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PROFESSOR: So today
we begin our study

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00:00:26,520 --> 00:00:29,500
of solids and in
particular of conductivity

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00:00:29,500 --> 00:00:34,270
in solids and
periodic potentials,

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00:00:34,270 --> 00:00:36,960
and that diagram will mean
something in a few weeks.

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00:00:36,960 --> 00:00:40,850
So before I get
started, questions

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00:00:40,850 --> 00:00:43,260
on everything previous?

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00:00:43,260 --> 00:00:43,760
Yeah.

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00:00:43,760 --> 00:00:48,490
AUDIENCE: So when you're talking
about fermions and bosons

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00:00:48,490 --> 00:00:48,990
[INAUDIBLE].

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00:00:48,990 --> 00:00:49,550
Statistics.

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00:00:49,550 --> 00:00:52,091
So we said That you had to write
the statement particular way

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00:00:52,091 --> 00:00:54,395
for fermions with a
minus sign particular.

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00:00:54,395 --> 00:00:55,020
PROFESSOR: Yes.

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00:00:55,020 --> 00:00:56,915
AUDIENCE: Well, we
were assuming that you

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00:00:56,915 --> 00:00:59,410
had to write the statement a
product statement of the two.

23
00:00:59,410 --> 00:01:01,297
Will it be that you
can't do that sometimes

24
00:01:01,297 --> 00:01:02,630
because they can't be separated.

25
00:01:02,630 --> 00:01:03,030
PROFESSOR: Absolutely.

26
00:01:03,030 --> 00:01:04,196
That's a wonderful question.

27
00:01:04,196 --> 00:01:07,592
We'll come to that in the second
next to the last lecture where

28
00:01:07,592 --> 00:01:09,800
we'll talk about something
called entanglement, which

29
00:01:09,800 --> 00:01:11,300
is really what
you're talking about.

30
00:01:11,300 --> 00:01:13,170
But here's the crucial
thing, for fermions,

31
00:01:13,170 --> 00:01:15,050
so what I wrote
last time, I said

32
00:01:15,050 --> 00:01:17,590
suppose we have two particles
that are identical one is

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00:01:17,590 --> 00:01:20,650
in the state we know one
is in the state chi of x

34
00:01:20,650 --> 00:01:23,660
and the other is in the
state described by phi of x.

35
00:01:23,660 --> 00:01:26,650
I'm going to call the position
of the first vertical x

36
00:01:26,650 --> 00:01:30,010
and the position of
the second particle y.

37
00:01:30,010 --> 00:01:35,210
And I can write the two states
without property, well, three.

38
00:01:35,210 --> 00:01:38,330
I could fill many, but two in
particular of the following

39
00:01:38,330 --> 00:01:46,366
from, chi of x phi of y plus
or minus chi of y phi of x.

40
00:01:49,700 --> 00:01:52,500
And the point of this
linear combination

41
00:01:52,500 --> 00:01:54,700
is that when they exchange
positions, the x and y

42
00:01:54,700 --> 00:01:58,287
So here this is the amplitude
for the first particle

43
00:01:58,287 --> 00:02:01,040
will be an x, and the
second particle to be a y.

44
00:02:01,040 --> 00:02:05,210
If I switch the sign or the
positions to psi plus minus

45
00:02:05,210 --> 00:02:07,590
of yx or if the first
particle is at y,

46
00:02:07,590 --> 00:02:12,120
the second particle is at x,
this is equal to just swap

47
00:02:12,120 --> 00:02:18,510
these two terms, plus or
minus psi plus minus of x,y.

48
00:02:18,510 --> 00:02:20,630
So this wave function
has been contracted

49
00:02:20,630 --> 00:02:23,260
to ensure that under the
exchange of the position

50
00:02:23,260 --> 00:02:26,830
of the two particles, the wave
function stays the same up

51
00:02:26,830 --> 00:02:28,290
to a sign.

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00:02:28,290 --> 00:02:36,170
This plus sign is associated
with systems we call bosons,

53
00:02:36,170 --> 00:02:39,730
and the minus sign is associated
with systems we call fermions.

54
00:02:39,730 --> 00:02:41,861
Now as I've
discussed previously,

55
00:02:41,861 --> 00:02:43,610
this is a persistent
property of a system.

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00:02:43,610 --> 00:02:45,630
If a system is bosonic,
at some moment in time,

57
00:02:45,630 --> 00:02:46,714
it will always be bosonic.

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00:02:46,714 --> 00:02:48,421
If it's fermionic in
some moment in time,

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00:02:48,421 --> 00:02:49,980
it will always be
fermionic, and that

60
00:02:49,980 --> 00:02:52,630
is a consequence of the fact
that the exchange operator that

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00:02:52,630 --> 00:02:56,450
swaps the two particles commutes
with the energy eigenfunction

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00:02:56,450 --> 00:02:58,770
if they're identical particles.

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00:02:58,770 --> 00:03:01,510
And so if it commutes with
the energy operator, then

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00:03:01,510 --> 00:03:04,430
its value is preserved
under time evolution

65
00:03:04,430 --> 00:03:06,460
with that energy operator.

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00:03:06,460 --> 00:03:09,740
So now the question
that was asked is,

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00:03:09,740 --> 00:03:11,861
if this is our definition
of bosons and fermions,

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00:03:11,861 --> 00:03:14,110
is it true that you can
always write the wave function

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00:03:14,110 --> 00:03:20,890
in this form of differences
of products of two states?

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00:03:20,890 --> 00:03:23,010
Well, the answer
is both yes and no.

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00:03:23,010 --> 00:03:28,380
The answer is yes for
sufficiently-simple systems.

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00:03:28,380 --> 00:03:31,346
The answer is no when you
have many, many particles,

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00:03:31,346 --> 00:03:33,970
and we can't always write it as
a simple product of two states.

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00:03:33,970 --> 00:03:36,800
We can't always write
as the sum of two terms.

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00:03:36,800 --> 00:03:39,710
What you will discover if you
studied multi-particle systems

76
00:03:39,710 --> 00:03:43,400
in 805 and 806 is that there's
a nice way of organizing

77
00:03:43,400 --> 00:03:45,376
this anti
symmetrization property,

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00:03:45,376 --> 00:03:47,125
and it doesn't necessarily
have two terms,

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00:03:47,125 --> 00:03:50,020
but it may have
many, many terms.

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00:03:50,020 --> 00:03:53,109
With that said, this is
not the defining property.

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00:03:53,109 --> 00:03:54,650
This is really the
defining property,

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00:03:54,650 --> 00:03:57,940
whether your a
boson or a fermion.

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00:03:57,940 --> 00:03:58,660
OK?

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00:03:58,660 --> 00:03:58,840
Cool.

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00:03:58,840 --> 00:03:59,340
Yeah.

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00:03:59,340 --> 00:04:05,706
AUDIENCE: So I had a question
about fermions. [INAUDIBLE]

87
00:04:05,706 --> 00:04:08,357
the explanation being
makes perfect sense

88
00:04:08,357 --> 00:04:10,554
about why two things can't
be in the same state.

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00:04:10,554 --> 00:04:11,220
PROFESSOR: Yeah.

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00:04:11,220 --> 00:04:13,700
AUDIENCE: But Used in
804-4, during recitation

91
00:04:13,700 --> 00:04:15,700
one of the instructors
said, well actually, this

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00:04:15,700 --> 00:04:18,628
takes like really hard
quantum field theory

93
00:04:18,628 --> 00:04:19,622
to actually prove this.

94
00:04:19,622 --> 00:04:20,620
What does that actually mean?

95
00:04:20,620 --> 00:04:20,870
PROFESSOR: OK.

96
00:04:20,870 --> 00:04:21,079
Good.

97
00:04:21,079 --> 00:04:22,454
So the question
is, look, there's

98
00:04:22,454 --> 00:04:25,720
this legendary
statement that this fact

99
00:04:25,720 --> 00:04:28,400
that the Pauli exclusion
principle is incredibly hard

100
00:04:28,400 --> 00:04:32,230
and requires the magical
details of quantum field theory

101
00:04:32,230 --> 00:04:34,840
to explain, and that
some surprising.

102
00:04:34,840 --> 00:04:37,470
So your recitation
lecturer said this to you.

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00:04:37,470 --> 00:04:39,785
AUDIENCE: Yes.

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00:04:39,785 --> 00:04:42,480
PROFESSOR: I'm sure I
love this person dearly,

105
00:04:42,480 --> 00:04:45,000
but I disagree with him
in an important way.

106
00:04:45,000 --> 00:04:47,400
So there are two
things are confusing.

107
00:04:47,400 --> 00:04:49,610
Two things that require
explanation I should say.

108
00:04:49,610 --> 00:04:52,445
One, we can explain
at the level of 804,

109
00:04:52,445 --> 00:04:54,900
and the other does
require machinery,

110
00:04:54,900 --> 00:04:56,851
but it's not crazy
complicated machinery.

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00:04:56,851 --> 00:04:58,850
So let me tell you what
we can explain with 804,

112
00:04:58,850 --> 00:05:00,690
and then let me
sketch for you how

113
00:05:00,690 --> 00:05:02,770
the more sophisticated
argument goes.

114
00:05:02,770 --> 00:05:06,394
So this fact, the basic
fact that wave functions can

115
00:05:06,394 --> 00:05:08,060
be symmetric or
anti-symmetric, and when

116
00:05:08,060 --> 00:05:09,480
you have identical
particles, they

117
00:05:09,480 --> 00:05:11,690
must be either symmetric
or anti-symmetric.

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00:05:11,690 --> 00:05:13,459
This follows from
the statement that we

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00:05:13,459 --> 00:05:14,500
have identical particles.

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00:05:14,500 --> 00:05:16,060
If you have identical
particles, it

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00:05:16,060 --> 00:05:20,860
must be true that
under the exchange,

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00:05:20,860 --> 00:05:22,284
they are either even or odd.

123
00:05:22,284 --> 00:05:24,700
They're either symmetric or
anti-symmetric under exchange.

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00:05:24,700 --> 00:05:27,930
So if we have an exchange
of identical particles,

125
00:05:27,930 --> 00:05:30,750
we need to have identical
symmetric or identical

126
00:05:30,750 --> 00:05:31,450
anti-symmetric.

127
00:05:31,450 --> 00:05:33,160
And now it's just an
empirical question.

128
00:05:33,160 --> 00:05:35,640
What particles in the
world are identical bosons

129
00:05:35,640 --> 00:05:37,800
and what particles are
identical fermions.

130
00:05:37,800 --> 00:05:38,700
Identical fermions?

131
00:05:38,700 --> 00:05:39,920
Well, how could you
tell the difference?

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00:05:39,920 --> 00:05:41,760
You can tell that
identical fermions do not

133
00:05:41,760 --> 00:05:43,720
want to be the same state,
and identical bosons

134
00:05:43,720 --> 00:05:46,760
do as we saw last time,
as an extra factor of two

135
00:05:46,760 --> 00:05:49,390
for two particles that are
actually n for n particles.

136
00:05:49,390 --> 00:05:51,281
So bosons want to be
in the same state.

137
00:05:51,281 --> 00:05:53,030
Fermions want to not
be in the same state,

138
00:05:53,030 --> 00:05:54,210
so if we just look
at the world and say,

139
00:05:54,210 --> 00:05:56,130
which particles are
well modeled by assuming

140
00:05:56,130 --> 00:05:57,830
we have bosons, which
particles are well

141
00:05:57,830 --> 00:05:59,460
modeled by assuming
we have fermions.

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00:05:59,460 --> 00:06:02,660
And answer turns out to be the
following, any particle that

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00:06:02,660 --> 00:06:05,250
has intrinsic angular momentum,
which is a half integer,

144
00:06:05,250 --> 00:06:07,780
any particle which
spin like the electron,

145
00:06:07,780 --> 00:06:09,390
which has a limit
to 1/2 and we'll

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00:06:09,390 --> 00:06:11,980
study that in more detail
later, any particle that

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00:06:11,980 --> 00:06:17,080
has half-integer intrinsic
angular momentum empirically

148
00:06:17,080 --> 00:06:18,196
turns out to be a fermion.

149
00:06:18,196 --> 00:06:20,070
Take two electrons, take
their wave function,

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00:06:20,070 --> 00:06:22,000
and swap the positions
of the two electrons,

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00:06:22,000 --> 00:06:23,708
the way function picks
up the minus sign.

152
00:06:23,708 --> 00:06:25,816
Take two neutrons,
also spin 1/2.

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00:06:25,816 --> 00:06:27,440
Swap them and you
pick up a minus sign.

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00:06:27,440 --> 00:06:28,420
It's a fermion.

155
00:06:28,420 --> 00:06:30,130
Take two particles of
light, two photons,

156
00:06:30,130 --> 00:06:32,120
swap them and you
get a plus sign,

157
00:06:32,120 --> 00:06:33,690
and these guys
have integer spin.

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00:06:33,690 --> 00:06:37,636
Angular momentum of light
is 1, and similarly,

159
00:06:37,636 --> 00:06:39,010
if we could, for
the Higgs boson,

160
00:06:39,010 --> 00:06:40,801
if you take two Higgs
bosons and swap them,

161
00:06:40,801 --> 00:06:42,430
you'll discover
that they're bosons

162
00:06:42,430 --> 00:06:46,120
and as a consequence there's
a plus sign under exchange.

163
00:06:46,120 --> 00:06:48,780
So that is not
surprising, it's just

164
00:06:48,780 --> 00:06:50,500
an observable
property of the world.

165
00:06:50,500 --> 00:06:53,440
The thing that's shocking
is why that's true, right?

166
00:06:53,440 --> 00:06:55,630
Why is it that things that
have half-integer angular

167
00:06:55,630 --> 00:06:59,031
momentum, which sounds like
an independent property

168
00:06:59,031 --> 00:07:01,530
from whether they're bosonic
or whether identical bosons are

169
00:07:01,530 --> 00:07:03,071
identified, why is
it that all things

170
00:07:03,071 --> 00:07:06,640
with half-integer angular
momentum or spin are fermions,

171
00:07:06,640 --> 00:07:09,730
and all things with integer
angular momentum like light

172
00:07:09,730 --> 00:07:15,620
or the Higgs boson or
me we're for bosonic,

173
00:07:15,620 --> 00:07:18,250
we're identical
with an even sign,

174
00:07:18,250 --> 00:07:20,500
and that requires
quantum field theory.

175
00:07:20,500 --> 00:07:24,100
But it's not insanely difficult.

176
00:07:24,100 --> 00:07:26,765
It requires the bearest minima
of quantum field theory.

177
00:07:26,765 --> 00:07:27,770
So anyone who wants
to understand,

178
00:07:27,770 --> 00:07:29,710
come to my office hours
and I happily explain that.

179
00:07:29,710 --> 00:07:31,460
It takes about 15
minutes just to give you

180
00:07:31,460 --> 00:07:34,135
the basics of quantum field
theory, just the bare basics.

181
00:07:34,135 --> 00:07:36,340
It's pretty straightforward.

182
00:07:36,340 --> 00:07:39,319
But it's a beautiful thing about
relativistic quantum mechanics,

183
00:07:39,319 --> 00:07:40,860
so I'm going to turn
this all around.

184
00:07:40,860 --> 00:07:42,430
So let's look at the history.

185
00:07:42,430 --> 00:07:44,760
The history of this was people
looked at atomic spectra

186
00:07:44,760 --> 00:07:46,165
and said, this is bizarre.

187
00:07:46,165 --> 00:07:48,450
It almost fits what we
get from central potential

188
00:07:48,450 --> 00:07:51,280
except it does buy this
factor of two integer indices.

189
00:07:51,280 --> 00:07:51,780
Ah-ha.

190
00:07:51,780 --> 00:07:54,650
There must be an
extra quantum number

191
00:07:54,650 --> 00:07:57,834
and on top of that, so the
spin, the two possible states

192
00:07:57,834 --> 00:07:58,500
of the electron.

193
00:07:58,500 --> 00:08:00,420
And secondly, it must
be that electrons

194
00:08:00,420 --> 00:08:01,679
can't be in the same state.

195
00:08:01,679 --> 00:08:03,345
So it sounds like
that's two hypotheses.

196
00:08:03,345 --> 00:08:05,650
Now, if you take
those two hypotheses,

197
00:08:05,650 --> 00:08:06,840
everything else follows.

198
00:08:06,840 --> 00:08:07,890
Fine.

199
00:08:07,890 --> 00:08:09,830
What Dirac discovered
when he studied

200
00:08:09,830 --> 00:08:12,590
the relativistic version of
the Schrodinger equation, which

201
00:08:12,590 --> 00:08:14,350
really is quantum
field theory, but when

202
00:08:14,350 --> 00:08:15,808
he studied the
relativistic version

203
00:08:15,808 --> 00:08:19,802
of electrons, quantum
mechanical relativistic version,

204
00:08:19,802 --> 00:08:21,760
he discovered that these
two things are linked.

205
00:08:26,470 --> 00:08:27,350
It is cool.

206
00:08:27,350 --> 00:08:27,851
I grant you.

207
00:08:27,851 --> 00:08:30,100
So you he discovered that
these two things are linked,

208
00:08:30,100 --> 00:08:32,200
and he identified what's
referred to as the spin

209
00:08:32,200 --> 00:08:33,146
statistics theorem.

210
00:08:33,146 --> 00:08:35,270
If you know the spin, you
determine the statistics.

211
00:08:35,270 --> 00:08:36,799
So exactly how that
works, that does

212
00:08:36,799 --> 00:08:38,424
require a little bit
of effort-- that's

213
00:08:38,424 --> 00:08:41,679
the 15 minutes-- but you can
dispense with that entirely

214
00:08:41,679 --> 00:08:45,000
and not be at all shocked by
anything if you just accept

215
00:08:45,000 --> 00:08:50,300
two principles instead of
one, first, that electrons are

216
00:08:50,300 --> 00:08:53,220
anti-fermionic, so they don't
want to be the same state,

217
00:08:53,220 --> 00:08:56,028
and secondly, they
have half-integer spin.

218
00:08:56,028 --> 00:09:01,330
AUDIENCE: Given that the only
thing that's really measurable

219
00:09:01,330 --> 00:09:06,574
is [? the square ?] or
non-square of the wave function

220
00:09:06,574 --> 00:09:10,211
and the way the function itself
is never completely actually,

221
00:09:10,211 --> 00:09:10,710
measurable.

222
00:09:13,436 --> 00:09:18,770
Is there a
mathematical reason why

223
00:09:18,770 --> 00:09:22,810
exchange can't transform
a wave function just

224
00:09:22,810 --> 00:09:23,830
by arbitrary phase?

225
00:09:23,830 --> 00:09:25,874
It has to be plus or minus 1?

226
00:09:25,874 --> 00:09:27,290
PROFESSOR: Well,
the reason it has

227
00:09:27,290 --> 00:09:30,060
to be plus or minus
1 for two particles,

228
00:09:30,060 --> 00:09:31,420
so you said a couple of things.

229
00:09:31,420 --> 00:09:32,628
Let me answer the first part.

230
00:09:32,628 --> 00:09:34,920
So what I will call the first
part of your question is,

231
00:09:34,920 --> 00:09:36,502
why did it have to
be plus or minus 1.

232
00:09:36,502 --> 00:09:38,390
Why couldn't it have
been an arbitrary phase?

233
00:09:38,390 --> 00:09:40,973
And the reason is that if you
do this exchange operation twice

234
00:09:40,973 --> 00:09:43,570
for two particles, you
get back the same thing.

235
00:09:43,570 --> 00:09:44,960
And that's what I mean by saying
I have identical particles.

236
00:09:44,960 --> 00:09:46,060
Because I could
quibble with this.

237
00:09:46,060 --> 00:09:46,810
You could say, well
look, you could

238
00:09:46,810 --> 00:09:48,560
define some different
notion of exchange,

239
00:09:48,560 --> 00:09:51,170
or under exchange, I
pick up an extra phase.

240
00:09:51,170 --> 00:09:53,470
I swap them and then I swap
them and I swap them again,

241
00:09:53,470 --> 00:09:54,053
I get a phase.

242
00:09:54,053 --> 00:09:55,430
But here's the
problem with that.

243
00:09:55,430 --> 00:09:57,990
The problem with that is,
and this can be dealt with,

244
00:09:57,990 --> 00:10:01,329
but an interesting
question is if it can only

245
00:10:01,329 --> 00:10:02,870
be dealt with in
two dimensions or it

246
00:10:02,870 --> 00:10:04,953
can be dealt with in general
number of dimensions.

247
00:10:04,953 --> 00:10:08,030
This is a story called antions,
which Frank Wilczek here

248
00:10:08,030 --> 00:10:09,550
has really pounded hard on.

249
00:10:09,550 --> 00:10:10,800
But here's the basic question.

250
00:10:13,480 --> 00:10:16,910
So does the wave function
have a single value?

251
00:10:16,910 --> 00:10:18,054
I'd like to think it does.

252
00:10:18,054 --> 00:10:20,220
The wave function really
should have a single value.

253
00:10:20,220 --> 00:10:21,490
If it has multiple
values, then it's

254
00:10:21,490 --> 00:10:23,784
ambiguous what the value of
the wave function is there.

255
00:10:23,784 --> 00:10:24,450
That's not good.

256
00:10:24,450 --> 00:10:26,530
That mean you have to specify
more than just the wave

257
00:10:26,530 --> 00:10:28,580
function, you have to
specify which of the values

258
00:10:28,580 --> 00:10:30,544
the wave function takes
in a particular point

259
00:10:30,544 --> 00:10:31,960
in the value of
the wave function.

260
00:10:31,960 --> 00:10:35,270
So if under double
exchange we've

261
00:10:35,270 --> 00:10:37,000
returned to the
original configuration,

262
00:10:37,000 --> 00:10:39,080
you pick up a phase,
something is screwed.

263
00:10:39,080 --> 00:10:43,384
You must be able to tell whether
or not you've exchanged twice,

264
00:10:43,384 --> 00:10:44,550
and here's why you can tell.

265
00:10:44,550 --> 00:10:46,850
Because while it's true that you
can't tell the overall phase,

266
00:10:46,850 --> 00:10:49,140
imagine if I take my system
and I put in a splitter

267
00:10:49,140 --> 00:10:52,662
and I have a beam
splitter I could think

268
00:10:52,662 --> 00:10:55,120
of it as a two-split experiment,
one component of that wave

269
00:10:55,120 --> 00:10:58,289
function I will double swap and
the other component I will not.

270
00:10:58,289 --> 00:10:59,580
And then I will interfere them.

271
00:10:59,580 --> 00:11:00,910
I will combine
them back together,

272
00:11:00,910 --> 00:11:02,160
do an interference experiment.

273
00:11:02,160 --> 00:11:04,580
So you measure that
relative phase.

274
00:11:04,580 --> 00:11:07,215
So that relative phase
definitely matters.

275
00:11:07,215 --> 00:11:09,040
Do you just see that point?

276
00:11:09,040 --> 00:11:10,800
So it's true that
they overall phase

277
00:11:10,800 --> 00:11:12,800
they can't measure, but
by doing a superposition

278
00:11:12,800 --> 00:11:17,610
and only exchanging one of
the superposition pairs,

279
00:11:17,610 --> 00:11:20,690
by physically
separating those, then

280
00:11:20,690 --> 00:11:23,290
you can see that in
interference experiments again.

281
00:11:23,290 --> 00:11:26,020
So you can still deal
with this, but it

282
00:11:26,020 --> 00:11:27,770
requires being able
to know whether or not

283
00:11:27,770 --> 00:11:28,830
you can exchange,
so there has to be

284
00:11:28,830 --> 00:11:31,330
some way of telling that you
sort of entwined these guys,

285
00:11:31,330 --> 00:11:32,740
and that's something you
could do in two dimensions.

286
00:11:32,740 --> 00:11:34,050
It turns out to
be difficult to do

287
00:11:34,050 --> 00:11:35,050
in general number of dimensions.

288
00:11:35,050 --> 00:11:36,630
Well, it's an act
of research topic.

289
00:11:36,630 --> 00:11:38,800
If you want to a more
detailed answer, ask me later.

290
00:11:38,800 --> 00:11:38,860
OK.

291
00:11:38,860 --> 00:11:39,370
One more question.

292
00:11:39,370 --> 00:11:39,580
Yeah.

293
00:11:39,580 --> 00:11:40,540
AUDIENCE: [INAUDIBLE].

294
00:11:45,820 --> 00:11:48,270
PROFESSOR: Yeah. it has
precisely one observable

295
00:11:48,270 --> 00:11:51,194
property, which is this sign,
this it's eigenvalue, value,

296
00:11:51,194 --> 00:11:53,610
and you can tell because the
wave function either vanishes

297
00:11:53,610 --> 00:11:57,010
when you take the two points
together or it doesn't.

298
00:11:57,010 --> 00:11:57,770
So that's it.

299
00:11:57,770 --> 00:12:01,722
AUDIENCE: So it's not like
the momentum [INAUDIBLE].

300
00:12:04,686 --> 00:12:07,650
So it's only basically
just [INAUDIBLE].

301
00:12:10,504 --> 00:12:11,170
PROFESSOR: Yeah.

302
00:12:11,170 --> 00:12:13,580
It's less interesting
than position or momentum.

303
00:12:13,580 --> 00:12:16,420
It has two eigenvalues
rather than many.

304
00:12:16,420 --> 00:12:19,630
But it's no less interesting
than, for example, spin.

305
00:12:19,630 --> 00:12:24,050
It contains information that
you can learn about system.

306
00:12:24,050 --> 00:12:24,550
OK.

307
00:12:24,550 --> 00:12:25,050
One more.

308
00:12:25,050 --> 00:12:30,061
AUDIENCE: I was wondering why
you can write this fermion

309
00:12:30,061 --> 00:12:34,265
as two electrons I guess when
it seems to me [INAUDIBLE].

310
00:12:34,265 --> 00:12:36,050
PROFESSOR: That's a
fantastic question.

311
00:12:36,050 --> 00:12:36,550
OK.

312
00:12:36,550 --> 00:12:36,770
Good.

313
00:12:36,770 --> 00:12:37,635
I was going to gloss over this.

314
00:12:37,635 --> 00:12:38,390
That's a fantastic question.

315
00:12:38,390 --> 00:12:39,390
So here's the question.

316
00:12:39,390 --> 00:12:42,420
Look, I have in my box here,
I have a hydrogen atom,

317
00:12:42,420 --> 00:12:44,930
and I've quantized
the cooling potential,

318
00:12:44,930 --> 00:12:47,570
and I know what the energy
eigenvalues are for the cooling

319
00:12:47,570 --> 00:12:49,007
potential, and I
put the electron

320
00:12:49,007 --> 00:12:49,965
in one of those states.

321
00:12:49,965 --> 00:12:52,550
And I say the wave function
describing my system as sign is

322
00:12:52,550 --> 00:12:54,830
equal to phi [? nlm ?] for
some particular [? nlm. ?]

323
00:12:54,830 --> 00:12:56,290
That's what we've been doing.

324
00:12:56,290 --> 00:12:58,740
But there are a lot of
electrons in the world.

325
00:12:58,740 --> 00:13:00,490
In fact, there are
electrons in the walls.

326
00:13:00,490 --> 00:13:01,870
There are electrons
in your nose.

327
00:13:01,870 --> 00:13:04,190
There are electrons everywhere,
and they are fermion.

328
00:13:04,190 --> 00:13:07,130
So the wave function
describing the entire system

329
00:13:07,130 --> 00:13:11,530
must be invariant up to a sign
under the exchange of any two

330
00:13:11,530 --> 00:13:12,030
electrons.

331
00:13:12,030 --> 00:13:13,571
If I take this
electron and I swap it

332
00:13:13,571 --> 00:13:16,070
with an electron
in Matt's ear-- Hi,

333
00:13:16,070 --> 00:13:18,450
Matt-- then the wave
function had better

334
00:13:18,450 --> 00:13:19,840
pick up a minus sign.

335
00:13:19,840 --> 00:13:22,830
So what business do I ever have
writing a single wave function

336
00:13:22,830 --> 00:13:23,924
for a single electron.

337
00:13:23,924 --> 00:13:24,840
Is that your question?

338
00:13:24,840 --> 00:13:25,524
AUDIENCE: Yes.

339
00:13:25,524 --> 00:13:27,190
PROFESSOR: That's an
excellent question.

340
00:13:27,190 --> 00:13:29,590
So let me answer that.

341
00:13:29,590 --> 00:13:34,840
So suppose I have
two electrons, and I

342
00:13:34,840 --> 00:13:38,749
have the wave function
for two electrons x and y.

343
00:13:38,749 --> 00:13:40,790
I mean, I write this as
if it's in one dimension.

344
00:13:40,790 --> 00:13:42,331
This is some position,
some position,

345
00:13:42,331 --> 00:13:45,450
but this generally is just an
arbitrary number of dimensions,

346
00:13:45,450 --> 00:13:48,500
and one is in the
state chi of x,

347
00:13:48,500 --> 00:13:50,344
and the other is in
the state phi of y.

348
00:13:50,344 --> 00:13:52,260
But I know that I can't
have this be the state

349
00:13:52,260 --> 00:13:54,051
because it must be
invariant under swapping

350
00:13:54,051 --> 00:13:55,190
x and y of two minus signs.

351
00:13:55,190 --> 00:13:58,950
So I need minus
chi at y phi at x,

352
00:13:58,950 --> 00:14:03,672
and let's normalize
this, so 1 over 2.

353
00:14:03,672 --> 00:14:04,630
So that's our electron.

354
00:14:04,630 --> 00:14:06,740
Now, at this point,
you might really worry,

355
00:14:06,740 --> 00:14:10,785
because say chi is the electron
is the wave function bound

356
00:14:10,785 --> 00:14:13,970
to my atom, and phi is the wave
function for the electron bound

357
00:14:13,970 --> 00:14:16,980
to an atom in Matt's ear.

358
00:14:16,980 --> 00:14:19,240
So we've just in writing this.

359
00:14:19,240 --> 00:14:20,232
Why can we do that?

360
00:14:20,232 --> 00:14:21,440
Well, let's think about this.

361
00:14:21,440 --> 00:14:22,920
What does the combination
of terms give us?

362
00:14:22,920 --> 00:14:24,860
What do they tell us is that
they're interference term.

363
00:14:24,860 --> 00:14:26,010
If I ask them what's
the probability

364
00:14:26,010 --> 00:14:28,120
to find an electron at
x and an electron at y,

365
00:14:28,120 --> 00:14:31,520
this is equal to, well,
it's this whole quantity

366
00:14:31,520 --> 00:14:36,162
norm squared sine norm
squared, which is equal to 1/2,

367
00:14:36,162 --> 00:14:38,370
and then there's a term
where this gets term squared,

368
00:14:38,370 --> 00:14:40,203
this term squared, and
then two cross terms.

369
00:14:40,203 --> 00:14:42,350
So the first term
gives us the chi

370
00:14:42,350 --> 00:14:47,210
at x norm squared phi
at y norm squared.

371
00:14:47,210 --> 00:14:52,815
The second term gives us plus
chi at y norm squared phi

372
00:14:52,815 --> 00:14:59,030
at x norm squared plus
twice the real part

373
00:14:59,030 --> 00:15:03,330
of I'm going to write
this subscript chi

374
00:15:03,330 --> 00:15:11,698
sub x phi y, complex conjugate,
complex conjugate chi y phi x.

375
00:15:11,698 --> 00:15:12,519
Everybody agree?

376
00:15:12,519 --> 00:15:14,310
That's just the norm
squared of everything.

377
00:15:14,310 --> 00:15:15,980
AUDIENCE: Shouldn't
that be minus?

378
00:15:15,980 --> 00:15:16,350
PROFESSOR: Oh, shoot.

379
00:15:16,350 --> 00:15:16,640
Sorry.

380
00:15:16,640 --> 00:15:17,140
Thank you.

381
00:15:17,140 --> 00:15:17,840
Minus.

382
00:15:17,840 --> 00:15:18,740
Yes, minus.

383
00:15:18,740 --> 00:15:20,420
Importantly minus.

384
00:15:20,420 --> 00:15:22,930
Now, let's be a little
more explicit about this.

385
00:15:22,930 --> 00:15:31,995
Saying that chi is the state
localized at the hydrogen,

386
00:15:31,995 --> 00:15:33,620
then that's as if
the hydrogen is here.

387
00:15:33,620 --> 00:15:35,790
If the proton is here, then
the wave function for chi

388
00:15:35,790 --> 00:15:37,470
has a norm squared
that looks like this.

389
00:15:40,200 --> 00:15:43,440
And if I say that phi is the
state corresponding to being

390
00:15:43,440 --> 00:15:54,110
localized, at Matt's ear,
then here is the ear,

391
00:15:54,110 --> 00:15:57,560
and here is the wave function
for it, so that's phi.

392
00:16:01,420 --> 00:16:05,310
So let's then ask, what's
the probability that I find

393
00:16:05,310 --> 00:16:10,410
the first electron here and
the second electron here?

394
00:16:10,410 --> 00:16:14,834
Well, if this is
x, and this is y,

395
00:16:14,834 --> 00:16:16,250
what's the probability
that I find

396
00:16:16,250 --> 00:16:17,380
a particle there and
a particle there?

397
00:16:17,380 --> 00:16:18,420
Well, what's this term?

398
00:16:18,420 --> 00:16:22,090
Well, it's chi of x, which
is not so small-- that's just

399
00:16:22,090 --> 00:16:26,930
some number-- times phi of
y, some value norm squared,

400
00:16:26,930 --> 00:16:28,680
so this is some number.

401
00:16:28,680 --> 00:16:29,720
What about this one?

402
00:16:29,720 --> 00:16:32,220
What's chi of y?

403
00:16:32,220 --> 00:16:35,230
Well, chi of y, this is all
sharply-localized wave function

404
00:16:35,230 --> 00:16:38,930
over here chi, and
it's 0 out here.

405
00:16:38,930 --> 00:16:41,260
The probability that an
electron bound to the hydrogen

406
00:16:41,260 --> 00:16:44,400
is found far away is
exponentially small

407
00:16:44,400 --> 00:16:46,790
as we've seen from the
wave functions of hydrogen.

408
00:16:46,790 --> 00:16:47,790
It's bound.

409
00:16:47,790 --> 00:16:51,650
So this is the negligibility
small as is phi of x.

410
00:16:51,650 --> 00:16:53,320
Phi of x is
negligibility do small.

411
00:16:53,320 --> 00:16:55,380
The wave function is
localized around the ear,

412
00:16:55,380 --> 00:16:57,070
but x is way over
there by the hydrogen.

413
00:16:57,070 --> 00:16:58,190
It's exponentially small.

414
00:16:58,190 --> 00:17:00,032
This term is also
exponentially small.

415
00:17:00,032 --> 00:17:01,240
Now in here, what about this?

416
00:17:01,240 --> 00:17:03,570
Chi of x phi of y, that's good.

417
00:17:03,570 --> 00:17:08,442
Those are both fine, but chi
of y phi of x phi of x, these

418
00:17:08,442 --> 00:17:10,650
are both negligibility small,
so these terms go away.

419
00:17:10,650 --> 00:17:18,119
So this is just equal to chi
of x norm squared phi of y norm

420
00:17:18,119 --> 00:17:18,650
squared.

421
00:17:18,650 --> 00:17:21,990
And this is what you
would get approximately,

422
00:17:21,990 --> 00:17:23,930
and that approximately
is fantastically good

423
00:17:23,930 --> 00:17:24,880
as they get far apart.

424
00:17:24,880 --> 00:17:31,020
This chi of x is
just the probability

425
00:17:31,020 --> 00:17:33,270
that the particle that x
given that it was in the wave

426
00:17:33,270 --> 00:17:35,850
function bound to the
hydrogen, and ditto given

427
00:17:35,850 --> 00:17:37,480
that it was bound to the ear.

428
00:17:37,480 --> 00:17:38,642
Cool?

429
00:17:38,642 --> 00:17:40,850
So the important thing is
when fermions are separated

430
00:17:40,850 --> 00:17:42,940
and the wave functions
are localized,

431
00:17:42,940 --> 00:17:45,300
you can treat them
independently.

432
00:17:45,300 --> 00:17:47,820
When the fermions
are not separated

433
00:17:47,820 --> 00:17:49,964
or when their wave
functions are not localized,

434
00:17:49,964 --> 00:17:51,380
you can't treat
them independently

435
00:17:51,380 --> 00:17:54,110
and you have to
include all the terms.

436
00:17:54,110 --> 00:17:55,201
That cool?

437
00:17:55,201 --> 00:17:55,700
OK.

438
00:17:55,700 --> 00:17:56,200
Yeah.

439
00:17:56,200 --> 00:17:58,142
AUDIENCE: That's
times 1/2, right?

440
00:17:58,142 --> 00:17:59,600
PROFESSOR: Yeah,
that is times 1/2.

441
00:17:59,600 --> 00:18:00,718
What did I do?

442
00:18:16,499 --> 00:18:18,540
I'm screwing something up
with the normalization.

443
00:18:18,540 --> 00:18:19,450
Ask me that after lecture.

444
00:18:19,450 --> 00:18:20,360
It's a good question.

445
00:18:20,360 --> 00:18:21,360
I'm screwing something up.

446
00:18:21,360 --> 00:18:23,568
I'm not going to get the
factor of 2 straightened out

447
00:18:23,568 --> 00:18:24,110
right now.

448
00:18:24,110 --> 00:18:24,610
OK.

449
00:18:24,610 --> 00:18:26,140
Does that answer your question?

450
00:18:26,140 --> 00:18:26,640
OK.

451
00:18:26,640 --> 00:18:27,140
Good.

452
00:18:27,140 --> 00:18:30,640
This is a really deep-- it's
important that this is true.

453
00:18:30,640 --> 00:18:32,350
I'm going to stop
questions because I

454
00:18:32,350 --> 00:18:34,210
need to pick up with
what we need to do.

455
00:18:34,210 --> 00:18:37,189
So this is a really
important question

456
00:18:37,189 --> 00:18:39,480
because if it weren't true
that you could independently

457
00:18:39,480 --> 00:18:41,030
deal with these electron,
then everything we

458
00:18:41,030 --> 00:18:42,600
did up until now in
quantum mechanics

459
00:18:42,600 --> 00:18:45,180
would have been totally useless
for any identical particles.

460
00:18:45,180 --> 00:18:48,150
And since as far as we can
tell, all fundamental particles

461
00:18:48,150 --> 00:18:53,366
are identical, this
would have been very bad.

462
00:18:56,780 --> 00:19:01,900
So with that, let's move
on to the study of solids.

463
00:19:01,900 --> 00:19:05,737
So I want to pick up where
your problem set left off.

464
00:19:05,737 --> 00:19:08,070
The last part of your problem
set, which I hope everyone

465
00:19:08,070 --> 00:19:11,850
did, the left part
of your problem set

466
00:19:11,850 --> 00:19:14,610
involved looking
at this simulation

467
00:19:14,610 --> 00:19:18,340
from the PhET
people of a particle

468
00:19:18,340 --> 00:19:20,800
in a series of n wells.

469
00:19:20,800 --> 00:19:23,610
And I want to look at the
result, at least the results

470
00:19:23,610 --> 00:19:27,810
that I got for that simulation.

471
00:19:27,810 --> 00:19:35,730
So let's see, here
are the data points

472
00:19:35,730 --> 00:19:38,930
that I got-- shall
I make that larger?

473
00:19:38,930 --> 00:19:40,050
Is that impossible to see?

474
00:19:42,700 --> 00:19:45,160
Those are the data
points that I pulled off

475
00:19:45,160 --> 00:19:47,102
of PhET just the
same way you did.

476
00:19:47,102 --> 00:19:48,810
You look, you point,
you move the cursor,

477
00:19:48,810 --> 00:19:50,185
and you pull off
the data points.

478
00:19:50,185 --> 00:19:53,210
So those are approximately then.

479
00:19:53,210 --> 00:19:58,780
And this is for 1, 2, 3, 4, 5,
6, 7, 8, 10 wells, and here's

480
00:19:58,780 --> 00:20:03,350
the plot of them as a function
of the index which state they

481
00:20:03,350 --> 00:20:04,020
are.

482
00:20:04,020 --> 00:20:05,373
So as a function, this
is the first state.

483
00:20:05,373 --> 00:20:07,500
This is the second state,
third, fourth, et cetera.

484
00:20:07,500 --> 00:20:10,610
Here are the
energies vertically.

485
00:20:10,610 --> 00:20:13,360
And hopefully, you all got
to plot some more of this

486
00:20:13,360 --> 00:20:15,720
and they seem to become
bunched together,

487
00:20:15,720 --> 00:20:21,030
and the energy of the ground
state without the potential

488
00:20:21,030 --> 00:20:21,800
was this.

489
00:20:25,260 --> 00:20:29,060
So I'm not going to add in
a parabola corresponding

490
00:20:29,060 --> 00:20:31,190
to a free particle with
the associated wavelengths.

491
00:20:31,190 --> 00:20:33,470
Remember I asked you to
say take that state that

492
00:20:33,470 --> 00:20:34,790
looks most like a momentum.

493
00:20:34,790 --> 00:20:36,580
eigenstate with a
definite wavelength

494
00:20:36,580 --> 00:20:39,100
and calculate the energy
of a free particle

495
00:20:39,100 --> 00:20:40,620
with the corresponding
wavelength.

496
00:20:40,620 --> 00:20:44,970
So this parabola-- I
should make this smaller--

497
00:20:44,970 --> 00:20:50,851
is a parabola describing
a free particle which

498
00:20:50,851 --> 00:20:53,350
agrees with those free particle
energies at the three points

499
00:20:53,350 --> 00:20:54,933
where you're supposed
to compute them.

500
00:20:54,933 --> 00:20:56,070
So there's the parabola.

501
00:20:56,070 --> 00:20:58,403
And so what I've done here
is I've grouped them together

502
00:20:58,403 --> 00:20:59,610
into the bands of states.

503
00:20:59,610 --> 00:21:01,860
So vertically we have energy.

504
00:21:01,860 --> 00:21:03,260
On the horizontal
side, we have n

505
00:21:03,260 --> 00:21:05,640
but remember that n is
also corresponding kind

506
00:21:05,640 --> 00:21:09,250
to a momentum because each
state wiggles and especially

507
00:21:09,250 --> 00:21:11,240
the states at the
top of each band

508
00:21:11,240 --> 00:21:12,865
have a reasonably
well-defined momenta,

509
00:21:12,865 --> 00:21:15,150
and so they look a lot
like momentum eigenstates.

510
00:21:15,150 --> 00:21:17,316
So I'm going to interpret
this horizontal direction,

511
00:21:17,316 --> 00:21:19,259
it's something like a momentum.

512
00:21:19,259 --> 00:21:21,550
And at the top of the states,
it certainly makes sense,

513
00:21:21,550 --> 00:21:24,719
but then what it actually is
is just the number the level,

514
00:21:24,719 --> 00:21:26,260
and what we see is
that the energy is

515
00:21:26,260 --> 00:21:31,240
a function of the level
is close-- near the top

516
00:21:31,240 --> 00:21:37,100
anyway-- to the energy
of a free particle,

517
00:21:37,100 --> 00:21:39,260
but the actual allowed
energies are bound.

518
00:21:39,260 --> 00:21:42,682
So which energies correspond
to allowed states,

519
00:21:42,682 --> 00:21:44,390
and which energies
have no allowed states

520
00:21:44,390 --> 00:21:46,348
associate with them is
encoded in this diagram.

521
00:21:46,348 --> 00:21:47,610
Are we cool with that?

522
00:21:47,610 --> 00:21:49,270
In this shaded region,
there are states

523
00:21:49,270 --> 00:21:52,430
with an energy in that region,
at least approximately,

524
00:21:52,430 --> 00:21:54,850
and in this unshaded
region, there aren't any.

525
00:21:54,850 --> 00:22:01,410
And in each band, there
are n energy eigenstates,

526
00:22:01,410 --> 00:22:04,040
and each band corresponds
to one of the bound

527
00:22:04,040 --> 00:22:05,880
states of the potential.

528
00:22:05,880 --> 00:22:06,380
OK.

529
00:22:06,380 --> 00:22:07,530
Everyone cool with that?

530
00:22:07,530 --> 00:22:09,655
So does this look more or
less like what y'all got?

531
00:22:09,655 --> 00:22:10,771
AUDIENCE: Yes.

532
00:22:10,771 --> 00:22:11,520
PROFESSOR: Good.

533
00:22:11,520 --> 00:22:12,020
OK.

534
00:22:12,020 --> 00:22:13,740
Questions about this
before I move on?

535
00:22:13,740 --> 00:22:15,715
It's an important one.

536
00:22:15,715 --> 00:22:18,540
And let me just remind you of a
couple of facts about this PhET

537
00:22:18,540 --> 00:22:24,120
simulation so here's what we
see if we have the single well.

538
00:22:24,120 --> 00:22:29,590
We have three states 1, 0, two
nodes, two zeroes, and they

539
00:22:29,590 --> 00:22:31,262
satisfy the node theorem.

540
00:22:31,262 --> 00:22:33,470
If we have many states, then
if we look at the bottom

541
00:22:33,470 --> 00:22:34,510
state in each well.

542
00:22:34,510 --> 00:22:39,200
So there's the bottom
state and the bottom band.

543
00:22:39,200 --> 00:22:39,990
It has no zeroes.

544
00:22:39,990 --> 00:22:43,550
It satisfies the node theorem,
and the top one by comparison

545
00:22:43,550 --> 00:22:46,020
looks extremely similar.

546
00:22:46,020 --> 00:22:49,540
It looks extremely similar
to the ground state, which

547
00:22:49,540 --> 00:22:52,910
is the orange one, the
lowest energy state.

548
00:22:52,910 --> 00:22:54,140
It is a little higher energy.

549
00:22:54,140 --> 00:22:56,560
You can see that because
the curvature is greater.

550
00:22:56,560 --> 00:23:00,529
If you look at the top and
bottom of each wave function,

551
00:23:00,529 --> 00:23:02,070
you see that the
curvature is greater

552
00:23:02,070 --> 00:23:04,860
for the yellow guy, which is
the guy at the top of the band,

553
00:23:04,860 --> 00:23:06,360
and the yellow guy,
meanwhile, looks

554
00:23:06,360 --> 00:23:09,120
like a wave with a
definite wavelength.

555
00:23:09,120 --> 00:23:11,020
But a general state
in here doesn't

556
00:23:11,020 --> 00:23:15,270
have any simple
symmetry properties.

557
00:23:15,270 --> 00:23:16,270
It's not periodic.

558
00:23:16,270 --> 00:23:17,720
It's not approximately periodic.

559
00:23:17,720 --> 00:23:19,720
Well, that one's kind of
approximately periodic.

560
00:23:19,720 --> 00:23:21,340
But the top guy
and the bottom guy

561
00:23:21,340 --> 00:23:23,230
look approximately periodic.

562
00:23:23,230 --> 00:23:25,270
They have some nice
symmetry properties.

563
00:23:25,270 --> 00:23:27,280
But in general, they're just
not translationally invariant.

564
00:23:27,280 --> 00:23:28,196
They're not symmetric.

565
00:23:28,196 --> 00:23:30,300
They don't have any
simple, nice structure.

566
00:23:30,300 --> 00:23:32,780
They are just some
horrible, ugly things.

567
00:23:32,780 --> 00:23:35,060
The bottom though
and the top guy

568
00:23:35,060 --> 00:23:38,680
always have some nice
symmetry property.

569
00:23:38,680 --> 00:23:40,160
Anyway to keep that in mind.

570
00:23:40,160 --> 00:23:43,340
And the last thing I want
you to note from all of this

571
00:23:43,340 --> 00:23:47,550
is that as we make the
potential barriers stronger,

572
00:23:47,550 --> 00:23:48,850
two things happen.

573
00:23:48,850 --> 00:23:51,920
First off, bands become very
narrow, the gaps between them

574
00:23:51,920 --> 00:23:57,880
become very large, and
secondly, the wave functions

575
00:23:57,880 --> 00:24:01,132
take now an even more sort
of complicated and messy,

576
00:24:01,132 --> 00:24:03,340
which is more obvious that
the wave functions are not

577
00:24:03,340 --> 00:24:04,180
periodic.

578
00:24:04,180 --> 00:24:05,980
This thing it looks
almost periodic,

579
00:24:05,980 --> 00:24:07,900
but is periodic with
some funny period,

580
00:24:07,900 --> 00:24:11,250
and it's not a single period by
any stretch of the imagination.

581
00:24:11,250 --> 00:24:12,120
OK.

582
00:24:12,120 --> 00:24:16,605
So any questions about the
wave functions in here?

583
00:24:16,605 --> 00:24:17,105
Yeah.

584
00:24:17,105 --> 00:24:23,886
AUDIENCE: In this one, there is
an overall [INAUDIBLE] period.

585
00:24:23,886 --> 00:24:24,386
[INAUDIBLE]

586
00:24:31,435 --> 00:24:32,450
PROFESSOR: Excellent.

587
00:24:32,450 --> 00:24:33,910
OK.

588
00:24:33,910 --> 00:24:36,940
So let me give you a little
bit of intuition for that.

589
00:24:36,940 --> 00:24:41,240
So suppose I have a big box.

590
00:24:41,240 --> 00:24:42,410
What's the ground state?

591
00:24:45,251 --> 00:24:46,792
What does the ground
state look like?

592
00:24:46,792 --> 00:24:47,140
Yeah.

593
00:24:47,140 --> 00:24:47,340
Good.

594
00:24:47,340 --> 00:24:48,130
Everyone's doing this.

595
00:24:48,130 --> 00:24:48,400
That's good.

596
00:24:48,400 --> 00:24:49,930
So the ground state
looks like this,

597
00:24:49,930 --> 00:24:51,930
and the first excited
state is going

598
00:24:51,930 --> 00:24:56,540
to do something like
this, and so on.

599
00:24:56,540 --> 00:24:57,040
Cool.

600
00:24:57,040 --> 00:25:02,250
And what's the third excited
state going to look like?

601
00:25:02,250 --> 00:25:04,852
I guess I shouldn't do that.

602
00:25:04,852 --> 00:25:07,310
The third excited state is
going to do something like this.

603
00:25:09,940 --> 00:25:12,200
Cool?

604
00:25:12,200 --> 00:25:15,280
Now, meanwhile, imagine
I take my periodic well

605
00:25:15,280 --> 00:25:20,370
and I add to it a bunch of
delta function scatterers

606
00:25:20,370 --> 00:25:21,730
with some strength.

607
00:25:21,730 --> 00:25:26,330
When the strength is zero, I
just recover my original well.

608
00:25:26,330 --> 00:25:28,180
When the strength is
non-zero though, we

609
00:25:28,180 --> 00:25:29,800
know that what a delta
function is going to do is,

610
00:25:29,800 --> 00:25:31,200
it's going to make
a little kink.

611
00:25:31,200 --> 00:25:32,840
It's going to induce
a little condition

612
00:25:32,840 --> 00:25:34,850
on the first derivative
at the delta function.

613
00:25:34,850 --> 00:25:36,350
So what's going to
this ground state

614
00:25:36,350 --> 00:25:40,070
when we turn a little tiny
bit of that delta function?

615
00:25:40,070 --> 00:25:40,570
Yeah.

616
00:25:40,570 --> 00:25:41,945
What we're going
to get is, we're

617
00:25:41,945 --> 00:25:44,869
going to get
something that kinks

618
00:25:44,869 --> 00:25:46,160
at each of the delta functions.

619
00:25:52,875 --> 00:25:55,000
But it still has to have
those boundary conditions.

620
00:25:55,000 --> 00:25:56,530
Everybody cool with that?

621
00:25:56,530 --> 00:25:59,512
And as we make the delta
function stronger and stronger,

622
00:25:59,512 --> 00:26:01,220
the effect of this is
going to eventually

623
00:26:01,220 --> 00:26:07,910
be to give us something
that looks like-- yeah?

624
00:26:07,910 --> 00:26:12,010
Now that should look
a lot like that.

625
00:26:12,010 --> 00:26:14,344
Everyone cool with that?

626
00:26:14,344 --> 00:26:16,510
Meanwhile, what's going to
happen to the second guy?

627
00:26:19,720 --> 00:26:21,340
Well the second guy, same thing.

628
00:26:21,340 --> 00:26:33,372
The effect on the second
state is going to be-- cool?

629
00:26:33,372 --> 00:26:35,080
I'm not sure of I need
the sound effects.

630
00:26:35,080 --> 00:26:41,170
So let's magnify and let's
look at the second state here.

631
00:26:41,170 --> 00:26:44,481
Ooh, it's going to be hard
to deal with that tiny energy

632
00:26:44,481 --> 00:26:44,980
splitting.

633
00:26:47,602 --> 00:26:49,824
Come on finger, you can do this.

634
00:26:49,824 --> 00:26:51,920
There we go.

635
00:26:51,920 --> 00:26:54,100
There is two.

636
00:26:54,100 --> 00:26:55,240
Ah-ha.

637
00:26:55,240 --> 00:26:56,690
That looks a lot like this.

638
00:26:56,690 --> 00:26:59,194
Yes, that second state looks
an awful lot like this.

639
00:26:59,194 --> 00:27:00,735
And let's go back
to that funny state

640
00:27:00,735 --> 00:27:02,026
that we were looking at before.

641
00:27:02,026 --> 00:27:04,040
Let' see if I can get it.

642
00:27:04,040 --> 00:27:05,970
Should be the third guy here.

643
00:27:05,970 --> 00:27:08,890
So there's a second
guy in the third band.

644
00:27:08,890 --> 00:27:11,720
Again, you see an envelope
and then the fluctuation

645
00:27:11,720 --> 00:27:15,300
from being in the third
state and in the second state

646
00:27:15,300 --> 00:27:16,270
in this band as well.

647
00:27:19,360 --> 00:27:23,790
So on the other hand, what's
the scale of the energy

648
00:27:23,790 --> 00:27:26,280
splitting due to a
well of it this width

649
00:27:26,280 --> 00:27:28,270
compared to the energy
splitting between states

650
00:27:28,270 --> 00:27:31,350
due to a well of this width?

651
00:27:31,350 --> 00:27:34,570
A wide well means that its
states are close together.

652
00:27:34,570 --> 00:27:37,980
An ENDOR well means the
states are far apart.

653
00:27:37,980 --> 00:27:40,179
So what state do we have?

654
00:27:40,179 --> 00:27:41,720
We have the state
that was the ground

655
00:27:41,720 --> 00:27:43,990
state of the whole
box with a correction

656
00:27:43,990 --> 00:27:46,060
due to the delta function
versus states which

657
00:27:46,060 --> 00:27:48,150
are, for example, excited
states in the box.

658
00:27:56,010 --> 00:28:02,125
The splitting between
this state and this state

659
00:28:02,125 --> 00:28:04,000
is going to be much
larger than the splitting

660
00:28:04,000 --> 00:28:06,145
between this state
and this state,

661
00:28:06,145 --> 00:28:07,777
just because this is
a tiny little well

662
00:28:07,777 --> 00:28:09,360
and so the link scale
is much shorter,

663
00:28:09,360 --> 00:28:10,630
the energy is much greater.

664
00:28:10,630 --> 00:28:12,900
That's the intuition you
should have from these guys.

665
00:28:12,900 --> 00:28:16,560
So when you see that there's
some approximate structure,

666
00:28:16,560 --> 00:28:20,310
let's see let's make this
nice and separated again.

667
00:28:20,310 --> 00:28:26,140
And Oh, yeah.

668
00:28:26,140 --> 00:28:28,579
these are kind of
difficult to control

669
00:28:28,579 --> 00:28:29,870
when they're so close together.

670
00:28:41,451 --> 00:28:42,617
The trouble with touch pads.

671
00:28:46,030 --> 00:28:46,600
There we go.

672
00:28:46,600 --> 00:28:46,900
OK.

673
00:28:46,900 --> 00:28:47,400
There we go.

674
00:28:47,400 --> 00:28:49,610
That was the guy we
were looking at before.

675
00:28:49,610 --> 00:28:52,670
And so you see that there
is an overall envelope which

676
00:28:52,670 --> 00:28:56,020
has three maxima, and then
there's the n inside each well.

677
00:28:56,020 --> 00:28:59,340
It's got two zeroes
inside each well.

678
00:28:59,340 --> 00:29:01,935
And it's that envelope which
is coming from the fact

679
00:29:01,935 --> 00:29:04,290
that you're in a box
and the two zeroes

680
00:29:04,290 --> 00:29:07,930
inside each well, which
is coming from this.

681
00:29:07,930 --> 00:29:11,910
All the states here in
this band are of the form

682
00:29:11,910 --> 00:29:13,170
two zeroes inside the well.

683
00:29:17,982 --> 00:29:20,190
All the states here are the
form two zeroes is inside

684
00:29:20,190 --> 00:29:25,970
each well in this
band except they're

685
00:29:25,970 --> 00:29:30,120
being modulated by an overall
sine wave due to the fact

686
00:29:30,120 --> 00:29:31,460
that they're in a box.

687
00:29:31,460 --> 00:29:32,990
And so they're
very closely split

688
00:29:32,990 --> 00:29:35,480
states with a different
overall modulation

689
00:29:35,480 --> 00:29:38,420
but with each state inside
each band corresponding

690
00:29:38,420 --> 00:29:40,589
to either the ground state
for the individual well

691
00:29:40,589 --> 00:29:42,630
or the first excited state
in the individual well

692
00:29:42,630 --> 00:29:44,290
or the second excited state.

693
00:29:44,290 --> 00:29:46,360
So each band corresponds
to which excited state

694
00:29:46,360 --> 00:29:48,565
you are inside the well,
and which state you

695
00:29:48,565 --> 00:29:50,530
are within the band
is your modulation

696
00:29:50,530 --> 00:29:55,950
of the overall wave
from being inside a box.

697
00:29:55,950 --> 00:29:56,750
That make sense?

698
00:29:56,750 --> 00:29:59,600
AUDIENCE: I guess so, but why
does the overall modulation

699
00:29:59,600 --> 00:30:02,450
have a smaller effect than the--

700
00:30:02,450 --> 00:30:03,320
PROFESSOR: Great.

701
00:30:03,320 --> 00:30:07,110
If I have a well
that's this wide,

702
00:30:07,110 --> 00:30:09,480
what are the ground
state energies?

703
00:30:09,480 --> 00:30:12,450
And let's say the
width is L. What

704
00:30:12,450 --> 00:30:15,457
are the energies in this square
well, an infinite well with L?

705
00:30:15,457 --> 00:30:16,915
Yeah, but what are
the eigenvalues?

706
00:30:16,915 --> 00:30:18,320
AUDIENCE: [INAUDIBLE].

707
00:30:18,320 --> 00:30:20,150
PROFESSOR: E sub n.

708
00:30:20,150 --> 00:30:23,120
AUDIENCE: [INAUDIBLE].

709
00:30:23,120 --> 00:30:23,900
PROFESSOR: Yeah.

710
00:30:23,900 --> 00:30:24,260
Exactly.

711
00:30:24,260 --> 00:30:26,634
So they go like, I'm just
going to write proportional to,

712
00:30:26,634 --> 00:30:30,230
n squared over L
squared, because it's

713
00:30:30,230 --> 00:30:37,490
a sine wave with
period n pi upon L.

714
00:30:37,490 --> 00:30:41,610
So the k is n upon
L with constants,

715
00:30:41,610 --> 00:30:44,040
and that means that the energy
again, constant h squared k

716
00:30:44,040 --> 00:30:47,145
upon 2n is k squared, which
is n square over L squared.

717
00:30:47,145 --> 00:30:49,020
But the important thing
is that the splitting

718
00:30:49,020 --> 00:30:51,530
between subsequent energy levels
is controlled by the width.

719
00:30:51,530 --> 00:30:52,870
It goes like 1 over L squared.

720
00:30:52,870 --> 00:30:54,840
So if you have a very
wide well, the splittings

721
00:30:54,840 --> 00:30:56,215
are very small,
the ground state.

722
00:30:59,450 --> 00:31:04,600
On the other hand, if you
have a thin, narrow well,

723
00:31:04,600 --> 00:31:06,540
then the splittings
also go like L squared,

724
00:31:06,540 --> 00:31:08,873
but L is very small here so
the splittings are gigantic.

725
00:31:12,740 --> 00:31:15,800
When we have a superposition
of a big box inside of which we

726
00:31:15,800 --> 00:31:18,400
have a bunch of barriers,
then the splitting

727
00:31:18,400 --> 00:31:22,310
due to being excited
in the individual wells

728
00:31:22,310 --> 00:31:24,267
goes like 1 over the
little distance squared.

729
00:31:24,267 --> 00:31:25,350
Let's call this a squared.

730
00:31:25,350 --> 00:31:28,600
So these go like 1 over a
squared, delta a delta E,

731
00:31:28,600 --> 00:31:31,210
goes like 1 over a
squared, and from here it

732
00:31:31,210 --> 00:31:37,170
goes like if this is L, delta
E goes like 1 over L squared.

733
00:31:37,170 --> 00:31:40,670
So the combined effect is that
you get splittings due to both.

734
00:31:40,670 --> 00:31:43,000
These splittings are
tiny, and these splittings

735
00:31:43,000 --> 00:31:45,416
are really large because this
is a much smaller distance,

736
00:31:45,416 --> 00:31:46,874
and this is a much
larger distance.

737
00:31:46,874 --> 00:31:48,200
Cool?

738
00:31:48,200 --> 00:31:51,336
So the only question
is how big in amplitude

739
00:31:51,336 --> 00:31:53,210
are these-- how effective
are these barriers.

740
00:31:53,210 --> 00:31:54,918
If they were strict
delta functions, then

741
00:31:54,918 --> 00:31:56,669
this is all we would get.

742
00:31:56,669 --> 00:31:58,210
And if there were
no delta functions,

743
00:31:58,210 --> 00:32:00,310
we would just get
precisely these.

744
00:32:00,310 --> 00:32:01,954
And when we have
some barrier, we

745
00:32:01,954 --> 00:32:03,620
get a combination of
the two, and that's

746
00:32:03,620 --> 00:32:05,502
what we're seeing here
with the splitting.

747
00:32:05,502 --> 00:32:07,710
So what I'm doing when I'm
tuning the separation here

748
00:32:07,710 --> 00:32:11,730
I'm controlling effectively
how strong is that barrier.

749
00:32:11,730 --> 00:32:13,846
And so as we make the
barriers stronger--

750
00:32:13,846 --> 00:32:15,220
there we go-- then
the splittings

751
00:32:15,220 --> 00:32:18,220
are controlled by
the individual wells,

752
00:32:18,220 --> 00:32:20,970
an as we make the
barrier very inefficient,

753
00:32:20,970 --> 00:32:25,630
then the splittings are
controlled by the overall box.

754
00:32:25,630 --> 00:32:26,410
Cool?

755
00:32:26,410 --> 00:32:27,080
Excellent.

756
00:32:27,080 --> 00:32:28,880
Other questions
about the simulation?

757
00:32:28,880 --> 00:32:29,380
Yeah.

758
00:32:29,380 --> 00:32:33,818
AUDIENCE: How close are
the actual [? wavelength ?]

759
00:32:33,818 --> 00:32:37,071
potentials or the combinations
of the original wavelength?

760
00:32:37,071 --> 00:32:38,320
PROFESSOR: Excellent question.

761
00:32:38,320 --> 00:32:41,570
So you can answer that
immediately from this.

762
00:32:41,570 --> 00:32:44,540
So the question is this,
look, if we had arbitrarily

763
00:32:44,540 --> 00:32:46,560
separated wells,
if we had something

764
00:32:46,560 --> 00:32:56,060
that looked like-- so they're
very, very far separated.

765
00:32:56,060 --> 00:32:58,310
Then the ground state would
be effectively degenerate,

766
00:32:58,310 --> 00:33:02,396
because the wave function
or the ground state

767
00:33:02,396 --> 00:33:04,270
would be the completely
symmetric combination

768
00:33:04,270 --> 00:33:05,240
of these guys--
you guys actually

769
00:33:05,240 --> 00:33:07,510
studied this in our problem
set-- then there's also

770
00:33:07,510 --> 00:33:12,172
a combination where you
have this, this, this.

771
00:33:12,172 --> 00:33:15,120
You could also take
this constant this.

772
00:33:15,120 --> 00:33:17,580
There are many
things you could do.

773
00:33:17,580 --> 00:33:21,119
I'm sorry, this,
many combinations.

774
00:33:21,119 --> 00:33:22,660
The point is since
each wave function

775
00:33:22,660 --> 00:33:24,990
for each well effectively
drops off to zero inside,

776
00:33:24,990 --> 00:33:26,710
we can just linear
combinations of these,

777
00:33:26,710 --> 00:33:31,860
and there's no penalty for using
the true ground seats in here

778
00:33:31,860 --> 00:33:35,120
because the potential is so
high, that the true solution

779
00:33:35,120 --> 00:33:37,620
has an exponential tail,
but that exponential tail is

780
00:33:37,620 --> 00:33:39,710
ridiculously small if
the barrier is big.

781
00:33:39,710 --> 00:33:41,790
So in the limit that the
barriers are gigantic,

782
00:33:41,790 --> 00:33:46,360
the true energy eigenstates
are just arbitrary

783
00:33:46,360 --> 00:33:48,500
linear combinations
of these guys,

784
00:33:48,500 --> 00:33:51,970
of the individual eigenstates
of each individual well.

785
00:33:51,970 --> 00:33:53,020
Agreed?

786
00:33:53,020 --> 00:33:57,042
However, when the barriers
a large but finite,

787
00:33:57,042 --> 00:33:59,500
then the true energetic states
have little tiny exponential

788
00:33:59,500 --> 00:34:02,700
tails, and the curvature of that
little tiny exponential tail

789
00:34:02,700 --> 00:34:05,890
will determine exactly
what the energy is.

790
00:34:05,890 --> 00:34:09,285
So a state that curves a
little bit versus a state that

791
00:34:09,285 --> 00:34:14,230
curves more will have
slightly different energies.

792
00:34:14,230 --> 00:34:16,860
So when the barriers
are very, very large,

793
00:34:16,860 --> 00:34:21,020
the linear combinations of the
individual well eigenstates

794
00:34:21,020 --> 00:34:23,134
should be very good
approximations, but not exact.

795
00:34:23,134 --> 00:34:25,989
And as the barriers become
less and less effective,

796
00:34:25,989 --> 00:34:28,290
they should become
less and less exact,

797
00:34:28,290 --> 00:34:30,040
and we can see that right here.

798
00:34:30,040 --> 00:34:34,290
So let's take the ground state.

799
00:34:34,290 --> 00:34:36,610
The bottom of the band is
the completely symmetric

800
00:34:36,610 --> 00:34:39,210
combination of the
ground seat of each well.

801
00:34:39,210 --> 00:34:40,450
Everyone see that?

802
00:34:40,450 --> 00:34:42,389
It's just well,
well, well, well.

803
00:34:42,389 --> 00:34:44,055
It's a completely
symmetric combination.

804
00:34:44,055 --> 00:34:47,350
At the top of the band
is the completely anti

805
00:34:47,350 --> 00:34:50,217
symmetric combination, and
let's put them for comparison.

806
00:34:50,217 --> 00:34:51,800
The top of the band
is the orange one,

807
00:34:51,800 --> 00:34:53,800
and the bottom of the
band is the yellow one.

808
00:34:53,800 --> 00:34:56,380
So the bottom of the band
is the completely symmetric,

809
00:34:56,380 --> 00:35:00,070
and the top of the band
is alternating combination

810
00:35:00,070 --> 00:35:02,560
of the ground
state in each well.

811
00:35:02,560 --> 00:35:05,340
And all the other states
inside here-- well,

812
00:35:05,340 --> 00:35:09,110
which are extremely
difficult to-- Let's see.

813
00:35:09,110 --> 00:35:09,670
There we go.

814
00:35:09,670 --> 00:35:12,105
These states are also linear
combinations of the ground

815
00:35:12,105 --> 00:35:14,640
state in each well with
different coefficients

816
00:35:14,640 --> 00:35:15,500
in front of them.

817
00:35:15,500 --> 00:35:17,041
There are the
different coefficients,

818
00:35:17,041 --> 00:35:19,260
and they are almost degenerate.

819
00:35:19,260 --> 00:35:21,049
But if we go to
higher energy states

820
00:35:21,049 --> 00:35:23,340
where the barrier is less
effective-- because they have

821
00:35:23,340 --> 00:35:25,215
high energy and the
ratio between the barrier

822
00:35:25,215 --> 00:35:27,770
height and their
energy is small--

823
00:35:27,770 --> 00:35:31,660
then you see that these
states are not particularly

824
00:35:31,660 --> 00:35:33,720
well approximated by
linear combinations.

825
00:35:33,720 --> 00:35:35,970
And the energies correspondingly
are not degenerating.

826
00:35:35,970 --> 00:35:38,270
AUDIENCE: So you said
linear combinations

827
00:35:38,270 --> 00:35:41,089
of the [INAUDIBLE].

828
00:35:41,089 --> 00:35:43,130
PROFESSOR: That gives you
a better approximation,

829
00:35:43,130 --> 00:35:44,600
but for the same reason.

830
00:35:44,600 --> 00:35:46,925
It's a good approximation,
but it's not exact.

831
00:35:46,925 --> 00:35:49,300
But it becomes excellent as
the barriers become infinite,

832
00:35:49,300 --> 00:35:53,950
even when they also go over
to the infinites very well.

833
00:35:53,950 --> 00:35:54,742
Very good question.

834
00:35:54,742 --> 00:35:55,241
Yeah.

835
00:35:55,241 --> 00:35:56,168
AUDIENCE: [INAUDIBLE].

836
00:36:05,371 --> 00:36:06,620
PROFESSOR: Very good question.

837
00:36:06,620 --> 00:36:08,080
We'll come to that
in little bit.

838
00:36:08,080 --> 00:36:09,940
We'll come to that shortly.

839
00:36:09,940 --> 00:36:12,555
So I'm done at the moment
with the PhET simulations.

840
00:36:14,970 --> 00:36:17,470
I'm motivated by all this, and
effectively by that question,

841
00:36:17,470 --> 00:36:19,011
I'm going to ask
the following thing.

842
00:36:19,011 --> 00:36:26,140
Look, real materials like metal
in my laptop, the real material

843
00:36:26,140 --> 00:36:28,070
is actually a
periodic potential.

844
00:36:28,070 --> 00:36:31,720
It's built out of a
crystal of metal atoms

845
00:36:31,720 --> 00:36:35,790
that are bound
together, and each atom

846
00:36:35,790 --> 00:36:37,680
is some
positively-charged beast,

847
00:36:37,680 --> 00:36:39,700
to which some
positively-charged nucleus,

848
00:36:39,700 --> 00:36:42,010
to which is bound an electron.

849
00:36:42,010 --> 00:36:44,100
And if I want to understand
properties of solids,

850
00:36:44,100 --> 00:36:46,380
like the fact that metals
conduct electricity,

851
00:36:46,380 --> 00:36:48,569
a basic fact I'd
like to explain,

852
00:36:48,569 --> 00:36:50,610
if I want to explain the
fact that metals conduct

853
00:36:50,610 --> 00:36:54,390
electricity, but
plastic doesn't, which

854
00:36:54,390 --> 00:36:58,780
we will be able to explain, and
diamond doesn't, which is cool,

855
00:36:58,780 --> 00:37:00,280
if we want to explain
that property,

856
00:37:00,280 --> 00:37:02,890
we probably ought to study
the physics of electrons

857
00:37:02,890 --> 00:37:04,937
in periodic systems where
the potential is, atom,

858
00:37:04,937 --> 00:37:06,270
I don't want to be stuck to you.

859
00:37:06,270 --> 00:37:07,310
Atom, I don't want
to be stuck to you.

860
00:37:07,310 --> 00:37:08,851
Atom I don't want
to be stuck to you.

861
00:37:08,851 --> 00:37:11,849
So it's a periodic
well a potential.

862
00:37:11,849 --> 00:37:13,640
So in order to study
the physics of solids,

863
00:37:13,640 --> 00:37:16,040
I need to understand first
the physics of electrons

864
00:37:16,040 --> 00:37:17,980
in periodic potentials.

865
00:37:17,980 --> 00:37:20,730
And these PhET simulations
were a first start at that.

866
00:37:20,730 --> 00:37:21,834
We did it n's wells.

867
00:37:21,834 --> 00:37:24,000
I want to now think about
what happens if I take not

868
00:37:24,000 --> 00:37:25,580
n, but an infinite
number of wells.

869
00:37:25,580 --> 00:37:29,160
What if I really strictly
periodic lattice on the line?

870
00:37:29,160 --> 00:37:30,930
What do we expect to happen?

871
00:37:30,930 --> 00:37:33,627
Well, looking at the results
of these PhET simulations,

872
00:37:33,627 --> 00:37:35,960
as you did this for different
numbers, n, what you found

873
00:37:35,960 --> 00:37:37,460
was the same sort
of band structure.

874
00:37:37,460 --> 00:37:39,292
You just find for
more and more wells,

875
00:37:39,292 --> 00:37:41,250
you find more and more
states inside each band.

876
00:37:41,250 --> 00:37:43,142
In fact, you find n
states within each band.

877
00:37:43,142 --> 00:37:44,600
Both the top and
bottom of the band

878
00:37:44,600 --> 00:37:48,577
quickly asymptote
to fixed values.

879
00:37:48,577 --> 00:37:50,660
As we take at large, what
do you expect to happen?

880
00:37:56,055 --> 00:37:57,930
Well, this was a problem
on your problem set.

881
00:37:57,930 --> 00:37:58,804
What did you predict?

882
00:37:58,804 --> 00:38:00,720
What should happen
when you take n large?

883
00:38:00,720 --> 00:38:02,905
AUDIENCE: [? It ?]
depends on the experiment.

884
00:38:02,905 --> 00:38:03,020
PROFESSOR: Yeah.

885
00:38:03,020 --> 00:38:04,940
How many states should
there be in each bend?

886
00:38:04,940 --> 00:38:05,760
AUDIENCE: N.

887
00:38:05,760 --> 00:38:07,230
PROFESSOR: N. So there
are arbitrarily many.

888
00:38:07,230 --> 00:38:07,620
Exactly.

889
00:38:07,620 --> 00:38:09,953
So there should be arbitrarily
many states in each band,

890
00:38:09,953 --> 00:38:12,810
but they all have to fit within
this energy band, the width,

891
00:38:12,810 --> 00:38:14,570
which is asymptoting
to a constant.

892
00:38:14,570 --> 00:38:17,050
So there has to be a
continuum of states,

893
00:38:17,050 --> 00:38:19,050
and they're not going to
be free particle states

894
00:38:19,050 --> 00:38:21,050
because the system
has a potential.

895
00:38:21,050 --> 00:38:22,300
What is the shape going to be?

896
00:38:22,300 --> 00:38:23,900
Well, a reasonable
guess is that the shape

897
00:38:23,900 --> 00:38:25,960
is going to be something
that fills out this curve.

898
00:38:25,960 --> 00:38:27,585
It's going to be a
little bit different

899
00:38:27,585 --> 00:38:30,420
from the free particle state.

900
00:38:30,420 --> 00:38:33,070
So let's find out.

901
00:38:33,070 --> 00:38:35,500
Let's just solve the
problem of an electron

902
00:38:35,500 --> 00:38:37,680
in a periodic
potential, and look,

903
00:38:37,680 --> 00:38:40,010
the periodic
potential is extremely

904
00:38:40,010 --> 00:38:41,790
similar to things
we've already solved.

905
00:38:41,790 --> 00:38:44,220
So I want to walk you through
just a basic argument.

906
00:38:48,300 --> 00:38:49,280
So periodic potentials.

907
00:38:54,420 --> 00:38:56,260
Instead of n wells,
consider a system

908
00:38:56,260 --> 00:39:01,560
of an infinite number
of wells, each one

909
00:39:01,560 --> 00:39:10,470
identical and just
for symmetry purposes.

910
00:39:10,470 --> 00:39:19,819
And let the width here be L, so
the period of the lattice is L.

911
00:39:19,819 --> 00:39:21,235
And what I mean
by that is saying,

912
00:39:21,235 --> 00:39:23,740
so this is V of x
seeing that it's

913
00:39:23,740 --> 00:39:29,080
periodic is the statement
as that V of x plus L

914
00:39:29,080 --> 00:39:30,070
is equal to V of x.

915
00:39:32,960 --> 00:39:37,704
Potential doesn't change if I
shift it by a lattice vector,

916
00:39:37,704 --> 00:39:39,120
Now there's a nice
way to say this

917
00:39:39,120 --> 00:39:42,560
which is that, if I take
the potential, v of x, and I

918
00:39:42,560 --> 00:39:45,990
translate it by L, I get the
same potential back V of x.

919
00:39:54,460 --> 00:39:56,770
And more to the point,
if I take the translation

920
00:39:56,770 --> 00:40:00,030
if I think of these as
operators expressions,

921
00:40:00,030 --> 00:40:03,040
if I take TL and
act on Vx and then

922
00:40:03,040 --> 00:40:11,450
the potential of Vx f of x,
then this is equal to V of x TL

923
00:40:11,450 --> 00:40:13,140
f of x.

924
00:40:13,140 --> 00:40:15,600
So if I think of these as
operators instead of just

925
00:40:15,600 --> 00:40:18,210
little functions
operator, operator,

926
00:40:18,210 --> 00:40:19,986
then if I take the
translator operator

927
00:40:19,986 --> 00:40:22,840
and I translate V of x f of x,
I'm going to get V of x plus L

928
00:40:22,840 --> 00:40:26,260
f of x plus L. And if I
do the right-hand side

929
00:40:26,260 --> 00:40:28,930
I get translator of L that's
f of x plus L but just times V

930
00:40:28,930 --> 00:40:29,970
of x.

931
00:40:29,970 --> 00:40:35,110
But if V of x is equal to V of x
plus L-- this is V of x plus L.

932
00:40:35,110 --> 00:40:38,350
So if I translate by L says take
this thing and translate it,

933
00:40:38,350 --> 00:40:42,140
plus L f of x plus L. And
on the right-hand side,

934
00:40:42,140 --> 00:40:45,623
we have V of x, not V of
x plus L translate by L f

935
00:40:45,623 --> 00:40:52,750
of x, f of x plus L And these
are equal because V of x plus L

936
00:40:52,750 --> 00:40:54,910
is equal to V of x.

937
00:40:54,910 --> 00:40:55,670
Yeah.

938
00:40:55,670 --> 00:41:00,250
So what that tells
you is as operators

939
00:41:00,250 --> 00:41:02,180
translate by L and
V of x commute.

940
00:41:08,060 --> 00:41:08,700
Equals 0.

941
00:41:12,644 --> 00:41:15,040
Everyone happy with that?

942
00:41:15,040 --> 00:41:18,610
So you should just be able
to immediately see this,

943
00:41:18,610 --> 00:41:22,304
the potential is periodic with
period L if by translated by L,

944
00:41:22,304 --> 00:41:23,220
nothing should happen.

945
00:41:23,220 --> 00:41:29,530
So the potential respects
T of L it commutes with it.

946
00:41:29,530 --> 00:41:31,510
So this tells you a neat thing.

947
00:41:31,510 --> 00:41:40,730
Remember that translate by L
is equal to E to the i PL upon

948
00:41:40,730 --> 00:41:46,440
H bar, so in particular,
TL commutes with P

949
00:41:46,440 --> 00:41:49,870
because this is just a
polynomial in P dL P,

950
00:41:49,870 --> 00:41:54,616
and in particular with
P squared is equal to 0,

951
00:41:54,616 --> 00:41:57,550
and that just follows
from the definition.

952
00:41:57,550 --> 00:42:01,290
So what that tells
us is that if we

953
00:42:01,290 --> 00:42:06,040
take the system with
our periodic potential,

954
00:42:06,040 --> 00:42:13,920
periodic V of x plus L
is equal to V of x, then

955
00:42:13,920 --> 00:42:17,540
the energy, which is P squared
upon 2 m of V of x commutes

956
00:42:17,540 --> 00:42:18,040
with TL.

957
00:42:27,950 --> 00:42:31,500
So in this system, with
a periodic potential,

958
00:42:31,500 --> 00:42:32,640
is momentum conserved?

959
00:42:37,292 --> 00:42:39,250
What must be true for
momentum to be conserved?

960
00:42:42,897 --> 00:42:45,230
So momentum conservation come
from a symmetry principle.

961
00:42:45,230 --> 00:42:46,604
AUDIENCE: Translation
invariance.

962
00:42:46,604 --> 00:42:48,112
PROFESSOR: Translation
invariance.

963
00:42:48,112 --> 00:42:49,570
In order for momentum
be conserved,

964
00:42:49,570 --> 00:42:53,170
the system must be
translationally invariant.

965
00:42:53,170 --> 00:42:56,640
Is the system
translationally invariant?

966
00:42:56,640 --> 00:42:57,140
No.

967
00:42:57,140 --> 00:42:59,660
If I shift it by a little
bit, it's not invariant.

968
00:42:59,660 --> 00:43:02,350
It is, however, invariant
under a certain subset

969
00:43:02,350 --> 00:43:08,240
of translations, which
is finite shifts by L.

970
00:43:08,240 --> 00:43:08,740
Yeah.

971
00:43:08,740 --> 00:43:10,240
So it's invariant
under shifts by L,

972
00:43:10,240 --> 00:43:12,140
which is a subset
of translations,

973
00:43:12,140 --> 00:43:15,720
and the legacy of that is
the fact that the energy

974
00:43:15,720 --> 00:43:18,830
commutes with translations by
L. The energy does not commute

975
00:43:18,830 --> 00:43:25,922
with P, Because P acting on V
of x, it takes the derivative

976
00:43:25,922 --> 00:43:27,130
and gives you the prime of x.

977
00:43:27,130 --> 00:43:30,070
It is not the same thing.

978
00:43:30,070 --> 00:43:32,640
But we have a subset
of translations

979
00:43:32,640 --> 00:43:35,880
under which the system is
invariant, and that's good.

980
00:43:35,880 --> 00:43:38,729
That's less powerful than
being a free particle,

981
00:43:38,729 --> 00:43:40,770
but it's more powerful
than not knowing anything.

982
00:43:40,770 --> 00:43:42,230
In particular, it
tells us that we

983
00:43:42,230 --> 00:43:45,450
can find energy eigenfunctions,
which are simultaneously

984
00:43:45,450 --> 00:43:48,970
eigenfunctions of the
energy, and, I'll call it

985
00:43:48,970 --> 00:43:52,790
Q for a moment, they have
an eigenvalue under TL.

986
00:43:52,790 --> 00:43:55,040
So we can find eigenfunctions
which are simultaneously

987
00:43:55,040 --> 00:43:57,120
eigenfunctions the
energy operator you might

988
00:43:57,120 --> 00:44:02,590
E of phi E is equal
to E phi Eq and which

989
00:44:02,590 --> 00:44:09,000
are eigenfunctions under
translate by L on phi Eq

990
00:44:09,000 --> 00:44:11,877
is equal to something
times phi Eq.

991
00:44:15,620 --> 00:44:18,290
So we're going to find
simultaneous eigenfunctions.

992
00:44:18,290 --> 00:44:19,290
Everyone cool with that?

993
00:44:19,290 --> 00:44:19,790
Yeah.

994
00:44:19,790 --> 00:44:24,460
AUDIENCE: Do you know if the
momentum operator is still

995
00:44:24,460 --> 00:44:25,722
the momentum operator?

996
00:44:25,722 --> 00:44:26,430
PROFESSOR: Sorry.

997
00:44:26,430 --> 00:44:27,030
Say it again.

998
00:44:27,030 --> 00:44:28,990
AUDIENCE: The momentum operator
is still in remission, right?

999
00:44:28,990 --> 00:44:29,090
PROFESSOR: Yeah.

1000
00:44:29,090 --> 00:44:31,298
The momentum operator is
still the momentum operator.

1001
00:44:31,298 --> 00:44:32,000
[INAUDIBLE].

1002
00:44:32,000 --> 00:44:33,760
AUDIENCE: Do you
know if [INAUDIBLE]?

1003
00:44:33,760 --> 00:44:34,310
PROFESSOR: Excellent.

1004
00:44:34,310 --> 00:44:34,640
Excellent.

1005
00:44:34,640 --> 00:44:35,020
Good Yes.

1006
00:44:35,020 --> 00:44:35,520
OK.

1007
00:44:35,520 --> 00:44:40,520
So if T of L is unitary,
note, T of L is unitary.

1008
00:44:42,542 --> 00:44:44,750
And you actually showed this
on a problem set before,

1009
00:44:44,750 --> 00:44:46,620
and let me remind you a
couple of facts about it.

1010
00:44:46,620 --> 00:44:48,119
The first is that
T of L is unitary,

1011
00:44:48,119 --> 00:44:54,920
and that says that TL dagger is
equal to the identity, right?

1012
00:44:54,920 --> 00:44:56,150
But what is TL dagger?

1013
00:44:56,150 --> 00:44:58,330
AUDIENCE: [INAUDIBLE].

1014
00:44:58,330 --> 00:44:59,210
PROFESSOR: Yeah.

1015
00:44:59,210 --> 00:45:01,180
TL dagger, well, P is
remission, so we just

1016
00:45:01,180 --> 00:45:04,090
pick up a minus
sign, so TL dagger

1017
00:45:04,090 --> 00:45:09,429
is equal to e to the
minus i PL over H bar,

1018
00:45:09,429 --> 00:45:10,970
and we can do think
of this two ways.

1019
00:45:10,970 --> 00:45:12,670
First thing you can
notice that it's clearly

1020
00:45:12,670 --> 00:45:15,086
the inverse of that guy, by
definition of the exponential,

1021
00:45:15,086 --> 00:45:18,250
but the other is, TL is the
thing that translates you by L,

1022
00:45:18,250 --> 00:45:20,311
and this operator,
e to the minus i PL,

1023
00:45:20,311 --> 00:45:22,810
well, I could put my H wit h
the and that's just translation

1024
00:45:22,810 --> 00:45:23,850
by minus L.

1025
00:45:23,850 --> 00:45:26,595
So this is translated by L and
then translated by minus L.

1026
00:45:26,595 --> 00:45:28,095
And of course, if
you translate by L

1027
00:45:28,095 --> 00:45:30,140
and then you
untranslate by L, you

1028
00:45:30,140 --> 00:45:32,759
haven't done anything
with the identity.

1029
00:45:32,759 --> 00:45:33,800
What are the eigenvalues?

1030
00:45:33,800 --> 00:45:35,120
What are the form
of the eigenvalues

1031
00:45:35,120 --> 00:45:36,115
of a unitary operator?

1032
00:45:36,115 --> 00:45:37,031
AUDIENCE: [INAUDIBLE].

1033
00:45:40,592 --> 00:45:41,800
PROFESSOR: Well, that's true.

1034
00:45:41,800 --> 00:45:43,258
The eigenfunctions
are orthonormal,

1035
00:45:43,258 --> 00:45:45,190
but the eigenvalues,
are they real numbers?

1036
00:45:45,190 --> 00:45:47,440
AUDIENCE: [INAUDIBLE].

1037
00:45:47,440 --> 00:45:48,330
PROFESSOR: Yeah.

1038
00:45:48,330 --> 00:45:52,665
So the eigenvalues of a unitary
operator are of the form,

1039
00:45:52,665 --> 00:45:54,970
well, imagine we're in
a moment eigenstate,

1040
00:45:54,970 --> 00:45:57,920
for a momentum eigenstate,
then P is a real number.

1041
00:45:57,920 --> 00:46:01,395
And then the eigenvalue of TL
is e to the i, the P over H bar,

1042
00:46:01,395 --> 00:46:02,270
that's a real number.

1043
00:46:02,270 --> 00:46:04,140
So is e to the i a real number?

1044
00:46:04,140 --> 00:46:05,390
That's a pure phase.

1045
00:46:05,390 --> 00:46:07,630
It's e to i a real number.

1046
00:46:07,630 --> 00:46:10,210
So the eigenvalues
are pure phases,

1047
00:46:10,210 --> 00:46:11,710
and this let's just
do a nice thing.

1048
00:46:11,710 --> 00:46:15,270
We can note that here the
translate by L of the phi,

1049
00:46:15,270 --> 00:46:17,350
if these are common
eigenfunctions of E

1050
00:46:17,350 --> 00:46:21,380
and of TL, well, TL is
unitary, its eigenvalue

1051
00:46:21,380 --> 00:46:24,620
must be a pure phase, e
to the i a real number.

1052
00:46:24,620 --> 00:46:30,550
So let's call that real number
alpha, e to the i alpha.

1053
00:46:30,550 --> 00:46:37,504
So maybe just label this
by alpha for the moment,

1054
00:46:37,504 --> 00:46:39,670
and you'll see why I want
to change this to q later,

1055
00:46:39,670 --> 00:46:43,634
but for the moment let's
just call it alpha.

1056
00:46:43,634 --> 00:46:45,302
Everyone cool with that?

1057
00:46:45,302 --> 00:46:47,510
So you actually show the
following thing on a problem

1058
00:46:47,510 --> 00:46:52,590
set, but I'm going to
re-drive it for you now.

1059
00:46:52,590 --> 00:46:55,830
We can say something
more about the form

1060
00:46:55,830 --> 00:46:58,280
of an eigenfunction of
the translation operator.

1061
00:46:58,280 --> 00:47:00,310
This is quite nice.

1062
00:47:00,310 --> 00:47:04,190
So suppose we have
a function phi

1063
00:47:04,190 --> 00:47:08,596
sub alpha such that TL and i is
equal to e to the i alpha phi

1064
00:47:08,596 --> 00:47:10,580
alpha.

1065
00:47:10,580 --> 00:47:12,210
I want to know what is the form.

1066
00:47:12,210 --> 00:47:15,240
What can I say about
the function phi f of x?

1067
00:47:15,240 --> 00:47:17,199
What can I say about the
form of this function?

1068
00:47:17,199 --> 00:47:19,406
We can actually say something
really useful for this.

1069
00:47:19,406 --> 00:47:21,532
This is going to turn out
to be necessary for us.

1070
00:47:21,532 --> 00:47:22,990
The first thing
I'm going to do is,

1071
00:47:22,990 --> 00:47:26,640
I'm going to just make
the following observation.

1072
00:47:26,640 --> 00:47:29,230
Define the function
u of x, which

1073
00:47:29,230 --> 00:47:35,820
is equal to e to the
minus i qx phi sub alpha.

1074
00:47:35,820 --> 00:47:37,980
So if phi sub alpha is
an eigenfunction of TL

1075
00:47:37,980 --> 00:47:40,612
with an eigenvalue
e to the alpha,

1076
00:47:40,612 --> 00:47:42,070
and I'm just defining
this function

1077
00:47:42,070 --> 00:47:45,200
u to be e to minus i
qx times phi sub alpha.

1078
00:47:45,200 --> 00:47:45,882
Cool?

1079
00:47:45,882 --> 00:47:46,840
It's just a definition.

1080
00:47:46,840 --> 00:47:48,820
I'm going to construct
some stupid u.

1081
00:47:48,820 --> 00:47:50,300
Note the following thing.

1082
00:47:50,300 --> 00:47:53,410
T sub L on u, if we
translate u of x, this

1083
00:47:53,410 --> 00:47:57,010
is u of x plus L.
Translating this,

1084
00:47:57,010 --> 00:48:00,800
I get e to the i qx goes to
e to the minus i qx plus L,

1085
00:48:00,800 --> 00:48:04,070
which is just the e to the
minus i qx times e to the minus

1086
00:48:04,070 --> 00:48:13,620
i qL e to the minus i qx
plus L phi alpha of x plus L.

1087
00:48:13,620 --> 00:48:19,460
But this is equal to from
here e to the minus i qL

1088
00:48:19,460 --> 00:48:22,140
e to the minus i qx.

1089
00:48:22,140 --> 00:48:25,360
And from phi of x plus a, I know
that if I translate by a to get

1090
00:48:25,360 --> 00:48:30,280
phi of x plus L, I just pick
up a phase, e to the phi alpha,

1091
00:48:30,280 --> 00:48:36,270
so e to the phi
alpha phi alpha of x.

1092
00:48:36,270 --> 00:48:42,190
But this is equal to e
to the i alpha minus qL,

1093
00:48:42,190 --> 00:48:43,590
putting these two
terms together,

1094
00:48:43,590 --> 00:48:46,540
and this is nothing but u of x.

1095
00:48:49,460 --> 00:48:50,877
Yeah.

1096
00:48:50,877 --> 00:48:52,210
So another following cool thing.

1097
00:48:52,210 --> 00:48:56,030
So here I define some function
u with some stupid value

1098
00:48:56,030 --> 00:48:57,760
q, which I just pulled
out of thin air,

1099
00:48:57,760 --> 00:48:59,310
and I'm just defining
this function.

1100
00:48:59,310 --> 00:49:02,040
What I've observed is that if
I translate this function by L,

1101
00:49:02,040 --> 00:49:05,460
it picks up an overall phase
times its original value

1102
00:49:05,460 --> 00:49:07,537
where the phase depends
on alpha and on q.

1103
00:49:07,537 --> 00:49:09,370
So then I'm going to
do the following thing.

1104
00:49:09,370 --> 00:49:12,230
This becomes really simple if
I pick a particular value of q,

1105
00:49:12,230 --> 00:49:20,140
for a special value of q, q is
equal to alpha divided by L,

1106
00:49:20,140 --> 00:49:25,890
then this is equal to e
to the i alpha minus qL.

1107
00:49:25,890 --> 00:49:28,896
That's going to be equal to e
to the i 0, which is just 1,

1108
00:49:28,896 --> 00:49:30,395
so I have the phase,
so this u of x.

1109
00:49:35,240 --> 00:49:37,310
So that means if I
translate by L on u

1110
00:49:37,310 --> 00:49:41,060
and I pick q is equal to
alpha over L, I get back u,

1111
00:49:41,060 --> 00:49:41,920
so u is periodic.

1112
00:49:50,610 --> 00:49:51,360
Everyone see that?

1113
00:49:51,360 --> 00:49:53,640
Question?

1114
00:49:53,640 --> 00:50:01,790
So as a result, I can always
write my wave function phi sub

1115
00:50:01,790 --> 00:50:05,190
e and we'll now call
this q or q is related

1116
00:50:05,190 --> 00:50:08,507
to alpha as q is
equal to alpha over L,

1117
00:50:08,507 --> 00:50:11,090
every energy eigenstate can find
a basis of energy eigenstates

1118
00:50:11,090 --> 00:50:13,270
with a definite eigenvalue
under the energy,

1119
00:50:13,270 --> 00:50:15,190
a definite eigenvalue
under T sub L,

1120
00:50:15,190 --> 00:50:17,720
and I can write
them in the form,

1121
00:50:17,720 --> 00:50:20,920
since u is equal to e
to the minus i qx phi,

1122
00:50:20,920 --> 00:50:25,010
then phi is equal
to e to the i qx u e

1123
00:50:25,010 --> 00:50:31,990
to the i qx u of
x where q, where

1124
00:50:31,990 --> 00:50:35,180
the eigenvalue, e
to the i phi alpha,

1125
00:50:35,180 --> 00:50:40,235
is equal to e to the i qL
and u of x is periodic.

1126
00:50:50,160 --> 00:50:51,760
Everyone cool with that?

1127
00:50:51,760 --> 00:50:54,500
It's not totally obvious how
much this is helping us here,

1128
00:50:54,500 --> 00:50:56,311
but what we've done
is, we've extracted,

1129
00:50:56,311 --> 00:50:58,060
we've observe that
there is some lingering

1130
00:50:58,060 --> 00:51:00,460
symmetry in the
system, and I've used

1131
00:51:00,460 --> 00:51:03,970
that symmetry to deduce
the form of the energy

1132
00:51:03,970 --> 00:51:05,785
eigenfunctions as
best as possible.

1133
00:51:05,785 --> 00:51:07,660
So I haven't completely
determined the energy

1134
00:51:07,660 --> 00:51:09,270
eigenfunctions,
I've just determined

1135
00:51:09,270 --> 00:51:12,240
that the energy eigenfunctions
are of the form, a phase,

1136
00:51:12,240 --> 00:51:15,220
e to the i qx, so
it varies as we

1137
00:51:15,220 --> 00:51:18,510
change x, times a
periodic function.

1138
00:51:18,510 --> 00:51:20,640
So the wave function
is periodic up

1139
00:51:20,640 --> 00:51:23,850
to an overall phase,
not a constant phase,

1140
00:51:23,850 --> 00:51:25,197
a position-dependent phase.

1141
00:51:27,949 --> 00:51:29,240
Does everybody agree with that?

1142
00:51:29,240 --> 00:51:30,115
Questions about that?

1143
00:51:35,220 --> 00:51:45,120
So a couple things
to note about this q,

1144
00:51:45,120 --> 00:51:51,560
the first is, the eigenvalue
under TL e to the i alpha, this

1145
00:51:51,560 --> 00:51:57,170
is the eigenvalue under L
also equal to e to the i qL,

1146
00:51:57,170 --> 00:52:06,640
but if I take q to
q plus 2 pi over L,

1147
00:52:06,640 --> 00:52:12,600
then nothing changes
to the eigenvalue,

1148
00:52:12,600 --> 00:52:15,580
because the 2 pi over
L times L is just 2 pi,

1149
00:52:15,580 --> 00:52:17,830
e to the 2 pi is 1, so we
don't change the eigenvalue.

1150
00:52:17,830 --> 00:52:21,270
So different ways
of q only correspond

1151
00:52:21,270 --> 00:52:25,100
to different
eigenvalues of translate

1152
00:52:25,100 --> 00:52:28,740
by L of a translation
by one lattice vector,

1153
00:52:28,740 --> 00:52:29,920
one lattice basing.

1154
00:52:29,920 --> 00:52:34,370
They only correspond to the
different eigenvalues of TL

1155
00:52:34,370 --> 00:52:35,982
if they don't differ
by 2 pi over L.

1156
00:52:35,982 --> 00:52:37,940
If different values of
q differ by 2 pi over L,

1157
00:52:37,940 --> 00:52:39,450
then they really
mean the same thing

1158
00:52:39,450 --> 00:52:44,580
because we're just talking
about translate by one period.

1159
00:52:44,580 --> 00:52:50,120
So q is equivalent to
q plus 2 pi over L.

1160
00:52:50,120 --> 00:52:52,228
So that's just the
first thing to know.

1161
00:52:55,980 --> 00:52:57,180
So what have we done so far?

1162
00:52:57,180 --> 00:53:00,450
What we've done so far
is nothing whatsoever

1163
00:53:00,450 --> 00:53:05,090
except extract, take advantage
of the translational symmetry

1164
00:53:05,090 --> 00:53:06,710
that's left, the
remaining lingering

1165
00:53:06,710 --> 00:53:08,370
little bit of
translational symmetry

1166
00:53:08,370 --> 00:53:10,460
to constrain the for of
the energetic functions.

1167
00:53:10,460 --> 00:53:13,220
What I want to do now is
observe some consequences

1168
00:53:13,220 --> 00:53:14,690
that follows
immediately from this.

1169
00:53:14,690 --> 00:53:16,106
So there are some
very nice things

1170
00:53:16,106 --> 00:53:17,560
to follow just from this.

1171
00:53:17,560 --> 00:53:20,920
We can learn something
great about the system

1172
00:53:20,920 --> 00:53:23,774
without knowing anything
else, without knowing anything

1173
00:53:23,774 --> 00:53:25,690
about the detailed
structure of the potential.

1174
00:53:25,690 --> 00:53:27,150
At this point, I've
made no assumptions

1175
00:53:27,150 --> 00:53:29,130
about the potential other than
the fact that it's periodic.

1176
00:53:29,130 --> 00:53:30,550
So the thing I'm
about to tell you

1177
00:53:30,550 --> 00:53:32,765
were to be true for
any periodic potential.

1178
00:53:32,765 --> 00:53:35,870
They follow only
from the structure.

1179
00:53:35,870 --> 00:53:38,980
So let's see what they are.

1180
00:53:38,980 --> 00:53:43,770
So the first one is
that the wave function

1181
00:53:43,770 --> 00:53:46,640
sine of x itself with
a definite value of e

1182
00:53:46,640 --> 00:53:55,660
and a definite value of
q is not periodic by L.

1183
00:53:55,660 --> 00:53:58,640
Because under a shift
by L, u is periodic,

1184
00:53:58,640 --> 00:54:01,435
but this part picks up
a phase e to the i qL.

1185
00:54:01,435 --> 00:54:04,060
That's what it is to say that an
eigenfunction, the translation

1186
00:54:04,060 --> 00:54:06,450
operator.

1187
00:54:06,450 --> 00:54:10,700
It's not periodic by L
unless q is equal to 0.

1188
00:54:13,620 --> 00:54:16,110
So for q equals 0,
there is precisely

1189
00:54:16,110 --> 00:54:19,010
one wave function
which is periodic by L.

1190
00:54:19,010 --> 00:54:23,500
Because if q is equal to
0, then this phase is 0.

1191
00:54:23,500 --> 00:54:28,130
So only if is equal to 0 is
the wave function periodic.

1192
00:54:28,130 --> 00:54:30,810
Cool?

1193
00:54:30,810 --> 00:54:34,850
So this should look familiar
because back in the band

1194
00:54:34,850 --> 00:54:39,581
structure, these guys
are not periodic.

1195
00:54:39,581 --> 00:54:42,080
So these states in the middle
of the band, they're horrible.

1196
00:54:42,080 --> 00:54:42,970
They're not periodic.

1197
00:54:42,970 --> 00:54:47,720
They're some horrible things,
but that top guy in the band

1198
00:54:47,720 --> 00:54:49,660
is periodic, and it
turns out that if we

1199
00:54:49,660 --> 00:54:51,940
had made the system
infinitely large,

1200
00:54:51,940 --> 00:54:53,490
it would become
exactly periodic.

1201
00:54:53,490 --> 00:54:55,677
The fact that it has an
envelope on it like this

1202
00:54:55,677 --> 00:54:58,010
is just the fact that we have
an finite number of wells,

1203
00:54:58,010 --> 00:54:59,969
we're in a box.

1204
00:54:59,969 --> 00:55:02,260
If we got rid of that box
and we made many, many wells,

1205
00:55:02,260 --> 00:55:03,981
we would find it
perfectly periodic.

1206
00:55:03,981 --> 00:55:04,480
Yeah.

1207
00:55:04,480 --> 00:55:07,056
AUDIENCE: Could q not be
something like 2 pi over L?

1208
00:55:07,056 --> 00:55:09,180
PROFESSOR: Yeah, but 2 pi
over L would be equal to,

1209
00:55:09,180 --> 00:55:10,970
it's equivalent to q
equals 0, because it

1210
00:55:10,970 --> 00:55:13,010
corresponds to the
same eigenvalue.

1211
00:55:13,010 --> 00:55:15,615
So q's that differ
by 2 pi over L,

1212
00:55:15,615 --> 00:55:16,990
I'm going to call
the same thing.

1213
00:55:16,990 --> 00:55:17,531
AUDIENCE: OK.

1214
00:55:17,531 --> 00:55:18,664
Got it.

1215
00:55:18,664 --> 00:55:19,330
PROFESSOR: Good.

1216
00:55:19,330 --> 00:55:22,240
Other questions?

1217
00:55:22,240 --> 00:55:26,160
So the wave function is
not periodic except for one

1218
00:55:26,160 --> 00:55:28,550
special case, and we already
have a guess as to which

1219
00:55:28,550 --> 00:55:29,340
special case.

1220
00:55:29,340 --> 00:55:31,548
It looks like it's the state
on the top of the energy

1221
00:55:31,548 --> 00:55:34,300
band, which is
approximately periodic.

1222
00:55:34,300 --> 00:55:36,930
Two, while the wave
function is not periodic,

1223
00:55:36,930 --> 00:55:40,020
the probability that you
find the particle at x plus L

1224
00:55:40,020 --> 00:55:41,520
is equal to the
probability that you

1225
00:55:41,520 --> 00:55:44,320
find the particle at x, and this
immediately falls in this form,

1226
00:55:44,320 --> 00:55:45,945
because when you take
the norm squared,

1227
00:55:45,945 --> 00:55:47,450
the phis cancels
out at every point

1228
00:55:47,450 --> 00:55:50,110
and we're left with
the periodic u of x.

1229
00:55:50,110 --> 00:55:52,660
So the probability
is equal to norm

1230
00:55:52,660 --> 00:55:56,070
squared of u of x, which is
equal to norm squared of u

1231
00:55:56,070 --> 00:55:57,780
of x plus L because
u is periodic.

1232
00:55:57,780 --> 00:56:01,660
So probability
distribution is periodic.

1233
00:56:01,660 --> 00:56:05,660
On the one hand,
this is reassuring,

1234
00:56:05,660 --> 00:56:07,770
because if you look
at the potential,

1235
00:56:07,770 --> 00:56:09,590
the potential is
perfectly periodic,

1236
00:56:09,590 --> 00:56:11,490
and it would be
really weird if you

1237
00:56:11,490 --> 00:56:13,582
could tell by some
probability, by some measure,

1238
00:56:13,582 --> 00:56:15,540
like how likely you would
find an electron here

1239
00:56:15,540 --> 00:56:17,020
as supposed to over here,
potential is the same,

1240
00:56:17,020 --> 00:56:18,650
the probability should
be the same find

1241
00:56:18,650 --> 00:56:21,240
the electron in each spot.

1242
00:56:21,240 --> 00:56:25,360
The problem with that logic
though, is that these are wells

1243
00:56:25,360 --> 00:56:27,500
and usually we think
that when we have wells,

1244
00:56:27,500 --> 00:56:31,310
we have bound states, and those
bound states are localized.

1245
00:56:31,310 --> 00:56:34,590
But the probability has to be
invariant under translations,

1246
00:56:34,590 --> 00:56:37,170
so it cannot be that the
electron wave functions are

1247
00:56:37,170 --> 00:56:38,335
delocalized.

1248
00:56:38,335 --> 00:56:40,560
The electrons wave
functions must

1249
00:56:40,560 --> 00:56:43,580
have equal overall amplitude
to be in any given well.

1250
00:56:47,050 --> 00:56:49,620
It must be invariant
under translations

1251
00:56:49,620 --> 00:56:51,902
by a lattice spacing.

1252
00:56:51,902 --> 00:56:52,860
So that's really weird.

1253
00:56:52,860 --> 00:56:54,530
We have a whole bunch
of quantum wells,

1254
00:56:54,530 --> 00:56:57,390
and yet there are
no localized states.

1255
00:56:57,390 --> 00:56:59,580
All the states are extended
just like free particle

1256
00:56:59,580 --> 00:57:00,413
states are extended.

1257
00:57:02,770 --> 00:57:05,190
It had to be this
way by periodicity,

1258
00:57:05,190 --> 00:57:07,240
but it should surprise
you a little bit.

1259
00:57:07,240 --> 00:57:08,250
Anyway it surprises me.

1260
00:57:08,250 --> 00:57:09,510
I shouldn't tell you
what to be surprised by.

1261
00:57:09,510 --> 00:57:10,010
Yeah.

1262
00:57:10,010 --> 00:57:12,634
AUDIENCE: To normalize
probability, what do you do?

1263
00:57:12,634 --> 00:57:13,300
PROFESSOR: Yeah.

1264
00:57:13,300 --> 00:57:13,670
Excellent.

1265
00:57:13,670 --> 00:57:14,020
OK.

1266
00:57:14,020 --> 00:57:16,450
So this is for the same
reasons that free particle wave

1267
00:57:16,450 --> 00:57:18,070
functions aren't
normalizable, these

1268
00:57:18,070 --> 00:57:20,407
aren't going to be
normalized either.

1269
00:57:20,407 --> 00:57:21,865
So what are we
going to have to do?

1270
00:57:21,865 --> 00:57:23,230
AUDIENCE: Build wave packets.

1271
00:57:23,230 --> 00:57:23,370
PROFESSOR: Yeah.

1272
00:57:23,370 --> 00:57:24,580
We've going to have
to build wave packets.

1273
00:57:24,580 --> 00:57:26,590
In order to really meaningfully
talk about that stuff,

1274
00:57:26,590 --> 00:57:28,339
we're going to have
to built wave packets.

1275
00:57:28,339 --> 00:57:33,480
We need wave packets to
make normalizable states.

1276
00:57:33,480 --> 00:57:37,620
But the localized wave
packets are necessarily

1277
00:57:37,620 --> 00:57:40,330
not going to be
energy eigenstates

1278
00:57:40,330 --> 00:57:41,667
just like for a free particle.

1279
00:57:41,667 --> 00:57:43,250
And so there's going
to be dispersion,

1280
00:57:43,250 --> 00:57:44,833
and the whole story
that you saw three

1281
00:57:44,833 --> 00:57:46,610
particles we're
going to see again

1282
00:57:46,610 --> 00:57:48,685
for the particles in
the periodic potential.

1283
00:57:48,685 --> 00:57:49,185
Cool?

1284
00:57:53,360 --> 00:57:54,260
So next thing.

1285
00:57:58,620 --> 00:58:04,460
Therefore, all phi
sub Eq extended.

1286
00:58:07,824 --> 00:58:09,990
This is going to have some
really cool consequences,

1287
00:58:09,990 --> 00:58:11,619
and understanding
this fact in detail

1288
00:58:11,619 --> 00:58:14,160
is going to explain to us the
difference between connectivity

1289
00:58:14,160 --> 00:58:16,510
in the middle and not
conductivity in plastic.

1290
00:58:19,667 --> 00:58:21,539
Although there's a
surprising hook in there.

1291
00:58:21,539 --> 00:58:23,580
Naively, it would have
gone the other way around.

1292
00:58:26,100 --> 00:58:29,070
So the last thing
to say three, is

1293
00:58:29,070 --> 00:58:35,770
that phi sub to Eq, so this is
the state with definite energy

1294
00:58:35,770 --> 00:58:42,070
and definite eigenvalue under
translation E to the i qL Yeah.

1295
00:58:42,070 --> 00:58:46,500
And you showed on a problem
set that if you take a wave

1296
00:58:46,500 --> 00:58:49,000
function, you multiply
it by E to the i qx, what

1297
00:58:49,000 --> 00:58:50,580
have you done to the expectation
value of the momentum?

1298
00:58:50,580 --> 00:58:51,980
AUDIENCE: You've
increased it by q.

1299
00:58:51,980 --> 00:58:53,170
PROFESSOR: You've
increased it by q, right?

1300
00:58:53,170 --> 00:58:55,180
This is the boost
it by q operation.

1301
00:58:55,180 --> 00:58:59,690
So q incidentally has units
of a wave number, so times

1302
00:58:59,690 --> 00:59:02,800
h bar has units of a momentum.
q is like a momentum.

1303
00:59:02,800 --> 00:59:03,660
Is q a momentum?

1304
00:59:06,310 --> 00:59:08,410
Well, we can answer
that question by asking,

1305
00:59:08,410 --> 00:59:12,100
is this an eigenfunction of P?

1306
00:59:12,100 --> 00:59:14,075
Is it an eigenfunction
a momentum operator?

1307
00:59:14,075 --> 00:59:15,527
And the answer is
definitely not.

1308
00:59:15,527 --> 00:59:17,485
That's an eigenfunction
of a momentum operator.

1309
00:59:17,485 --> 00:59:21,370
I take a derivative, I get q
times i and I multiply by h

1310
00:59:21,370 --> 00:59:24,050
bar, I get h bar q,
momentum, but when

1311
00:59:24,050 --> 00:59:26,210
I have this extra
periodic function,

1312
00:59:26,210 --> 00:59:28,796
this thing together is no
longer an eigenfunction of q,

1313
00:59:28,796 --> 00:59:31,170
because u is some periodic
function, which is necessarily

1314
00:59:31,170 --> 00:59:38,650
not a momentum eigenfunction.

1315
00:59:38,650 --> 00:59:40,740
So this guy is not a
momentum eigenfunction.

1316
00:59:40,740 --> 00:59:42,040
You can see that by just
taking the derivative.

1317
00:59:42,040 --> 00:59:43,850
I get one term that
picks up a q from this,

1318
00:59:43,850 --> 00:59:46,516
but I get another term that gets
a derivative with respect to u,

1319
00:59:46,516 --> 00:59:52,220
and that's not proportional with
the constant to u e to the i qx

1320
00:59:52,220 --> 00:59:54,110
unless u is in the
form e to the i kx.

1321
00:59:54,110 --> 00:59:55,950
But if u is in the
form e to the i kx,

1322
00:59:55,950 --> 00:59:57,741
this is just a free
particle wave function,

1323
00:59:57,741 --> 01:00:01,570
which it can't be if we
have a periodic potential.

1324
01:00:01,570 --> 01:00:05,380
So finally q is not a
momentum eigenstate.

1325
01:00:08,820 --> 01:00:13,705
And correspondingly,
q and more important

1326
01:00:13,705 --> 01:00:27,300
h bar q is not a momentum, I'll
say the momentum in the sense

1327
01:00:27,300 --> 01:00:30,910
that it's not the eigenvalue
of the momentum operator.

1328
01:00:30,910 --> 01:00:38,370
However, it has-- semi colon--
it has units of momentum.

1329
01:00:38,370 --> 01:00:40,371
And not only does it
have units of momentum,

1330
01:00:40,371 --> 01:00:42,620
it's the thing that tells
you how the state transforms

1331
01:00:42,620 --> 01:00:43,411
under translations.

1332
01:00:43,411 --> 01:00:46,300
It translates by an e to the
i qL under translations by L.

1333
01:00:46,300 --> 01:00:47,980
And there is usually
what momentum is.

1334
01:00:47,980 --> 01:00:49,480
it's the constant
that tells you how

1335
01:00:49,480 --> 01:00:51,530
the state transfers
under momentum.

1336
01:00:51,530 --> 01:00:53,740
So it's kind of like a
momentum, but it's not

1337
01:00:53,740 --> 01:00:55,407
the eigenvalue of the
momentum operator.

1338
01:00:55,407 --> 01:00:56,323
We have a name for it.

1339
01:00:56,323 --> 01:00:58,080
We call it the crystal
momentum, and we'll

1340
01:00:58,080 --> 01:00:59,700
understand what it does for us.

1341
01:00:59,700 --> 01:01:00,970
Part of the goal for
the next two lectures

1342
01:01:00,970 --> 01:01:02,620
is going to be to understand
exactly what this crystal

1343
01:01:02,620 --> 01:01:03,415
moment is.

1344
01:01:09,609 --> 01:01:11,150
So for the moment,
I'm just giving it

1345
01:01:11,150 --> 01:01:12,525
a name sort of
like the beginning

1346
01:01:12,525 --> 01:01:14,390
we give super position
a name, and we'll

1347
01:01:14,390 --> 01:01:17,824
exploit its properties
basically now.

1348
01:01:17,824 --> 01:01:20,240
So these are the things that
follow just from periodicity.

1349
01:01:20,240 --> 01:01:22,031
I didn't know anything
about the potential.

1350
01:01:22,031 --> 01:01:24,730
I just used periodicity.

1351
01:01:24,730 --> 01:01:27,310
So before we move on to talking
about a specific potential,

1352
01:01:27,310 --> 01:01:30,170
you all have questions
about the general structure

1353
01:01:30,170 --> 01:01:31,920
of periodic systems like this.

1354
01:01:35,030 --> 01:01:35,530
Yeah.

1355
01:01:35,530 --> 01:01:37,107
AUDIENCE: What's the
difference between the psi

1356
01:01:37,107 --> 01:01:37,386
and the [INAUDIBLE]?

1357
01:01:37,386 --> 01:01:38,660
PROFESSOR: Oh, excellent.

1358
01:01:38,660 --> 01:01:42,381
I shouldn't have used psi,
I should have used phi.

1359
01:01:42,381 --> 01:01:42,880
Thanks.

1360
01:01:47,120 --> 01:01:51,190
I'll use psi only when we talk
about general super positions

1361
01:01:51,190 --> 01:01:53,214
and wave packets.

1362
01:01:53,214 --> 01:01:53,880
Other questions?

1363
01:01:57,040 --> 01:01:58,120
All right.

1364
01:01:58,120 --> 01:02:04,550
So with all that, let's talk
about a specific potential.

1365
01:02:04,550 --> 01:02:06,050
So far we've extracted
about as much

1366
01:02:06,050 --> 01:02:07,508
as of the physics
out of the system

1367
01:02:07,508 --> 01:02:09,437
that we can just
from periodicity.

1368
01:02:09,437 --> 01:02:11,103
In order to make more
progress, in order

1369
01:02:11,103 --> 01:02:14,410
to talk about, for example,
what are the allowed energy

1370
01:02:14,410 --> 01:02:17,920
eigenvalues and for that matter,
how does that allowed energy

1371
01:02:17,920 --> 01:02:20,404
eigenvalue depend on q,
the crystal momentum,

1372
01:02:20,404 --> 01:02:22,070
in order to answer
that, we have to talk

1373
01:02:22,070 --> 01:02:23,195
about a specific system.

1374
01:02:23,195 --> 01:02:25,320
So let's go ahead and talk
about a specific system.

1375
01:02:25,320 --> 01:02:26,365
I'm going to pick my favorite.

1376
01:02:26,365 --> 01:02:27,893
Now the problem
set, you're going

1377
01:02:27,893 --> 01:02:30,000
to do something really
cool about it, which

1378
01:02:30,000 --> 01:02:31,330
I'll explain in a moment.

1379
01:02:31,330 --> 01:02:32,890
So you'll do a general
case in the problem set,

1380
01:02:32,890 --> 01:02:34,890
but for the moment, let's
work with a simple example.

1381
01:02:34,890 --> 01:02:36,264
And the simplest
example is going

1382
01:02:36,264 --> 01:02:38,540
to be a periodic potential
with the simplest

1383
01:02:38,540 --> 01:02:40,580
possible barriers
in the potential.

1384
01:02:40,580 --> 01:02:43,330
What's the simplest barrier?

1385
01:02:43,330 --> 01:02:44,030
Delta function.

1386
01:02:44,030 --> 01:02:44,530
Yes.

1387
01:02:44,530 --> 01:02:48,820
So the potential is going
to be for this example

1388
01:02:48,820 --> 01:02:56,930
V of x is equal to sum from n
is equal to minus infinity of h

1389
01:02:56,930 --> 01:03:05,560
bar squared over 2mL g
naught delta of x minus nL.

1390
01:03:05,560 --> 01:03:06,350
So what is this?

1391
01:03:06,350 --> 01:03:08,944
So this is just some
overall constant out front.

1392
01:03:08,944 --> 01:03:10,610
Normally we call this
V naught, but I've

1393
01:03:10,610 --> 01:03:12,390
made g naught dimensionless
by pulling out

1394
01:03:12,390 --> 01:03:13,715
an h bar squared over 2nL.

1395
01:03:13,715 --> 01:03:16,760
L is the spacing
between delta barriers.

1396
01:03:16,760 --> 01:03:19,710
So the potential
looks like this.

1397
01:03:19,710 --> 01:03:20,436
Here is 0.

1398
01:03:20,436 --> 01:03:21,810
We have a delta
function barrier.

1399
01:03:21,810 --> 01:03:25,805
We have a delta function barrier
L. We have a delta function

1400
01:03:25,805 --> 01:03:31,019
barrier at 2L, dot, dot,
dot, minus L, dot, dot, dot.

1401
01:03:31,019 --> 01:03:32,060
So there's our potential.

1402
01:03:34,522 --> 01:03:36,730
And I want to know what are
the energy eigenfunctions

1403
01:03:36,730 --> 01:03:38,396
for a single particle
in this potential.

1404
01:03:41,380 --> 01:03:51,138
And again, g naught is
the dimensionless strength

1405
01:03:51,138 --> 01:03:52,110
of the potential.

1406
01:03:57,240 --> 01:04:00,520
So how do we solve this problem?

1407
01:04:00,520 --> 01:04:03,490
Well, we've done
this so many times.

1408
01:04:03,490 --> 01:04:06,300
What we see is that in
between each barrier,

1409
01:04:06,300 --> 01:04:08,730
the particle is just free, so
we know the form of the wave

1410
01:04:08,730 --> 01:04:10,420
function between each
barrier, it's just

1411
01:04:10,420 --> 01:04:12,430
e to the i kx plus 2
minus e to the minus i kx,

1412
01:04:12,430 --> 01:04:14,730
where k is defined
from the energy h bar

1413
01:04:14,730 --> 01:04:17,479
squared k squared upon 2n is e.

1414
01:04:17,479 --> 01:04:19,770
At the barrier, we have to
satisfy appropriate matching

1415
01:04:19,770 --> 01:04:20,660
conditions.

1416
01:04:20,660 --> 01:04:23,287
So I could put a and b here
and c and d and e and f.

1417
01:04:23,287 --> 01:04:25,620
And under each one of these
imposed boundary conditions,

1418
01:04:25,620 --> 01:04:27,390
and this is going to be an
infinite number of coefficients

1419
01:04:27,390 --> 01:04:29,560
with an infinite number
of boundary conditions.

1420
01:04:29,560 --> 01:04:31,820
And that sounds like it's
going to take some time.

1421
01:04:31,820 --> 01:04:33,870
But we can use
something really nice.

1422
01:04:33,870 --> 01:04:36,160
We already know that the
wave function is periodic.

1423
01:04:36,160 --> 01:04:43,020
So suppose between 0 and
L, let's say less than x,

1424
01:04:43,020 --> 01:04:48,100
less than L the wave
function phi takes the form,

1425
01:04:48,100 --> 01:04:52,480
and we know that phi of eq
takes the form e to the i qx,

1426
01:04:52,480 --> 01:04:54,220
and now some periodic
function, which

1427
01:04:54,220 --> 01:04:59,474
would write as a e to
the i kx-- actually,

1428
01:04:59,474 --> 01:05:00,640
I'm not going to write that.

1429
01:05:03,340 --> 01:05:06,330
So in between 0 and L,
it's a free particle,

1430
01:05:06,330 --> 01:05:10,680
and it's equal to A e to the
i kx plus B e to the minus

1431
01:05:10,680 --> 01:05:14,150
akx, where k is defined
purely from the energy h bar

1432
01:05:14,150 --> 01:05:17,942
squared k squared
upon 2m is the energy.

1433
01:05:17,942 --> 01:05:19,150
So this is what we mean by k.

1434
01:05:22,620 --> 01:05:25,910
And in between 2L it will
take the similar form,

1435
01:05:25,910 --> 01:05:28,520
but what we can do
is, we can define,

1436
01:05:28,520 --> 01:05:33,090
so this is sine of x,
between L less than x less

1437
01:05:33,090 --> 01:05:38,020
than 2L phi eq is
equal to-- well,

1438
01:05:38,020 --> 01:05:39,620
I could define it
with new constants,

1439
01:05:39,620 --> 01:05:42,050
but I know what it has to be
because whatever the value is

1440
01:05:42,050 --> 01:05:46,120
here it's the same as the
value here up to a shift by L

1441
01:05:46,120 --> 01:05:48,390
the translation to
phase e to the i qL.

1442
01:05:48,390 --> 01:05:50,580
So this is equal
between L and 2L,

1443
01:05:50,580 --> 01:05:54,480
it's equal to e to the
i qL and periodicity

1444
01:05:54,480 --> 01:05:57,750
condition, the same
thing, A plus B,

1445
01:05:57,750 --> 01:05:59,672
with the same coefficients.

1446
01:05:59,672 --> 01:06:00,910
Everyone cool with that?

1447
01:06:04,720 --> 01:06:06,367
So I can now translate,
so let's think

1448
01:06:06,367 --> 01:06:07,950
about what the
boundary conditions are

1449
01:06:07,950 --> 01:06:09,175
going to have to be.

1450
01:06:09,175 --> 01:06:12,200
At each of the delta functions,
the delta function boundary

1451
01:06:12,200 --> 01:06:15,120
condition is going to say that
the slope here minus the slope

1452
01:06:15,120 --> 01:06:17,880
here is something proportional
to the amplitude here.

1453
01:06:17,880 --> 01:06:20,992
And the slope here is going to
be minus something proportional

1454
01:06:20,992 --> 01:06:21,700
to the amplitude.

1455
01:06:24,430 --> 01:06:28,260
But the slope here is the
same as the slope here up

1456
01:06:28,260 --> 01:06:30,110
to an overall phase
from the translation.

1457
01:06:30,110 --> 01:06:34,055
And the slope here is the same
as the slope here up to a phase

1458
01:06:34,055 --> 01:06:35,620
coming from the translation.

1459
01:06:35,620 --> 01:06:37,800
So we can now turn this
into instead of this,

1460
01:06:37,800 --> 01:06:38,807
the condition between
the slopes here

1461
01:06:38,807 --> 01:06:41,348
and here, I can turn that into
a condition between the slopes

1462
01:06:41,348 --> 01:06:42,300
here and here.

1463
01:06:42,300 --> 01:06:42,800
Wow.

1464
01:06:42,800 --> 01:06:43,617
This is horrible.

1465
01:06:43,617 --> 01:06:44,950
Here are my two delta functions.

1466
01:06:44,950 --> 01:06:48,310
The slope here and here, the
boundary condition I put at 0,

1467
01:06:48,310 --> 01:06:50,670
I can translate this
into the slope here up

1468
01:06:50,670 --> 01:06:54,660
to an overall phase
e to the i qL.

1469
01:06:54,660 --> 01:06:56,510
So I can turn the
boundary condition here

1470
01:06:56,510 --> 01:06:58,301
into a boundary condition
between these two

1471
01:06:58,301 --> 01:07:00,630
guys inside one domain.

1472
01:07:00,630 --> 01:07:02,380
And so I can turn this
into a problem that

1473
01:07:02,380 --> 01:07:04,940
just involved A and
B and that's it.

1474
01:07:04,940 --> 01:07:06,190
Everyone see that?

1475
01:07:06,190 --> 01:07:07,529
So let's do it.

1476
01:07:07,529 --> 01:07:09,570
So questions on the basic
strategy at that point?

1477
01:07:13,490 --> 01:07:14,320
So let's do it.

1478
01:07:14,320 --> 01:07:16,570
So what we need is we got
the general form of the wave

1479
01:07:16,570 --> 01:07:19,470
function, and I'm going to erase
this because we don't need it.

1480
01:07:19,470 --> 01:07:21,220
Got the general form
of the wave function,

1481
01:07:21,220 --> 01:07:22,450
and then we need to
impose the boundary

1482
01:07:22,450 --> 01:07:24,074
conditions of the
delta function.

1483
01:07:24,074 --> 01:07:25,990
And in particular, let's
just impose them at 0

1484
01:07:25,990 --> 01:07:27,450
because imposing
them at 0 is going

1485
01:07:27,450 --> 01:07:29,120
to be equivalent to
imposing them everywhere

1486
01:07:29,120 --> 01:07:31,453
else by periodicity if I use
the periodicity of the wave

1487
01:07:31,453 --> 01:07:32,840
function.

1488
01:07:32,840 --> 01:07:39,440
So the boundary conditions
at the delta functions.

1489
01:07:39,440 --> 01:07:42,010
So the first is
that phi at 0 plus

1490
01:07:42,010 --> 01:07:43,895
is equal to phi at
0 minus, so this

1491
01:07:43,895 --> 01:07:46,550
is a statement that the
wave function is continuous.

1492
01:07:46,550 --> 01:07:50,767
But phi at 0 minus is equal
to-- well, that's this point.

1493
01:07:50,767 --> 01:07:53,100
But this point is the same
as this point except the wave

1494
01:07:53,100 --> 01:07:55,190
function picks up a
phase e to the i qL.

1495
01:07:55,190 --> 01:07:59,440
This is equal to phi
at L minus, right here,

1496
01:07:59,440 --> 01:08:01,602
times e to the i qL.

1497
01:08:01,602 --> 01:08:03,686
And we have to be careful
because the statement is

1498
01:08:03,686 --> 01:08:05,851
at the wave function here,
is the wave function here

1499
01:08:05,851 --> 01:08:06,660
time e to the i qL.

1500
01:08:06,660 --> 01:08:09,243
So the wave function here times
the wave function here times e

1501
01:08:09,243 --> 01:08:10,400
to the i qL minus.

1502
01:08:23,740 --> 01:08:28,630
And this leads to phi zero
plus-- the value right here--

1503
01:08:28,630 --> 01:08:32,589
is A plus B because it's just
this wave function evaluated

1504
01:08:32,589 --> 01:08:34,779
at x equal 0, and
this guy at L is

1505
01:08:34,779 --> 01:08:36,649
going to be this
evaluated at x is

1506
01:08:36,649 --> 01:08:39,520
equal to L times e
to the minus i qL.

1507
01:08:39,520 --> 01:08:49,062
So this is equal to A e to
the i k minus qL plus B e

1508
01:08:49,062 --> 01:08:56,040
to the minus i k plus L, So
there's my first condition.

1509
01:08:56,040 --> 01:08:58,700
Second boundary condition
is the derivative condition,

1510
01:08:58,700 --> 01:09:01,120
and the derivative
condition is that phi prime

1511
01:09:01,120 --> 01:09:06,930
at 0 plus minus phi
prime at 0 minus

1512
01:09:06,930 --> 01:09:11,545
is equal to g naught
upon L phi at 0.

1513
01:09:15,910 --> 01:09:23,540
And this gives me that ik A
minus B. So that's this guy,

1514
01:09:23,540 --> 01:09:25,870
and now the derivative
of the second guy

1515
01:09:25,870 --> 01:09:30,359
picked up our extra
phase minus ik A

1516
01:09:30,359 --> 01:09:36,340
e to the ik minus q L
minus B e to the minus ik

1517
01:09:36,340 --> 01:09:39,734
plus qL-- this is
all in the notes

1518
01:09:39,734 --> 01:09:41,609
so I'm not going to
worry too much about it--

1519
01:09:41,609 --> 01:09:45,580
is equal to, on the right-hand
side is g naught upon L A

1520
01:09:45,580 --> 01:09:50,520
plus B.

1521
01:09:50,520 --> 01:09:53,225
So now note the following
properties of these equations.

1522
01:09:53,225 --> 01:09:55,600
These equations determine an
equation, so from the first,

1523
01:09:55,600 --> 01:09:59,760
I'll call this 1,
and I'll call this 2.

1524
01:09:59,760 --> 01:10:05,570
So 1 implies that A is
equal to B times something.

1525
01:10:05,570 --> 01:10:06,920
It's just a linear equation.

1526
01:10:06,920 --> 01:10:10,107
So if we pull the side
over, put this over here,

1527
01:10:10,107 --> 01:10:11,440
A is equal to B times something.

1528
01:10:11,440 --> 01:10:12,580
Everybody agree with that?

1529
01:10:12,580 --> 01:10:20,070
Some horrible expression
of k and q times.

1530
01:10:20,070 --> 01:10:25,680
And 2 also gives an expression
form A is equal to B times

1531
01:10:25,680 --> 01:10:31,350
a horrible expression in terms
of k and q but a different one.

1532
01:10:31,350 --> 01:10:34,360
Call this expression 1,
I'll call this expression 2.

1533
01:10:34,360 --> 01:10:37,480
So as usual, so this is
two different expressions

1534
01:10:37,480 --> 01:10:39,490
for A given B and
this only makes

1535
01:10:39,490 --> 01:10:42,760
sense it's only a solution if
these two horrible expressions

1536
01:10:42,760 --> 01:10:44,374
are equal to each other.

1537
01:10:44,374 --> 01:10:46,415
And so this is what we've
done many times before,

1538
01:10:46,415 --> 01:10:48,210
and if you've set these two
expressions equal to each other

1539
01:10:48,210 --> 01:10:49,780
and do a little bit
of trigonometry,

1540
01:10:49,780 --> 01:10:51,530
you get the following relation.

1541
01:10:51,530 --> 01:11:03,250
Cosine of qL is equal to
cosine of kL plus g naught

1542
01:11:03,250 --> 01:11:06,010
upon 2kL sine kL.

1543
01:11:10,950 --> 01:11:13,770
Where what k means is nothing
other than the square root

1544
01:11:13,770 --> 01:11:16,780
of 2ne upon h bar.

1545
01:11:16,780 --> 01:11:20,030
k is defined from that
relation, and what q means is q

1546
01:11:20,030 --> 01:11:23,170
is the eigenfunction or
e to the i qL anyway,

1547
01:11:23,170 --> 01:11:25,230
is the eigenfunction
of the wave function

1548
01:11:25,230 --> 01:11:29,534
under translation by L.

1549
01:11:29,534 --> 01:11:31,450
So k is really playing
the role of the energy.

1550
01:11:37,710 --> 01:11:38,710
Everyone cool with that?

1551
01:11:38,710 --> 01:11:40,720
I just skipped the algebra, but
if you do the horrible algebra

1552
01:11:40,720 --> 01:11:42,070
from setting these
two expressions

1553
01:11:42,070 --> 01:11:43,390
equal to each other,
then you get this.

1554
01:11:43,390 --> 01:11:43,889
Yeah.

1555
01:11:43,889 --> 01:11:46,257
AUDIENCE: Is q some like
undetermined thing as of right

1556
01:11:46,257 --> 01:11:47,430
now, or do we know--

1557
01:11:47,430 --> 01:11:48,740
PROFESSOR: Excellent.

1558
01:11:48,740 --> 01:11:53,670
What values of q are
allowed by this expression?

1559
01:11:53,670 --> 01:11:55,420
Is exactly the right
question to ask here.

1560
01:11:55,420 --> 01:12:00,530
What values of q's are
allowed for our wave function?

1561
01:12:00,530 --> 01:12:03,470
Is there any condition on
what the value q is so far?

1562
01:12:06,350 --> 01:12:06,850
No.

1563
01:12:06,850 --> 01:12:08,640
It could have been
absolutely any number.

1564
01:12:08,640 --> 01:12:11,330
For any number q, we
can find a eigenfunction

1565
01:12:11,330 --> 01:12:15,860
to translate by L, e to the i
qL as the eigenvalue, any value

1566
01:12:15,860 --> 01:12:16,360
of q.

1567
01:12:16,360 --> 01:12:20,720
However, q is equivalent
to q plus 2 pi upon L.

1568
01:12:20,720 --> 01:12:22,386
If you skipped by
2 pi over L, then

1569
01:12:22,386 --> 01:12:24,010
it's not really a
different eigenvalue.

1570
01:12:24,010 --> 01:12:25,590
It's the same eigenvalue.

1571
01:12:25,590 --> 01:12:26,625
So q is defined.

1572
01:12:26,625 --> 01:12:30,000
It's any continuous number
between let's say minus

1573
01:12:30,000 --> 01:12:32,030
pi over L and pi over
L for simplicity,

1574
01:12:32,030 --> 01:12:34,635
make it nice and small. i
could have said 0 and 2 pi.

1575
01:12:34,635 --> 01:12:37,860
It doesn't make any difference,
minus pi and pi over L.

1576
01:12:37,860 --> 01:12:40,800
So q is a free parameter.

1577
01:12:40,800 --> 01:12:42,210
So there exists a state.

1578
01:12:42,210 --> 01:12:50,470
Another way to say this
is that there exist states

1579
01:12:50,470 --> 01:12:53,930
and in particular eigenstates
for any value of q,

1580
01:12:53,930 --> 01:12:59,740
for any q, and any e, so
for any value of q and e

1581
01:12:59,740 --> 01:13:04,075
which satisfy this equation
corresponds to a state.

1582
01:13:04,075 --> 01:13:06,450
There exists states for any
q E satisfying this equation.

1583
01:13:11,260 --> 01:13:11,810
Cool?

1584
01:13:11,810 --> 01:13:14,614
This is just like
the finite well.

1585
01:13:14,614 --> 01:13:16,030
In the case of the
finite well, we

1586
01:13:16,030 --> 01:13:18,790
go through exactly
the same analysis

1587
01:13:18,790 --> 01:13:20,960
the only difference is
that we don't have the qL.

1588
01:13:20,960 --> 01:13:23,672
What happens in the
finite well case

1589
01:13:23,672 --> 01:13:25,630
we impose a boundary
condition off in infinity.

1590
01:13:25,630 --> 01:13:27,140
So the thing has normalized,
and we put another boundary

1591
01:13:27,140 --> 01:13:28,870
condition in infinity,
and what we ended up

1592
01:13:28,870 --> 01:13:31,536
getting is something of the form
roughly 1 is equal to something

1593
01:13:31,536 --> 01:13:32,960
like this.

1594
01:13:32,960 --> 01:13:35,240
It's actually not exactly
that, but here we have kL,

1595
01:13:35,240 --> 01:13:36,823
and if you combine
these two together,

1596
01:13:36,823 --> 01:13:39,690
you get something kL plus
a phase shift over kL

1597
01:13:39,690 --> 01:13:40,470
and then a 1.

1598
01:13:40,470 --> 01:13:42,830
If you multiply, you get
kL is equal to cosine

1599
01:13:42,830 --> 01:13:45,410
of kL plus a phase
shift, which is exactly

1600
01:13:45,410 --> 01:13:48,510
the form for the
delta potential.

1601
01:13:52,640 --> 01:13:53,210
Good.

1602
01:13:53,210 --> 01:13:54,160
So this is similar.

1603
01:13:54,160 --> 01:13:57,630
However, here we have
an extra parameter q,

1604
01:13:57,630 --> 01:13:59,790
which is free to vary.

1605
01:13:59,790 --> 01:14:04,390
q could be anything between
minus pi over L and pi over L

1606
01:14:04,390 --> 01:14:07,020
So what does cosine
of qL turn into?

1607
01:14:07,020 --> 01:14:10,725
Well, it's anything qL can
take q is 2 pi upon L or pi

1608
01:14:10,725 --> 01:14:14,560
upon L to minus pi over L. So qL
can go between pi and minus pi,

1609
01:14:14,560 --> 01:14:16,620
so cosine of qL
varies between cosine

1610
01:14:16,620 --> 01:14:20,180
of pi, which is 1 and cosine
of minus pi, which is minus 1.

1611
01:14:20,180 --> 01:14:23,320
So any value of cosine
of qL between 1 minus 1

1612
01:14:23,320 --> 01:14:26,200
is a valid value of
qL On the other hand,

1613
01:14:26,200 --> 01:14:28,780
if this is equal to
7, is there a q such

1614
01:14:28,780 --> 01:14:30,123
that this is equal to 7?

1615
01:14:30,123 --> 01:14:31,379
No.

1616
01:14:31,379 --> 01:14:33,920
So what we need to do now is,
we need to solve this equation,

1617
01:14:33,920 --> 01:14:36,270
but it's obviously a god
awful transcendental equation.

1618
01:14:36,270 --> 01:14:38,669
So how do we solve horrible
transcendental equations?

1619
01:14:38,669 --> 01:14:39,585
AUDIENCE: Graphically.

1620
01:14:39,585 --> 01:14:40,660
PROFESSOR: Graphically.

1621
01:14:40,660 --> 01:14:41,909
So let's solve it graphically.

1622
01:14:47,970 --> 01:14:54,400
So to solve it graphically, that
lets plot the left-hand side

1623
01:14:54,400 --> 01:14:57,670
and the right-hand side
as a function of qL

1624
01:14:57,670 --> 01:15:01,510
So in this direction, I'm
going to plot cosine of qL.

1625
01:15:01,510 --> 01:15:04,170
That's the left side.

1626
01:15:04,170 --> 01:15:07,250
And cosine of qL we know is
going to vary between plus 1

1627
01:15:07,250 --> 01:15:08,890
and minus 1.

1628
01:15:12,790 --> 01:15:17,760
Because q between q
is equal to pi over L,

1629
01:15:17,760 --> 01:15:24,020
and q is equal to
minus pi over L.

1630
01:15:24,020 --> 01:15:27,860
And in the horizontal
direction, I'm going to plot kL.

1631
01:15:31,780 --> 01:15:35,990
And when kL is 0, remember that
the definition of kL is that E,

1632
01:15:35,990 --> 01:15:37,695
the energy eigenvalue
is equal to h

1633
01:15:37,695 --> 01:15:41,990
bar squared k squared upon 2m.

1634
01:15:41,990 --> 01:15:44,260
And what we're looking
for are points,

1635
01:15:44,260 --> 01:15:46,510
are any common solution to
this equation where there's

1636
01:15:46,510 --> 01:15:48,165
a value of q and a
value of E of k such

1637
01:15:48,165 --> 01:15:50,290
that these two expressions
are equal to each other.

1638
01:15:50,290 --> 01:15:50,758
Yeah.

1639
01:15:50,758 --> 01:15:51,694
AUDIENCE: [INAUDIBLE].

1640
01:15:55,906 --> 01:15:56,650
PROFESSOR: Sorry?

1641
01:15:56,650 --> 01:15:59,325
AUDIENCE: [INAUDIBLE].

1642
01:15:59,325 --> 01:16:01,010
PROFESSOR: Oh, sorry.

1643
01:16:01,010 --> 01:16:01,960
Thank you.

1644
01:16:01,960 --> 01:16:06,030
This is qL is equal to
0 and qL is equal to pi.

1645
01:16:09,420 --> 01:16:10,210
Thank you.

1646
01:16:10,210 --> 01:16:10,710
Yes.

1647
01:16:14,630 --> 01:16:17,530
Are there types of
other questions?

1648
01:16:17,530 --> 01:16:22,802
So kL equals 0 is going to
be equal to 0, k equals 0,

1649
01:16:22,802 --> 01:16:24,510
and this is going to
be a function of kL.

1650
01:16:24,510 --> 01:16:25,570
I'm going to plot it
as a function of kL

1651
01:16:25,570 --> 01:16:26,778
because that's what shows up.

1652
01:16:26,778 --> 01:16:30,040
But remember that k is
nothing but code for E.

1653
01:16:30,040 --> 01:16:32,410
And so what we're looking
for a common solutions

1654
01:16:32,410 --> 01:16:33,510
of that equation.

1655
01:16:33,510 --> 01:16:35,110
Now let's first just plot.

1656
01:16:35,110 --> 01:16:39,010
So if this is 0, let's
just plot cosine of kL,

1657
01:16:39,010 --> 01:16:43,372
the right-hand side cosine of kL
plus g naught over 2kL sine kL.

1658
01:16:43,372 --> 01:16:47,230
And let's do this first
for the free particle.

1659
01:16:47,230 --> 01:16:50,030
Oh, I wish I had colored chalk.

1660
01:16:50,030 --> 01:16:56,051
For the free particle, free is
going to be g naught equals 0.

1661
01:16:56,051 --> 01:16:57,634
There is a potential
and the potential

1662
01:16:57,634 --> 01:16:59,220
is 0 times the delta
function, which

1663
01:16:59,220 --> 01:17:01,400
is not so much delta function.

1664
01:17:01,400 --> 01:17:03,660
So for free particle
when g is equal to 0,

1665
01:17:03,660 --> 01:17:05,190
that second term
vanishes, and we

1666
01:17:05,190 --> 01:17:13,420
have cosine of qL is
equal to cosine of kL.

1667
01:17:13,420 --> 01:17:16,670
Kale, if you fry
it, it's crispy.

1668
01:17:16,670 --> 01:17:18,544
So what is that going to be?

1669
01:17:18,544 --> 01:17:19,960
Well, this has an
obvious solution

1670
01:17:19,960 --> 01:17:27,440
which is that q is equal
to k, but the problem

1671
01:17:27,440 --> 01:17:33,860
is, q is only defined up to 2
pi over L, so mod 2 pi over L.

1672
01:17:33,860 --> 01:17:35,990
Well, let's see what
this looks like in here.

1673
01:17:35,990 --> 01:17:38,230
So for the free particle
what does this tell us?

1674
01:17:38,230 --> 01:17:41,320
So for the free particle,
let me just write this

1675
01:17:41,320 --> 01:17:45,270
as therefore, q is equal
to L or q is equal to k,

1676
01:17:45,270 --> 01:17:48,010
and therefore E is equal
to h bar squared is

1677
01:17:48,010 --> 01:17:50,430
equal to q squared.

1678
01:17:50,430 --> 01:17:54,850
So any q or E are allowed
such that q squared

1679
01:17:54,850 --> 01:17:58,192
is equal E times 2
upon h bar squared.

1680
01:17:58,192 --> 01:17:59,650
So that's just the
usual condition.

1681
01:17:59,650 --> 01:18:03,030
It's the free particle has
energy h bar squared k squared

1682
01:18:03,030 --> 01:18:06,245
upon 2n, and we just chose
to call it k equals q.

1683
01:18:06,245 --> 01:18:08,370
So what we're doing is
instead of organizing things

1684
01:18:08,370 --> 01:18:09,800
as momentum eigenstates
of the free particle

1685
01:18:09,800 --> 01:18:12,310
or organizing the free particle
energy eigenstates in terms

1686
01:18:12,310 --> 01:18:13,570
of transient by L eigenstates.

1687
01:18:13,570 --> 01:18:14,236
Perfectly valid.

1688
01:18:14,236 --> 01:18:16,750
It commutes with
the energy operator.

1689
01:18:16,750 --> 01:18:18,250
What does this look like?

1690
01:18:18,250 --> 01:18:22,060
Well, cosine of kL is cosine
of qL, when k is equal to 0,

1691
01:18:22,060 --> 01:18:25,080
q is equal to 0.

1692
01:18:25,080 --> 01:18:25,850
So we get this.

1693
01:18:29,080 --> 01:18:40,630
And the points, when the
points where qL is equal to pi

1694
01:18:40,630 --> 01:18:44,820
or minus 1, these
are the points where

1695
01:18:44,820 --> 01:18:55,160
we get the qL is equal to n pi
or equivalently plus or minus

1696
01:18:55,160 --> 01:18:59,060
kL is equal to n pi
plus or minus n pi.

1697
01:18:59,060 --> 01:19:02,310
So this is n is
1, n is 2, n is 3.

1698
01:19:02,310 --> 01:19:04,210
Everyone cool with that?

1699
01:19:04,210 --> 01:19:14,996
And these guys are 2 pi
times n or 2n plus 1 2n pi.

1700
01:19:14,996 --> 01:19:19,410
And I really want this
to be odd, so n odd.

1701
01:19:19,410 --> 01:19:24,400
And this is n pi kL
so n pi for n even.

1702
01:19:31,013 --> 01:19:32,940
Everyone cool with that?

1703
01:19:32,940 --> 01:19:36,194
So that's what it would look
like for a free particle.

1704
01:19:36,194 --> 01:19:38,110
What happens if we don't
have a free particle?

1705
01:19:38,110 --> 01:19:42,060
What happens if we have
the interacting potential?

1706
01:19:42,060 --> 01:19:46,840
So now let's have g
naught equal to 1 or 0.

1707
01:19:46,840 --> 01:19:50,560
And again, here is plus 1.

1708
01:19:50,560 --> 01:19:51,310
Here is minus 1.

1709
01:19:55,340 --> 01:19:58,570
And we're plotting
as a function of kL.

1710
01:19:58,570 --> 01:20:01,026
And now it changes,
so let g naught

1711
01:20:01,026 --> 01:20:02,990
be a small, positive number.

1712
01:20:02,990 --> 01:20:05,980
If g naught is a small, positive
number, then near k goes to 0.

1713
01:20:05,980 --> 01:20:09,720
Sine goes like it's argument,
and so that means kL over kL

1714
01:20:09,720 --> 01:20:15,730
we get cosine of 0, which
is 1, plus g naught over 2.

1715
01:20:15,730 --> 01:20:18,730
So at 0, the value of
the right-hand side

1716
01:20:18,730 --> 01:20:21,150
is greater than 1.

1717
01:20:21,150 --> 01:20:23,950
Is there an energy eigenvalue
with energy equal to 0?

1718
01:20:26,690 --> 01:20:28,630
No.

1719
01:20:28,630 --> 01:20:30,880
We have to wait until
cosine gets sufficiently

1720
01:20:30,880 --> 01:20:34,670
small and sine is increasing,
so what does this curve do?

1721
01:20:34,670 --> 01:20:36,950
It looks like this.

1722
01:20:42,920 --> 01:20:47,910
It's again periodic with
the appropriate period,

1723
01:20:47,910 --> 01:20:49,760
but the amplitude
of the deviation

1724
01:20:49,760 --> 01:20:52,109
from the free particle
is falling off

1725
01:20:52,109 --> 01:20:53,650
as we go to higher
and higher values.

1726
01:20:53,650 --> 01:20:58,200
So we've taken this
point and pushed it out.

1727
01:20:58,200 --> 01:20:59,450
So what does this tell us?

1728
01:20:59,450 --> 01:21:01,830
Well, remember that there
is an energy eigenfunction

1729
01:21:01,830 --> 01:21:07,290
with a given q and remember
this is cosine of qL,

1730
01:21:07,290 --> 01:21:08,540
there is an energy eigenvalue.

1731
01:21:08,540 --> 01:21:11,360
There exists an
energy eigenstate

1732
01:21:11,360 --> 01:21:24,630
with an e and TL eigenstate
for any E and q such

1733
01:21:24,630 --> 01:21:30,386
that there exists a solution.

1734
01:21:30,386 --> 01:21:31,510
And so what does that mean?

1735
01:21:31,510 --> 01:21:35,000
Well, is there an energy
eigenvalue with this value of E

1736
01:21:35,000 --> 01:21:37,630
or this value of k?

1737
01:21:37,630 --> 01:21:38,340
No.

1738
01:21:38,340 --> 01:21:38,839
Right?

1739
01:21:38,839 --> 01:21:41,880
Because here's the right-hand
side and the left-hand side

1740
01:21:41,880 --> 01:21:43,254
is any value in here.

1741
01:21:43,254 --> 01:21:44,920
There's a q for which
any left-hand side

1742
01:21:44,920 --> 01:21:47,367
is any value between
1 and minus 1.

1743
01:21:47,367 --> 01:21:49,950
So there's no value of q you can
pick so the left-hand side is

1744
01:21:49,950 --> 01:21:54,680
equal to the right-hand
side for the value of k.

1745
01:21:54,680 --> 01:21:56,990
And similarly here,
here, here, here, here,

1746
01:21:56,990 --> 01:21:59,286
specifically for this
value of k or this value

1747
01:21:59,286 --> 01:22:04,160
of the energy, E minimum,
there's one value of q.

1748
01:22:04,160 --> 01:22:09,910
This one. q is equal to 0 where
there is an energy eigenstate.

1749
01:22:09,910 --> 01:22:12,520
So there's a minimum
energy in the system.

1750
01:22:12,520 --> 01:22:15,900
What about for this
value of energy?

1751
01:22:15,900 --> 01:22:16,400
Yeah.

1752
01:22:16,400 --> 01:22:17,940
There's exactly
one solution there,

1753
01:22:17,940 --> 01:22:19,170
and actually if
you're careful, it's

1754
01:22:19,170 --> 01:22:20,595
2 because it's the
cosine of qL. q

1755
01:22:20,595 --> 01:22:22,150
could have been
positive or negative.

1756
01:22:25,180 --> 01:22:29,480
So we have a solution here
with this value of energy

1757
01:22:29,480 --> 01:22:32,149
and with this
value of cosine qL.

1758
01:22:32,149 --> 01:22:34,690
Any q that gives us this value
of cosine qL is going to work,

1759
01:22:34,690 --> 01:22:36,539
and that's one of two values.

1760
01:22:36,539 --> 01:22:38,080
And similarly for
each of these guys,

1761
01:22:38,080 --> 01:22:42,570
there's a state for every
value of energy in this region

1762
01:22:42,570 --> 01:22:48,840
until we get here, and when we
get here, this is E maximum.

1763
01:22:48,840 --> 01:22:51,605
If we go to any higher
energy, any higher k,

1764
01:22:51,605 --> 01:22:54,230
then there's no solution for any
value of q and any value of E.

1765
01:22:54,230 --> 01:22:56,900
So what we get are these
bands, continuous bands

1766
01:22:56,900 --> 01:23:00,550
for any energy separated
by gaps and then again

1767
01:23:00,550 --> 01:23:07,600
continuous bands for any energy
and bands for any energy.

1768
01:23:11,584 --> 01:23:13,000
So all of these
shaded in regions,

1769
01:23:13,000 --> 01:23:15,770
any energy in this
region corresponds

1770
01:23:15,770 --> 01:23:18,010
to a state and similarly here.

1771
01:23:23,230 --> 01:23:25,760
So here's the
upshot of all this.

1772
01:23:25,760 --> 01:23:29,230
We're going to study the details
of this in detail next time.

1773
01:23:29,230 --> 01:23:32,120
When we have a
periodic potential,

1774
01:23:32,120 --> 01:23:35,510
every energy
eigenfunction is extended

1775
01:23:35,510 --> 01:23:37,150
through the entire material.

1776
01:23:37,150 --> 01:23:38,970
Every energy
eigenfunction is extended

1777
01:23:38,970 --> 01:23:40,090
across the entire lattice.

1778
01:23:40,090 --> 01:23:41,900
None of them are localized.

1779
01:23:41,900 --> 01:23:43,410
And that's like a free particle.

1780
01:23:43,410 --> 01:23:46,540
However, not every energy
is an allowed energy.

1781
01:23:46,540 --> 01:23:49,150
Only some energies are allowed.

1782
01:23:49,150 --> 01:23:51,250
Some energies do not
correspond to allowed energy

1783
01:23:51,250 --> 01:23:54,670
eigenfunctions eigenvalues,
and some energies do.

1784
01:23:54,670 --> 01:23:57,100
And they come in continuous
bands of allowed energies

1785
01:23:57,100 --> 01:24:00,860
and continuous gaps of
disallowed energies.

1786
01:24:00,860 --> 01:24:03,760
And it's going to turn out to be
exactly this structure of bands

1787
01:24:03,760 --> 01:24:08,480
and gaps, or band gaps,
that is going to give us

1788
01:24:08,480 --> 01:24:10,170
the structure of
conductivity in metals

1789
01:24:10,170 --> 01:24:12,670
and explain to us why we don't
have conductivity in plastic.

1790
01:24:12,670 --> 01:24:15,280
We'll pick up on that next time.