
So, is u of x a real function?

225
00:11:41,800 --> 00:11:47,380
Well, so when we
started out asking

226
00:11:47,380 --> 00:11:50,220
what are the eigenfunctions
of the transit by l operator,

227
00:11:50,220 --> 00:11:52,330
all we showed was
that, and I'm going

228
00:11:52,330 --> 00:11:54,740
to do this on a separate
board just to make it clearer.

229
00:11:54,740 --> 00:11:57,010
Tell me if this turns off,
because it kept bumping.

230
00:11:57,010 --> 00:11:58,410
OK.

231
00:11:58,410 --> 00:12:01,785
So when we started
with translate by l,

232
00:12:01,785 --> 00:12:03,410
and we constructed
it's eigenfunctions,

233
00:12:03,410 --> 00:12:07,690
we said that translate
by l q Phi sub cubed

234
00:12:07,690 --> 00:12:09,730
is equal to some phase,
and this is unitary,

235
00:12:09,730 --> 00:12:12,230
so we're talking about they
must be an actual phase in the i

236
00:12:12,230 --> 00:12:15,300
alpha of Phi sub q of x.

237
00:12:15,300 --> 00:12:17,520
And let's just suppose
that this is true.

238
00:12:17,520 --> 00:12:26,810
Then this tells us that Phi sub
q times e to the minus i q l

239
00:12:26,810 --> 00:12:27,837
equals u.

240
00:12:27,837 --> 00:12:30,420
So, I'm just going to use this
to define a new function, u sub

241
00:12:30,420 --> 00:12:31,520
q.

242
00:12:31,520 --> 00:12:32,313
Or just u.

243
00:12:32,313 --> 00:12:33,080
I'll use sub q.

244
00:12:33,080 --> 00:12:34,240
Fine. of x.

245
00:12:34,240 --> 00:12:35,940
So this defines a
new function, sub q.

246
00:12:35,940 --> 00:12:38,990
I take an eigenfunction, I
multiply it by some phase.

247
00:12:38,990 --> 00:12:42,580
Sorry, minus i q x.

248
00:12:42,580 --> 00:12:50,720
If we choose q l to be equal to
alpha, then acting on u sub q,

249
00:12:50,720 --> 00:12:55,470
by translate by l,
on u sub q, of x,

250
00:12:55,470 --> 00:12:58,776
is equal to-- well, if we act
on Phi sub q with translate

251
00:12:58,776 --> 00:13:02,390
by l, what happens to Phi sub q
we pick up a phase e d i alpha.

252
00:13:02,390 --> 00:13:04,040
What happens to e
to the minus i q x?

253
00:13:04,040 --> 00:13:05,160
x goes to x plus l.

254
00:13:05,160 --> 00:13:07,240
We pick up a phase e
to the minus i q l.

255
00:13:07,240 --> 00:13:09,790
So if q l is equal to alpha,
those two phases cancel,

256
00:13:09,790 --> 00:13:14,150
and we just get u
back. u sub q of x.

257
00:13:14,150 --> 00:13:17,000
But translate by l, if
u sub q, by definition,

258
00:13:17,000 --> 00:13:21,370
is equal to u sub q of x plus l.

259
00:13:21,370 --> 00:13:25,920
So we've determined is that if
we take q l is equal to alpha,

260
00:13:25,920 --> 00:13:29,850
then Phi sub q if eigenvalue
label by its eigenvalue, q,

261
00:13:29,850 --> 00:13:32,150
can be written in
the form e to the i q

262
00:13:32,150 --> 00:13:38,420
x u sub q of x, where
this is periodic.

263
00:13:42,690 --> 00:13:45,260
Everybody agree with that?

264
00:13:45,260 --> 00:13:46,130
OK.

265
00:13:46,130 --> 00:13:48,140
So that's step one.

266
00:13:48,140 --> 00:13:52,080
Step two is to say well
look, since the eigenvalue

267
00:13:52,080 --> 00:13:58,830
of this guy, under t
sub l, e d i alpha is

268
00:13:58,830 --> 00:14:01,700
equal to e to the i q l.

269
00:14:01,700 --> 00:14:06,590
Since this is periodic under
shifts of q, by 2 pi upon l,

270
00:14:06,590 --> 00:14:12,220
I can just choose to
define q up to 2 pi over l.

271
00:14:12,220 --> 00:14:15,840
So 2 q, I will take
to be equivalent to q

272
00:14:15,840 --> 00:14:17,040
plus 2 pi over l.

273
00:14:20,142 --> 00:14:21,600
And the reason I'm
going to do that

274
00:14:21,600 --> 00:14:23,130
is because it gives
the same eigenvalue,

275
00:14:23,130 --> 00:14:25,080
and if I want to label
things by eigenvalues,

276
00:14:25,080 --> 00:14:27,180
it's sort of redundant
to give multiple values

277
00:14:27,180 --> 00:14:29,540
to the same eigenvalue.

278
00:14:29,540 --> 00:14:32,580
Now there's a
subtlety, here though.

279
00:14:32,580 --> 00:14:34,480
And this little
thing here is this.

280
00:14:34,480 --> 00:14:36,560
Suppose we have a free particle.

281
00:14:36,560 --> 00:14:40,382
Does a free particle
respect translation by l?

282
00:14:40,382 --> 00:14:42,590
So if we have a free particle,
the potential is zero.

283
00:14:42,590 --> 00:14:46,371
That constant function is also
periodic under shifts by l.

284
00:14:46,371 --> 00:14:46,870
Right?

285
00:14:46,870 --> 00:14:48,150
Because it's just zero.

286
00:14:48,150 --> 00:14:51,490
So it's stupidly periodic,
but it's periodic nonetheless.

287
00:14:51,490 --> 00:14:53,540
So now I'm going to ask
the following question.

288
00:14:53,540 --> 00:14:55,450
What are the common
eigenfunctions

289
00:14:55,450 --> 00:14:58,030
of the energy and translate
by l for the free particle?

290
00:14:58,030 --> 00:14:59,120
We did this last time.

291
00:14:59,120 --> 00:15:06,150
So the common eigenfunctions of
translate by l and the energy

292
00:15:06,150 --> 00:15:11,162
are the wave functions
Phi sub q, comma e,

293
00:15:11,162 --> 00:15:16,130
are equal to e to the i
q x times some function

294
00:15:16,130 --> 00:15:19,331
u of x, on general grounds.

295
00:15:19,331 --> 00:15:21,080
But we know what these
eigenfunctions are.

296
00:15:21,080 --> 00:15:23,060
They're just e to the i k x.

297
00:15:23,060 --> 00:15:26,640
Where k squared upon 2 m is e.

298
00:15:26,640 --> 00:15:28,760
[INAUDIBLE] So we
know that these

299
00:15:28,760 --> 00:15:31,430
are the correct eigenfunctions,
but we're writing them

300
00:15:31,430 --> 00:15:33,800
in the form e v i q x u.

301
00:15:33,800 --> 00:15:35,837
Now you say that's fine.

302
00:15:35,837 --> 00:15:37,170
There's nothing wrong with this.

303
00:15:37,170 --> 00:15:40,140
We just say u is constant
and q is equal to k.

304
00:15:42,920 --> 00:15:44,295
These functions
are of this form,

305
00:15:44,295 --> 00:15:46,520
but they're of this
form with e v i q x

306
00:15:46,520 --> 00:15:49,760
being e d i k x and u
of x being constant.

307
00:15:49,760 --> 00:15:50,260
Right?

308
00:15:50,260 --> 00:15:51,593
There's nothing wrong with that.

309
00:15:51,593 --> 00:15:52,360
Everyone agree?

310
00:15:52,360 --> 00:15:53,430
Perfectly consistent.

311
00:15:53,430 --> 00:15:59,730
However, I thought we said
that q is periodic by 2 pi?

312
00:15:59,730 --> 00:16:01,400
If q is periodic
by 2 pi, then that

313
00:16:01,400 --> 00:16:02,983
would seem to imply that
k is periodic by 2 pi,

314
00:16:02,983 --> 00:16:04,720
and we know that's not
true because any k is

315
00:16:04,720 --> 00:16:05,886
allowed for a free particle.

316
00:16:08,560 --> 00:16:13,230
So if we want to think about
q is periodic by 2 pi upon l,

317
00:16:13,230 --> 00:16:17,870
then we cannot require
that u is real.

318
00:16:17,870 --> 00:16:19,870
Because it must be the
phase that makes this up.

319
00:16:19,870 --> 00:16:24,520
It must be, so I can always
write this as e to the i q x

320
00:16:24,520 --> 00:16:27,290
where q is less
than 2 pi upon l.

321
00:16:27,290 --> 00:16:33,430
I'm sorry, where q is between
2 pi or pi upon l and minus pi

322
00:16:33,430 --> 00:16:34,897
upon l.

323
00:16:34,897 --> 00:16:37,230
So that it's defined only
after this periodically thing.

324
00:16:37,230 --> 00:16:41,710
But times some additional
phase, e to the i k minus q

325
00:16:41,710 --> 00:16:46,870
x This is trivially
equal to e to the i k x.

326
00:16:46,870 --> 00:16:49,899
But now u is not
a real function.

327
00:16:49,899 --> 00:16:52,190
On the other hand, if we
hadn't imposed the requirement

328
00:16:52,190 --> 00:16:56,580
that q is periodic, we wouldn't
have needed to make u real.

329
00:16:56,580 --> 00:16:59,470
We could just taken q to be
equal to k, for any value k,

330
00:16:59,470 --> 00:17:02,020
and then u would be
constant. u would be real.

331
00:17:02,020 --> 00:17:04,040
So this is important
for answering

332
00:17:04,040 --> 00:17:07,810
the excellent question
that our fearless restation

333
00:17:07,810 --> 00:17:10,569
instructor provoked
me to answer.

334
00:17:10,569 --> 00:17:13,464
Which is that so
what-- we'll come back

335
00:17:13,464 --> 00:17:14,839
to the question
in just a second.

336
00:17:14,839 --> 00:17:16,255
But what I want
to emphasize this,

337
00:17:16,255 --> 00:17:18,907
that if we're going to take
q to be not periodic, Sorry.

338
00:17:18,907 --> 00:17:21,240
If we're going to take q to
be defined only up to shifts

339
00:17:21,240 --> 00:17:25,530
by 2 pi over l, it's important
that we allow u to be not real.

340
00:17:25,530 --> 00:17:29,330
It must be able to
be an overall phase.

341
00:17:29,330 --> 00:17:33,860
But if we want u to be
always real, we can do that.

342
00:17:33,860 --> 00:17:36,270
We just can't impose
this periodicity.

343
00:17:36,270 --> 00:17:38,720
Different values of q mean
different wave functions.

344
00:17:38,720 --> 00:17:40,710
And this is really what's going
on when you see those plots,

345
00:17:40,710 --> 00:17:42,490
sometimes you see the
plots as parabolas.

346
00:17:42,490 --> 00:17:44,810
The bands are represented
by parabolas with wiggles,

347
00:17:44,810 --> 00:17:47,077
and sometimes they're folded up.

348
00:17:47,077 --> 00:17:48,160
And that's the difference.

349
00:17:48,160 --> 00:17:49,360
The difference is that
when you fold them up,

350
00:17:49,360 --> 00:17:51,190
you're imposing this
periodicity and you're

351
00:17:51,190 --> 00:17:53,610
labeling the
eigenfunctions by q,

352
00:17:53,610 --> 00:17:58,740
and the overall amount of the
number effectively of k phases

353
00:17:58,740 --> 00:18:01,444
that you're subtracting off.

354
00:18:01,444 --> 00:18:01,944
Yeah?

355
00:18:01,944 --> 00:18:04,860
AUDIENCE: So is this an
arbitrary choice? [INAUDIBLE]

356
00:18:04,860 --> 00:18:05,820
PROFESSOR: Yeah.

357
00:18:05,820 --> 00:18:06,770
I mean, how to say?

358
00:18:06,770 --> 00:18:10,570
It's exactly akin to
a choice of variables.

359
00:18:10,570 --> 00:18:12,860
In describing the
position of this particle,

360
00:18:12,860 --> 00:18:14,360
should we use
Cartesian coordinates,

361
00:18:14,360 --> 00:18:15,985
or should we use
Spherical coordinates?

362
00:18:15,985 --> 00:18:18,540
Well it can't possibly matter.

363
00:18:18,540 --> 00:18:22,200
And so you'd better make sure in
any description of your system,

364
00:18:22,200 --> 00:18:26,082
that changing your coordinates
doesn't change your results.

365
00:18:26,082 --> 00:18:27,790
And here, that's
exactly what's going on.

366
00:18:27,790 --> 00:18:30,123
Do we want to define our
variable to be periodic by 2 pi

367
00:18:30,123 --> 00:18:31,330
upon l?

368
00:18:31,330 --> 00:18:32,080
Well, OK then.

369
00:18:32,080 --> 00:18:33,410
But u can't be real.

370
00:18:33,410 --> 00:18:35,830
Or we could take q to
be not periodic by 2 pi

371
00:18:35,830 --> 00:18:37,147
l and impose that u is real.

372
00:18:37,147 --> 00:18:38,480
It's just a choice of variables.

373
00:18:38,480 --> 00:18:40,750
But it can't possibly
give different answers.

374
00:18:40,750 --> 00:18:43,104
The point is, this is a
subtle little distinction it

375
00:18:43,104 --> 00:18:45,270
we gloss over, and is glossed
over into my knowledge

376
00:18:45,270 --> 00:18:46,936
every book on intro
to quantum mechanics

377
00:18:46,936 --> 00:18:49,110
that even covers
periodic potentials.

378
00:18:49,110 --> 00:18:50,510
It can be very confusing.

379
00:18:50,510 --> 00:18:52,770
Anyway, the reason that I
had to go through all this,

380
00:18:52,770 --> 00:18:55,330
is that in order to answer
the very, very good question

381
00:18:55,330 --> 00:18:59,040
professor Evans posed, I'm going
to need to deal with this fact.

382
00:18:59,040 --> 00:19:04,680
So for the moment, let me deal
with-- let's work with u real.

383
00:19:04,680 --> 00:19:09,630
And q, q an unconstrained,
real number.

384
00:19:09,630 --> 00:19:10,130
OK.

385
00:19:10,130 --> 00:19:10,915
So not periodic.

386
00:19:16,239 --> 00:19:17,780
Are we cool with
that for the moment?

387
00:19:20,480 --> 00:19:22,875
So if we do that, then notice
the falling nice property

388
00:19:22,875 --> 00:19:23,750
of our wave function.

389
00:19:23,750 --> 00:19:26,800
Our wave function, Phi sub
q, is equal e to the i q

390
00:19:26,800 --> 00:19:29,617
x times u of q, or u of x.

391
00:19:29,617 --> 00:19:30,200
Which is real.

392
00:19:43,060 --> 00:19:45,640
So when we can construct
the current-- remember

393
00:19:45,640 --> 00:19:52,890
that j boils down to
the imaginary part,

394
00:19:52,890 --> 00:19:57,070
h bar over 2 m i.

395
00:19:57,070 --> 00:20:01,410
Well, h bar over m times
the imaginary part of SI

396
00:20:01,410 --> 00:20:04,740
complex conjugate
derivative, with respect

397
00:20:04,740 --> 00:20:10,090
to x, which is the current,
in the x direction of SI.

398
00:20:10,090 --> 00:20:12,180
And we need this
to be imaginary,

399
00:20:12,180 --> 00:20:13,540
or we will get no current.

400
00:20:13,540 --> 00:20:15,570
You show this in a problem set,
if you have a pure, real wave

401
00:20:15,570 --> 00:20:16,300
function, for example.

402
00:20:16,300 --> 00:20:17,680
A single real
exponential, that's

403
00:20:17,680 --> 00:20:21,880
decaying, as on the
wrong side of a barrier.

404
00:20:21,880 --> 00:20:23,200
Then you get no current.

405
00:20:23,200 --> 00:20:23,842
Nothing flows.

406
00:20:23,842 --> 00:20:24,675
And that make sense.

407
00:20:24,675 --> 00:20:26,520
It's exponentially decaying.

408
00:20:26,520 --> 00:20:27,420
Nothing gets across.

409
00:20:27,420 --> 00:20:29,086
So we need the wave
function to be real.

410
00:20:29,086 --> 00:20:32,079
So if q were zero
we would get zero.

411
00:20:32,079 --> 00:20:34,620
And what you can immediately do
from this, compute from this,

412
00:20:34,620 --> 00:20:36,911
is that while the derivative,
if the derivative doesn't

413
00:20:36,911 --> 00:20:39,970
hit e to the i q x, if it hits
u, than the phase e to the i q

414
00:20:39,970 --> 00:20:40,960
x cancels.

415
00:20:40,960 --> 00:20:43,306
And so the contribution
from that term vanishes.

416
00:20:43,306 --> 00:20:45,430
So the only term that's
going to contribute in here

417
00:20:45,430 --> 00:20:47,855
is when the derivative
hits the e to the i q x.

418
00:20:47,855 --> 00:20:54,991
But then this is going
to be equal to h bar q.

419
00:20:54,991 --> 00:20:56,615
And we want the
imaginary parts, that's

420
00:20:56,615 --> 00:21:00,910
going to be e to the I over m.

421
00:21:00,910 --> 00:21:03,390
And then we're left
with u squared of x.

422
00:21:06,430 --> 00:21:09,330
So this is the
current, but we have

423
00:21:09,330 --> 00:21:11,670
to do it-- we had take
advantage in order for this

424
00:21:11,670 --> 00:21:15,340
to be sort of clean, we had to
take advantage of u being real.

425
00:21:15,340 --> 00:21:16,510
Everybody cool with that?

426
00:21:16,510 --> 00:21:18,600
Now there's one
last twist on this,

427
00:21:18,600 --> 00:21:21,260
which is that if I
have k-- if I have q.

428
00:21:21,260 --> 00:21:22,610
So this is a side note.

429
00:21:22,610 --> 00:21:24,700
Going back up here,
to this logic.

430
00:21:24,700 --> 00:21:28,080
If I have q, and I
want, I can always

431
00:21:28,080 --> 00:21:34,390
write it as some q
naught plus n pi over l.

432
00:21:37,176 --> 00:21:38,550
And so now what
I want to do is I

433
00:21:38,550 --> 00:21:41,060
want to take sort of a
hybrid of these two pictures.

434
00:21:41,060 --> 00:21:43,500
And I want to say
Phi sub q is going

435
00:21:43,500 --> 00:21:49,400
to be equal to e to
the i q naught x.

436
00:21:49,400 --> 00:21:52,480
Where this is the value
that's periodic by 2 pi.

437
00:21:52,480 --> 00:21:59,809
e to the I n pi over l x u.

438
00:21:59,809 --> 00:22:02,350
And so now really what's going
to happen, what I'm doing here

439
00:22:02,350 --> 00:22:05,270
is I'm labeling q, not
by a single number.

440
00:22:05,270 --> 00:22:07,570
I'm labeling my wave function
not by single number q,

441
00:22:07,570 --> 00:22:10,420
but by q naught
and an integer n.

442
00:22:10,420 --> 00:22:11,240
Comma n.

443
00:22:11,240 --> 00:22:12,320
So q naught and n.

444
00:22:12,320 --> 00:22:14,510
So now q naught is periodic.

445
00:22:14,510 --> 00:22:17,510
It's defined up
to shifts by 2 pi.

446
00:22:17,510 --> 00:22:19,630
n is an additional integer,
and what it's telling

447
00:22:19,630 --> 00:22:21,730
you is how many times did
you have to shift back

448
00:22:21,730 --> 00:22:25,810
to get into that fundamental
zone between pi and minus pi.

449
00:22:25,810 --> 00:22:28,276
And this fits nicely
into this story,

450
00:22:28,276 --> 00:22:29,900
because now all we're
going to get here

451
00:22:29,900 --> 00:22:31,930
is q, which is q
naught plus n pi.

452
00:22:31,930 --> 00:22:37,960
So the current depends on both
the part defined mod 2 pi over

453
00:22:37,960 --> 00:22:39,420
l, and the integer,
which tells you

454
00:22:39,420 --> 00:22:42,340
how many factors of 2 pi over
l did you have to subtract off

455
00:22:42,340 --> 00:22:45,280
to get into that
fundamental domain.

456
00:22:45,280 --> 00:22:47,650
So let's think back
to our band structure.

457
00:22:47,650 --> 00:22:48,965
So what is this n quantity?

458
00:22:48,965 --> 00:22:50,590
Let's think back to
our band structure.

459
00:22:50,590 --> 00:22:53,500
In our band structure, we had
something that looks like this.

460
00:22:59,300 --> 00:23:01,700
And here's the value of q.

461
00:23:01,700 --> 00:23:03,000
But am I plotting q?

462
00:23:03,000 --> 00:23:03,500
No.

463
00:23:03,500 --> 00:23:04,624
I'm plotting here q naught.

464
00:23:04,624 --> 00:23:07,440
I'm plotting the part
that's periodically

465
00:23:07,440 --> 00:23:08,730
defined up to 2 pi over l.

466
00:23:08,730 --> 00:23:09,822
So this is pi over l.

467
00:23:09,822 --> 00:23:10,905
This is minus 2 pi over l.

468
00:23:10,905 --> 00:23:13,530
Or minus pi over l.

469
00:23:13,530 --> 00:23:14,170
OK.

470
00:23:14,170 --> 00:23:17,207
And what we see is that
there isn't a single energy.

471
00:23:17,207 --> 00:23:19,790
Because this is the energy the
vertical direction for the band

472
00:23:19,790 --> 00:23:20,320
pictures.

473
00:23:20,320 --> 00:23:23,630
There isn't a single energy
for a given value of q.

474
00:23:23,630 --> 00:23:25,770
In fact, the set of
energy eigenvalue--

475
00:23:25,770 --> 00:23:27,996
or the set of allowed
states or energy eigenvalues

476
00:23:27,996 --> 00:23:30,370
for an allowed value of q
would say this particular value

477
00:23:30,370 --> 00:23:33,850
of q naught, how many
of them are there.

478
00:23:33,850 --> 00:23:36,220
Well, there are as many
as there are integers.

479
00:23:36,220 --> 00:23:37,910
One, two, three, four, count.

480
00:23:37,910 --> 00:23:41,640
So to specify a state, I don't
just have to specify q NAUGHT,

481
00:23:41,640 --> 00:23:43,600
I also have to specify N.

482
00:23:43,600 --> 00:23:47,420
Which one of these
guys I'm hitting.

483
00:23:47,420 --> 00:23:49,789
And when you unfold this
into the parabola picture,

484
00:23:49,789 --> 00:23:51,080
remember where these came from.

485
00:23:51,080 --> 00:23:52,370
These came from these curves.

486
00:23:52,370 --> 00:23:54,230
Came from shifting over.

487
00:23:54,230 --> 00:23:57,350
And the higher up you go, the
more you had to shift over.

488
00:23:57,350 --> 00:24:00,891
And that's exactly the
integer piece in n pi over l.

489
00:24:00,891 --> 00:24:02,390
And so we can write
the current now,

490
00:24:02,390 --> 00:24:08,050
in terms of h bar q naught
upon m, u squared-- I'm sorry.

491
00:24:08,050 --> 00:24:17,947
h bar q naught upon m plus n
pi h bar upon m u squared of x.

492
00:24:17,947 --> 00:24:20,030
So we get a contribution
from the crystal momentum

493
00:24:20,030 --> 00:24:23,000
and from which we're in.

494
00:24:23,000 --> 00:24:25,010
OK?

495
00:24:25,010 --> 00:24:29,110
So sort of an elaborate story
to answer the phase question.

496
00:24:29,110 --> 00:24:29,610
Yeah?

497
00:24:29,610 --> 00:24:32,935
AUDIENCE: [INAUDIBLE]

498
00:24:32,935 --> 00:24:33,890
PROFESSOR: Good.

499
00:24:33,890 --> 00:24:36,580
So here we had SI--
so SI-- I'm sorry.

500
00:24:36,580 --> 00:24:38,640
I should have done this for Phi.

501
00:24:38,640 --> 00:24:41,200
But I meant this
wave function, right.

502
00:24:41,200 --> 00:24:43,380
This is Phi, this is Phi q.

503
00:24:43,380 --> 00:24:46,697
So from here we're going
to get the imaginary part.

504
00:24:46,697 --> 00:24:48,780
So we get the imaginary
part of this wave function

505
00:24:48,780 --> 00:24:54,227
which is u to the minus i
q x u of x derivative of e

506
00:24:54,227 --> 00:24:55,265
to the i q x u of x.

507
00:24:55,265 --> 00:24:57,640
Now the term that contributes
is when the derivative hits

508
00:24:57,640 --> 00:24:59,450
the e to the i q.

509
00:24:59,450 --> 00:25:01,890
x pulls down a factor of
i q, and the two phases

510
00:25:01,890 --> 00:25:04,180
cancel from these guys,
leaving us with a u of x here,

511
00:25:04,180 --> 00:25:05,060
and a u of x here.

512
00:25:05,060 --> 00:25:05,952
AUDIENCE: [INAUDIBLE]

513
00:25:09,565 --> 00:25:10,460
PROFESSOR: Oh sorry.

514
00:25:10,460 --> 00:25:11,740
This is a potential.

515
00:25:11,740 --> 00:25:12,240
Good.

516
00:25:12,240 --> 00:25:12,990
That's the point.

517
00:25:12,990 --> 00:25:14,040
So this is the potential.

518
00:25:14,040 --> 00:25:19,240
So in this statement that what
we have this translation by x.

519
00:25:19,240 --> 00:25:20,490
So this is just some function.

520
00:25:20,490 --> 00:25:22,410
It has nothing to the potential.

521
00:25:22,410 --> 00:25:24,480
It's defined in terms
of the wave function.

522
00:25:24,480 --> 00:25:27,184
The eigenfunction
of translate by l.

523
00:25:27,184 --> 00:25:28,600
So the logic here
goes, if we know

524
00:25:28,600 --> 00:25:30,890
we have a function
of translate by l,

525
00:25:30,890 --> 00:25:33,240
then I construct
a new function u.

526
00:25:33,240 --> 00:25:35,510
Nothing to do with the
potential, just a new function.

527
00:25:35,510 --> 00:25:38,290
Which is e to the
minus i q x times it.

528
00:25:38,290 --> 00:25:39,420
You can't stop me.

529
00:25:39,420 --> 00:25:42,070
You hand me a function, I will
hand you a different function.

530
00:25:42,070 --> 00:25:46,822
And then we pick q felicitously,
to show that u is periodic.

531
00:25:46,822 --> 00:25:48,280
So u is just some
periodic function

532
00:25:48,280 --> 00:25:51,330
which is contained which is
defined from the wave function.

533
00:25:51,330 --> 00:25:52,050
From the energy.

534
00:25:52,050 --> 00:25:54,720
From eigenfunction of t l.

535
00:25:54,720 --> 00:25:58,070
Did that answer your question?

536
00:25:58,070 --> 00:25:59,980
OK.

537
00:25:59,980 --> 00:26:01,640
So here, it just
came from the fact

538
00:26:01,640 --> 00:26:04,167
that u is Phi is the e to
the i q x u, x and then

539
00:26:04,167 --> 00:26:05,500
a factor of u for each of these.

540
00:26:08,770 --> 00:26:10,930
Other questions.

541
00:26:10,930 --> 00:26:11,430
Yeah.

542
00:26:11,430 --> 00:26:12,891
AUDIENCE: [INAUDIBLE]

543
00:26:19,032 --> 00:26:21,506
PROFESSOR: So this picture,
when it's unfolded,

544
00:26:21,506 --> 00:26:23,630
first off, you know what
it is for a free particle.

545
00:26:23,630 --> 00:26:26,440
So we want the energy
as a function of q.

546
00:26:30,300 --> 00:26:32,771
So what is it for
a free particle?

547
00:26:32,771 --> 00:26:33,270
Parabola.

548
00:26:33,270 --> 00:26:33,853
Yeah, exactly.

549
00:26:38,930 --> 00:26:42,750
And now let's add in-- let's
make this a function of q,

550
00:26:42,750 --> 00:26:47,640
not q naught, but
so here's pi over l.

551
00:26:47,640 --> 00:26:50,410
Here's 2 pi over l.

552
00:26:50,410 --> 00:26:52,220
Here's 3 pi over l.

553
00:26:52,220 --> 00:26:54,220
And I need to do this
carefully, because it's

554
00:26:54,220 --> 00:26:55,928
incredibly difficult
to get the straight.

555
00:27:00,540 --> 00:27:01,040
OK.

556
00:27:07,345 --> 00:27:11,860
My artistic skills are not
exactly the thing of legend.

557
00:27:11,860 --> 00:27:13,417
OK.

558
00:27:13,417 --> 00:27:15,250
So here's the parabola
that would have been,

559
00:27:15,250 --> 00:27:17,409
if we had not turned on
a periodic potential.

560
00:27:17,409 --> 00:27:18,950
As we turn on the
periodic potential,

561
00:27:18,950 --> 00:27:21,800
we know that the
energies change.

562
00:27:21,800 --> 00:27:24,000
And so in the first
band it's easy to see,

563
00:27:24,000 --> 00:27:25,940
because for minus pi
over l, it's pi over l.

564
00:27:25,940 --> 00:27:27,148
We don't have to do anything.

565
00:27:29,850 --> 00:27:32,730
So it look exactly the same
as the lowest band over here.

566
00:27:32,730 --> 00:27:38,680
So in particular-- OK?

567
00:27:38,680 --> 00:27:40,430
So what about this second band?

568
00:27:40,430 --> 00:27:43,040
Well what I want to know what's
the allowed, the other allowed

569
00:27:43,040 --> 00:27:44,830
energy that's say,
plus pi over l.

570
00:27:44,830 --> 00:27:46,980
Plus pi over l, it's going
to be something greater

571
00:27:46,980 --> 00:27:48,590
than this value.

572
00:27:48,590 --> 00:27:50,630
But plus pi over l, we
already know the answer

573
00:27:50,630 --> 00:27:52,620
from that diagram,
because plus pi over l is

574
00:27:52,620 --> 00:27:55,170
the same as minus pi over l,
so what's the value over here?

575
00:27:55,170 --> 00:27:57,150
Well, the value over
there for the second band

576
00:27:57,150 --> 00:28:00,160
is slightly above, and then
it increases and decreases.

577
00:28:00,160 --> 00:28:02,700
So slightly above,
and then it increases.

578
00:28:06,470 --> 00:28:08,195
Shift by pi over l.

579
00:28:11,020 --> 00:28:11,520
Whoops.

580
00:28:11,520 --> 00:28:13,780
Did I shift by pi
over l for this guy?

581
00:28:13,780 --> 00:28:14,804
That's one, two.

582
00:28:17,750 --> 00:28:18,250
Yes.

583
00:28:18,250 --> 00:28:18,560
I did.

584
00:28:18,560 --> 00:28:19,070
Good.

585
00:28:19,070 --> 00:28:20,050
And it goes the other way.

586
00:28:20,050 --> 00:28:21,966
So just noting that it
goes away from the top.

587
00:28:26,634 --> 00:28:28,300
I have a hard time
drawing these things.

588
00:28:32,250 --> 00:28:36,689
So for every value of q,
there's an allowed energy.

589
00:28:36,689 --> 00:28:38,230
But it's different
than it would have

590
00:28:38,230 --> 00:28:39,380
been for the free particle.

591
00:28:39,380 --> 00:28:42,080
And then we do the same
thing for the next state.

592
00:28:42,080 --> 00:28:44,390
And it looks like this.

593
00:28:47,414 --> 00:28:49,330
So now imagine what
happens when we take this,

594
00:28:49,330 --> 00:28:51,975
and we it over one two.

595
00:28:51,975 --> 00:28:53,350
What we get is a
band the looks--

596
00:28:53,350 --> 00:28:54,870
that should look like this.

597
00:28:54,870 --> 00:28:56,744
That's what the second
band should look like.

598
00:29:00,680 --> 00:29:02,970
And indeed, when we put it
in the fundamental domain,

599
00:29:02,970 --> 00:29:04,200
this is what we get.

600
00:29:04,200 --> 00:29:05,620
This is what the first
band and the second band

601
00:29:05,620 --> 00:29:06,440
together look like.

602
00:29:06,440 --> 00:29:08,106
And then the third
band, we'll move this

603
00:29:08,106 --> 00:29:11,790
over once, and then twice,
it's going to look like this.

604
00:29:11,790 --> 00:29:15,030
And this guy, move it over
once, twice, looks like whoops.

605
00:29:21,401 --> 00:29:21,900
Yeah?

606
00:29:21,900 --> 00:29:25,850
AUDIENCE: If we wanted to plot
u with respect to k instead,

607
00:29:25,850 --> 00:29:28,285
would that just be a
parabola dotted line?

608
00:29:28,285 --> 00:29:30,070
If so, why do we
not have really--

609
00:29:30,070 --> 00:29:31,849
PROFESSOR: If we
just wanted-- sorry.

610
00:29:31,849 --> 00:29:32,390
Say it again?

611
00:29:32,390 --> 00:29:34,503
AUDIENCE: E as a function
of k instead of q.

612
00:29:34,503 --> 00:29:34,750
PROFESSOR: Oh.

613
00:29:34,750 --> 00:29:35,010
Yeah.

614
00:29:35,010 --> 00:29:37,301
E as a function of k is always
going to look like that.

615
00:29:37,301 --> 00:29:39,000
But k is not a
well-- so what is k?

616
00:29:39,000 --> 00:29:43,660
K is just defined as h bar
squared, k squared upon 2 m

617
00:29:43,660 --> 00:29:45,200
is equal to e.

618
00:29:45,200 --> 00:29:47,220
So this doesn't
tell you anything.

619
00:29:47,220 --> 00:29:47,720
Right.

620
00:29:47,720 --> 00:29:48,930
Because any allowed k.

621
00:29:48,930 --> 00:29:51,570
Sure any allowed k is
some valid value of e.

622
00:29:51,570 --> 00:29:54,750
But this didn't tell you
which values of e are allowed.

623
00:29:54,750 --> 00:29:56,520
Only some values of
e are allowed, right?

624
00:29:56,520 --> 00:29:59,100
There are no values of e--
there are no energy eigenstates

625
00:29:59,100 --> 00:30:01,657
with energy in between
here and here, right?

626
00:30:01,657 --> 00:30:04,240
And so that tells you they're
no allowed k's because k is just

627
00:30:04,240 --> 00:30:06,664
defined, it's just
completely defined by e.

628
00:30:06,664 --> 00:30:08,830
So this doesn't tell you
anything about which states

629
00:30:08,830 --> 00:30:08,880
you're at.

630
00:30:08,880 --> 00:30:11,300
It just that given an e,
there's some quantity that

631
00:30:11,300 --> 00:30:12,280
could define k.

632
00:30:12,280 --> 00:30:14,520
This is a definition
of k, in terms of e.

633
00:30:14,520 --> 00:30:19,140
What this diagram is telling
you is which e's are allowed.

634
00:30:19,140 --> 00:30:20,072
AUDIENCE: [INAUDIBLE]

635
00:30:27,800 --> 00:30:28,710
PROFESSOR: Yes.

636
00:30:28,710 --> 00:30:29,210
Yes.

637
00:30:29,210 --> 00:30:30,160
There should be.

638
00:30:30,160 --> 00:30:30,710
Let's see.

639
00:30:30,710 --> 00:30:31,050
What's

640
00:30:31,050 --> 00:30:31,924
AUDIENCE: [INAUDIBLE]

641
00:30:36,957 --> 00:30:37,790
PROFESSOR: Oh, here.

642
00:30:37,790 --> 00:30:38,460
Yes.

643
00:30:38,460 --> 00:30:41,330
Yes, you're absolutely right.

644
00:30:41,330 --> 00:30:42,460
Over out.

645
00:30:42,460 --> 00:30:43,520
Thank you.

646
00:30:43,520 --> 00:30:44,120
Excellent.

647
00:30:44,120 --> 00:30:45,030
That's exactly right.

648
00:30:45,030 --> 00:30:46,210
Yeah.

649
00:30:46,210 --> 00:30:46,710
Oh man.

650
00:30:46,710 --> 00:30:47,918
I made a dimensional mistake.

651
00:30:47,918 --> 00:30:49,790
Thank you.

652
00:30:49,790 --> 00:30:50,520
Jesus.

653
00:30:50,520 --> 00:30:51,020
OK.

654
00:30:54,020 --> 00:30:55,626
Good.

655
00:30:55,626 --> 00:30:56,602
Yeah.

656
00:30:56,602 --> 00:30:58,930
AUDIENCE: Could
you like re-explain

657
00:30:58,930 --> 00:31:01,845
how imperfections and a lattice
leads to actual conduction?

658
00:31:01,845 --> 00:31:02,800
PROFESSOR: Yeah.

659
00:31:02,800 --> 00:31:03,675
I'm going to do that.

660
00:31:03,675 --> 00:31:05,008
So that's an excellent question.

661
00:31:05,008 --> 00:31:06,890
The question is could
you explain again

662
00:31:06,890 --> 00:31:09,800
how imperfections and a lattice
leads to actual conduction.

663
00:31:09,800 --> 00:31:11,620
As we talked about
last time, when

664
00:31:11,620 --> 00:31:13,080
you have a perfect
lattice, there

665
00:31:13,080 --> 00:31:15,670
is actually no current
flowing in response

666
00:31:15,670 --> 00:31:19,310
to an applied
electromagnetic field.

667
00:31:19,310 --> 00:31:20,950
If you put on a
capacitor, played

668
00:31:20,950 --> 00:31:23,880
across your perfect lattice,
you don't get any current.

669
00:31:23,880 --> 00:31:26,917
So the particle, the charged
particle in your lattice,

670
00:31:26,917 --> 00:31:29,500
just oscillates back and forth
in a block oscillation, running

671
00:31:29,500 --> 00:31:31,250
up the band, and down the
band, and up the band,

672
00:31:31,250 --> 00:31:32,070
and down the band.

673
00:31:34,840 --> 00:31:37,639
So, let me slightly
change your question,

674
00:31:37,639 --> 00:31:39,180
and turn it into
two other questions.

675
00:31:39,180 --> 00:31:42,250
The first question is given
that that's obviously not what

676
00:31:42,250 --> 00:31:45,253
happens in real
materials, why don't we

677
00:31:45,253 --> 00:31:46,627
just give up on
quantum mechanics

678
00:31:46,627 --> 00:31:49,110
and say it totally failed?

679
00:31:49,110 --> 00:31:51,126
And so this is a totally
reasonable question,

680
00:31:51,126 --> 00:31:53,250
and I want to emphasize
something important to you.

681
00:31:53,250 --> 00:31:54,208
Which is the following.

682
00:31:54,208 --> 00:31:56,420
That model led to
a prediction, which

683
00:31:56,420 --> 00:31:59,606
is that if you put a capacitor
plate across a perfect crystal,

684
00:31:59,606 --> 00:32:01,480
then you would get no
current flowing across,

685
00:32:01,480 --> 00:32:03,188
you would just see
that the electron wave

686
00:32:03,188 --> 00:32:04,230
packets oscillate.

687
00:32:04,230 --> 00:32:06,920
Or block oscillations as
we discussed last time.

688
00:32:06,920 --> 00:32:09,410
And that is manifestly
what happens with copper.

689
00:32:09,410 --> 00:32:12,260
But the experimentalist comes
back to you and says look dude.

690
00:32:12,260 --> 00:32:14,790
That is a ridiculous model
because the copper isn't

691
00:32:14,790 --> 00:32:17,110
in fact perfect, it's messy.

692
00:32:17,110 --> 00:32:18,770
So how do you test the model?

693
00:32:18,770 --> 00:32:20,180
Well there are
two ways to test--

694
00:32:20,180 --> 00:32:21,510
to deal with the situation.

695
00:32:21,510 --> 00:32:24,330
One is you improve the model
to incorporate properties

696
00:32:24,330 --> 00:32:25,390
that copper actually has.

697
00:32:25,390 --> 00:32:27,370
And see if you can actually
get the same conductivity

698
00:32:27,370 --> 00:32:28,010
that you see.

699
00:32:28,010 --> 00:32:31,900
But the other is you could
improve the material, instead

700
00:32:31,900 --> 00:32:34,880
of improving the theory.

701
00:32:34,880 --> 00:32:36,770
So let's make up what--
can we actually build

702
00:32:36,770 --> 00:32:38,192
a perfect crystal?

703
00:32:38,192 --> 00:32:40,650
This is actually something that
I'm doing research on right

704
00:32:40,650 --> 00:32:40,850
now.

705
00:32:40,850 --> 00:32:42,120
Not on the building side,
but on the theory side,

706
00:32:42,120 --> 00:32:44,536
because I'm a theorist and you
should not let me in a lab.

707
00:32:46,694 --> 00:32:48,360
But I collaborate
with experimentalists,

708
00:32:48,360 --> 00:32:52,680
so they're nice people.

709
00:32:52,680 --> 00:32:54,010
They're very good physicists.

710
00:32:54,010 --> 00:32:56,590
So here's something you can do.

711
00:32:56,590 --> 00:33:00,867
You can build a system that has
exactly a periodic potential.

712
00:33:00,867 --> 00:33:02,450
It turns out it's
very difficult to do

713
00:33:02,450 --> 00:33:04,810
this with quantum systems.

714
00:33:04,810 --> 00:33:08,470
But what you can do is you
can do it with lattices

715
00:33:08,470 --> 00:33:15,130
not of atoms, but
lattices of dielectric.

716
00:33:15,130 --> 00:33:15,980
So the equation.

717
00:33:15,980 --> 00:33:17,810
Here's a cool fact,
the equation for light

718
00:33:17,810 --> 00:33:19,976
going through a dielectric,
where the dielectric has

719
00:33:19,976 --> 00:33:21,571
different constants,
like wave guides.

720
00:33:21,571 --> 00:33:22,820
You've got glass, you got air.

721
00:33:22,820 --> 00:33:24,180
You've got glass, you got air.

722
00:33:24,180 --> 00:33:26,650
That equation can be put
in exactly the same form

723
00:33:26,650 --> 00:33:29,210
as the Schrodinger equation for
the time evolution of a wave

724
00:33:29,210 --> 00:33:30,020
function.

725
00:33:30,020 --> 00:33:31,370
They're both waves.

726
00:33:31,370 --> 00:33:33,870
And so it's not so surprising
these two wave equations are

727
00:33:33,870 --> 00:33:35,730
related to each
other a nice way.

728
00:33:35,730 --> 00:33:38,200
Meanwhile, the index
of the dielectric

729
00:33:38,200 --> 00:33:42,180
turns into the potential for
the quantum mechanical problem.

730
00:33:42,180 --> 00:33:44,530
So if you have a periodic
potential, what do you want?

731
00:33:44,530 --> 00:33:47,060
You want a periodic
dielectric constant.

732
00:33:47,060 --> 00:33:48,080
Yeah.

733
00:33:48,080 --> 00:33:50,590
And so you can build a
system which incredibly,

734
00:33:50,590 --> 00:33:53,220
cleanly, has a periodic
dielectric constant

735
00:33:53,220 --> 00:33:54,450
and no disorder.

736
00:33:54,450 --> 00:33:56,630
And then you can put
light into the system,

737
00:33:56,630 --> 00:33:59,300
and you can ask what
happens to this system.

738
00:33:59,300 --> 00:34:00,510
So here's the idea,

739
00:34:00,510 --> 00:34:03,730
I take a system which
is a periodic-- I'm

740
00:34:03,730 --> 00:34:07,660
going to draw the
potential here.

741
00:34:07,660 --> 00:34:09,760
So I'm going to draw
the dielectric constant.

742
00:34:09,760 --> 00:34:15,100
So small, large, small, large,
small, large, small, large, et

743
00:34:15,100 --> 00:34:17,095
cetera.

744
00:34:17,095 --> 00:34:19,345
But instead of having it be
a one dimensional lattice,

745
00:34:19,345 --> 00:34:22,110
I'm going to make it a
two dimensional lattice.

746
00:34:22,110 --> 00:34:25,230
So now, basically, I've
got a set of wave guides.

747
00:34:25,230 --> 00:34:26,540
Let me draw this differently.

748
00:34:41,207 --> 00:34:42,790
So does everyone get
the picture here?

749
00:34:42,790 --> 00:34:44,206
So literally what
you have, is you

750
00:34:44,206 --> 00:34:46,120
have glass, glass with
a different index,

751
00:34:46,120 --> 00:34:47,360
glass, glass with a
different-- if you

752
00:34:47,360 --> 00:34:49,300
can think of those as
a line of glass fibers.

753
00:34:49,300 --> 00:34:50,830
Optical fibers.

754
00:34:50,830 --> 00:34:54,179
And you shine your light that's
reasonably well localized,

755
00:34:54,179 --> 00:34:58,840
in both position, and in phase
variation, or crystal momentum.

756
00:34:58,840 --> 00:35:01,770
Because you can control
the phase of the light.

757
00:35:01,770 --> 00:35:06,040
So you send this wave packet
in and you ask what happens.

758
00:35:06,040 --> 00:35:07,900
Well not a whole lot happens.

759
00:35:07,900 --> 00:35:09,330
It's a wave packet.

760
00:35:09,330 --> 00:35:11,320
It's going through
a wave guide, but we

761
00:35:11,320 --> 00:35:13,035
haven't implemented
an electric field.

762
00:35:13,035 --> 00:35:15,160
To handle an electric field,
you need the potential

763
00:35:15,160 --> 00:35:15,990
to be constantly varying.

764
00:35:15,990 --> 00:35:16,490
Uh huh.

765
00:35:16,490 --> 00:35:18,365
So it's at a linear
ramp into the potential.

766
00:35:18,365 --> 00:35:20,240
Instead of making it
just perfectly periodic,

767
00:35:20,240 --> 00:35:23,160
let's make the index
ramp just a little bit.

768
00:35:23,160 --> 00:35:26,010
And this experiment
has been done.

769
00:35:26,010 --> 00:35:28,260
In this experiment, so as
the wave packet moves along,

770
00:35:28,260 --> 00:35:30,343
what's discovered is that
the position-- if I draw

771
00:35:30,343 --> 00:35:34,281
the x as a function of
t, so now the role of t

772
00:35:34,281 --> 00:35:35,780
is being played by
the distance it's

773
00:35:35,780 --> 00:35:41,575
moved along the wave guide, what
you find is that it does this.

774
00:35:41,575 --> 00:35:45,540
It exhibits beautiful
block oscillations.

775
00:35:45,540 --> 00:35:50,690
And this has been proved
in a very small number

776
00:35:50,690 --> 00:35:53,330
of real honest quantum
mechanical systems.

777
00:35:53,330 --> 00:35:55,080
The most elegant
experiment that I know of

778
00:35:55,080 --> 00:35:58,240
was done by Wolfgang
Ketterle, who's here at MIT.

779
00:35:58,240 --> 00:36:01,020
And he got three data points
because it was preposterously

780
00:36:01,020 --> 00:36:02,650
difficult and declared victory.

781
00:36:02,650 --> 00:36:04,870
So I talked to him about
this in the hallway one day.

782
00:36:04,870 --> 00:36:06,150
And he said yes,
this was ridiculous,

783
00:36:06,150 --> 00:36:07,358
but we got three data points.

784
00:36:07,358 --> 00:36:08,980
We got small, we got large.

785
00:36:08,980 --> 00:36:09,480
Victory.

786
00:36:09,480 --> 00:36:10,380
We declared victory.

787
00:36:10,380 --> 00:36:11,880
But it really needs
to be done well.

788
00:36:11,880 --> 00:36:13,520
So one of the
interesting questions

789
00:36:13,520 --> 00:36:16,564
in this part of the field right
now is we know that it's true.

790
00:36:16,564 --> 00:36:17,480
But we want to see it.

791
00:36:17,480 --> 00:36:20,542
We want to feel it, so various
people around the world

792
00:36:20,542 --> 00:36:22,750
are working on making a
truly beautiful demonstration

793
00:36:22,750 --> 00:36:23,881
of this bit of physics.

794
00:36:23,881 --> 00:36:24,380
Yeah.

795
00:36:24,380 --> 00:36:25,372
AUDIENCE: [INAUDIBLE]

796
00:36:33,804 --> 00:36:35,705
PROFESSOR: It's
totally impractical,

797
00:36:35,705 --> 00:36:39,940
because any interference
is just going to kill you.

798
00:36:39,940 --> 00:36:40,806
Unfortunately.

799
00:36:40,806 --> 00:36:42,180
So, you have to
work ridiculously

800
00:36:42,180 --> 00:36:43,690
hard to make systems clean.

801
00:36:43,690 --> 00:36:45,190
So the question is
really a question

802
00:36:45,190 --> 00:36:47,950
about quantum computation,
which we'll come to next week.

803
00:36:47,950 --> 00:36:50,322
But, the basic question
is how robust is this.

804
00:36:50,322 --> 00:36:52,030
And the answer is it's
not robust at all.

805
00:36:52,030 --> 00:36:54,840
But which you can tell because
everything in the real world

806
00:36:54,840 --> 00:36:57,509
has enough impurity
that it conducts.

807
00:36:57,509 --> 00:36:58,300
Or as an insulator.

808
00:36:58,300 --> 00:36:58,800
Yeah.

809
00:36:58,800 --> 00:37:02,110
AUDIENCE: What place sort of
like the larger role in sort

810
00:37:02,110 --> 00:37:04,875
of like the perfection of
a lattice like temperature

811
00:37:04,875 --> 00:37:05,590
or impurities.

812
00:37:05,590 --> 00:37:07,214
PROFESSOR: That's a
very good question.

813
00:37:07,214 --> 00:37:11,086
So the question is what's
the most important property?

814
00:37:11,086 --> 00:37:12,460
What's most
important disordering

815
00:37:12,460 --> 00:37:14,810
property that leads
to conduction?

816
00:37:14,810 --> 00:37:17,390
And there's temperature
fluctuations,

817
00:37:17,390 --> 00:37:19,920
there are impurities
in the lattice.

818
00:37:19,920 --> 00:37:23,434
There are decohereing
effects which

819
00:37:23,434 --> 00:37:24,600
is a more complicated story.

820
00:37:24,600 --> 00:37:26,641
And that's actually, it
depends on the situation,

821
00:37:26,641 --> 00:37:27,790
it depends on the system.

822
00:37:27,790 --> 00:37:30,060
And exactly how it
depends is something

823
00:37:30,060 --> 00:37:32,170
that is an active
area of research.

824
00:37:32,170 --> 00:37:34,580
Now there are many, many
ways to probe this physics.

825
00:37:34,580 --> 00:37:36,260
So we know that these block
oscillations are true.

826
00:37:36,260 --> 00:37:38,170
We see them in all sorts
of different systems

827
00:37:38,170 --> 00:37:39,970
that are analogous.

828
00:37:39,970 --> 00:37:41,636
So there's lots of
[INAUDIBLE], it's not

829
00:37:41,636 --> 00:37:43,345
like this is an
ambiguous bit of physics.

830
00:37:43,345 --> 00:37:45,261
But it's one that turns
out to be surprisingly

831
00:37:45,261 --> 00:37:46,340
difficult to tease apart.

832
00:37:46,340 --> 00:37:48,560
The reason I bring all this up
is to emphasize the following,

833
00:37:48,560 --> 00:37:50,770
our model made a prediction
that disagreed explicitly

834
00:37:50,770 --> 00:37:54,780
with the connectivity property
of copper and other materials.

835
00:37:54,780 --> 00:37:57,820
So don't throw away the model.

836
00:37:57,820 --> 00:38:01,040
Observe that you've
modeled the wrong system.

837
00:38:01,040 --> 00:38:03,180
If you find a system
that fits your--

838
00:38:03,180 --> 00:38:05,440
that is-- that shares
the assumptions

839
00:38:05,440 --> 00:38:07,730
of your model, that's
when you ask did it work.

840
00:38:07,730 --> 00:38:09,360
And it worked like a champ.

841
00:38:09,360 --> 00:38:09,860
OK.

842
00:38:09,860 --> 00:38:11,760
So now let's talk
about real materials.

843
00:38:11,760 --> 00:38:14,504
This is going to close up our
discussion bands and solids.

844
00:38:14,504 --> 00:38:15,920
And this is actually
what I wanted

845
00:38:15,920 --> 00:38:19,270
to get to at the
beginning of the lecture.

846
00:38:19,270 --> 00:38:20,820
But that's OK.

847
00:38:20,820 --> 00:38:24,370
There are lots of questions
and they were good questions.

848
00:38:24,370 --> 00:38:26,100
So this is an extremely brief.

849
00:38:26,100 --> 00:38:27,975
But I want to ask you
the following question.

850
00:38:27,975 --> 00:38:31,370
What happens in the
following three systems?

851
00:38:31,370 --> 00:38:38,926
So first, imagine
we take why don't we

852
00:38:38,926 --> 00:38:41,860
take a system with built
out of single wells,

853
00:38:41,860 --> 00:38:47,030
which have some set
of energy eigenstates,

854
00:38:47,030 --> 00:38:51,010
and then we build the
periodic array out of them.

855
00:38:51,010 --> 00:38:51,900
What do we expect?

856
00:38:51,900 --> 00:38:53,870
And let me draw this bigger.

857
00:39:02,270 --> 00:39:04,319
What do we expect to see
when we build a lattice?

858
00:39:04,319 --> 00:39:06,610
We expect that this is going
to-- that these states are

859
00:39:06,610 --> 00:39:08,970
going to spread out
into bands a funny way

860
00:39:08,970 --> 00:39:11,040
Yeah and let's just talk
about the 1 d potential.

861
00:39:11,040 --> 00:39:14,522
So what we'll find is that this
band turns into-- I'm sorry.

862
00:39:14,522 --> 00:39:15,980
This state, this
single state turns

863
00:39:15,980 --> 00:39:17,930
into a band of allowed
energy eigenstates.

864
00:39:23,260 --> 00:39:26,620
There's now a plot
of the energy.

865
00:39:26,620 --> 00:39:30,260
And similarly,
this state is going

866
00:39:30,260 --> 00:39:32,760
to lead to another
band with some width.

867
00:39:37,940 --> 00:39:41,830
And this state is going to
lead to another band, which

868
00:39:41,830 --> 00:39:42,460
is even wider.

869
00:39:47,100 --> 00:39:48,100
Everyone cool with that?

870
00:39:51,965 --> 00:39:52,590
Quick question?

871
00:39:52,590 --> 00:39:56,320
In 1 d, do these
bands ever overlap?

872
00:39:56,320 --> 00:39:56,820
No.

873
00:39:56,820 --> 00:39:57,780
By the node theorem.

874
00:39:57,780 --> 00:39:58,290
Right?

875
00:39:58,290 --> 00:39:58,789
OK.

876
00:39:58,789 --> 00:40:00,700
Now let's take a single
electron, and let's

877
00:40:00,700 --> 00:40:02,780
put in-- let's take
a single electron,

878
00:40:02,780 --> 00:40:04,090
and let's put it in the system.

879
00:40:04,090 --> 00:40:04,970
What will happen?

880
00:40:04,970 --> 00:40:06,761
Well if we put it in
the system, what state

881
00:40:06,761 --> 00:40:10,690
will this single
electron fall into?

882
00:40:10,690 --> 00:40:11,340
Yeah one event.

883
00:40:11,340 --> 00:40:12,676
But which state?

884
00:40:12,676 --> 00:40:13,550
AUDIENCE: [INAUDIBLE]

885
00:40:13,550 --> 00:40:15,360
PROFESSOR: Yeah, if you
kick the system around,

886
00:40:15,360 --> 00:40:16,460
you let it relax a little bit.

887
00:40:16,460 --> 00:40:18,210
It's going to fall down
to the ground state.

888
00:40:18,210 --> 00:40:20,030
You have to couple to
something else like hydrogen

889
00:40:20,030 --> 00:40:22,405
has to be coupled with an
electromagnetic field to decay.

890
00:40:22,405 --> 00:40:24,130
But couple it, kick
it, and let it decay.

891
00:40:24,130 --> 00:40:25,714
It'll settle down
to its ground state.

892
00:40:25,714 --> 00:40:27,921
So you get an electron down
here in the ground state,

893
00:40:27,921 --> 00:40:29,470
and looking back
at that band, we

894
00:40:29,470 --> 00:40:31,845
know that the band for that
ground state looks like this.

895
00:40:34,900 --> 00:40:36,576
So, here it is.

896
00:40:36,576 --> 00:40:37,450
There's our electron.

897
00:40:37,450 --> 00:40:39,400
It's sitting in the
lowest energy eigenstate.

898
00:40:39,400 --> 00:40:40,030
Is it moving?

899
00:40:44,100 --> 00:40:47,530
Well, it's in a
stationary state.

900
00:40:47,530 --> 00:40:50,880
Is the expectation value of
the position changing in time?

901
00:40:50,880 --> 00:40:51,380
No.

902
00:40:51,380 --> 00:40:53,190
The expectation values
don't change in time,

903
00:40:53,190 --> 00:40:54,020
in the stationary state.

904
00:40:54,020 --> 00:40:55,700
That's part of what it
is to be a stationery

905
00:40:55,700 --> 00:40:57,020
state, to be an
energy eigenstate.

906
00:40:57,020 --> 00:40:57,230
OK.

907
00:40:57,230 --> 00:40:58,188
Great. it's not moving.

908
00:40:58,188 --> 00:41:02,350
Now, in order to make it
move, what do you have to do?

909
00:41:02,350 --> 00:41:05,250
What kind of state corresponds
to the position changing

910
00:41:05,250 --> 00:41:07,410
in time?

911
00:41:07,410 --> 00:41:07,910
Yes.

912
00:41:07,910 --> 00:41:08,700
Superposition.

913
00:41:08,700 --> 00:41:08,990
Right?

914
00:41:08,990 --> 00:41:10,970
From the superpositions
we'll get interference terms.

915
00:41:10,970 --> 00:41:13,270
So if we put in a superposition
of say, this state,

916
00:41:13,270 --> 00:41:15,020
and this state,
which corresponds

917
00:41:15,020 --> 00:41:17,734
to different energies.

918
00:41:17,734 --> 00:41:19,650
If we put it in a
superposition of these guys,

919
00:41:19,650 --> 00:41:22,640
then it's meaningfully moving.

920
00:41:22,640 --> 00:41:25,690
It has some meaningful,
well defined time variation

921
00:41:25,690 --> 00:41:28,680
of its position
expectation value.

922
00:41:28,680 --> 00:41:32,330
So in order to induce
a current, in order

923
00:41:32,330 --> 00:41:34,767
to induce a current
of this system

924
00:41:34,767 --> 00:41:36,350
where the electron
wave packet carries

925
00:41:36,350 --> 00:41:39,170
a little bit of momentum is
changing in time it's position,

926
00:41:39,170 --> 00:41:41,545
what do I have to do to the
electron in the ground state?

927
00:41:41,545 --> 00:41:42,919
I have to excite
it, so that it's

928
00:41:42,919 --> 00:41:44,560
in a superposition
of the grounds state

929
00:41:44,560 --> 00:41:45,880
and some excited state.

930
00:41:45,880 --> 00:41:48,730
Or more generally, into a
superposition of other states.

931
00:41:48,730 --> 00:41:49,610
Yes?

932
00:41:49,610 --> 00:41:51,515
In order to induce
the current, I

933
00:41:51,515 --> 00:41:54,567
must put the electron
into a higher energy state

934
00:41:54,567 --> 00:41:56,650
and in a particular
superposition of higher energy

935
00:41:56,650 --> 00:41:58,140
states.

936
00:41:58,140 --> 00:41:59,860
Everyone down with that?

937
00:41:59,860 --> 00:42:03,240
Here's why this is so important.

938
00:42:03,240 --> 00:42:05,706
Imagine each one of
these wells is actually

939
00:42:05,706 --> 00:42:07,330
not some square well,
but it's an atom.

940
00:42:09,970 --> 00:42:11,460
And let's say the
atom is hydrogen,

941
00:42:11,460 --> 00:42:14,034
just for-- this doesn't
actually happen,

942
00:42:14,034 --> 00:42:15,950
but just imagine-- in
particular what it means

943
00:42:15,950 --> 00:42:19,156
is it has the ion, the
nucleus is charge plus 1.

944
00:42:19,156 --> 00:42:21,030
And so in order for the
system to be neutral,

945
00:42:21,030 --> 00:42:26,110
I must have one
electron for every well.

946
00:42:26,110 --> 00:42:29,387
So if I have n wells, I must
have n electrons in the system.

947
00:42:29,387 --> 00:42:30,470
Everybody agree with that?

948
00:42:30,470 --> 00:42:31,250
In order to be neutral.

949
00:42:31,250 --> 00:42:32,500
Otherwise, the thing's
charged and all sorts

950
00:42:32,500 --> 00:42:33,958
of terrible things--
electrons will

951
00:42:33,958 --> 00:42:36,679
get ripped off from nearby cad.

952
00:42:36,679 --> 00:42:38,220
So we must have an
electron per well.

953
00:42:38,220 --> 00:42:41,430
How many states
are in this band?

954
00:42:41,430 --> 00:42:44,310
For n wells?

955
00:42:44,310 --> 00:42:44,820
n.

956
00:42:44,820 --> 00:42:45,410
Right?

957
00:42:45,410 --> 00:42:45,570
OK.

958
00:42:45,570 --> 00:42:48,111
So if I put in the n electrons
I need to neutralize a system,

959
00:42:48,111 --> 00:42:51,380
where do those n electrons go?

960
00:42:51,380 --> 00:42:53,240
Yeah, they fill
up the first band.

961
00:42:53,240 --> 00:42:55,490
And if we let the system
relax with lowest energy

962
00:42:55,490 --> 00:42:58,610
configuration, every
state in this lowest band

963
00:42:58,610 --> 00:43:02,260
will be filled, and none of
these states will be filled.

964
00:43:02,260 --> 00:43:04,790
Everyone down with that?

965
00:43:04,790 --> 00:43:06,310
So here's my question.

966
00:43:06,310 --> 00:43:08,320
When I've got that ground
state configuration

967
00:43:08,320 --> 00:43:13,570
of this lattice of atoms
with one electron per well,

968
00:43:13,570 --> 00:43:15,180
in these distributed
wave functions,

969
00:43:15,180 --> 00:43:17,910
filling out these bands,
is anything moving?

970
00:43:20,590 --> 00:43:23,260
Wow, you guys are
so quiet today.

971
00:43:23,260 --> 00:43:24,950
Is anything moving?

972
00:43:24,950 --> 00:43:26,950
This system is in an
energy eigenstate.

973
00:43:26,950 --> 00:43:29,097
In particular, it's in a
completely antisymmetrized

974
00:43:29,097 --> 00:43:31,180
configuration, because
they're identical fermions.

975
00:43:34,140 --> 00:43:36,580
So, nothing is moving.

976
00:43:36,580 --> 00:43:38,870
If we want to induce a
current, what do we have to do?

977
00:43:41,790 --> 00:43:42,290
Yeah.

978
00:43:42,290 --> 00:43:43,800
We have put them
in a superposition.

979
00:43:43,800 --> 00:43:47,710
But where's the next
allowed energy eigenstate?

980
00:43:47,710 --> 00:43:49,150
Next band.

981
00:43:49,150 --> 00:43:50,382
So it's in the next band.

982
00:43:50,382 --> 00:43:51,840
The next allowed
energy eigenstate.

983
00:43:51,840 --> 00:43:53,298
So the configuration
we have now is

984
00:43:53,298 --> 00:43:56,610
that these guys are all filled,
these guys are all empty,

985
00:43:56,610 --> 00:43:58,630
but in order to take
an electron from here

986
00:43:58,630 --> 00:44:00,880
and put it into
this excited state,

987
00:44:00,880 --> 00:44:03,460
we have to put in a minimum
amount of energy, which

988
00:44:03,460 --> 00:44:08,410
is the gap between
those two bands.

989
00:44:08,410 --> 00:44:09,510
Right?

990
00:44:09,510 --> 00:44:10,870
So now think about it this way.

991
00:44:10,870 --> 00:44:15,140
Suppose I take light and I
send my light at this crystal.

992
00:44:15,140 --> 00:44:18,570
In order for the light to
scatter off the crystal,

993
00:44:18,570 --> 00:44:21,460
you must have electrons
in superposition states

994
00:44:21,460 --> 00:44:25,370
so that they can have a
dipole and absorb and radiate

995
00:44:25,370 --> 00:44:27,090
that energy.

996
00:44:27,090 --> 00:44:27,590
Yeah.

997
00:44:30,380 --> 00:44:32,130
But in order for that
to happen, the light

998
00:44:32,130 --> 00:44:35,114
has to excite an
electron across the gap.

999
00:44:35,114 --> 00:44:37,280
It has to give it this
macroscopic amount of energy.

1000
00:44:37,280 --> 00:44:38,090
Well, it's not macroscopic.

1001
00:44:38,090 --> 00:44:38,595
it's large.

1002
00:44:38,595 --> 00:44:41,710
It's not infinitesimally small.

1003
00:44:41,710 --> 00:44:43,920
That means that there's a
minimum amount of energy

1004
00:44:43,920 --> 00:44:46,230
that that incident
light must have in order

1005
00:44:46,230 --> 00:44:49,140
to excite the electron
in the first place.

1006
00:44:49,140 --> 00:44:54,100
So very long wavelength
light will never do that.

1007
00:44:54,100 --> 00:44:56,420
Light along wavelength
will not have enough energy

1008
00:44:56,420 --> 00:44:59,580
to excite an electron across
this gap into the next band

1009
00:44:59,580 --> 00:45:01,370
to allow there to
be a current, which

1010
00:45:01,370 --> 00:45:03,880
could oppose the electric field.

1011
00:45:03,880 --> 00:45:07,990
So the only for light to
scatter off of this crystal,

1012
00:45:07,990 --> 00:45:12,670
is if the energy, h
bar omega, of the light

1013
00:45:12,670 --> 00:45:14,930
is greater than or
equal to, let's say

1014
00:45:14,930 --> 00:45:18,124
greater than approximately,
the band gap delta e.

1015
00:45:21,210 --> 00:45:22,310
That cool?

1016
00:45:22,310 --> 00:45:24,210
We've just discovered something.

1017
00:45:24,210 --> 00:45:25,790
Crystals are
transparent unless you

1018
00:45:25,790 --> 00:45:28,530
look at sufficiently
high frequencies.

1019
00:45:28,530 --> 00:45:30,201
That's cool.

1020
00:45:30,201 --> 00:45:30,700
Right?

1021
00:45:30,700 --> 00:45:32,450
A crystal is
transparent unless you

1022
00:45:32,450 --> 00:45:34,120
look at sufficiently
high frequencies.

1023
00:45:34,120 --> 00:45:36,220
If you look at low
frequencies, your crystal

1024
00:45:36,220 --> 00:45:39,050
should be transparent.

1025
00:45:39,050 --> 00:45:40,380
Well that's really interesting.

1026
00:45:40,380 --> 00:45:42,740
In particular, we immediately
learn something cool

1027
00:45:42,740 --> 00:45:44,420
about two different materials.

1028
00:45:44,420 --> 00:45:48,310
Consider diamond and copper.

1029
00:45:50,890 --> 00:45:53,380
These are both crystals.

1030
00:45:53,380 --> 00:45:56,570
They're solids made
out of a regular array,

1031
00:45:56,570 --> 00:45:58,730
perhaps not perfect, but
extraordinarily good,

1032
00:45:58,730 --> 00:46:02,167
regular array of atoms
of the same time.

1033
00:46:02,167 --> 00:46:03,500
Array in a particular structure.

1034
00:46:03,500 --> 00:46:05,210
Diamond, anything and I
think face inner cubic.

1035
00:46:05,210 --> 00:46:05,700
I don't remember.

1036
00:46:05,700 --> 00:46:06,360
I really should know that.

1037
00:46:06,360 --> 00:46:06,950
Anyway, copper.

1038
00:46:06,950 --> 00:46:07,575
It's a lattice.

1039
00:46:11,317 --> 00:46:12,150
That's embarrassing.

1040
00:46:12,150 --> 00:46:14,530
I really should know that.

1041
00:46:14,530 --> 00:46:17,810
So we have these
two materials which

1042
00:46:17,810 --> 00:46:22,244
one has the larger band gap?

1043
00:46:22,244 --> 00:46:23,660
Diamond, because
it's transparent.

1044
00:46:23,660 --> 00:46:24,910
At in the visible.

1045
00:46:24,910 --> 00:46:28,060
So the band gap,
delta e of diamond

1046
00:46:28,060 --> 00:46:33,620
is much larger than the
band gap for copper.

1047
00:46:33,620 --> 00:46:35,760
But in fact, this is
a little more subtle,

1048
00:46:35,760 --> 00:46:39,308
because copper in fact,
doesn't even have a band gap.

1049
00:46:43,530 --> 00:46:45,105
We made an important
assumption here.

1050
00:46:45,105 --> 00:46:47,230
So I want to think about--
we're going to come back

1051
00:46:47,230 --> 00:46:48,605
to copper in a
second, but I want

1052
00:46:48,605 --> 00:46:50,030
to point out the
nice thing here.

1053
00:46:50,030 --> 00:46:53,910
Which is that diamond
has to have a band gap.

1054
00:46:53,910 --> 00:46:54,910
It's transparent.

1055
00:46:54,910 --> 00:46:56,310
It must have band gap.

1056
00:46:56,310 --> 00:46:59,740
It must be such that when you
fill up all the electrons you

1057
00:46:59,740 --> 00:47:02,830
need for it to be neutral, there
is a gap to the next energy

1058
00:47:02,830 --> 00:47:03,330
states.

1059
00:47:03,330 --> 00:47:09,670
And that gap must be larger than
visible wavelengths of light.

1060
00:47:09,670 --> 00:47:10,330
Yeah.

1061
00:47:10,330 --> 00:47:10,830
That's cool.

1062
00:47:10,830 --> 00:47:13,038
And that must be true of
all the transparent crystals

1063
00:47:13,038 --> 00:47:14,500
that you see.

1064
00:47:14,500 --> 00:47:16,830
Otherwise, they
wouldn't be transparent.

1065
00:47:16,830 --> 00:47:18,830
They would respond by
having free electrons that

1066
00:47:18,830 --> 00:47:20,190
could respond like a metal.

1067
00:47:20,190 --> 00:47:20,690
Yeah.

1068
00:47:20,690 --> 00:47:21,914
AUDIENCE: So, [INAUDIBLE]

1069
00:47:25,400 --> 00:47:26,210
PROFESSOR: Yeah.

1070
00:47:29,190 --> 00:47:31,210
Are diamonds good conductors?

1071
00:47:31,210 --> 00:47:31,710
No.

1072
00:47:31,710 --> 00:47:33,090
They're terrible conductors.

1073
00:47:33,090 --> 00:47:36,090
In fact, there
preposterously-- if you

1074
00:47:36,090 --> 00:47:38,573
compare the number of--
I'll get into this later.

1075
00:47:38,573 --> 00:47:39,656
But yes, they're terrible.

1076
00:47:39,656 --> 00:47:42,040
AUDIENCE: [INAUDIBLE]

1077
00:47:42,040 --> 00:47:45,547
PROFESSOR: Uh, that's a slightly
more complicated story, which

1078
00:47:45,547 --> 00:47:46,380
let me come back to.

1079
00:47:46,380 --> 00:47:48,255
Hold on to that, and if
I don't answer today,

1080
00:47:48,255 --> 00:47:50,652
ask me after in office hours
because it's a little more--

1081
00:47:50,652 --> 00:47:52,500
what?

1082
00:47:52,500 --> 00:47:54,530
Really?

1083
00:47:54,530 --> 00:47:55,030
Wow.

1084
00:48:01,130 --> 00:48:02,510
Well, MIT.

1085
00:48:06,000 --> 00:48:07,920
It's all about the intellect.

1086
00:48:07,920 --> 00:48:09,550
And everything else has to-- OK.

1087
00:48:09,550 --> 00:48:11,710
So, this is pretty good,
but here's the thing.

1088
00:48:11,710 --> 00:48:14,780
In one dimensional crystals,
the only thing that can happen

1089
00:48:14,780 --> 00:48:17,920
is, look if you have each
band come from allowed energy

1090
00:48:17,920 --> 00:48:22,030
state and each energy
state, each well

1091
00:48:22,030 --> 00:48:25,140
comes with one electron, or two
electrons, or three electrons,

1092
00:48:25,140 --> 00:48:27,190
you will always have filled
bands, and then a gap

1093
00:48:27,190 --> 00:48:28,524
and filled bands and then a gap.

1094
00:48:28,524 --> 00:48:29,814
Does everybody agree with that?

1095
00:48:29,814 --> 00:48:31,570
You can't have a
partially filled band

1096
00:48:31,570 --> 00:48:34,590
if each band comes from a
bouncy, in a single well,

1097
00:48:34,590 --> 00:48:38,540
and each well comes with an
integer number of electrons.

1098
00:48:38,540 --> 00:48:40,512
You just-- you're stuck.

1099
00:48:40,512 --> 00:48:41,012
Yeah.

1100
00:48:41,012 --> 00:48:42,004
AUDIENCE: [INAUDIBLE]

1101
00:48:42,004 --> 00:48:44,050
PROFESSOR: Oh.

1102
00:48:44,050 --> 00:48:45,330
I'm lying about spin.

1103
00:48:45,330 --> 00:48:50,530
But spin in one dimension is
little-- I'm lying about spin.

1104
00:48:53,200 --> 00:48:56,310
But do you really want
me to get in spin?

1105
00:48:56,310 --> 00:48:56,810
Man.

1106
00:48:56,810 --> 00:48:57,309
OK.

1107
00:48:57,309 --> 00:48:59,310
So if we include spin,
and we have splitting,

1108
00:48:59,310 --> 00:49:00,550
then it becomes a
more subtle story.

1109
00:49:00,550 --> 00:49:02,425
If we include spin, then
there are two states

1110
00:49:02,425 --> 00:49:04,680
for every allowed energy
eigenstate of the potential.

1111
00:49:04,680 --> 00:49:06,300
However, there are
generically going

1112
00:49:06,300 --> 00:49:09,525
to be interactions between
the-- there are generically

1113
00:49:09,525 --> 00:49:11,150
going to be magnetic
interactions which

1114
00:49:11,150 --> 00:49:13,319
split the energy of
those two spin states.

1115
00:49:13,319 --> 00:49:15,110
Electrons spin up, and
electrons spin down,

1116
00:49:15,110 --> 00:49:17,410
will generically have
different energies.

1117
00:49:17,410 --> 00:49:19,884
Now in 3D, this isn't
such a big deal,

1118
00:49:19,884 --> 00:49:21,300
because those
splittings are tiny,

1119
00:49:21,300 --> 00:49:22,883
and so the states
can sort of overlap.

1120
00:49:22,883 --> 00:49:24,880
But in 1D they can't.

1121
00:49:24,880 --> 00:49:28,080
So I mean, that's
also not exactly true,

1122
00:49:28,080 --> 00:49:30,010
but it depends on
exactly the details.

1123
00:49:30,010 --> 00:49:31,890
It depends on the
details of the system,

1124
00:49:31,890 --> 00:49:33,015
is what I wanted to get to.

1125
00:49:36,690 --> 00:49:38,200
Curse you.

1126
00:49:38,200 --> 00:49:40,340
So let me talk about
the same phenomena

1127
00:49:40,340 --> 00:49:41,846
in an easier context,
where we don't

1128
00:49:41,846 --> 00:49:44,470
have to worry about spin, which
we haven't discussed in detail,

1129
00:49:44,470 --> 00:49:45,080
in the class.

1130
00:49:45,080 --> 00:49:46,501
Which is in three dimensions.

1131
00:49:46,501 --> 00:49:48,250
Where the story changes
in a dramatic way.

1132
00:49:48,250 --> 00:49:48,749
Yeah.

1133
00:49:48,749 --> 00:49:49,651
AUDIENCE: [INAUDIBLE]

1134
00:49:54,124 --> 00:49:54,930
PROFESSOR: Oh.

1135
00:49:54,930 --> 00:49:55,497
It's not.

1136
00:49:55,497 --> 00:49:56,580
Generically, no, it's not.

1137
00:49:56,580 --> 00:49:57,165
It will depend.

1138
00:49:57,165 --> 00:49:58,039
AUDIENCE: [INAUDIBLE]

1139
00:50:03,138 --> 00:50:06,277
PROFESSOR: You say it happens
to be the salient one.

1140
00:50:06,277 --> 00:50:07,614
Yeah, exactly.

1141
00:50:07,614 --> 00:50:08,530
That is exactly right.

1142
00:50:08,530 --> 00:50:09,863
There the gaps are not the same.

1143
00:50:09,863 --> 00:50:13,370
That they do not
remain constant.

1144
00:50:13,370 --> 00:50:13,870
OK.

1145
00:50:13,870 --> 00:50:15,220
So let's talk more
about this system,

1146
00:50:15,220 --> 00:50:16,780
but let's talk about
it in three dimensions.

1147
00:50:16,780 --> 00:50:19,196
So in three dimensions, you
guys did an interesting thing,

1148
00:50:19,196 --> 00:50:21,270
when you studied,
you didn't know

1149
00:50:21,270 --> 00:50:24,030
this was about the structure
of solids, but it really was.

1150
00:50:24,030 --> 00:50:25,830
When you studied
the rigid rotor.

1151
00:50:25,830 --> 00:50:27,371
And when you studied
the rigid rotor,

1152
00:50:27,371 --> 00:50:29,320
you found that you
had energy eigenstates

1153
00:50:29,320 --> 00:50:33,810
and they were degenerate
with degeneracy 2 l plus 1.

1154
00:50:33,810 --> 00:50:35,800
The various different
l z eigenstates.

1155
00:50:35,800 --> 00:50:36,620
Yeah.

1156
00:50:36,620 --> 00:50:38,120
And then we turned
on an interaction

1157
00:50:38,120 --> 00:50:41,110
which was the energy costs,
the energy penalty for having

1158
00:50:41,110 --> 00:50:42,570
angle momentum in z direction.

1159
00:50:42,570 --> 00:50:45,110
Which added an l z
term to the energy.

1160
00:50:45,110 --> 00:50:48,140
And what you found is that as
a function of the coefficient,

1161
00:50:48,140 --> 00:50:52,155
which I think we called
epsilon, of that perturbation

1162
00:50:52,155 --> 00:50:59,900
of the energies of the energy
was equal to l squared over 2

1163
00:50:59,900 --> 00:51:04,004
i plus epsilon l z.

1164
00:51:04,004 --> 00:51:05,670
What you found is
that these guys split.

1165
00:51:05,670 --> 00:51:07,180
So this remained constant.

1166
00:51:07,180 --> 00:51:11,720
And this split into, so this
is the [INAUDIBLE] l equals 1.

1167
00:51:11,720 --> 00:51:12,890
So l equals zero.

1168
00:51:12,890 --> 00:51:15,170
So this is one, this
is three, this is five.

1169
00:51:15,170 --> 00:51:17,630
So the l equals zero
state, nothing happens.

1170
00:51:17,630 --> 00:51:18,230
l equals 1.

1171
00:51:18,230 --> 00:51:20,780
There's one that changes,
one that doesn't.

1172
00:51:20,780 --> 00:51:22,280
And then this guy has five.

1173
00:51:22,280 --> 00:51:26,161
One, two, three, four, five.

1174
00:51:26,161 --> 00:51:26,660
OK.

1175
00:51:26,660 --> 00:51:29,824
And what we found here is
that these guys could cross.

1176
00:51:29,824 --> 00:51:31,240
States from different
multiplates,

1177
00:51:31,240 --> 00:51:32,910
with different values
of l, had energies

1178
00:51:32,910 --> 00:51:35,550
that could cross as a
function of the strength

1179
00:51:35,550 --> 00:51:37,691
of the deformation
of your system.

1180
00:51:37,691 --> 00:51:38,190
Right?

1181
00:51:38,190 --> 00:51:39,610
The deformation is
where you have a sphere

1182
00:51:39,610 --> 00:51:40,920
and you stick out your arm.

1183
00:51:40,920 --> 00:51:42,330
So it's no longer symmetric top.

1184
00:51:45,530 --> 00:51:49,125
So here we can have
states crossing.

1185
00:51:49,125 --> 00:51:50,875
There's no nodes here
in three dimensions.

1186
00:51:53,720 --> 00:51:56,830
So as a consequence, when
you have a three dimensional

1187
00:51:56,830 --> 00:51:59,640
material built out of atoms.

1188
00:51:59,640 --> 00:52:01,390
So here's my sort of
pictorial description

1189
00:52:01,390 --> 00:52:03,348
of three dimensional
system built out of atoms.

1190
00:52:03,348 --> 00:52:05,570
You have a potential well,
potential, potential well.

1191
00:52:05,570 --> 00:52:08,670
Now, if the energy in one
particular potential well,

1192
00:52:08,670 --> 00:52:12,600
is like this, and like
this, and like this,

1193
00:52:12,600 --> 00:52:16,350
then when we add in a
lattice we get bands again.

1194
00:52:16,350 --> 00:52:18,480
The structure's a
little more intricate

1195
00:52:18,480 --> 00:52:20,180
because it depends
on the momentum.

1196
00:52:20,180 --> 00:52:24,340
But these bands now can overlap.

1197
00:52:28,700 --> 00:52:29,200
OK.

1198
00:52:33,230 --> 00:52:34,104
Everybody see that?

1199
00:52:34,104 --> 00:52:35,520
Because there's
nothing preventing

1200
00:52:35,520 --> 00:52:39,040
states from different--
in different multiplates

1201
00:52:39,040 --> 00:52:41,630
from having the same
energy in three dimensions.

1202
00:52:41,630 --> 00:52:43,190
There's no nodes
here that tells you

1203
00:52:43,190 --> 00:52:44,620
have to keep the
ordering constant

1204
00:52:44,620 --> 00:52:45,990
as you turn on the potential.

1205
00:52:45,990 --> 00:52:48,300
Now we turn on the multiple
particle potential,

1206
00:52:48,300 --> 00:52:50,940
and they can interact,
they can overlap.

1207
00:52:50,940 --> 00:52:52,750
As a consequence,
when we fill up,

1208
00:52:52,750 --> 00:52:55,095
let's say we two
electrons per potential,

1209
00:52:55,095 --> 00:53:00,720
or per well, when we filled
those first two bands, well,

1210
00:53:00,720 --> 00:53:04,100
there is the first-- so the
first band is now filled.

1211
00:53:04,100 --> 00:53:07,745
The second band and part of the
first-- part of the third band

1212
00:53:07,745 --> 00:53:09,870
and most of the second band
are going to be filled.

1213
00:53:09,870 --> 00:53:12,030
But part of the second
band is now available,

1214
00:53:12,030 --> 00:53:15,610
and much of the third
band is now available.

1215
00:53:15,610 --> 00:53:19,250
We filled in 2n electrons, but
we haven't filled up this band,

1216
00:53:19,250 --> 00:53:21,850
because it's really two
bands jammed together.

1217
00:53:21,850 --> 00:53:24,120
Or really bands from
two different orbitals

1218
00:53:24,120 --> 00:53:25,004
jammed together.

1219
00:53:25,004 --> 00:53:26,045
They happened to overlap.

1220
00:53:28,690 --> 00:53:30,770
So as a consequence
here, if this

1221
00:53:30,770 --> 00:53:33,760
is the length of the
energy of the last electron

1222
00:53:33,760 --> 00:53:38,397
that you put in, how much energy
do you have to give the system,

1223
00:53:38,397 --> 00:53:40,605
do you have to add the
system, to excite the energy--

1224
00:53:40,605 --> 00:53:43,100
or to excite the electrons
into excited states,

1225
00:53:43,100 --> 00:53:44,480
in particular into
superpositions

1226
00:53:44,480 --> 00:53:45,771
so that the electrons can move?

1227
00:53:45,771 --> 00:53:47,300
AUDIENCE: [INAUDIBLE]

1228
00:53:47,300 --> 00:53:48,250
PROFESSOR: Yeah.

1229
00:53:48,250 --> 00:53:49,470
Preposterously small amount.

1230
00:53:49,470 --> 00:53:52,440
An amount that goes like one
over the number of particles.

1231
00:53:52,440 --> 00:53:54,670
So in the continuum
limit, it's zero.

1232
00:53:54,670 --> 00:53:56,792
There's an arbitrarily
nearby energy.

1233
00:53:56,792 --> 00:53:59,000
So how much energy does it
take to excite an electron

1234
00:53:59,000 --> 00:54:03,650
and cause a current that opposes
the induced electric field?

1235
00:54:03,650 --> 00:54:04,232
Nothing.

1236
00:54:04,232 --> 00:54:05,690
Any electric field
that you send in

1237
00:54:05,690 --> 00:54:09,909
will be opposed by
an induced current.

1238
00:54:09,909 --> 00:54:11,700
So this behaves like
a classical conductor.

1239
00:54:11,700 --> 00:54:13,616
You turn on an electric
field, and the charges

1240
00:54:13,616 --> 00:54:16,390
will flow to oppose
that externally

1241
00:54:16,390 --> 00:54:18,610
imposed electric field.

1242
00:54:18,610 --> 00:54:21,340
You get charges then building up
on the walls of your capacitor

1243
00:54:21,340 --> 00:54:24,270
plates.

1244
00:54:24,270 --> 00:54:26,855
So, this is where
we have a conductor.

1245
00:54:30,460 --> 00:54:32,230
Because there's
an unfilled band.

1246
00:54:37,820 --> 00:54:46,230
And back here , we had an
insulator because we had filled

1247
00:54:46,230 --> 00:54:47,803
bands separated by gap.

1248
00:54:56,020 --> 00:54:59,760
The gap between the filled band
and the next available band.

1249
00:54:59,760 --> 00:55:01,570
This is actually called
a band insulator.

1250
00:55:01,570 --> 00:55:03,694
Because there are other
ways of being an insulator.

1251
00:55:05,770 --> 00:55:08,805
So from this so far, just from
the basic quantum mechanics

1252
00:55:08,805 --> 00:55:10,430
of a particle and a
periodic potential,

1253
00:55:10,430 --> 00:55:15,650
we now understand why some
crystals are transparent.

1254
00:55:15,650 --> 00:55:17,510
Why some materials conduct.

1255
00:55:17,510 --> 00:55:20,370
Why the materials that
are transparency are also

1256
00:55:20,370 --> 00:55:22,200
insulators.

1257
00:55:22,200 --> 00:55:27,241
And the things that conduct
are not transparent, generally.

1258
00:55:27,241 --> 00:55:27,740
Yeah.

1259
00:55:27,740 --> 00:55:28,698
AUDIENCE: [INAUDIBLE]

1260
00:55:35,883 --> 00:55:36,809
PROFESSOR: Excellent.

1261
00:55:36,809 --> 00:55:37,600
Excellent question.

1262
00:55:37,600 --> 00:55:40,375
So what's so special about
diamond and differ from copper?

1263
00:55:40,375 --> 00:55:41,750
And so the answer
goes like this.

1264
00:55:41,750 --> 00:55:44,000
So what determined the
exact band structure

1265
00:55:44,000 --> 00:55:46,640
in for a 1D periodic potential?

1266
00:55:46,640 --> 00:55:47,940
Two properties.

1267
00:55:47,940 --> 00:55:50,440
One was l, the periodicity.

1268
00:55:50,440 --> 00:55:52,610
And that came in
the q l and k l.

1269
00:55:52,610 --> 00:55:56,600
And the second is the detailed
shape of the potential.

1270
00:55:56,600 --> 00:55:58,210
Now in three
dimensions, the story's

1271
00:55:58,210 --> 00:55:59,900
going be a little
more complicated.

1272
00:55:59,900 --> 00:56:01,010
In three dimensions,
the things that

1273
00:56:01,010 --> 00:56:02,510
are going to determine
the potential

1274
00:56:02,510 --> 00:56:04,590
are not just the
distance between atoms,

1275
00:56:04,590 --> 00:56:06,510
but you have a three
dimensional lattice.

1276
00:56:06,510 --> 00:56:07,470
And the three
dimensional lattice

1277
00:56:07,470 --> 00:56:08,636
could have different shapes.

1278
00:56:08,636 --> 00:56:10,800
It could be cubic, it
could be hexagonal,

1279
00:56:10,800 --> 00:56:13,110
could be complicating in
all sorts of different ways.

1280
00:56:13,110 --> 00:56:13,210
Right?

1281
00:56:13,210 --> 00:56:14,790
It could be bent, it
could be rhomboidal,

1282
00:56:14,790 --> 00:56:17,165
and it could have all sorts
of different crystallographic

1283
00:56:17,165 --> 00:56:18,024
structures.

1284
00:56:18,024 --> 00:56:20,440
So that's going to go into it,
in the same way that l went

1285
00:56:20,440 --> 00:56:22,800
into it, which is the only
parameter in one dimension.

1286
00:56:22,800 --> 00:56:24,299
In the same way
that l goes into it.

1287
00:56:24,299 --> 00:56:26,470
So the crystal structure,
the shape of the lattice,

1288
00:56:26,470 --> 00:56:27,511
is going to determine it.

1289
00:56:27,511 --> 00:56:29,790
Secondly, the structure of
the orbitals is different.

1290
00:56:29,790 --> 00:56:30,630
Different atoms are
different wells,

1291
00:56:30,630 --> 00:56:32,504
so they'll give you
different band structure.

1292
00:56:32,504 --> 00:56:35,860
So different materials for
example, diamond versus copper,

1293
00:56:35,860 --> 00:56:38,440
are going to give you different
bands allowed energies,

1294
00:56:38,440 --> 00:56:40,100
because the potential
is different.

1295
00:56:40,100 --> 00:56:41,090
It has different shape.

1296
00:56:41,090 --> 00:56:42,790
And so when you
solve the problem

1297
00:56:42,790 --> 00:56:45,160
for the energy eigenvalues
is a function of now

1298
00:56:45,160 --> 00:56:47,630
the three different components
of the crystal momentum,

1299
00:56:47,630 --> 00:56:49,540
you'll just get a
different set of equations.

1300
00:56:49,540 --> 00:56:53,070
And working those out
is not terribly hard.

1301
00:56:53,070 --> 00:56:58,370
But it's a computation that must
be done, and it is not trivial.

1302
00:56:58,370 --> 00:57:00,992
And so one of the sort of, I
don't know if I'd say exciting,

1303
00:57:00,992 --> 00:57:02,450
but one of the
things that one does

1304
00:57:02,450 --> 00:57:04,015
when one takes a
course in solids,

1305
00:57:04,015 --> 00:57:05,640
is you go through a
bunch of materials.

1306
00:57:05,640 --> 00:57:07,098
And you understand
the relationship

1307
00:57:07,098 --> 00:57:10,000
between the potential, at
the atomic orbital structure

1308
00:57:10,000 --> 00:57:12,540
of the individual atom,
the crystal structure,

1309
00:57:12,540 --> 00:57:13,970
and the resulting
band structure.

1310
00:57:13,970 --> 00:57:15,340
And there's some sort
of nice mnemonics,

1311
00:57:15,340 --> 00:57:17,390
and there are calculations
you do to get the answer.

1312
00:57:17,390 --> 00:57:18,360
AUDIENCE: [INAUDIBLE]

1313
00:57:25,635 --> 00:57:27,390
PROFESSOR: You
will almost always

1314
00:57:27,390 --> 00:57:29,140
find overlapping bands
in three dimensions

1315
00:57:29,140 --> 00:57:30,306
in sufficiently high energy.

1316
00:57:30,306 --> 00:57:32,922
I can't off the top of my head
give you a theorem about that,

1317
00:57:32,922 --> 00:57:33,880
but yeah, it's generic.

1318
00:57:36,565 --> 00:57:37,065
Yeah.

1319
00:57:37,065 --> 00:57:40,190
AUDIENCE: --analog to
conductor in one dimension?

1320
00:57:40,190 --> 00:57:44,034
You have these like,
non-zero band depths?

1321
00:57:44,034 --> 00:57:44,700
PROFESSOR: Yeah.

1322
00:57:44,700 --> 00:57:47,060
And this is why Matt
was barfing at me.

1323
00:57:47,060 --> 00:57:49,470
So the answer to that is yeah.

1324
00:57:49,470 --> 00:57:51,172
There aren't [INAUDIBLE].

1325
00:57:51,172 --> 00:57:52,130
But what would we need?

1326
00:57:52,130 --> 00:57:54,390
What we need is
one of two things.

1327
00:57:54,390 --> 00:57:56,920
We need either the
band gap coincidentally

1328
00:57:56,920 --> 00:57:58,410
is ridiculously small.

1329
00:57:58,410 --> 00:58:01,660
What's a good example of that?

1330
00:58:01,660 --> 00:58:03,590
A free particle.

1331
00:58:03,590 --> 00:58:06,690
In the case of a free particle,
these band gaps go to 0.

1332
00:58:06,690 --> 00:58:07,190
Right?

1333
00:58:07,190 --> 00:58:08,940
And so that's a conductor.

1334
00:58:08,940 --> 00:58:11,370
Just an electron.

1335
00:58:11,370 --> 00:58:12,110
Right.

1336
00:58:12,110 --> 00:58:13,010
It conducts, right?

1337
00:58:13,010 --> 00:58:13,380
OK.

1338
00:58:13,380 --> 00:58:14,588
So that can certainly happen.

1339
00:58:14,588 --> 00:58:15,997
But that's sort of stupid.

1340
00:58:15,997 --> 00:58:17,330
I mean, it's not totally stupid.

1341
00:58:17,330 --> 00:58:19,052
But it's sort of stupid.

1342
00:58:19,052 --> 00:58:20,510
But a better answer
would be, well,

1343
00:58:20,510 --> 00:58:22,450
can you have a system
where there are bands

1344
00:58:22,450 --> 00:58:26,020
but you didn't have one
electron per potential well?

1345
00:58:26,020 --> 00:58:26,520
And yeah.

1346
00:58:26,520 --> 00:58:29,070
You could orchestrate
that in lots of ways.

1347
00:58:29,070 --> 00:58:31,477
Now it involves orchestration.

1348
00:58:31,477 --> 00:58:33,060
So it's not the
generic system that we

1349
00:58:33,060 --> 00:58:33,910
were talking about here.

1350
00:58:33,910 --> 00:58:34,680
But you can't orchestrate it.

1351
00:58:34,680 --> 00:58:37,140
So spin is a useful thing that
gives you an extra handle.

1352
00:58:37,140 --> 00:58:39,390
If you have twice as
many states per well then

1353
00:58:39,390 --> 00:58:42,030
you can have half a band filled.

1354
00:58:42,030 --> 00:58:43,876
So that's one way to do it.

1355
00:58:43,876 --> 00:58:46,250
Then it becomes dependent on
details of the system, which

1356
00:58:46,250 --> 00:58:47,745
is what I didn't
want to get into.

1357
00:58:47,745 --> 00:58:49,120
But yeah, you can
orchestrate it.

1358
00:58:49,120 --> 00:58:51,245
It's just not a generic
thing from what we've done.

1359
00:58:51,245 --> 00:58:53,410
And it's really not
for spin-less systems.

1360
00:58:53,410 --> 00:58:57,280
On the other hand,
accidental small gaps.

1361
00:58:57,280 --> 00:58:57,780
Easy.

1362
00:58:57,780 --> 00:58:58,410
That happens.

1363
00:58:58,410 --> 00:59:00,252
That certainly happens.

1364
00:59:00,252 --> 00:59:01,710
So that brings me
to the last thing

1365
00:59:01,710 --> 00:59:03,960
I wanted to talk about before
getting to entanglement,

1366
00:59:03,960 --> 00:59:07,680
which is accidental small gaps.

1367
00:59:07,680 --> 00:59:09,872
So what happens
to a system which

1368
00:59:09,872 --> 00:59:11,330
is-- so there are
some systems that

1369
00:59:11,330 --> 00:59:14,220
are neither conductors
nor insulators.

1370
00:59:14,220 --> 00:59:17,560
They are reasonably
good conductors

1371
00:59:17,560 --> 00:59:19,741
and reasonably bad insulators.

1372
00:59:19,741 --> 00:59:20,740
But they're not perfect.

1373
00:59:24,354 --> 00:59:26,270
And these materials are
called semiconductors.

1374
00:59:26,270 --> 00:59:29,340
I want to talk about why
they're called semiconductors

1375
00:59:29,340 --> 00:59:31,574
and what that means.

1376
00:59:31,574 --> 00:59:32,990
So this is going
to be very brief.

1377
00:59:32,990 --> 00:59:34,406
Then I'm going to
give you-- we're

1378
00:59:34,406 --> 00:59:35,990
going to get into entanglement.

1379
00:59:35,990 --> 00:59:40,390
So consider a
system exactly using

1380
00:59:40,390 --> 00:59:42,515
the same logic we've
used so far which

1381
00:59:42,515 --> 00:59:43,640
has the following property.

1382
00:59:43,640 --> 00:59:45,889
We have two bands.

1383
00:59:45,889 --> 00:59:47,680
And the bottom band is
filled because we've

1384
00:59:47,680 --> 00:59:49,920
got just the right number
of charged particles.

1385
00:59:49,920 --> 00:59:51,035
Bottom band is filled.

1386
00:59:51,035 --> 00:59:53,370
And this guy is empty,
but the gap is tiny.

1387
00:59:53,370 --> 00:59:53,870
OK.

1388
00:59:53,870 --> 00:59:55,850
Delta e is very small.

1389
00:59:55,850 --> 00:59:57,270
Now delta e has dimensions.

1390
00:59:57,270 --> 00:59:58,210
It has units, right?

1391
00:59:58,210 --> 01:00:00,210
So when I say small, that
doesn't mean anything.

1392
01:00:00,210 --> 01:00:01,959
I need to tell you
small compared to what.

1393
01:00:01,959 --> 01:00:03,640
So what's a salient
thing that controls

1394
01:00:03,640 --> 01:00:06,000
an energy scale for
a real material?

1395
01:00:06,000 --> 01:00:07,330
Well the temperature.

1396
01:00:07,330 --> 01:00:09,950
If you have a hot
piece of copper,

1397
01:00:09,950 --> 01:00:11,740
then the lattice
is wiggling around.

1398
01:00:11,740 --> 01:00:13,450
And every once in
a while, an ion

1399
01:00:13,450 --> 01:00:15,510
can hit one of the
electrons and excite it,

1400
01:00:15,510 --> 01:00:16,570
give it some momentum.

1401
01:00:16,570 --> 01:00:19,480
And so there's an
available reservoir

1402
01:00:19,480 --> 01:00:22,050
of energy for exciting
individual electrons.

1403
01:00:22,050 --> 01:00:23,642
You have it really
hot, what happens

1404
01:00:23,642 --> 01:00:25,350
is every once in a
while an electron will

1405
01:00:25,350 --> 01:00:27,683
get nailed by a little thermal
fluctuation in the system

1406
01:00:27,683 --> 01:00:30,222
and get excited above the gap.

1407
01:00:30,222 --> 01:00:32,180
And now it's in a super--
and generically, it's

1408
01:00:32,180 --> 01:00:33,990
going to be in a superposition
state of one of these excited

1409
01:00:33,990 --> 01:00:34,102
states.

1410
01:00:34,102 --> 01:00:35,685
So it's in general
going to be moving.

1411
01:00:35,685 --> 01:00:36,330
It can radiate.

1412
01:00:36,330 --> 01:00:38,189
It will eventually
fall back down.

1413
01:00:38,189 --> 01:00:39,730
But you're constantly
being buffeted.

1414
01:00:39,730 --> 01:00:41,146
The sea of electrons
is constantly

1415
01:00:41,146 --> 01:00:43,800
being buffeted by this
thermal fluctuation.

1416
01:00:43,800 --> 01:00:45,657
And as a result,
you constantly have

1417
01:00:45,657 --> 01:00:47,490
electrons being excited
up, cruising around,

1418
01:00:47,490 --> 01:00:49,070
falling back down.

1419
01:00:49,070 --> 01:00:52,380
So you end up with some
population of electrons.

1420
01:00:52,380 --> 01:00:54,390
And they can ask--
and both when asked,

1421
01:00:54,390 --> 01:00:55,960
although not quite
in this language,

1422
01:00:55,960 --> 01:00:58,210
how likely are you to
get an electron up here?

1423
01:00:58,210 --> 01:01:01,030
How likely is an electron
to be excited up thermally?

1424
01:01:01,030 --> 01:01:03,780
And those of you taking 8.04
will know the answer to this.

1425
01:01:03,780 --> 01:01:10,361
The probability goes as e to
the minus delta e over kt.

1426
01:01:10,361 --> 01:01:12,860
So let's think of this where
this is the Boltzmann constant.

1427
01:01:12,860 --> 01:01:13,818
So what does this mean?

1428
01:01:13,818 --> 01:01:17,132
At very low temperatures, if
the gap isn't 0, then this is 0.

1429
01:01:17,132 --> 01:01:18,116
It doesn't happen.

1430
01:01:18,116 --> 01:01:20,490
But at large temperatures,
the denominator here is large.

1431
01:01:20,490 --> 01:01:21,990
If the temperature
is large compared

1432
01:01:21,990 --> 01:01:24,750
to the width of the gap,
then this is a small number.

1433
01:01:24,750 --> 01:01:26,990
And e to the minus of a
small number is close to 1.

1434
01:01:26,990 --> 01:01:28,750
So at high temperature,
you're very

1435
01:01:28,750 --> 01:01:30,542
likely to excite
electrons up here.

1436
01:01:30,542 --> 01:01:32,125
And now if you have
electrons up here,

1437
01:01:32,125 --> 01:01:33,624
you have a bunch
of available states

1438
01:01:33,624 --> 01:01:35,230
down here-- also
known as holes--

1439
01:01:35,230 --> 01:01:37,021
and you have a bunch
of available electrons

1440
01:01:37,021 --> 01:01:38,744
up here with lots
of available states.

1441
01:01:38,744 --> 01:01:41,035
So at a high temperature, a
material with a small gap--

1442
01:01:41,035 --> 01:01:42,743
or at least at
temperatures high compared

1443
01:01:42,743 --> 01:01:45,570
to the size of the gap--
it's basically a conductor.

1444
01:01:45,570 --> 01:01:48,580
And at low temperatures,
it's basically an insulator.

1445
01:01:48,580 --> 01:01:50,220
This is called a semiconductor.

1446
01:01:50,220 --> 01:01:58,350
And there are notes on
the Stellar web page

1447
01:01:58,350 --> 01:02:01,560
that discuss in a little more
detail what I just went through

1448
01:02:01,560 --> 01:02:04,230
and show you how you
build a transistor out

1449
01:02:04,230 --> 01:02:05,570
of a semiconductor.

1450
01:02:05,570 --> 01:02:09,340
And the important bit
of physics is just this.

1451
01:02:09,340 --> 01:02:10,330
OK.

1452
01:02:10,330 --> 01:02:13,520
So that finishes us
up for the band gap

1453
01:02:13,520 --> 01:02:14,934
systems for periodic potentials.

1454
01:02:14,934 --> 01:02:16,350
We've done something
kind of cool.

1455
01:02:16,350 --> 01:02:19,220
We've explained why
diamonds are transparent.

1456
01:02:19,220 --> 01:02:21,470
We've explained why
they don't conduct.

1457
01:02:21,470 --> 01:02:24,130
We've explained why copper
does and it's opaque.

1458
01:02:24,130 --> 01:02:26,942
And that's pretty good
for 15 minutes of work.

1459
01:02:26,942 --> 01:02:28,710
It's not bad.

1460
01:02:28,710 --> 01:02:31,055
But along the way, we also
talked about the analogous

1461
01:02:31,055 --> 01:02:34,390
system of what are
called photonic crystals.

1462
01:02:34,390 --> 01:02:37,640
Systems of periodic
arrays of dielectrics.

1463
01:02:37,640 --> 01:02:38,624
Like wave guides.

1464
01:02:38,624 --> 01:02:40,040
And those have the
same structure.

1465
01:02:40,040 --> 01:02:42,860
They have bands of
allowed energy and gaps

1466
01:02:42,860 --> 01:02:46,107
of disallowed energies where
no waves propagate through.

1467
01:02:46,107 --> 01:02:47,690
So you might think
that's a little bit

1468
01:02:47,690 --> 01:02:48,750
of a ridiculous example.

1469
01:02:48,750 --> 01:02:50,700
So just to close
this off, you've

1470
01:02:50,700 --> 01:02:53,110
all seen a good example of
a photonic crystal flying

1471
01:02:53,110 --> 01:02:55,990
past you.

1472
01:02:55,990 --> 01:03:01,580
You know that highly reflective
at very specific frequency

1473
01:03:01,580 --> 01:03:03,780
structure on the
surface of a butterfly

1474
01:03:03,780 --> 01:03:05,370
wing that makes
it shiny and blue?

1475
01:03:05,370 --> 01:03:06,200
It looks metallic.

1476
01:03:06,200 --> 01:03:08,620
It looks like it's
a crystal reflecting

1477
01:03:08,620 --> 01:03:09,620
in a specific frequency.

1478
01:03:09,620 --> 01:03:10,890
At some sharp blue.

1479
01:03:10,890 --> 01:03:12,850
And the reason is, it's
a photonic crystal.

1480
01:03:12,850 --> 01:03:14,129
It is exactly this form.

1481
01:03:14,129 --> 01:03:15,670
If you look at it
under a microscope,

1482
01:03:15,670 --> 01:03:18,300
you see little rays
of protein which

1483
01:03:18,300 --> 01:03:20,720
have different
dielectric than air.

1484
01:03:20,720 --> 01:03:24,270
And they form exact crystals--
or not exact, but very good

1485
01:03:24,270 --> 01:03:27,780
crystals-- that reflect at
very specific wavelengths.

1486
01:03:27,780 --> 01:03:30,980
And as a consequence, they
have a metallic sheen.

1487
01:03:30,980 --> 01:03:34,560
So why would a butterfly
put a photonic crystal

1488
01:03:34,560 --> 01:03:35,690
on its surface?

1489
01:03:35,690 --> 01:03:36,940
Well it's extremely light.

1490
01:03:36,940 --> 01:03:37,827
It's fairly rigid.

1491
01:03:37,827 --> 01:03:39,660
It looks shiny and
metallic without actually

1492
01:03:39,660 --> 01:03:41,080
being shiny and metallic.

1493
01:03:41,080 --> 01:03:44,260
And it's not a pigment, so it
doesn't absorb light and decay

1494
01:03:44,260 --> 01:03:44,966
over time.

1495
01:03:44,966 --> 01:03:46,590
It's like the best
thing you could ever

1496
01:03:46,590 --> 01:03:48,881
do if you wanted to be a
shiny, fluttery, flying thing.

1497
01:03:52,241 --> 01:03:52,740
Anyway.

1498
01:03:52,740 --> 01:03:54,960
So there's an incredible
amount of physics

1499
01:03:54,960 --> 01:03:56,920
in this story of the band gaps.

1500
01:03:56,920 --> 01:04:00,180
And consider this an
introduction to the topic.

1501
01:04:00,180 --> 01:04:00,940
OK.

1502
01:04:00,940 --> 01:04:02,730
So that's it for band gaps.

1503
01:04:02,730 --> 01:04:04,845
And I want to move
on to the remainder,

1504
01:04:04,845 --> 01:04:06,380
the last topic of our course.

1505
01:04:06,380 --> 01:04:08,255
Which is going to be
entanglement and quantum

1506
01:04:08,255 --> 01:04:08,870
computation.

1507
01:04:08,870 --> 01:04:12,010
And here I need to give
you one quick observation

1508
01:04:12,010 --> 01:04:14,584
and then move on to
the punchline of today.

1509
01:04:14,584 --> 01:04:16,000
The one quick
observation is this.

1510
01:04:16,000 --> 01:04:17,992
We've talked about
identical particles before.

1511
01:04:17,992 --> 01:04:20,450
And we've talked about identical
particles in funny states.

1512
01:04:20,450 --> 01:04:23,500
So for example, imagine I
have two particles described

1513
01:04:23,500 --> 01:04:25,570
by a wave function where
the first particle could

1514
01:04:25,570 --> 01:04:30,500
be in the state a and the
second particles in the state b.

1515
01:04:30,500 --> 01:04:33,189
And I can build a wave function
for the first particle being

1516
01:04:33,189 --> 01:04:34,980
in state a and the
second particle in state

1517
01:04:34,980 --> 01:04:35,980
b in the following way.

1518
01:04:35,980 --> 01:04:38,780
So let's say position
a and position b.

1519
01:04:38,780 --> 01:04:42,730
I could take a single particle
wave function, chi of a,

1520
01:04:42,730 --> 01:04:45,851
and a single particle
wave functions phi of b.

1521
01:04:45,851 --> 01:04:47,600
And we've talked about
what this tells us.

1522
01:04:47,600 --> 01:04:49,370
And you've studied this
on your problem set.

1523
01:04:49,370 --> 01:04:51,620
What this tells you is that
the probability of finding

1524
01:04:51,620 --> 01:04:53,717
the particle at point A
is given by chi a squared.

1525
01:04:53,717 --> 01:04:56,050
And this is normalized, so
when we integrate against it,

1526
01:04:56,050 --> 01:04:57,517
we get 1.

1527
01:04:57,517 --> 01:04:59,100
And similarly, the
probability that we

1528
01:04:59,100 --> 01:05:02,190
find the second particle at
b is this thing norm squared.

1529
01:05:02,190 --> 01:05:05,296
And it's independent
of what a is.

1530
01:05:05,296 --> 01:05:06,670
But we also
studied-- and so this

1531
01:05:06,670 --> 01:05:07,961
was called the distinguishable.

1532
01:05:07,961 --> 01:05:12,400
We also studied the symmetric
configuration, which

1533
01:05:12,400 --> 01:05:16,790
was equal to 1 over
root phi, root 2.

1534
01:05:16,790 --> 01:05:18,016
Chi of a.

1535
01:05:18,016 --> 01:05:21,650
Phi of b.

1536
01:05:21,650 --> 01:05:26,890
Symmetric, plus
chi of b phi of a.

1537
01:05:26,890 --> 01:05:29,140
And this tells us
something totally awesome.

1538
01:05:29,140 --> 01:05:36,990
What's the probability that I
find the first particle at a?

1539
01:05:36,990 --> 01:05:41,070
It's the norm squared of
chi of a phi of b, right?

1540
01:05:41,070 --> 01:05:42,730
If we integrate
over all phi b, this

1541
01:05:42,730 --> 01:05:44,460
is the norm squared
integrates to 1.

1542
01:05:44,460 --> 01:05:45,277
So it's fine.

1543
01:05:45,277 --> 01:05:46,610
So there's a factor of one half.

1544
01:05:46,610 --> 01:05:50,400
We either find it at
chi of a or chi of b.

1545
01:05:50,400 --> 01:05:54,810
However if I tell you that I've
measured the first particle

1546
01:05:54,810 --> 01:05:56,697
and I find it in
the state chi, what

1547
01:05:56,697 --> 01:05:58,280
can you say about
the second particle?

1548
01:06:01,010 --> 01:06:02,890
It's in the state phi.

1549
01:06:02,890 --> 01:06:05,490
If you know the first
particle's in the state chi,

1550
01:06:05,490 --> 01:06:06,990
the second part is
in the state phi.

1551
01:06:06,990 --> 01:06:09,490
Because we measured it and
it's not in the state--

1552
01:06:09,490 --> 01:06:12,340
the first particle's
not in the state phi.

1553
01:06:12,340 --> 01:06:13,870
So measuring one
particle tells you

1554
01:06:13,870 --> 01:06:15,536
something about the
second particle.

1555
01:06:15,536 --> 01:06:16,910
And this is deeply
disconcerting,

1556
01:06:16,910 --> 01:06:18,618
because I could've
taken these particles,

1557
01:06:18,618 --> 01:06:20,040
put them in this
entangled state,

1558
01:06:20,040 --> 01:06:23,120
and sent one particle
off to a distant planet

1559
01:06:23,120 --> 01:06:26,250
and the second particle
to my sister in DC.

1560
01:06:26,250 --> 01:06:28,830
And my sister measures
this second particle

1561
01:06:28,830 --> 01:06:31,110
and determines what state
it's in and is immediately

1562
01:06:31,110 --> 01:06:34,940
determined what state
the first particle is

1563
01:06:34,940 --> 01:06:37,990
in over in this distant
planet Zorg, right?

1564
01:06:37,990 --> 01:06:39,817
So that's deeply disconcerting.

1565
01:06:39,817 --> 01:06:42,150
And to those of us who have
studied quantum mechanics up

1566
01:06:42,150 --> 01:06:43,858
to this point-- which
we all in this room

1567
01:06:43,858 --> 01:06:45,950
have-- to those of us
who have studied quantum

1568
01:06:45,950 --> 01:06:47,562
mechanics to this
level of development

1569
01:06:47,562 --> 01:06:49,520
and understand that it
is a correct description

1570
01:06:49,520 --> 01:06:52,870
of many experiments, this
should be yet another moment

1571
01:06:52,870 --> 01:06:54,112
of serious discomfort.

1572
01:06:54,112 --> 01:06:56,195
We've run into a bunch of
these over the semester.

1573
01:06:56,195 --> 01:06:57,861
But this one should
be troubling to you.

1574
01:06:57,861 --> 01:06:59,210
Because look.

1575
01:06:59,210 --> 01:07:01,370
How can something
here dramatically

1576
01:07:01,370 --> 01:07:03,460
change the state,
the configuration,

1577
01:07:03,460 --> 01:07:06,222
the initial configuration, of a
particle arbitrarily far away?

1578
01:07:06,222 --> 01:07:07,430
Isn't that deeply concerning?

1579
01:07:07,430 --> 01:07:08,700
And if you think
about relativity,

1580
01:07:08,700 --> 01:07:10,750
this should be all the
more deeply disconcerting.

1581
01:07:10,750 --> 01:07:13,280
Because how does relativistic
causality fit into this?

1582
01:07:13,280 --> 01:07:15,270
So there was a person
that roughly this time,

1583
01:07:15,270 --> 01:07:17,840
a little earlier, who was
troubled by this problem.

1584
01:07:17,840 --> 01:07:19,240
And his name was Einstein.

1585
01:07:19,240 --> 01:07:21,350
And so one of the things
that's kind of amazing

1586
01:07:21,350 --> 01:07:25,340
is that he created a thought
experiment which we're

1587
01:07:25,340 --> 01:07:28,440
going to study in detail next
week called the EPR experiment.

1588
01:07:28,440 --> 01:07:30,380
And there's a beautiful
historical story

1589
01:07:30,380 --> 01:07:32,510
about the setting
and the meaning

1590
01:07:32,510 --> 01:07:34,000
and the particular person.

1591
01:07:34,000 --> 01:07:35,583
And unfortunately,
I'm not a historian

1592
01:07:35,583 --> 01:07:37,866
so I can't tell you that story.

1593
01:07:37,866 --> 01:07:39,490
It sure would be nice
if we had someone

1594
01:07:39,490 --> 01:07:41,360
who wrote a
biography of Einstein

1595
01:07:41,360 --> 01:07:43,800
to tell you a little
bit about that story.

1596
01:07:43,800 --> 01:07:45,300
Oh look, it's Tom
Levenson who wrote

1597
01:07:45,300 --> 01:07:46,520
a biography about Einstein.

1598
01:07:46,520 --> 01:07:47,858
So Tom is--

1599
01:07:58,140 --> 01:08:00,109
TOM LEVENSON: Oh, I
need a microphone.

1600
01:08:04,800 --> 01:08:06,850
Those of who have taken
courses in [INAUDIBLE]--

1601
01:08:06,850 --> 01:08:11,190
and I'm sure that's all of
you because of the GIRs--

1602
01:08:11,190 --> 01:08:15,690
know this is larger than
the usual [INAUDIBLE] class.

1603
01:08:15,690 --> 01:08:20,270
So I'm very used to microphones,
but not in this context.

1604
01:08:20,270 --> 01:08:20,996
OK.

1605
01:08:20,996 --> 01:08:22,810
Is this-- yeah, it's on.

1606
01:08:22,810 --> 01:08:23,840
Can you hear me?

1607
01:08:23,840 --> 01:08:26,319
All right.

1608
01:08:26,319 --> 01:08:32,200
So there are lots of ways to
slice the story of Einstein

1609
01:08:32,200 --> 01:08:35,602
by the time he reaches the EPR
experiment, which is Einstein,

1610
01:08:35,602 --> 01:08:38,060
Podolsky, and Rosen for the
three people who actually wrote

1611
01:08:38,060 --> 01:08:38,560
the paper.

1612
01:08:42,520 --> 01:08:47,740
Just to dot the I's and cross
the T's on the paper itself,

1613
01:08:47,740 --> 01:08:51,260
Rosen is apparently for person
who first came to Einstein.

1614
01:08:51,260 --> 01:08:55,760
Podolsky and Rosen were two
young physicists in Princeton

1615
01:08:55,760 --> 01:08:57,350
after Einstein
moved to Princeton.

1616
01:08:57,350 --> 01:09:00,899
Einstein moved to
Princeton in 1933.

1617
01:09:00,899 --> 01:09:03,540
About three weeks
before-- I'm sorry,

1618
01:09:03,540 --> 01:09:05,290
he moved to Princeton '33.

1619
01:09:05,290 --> 01:09:08,229
He left Germany
in 1932 December,

1620
01:09:08,229 --> 01:09:11,800
about three weeks before
Hitler took power.

1621
01:09:11,800 --> 01:09:17,390
And he did so with decisiveness
and dispatch and a head

1622
01:09:17,390 --> 01:09:20,029
of almost all of
his-- in fact, I

1623
01:09:20,029 --> 01:09:24,359
think all of his German-Jewish
physicist colleagues

1624
01:09:24,359 --> 01:09:27,850
and those German physicists
for whom the Hitler

1625
01:09:27,850 --> 01:09:29,297
regime was unacceptable.

1626
01:09:29,297 --> 01:09:31,130
Which shows that Einstein
really was smarter

1627
01:09:31,130 --> 01:09:32,520
than most of his peers.

1628
01:09:32,520 --> 01:09:37,330
That's one of many different
ways you can ascertain that.

1629
01:09:37,330 --> 01:09:39,370
And so he came to
Princeton in '33.

1630
01:09:39,370 --> 01:09:42,960
He actually went to Caltech
before we went to Princeton.

1631
01:09:42,960 --> 01:09:45,750
As part of an ongoing
visitor-ship he had there.

1632
01:09:45,750 --> 01:09:48,910
Came back to Europe, hung
with the queen of Belgium who

1633
01:09:48,910 --> 01:09:50,880
was a friend of his.

1634
01:09:50,880 --> 01:09:53,640
Went to England.

1635
01:09:53,640 --> 01:09:56,750
And then headed
across the Atlantic

1636
01:09:56,750 --> 01:09:59,200
and took up residency
in Princeton

1637
01:09:59,200 --> 01:10:01,300
at the Institute
for Advanced Studies

1638
01:10:01,300 --> 01:10:04,390
where he stayed for
the rest of his life.

1639
01:10:04,390 --> 01:10:09,280
And over the course of the--
that was '33, he died in '55,

1640
01:10:09,280 --> 01:10:10,470
I think.

1641
01:10:10,470 --> 01:10:12,470
I should know that, but
I think that's right.

1642
01:10:12,470 --> 01:10:13,790
22 years.

1643
01:10:13,790 --> 01:10:16,460
He worked with a lot
of different, mostly

1644
01:10:16,460 --> 01:10:18,090
younger physicists.

1645
01:10:18,090 --> 01:10:21,355
And Podolsky and Rosen were
early members of that chain.

1646
01:10:21,355 --> 01:10:24,280
So Rosen was talking
with him some day

1647
01:10:24,280 --> 01:10:25,900
and starts to frame
this experiment.

1648
01:10:25,900 --> 01:10:27,347
Einstein develops it.

1649
01:10:27,347 --> 01:10:28,680
The three of them talk about it.

1650
01:10:28,680 --> 01:10:31,530
They write the paper
and they put it out.

1651
01:10:31,530 --> 01:10:34,710
And I want to share
with you, actually,

1652
01:10:34,710 --> 01:10:39,950
a really lovely description
of the way the problem was

1653
01:10:39,950 --> 01:10:50,949
represented in a way
by-- this is from a book

1654
01:10:50,949 --> 01:10:52,240
that I recommend to all of you.

1655
01:10:52,240 --> 01:10:54,370
It's actually
really hard to find.

1656
01:10:54,370 --> 01:10:55,120
It's really sweet.

1657
01:10:55,120 --> 01:10:59,130
Jeremy Bernstein,
who is a physicist.

1658
01:10:59,130 --> 01:11:00,202
He's sort of been around.

1659
01:11:00,202 --> 01:11:01,160
A physicist and writer.

1660
01:11:01,160 --> 01:11:03,170
He's in his eighties now.

1661
01:11:03,170 --> 01:11:04,600
He lives in Aspen.

1662
01:11:04,600 --> 01:11:07,022
He worked with CERN
for a number of years.

1663
01:11:07,022 --> 01:11:08,230
He's always been independent.

1664
01:11:08,230 --> 01:11:11,320
He wrote for the New Yorker.

1665
01:11:11,320 --> 01:11:12,860
Anyway.

1666
01:11:12,860 --> 01:11:17,210
So you've all heard of the
physicist Bell, I assume?

1667
01:11:17,210 --> 01:11:18,580
Bell's inequality?

1668
01:11:18,580 --> 01:11:19,860
OK.

1669
01:11:19,860 --> 01:11:25,460
So Bell had a lovely way to
describe-- I'm trying to find.

1670
01:11:25,460 --> 01:11:27,840
I had this marked and then
I lost my piece of paper.

1671
01:11:32,410 --> 01:11:34,414
I have already lost it.

1672
01:11:34,414 --> 01:11:35,080
That's terrible.

1673
01:11:35,080 --> 01:11:39,730
So Bell has this wonderful
way of describing

1674
01:11:39,730 --> 01:11:41,186
this problem of entanglement.

1675
01:11:41,186 --> 01:11:42,560
And it's based on
his description

1676
01:11:42,560 --> 01:11:43,470
of an actual person.

1677
01:11:43,470 --> 01:11:44,670
I was going to read
you his actual quote.

1678
01:11:44,670 --> 01:11:46,560
Now I'm just going to
paraphrase it for you.

1679
01:11:46,560 --> 01:11:50,020
He had a friend named,
I think, Bartelstein.

1680
01:11:50,020 --> 01:11:53,650
Or at least someone
known to him.

1681
01:11:53,650 --> 01:11:55,660
Who had two quirks.

1682
01:11:55,660 --> 01:12:01,810
An unusual color sense and a
taste for mismatched socks.

1683
01:12:01,810 --> 01:12:05,060
And so Bell used to say, if you
saw Bartelstein and you could

1684
01:12:05,060 --> 01:12:08,740
only see one leg and
that sock was pink,

1685
01:12:08,740 --> 01:12:15,580
you knew to a certainty that
the other sock was not pink.

1686
01:12:15,580 --> 01:12:19,490
He comes up I think--
I'm trying to remember

1687
01:12:19,490 --> 01:12:21,690
who this is originally
attributed to.

1688
01:12:21,690 --> 01:12:22,190
Same thing.

1689
01:12:22,190 --> 01:12:30,320
If you have a coin and you
cut it in half down the-- so

1690
01:12:30,320 --> 01:12:32,110
you've got two coin shape disks.

1691
01:12:32,110 --> 01:12:33,979
You cut the disk in
half, not-- and you

1692
01:12:33,979 --> 01:12:36,270
have one side that's the head
and the other side that's

1693
01:12:36,270 --> 01:12:38,120
the tail.

1694
01:12:38,120 --> 01:12:40,120
And they're separated.

1695
01:12:40,120 --> 01:12:41,966
They get handed to two
different gamblers.

1696
01:12:41,966 --> 01:12:44,340
And one gambler tries to cheat
the gambling establishment

1697
01:12:44,340 --> 01:12:46,404
by tossing in his half coin.

1698
01:12:46,404 --> 01:12:47,820
And you see the
head that you know

1699
01:12:47,820 --> 01:12:50,890
somebody-- somebody
at some other casino

1700
01:12:50,890 --> 01:12:53,660
is cheating by tossing
in the half coin that

1701
01:12:53,660 --> 01:12:54,820
only has a tail on it.

1702
01:12:54,820 --> 01:12:56,653
So there are lots of
ways to represent this.

1703
01:12:56,653 --> 01:12:59,795
And many physicists
being very witty indeed

1704
01:12:59,795 --> 01:13:03,640
have come up with
different metaphors for it.

1705
01:13:06,240 --> 01:13:13,930
So Allan just described for
you the basic claim in EPR.

1706
01:13:13,930 --> 01:13:14,650
Its weirdness.

1707
01:13:14,650 --> 01:13:16,930
That you have two
particles that are

1708
01:13:16,930 --> 01:13:19,560
entangled in some way and
then go their separate ways.

1709
01:13:19,560 --> 01:13:25,650
And thus you have--
if you have knowledge

1710
01:13:25,650 --> 01:13:28,410
of what's the state of one,
you have certain knowledge

1711
01:13:28,410 --> 01:13:32,420
of the state of the
other, violating

1712
01:13:32,420 --> 01:13:34,610
relativistic ideas of locality.

1713
01:13:34,610 --> 01:13:37,950
And just kind of
making you queasy

1714
01:13:37,950 --> 01:13:41,050
if you're sort of
approaching it naively.

1715
01:13:41,050 --> 01:13:42,860
What Einstein,
Podolsky, and Rosen

1716
01:13:42,860 --> 01:13:47,490
argued was actually something
a little bit-- in fact,

1717
01:13:47,490 --> 01:13:51,110
the paper comes to an end
on that note of queasiness.

1718
01:13:51,110 --> 01:13:53,440
But what they argue is a
little bit more subtle.

1719
01:13:53,440 --> 01:13:55,205
Because what they said is, OK.

1720
01:13:57,546 --> 01:13:59,045
You perform this
thought experiment.

1721
01:13:59,045 --> 01:14:00,990
You send the two particles off.

1722
01:14:00,990 --> 01:14:03,600
You measure position of one,
you know absolutely the position

1723
01:14:03,600 --> 01:14:05,370
of the other.

1724
01:14:05,370 --> 01:14:08,220
You've conferred--
and the paper turns

1725
01:14:08,220 --> 01:14:11,960
on a discussion
of the connection

1726
01:14:11,960 --> 01:14:15,120
between a measurement--
a physical measurement--

1727
01:14:15,120 --> 01:14:17,630
and a property of
physical reality.

1728
01:14:17,630 --> 01:14:20,050
And they have definition
for what reality is.

1729
01:14:20,050 --> 01:14:25,040
And that is something whose-- if
you can perform a measurement,

1730
01:14:25,040 --> 01:14:26,985
you know that
quantity absolutely.

1731
01:14:30,680 --> 01:14:33,480
I don't have the mathematics
to express that properly.

1732
01:14:33,480 --> 01:14:36,780
But that'll do for
this hand waving.

1733
01:14:36,780 --> 01:14:38,320
You can then do
another experiment

1734
01:14:38,320 --> 01:14:40,660
and measure a
complementary property.

1735
01:14:40,660 --> 01:14:42,462
And you know that
piece of reality.

1736
01:14:42,462 --> 01:14:46,520
But you can't do the--
so on the one hand,

1737
01:14:46,520 --> 01:14:52,410
quantum mechanics says you
can't know physical reality

1738
01:14:52,410 --> 01:14:54,300
to this level of precision.

1739
01:14:54,300 --> 01:15:00,590
And on the other hand,
the fact that you

1740
01:15:00,590 --> 01:15:07,640
can do that measurement violates
the relativistic picture

1741
01:15:07,640 --> 01:15:08,880
of reality.

1742
01:15:08,880 --> 01:15:13,090
So you have what they
claimed was a paradox.

1743
01:15:13,090 --> 01:15:15,860
And this paper was published.

1744
01:15:15,860 --> 01:15:22,010
And it received a range of
reactions from indifference

1745
01:15:22,010 --> 01:15:25,034
by younger physicists who said,
we don't care that it's weird.

1746
01:15:25,034 --> 01:15:26,950
We're going to keep on
doing quantum mechanics

1747
01:15:26,950 --> 01:15:29,150
and performing experiments
and making measurements.

1748
01:15:29,150 --> 01:15:30,160
And just see where
this leads us.

1749
01:15:30,160 --> 01:15:31,993
Remember, this is
happening in the mid '30s.

1750
01:15:31,993 --> 01:15:35,760
1935.

1751
01:15:35,760 --> 01:15:39,730
One of these three books will
tell me precisely in a moment.

1752
01:15:39,730 --> 01:15:45,930
And the quantum theory, as it
turned into quantum mechanics,

1753
01:15:45,930 --> 01:15:49,250
developed in its first
period between '23 and '27.

1754
01:15:49,250 --> 01:15:53,120
And by '35, you have enormous
numbers of productive results

1755
01:15:53,120 --> 01:15:55,260
and unexpected things
and the prediction

1756
01:15:55,260 --> 01:15:58,770
of the positron and
then its observation.

1757
01:15:58,770 --> 01:16:02,990
And I mean, the theory is
enormously, dramatically,

1758
01:16:02,990 --> 01:16:05,030
excitingly productive.

1759
01:16:05,030 --> 01:16:08,780
So those who are really
heads down doing the work

1760
01:16:08,780 --> 01:16:13,810
are, for the most part,
saying, this is fine.

1761
01:16:13,810 --> 01:16:16,390
We'll get back to it when we're
old and retired and bored.

1762
01:16:18,920 --> 01:16:20,360
But that wasn't
the uniform case.

1763
01:16:20,360 --> 01:16:24,180
And most notably
Niels Bohr found

1764
01:16:24,180 --> 01:16:26,780
this paper really troubling.

1765
01:16:26,780 --> 01:16:29,600
And spent about six
weeks, apparently,

1766
01:16:29,600 --> 01:16:32,310
discussing this and trying to
come up with a response to it.

1767
01:16:32,310 --> 01:16:34,010
And what he responded
was essentially

1768
01:16:34,010 --> 01:16:40,540
that-- in some ways, it
was the same reaction

1769
01:16:40,540 --> 01:16:42,280
as his younger colleagues.

1770
01:16:42,280 --> 01:16:44,420
Get over it.

1771
01:16:44,420 --> 01:16:47,530
But more precisely,
it was he said,

1772
01:16:47,530 --> 01:16:49,050
there's no
description of reality

1773
01:16:49,050 --> 01:16:51,530
that excludes the measuring
apparatus anymore.

1774
01:16:51,530 --> 01:16:56,521
You can't make statements
about physical reality

1775
01:16:56,521 --> 01:16:59,020
unless you include a description
of the measuring apparatus.

1776
01:16:59,020 --> 01:17:01,304
And you've said that we can
measure this one quantity

1777
01:17:01,304 --> 01:17:02,970
with precision and
know the other thing.

1778
01:17:02,970 --> 01:17:05,303
And then we can subsequently,
in a separate observation,

1779
01:17:05,303 --> 01:17:08,810
measure a complimentary
quality and know the other one.

1780
01:17:08,810 --> 01:17:12,650
You still can't know much
them at the same time.

1781
01:17:12,650 --> 01:17:16,980
It's still true that the
complementarity in essence

1782
01:17:16,980 --> 01:17:20,730
means that once you know
one part of the picture,

1783
01:17:20,730 --> 01:17:23,550
you know some other
part the picture.

1784
01:17:23,550 --> 01:17:27,570
And that's just the nature
of the quantum world.

1785
01:17:27,570 --> 01:17:30,940
Einstein had argued that
the EPR paradox suggested

1786
01:17:30,940 --> 01:17:33,140
that quantum mechanics
was incomplete.

1787
01:17:33,140 --> 01:17:35,740
And Bohr essentially
responded in effect

1788
01:17:35,740 --> 01:17:38,935
that Einstein's description of
quantum mechanical explanation

1789
01:17:38,935 --> 01:17:39,560
was inadequate.

1790
01:17:43,686 --> 01:17:45,060
The important
thing to remember--

1791
01:17:45,060 --> 01:17:49,480
and I want to just spend a
couple minutes going back

1792
01:17:49,480 --> 01:17:51,890
into the pre-history
of all this,

1793
01:17:51,890 --> 01:17:53,530
and then a couple
minutes speculating

1794
01:17:53,530 --> 01:17:56,760
on why Einstein reached
the position he did.

1795
01:17:56,760 --> 01:17:59,950
And what that might tell you
about the practice of science

1796
01:17:59,950 --> 01:18:04,470
as a lived experience
as opposed to one

1797
01:18:04,470 --> 01:18:07,730
reflected in your textbooks.

1798
01:18:07,730 --> 01:18:10,420
But the thing to remember
is that there's nothing

1799
01:18:10,420 --> 01:18:13,600
logically wrong
with the EPR paper.

1800
01:18:13,600 --> 01:18:14,100
Right?

1801
01:18:14,100 --> 01:18:14,600
You know.

1802
01:18:17,420 --> 01:18:21,370
It does what it says it does and
there's no overt error in it.

1803
01:18:21,370 --> 01:18:24,610
And there's nothing wrong
with Bohr's response.

1804
01:18:24,610 --> 01:18:26,540
And in fact, when the
experiments were--

1805
01:18:26,540 --> 01:18:32,910
Bell formalized the-- what
Bell's inequality really does

1806
01:18:32,910 --> 01:18:34,840
is it formalized
the two arguments.

1807
01:18:34,840 --> 01:18:37,300
It says, if Bohr
is right, you will

1808
01:18:37,300 --> 01:18:38,720
observe this in the experiment.

1809
01:18:38,720 --> 01:18:40,730
And if Einstein is
right, you would

1810
01:18:40,730 --> 01:18:41,970
observe something different.

1811
01:18:41,970 --> 01:18:44,210
The experiments were
done, and I imagine

1812
01:18:44,210 --> 01:18:46,530
are still being done, as
sort of demonstrations.

1813
01:18:46,530 --> 01:18:48,780
And they showed that Bohr's
interpretation was correct

1814
01:18:48,780 --> 01:18:52,490
and that yes, quantum
mechanics produces results

1815
01:18:52,490 --> 01:18:56,150
that are non-local just
as Allan described to you.

1816
01:18:56,150 --> 01:18:58,980
And that the world really
is as strange as people

1817
01:18:58,980 --> 01:19:01,990
first glimpsed in
1925, '26, and '27.

1818
01:19:01,990 --> 01:19:07,660
And the question of whether
or not that strangeness

1819
01:19:07,660 --> 01:19:10,244
is adequately explained
without the explanations

1820
01:19:10,244 --> 01:19:11,910
that you're going to
learn in this class

1821
01:19:11,910 --> 01:19:14,285
and subsequent ones are
quote "complete" or not.

1822
01:19:14,285 --> 01:19:18,490
And completeness is a very
funny, very, very tricky

1823
01:19:18,490 --> 01:19:19,220
concept.

1824
01:19:19,220 --> 01:19:23,880
But the question whether or
not the framework of quantum

1825
01:19:23,880 --> 01:19:28,830
mechanics is somehow
unsatisfactory in any kind

1826
01:19:28,830 --> 01:19:31,470
of formal a technical
sense is one

1827
01:19:31,470 --> 01:19:34,891
that's at least partly dependent
on your scientific temperament,

1828
01:19:34,891 --> 01:19:35,390
I think.

1829
01:19:38,570 --> 01:19:43,020
So that's the cartoon version
of what happened in '35.

1830
01:19:43,020 --> 01:19:44,710
Einstein with his
two young colleagues

1831
01:19:44,710 --> 01:19:49,140
proposes-- really you
should understand the EPR

1832
01:19:49,140 --> 01:19:52,770
paper as a description in detail
of a consequence of quantum

1833
01:19:52,770 --> 01:19:55,934
theory as it was then
expressed with the conclusion--

1834
01:19:55,934 --> 01:19:57,600
and I just want to
read you this thing--

1835
01:19:57,600 --> 01:20:00,270
no reasonable
definition of reality

1836
01:20:00,270 --> 01:20:03,020
could be expected to
permit such a result.

1837
01:20:06,100 --> 01:20:10,220
In fact, it's called a
paradox, but it isn't.

1838
01:20:10,220 --> 01:20:11,840
It's a complaint.

1839
01:20:11,840 --> 01:20:15,250
You know, it's a memo to
the Flying Spaghetti Monster

1840
01:20:15,250 --> 01:20:17,584
that the universe
shouldn't be this way

1841
01:20:17,584 --> 01:20:19,750
if, in fact, experiments
turn out to show that it is

1842
01:20:19,750 --> 01:20:20,430
and they have.

1843
01:20:25,850 --> 01:20:28,420
The oddity here for a
biographer of Einstein

1844
01:20:28,420 --> 01:20:30,430
opposed to a physicist
is given what

1845
01:20:30,430 --> 01:20:34,400
you know about Einstein between
1879 when he's born and say,

1846
01:20:34,400 --> 01:20:37,060
1925 or so when he
completes the last

1847
01:20:37,060 --> 01:20:38,226
of his really great physics.

1848
01:20:41,100 --> 01:20:46,059
How could-- I mean,
I actually keep--

1849
01:20:46,059 --> 01:20:48,600
I've been working on Einstein
off and on for years and years.

1850
01:20:48,600 --> 01:20:54,880
I keep finding out new ways in
which he's just inconceivably

1851
01:20:54,880 --> 01:20:58,030
bright and on target and with
a nose for the right problem

1852
01:20:58,030 --> 01:20:59,490
and insightful.

1853
01:20:59,490 --> 01:21:02,820
And yet by the 1935,
10 years later,

1854
01:21:02,820 --> 01:21:05,070
he's still a
relatively young man.

1855
01:21:05,070 --> 01:21:06,920
He's in his '50s.

1856
01:21:06,920 --> 01:21:11,390
Which being in my '50s I think
is an extremely young man.

1857
01:21:11,390 --> 01:21:14,110
Just 10 years after
doing work that's

1858
01:21:14,110 --> 01:21:19,160
right on the edge of
modern quantum mechanics

1859
01:21:19,160 --> 01:21:21,230
that is essential
to its foundation.

1860
01:21:21,230 --> 01:21:23,290
That's really extraordinary.

1861
01:21:23,290 --> 01:21:24,950
10 years after
that, he's saying,

1862
01:21:24,950 --> 01:21:26,820
no reasonable
definition of reality

1863
01:21:26,820 --> 01:21:31,020
should be permitted
to behave this way.

1864
01:21:31,020 --> 01:21:32,621
Where does that come from?

1865
01:21:32,621 --> 01:21:34,620
Well the first thing I
want to tell you-- again,

1866
01:21:34,620 --> 01:21:37,410
this is all going to be
a really cartoon version.

1867
01:21:37,410 --> 01:21:40,040
Because there's not
much time, I understand.

1868
01:21:40,040 --> 01:21:44,720
Is that Einstein-- I
mean, how much are you

1869
01:21:44,720 --> 01:21:48,590
aware of Einstein's role in the
creation of the quantum theory?

1870
01:21:48,590 --> 01:21:49,090
A lot.

1871
01:21:49,090 --> 01:21:49,831
I mean, a lot?

1872
01:21:49,831 --> 01:21:50,330
None?

1873
01:21:50,330 --> 01:21:50,829
OK.

1874
01:21:53,590 --> 01:21:55,240
I mean, I'm going
to make a claim

1875
01:21:55,240 --> 01:21:59,790
that except for Heisenberg,
Schrodinger, maybe Bohr.

1876
01:21:59,790 --> 01:22:00,700
Maybe Born.

1877
01:22:00,700 --> 01:22:02,197
Maybe a couple of others.

1878
01:22:02,197 --> 01:22:04,030
There's no one more
important to the quantum

1879
01:22:04,030 --> 01:22:05,720
theory than Einstein.

1880
01:22:05,720 --> 01:22:07,250
And you could maybe
even argue that

1881
01:22:07,250 --> 01:22:10,480
from a sort of
foundational point of view

1882
01:22:10,480 --> 01:22:13,780
that without Einstein,
rigorous thinking about quantum

1883
01:22:13,780 --> 01:22:16,210
mechanics would have
taken much, much longer.

1884
01:22:16,210 --> 01:22:19,540
I mean, he's really
central to it.

1885
01:22:19,540 --> 01:22:24,200
Planck in 1900 publishes
as an ad hoc solution

1886
01:22:24,200 --> 01:22:29,080
to the black body problem
the first quantum theory.

1887
01:22:29,080 --> 01:22:33,350
In 1905, Einstein says
it's not an ad hoc thing.

1888
01:22:33,350 --> 01:22:37,730
If you look at the
photoelectric effect

1889
01:22:37,730 --> 01:22:41,660
is the particular problem he's
dealing with in explanation.

1890
01:22:41,660 --> 01:22:44,340
But that's really-- the behavior
of the photoelectric effect

1891
01:22:44,340 --> 01:22:47,360
is really presented as the
confirmation of this idea

1892
01:22:47,360 --> 01:22:51,520
that light exists as quanta with
particular kinds of behavior.

1893
01:22:51,520 --> 01:22:54,920
And from 1905 on, he
spends probably more time

1894
01:22:54,920 --> 01:22:58,260
on quantum problems than he did
on any other physics problems.

1895
01:22:58,260 --> 01:23:00,060
Certainly more
than on relativity.

1896
01:23:00,060 --> 01:23:01,590
Though he spent
enormous energy on

1897
01:23:01,590 --> 01:23:03,406
special and general relativity.

1898
01:23:03,406 --> 01:23:05,530
One of most amazing things
about Einstein, in fact,

1899
01:23:05,530 --> 01:23:10,260
is that despite the fact
that he's seen and appears

1900
01:23:10,260 --> 01:23:13,420
by 1935 to be this hidebound
old guy who can't accommodate

1901
01:23:13,420 --> 01:23:17,064
himself to the new world is he
had an extraordinary capacity

1902
01:23:17,064 --> 01:23:19,230
to do what the Red Queen
did in Alice in Wonderland.

1903
01:23:19,230 --> 01:23:21,800
and believe two impossible
things before breakfast.

1904
01:23:21,800 --> 01:23:23,220
Just think in 1905.

1905
01:23:23,220 --> 01:23:27,290
April, he publishes The
Quantum Theory Of Light.

1906
01:23:27,290 --> 01:23:29,850
June he publishes
Special Relativity,

1907
01:23:29,850 --> 01:23:32,470
which treats light as a wave.

1908
01:23:32,470 --> 01:23:36,310
And makes no mention of
his revolutionary-- I mean,

1909
01:23:36,310 --> 01:23:38,740
he called it revolutionary
in a private letter in 1905.

1910
01:23:38,740 --> 01:23:42,180
So he knew what he had in
the quantum theory of light.

1911
01:23:42,180 --> 01:23:45,301
But in special relativity, go
read the special relativity

1912
01:23:45,301 --> 01:23:45,800
paper.

1913
01:23:45,800 --> 01:23:48,210
It's actually lovely reading.

1914
01:23:48,210 --> 01:23:51,910
And you'll see he doesn't
even nod in that direction.

1915
01:23:51,910 --> 01:23:55,150
He doesn't say, you know this
is a hero-- he says nothing.

1916
01:23:55,150 --> 01:23:56,736
So he's capable of
doing excellent--

1917
01:23:56,736 --> 01:23:58,110
and there's a
reason that year is

1918
01:23:58,110 --> 01:24:00,760
called the annus mirabilis,
the year of miracles.

1919
01:24:00,760 --> 01:24:02,190
And in part it's
because Einstein

1920
01:24:02,190 --> 01:24:05,790
is able to actually really
focus on these things.

1921
01:24:05,790 --> 01:24:08,320
And I realize class
is almost over.

1922
01:24:08,320 --> 01:24:12,950
So there's several more steps
in Einstein's quantum journey.

1923
01:24:12,950 --> 01:24:19,170
But you know, what
you should take away

1924
01:24:19,170 --> 01:24:23,260
is that Einstein's
ability to deal

1925
01:24:23,260 --> 01:24:25,837
with the problems
of quantum pictures

1926
01:24:25,837 --> 01:24:27,670
extends to the point--
he's the first person

1927
01:24:27,670 --> 01:24:29,600
to suggest there might be
a problem with causality

1928
01:24:29,600 --> 01:24:30,474
in quantum mechanics.

1929
01:24:30,474 --> 01:24:32,540
He does this in
1917, eight years

1930
01:24:32,540 --> 01:24:34,600
before quantum
mechanics is invented.

1931
01:24:34,600 --> 01:24:37,770
When he starts looking at what
the quote "classical" quantum

1932
01:24:37,770 --> 01:24:40,100
theory tells you
about the emission

1933
01:24:40,100 --> 01:24:41,860
of radiation from
an excited atom.

1934
01:24:41,860 --> 01:24:43,990
He realized you can't
predict it precisely.

1935
01:24:43,990 --> 01:24:45,930
Radioactive decay
has the same problem.

1936
01:24:45,930 --> 01:24:47,530
He says, well-- he
writes in a letter.

1937
01:24:47,530 --> 01:24:50,170
I don't want to give up
causality, but we may have to.

1938
01:24:50,170 --> 01:24:52,440
So he's aware of these things.

1939
01:24:52,440 --> 01:24:55,380
So I see class is
over now at 12:30.

1940
01:24:55,380 --> 01:24:56,150
OK, sorry.

1941
01:24:56,150 --> 01:24:58,820
So juts to finish
off, the question

1942
01:24:58,820 --> 01:25:02,160
here is why does
Einstein give up on this.

1943
01:25:02,160 --> 01:25:06,470
And the answer, I think, is
because in addition to his-- as

1944
01:25:06,470 --> 01:25:09,150
he started at the
beginning of his career,

1945
01:25:09,150 --> 01:25:14,470
he says with the
quantum theory of light

1946
01:25:14,470 --> 01:25:17,520
and with special relativity,
ignore your physical pictures.

1947
01:25:17,520 --> 01:25:21,120
Try and look at the
phenomena and explain those.

1948
01:25:21,120 --> 01:25:25,710
And by 1935, that becomes
very difficult for him.

1949
01:25:25,710 --> 01:25:28,545
Because the phenomenology
becomes too strange.

1950
01:25:28,545 --> 01:25:30,420
One of the things that
quantum mechanics does

1951
01:25:30,420 --> 01:25:38,590
is it takes away the
immediate ability

1952
01:25:38,590 --> 01:25:39,990
to visualize physical systems.

1953
01:25:39,990 --> 01:25:40,490
There.

1954
01:25:40,490 --> 01:25:44,530
English is my first
language sometimes.

1955
01:25:44,530 --> 01:25:49,010
And that's an aesthetic
failure on Einstein's part.

1956
01:25:49,010 --> 01:25:52,360
He had the intellectual
capacity and explicitly said,

1957
01:25:52,360 --> 01:25:55,080
quantum mechanics is a
logically consistent theory

1958
01:25:55,080 --> 01:25:58,280
that incredibly powerfully
describes lots of problems.

1959
01:25:58,280 --> 01:25:59,400
He said that in print.

1960
01:25:59,400 --> 01:26:02,700
He nominated Schrodinger and
Heisenberg for Nobel Prizes

1961
01:26:02,700 --> 01:26:03,400
twice.

1962
01:26:03,400 --> 01:26:04,710
I mean, he wasn't stupid.

1963
01:26:04,710 --> 01:26:05,940
He was Albert Einstein.

1964
01:26:05,940 --> 01:26:10,090
But he was aesthetically
incapable of pursuing

1965
01:26:10,090 --> 01:26:12,800
this new physics
in ways that were

1966
01:26:12,800 --> 01:26:17,874
possible under the research
possibilities of the time.

1967
01:26:17,874 --> 01:26:19,540
And that is what I
would leave you with.

1968
01:26:19,540 --> 01:26:22,632
Physics is an aesthetic as well
as an intellectual pursuit.

1969
01:26:22,632 --> 01:26:23,340
So thank you all.

1970
01:26:23,340 --> 01:26:24,890
[APPLAUSE]