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PROFESSOR: So today we're
going to do our last lecture

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00:00:26,690 --> 00:00:30,089
on scattering in 1D
quantum mechanics,

10
00:00:30,089 --> 00:00:32,130
and we're going to introduce
some powerful ideas,

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00:00:32,130 --> 00:00:34,129
in particular, the phase
shift and the S matrix,

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00:00:34,129 --> 00:00:35,270
and they're cool.

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00:00:35,270 --> 00:00:36,570
We'll use them for good.

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00:00:36,570 --> 00:00:40,660
Before we get started,
questions from last time?

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00:00:43,424 --> 00:00:45,340
AUDIENCE: What was the
music that was playing?

16
00:00:45,340 --> 00:00:46,180
PROFESSOR: Just now?

17
00:00:46,180 --> 00:00:48,820
It's a band called
Saint Germain.

18
00:00:48,820 --> 00:00:52,280
It's actually a guy,
but he refers to himself

19
00:00:52,280 --> 00:00:53,720
as a band called Saint Germain.

20
00:00:53,720 --> 00:00:59,200
Anyway, it's from an album
I think called "Traveler."

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00:00:59,200 --> 00:01:02,480
Physics questions?

22
00:01:02,480 --> 00:01:04,530
Anyone?

23
00:01:04,530 --> 00:01:05,817
OK, good.

24
00:01:05,817 --> 00:01:06,650
Well, bad, actually.

25
00:01:06,650 --> 00:01:08,316
I'd much prefer it
if you had questions,

26
00:01:08,316 --> 00:01:11,296
but I'll take that as a sign
of knowledge, competence,

27
00:01:11,296 --> 00:01:11,795
and mastery.

28
00:01:16,680 --> 00:01:19,250
So last time, we
talked about scattering

29
00:01:19,250 --> 00:01:21,870
past a barrier
with some height L

30
00:01:21,870 --> 00:01:24,777
and some height V, which
I think we called V0,

31
00:01:24,777 --> 00:01:26,860
and we observed a bunch
of nice things about this.

32
00:01:26,860 --> 00:01:29,570
First off, we computed the
probability of transmission

33
00:01:29,570 --> 00:01:34,400
across this barrier as a
function of the energy,

34
00:01:34,400 --> 00:01:35,900
and it had a bunch
of nice features.

35
00:01:35,900 --> 00:01:39,160
First off, it asymptoted
to 1 at high energies,

36
00:01:39,160 --> 00:01:39,910
which makes sense.

37
00:01:39,910 --> 00:01:43,700
Things shouldn't really care if
you have a little tiny barrier.

38
00:01:43,700 --> 00:01:45,215
It went to 0 at 0 energy.

39
00:01:47,894 --> 00:01:49,560
I think we should all
find that obvious.

40
00:01:49,560 --> 00:01:50,430
If you have
extremely low energy,

41
00:01:50,430 --> 00:01:52,720
you're just going to bounce
off this very hard wall.

42
00:01:52,720 --> 00:01:55,200
In between, though,
there's some structure.

43
00:01:55,200 --> 00:01:57,950
In particular, at certain
values of the energy

44
00:01:57,950 --> 00:02:00,540
corresponding to certain
values of the wave

45
00:02:00,540 --> 00:02:04,280
number in the barrier region,
at certain values of the energy,

46
00:02:04,280 --> 00:02:06,841
we saw that the
transmission was perfect.

47
00:02:06,841 --> 00:02:07,340
Oops.

48
00:02:07,340 --> 00:02:08,464
This should have been here.

49
00:02:08,464 --> 00:02:09,740
Sorry.

50
00:02:09,740 --> 00:02:18,220
The transition was perfect at
special values of the energy,

51
00:02:18,220 --> 00:02:21,980
and that the reflection
at those points was 0.

52
00:02:21,980 --> 00:02:25,260
You transmitted perfectly.

53
00:02:25,260 --> 00:02:27,100
But meanwhile, the
reflection, which

54
00:02:27,100 --> 00:02:30,360
is 1 minus the transmission,
hit maxima at special points,

55
00:02:30,360 --> 00:02:31,740
and those special
points turn out

56
00:02:31,740 --> 00:02:37,440
to be half integer shifted
away from the special points

57
00:02:37,440 --> 00:02:38,875
for perfect transmission.

58
00:02:38,875 --> 00:02:41,000
So at certain points, we
have perfect transmission.

59
00:02:41,000 --> 00:02:45,049
At certain points, we have
extremely efficient reflection.

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00:02:45,049 --> 00:02:46,590
One of our goals
today is going to be

61
00:02:46,590 --> 00:02:50,430
to understand this physics,
the physics of resonance

62
00:02:50,430 --> 00:02:53,667
in scattering off
of a potential.

63
00:02:53,667 --> 00:02:55,000
So the asymptotes we understand.

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00:02:55,000 --> 00:02:56,010
Classically, that makes sense.

65
00:02:56,010 --> 00:02:57,259
Classically, that makes sense.

66
00:02:57,259 --> 00:03:00,732
But classically, this is a
really weird structure to see.

67
00:03:00,732 --> 00:03:02,690
And notice that it's also
not something that we

68
00:03:02,690 --> 00:03:05,970
saw when we looked at
scattering off of a simple step.

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00:03:05,970 --> 00:03:08,130
When we looked at a
simple step, what we got

70
00:03:08,130 --> 00:03:13,582
was something that looked
like 0 and then this.

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00:03:13,582 --> 00:03:14,540
There was no structure.

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00:03:14,540 --> 00:03:16,740
It was just a
nice, simple curve.

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00:03:16,740 --> 00:03:18,680
So something is
happening when we

74
00:03:18,680 --> 00:03:22,352
have a barrier as
opposed to a step.

75
00:03:22,352 --> 00:03:24,060
So our job is going
to be, in some sense,

76
00:03:24,060 --> 00:03:26,590
to answer why are
they so different.

77
00:03:26,590 --> 00:03:32,060
Before we get going, questions
about the step barrier.

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00:03:32,060 --> 00:03:32,680
Yeah?

79
00:03:32,680 --> 00:03:36,408
AUDIENCE: That little line that
you drew over your other graph,

80
00:03:36,408 --> 00:03:37,810
is that to scale?

81
00:03:37,810 --> 00:03:39,310
PROFESSOR: Oh, sorry.

82
00:03:39,310 --> 00:03:40,212
This one?

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00:03:40,212 --> 00:03:43,570
AUDIENCE: No, the little line
that you drew right there.

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00:03:43,570 --> 00:03:44,530
PROFESSOR: This?

85
00:03:44,530 --> 00:03:46,030
Sorry, I should
drawn it separately.

86
00:03:46,030 --> 00:03:49,570
That was the transmission
as a function of energy

87
00:03:49,570 --> 00:03:54,310
across a single step, and
so that looked like this.

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00:03:54,310 --> 00:03:55,937
AUDIENCE: So if we
were to overlap that

89
00:03:55,937 --> 00:03:57,260
onto our resonance one--

90
00:03:57,260 --> 00:03:58,051
PROFESSOR: Exactly.

91
00:03:58,051 --> 00:04:00,150
AUDIENCE: Would
it [INAUDIBLE] to?

92
00:04:00,150 --> 00:04:01,600
PROFESSOR: Oh, sorry.

93
00:04:01,600 --> 00:04:03,590
So I just meant to
compare the two.

94
00:04:03,590 --> 00:04:05,923
I just wanted to think of
them as two different systems.

95
00:04:05,923 --> 00:04:08,820
In this system, the transmission
curve is a nice, simple curve.

96
00:04:08,820 --> 00:04:09,920
It has no structure.

97
00:04:09,920 --> 00:04:13,120
In the case of a step barrier,
we get non-zero transmission

98
00:04:13,120 --> 00:04:15,370
at low energies
instead of going to 0

99
00:04:15,370 --> 00:04:17,431
and we get this
resonant structure.

100
00:04:17,431 --> 00:04:18,514
So it's just for contrast.

101
00:04:22,540 --> 00:04:24,460
Before going into
that in detail,

102
00:04:24,460 --> 00:04:27,820
I want to do one slight
variation of this problem.

103
00:04:27,820 --> 00:04:29,680
I'm not going to do any
of the computations.

104
00:04:29,680 --> 00:04:31,221
I'm just going to
tell you how to get

105
00:04:31,221 --> 00:04:33,600
the answer from the answers
we've already computed.

106
00:04:33,600 --> 00:04:36,160
So consider a square well.

107
00:04:36,160 --> 00:04:38,015
You guys have solved
the finite square well

108
00:04:38,015 --> 00:04:39,104
on your problem sets.

109
00:04:39,104 --> 00:04:41,020
Consider a finite square
well again of width L

110
00:04:41,020 --> 00:04:42,990
and of depth now minus V0.

111
00:04:42,990 --> 00:04:46,030
It's the same thing
with V goes to minus V.

112
00:04:46,030 --> 00:04:48,030
And again, I want to
consider scattering states.

113
00:04:48,030 --> 00:04:50,480
So I want to consider states
with positive energy, energy

114
00:04:50,480 --> 00:04:51,950
above the asymptotic potential.

115
00:04:54,530 --> 00:04:57,270
Now, at this point, by now we
all know how to solve this.

116
00:04:57,270 --> 00:04:59,755
We write plain waves
here, plain waves here,

117
00:04:59,755 --> 00:05:00,880
and plain waves in between.

118
00:05:00,880 --> 00:05:03,750
We solve the potential in each
region where it's constant,

119
00:05:03,750 --> 00:05:06,392
and then we impose continuity
of the wave function

120
00:05:06,392 --> 00:05:08,600
and continuity of the
derivative of the wave function

121
00:05:08,600 --> 00:05:10,080
at the matching points.

122
00:05:10,080 --> 00:05:11,580
So we know how to
solve this problem

123
00:05:11,580 --> 00:05:13,890
and deduce the transmission
reflection coefficients.

124
00:05:13,890 --> 00:05:15,890
We've done it for this
problem, and it's exactly

125
00:05:15,890 --> 00:05:16,760
the same algebra.

126
00:05:16,760 --> 00:05:18,712
And in fact, it's so
exactly the same algebra

127
00:05:18,712 --> 00:05:20,670
that we can just take
the results from this one

128
00:05:20,670 --> 00:05:23,840
and take V to minus V and
we'll get the right answer.

129
00:05:23,840 --> 00:05:24,970
You kind of have to.

130
00:05:24,970 --> 00:05:32,360
So if we do, what we get for
the transition probability--

131
00:05:32,360 --> 00:05:35,920
and now this is the transmission
probability for a square well--

132
00:05:35,920 --> 00:05:39,650
and again, I'm going to use the
same dimensionless constants.

133
00:05:39,650 --> 00:05:45,530
g0 squared is equal to 2mL
squared over h bar squared

134
00:05:45,530 --> 00:05:46,660
times V0.

135
00:05:46,660 --> 00:05:48,180
This is 1 over an energy.

136
00:05:48,180 --> 00:05:48,930
This is an energy.

137
00:05:48,930 --> 00:05:49,960
This is dimensionless.

138
00:05:49,960 --> 00:05:55,390
And the dimensionless energy
epsilon is equal to E over V.

139
00:05:55,390 --> 00:05:57,580
To express my
transmission probability,

140
00:05:57,580 --> 00:05:59,940
life is better when
it's dimensionless.

141
00:05:59,940 --> 00:06:02,140
So T, the transmission
probability,

142
00:06:02,140 --> 00:06:04,190
again, it's one of
these horrible 1 overs.

143
00:06:04,190 --> 00:06:13,780
1 over 1 plus 1 over 4
epsilon, epsilon plus 1,

144
00:06:13,780 --> 00:06:18,240
sine squared of g0, square
root of epsilon plus 1.

145
00:06:24,840 --> 00:06:27,580
This is again the same.

146
00:06:27,580 --> 00:06:30,081
You can see what we got last
time by now taking V to minus V

147
00:06:30,081 --> 00:06:30,580
again.

148
00:06:30,580 --> 00:06:32,160
That takes epsilon
to minus epsilon.

149
00:06:32,160 --> 00:06:36,360
So we get epsilon, 1 minus
epsilon, or epsilon minus 1

150
00:06:36,360 --> 00:06:38,894
picking up the minus
sign from this epsilon,

151
00:06:38,894 --> 00:06:40,810
and here we get a 1 minus
epsilon instead of 1

152
00:06:40,810 --> 00:06:41,450
plus epsilon.

153
00:06:41,450 --> 00:06:43,390
That's precisely what
we got last time.

154
00:06:43,390 --> 00:06:46,609
So if you're feeling punchy
at home tonight, check this.

155
00:06:46,609 --> 00:06:48,400
In some sense, you're
going to re-derive it

156
00:06:48,400 --> 00:06:49,486
on the problem set.

157
00:06:52,690 --> 00:06:55,290
So it looks basically
the same as before.

158
00:06:55,290 --> 00:06:57,480
We have a sine
function downstairs,

159
00:06:57,480 --> 00:07:00,011
which again will sometimes be 0.

160
00:07:00,011 --> 00:07:01,510
The sine will
occasionally be 0 when

161
00:07:01,510 --> 00:07:03,250
its argument is
a multiple of pi.

162
00:07:03,250 --> 00:07:05,290
And when that sine
function is 0,

163
00:07:05,290 --> 00:07:07,940
then the transmission
probability is 1 over 1 plus 0.

164
00:07:07,940 --> 00:07:09,880
It's also known as 1.

165
00:07:09,880 --> 00:07:10,900
Transmission is perfect.

166
00:07:10,900 --> 00:07:12,480
So we again get a
resonant structure,

167
00:07:12,480 --> 00:07:14,900
and it's in fact exactly the
same plot, or almost exactly

168
00:07:14,900 --> 00:07:17,100
the same plot.

169
00:07:17,100 --> 00:07:19,885
I'm going to plot
transmission-- let's go ahead

170
00:07:19,885 --> 00:07:22,090
and do this--
transmission probability

171
00:07:22,090 --> 00:07:25,870
as a function, again, of
the dimensionless energy.

172
00:07:25,870 --> 00:07:37,450
And what we get is, again,
not very well drawn resonances

173
00:07:37,450 --> 00:07:39,720
where transmission
goes to 1 thanks

174
00:07:39,720 --> 00:07:43,030
to my beautiful artistic skills.

175
00:07:43,030 --> 00:07:46,690
Points where the reflection
hits a local maximum.

176
00:07:50,470 --> 00:07:52,540
But we know one more
thing about this system,

177
00:07:52,540 --> 00:07:55,930
which is that, in addition to
having scattering states whose

178
00:07:55,930 --> 00:07:58,160
transition probability are
indicated by this plot,

179
00:07:58,160 --> 00:08:02,480
we know we also have, for
negative energy, bound states.

180
00:08:02,480 --> 00:08:04,910
Unlike the step barrier,
for the step well,

181
00:08:04,910 --> 00:08:06,430
we also have bound states.

182
00:08:06,430 --> 00:08:07,960
So here's the
transmission curve,

183
00:08:07,960 --> 00:08:10,660
but I just want to remind
you that we have energies.

184
00:08:10,660 --> 00:08:14,650
At special values of energies,
we also have bound states.

185
00:08:14,650 --> 00:08:17,190
And precisely what
energies depends

186
00:08:17,190 --> 00:08:19,260
on the structure of
the well and the depth,

187
00:08:19,260 --> 00:08:24,240
but if the depth of the well
is, say, V0 so this is minus 1,

188
00:08:24,240 --> 00:08:27,230
we know that the lowest
bound state is always

189
00:08:27,230 --> 00:08:29,770
greater energy than
the bottom of the well.

190
00:08:29,770 --> 00:08:31,320
Cool?

191
00:08:31,320 --> 00:08:33,669
So epsilon, remember,
is E over V0,

192
00:08:33,669 --> 00:08:36,289
and in the square
well of depth V0,

193
00:08:36,289 --> 00:08:38,830
the lowest bound state cannot
possibly have energy lower than

194
00:08:38,830 --> 00:08:41,299
the bottom of the well, so
its epsilon must have epsilon

195
00:08:41,299 --> 00:08:44,864
greater than minus 1.

196
00:08:44,864 --> 00:08:46,280
Anyway, this is
just to remind you

197
00:08:46,280 --> 00:08:49,360
that there are bound states.

198
00:08:49,360 --> 00:08:51,030
Think of this like
a gun put down

199
00:08:51,030 --> 00:08:52,920
on a table in a play in Act One.

200
00:08:56,600 --> 00:08:58,744
It's that dramatic.

201
00:08:58,744 --> 00:08:59,660
It will show up again.

202
00:08:59,660 --> 00:09:03,677
It will come back and
play well with us.

203
00:09:03,677 --> 00:09:05,260
I want to talk about
these resonances.

204
00:09:05,260 --> 00:09:07,280
Let's understand
why they're there.

205
00:09:07,280 --> 00:09:09,030
There are a bunch of
ways of understanding

206
00:09:09,030 --> 00:09:11,770
why these resonances
are present, and let

207
00:09:11,770 --> 00:09:15,040
me just give you a couple.

208
00:09:15,040 --> 00:09:16,790
First, a heuristic
picture, and then I

209
00:09:16,790 --> 00:09:19,560
want to give you a very precise
computational picture of why

210
00:09:19,560 --> 00:09:21,300
these resonances are happening.

211
00:09:21,300 --> 00:09:23,600
So the first is imagine
we have a state where

212
00:09:23,600 --> 00:09:24,960
the transmission is perfect.

213
00:09:24,960 --> 00:09:26,980
What that tells you is
KL is a multiple of pi.

214
00:09:29,520 --> 00:09:32,060
If this is the
distance L of the well,

215
00:09:32,060 --> 00:09:35,240
in here the wave is
exactly one period

216
00:09:35,240 --> 00:09:38,960
if its KL is equal
to a multiple of pi.

217
00:09:47,150 --> 00:09:49,810
Let me simplify my life and
consider the case KL is 2 pi.

218
00:10:01,550 --> 00:10:02,710
Sorry.

219
00:10:02,710 --> 00:10:04,300
4 pi.

220
00:10:04,300 --> 00:10:05,950
I switched notations for you.

221
00:10:05,950 --> 00:10:10,120
I was using in my head, rather
in my notes, width of the well

222
00:10:10,120 --> 00:10:16,070
is 2L for reasons
you'll see on the notes

223
00:10:16,070 --> 00:10:18,380
if you look at the notes,
but let's ignore that.

224
00:10:18,380 --> 00:10:19,820
Let's focus on these guys.

225
00:10:19,820 --> 00:10:24,280
So consider the state,
configuration in energy,

226
00:10:24,280 --> 00:10:28,020
such that in the well, we have
exactly one period of the wave

227
00:10:28,020 --> 00:10:28,740
function.

228
00:10:28,740 --> 00:10:31,780
That means that the value of the
wave function at the two ends

229
00:10:31,780 --> 00:10:35,400
is the same and the
slope is the same.

230
00:10:35,400 --> 00:10:39,420
So whatever the
energy is out here,

231
00:10:39,420 --> 00:10:41,820
if it matches smoothly
and continuously

232
00:10:41,820 --> 00:10:44,485
and its derivative is continuous
here, it will match smoothly

233
00:10:44,485 --> 00:10:49,480
and continuously out here as
well with the same amplitude

234
00:10:49,480 --> 00:10:53,954
and the same period
inside and outside.

235
00:10:53,954 --> 00:10:56,120
The amplitude is the same
and the phase is the same.

236
00:10:56,120 --> 00:10:58,161
That means this wave must
have the same amplitude

237
00:10:58,161 --> 00:10:59,700
and the same period.

238
00:10:59,700 --> 00:11:01,210
It has to have the same period
because it's the same energy,

239
00:11:01,210 --> 00:11:03,460
but it must have the same
amplitude and the same slope

240
00:11:03,460 --> 00:11:05,100
at that point.

241
00:11:05,100 --> 00:11:07,450
And that means
that this wave has

242
00:11:07,450 --> 00:11:09,280
the same amplitude as this wave.

243
00:11:09,280 --> 00:11:11,449
The norm squared is the
transmission probability,

244
00:11:11,449 --> 00:11:13,990
the norm squared of this divided
by the norm squared of this.

245
00:11:13,990 --> 00:11:16,430
That means the transmission
probability has to be 1.

246
00:11:16,430 --> 00:11:18,730
What would have
happened had the system,

247
00:11:18,730 --> 00:11:21,899
instead of being perfectly
periodic inside the well--

248
00:11:21,899 --> 00:11:23,190
actually, let me leave that up.

249
00:11:28,207 --> 00:11:29,040
Let's do it as this.

250
00:11:36,501 --> 00:11:37,000
Shoot.

251
00:11:37,000 --> 00:11:39,208
I'm even getting my qualitative
wave functions wrong.

252
00:11:39,208 --> 00:11:40,970
Let's try this again.

253
00:11:40,970 --> 00:11:45,710
Start at the top,
go down, and then

254
00:11:45,710 --> 00:11:49,367
its deeper inside the well
so the amplitude out here,

255
00:11:49,367 --> 00:11:51,950
the difference in energy between
the energy and the potential,

256
00:11:51,950 --> 00:11:54,390
is less, which means
the period is longer

257
00:11:54,390 --> 00:11:56,176
and the amplitude is greater.

258
00:11:56,176 --> 00:11:58,300
The way I've drawn it, it's
got a particular value.

259
00:11:58,300 --> 00:12:07,510
It's got zero derivative
at this point,

260
00:12:07,510 --> 00:12:15,547
so it's got to-- so
there's our wave function.

261
00:12:15,547 --> 00:12:16,380
Same thing out here.

262
00:12:20,162 --> 00:12:21,870
The important thing
is the amplitude here

263
00:12:21,870 --> 00:12:23,328
has to be the same
as the amplitude

264
00:12:23,328 --> 00:12:26,080
here because the amplitude and
the amplitude and the phase

265
00:12:26,080 --> 00:12:29,760
were exactly as if there had
been no intervening region.

266
00:12:29,760 --> 00:12:31,670
Everyone agree with that?

267
00:12:31,670 --> 00:12:35,280
By contrast, if we had
looked at a situation where

268
00:12:35,280 --> 00:12:42,080
inside the well, it was not the
same amplitude, so for example,

269
00:12:42,080 --> 00:12:44,780
something like this, came
up with some slope which

270
00:12:44,780 --> 00:12:46,750
is different and a value
which is different,

271
00:12:46,750 --> 00:12:49,820
then this is going to match onto
something with the same period

272
00:12:49,820 --> 00:12:53,165
but with a different amplitude
than it would have over here.

273
00:12:56,015 --> 00:12:57,890
Same period because it's
got the same energy,

274
00:12:57,890 --> 00:12:59,680
but a different
amplitude because it

275
00:12:59,680 --> 00:13:03,810
has to match on with the
value and the derivative.

276
00:13:03,810 --> 00:13:06,850
So you can think through
this and pretty quickly

277
00:13:06,850 --> 00:13:10,790
convince yourself of the
necessity of the transmission

278
00:13:10,790 --> 00:13:14,050
aptitude being 1 if this
is exactly periodic.

279
00:13:14,050 --> 00:13:16,220
Again, if this is exactly
one period inside,

280
00:13:16,220 --> 00:13:17,770
you can just
imagine this is gone

281
00:13:17,770 --> 00:13:19,450
and you get a continuous
wave function,

282
00:13:19,450 --> 00:13:20,908
so the amplitude
and the derivative

283
00:13:20,908 --> 00:13:22,701
must be the same
on both sides as

284
00:13:22,701 --> 00:13:23,950
if there was no barrier there.

285
00:13:27,110 --> 00:13:29,660
When the wave function
doesn't have exactly one

286
00:13:29,660 --> 00:13:33,620
period inside the well,
you can't do that,

287
00:13:33,620 --> 00:13:37,120
so the amplitudes can't
be the same on both sides.

288
00:13:37,120 --> 00:13:38,996
But that's not a very
satisfying explanation.

289
00:13:38,996 --> 00:13:40,828
That's really an
explanation about solutions

290
00:13:40,828 --> 00:13:42,110
to the differential equation.

291
00:13:42,110 --> 00:13:44,609
I'm just telling you properties
of second order differential

292
00:13:44,609 --> 00:13:45,120
equations.

293
00:13:45,120 --> 00:13:46,440
Let's think of a
more physical, more

294
00:13:46,440 --> 00:13:47,731
quantum mechanical explanation.

295
00:13:47,731 --> 00:13:50,360
Why are we getting
these resonances?

296
00:13:50,360 --> 00:13:52,500
Well, I want to think
about this in the same way

297
00:13:52,500 --> 00:13:55,070
as we thought about the boxes
in the very first lecture.

298
00:13:57,740 --> 00:14:03,620
Suppose we have
this square well,

299
00:14:03,620 --> 00:14:06,889
and I know I have
some amplitude here.

300
00:14:06,889 --> 00:14:07,930
I've got a wave function.

301
00:14:07,930 --> 00:14:09,157
It's got some amplitude here.

302
00:14:09,157 --> 00:14:10,740
It's got some momentum
going this way,

303
00:14:10,740 --> 00:14:12,160
some positive momentum.

304
00:14:12,160 --> 00:14:15,770
And I want to ask, what's the
probability that I will scatter

305
00:14:15,770 --> 00:14:18,020
past the potential, let's
say to this point right here

306
00:14:18,020 --> 00:14:19,686
just on the other
side of the potential?

307
00:14:19,686 --> 00:14:22,849
What's the probability
that I will scatter across?

308
00:14:22,849 --> 00:14:24,640
Well, you say, we've
done this calculation.

309
00:14:24,640 --> 00:14:27,000
We know the probability to
transmit across this step

310
00:14:27,000 --> 00:14:27,870
potential.

311
00:14:27,870 --> 00:14:29,520
We did that last time.

312
00:14:29,520 --> 00:14:30,710
So that's T step.

313
00:14:35,410 --> 00:14:37,170
We know the probability
of scattering

314
00:14:37,170 --> 00:14:43,134
across this potential step,
and that's transmission up.

315
00:14:43,134 --> 00:14:45,550
And so the probability that
you transmit from here to here

316
00:14:45,550 --> 00:14:47,883
is the probability that you
transmit here first and then

317
00:14:47,883 --> 00:14:51,090
the probability that
you transmit here,

318
00:14:51,090 --> 00:14:52,757
the product of
the probabilities.

319
00:14:52,757 --> 00:14:53,465
Sound reasonable?

320
00:14:56,491 --> 00:14:56,990
Let's vote.

321
00:14:56,990 --> 00:14:59,700
How many people think that this
is equal to the transmission

322
00:14:59,700 --> 00:15:04,170
probability across
the potential well?

323
00:15:04,170 --> 00:15:05,047
Any votes?

324
00:15:05,047 --> 00:15:06,630
You have to vote one
way or the other.

325
00:15:06,630 --> 00:15:07,370
No, it is not.

326
00:15:07,370 --> 00:15:09,000
How many people vote?

327
00:15:09,000 --> 00:15:09,500
OK.

328
00:15:09,500 --> 00:15:11,500
Yes, it is.

329
00:15:11,500 --> 00:15:12,000
OK.

330
00:15:12,000 --> 00:15:13,950
The no's have it, and
that's not terribly

331
00:15:13,950 --> 00:15:16,660
surprising because, of course,
these have no resonance

332
00:15:16,660 --> 00:15:18,035
structure, so
where did that come

333
00:15:18,035 --> 00:15:20,080
from if it's just
that thing squared?

334
00:15:20,080 --> 00:15:23,440
So that probably can't be it,
but here's the bigger problem.

335
00:15:23,440 --> 00:15:24,910
Why is this the wrong argument?

336
00:15:27,379 --> 00:15:28,920
AUDIENCE: Because
there's reflection.

337
00:15:28,920 --> 00:15:29,640
PROFESSOR: Yeah, exactly.

338
00:15:29,640 --> 00:15:31,810
There's reflection, but that's
only one step in the answer

339
00:15:31,810 --> 00:15:33,018
or why it's the wrong answer.

340
00:15:33,018 --> 00:15:34,007
Why else?

341
00:15:34,007 --> 00:15:35,590
AUDIENCE: Your
transmission operations

342
00:15:35,590 --> 00:15:37,197
have to do it far away?

343
00:15:37,197 --> 00:15:39,530
PROFESSOR: That's, true but
I just want the probability,

344
00:15:39,530 --> 00:15:41,280
and if I got here with
positive momentum,

345
00:15:41,280 --> 00:15:43,113
I'm eventually going
to get out to infinity,

346
00:15:43,113 --> 00:15:45,080
so it's the same
probability because it's

347
00:15:45,080 --> 00:15:46,330
just an E to the ikx out here.

348
00:15:46,330 --> 00:15:48,300
The wave function
is just E to the ikx

349
00:15:48,300 --> 00:15:51,490
so the probability is
going to be the same.

350
00:15:51,490 --> 00:15:53,439
Other reasons?

351
00:15:53,439 --> 00:15:55,772
AUDIENCE: Well, the width of
the well is very important,

352
00:15:55,772 --> 00:15:57,272
but the first
argument ignores that.

353
00:15:57,272 --> 00:15:58,063
PROFESSOR: Exactly.

354
00:15:58,063 --> 00:15:59,090
That's also true.

355
00:15:59,090 --> 00:16:01,673
So far, the reasons we have are
we need the width of the well,

356
00:16:01,673 --> 00:16:03,120
doesn't appear.

357
00:16:03,120 --> 00:16:04,405
That seems probably wrong.

358
00:16:07,024 --> 00:16:09,440
The second is it's possible
that you could have reflected.

359
00:16:09,440 --> 00:16:10,898
We haven't really
incorporated that

360
00:16:10,898 --> 00:16:12,480
in any sort of elegant way.

361
00:16:12,480 --> 00:16:14,420
AUDIENCE: There are
other ways to transmit

362
00:16:14,420 --> 00:16:14,905
by reflecting twice.

363
00:16:14,905 --> 00:16:16,130
PROFESSOR: That's
absolutely true.

364
00:16:16,130 --> 00:16:17,160
There are other
ways to transmit,

365
00:16:17,160 --> 00:16:19,320
so you could transmit,
then you could reflect.

366
00:16:19,320 --> 00:16:22,581
So we could transmit then
reflect, and reflect again,

367
00:16:22,581 --> 00:16:23,330
and then transmit.

368
00:16:23,330 --> 00:16:26,190
We could transmit,
reflect, reflect, transmit.

369
00:16:26,190 --> 00:16:28,170
What else?

370
00:16:28,170 --> 00:16:30,590
Do probabilities add
in quantum mechanics?

371
00:16:33,929 --> 00:16:35,470
And when you have
products of events,

372
00:16:35,470 --> 00:16:37,470
do probabilities multiply?

373
00:16:37,470 --> 00:16:39,210
What adds in quantum mechanics?

374
00:16:39,210 --> 00:16:40,479
AUDIENCE: [INAUDIBLE].

375
00:16:40,479 --> 00:16:41,520
PROFESSOR: The amplitude.

376
00:16:41,520 --> 00:16:42,740
The wave function.

377
00:16:42,740 --> 00:16:45,520
We do not take the
product of probabilities.

378
00:16:45,520 --> 00:16:47,410
What we do is we ask,
what's the amplitude

379
00:16:47,410 --> 00:16:49,237
to get here from
there, and we take

380
00:16:49,237 --> 00:16:51,320
the amplitude norm squared
to get the probability.

381
00:16:54,290 --> 00:16:56,830
So the correct
question is what's

382
00:16:56,830 --> 00:16:59,099
the amplitude to get
from here to here?

383
00:16:59,099 --> 00:17:00,890
How does the wave
function of the amplitude

384
00:17:00,890 --> 00:17:03,610
change as you move
from here to here?

385
00:17:03,610 --> 00:17:06,534
And for that, think back
to the two slit experiment

386
00:17:06,534 --> 00:17:08,160
or think back to the boxes.

387
00:17:08,160 --> 00:17:11,134
We asked the following question.

388
00:17:11,134 --> 00:17:15,069
The amplitude that you should
transmit across this well

389
00:17:15,069 --> 00:17:18,740
has a bunch of components,
is a sum of a bunch of terms.

390
00:17:18,740 --> 00:17:22,564
You could transmit
down this well.

391
00:17:22,564 --> 00:17:24,230
Inside here, you know
your wave function

392
00:17:24,230 --> 00:17:30,910
is e to the i k prime L-- or
I think I'm calling this k2 x.

393
00:17:30,910 --> 00:17:34,220
And in moving across the
well, your wave function

394
00:17:34,220 --> 00:17:37,850
evolves by an e to the
i k2 L, and then you

395
00:17:37,850 --> 00:17:41,010
could transmit again with
some transmission amplitude.

396
00:17:41,010 --> 00:17:43,800
So this would be the
transmission down times e

397
00:17:43,800 --> 00:17:49,114
to the i k2 L times
the transmit up.

398
00:17:49,114 --> 00:17:50,780
As we saw last time,
these are the same,

399
00:17:50,780 --> 00:17:51,900
but I just want to
keep them separate

400
00:17:51,900 --> 00:17:53,520
so you know which
one talking about.

401
00:17:53,520 --> 00:17:54,330
This is a contribution.

402
00:17:54,330 --> 00:17:56,360
This is something that could
contribute to the amplitude.

403
00:17:56,360 --> 00:17:58,790
Is it the only thing that could
contribute to the amplitude?

404
00:17:58,790 --> 00:17:59,289
No.

405
00:17:59,289 --> 00:18:01,590
What else could contribute?

406
00:18:01,590 --> 00:18:02,450
Bounce, right?

407
00:18:02,450 --> 00:18:06,120
So to get from here, to here
I could transmit, evolve,

408
00:18:06,120 --> 00:18:07,030
transmit.

409
00:18:07,030 --> 00:18:11,280
I could also transmit,
evolve, reflect, evolve,

410
00:18:11,280 --> 00:18:13,840
reflect, evolve, transmit.

411
00:18:13,840 --> 00:18:18,490
So there's also a term
that's t, e to the ikL, r,

412
00:18:18,490 --> 00:18:24,030
e to the minus ikL,
r, e to the iKL.

413
00:18:26,697 --> 00:18:28,280
Sorry, e to the plus
ikL because we're

414
00:18:28,280 --> 00:18:30,509
increasing the
evolution of the phase.

415
00:18:30,509 --> 00:18:32,050
And then transmit
finally at the end.

416
00:18:36,060 --> 00:18:38,370
And these k's are
all k2, but I could

417
00:18:38,370 --> 00:18:41,242
have done that many times.

418
00:18:41,242 --> 00:18:43,450
But notice that each time
what I'm going to do is I'm

419
00:18:43,450 --> 00:18:47,110
going to transmit, reflect,
reflect, transmit or transmit,

420
00:18:47,110 --> 00:18:50,950
reflect, reflect, reflect,
reflect, transmit.

421
00:18:50,950 --> 00:18:58,890
So I'm always going to do
this some number of times.

422
00:18:58,890 --> 00:19:01,420
I do this once, I do this
twice, I do it thrice.

423
00:19:01,420 --> 00:19:03,080
This gives me a
geometric series.

424
00:19:03,080 --> 00:19:09,560
This is t e to
the i k2 L t times

425
00:19:09,560 --> 00:19:13,370
1 plus this quantity plus
this quantity squared.

426
00:19:13,370 --> 00:19:16,900
That's a geometric series,
1 over 1 plus this quantity

427
00:19:16,900 --> 00:19:19,610
squared.

428
00:19:19,610 --> 00:19:22,017
Sorry, 1 minus because
it's a geometric sum.

429
00:19:22,017 --> 00:19:23,100
And what is this quantity?

430
00:19:23,100 --> 00:19:24,900
Well, it's r squared,
and remember the r's

431
00:19:24,900 --> 00:19:26,733
in both directions are
the same, so I'm just

432
00:19:26,733 --> 00:19:30,452
going to write it as
r squared e to the 2i.

433
00:19:33,826 --> 00:19:34,686
r squared.

434
00:19:34,686 --> 00:19:36,060
It's a real number,
but I'm going

435
00:19:36,060 --> 00:19:38,259
to put the absolute
value on anyway.

436
00:19:38,259 --> 00:19:39,550
It's going to simplify my life.

437
00:19:39,550 --> 00:19:47,750
e to the 2i k2 L. So
this is our prediction

438
00:19:47,750 --> 00:19:50,002
from multiple bounces.

439
00:19:50,002 --> 00:19:51,960
What we're doing here is
we're taking seriously

440
00:19:51,960 --> 00:19:55,870
the superposition principle that
says given any process, any way

441
00:19:55,870 --> 00:19:58,440
that that process could happen,
you sum up the amplitudes

442
00:19:58,440 --> 00:20:00,106
and the probability
is the norm squared.

443
00:20:00,106 --> 00:20:03,100
If we have a source
and we have two slits

444
00:20:03,100 --> 00:20:06,226
and I ask you, what's the
probability that you land here,

445
00:20:06,226 --> 00:20:08,350
the probability is not the
sum of the probabilities

446
00:20:08,350 --> 00:20:09,690
for each individual transit.

447
00:20:09,690 --> 00:20:13,230
The probability is the
square of the amplitude where

448
00:20:13,230 --> 00:20:15,710
the amplitude is the
sum, amplitude top

449
00:20:15,710 --> 00:20:17,730
plus amplitude bottom.

450
00:20:17,730 --> 00:20:19,395
Here, there are
many, many slits.

451
00:20:19,395 --> 00:20:21,420
There are many different
ways this could happen.

452
00:20:21,420 --> 00:20:23,345
You could reflect
multiple times.

453
00:20:23,345 --> 00:20:25,809
Everyone cool with that?

454
00:20:25,809 --> 00:20:27,350
By the same token,
we could have done

455
00:20:27,350 --> 00:20:28,780
the same thing for
reflection, but let's stick

456
00:20:28,780 --> 00:20:30,154
with transmission
for the moment.

457
00:20:30,154 --> 00:20:31,930
This is what we get
for the transmission,

458
00:20:31,930 --> 00:20:34,800
and again, the transmission
amplitudes across the step,

459
00:20:34,800 --> 00:20:36,470
as we saw last
time, are the same.

460
00:20:36,470 --> 00:20:37,855
So this is, in fact, t squared.

461
00:20:42,720 --> 00:20:47,590
And this gives us a result for
transmission down the potential

462
00:20:47,590 --> 00:20:58,930
well, and if we use what the
reflection and transmission

463
00:20:58,930 --> 00:21:03,290
amplitudes were
for our step wells,

464
00:21:03,290 --> 00:21:10,040
the answer that this gives is
1 over e to the i k2 L minus 2i

465
00:21:10,040 --> 00:21:19,390
upon the transmission for
a step times sine of k2 L.

466
00:21:19,390 --> 00:21:24,799
Now, this isn't the
same as the probability

467
00:21:24,799 --> 00:21:26,340
that we derived over
here, but that's

468
00:21:26,340 --> 00:21:27,440
because this isn't
the probability.

469
00:21:27,440 --> 00:21:28,398
This was the amplitude.

470
00:21:28,398 --> 00:21:30,030
We just computed
the total amplitude.

471
00:21:30,030 --> 00:21:31,920
To get the probability,
we have to take

472
00:21:31,920 --> 00:21:34,300
the norm squared
of the amplitude.

473
00:21:34,300 --> 00:21:36,175
And when we take
the norm squared,

474
00:21:36,175 --> 00:21:43,940
what we get is 1 upon 1 plus 1
over 4 epsilon, epsilon plus 1,

475
00:21:43,940 --> 00:21:48,170
sine squared of g0
root epsilon plus 1.

476
00:21:52,892 --> 00:21:53,850
We get the same answer.

477
00:21:59,317 --> 00:22:02,310
AUDIENCE: Is that an equality?

478
00:22:02,310 --> 00:22:04,480
PROFESSOR: This is an equality.

479
00:22:04,480 --> 00:22:05,010
Oh, sorry.

480
00:22:08,530 --> 00:22:10,297
Thank you, Barton.

481
00:22:10,297 --> 00:22:11,880
We get to the
transmission probability

482
00:22:11,880 --> 00:22:13,230
when we take the norm
squared of the amplitude.

483
00:22:13,230 --> 00:22:13,860
Thank you.

484
00:22:13,860 --> 00:22:17,050
Is equal to this, which is
the same as we got before.

485
00:22:17,050 --> 00:22:17,550
Thanks.

486
00:22:21,280 --> 00:22:22,445
Yeah, please?

487
00:22:22,445 --> 00:22:25,770
AUDIENCE: Tell me why kL
equals pi doesn't work.

488
00:22:28,269 --> 00:22:29,810
PROFESSOR: kL equals
pi doesn't work.

489
00:22:29,810 --> 00:22:30,350
It does.

490
00:22:30,350 --> 00:22:31,891
You just have to be
careful what kL--

491
00:22:35,554 --> 00:22:37,012
AUDIENCE: On the
first drawing, you

492
00:22:37,012 --> 00:22:38,678
changed your kL [INAUDIBLE].

493
00:22:43,247 --> 00:22:44,580
PROFESSOR: Because I'm an idiot.

494
00:22:44,580 --> 00:22:45,810
Because I got a
factor of 2 wrong.

495
00:22:45,810 --> 00:22:46,309
Thank you.

496
00:22:52,280 --> 00:22:55,200
Thank you, Matt.

497
00:22:55,200 --> 00:22:57,750
Thank you.

498
00:22:57,750 --> 00:22:58,422
Answer analysis.

499
00:22:58,422 --> 00:22:59,380
It's a wonderful thing.

500
00:23:06,800 --> 00:23:10,140
This does something
really nice for us.

501
00:23:10,140 --> 00:23:15,230
Why are we getting a
resonance at special values

502
00:23:15,230 --> 00:23:16,630
at the energy?

503
00:23:16,630 --> 00:23:17,890
What's happening?

504
00:23:17,890 --> 00:23:22,090
Well, in this quantum
mechanical process

505
00:23:22,090 --> 00:23:25,250
of multiple interactions,
multiple scatterings,

506
00:23:25,250 --> 00:23:29,659
there are many terms in the
amplitude for transmitting.

507
00:23:29,659 --> 00:23:31,450
There are terms that
involve no reflection,

508
00:23:31,450 --> 00:23:33,425
there are terms that
involve two reflections,

509
00:23:33,425 --> 00:23:36,570
there are terms that
involve four reflections,

510
00:23:36,570 --> 00:23:41,170
and they all come with an
actual magnitude and a phase.

511
00:23:41,170 --> 00:23:47,270
And when the phase is the
same, they add constructively,

512
00:23:47,270 --> 00:23:51,200
and when the phases are not
the same, they interfere.

513
00:23:51,200 --> 00:23:53,850
And when the phases
are exactly off,

514
00:23:53,850 --> 00:23:57,500
they interfere
destructively, and that

515
00:23:57,500 --> 00:23:59,980
is why you're
getting a resonance.

516
00:23:59,980 --> 00:24:02,760
Multiple terms in
your superposition

517
00:24:02,760 --> 00:24:04,620
interfere with each
other, something

518
00:24:04,620 --> 00:24:07,060
that does not
happen classically.

519
00:24:07,060 --> 00:24:09,120
Classically, the
probabilities are products.

520
00:24:09,120 --> 00:24:11,440
Quantum mechanically,
we have superposition

521
00:24:11,440 --> 00:24:14,110
and probabilities are the
squares of the amplitude,

522
00:24:14,110 --> 00:24:16,630
and we get interference
effects in the probabilities.

523
00:24:16,630 --> 00:24:18,590
Cool?

524
00:24:18,590 --> 00:24:22,530
To me, this is a nice, glorious
version of the two slit

525
00:24:22,530 --> 00:24:25,260
experiment, and we're going
to bump up again into it later

526
00:24:25,260 --> 00:24:31,610
when we talk about the physics
of solids in the real world.

527
00:24:31,610 --> 00:24:33,870
Questions at this point?

528
00:24:33,870 --> 00:24:34,689
Yeah?

529
00:24:34,689 --> 00:24:36,730
AUDIENCE: Question on
something you said earlier.

530
00:24:36,730 --> 00:24:38,399
What is the k2 L
equals [INAUDIBLE]?

531
00:24:38,399 --> 00:24:40,190
PROFESSOR: Yeah, what's
special about that?

532
00:24:40,190 --> 00:24:46,660
What's special about that is at
this point, where kL is n pi,

533
00:24:46,660 --> 00:24:48,670
we get perfect transmission.

534
00:24:48,670 --> 00:24:52,200
When kL is equal
to n plus 1/2 pi,

535
00:24:52,200 --> 00:24:55,054
the reflection is
as good as it gets.

536
00:24:55,054 --> 00:24:56,720
What that's really
doing is it's saying,

537
00:24:56,720 --> 00:25:00,550
when is this locally largest?

538
00:25:00,550 --> 00:25:03,324
So that's the special
points when the transmission

539
00:25:03,324 --> 00:25:05,740
is as small as possible, which
means the reflection, which

540
00:25:05,740 --> 00:25:07,406
is 1 minus the
transmission probability,

541
00:25:07,406 --> 00:25:09,620
is as large as possible.

542
00:25:09,620 --> 00:25:13,659
And you can get that, again,
from these expressions.

543
00:25:13,659 --> 00:25:14,325
Other questions?

544
00:25:28,790 --> 00:25:31,780
So I want to think a little more
about these square barriers.

545
00:25:31,780 --> 00:25:34,080
And in particular, in
thinking about the square well

546
00:25:34,080 --> 00:25:36,360
barrier, what we've
been talking about

547
00:25:36,360 --> 00:25:39,650
all along are
monochromatic wave packets.

548
00:25:39,650 --> 00:25:42,180
We've been talking about
plain waves, just simply

549
00:25:42,180 --> 00:25:48,990
e to the ikx, but you
can't put a single particle

550
00:25:48,990 --> 00:25:52,630
in a state which
is a plain wave.

551
00:25:52,630 --> 00:25:53,710
It's not normalizable.

552
00:25:53,710 --> 00:25:55,490
What we really mean
at the end of the day

553
00:25:55,490 --> 00:25:57,198
when we talk about
single particles is we

554
00:25:57,198 --> 00:26:00,940
put them in some well localized
wave packet, which at time 0,

555
00:26:00,940 --> 00:26:05,200
let's say, is at position x0,
which in this case is negative,

556
00:26:05,200 --> 00:26:09,840
and which has some well
defined average momentum, k0.

557
00:26:09,840 --> 00:26:12,340
I'll say it's the expectation
value of momentum in this wave

558
00:26:12,340 --> 00:26:14,077
packet.

559
00:26:14,077 --> 00:26:15,660
And the question we
really want to ask

560
00:26:15,660 --> 00:26:18,970
when we talk about scattering
is, what happens to this beast

561
00:26:18,970 --> 00:26:23,310
as it hits the
barrier, which I'm

562
00:26:23,310 --> 00:26:24,810
going to put the
left hand side at 0

563
00:26:24,810 --> 00:26:27,170
and the right hand
side at L and let

564
00:26:27,170 --> 00:26:29,810
the depth be minus v0 again.

565
00:26:29,810 --> 00:26:32,740
What happens as this incident
wave packet hits the barrier

566
00:26:32,740 --> 00:26:33,859
and then scatters off?

567
00:26:33,859 --> 00:26:36,150
Well, we know what would
happen if it was a plain wave,

568
00:26:36,150 --> 00:26:37,520
but a plain wave
wouldn't be localized.

569
00:26:37,520 --> 00:26:39,160
So this is the
question I want to ask,

570
00:26:39,160 --> 00:26:42,530
and I want to use the
results that we already have.

571
00:26:42,530 --> 00:26:43,750
Now, here's the key thing.

572
00:26:46,490 --> 00:26:49,100
Consider to begin
with just a wave

573
00:26:49,100 --> 00:26:51,890
packet for a free
particle centered

574
00:26:51,890 --> 00:26:58,520
at x0 and with momentum k0,
just look for a free particle.

575
00:26:58,520 --> 00:27:01,350
We know how to write this.

576
00:27:01,350 --> 00:27:02,980
We can put the
system, for example,

577
00:27:02,980 --> 00:27:04,650
we can take our wave
function at time 0

578
00:27:04,650 --> 00:27:07,560
to be a Gaussian, some
normalization times

579
00:27:07,560 --> 00:27:13,270
e to the minus x minus x0
squared over 2a squared.

580
00:27:13,270 --> 00:27:15,309
And we want to give
it some momentum k0,

581
00:27:15,309 --> 00:27:17,600
and you know how to do that
after the last problem set,

582
00:27:17,600 --> 00:27:20,066
e to the i k0 x.

583
00:27:20,066 --> 00:27:22,830
Everyone cool with that?

584
00:27:22,830 --> 00:27:26,995
So there's our
initial wave function,

585
00:27:26,995 --> 00:27:28,870
and we want to know how
it evolves with time,

586
00:27:28,870 --> 00:27:30,130
and we know how to do that.

587
00:27:30,130 --> 00:27:31,930
To evolve it in time,
we first expand it

588
00:27:31,930 --> 00:27:33,820
in energy eigenstates.

589
00:27:33,820 --> 00:27:39,260
So psi of x0 is equal to--
well, the energy eigenstates

590
00:27:39,260 --> 00:27:46,150
in this case are plain waves,
dk e to the ikx over root 2 pi

591
00:27:46,150 --> 00:27:49,990
times some coefficients, f of
k, the expansion coefficients.

592
00:27:49,990 --> 00:27:52,400
But these are just
the Fourier transform

593
00:27:52,400 --> 00:27:55,550
of our initial
Gaussian wave packet,

594
00:27:55,550 --> 00:28:02,740
and we know what the form
of f of k is equal to.

595
00:28:02,740 --> 00:28:05,560
Well, it's a Gaussian
of width 1 upon alpha

596
00:28:05,560 --> 00:28:10,130
e to the minus k minus k0
because it has momentum k0,

597
00:28:10,130 --> 00:28:13,460
so it's centered around k0,
squared over 2 times a squared,

598
00:28:13,460 --> 00:28:15,310
and the a goes upstairs.

599
00:28:15,310 --> 00:28:19,700
And the position of
the initial wave packet

600
00:28:19,700 --> 00:28:21,300
is encoded in the
Fourier transform,

601
00:28:21,300 --> 00:28:23,050
and I'm going to put
a normalization here,

602
00:28:23,050 --> 00:28:26,560
which I'm not going to worry
about, with an overall phase

603
00:28:26,560 --> 00:28:31,900
e to the minus ik x0.

604
00:28:31,900 --> 00:28:34,360
So in just the same say
that adding on a phase,

605
00:28:34,360 --> 00:28:36,667
e to the i k0 x, in
the position space

606
00:28:36,667 --> 00:28:39,250
wave function tells you what the
expectation value of momentum

607
00:28:39,250 --> 00:28:41,285
is, tacking on the phase--
and we can get this

608
00:28:41,285 --> 00:28:43,368
from just Fourier transform--
tacking on the phase

609
00:28:43,368 --> 00:28:45,470
e to the ik x0 in
the Fourier transform

610
00:28:45,470 --> 00:28:48,210
tells you the center point,
the central position,

611
00:28:48,210 --> 00:28:49,292
of the wave packet.

612
00:28:49,292 --> 00:28:50,750
So this you did on
the problem set,

613
00:28:50,750 --> 00:28:52,650
and this is a trivial
momentum space

614
00:28:52,650 --> 00:28:55,170
version of the same thing.

615
00:28:55,170 --> 00:28:56,890
So here's our wave
packet expanded

616
00:28:56,890 --> 00:28:58,890
in plain waves, which
are energy eigenstates.

617
00:28:58,890 --> 00:29:01,015
And the statement that
these are energy eigenstates

618
00:29:01,015 --> 00:29:03,650
is equivalent to the statement
that under time evolution,

619
00:29:03,650 --> 00:29:06,207
they do nothing but
rotate by a phase.

620
00:29:06,207 --> 00:29:08,165
So if we want to know
what the wave function is

621
00:29:08,165 --> 00:29:12,540
as a function of time, psi of x
and t is equal to the integral,

622
00:29:12,540 --> 00:29:18,470
dk, the Fourier mode f of k.

623
00:29:18,470 --> 00:29:20,052
I'll write it out.

624
00:29:20,052 --> 00:29:22,610
Actually, I'm going to
write this out explicitly.

625
00:29:22,610 --> 00:29:28,540
f of k 1 over root
2 pi e to the ikx

626
00:29:28,540 --> 00:29:33,410
minus omega t where omega, of
course, is a function of t.

627
00:29:33,410 --> 00:29:39,750
It's a free particle, so
for our free particle,

628
00:29:39,750 --> 00:29:45,462
h bar squared k squared upon
2m is equal to h bar omega.

629
00:29:48,410 --> 00:29:50,300
We've done this before.

630
00:29:50,300 --> 00:29:52,680
But now what I want to do
is I want to take exactly

631
00:29:52,680 --> 00:29:54,910
the same system and I
want to add, at position

632
00:29:54,910 --> 00:30:00,120
zero, a well of
depth v0 and width L.

633
00:30:00,120 --> 00:30:03,553
How is our story
going to change?

634
00:30:03,553 --> 00:30:05,886
Well, we want our initial
wave packet, which is up to us

635
00:30:05,886 --> 00:30:08,330
to choose, we want our initial
wave packet to be the same.

636
00:30:08,330 --> 00:30:09,871
We want to start
with a Gaussian far,

637
00:30:09,871 --> 00:30:12,359
far, far away from the barrier.

638
00:30:12,359 --> 00:30:14,150
We want it to be well
localized in position

639
00:30:14,150 --> 00:30:17,090
and well localized in momentum
space, not perfectly localized,

640
00:30:17,090 --> 00:30:17,659
of course.

641
00:30:17,659 --> 00:30:19,700
It's a finite Gaussian to
satisfy the uncertainty

642
00:30:19,700 --> 00:30:22,520
principle, but it's
well localized.

643
00:30:22,520 --> 00:30:24,096
So how does this story change?

644
00:30:24,096 --> 00:30:25,470
Well, this doesn't
change at all.

645
00:30:25,470 --> 00:30:27,080
It's the same wave function.

646
00:30:27,080 --> 00:30:29,980
However, when we expand
in energy eigenstates,

647
00:30:29,980 --> 00:30:37,350
the energy eigenstates are
no longer simple plain waves.

648
00:30:37,350 --> 00:30:40,600
The energy eigenstates,
as we know for the system,

649
00:30:40,600 --> 00:30:43,570
take a different form.

650
00:30:43,570 --> 00:30:48,990
For the square well and positive
energy scattering states,

651
00:30:48,990 --> 00:30:51,170
the plain wave, or the
energy eigenstates,

652
00:30:51,170 --> 00:30:55,470
which I will label by k, just
because I'm going to call e

653
00:30:55,470 --> 00:31:01,121
is equal to h bar squared
k squared upon 2m--

654
00:31:01,121 --> 00:31:03,620
the energy is a constant-- which
is the k asymptotically far

655
00:31:03,620 --> 00:31:04,620
away from the potential.

656
00:31:06,850 --> 00:31:12,050
The wave function I can write
as 1 over root 2 pi times

657
00:31:12,050 --> 00:31:16,300
e to the ikx when we're
on the left hand side,

658
00:31:16,300 --> 00:31:18,920
but being on the left hand side
is equivalent to multiplying

659
00:31:18,920 --> 00:31:21,870
by a theta function of minus x.

660
00:31:21,870 --> 00:31:24,750
This is the function which is
0 when its argument is negative

661
00:31:24,750 --> 00:31:30,350
and 1 when its
argument is positive.

662
00:31:33,530 --> 00:31:36,930
Plus we have the
reflected term, which

663
00:31:36,930 --> 00:31:39,680
has an amplitude, r, which
is again a function of k.

664
00:31:39,680 --> 00:31:41,250
This is the
reflection amplitude,

665
00:31:41,250 --> 00:31:43,850
also known as c upon a.

666
00:31:43,850 --> 00:31:48,020
e to the minus ikx-- again,
on the left hand side--

667
00:31:48,020 --> 00:31:53,980
theta of minus x plus a
transition amplitude whose

668
00:31:53,980 --> 00:31:57,070
norm squared is transmission
probability e to the ikx when

669
00:31:57,070 --> 00:32:01,470
we're on the right, theta of x.

670
00:32:06,280 --> 00:32:08,930
So this is just a slightly
different notation

671
00:32:08,930 --> 00:32:11,810
than what we usually write
with left and right separated.

672
00:32:17,950 --> 00:32:20,565
So what we want to do now is
we want to decompose our wave

673
00:32:20,565 --> 00:32:22,690
function in terms of the
actual energy eigenstates.

674
00:32:29,440 --> 00:32:33,860
The way we're going to do that,
and having done that, having

675
00:32:33,860 --> 00:32:38,275
expanded our wave
function in this basis,

676
00:32:38,275 --> 00:32:40,650
we can determine the time
evolution in the following way.

677
00:32:40,650 --> 00:32:43,420
First, we expand
the wave function

678
00:32:43,420 --> 00:32:48,230
at time 0 as an
integral, dk, and I'm

679
00:32:48,230 --> 00:32:50,176
going to pull the root 2 pi out.

680
00:32:53,880 --> 00:32:55,340
And we have some
Fourier transform,

681
00:32:55,340 --> 00:32:56,890
which I'm now going
to call f tilde

682
00:32:56,890 --> 00:32:59,223
because is slightly different
than the f we used before,

683
00:32:59,223 --> 00:33:01,500
but it's what the expansion
coefficients have to be,

684
00:33:01,500 --> 00:33:11,260
f tilde of k times
this beast, phi of k.

685
00:33:24,745 --> 00:33:26,120
Let me actually
take this product

686
00:33:26,120 --> 00:33:27,995
and write it out in
terms of the three terms.

687
00:33:27,995 --> 00:33:33,680
So those three terms are going
to be f of k, again tilde,

688
00:33:33,680 --> 00:33:43,340
e to the ikx, theta of minus x,
plus f tilde r e to the minus

689
00:33:43,340 --> 00:33:53,272
ikx, theta of minus
x plus f tilde t

690
00:33:53,272 --> 00:33:57,100
e to the ikx, theta of x.

691
00:34:02,490 --> 00:34:03,590
Let's look at these terms.

692
00:34:06,510 --> 00:34:08,639
Finally, we want to look
at the time evolution,

693
00:34:08,639 --> 00:34:11,300
but we started out as a
superposition of states

694
00:34:11,300 --> 00:34:13,159
with definite
energy labeled by k,

695
00:34:13,159 --> 00:34:17,346
so we know the time evolution
is e to the ikx minus omega t, e

696
00:34:17,346 --> 00:34:22,000
to the i minus ikx plus
omega t, so minus omega t,

697
00:34:22,000 --> 00:34:24,130
and kx minus omega t.

698
00:34:24,130 --> 00:34:27,310
So we can immediately, from this
time evolving wave function,

699
00:34:27,310 --> 00:34:29,699
identify these
two terms as terms

700
00:34:29,699 --> 00:34:33,350
with a central peak
moving to the right,

701
00:34:33,350 --> 00:34:36,588
and this, central peak moving
to the left, kx plus omega t.

702
00:34:40,040 --> 00:34:41,040
Everyone cool with that?

703
00:34:44,530 --> 00:34:45,030
Questions?

704
00:34:49,326 --> 00:34:50,826
AUDIENCE: Can you
explain real quick

705
00:34:50,826 --> 00:34:53,666
one more time how
that's the [INAUDIBLE]?

706
00:34:53,666 --> 00:34:55,290
PROFESSOR: How that--
sorry, say again.

707
00:34:55,290 --> 00:34:57,166
AUDIENCE: How the top
equation [INAUDIBLE].

708
00:34:57,166 --> 00:35:00,450
PROFESSOR: How the top
equation led to this,

709
00:35:00,450 --> 00:35:01,697
or just where this came from?

710
00:35:01,697 --> 00:35:02,950
AUDIENCE: [INAUDIBLE].

711
00:35:02,950 --> 00:35:04,800
PROFESSOR: Good.

712
00:35:04,800 --> 00:35:06,710
This is just a notational thing.

713
00:35:06,710 --> 00:35:11,900
Usually when I say phi sub
k is equal to, on the left,

714
00:35:11,900 --> 00:35:16,560
e to the ikx with some
overall amplitude.

715
00:35:16,560 --> 00:35:21,870
Let's say 1 over
2 pi e to the ikx

716
00:35:21,870 --> 00:35:28,040
plus c over a e to the
minus ikx on the left,

717
00:35:28,040 --> 00:35:30,770
and because this is
the reflected wave,

718
00:35:30,770 --> 00:35:35,460
I'm just going to call this
R. And then on the right,

719
00:35:35,460 --> 00:35:38,099
we have e to the ikx.

720
00:35:38,099 --> 00:35:39,890
There's only a wave
travelling to the right

721
00:35:39,890 --> 00:35:42,162
and the coefficient is the
transmission amplitude.

722
00:35:42,162 --> 00:35:44,120
So this is what we normally
write, but then I'm

723
00:35:44,120 --> 00:35:46,760
using the so-called
theta function.

724
00:35:46,760 --> 00:35:51,500
Theta of x is defined as 0
when x is less than 0 and 1

725
00:35:51,500 --> 00:35:53,680
when x is greater than 0.

726
00:35:53,680 --> 00:35:55,520
This is a function 1.

727
00:35:55,520 --> 00:35:57,310
Using theta function
allows me to write

728
00:35:57,310 --> 00:35:59,510
this thing as a single
function without having

729
00:35:59,510 --> 00:36:02,490
to goose around
with lots of terms.

730
00:36:02,490 --> 00:36:03,440
Is that cool?

731
00:36:03,440 --> 00:36:04,035
Great.

732
00:36:04,035 --> 00:36:06,460
AUDIENCE: Are we just
thinking about one

733
00:36:06,460 --> 00:36:09,612
step here or the whole
well, because always, we

734
00:36:09,612 --> 00:36:11,320
should have something
[INAUDIBLE].

735
00:36:11,320 --> 00:36:11,900
PROFESSOR: Fantastic.

736
00:36:11,900 --> 00:36:12,640
Excellent, excellent.

737
00:36:12,640 --> 00:36:13,390
Thank you so much.

738
00:36:16,180 --> 00:36:19,130
What I want to do is I want
to think of the wave function.

739
00:36:19,130 --> 00:36:22,539
This is a good description when
the particle is far, far away,

740
00:36:22,539 --> 00:36:24,830
and this is a good description
when the particle is not

741
00:36:24,830 --> 00:36:27,240
in the potential.

742
00:36:27,240 --> 00:36:27,740
Sorry.

743
00:36:27,740 --> 00:36:28,050
Thank you.

744
00:36:28,050 --> 00:36:29,520
I totally glossed
over this step.

745
00:36:29,520 --> 00:36:31,860
I want to imagine
this as a potential

746
00:36:31,860 --> 00:36:40,422
where all of the matching is
implemented at x equals 0,

747
00:36:40,422 --> 00:36:41,880
so when I write it
in this fashion.

748
00:36:44,816 --> 00:36:46,565
Another equivalent way
to think about this

749
00:36:46,565 --> 00:36:48,648
is this is a good description
of the wave function

750
00:36:48,648 --> 00:36:49,940
when we're not inside the well.

751
00:36:49,940 --> 00:36:52,273
So for the purposes of the
rest of the analysis that I'm

752
00:36:52,273 --> 00:36:53,970
going to do, this
is exactly what

753
00:36:53,970 --> 00:36:59,010
the form of the wave function is
when we're not inside the well,

754
00:36:59,010 --> 00:37:01,060
and then let's just
not use this to ask

755
00:37:01,060 --> 00:37:03,380
about questions inside the well.

756
00:37:03,380 --> 00:37:04,849
When I write it in
this theta form,

757
00:37:04,849 --> 00:37:06,890
or for that matter, when
I write it in this form,

758
00:37:06,890 --> 00:37:08,130
this is not the form.

759
00:37:08,130 --> 00:37:13,780
What I mean is left of the
well and right of the well,

760
00:37:13,780 --> 00:37:20,180
and inside-- t e to
the ikx-- inside,

761
00:37:20,180 --> 00:37:23,789
it's doing something
else, but we

762
00:37:23,789 --> 00:37:25,330
don't want to ask
questions about it.

763
00:37:28,080 --> 00:37:31,140
That's just going
to simplify my life.

764
00:37:31,140 --> 00:37:31,810
Yeah?

765
00:37:31,810 --> 00:37:34,250
AUDIENCE: And is
the lowercase t then

766
00:37:34,250 --> 00:37:35,977
just the square
root of capital T?

767
00:37:35,977 --> 00:37:37,810
PROFESSOR: It's the
square root of capital T

768
00:37:37,810 --> 00:37:40,060
but you've got to be careful
because there can be a phase.

769
00:37:40,060 --> 00:37:42,185
Remember, this is the
amplitude, and what it really

770
00:37:42,185 --> 00:37:45,430
is is this reflection
is B over A,

771
00:37:45,430 --> 00:37:47,240
and these guys are
complex numbers.

772
00:37:47,240 --> 00:37:48,990
And it's true that B
over A norm squared

773
00:37:48,990 --> 00:37:52,660
is the transmission
probability, but B over A

774
00:37:52,660 --> 00:37:55,090
has a phase, and that
quantity I'm going to call t,

775
00:37:55,090 --> 00:37:57,500
and we'll interpret that in
more detail in a few minutes.

776
00:38:01,610 --> 00:38:03,520
So it's of this form.

777
00:38:03,520 --> 00:38:06,420
I just want to look at
each of these three terms.

778
00:38:06,420 --> 00:38:09,040
In particular, I want to
focus on them at time 0.

779
00:38:09,040 --> 00:38:11,380
So at t equals 0, what
does this look like?

780
00:38:11,380 --> 00:38:17,740
Well, that first term
is integral dk of f of k

781
00:38:17,740 --> 00:38:25,630
e to the ikx minus omega
t, theta of minus x.

782
00:38:25,630 --> 00:38:28,065
Notice that theta of minus
x is independent of k.

783
00:38:28,065 --> 00:38:29,440
It's independent
of the integral,

784
00:38:29,440 --> 00:38:33,930
so this is just a function
times theta of minus x.

785
00:38:33,930 --> 00:38:35,917
And this function
was constructed

786
00:38:35,917 --> 00:38:37,250
to give us the initial Gaussian.

787
00:38:42,210 --> 00:38:43,650
And this was at t equals 0.

788
00:38:43,650 --> 00:38:49,090
So this is just our Gaussian
at position x0 centered

789
00:38:49,090 --> 00:38:55,292
around value k0 at time
equals 0, theta of minus x.

790
00:39:01,555 --> 00:39:07,830
Sorry, at position x0.

791
00:39:07,830 --> 00:39:10,880
All this function is, this
is the Fourier transform

792
00:39:10,880 --> 00:39:13,270
of our Gaussian and we're
undoing the Fourier transform.

793
00:39:13,270 --> 00:39:16,090
This is just giving
us our Gaussian back.

794
00:39:16,090 --> 00:39:18,830
And as long as the particle
is far away from the well--

795
00:39:18,830 --> 00:39:21,992
so here's the well and
here's our wave packet--

796
00:39:21,992 --> 00:39:23,700
that theta function
is totally irrelevant

797
00:39:23,700 --> 00:39:26,272
because the Gaussian makes
it 0 away from the center

798
00:39:26,272 --> 00:39:27,230
of the Gaussian anyway.

799
00:39:34,800 --> 00:39:36,520
And so just as a
quick question, if we

800
00:39:36,520 --> 00:39:39,260
look at this is a function of
t so we put back in the minus

801
00:39:39,260 --> 00:39:49,760
omega t, so if we put back
in the time dependence,

802
00:39:49,760 --> 00:39:52,870
this is a wave that's
moving to the right,

803
00:39:52,870 --> 00:39:54,705
and to be more
precise about that,

804
00:39:54,705 --> 00:39:57,080
what does it mean to say it's
a wave moving to the right?

805
00:39:57,080 --> 00:40:02,700
Well, this is an envelope
on this set of plain waves,

806
00:40:02,700 --> 00:40:04,890
and the envelope,
by construction,

807
00:40:04,890 --> 00:40:06,770
was well localized
around position x

808
00:40:06,770 --> 00:40:09,994
but it was also well
localized in momentum,

809
00:40:09,994 --> 00:40:11,660
and in particular,
the Fourier transform

810
00:40:11,660 --> 00:40:14,130
is well localized
around the momentum k0.

811
00:40:17,930 --> 00:40:20,370
And so using the method
of stationary phase,

812
00:40:20,370 --> 00:40:27,660
or just asking, where is the
phase constant stationary,

813
00:40:27,660 --> 00:40:33,440
we get that the central
peak of the wave function

814
00:40:33,440 --> 00:40:36,890
satisfies the equation d d k0.

815
00:40:36,890 --> 00:40:38,940
If you're not familiar
with stationary phase,

816
00:40:38,940 --> 00:40:42,240
let the recitation
instructors know

817
00:40:42,240 --> 00:40:45,110
and they will
discuss it for you.

818
00:40:45,110 --> 00:40:49,890
The points of stationary
phase of this superposition

819
00:40:49,890 --> 00:40:58,690
of this wave packet lie
at the position d dk of kx

820
00:40:58,690 --> 00:41:04,930
minus omega of kt
evaluated at the peak, k0,

821
00:41:04,930 --> 00:41:06,530
of the distribution.

822
00:41:06,530 --> 00:41:10,860
But d dk of kx minus omega
t is, for the first term x,

823
00:41:10,860 --> 00:41:13,350
and for the second
term, d dk of omega.

824
00:41:13,350 --> 00:41:16,650
Well, we know that omega, we're
in the free particle regime.

825
00:41:16,650 --> 00:41:19,490
Omega, we all know what it is.

826
00:41:19,490 --> 00:41:21,952
It's h bar k squared upon 2m.

827
00:41:21,952 --> 00:41:23,660
We take the derivative
with respect to k.

828
00:41:23,660 --> 00:41:25,300
We get h bar k over m.

829
00:41:25,300 --> 00:41:32,860
The 2's cancel, so x
minus h bar k over m t.

830
00:41:32,860 --> 00:41:37,140
And a place where
this phase is 0,

831
00:41:37,140 --> 00:41:40,920
where the phase is
stationary, moves over time

832
00:41:40,920 --> 00:41:46,190
as x is equal to
h bar k over m t.

833
00:41:46,190 --> 00:41:49,880
But h bar k over m,
that's the momentum.

834
00:41:49,880 --> 00:41:52,930
p over m, that's the
classical velocity.

835
00:41:52,930 --> 00:41:58,140
So this is v0 t, the velocity
associated with that momentum.

836
00:41:58,140 --> 00:41:58,919
So this is good.

837
00:41:58,919 --> 00:42:01,210
We've done the right job of
setting up our wave packet.

838
00:42:01,210 --> 00:42:03,120
We built a Gaussian
that was far away that

839
00:42:03,120 --> 00:42:05,794
was moving in with a fixed
velocity towards the barrier,

840
00:42:05,794 --> 00:42:08,210
and now we want to know what
happens after it collides off

841
00:42:08,210 --> 00:42:09,300
the barrier.

842
00:42:09,300 --> 00:42:10,012
Cool?

843
00:42:10,012 --> 00:42:11,970
So what we really want
to ask is at late times,

844
00:42:11,970 --> 00:42:13,553
what does the wave
function look like?

845
00:42:15,794 --> 00:42:17,460
Again, we're focusing
on the first term.

846
00:42:17,460 --> 00:42:20,450
The position of this wave packet
at late times, a positive t,

847
00:42:20,450 --> 00:42:22,932
is positive, and
when it's positive,

848
00:42:22,932 --> 00:42:24,890
then this theta function
kills its contribution

849
00:42:24,890 --> 00:42:27,860
to the overall wave function.

850
00:42:27,860 --> 00:42:30,410
This theta is now theta
of minus a positive number

851
00:42:30,410 --> 00:42:34,060
and this Gaussian is gone.

852
00:42:34,060 --> 00:42:35,510
What is it replaced by?

853
00:42:35,510 --> 00:42:38,060
Well, these two terms
aren't necessarily 0.

854
00:42:38,060 --> 00:42:40,350
In particular, this one
is moving to the left.

855
00:42:40,350 --> 00:42:43,120
So as time goes forward, x
is moving further and further

856
00:42:43,120 --> 00:42:47,190
to the left, and so this theta
function starts turning on.

857
00:42:47,190 --> 00:42:49,490
And similarly, as
x goes positive,

858
00:42:49,490 --> 00:42:53,290
this theta function
starts turning on as well.

859
00:42:53,290 --> 00:42:59,959
Let's focus on this third
term, the transmitted term.

860
00:42:59,959 --> 00:43:01,250
Let's focus on this third term.

861
00:43:08,120 --> 00:43:12,400
In particular, that term looks
like integral dk over root 2

862
00:43:12,400 --> 00:43:18,130
pi, f times the transmission
amplitude times e

863
00:43:18,130 --> 00:43:21,590
to the ikx minus omega t.

864
00:43:24,540 --> 00:43:26,680
And there's an overall
theta of x outside,

865
00:43:26,680 --> 00:43:28,810
but for late times where
the center of the wave

866
00:43:28,810 --> 00:43:31,510
packet where the transmitted
wave packet should be positive,

867
00:43:31,510 --> 00:43:34,255
this theta is just going to be
1 so we can safely ignore it.

868
00:43:34,255 --> 00:43:37,740
It's just saying we're
far off to the right.

869
00:43:37,740 --> 00:43:41,700
And now I want to
do one last thing.

870
00:43:41,700 --> 00:43:44,310
This was an overall envelope.

871
00:43:44,310 --> 00:43:46,025
This t was our
scattering amplitude,

872
00:43:46,025 --> 00:43:47,900
and I want to write it
in the following form.

873
00:43:47,900 --> 00:43:53,410
I want to write it as root
T e to the minus i phi

874
00:43:53,410 --> 00:43:54,287
where phi is k.

875
00:43:54,287 --> 00:43:55,870
So what this is
saying is that indeed,

876
00:43:55,870 --> 00:43:57,703
as was pointed out
earlier, the norm squared

877
00:43:57,703 --> 00:44:01,010
of this coefficient T is the
transmission probability,

878
00:44:01,010 --> 00:44:02,540
but it has a phase.

879
00:44:02,540 --> 00:44:05,520
And what I want to know is
what does this phase mean?

880
00:44:05,520 --> 00:44:08,360
What information is
contained in this phase?

881
00:44:08,360 --> 00:44:10,110
And that's what
we're about to find.

882
00:44:10,110 --> 00:44:11,470
So let's put that in.

883
00:44:11,470 --> 00:44:18,030
Here we have root
T and minus phi

884
00:44:18,030 --> 00:44:20,952
where omega and phi
are both functions of k

885
00:44:20,952 --> 00:44:22,410
because the
transmission amplitudes

886
00:44:22,410 --> 00:44:27,260
depend on the momentum,
or the energy.

887
00:44:27,260 --> 00:44:30,840
And now I again want to know,
how does this wave packet move?

888
00:44:30,840 --> 00:44:33,930
If I look at this wave packet,
how does it move en masse?

889
00:44:33,930 --> 00:44:37,510
As a group of waves, how
does this wave packet move?

890
00:44:37,510 --> 00:44:40,130
In particular,
with what velocity?

891
00:44:40,130 --> 00:44:42,560
I'll again make an argument
by stationary phase.

892
00:44:42,560 --> 00:44:44,960
I'll look at a point
of phase equals 0

893
00:44:44,960 --> 00:44:46,565
and ask how it moves over time.

894
00:44:46,565 --> 00:44:47,940
And the point of
stationary phase

895
00:44:47,940 --> 00:44:54,860
is again given by d dk of
the phase kx minus omega

896
00:44:54,860 --> 00:45:01,220
of kt minus phi of k is equal
to 0 evaluated at k0, which

897
00:45:01,220 --> 00:45:03,200
is where our envelope
is sharply peaked.

898
00:45:07,470 --> 00:45:09,420
This expression
is equal to-- this

899
00:45:09,420 --> 00:45:12,730
is, again, x minus d omega dk.

900
00:45:12,730 --> 00:45:22,250
That's the classical velocity,
v0 t-- minus d phi dk.

901
00:45:22,250 --> 00:45:28,495
But just as a note, d phi
dk is equal to d omega dk,

902
00:45:28,495 --> 00:45:34,290
d phi d omega, but
this is equal to-- this

903
00:45:34,290 --> 00:45:36,110
is just the chain rule.
d omega dk, that's

904
00:45:36,110 --> 00:45:38,310
the classical velocity.

905
00:45:38,310 --> 00:45:45,230
d phi d omega, that's
d phi dE times h bar.

906
00:45:45,230 --> 00:45:48,510
I just multiplied by h
bar on the top and bottom.

907
00:45:48,510 --> 00:45:57,280
So this is equal to x minus
v0 t plus h bar d phi dE.

908
00:46:00,480 --> 00:46:03,100
So first off, let's just make
sure that the units make sense.

909
00:46:03,100 --> 00:46:04,997
That's a length,
that's a velocity,

910
00:46:04,997 --> 00:46:06,580
so this had better
have units of time.

911
00:46:06,580 --> 00:46:07,950
Time, that's good.

912
00:46:07,950 --> 00:46:10,890
So h bar times d phase over dE.

913
00:46:10,890 --> 00:46:13,970
Well, phase is dimensionless,
energy is units of energy,

914
00:46:13,970 --> 00:46:17,390
h is energy times time, so
this dimensionally works out.

915
00:46:17,390 --> 00:46:18,500
So this is 0.

916
00:46:18,500 --> 00:46:20,630
So the claim is the
point of stationary phase

917
00:46:20,630 --> 00:46:24,260
has this derivative equal to
0, so setting this equal to 0

918
00:46:24,260 --> 00:46:26,580
tells us that the peak
of the wave function

919
00:46:26,580 --> 00:46:29,650
moves according
to this equation.

920
00:46:33,420 --> 00:46:34,915
So this is really satisfying.

921
00:46:34,915 --> 00:46:37,290
This should be really satisfying
for a couple of reasons.

922
00:46:37,290 --> 00:46:41,440
First off, it tells you that
the peak of the transmitted wave

923
00:46:41,440 --> 00:46:44,460
packet, not just a plain wave,
the peak of the actual wave

924
00:46:44,460 --> 00:46:48,100
packet, well localized,
moves with overall velocity

925
00:46:48,100 --> 00:46:50,860
v0, the constant velocity
that we started with,

926
00:46:50,860 --> 00:46:51,530
and that's good.

927
00:46:51,530 --> 00:46:53,280
If it was moving with
some other velocity,

928
00:46:53,280 --> 00:46:54,696
we would have lost
energy somehow.

929
00:46:54,696 --> 00:46:57,600
That would be not so sensible.

930
00:46:57,600 --> 00:47:00,240
So this wave packet is moving
with an overall velocity v0.

931
00:47:00,240 --> 00:47:06,650
However, it doesn't
move along just

932
00:47:06,650 --> 00:47:09,300
as the original wave
packet's peak had.

933
00:47:09,300 --> 00:47:11,550
It moves as if shifted in time.

934
00:47:14,560 --> 00:47:16,920
So the phase, and more to
the point, the gradient

935
00:47:16,920 --> 00:47:18,420
of the phase with
respect to energy,

936
00:47:18,420 --> 00:47:22,150
the rate of change with energy
of the phase times h bar

937
00:47:22,150 --> 00:47:28,690
is giving us a shift in the time
of where the wave packet is.

938
00:47:28,690 --> 00:47:29,530
What does that mean?

939
00:47:32,730 --> 00:47:33,910
Let's be precise about this.

940
00:47:43,120 --> 00:47:45,545
Let's interpret this.

941
00:47:45,545 --> 00:47:47,420
Here's the interpretation
I want to give you.

942
00:47:51,160 --> 00:47:53,070
Consider classically the system.

943
00:47:53,070 --> 00:47:57,180
Classically, we have an
object with some energy,

944
00:47:57,180 --> 00:48:01,390
and it rolls along and it finds
a potential well, and what

945
00:48:01,390 --> 00:48:04,200
happens when it gets
into the potential well?

946
00:48:04,200 --> 00:48:05,494
It speeds up.

947
00:48:05,494 --> 00:48:06,910
because it's got
a lot more energy

948
00:48:06,910 --> 00:48:07,790
relative to the potential.

949
00:48:07,790 --> 00:48:10,290
So it goes much faster in here
and it gets to the other side

950
00:48:10,290 --> 00:48:11,460
and it slows down again.

951
00:48:11,460 --> 00:48:13,571
So if I had taken a
particle with velocity--

952
00:48:13,571 --> 00:48:15,070
let's call this
position 0 and let's

953
00:48:15,070 --> 00:48:17,680
say it gets to this
wall at time 0,

954
00:48:17,680 --> 00:48:21,390
so it's moving with
x is equal to v0 t,

955
00:48:21,390 --> 00:48:23,060
if there had been
no barrier there,

956
00:48:23,060 --> 00:48:25,018
then at subsequent times,
it would get out here

957
00:48:25,018 --> 00:48:29,160
in a time that distance
over v0, right?

958
00:48:29,160 --> 00:48:32,570
However, imagine this well
was extraordinarily deep.

959
00:48:32,570 --> 00:48:34,330
If this well were
extraordinarily deep,

960
00:48:34,330 --> 00:48:35,530
what would happen?

961
00:48:35,530 --> 00:48:38,840
Well basically, in here, its
velocity is arbitrarily large,

962
00:48:38,840 --> 00:48:42,145
and it would just immediately
jump across this well.

963
00:48:45,180 --> 00:48:48,460
This would be a perfectly
good description of the motion

964
00:48:48,460 --> 00:48:51,020
before it gets to the well,
but after it leaves the well,

965
00:48:51,020 --> 00:48:57,290
the position is going
to be v0 t plus-- well,

966
00:48:57,290 --> 00:48:58,520
what's the time shift?

967
00:48:58,520 --> 00:49:00,430
The time shift is
the time that we

968
00:49:00,430 --> 00:49:04,080
didn't need to cross this gap.

969
00:49:04,080 --> 00:49:06,540
And how much would
that time have been?

970
00:49:06,540 --> 00:49:10,220
Well, it's the distance
divided by the velocity.

971
00:49:10,220 --> 00:49:13,300
That's the time we didn't need.

972
00:49:13,300 --> 00:49:16,740
So t plus-- it moves as
if it's at a later time--

973
00:49:16,740 --> 00:49:20,430
t plus L over v0.

974
00:49:20,430 --> 00:49:22,944
Cool?

975
00:49:22,944 --> 00:49:25,360
So if we had a really deep
well and we watched a particle,

976
00:49:25,360 --> 00:49:29,120
we would watch it move x0,
x v0 t, v0 t, v0 t, v0 t

977
00:49:29,120 --> 00:49:33,550
plus L over v, v0
t plus L over v.

978
00:49:33,550 --> 00:49:36,560
So it's the time that we made
up by being deep in the well.

979
00:49:36,560 --> 00:49:40,450
So there's a classical picture
because it goes faster inside.

980
00:49:40,450 --> 00:49:44,520
So comparing these, here
we've done a calculation

981
00:49:44,520 --> 00:49:48,760
of the time shift due
to the quantum particle

982
00:49:48,760 --> 00:49:54,390
quantum mechanically
transiting the potential well.

983
00:49:54,390 --> 00:49:55,430
So let's compare these.

984
00:49:58,420 --> 00:50:02,530
This says the classical
prediction is the time it took,

985
00:50:02,530 --> 00:50:08,130
delta t, classical is
equal to L over v0,

986
00:50:08,130 --> 00:50:10,940
and the question is,
is this the same?

987
00:50:10,940 --> 00:50:13,880
One way to phrase this question
is, is it the same as h bar d

988
00:50:13,880 --> 00:50:21,350
phi dE evaluated at k0?

989
00:50:28,100 --> 00:50:35,290
From our results last time,
for the amplitude c over a,

990
00:50:35,290 --> 00:50:39,860
we get that phi is equal
to-- which is just minus

991
00:50:39,860 --> 00:50:46,430
the argument, or the phase, of
c over a, or of the transmission

992
00:50:46,430 --> 00:50:48,254
amplitude little
t-- phi turns out

993
00:50:48,254 --> 00:50:58,670
to be equal to k2 L minus arctan
of k1 squared plus k2 squared

994
00:50:58,670 --> 00:51:06,290
over 2 k1 k2, tan of
k2 L. I look at this

995
00:51:06,290 --> 00:51:07,870
and it doesn't tell
me all that much.

996
00:51:07,870 --> 00:51:10,444
It's a little bit bewildering,
so let's unpack this.

997
00:51:10,444 --> 00:51:11,860
What we really
want to know is, is

998
00:51:11,860 --> 00:51:13,318
this close to the
classical result?

999
00:51:20,050 --> 00:51:22,120
Here's a quick way to check.

1000
00:51:22,120 --> 00:51:24,630
We know this expression is
going to simplify near resonance

1001
00:51:24,630 --> 00:51:27,100
where the sine vanishes,
so let's look, just

1002
00:51:27,100 --> 00:51:31,130
for simplicity,
near the resonance.

1003
00:51:31,130 --> 00:51:34,380
And in particular, let's
look near the resonance k2 L

1004
00:51:34,380 --> 00:51:36,974
is equal to n pi.

1005
00:51:36,974 --> 00:51:39,140
Then it turns out that a
quick calculation gives you

1006
00:51:39,140 --> 00:51:43,840
that h bar d phi dE
at this value of k,

1007
00:51:43,840 --> 00:51:50,540
at that value of the energy,
goes as L over 2 v0 times

1008
00:51:50,540 --> 00:51:56,840
1 plus the energy over
the depth of the well, v0.

1009
00:51:56,840 --> 00:52:00,070
Now remember, the classical
approximation was L over v0.

1010
00:52:00,070 --> 00:52:01,330
We just did this very quickly.

1011
00:52:01,330 --> 00:52:03,310
We did it assuming an
arbitrarily deep well.

1012
00:52:03,310 --> 00:52:05,820
So v0 is arbitrarily
larger in magnitude than E,

1013
00:52:05,820 --> 00:52:07,222
so this term is negligible.

1014
00:52:07,222 --> 00:52:09,180
Where we should compare
that very simple, naive

1015
00:52:09,180 --> 00:52:12,510
classical result is
here, L over 2 v0.

1016
00:52:12,510 --> 00:52:15,180
And what we see is that the
quantum mechanical result

1017
00:52:15,180 --> 00:52:18,030
gives a time shift which
is down by a factor of 2.

1018
00:52:21,920 --> 00:52:22,880
So what's going on?

1019
00:52:22,880 --> 00:52:26,950
Well, apparently, the
things slow down inside.

1020
00:52:26,950 --> 00:52:29,630
The time that it
took us to cross

1021
00:52:29,630 --> 00:52:31,810
was greater than you
would have naively guessed

1022
00:52:31,810 --> 00:52:34,530
by making it arbitrarily deep.

1023
00:52:34,530 --> 00:52:37,590
And we can make that
a little more sharp

1024
00:52:37,590 --> 00:52:46,380
by plotting, as a function of
E over v0, the actual phase

1025
00:52:46,380 --> 00:52:47,984
shift.

1026
00:52:47,984 --> 00:52:50,150
If do a better job than
saying it's infinitely deep,

1027
00:52:50,150 --> 00:52:55,720
the classical prediction
looks something like this,

1028
00:52:55,720 --> 00:53:01,109
and this is for delta t,
the time shift, classical.

1029
00:53:01,109 --> 00:53:03,400
When you look at the correct
quantum mechanical result,

1030
00:53:03,400 --> 00:53:19,920
here's what you find,
where the difference is

1031
00:53:19,920 --> 00:53:24,570
a factor of 2, 1/2 the
height down, and again,

1032
00:53:24,570 --> 00:53:26,450
1/2 the height down.

1033
00:53:26,450 --> 00:53:28,810
So this is that factor
of 2 downstairs.

1034
00:53:28,810 --> 00:53:32,442
So the wave packet goes
actually a little bit faster

1035
00:53:32,442 --> 00:53:33,900
than the classical
prediction would

1036
00:53:33,900 --> 00:53:36,440
guess except near
resonance, and these

1037
00:53:36,440 --> 00:53:38,815
are at the resonant
values of the momentum.

1038
00:53:38,815 --> 00:53:40,440
At the resonant values
of the momentum,

1039
00:53:40,440 --> 00:53:41,971
it takes much longer
to get across.

1040
00:53:41,971 --> 00:53:43,470
Instead of going a
little bit faster

1041
00:53:43,470 --> 00:53:45,430
than the classical
result, it goes a factor

1042
00:53:45,430 --> 00:53:48,570
of 2 slower than the
classical result.

1043
00:53:48,570 --> 00:53:50,050
And so now I ask
you the question,

1044
00:53:50,050 --> 00:53:51,450
why is it going more slowly?

1045
00:53:51,450 --> 00:53:55,190
Why does it spend so much more
time inside the well quantum

1046
00:53:55,190 --> 00:53:57,810
mechanically than it
would have classically?

1047
00:53:57,810 --> 00:54:00,830
Why is the particle effectively
taking so much longer

1048
00:54:00,830 --> 00:54:03,990
to transit the well
near resonance?

1049
00:54:03,990 --> 00:54:06,449
AUDIENCE: Because it can
reflect and it can keep going

1050
00:54:06,449 --> 00:54:08,490
and a classical particle
is not going to do that.

1051
00:54:08,490 --> 00:54:09,280
PROFESSOR: Yeah, exactly.

1052
00:54:09,280 --> 00:54:10,950
So the classical particle
just goes across.

1053
00:54:10,950 --> 00:54:13,160
The quantum mechanical
particle has a superposition

1054
00:54:13,160 --> 00:54:14,534
of contributions
to its amplitude

1055
00:54:14,534 --> 00:54:17,792
where it transits-- transit,
bounce, bounce, transit,

1056
00:54:17,792 --> 00:54:20,560
transit, bounce, bounce,
bounce, bounce, transit.

1057
00:54:20,560 --> 00:54:22,250
And now you can
ask, how much time

1058
00:54:22,250 --> 00:54:25,540
was spent by each of those
imaginary particles imaginarily

1059
00:54:25,540 --> 00:54:27,005
moving across?

1060
00:54:27,005 --> 00:54:29,380
And if you're careful about
how you set up that question,

1061
00:54:29,380 --> 00:54:33,740
you can recover this factor of
2, which is kind of beautiful.

1062
00:54:33,740 --> 00:54:36,640
But the important thing here is
when you're hitting resonance,

1063
00:54:36,640 --> 00:54:39,780
the multiple scattering
processes are important.

1064
00:54:39,780 --> 00:54:41,040
They're not canceling out.

1065
00:54:41,040 --> 00:54:42,270
They're not at random phase.

1066
00:54:42,270 --> 00:54:44,030
They're not interfering
destructively with each other.

1067
00:54:44,030 --> 00:54:45,488
They're interfering
constructively,

1068
00:54:45,488 --> 00:54:47,700
and you get perfect
transmission precisely

1069
00:54:47,700 --> 00:54:51,160
because of the
constructive interference

1070
00:54:51,160 --> 00:54:53,050
of an infinite number
of contributions

1071
00:54:53,050 --> 00:54:54,882
to the quantum
mechanical amplitude.

1072
00:54:54,882 --> 00:54:56,840
And this is, again, we're
seeing the same thing

1073
00:54:56,840 --> 00:54:59,550
in this annoying slow down.

1074
00:54:59,550 --> 00:55:01,150
This tells us another
thing, though,

1075
00:55:01,150 --> 00:55:04,110
which is that the
scattering phase, the phase

1076
00:55:04,110 --> 00:55:06,650
in the transmission
amplitude, contains

1077
00:55:06,650 --> 00:55:08,400
an awful lot of the
physics of the system.

1078
00:55:08,400 --> 00:55:12,350
It's telling us about how long
it takes for the wave packet

1079
00:55:12,350 --> 00:55:14,680
to transit across the
potential, effectively.

1080
00:55:14,680 --> 00:55:15,347
Yeah?

1081
00:55:15,347 --> 00:55:16,600
AUDIENCE: What's the
vertical axis being used for?

1082
00:55:16,600 --> 00:55:17,308
PROFESSOR: Sorry.

1083
00:55:17,308 --> 00:55:21,620
The vertical axis
here is the shift

1084
00:55:21,620 --> 00:55:24,300
in the time due to the fact
that it went across this well

1085
00:55:24,300 --> 00:55:26,700
and went a little
bit faster inside.

1086
00:55:26,700 --> 00:55:30,580
So empirically, what it means is
when you get out very far away

1087
00:55:30,580 --> 00:55:33,070
and you watch the motion
of the wave packet,

1088
00:55:33,070 --> 00:55:35,794
and you ask, how long
has it been since it got

1089
00:55:35,794 --> 00:55:37,210
to the barrier in
the first place,

1090
00:55:37,210 --> 00:55:38,640
it took less time than
you would have guessed

1091
00:55:38,640 --> 00:55:41,400
by knowing that its velocity is
v0, and the amount of time less

1092
00:55:41,400 --> 00:55:42,757
is this much time.

1093
00:55:42,757 --> 00:55:43,840
That answer your question?

1094
00:55:43,840 --> 00:55:44,720
Good.

1095
00:55:44,720 --> 00:55:45,972
Yeah?

1096
00:55:45,972 --> 00:55:48,710
AUDIENCE: Why is each
of the amplitudes 1/2?

1097
00:55:48,710 --> 00:55:50,880
Why is the first one, and
it bounces around twice.

1098
00:55:54,847 --> 00:55:56,930
PROFESSOR: When we compare
the classical amplitude

1099
00:55:56,930 --> 00:56:00,140
and the limit that
v0 goes to infinity?

1100
00:56:00,140 --> 00:56:03,660
The comparison is
just a factor of 2.

1101
00:56:03,660 --> 00:56:07,150
It's more complicated out here.

1102
00:56:07,150 --> 00:56:10,428
In the limit that v0 is
large, this is exactly 1/2.

1103
00:56:10,428 --> 00:56:11,344
AUDIENCE: [INAUDIBLE].

1104
00:56:13,779 --> 00:56:15,320
PROFESSOR: The
resonances are leading

1105
00:56:15,320 --> 00:56:16,486
to this extra factor of 1/2.

1106
00:56:18,907 --> 00:56:20,490
I have to say I don't
remember exactly

1107
00:56:20,490 --> 00:56:24,020
whether, as you include
the sub-leading terms of 1

1108
00:56:24,020 --> 00:56:26,590
over v0, whether it stays
1/2 or whether it doesn't,

1109
00:56:26,590 --> 00:56:28,480
but in the limit
that v0 is large,

1110
00:56:28,480 --> 00:56:30,500
it remains either close
to 1/2 or exactly 1/2,

1111
00:56:30,500 --> 00:56:31,750
I just don't remember exactly.

1112
00:56:31,750 --> 00:56:34,000
The important thing is
that there's a sharp dip.

1113
00:56:34,000 --> 00:56:38,680
It takes much longer to transit,
and so you get less bonus time,

1114
00:56:38,680 --> 00:56:40,240
as it were.

1115
00:56:40,240 --> 00:56:43,370
You've gained less time in
the quantum mechanical model

1116
00:56:43,370 --> 00:56:44,822
than the classical model.

1117
00:56:44,822 --> 00:56:47,280
And when you do the experiment,
you get the quantum result.

1118
00:56:50,940 --> 00:56:52,580
That's the crucial point.

1119
00:56:52,580 --> 00:56:53,260
Other questions?

1120
00:56:56,680 --> 00:56:59,290
So the phase contains an
awful lot of the physics.

1121
00:57:04,240 --> 00:57:08,440
So I want to generalize
this whole story

1122
00:57:08,440 --> 00:57:11,920
in a very particular way,
this way of reorganizing

1123
00:57:11,920 --> 00:57:13,140
the scattering in 1D.

1124
00:57:13,140 --> 00:57:14,060
What we're doing
right now is we're

1125
00:57:14,060 --> 00:57:16,060
studying scattering
problems in one dimension,

1126
00:57:16,060 --> 00:57:17,900
but we live in three dimensions.

1127
00:57:17,900 --> 00:57:19,983
The story is going to be
more complicated in three

1128
00:57:19,983 --> 00:57:20,483
dimensions.

1129
00:57:20,483 --> 00:57:22,649
It's going to be more
complicated in two dimensions,

1130
00:57:22,649 --> 00:57:24,220
but the basic ideas
are all the same.

1131
00:57:24,220 --> 00:57:26,219
It's just the details are
going to be different.

1132
00:57:26,219 --> 00:57:29,420
And one thing that turns out
to be very useful in organizing

1133
00:57:29,420 --> 00:57:31,449
scattering, both
in one dimension

1134
00:57:31,449 --> 00:57:32,990
and in three
dimensions, is something

1135
00:57:32,990 --> 00:57:34,198
called the scattering matrix.

1136
00:57:34,198 --> 00:57:35,870
I'm going to talk
about that now.

1137
00:57:35,870 --> 00:57:39,380
In three dimensions,
it's essential,

1138
00:57:39,380 --> 00:57:41,720
but even in one dimension,
where it's usually not used,

1139
00:57:41,720 --> 00:57:43,360
it's a very powerful
way to organize

1140
00:57:43,360 --> 00:57:49,350
our knowledge of the system as
encoded by the scattering data.

1141
00:57:49,350 --> 00:57:51,000
So here's the basic idea.

1142
00:57:51,000 --> 00:57:52,800
As we discussed
before, what we really

1143
00:57:52,800 --> 00:57:55,420
want to do in the ideal world
is take some unknown potential

1144
00:57:55,420 --> 00:57:58,510
in some bounded
region, some region,

1145
00:57:58,510 --> 00:58:01,660
and outside, we have the
potential is constant.

1146
00:58:01,660 --> 00:58:03,300
[INAUDIBLE] my bad
artistic skills.

1147
00:58:03,300 --> 00:58:06,740
So potential is
constant out here.

1148
00:58:06,740 --> 00:58:08,990
And the potential could be
some horrible thing in here

1149
00:58:08,990 --> 00:58:10,490
that we don't happen
to know, and we

1150
00:58:10,490 --> 00:58:12,530
want to read off of
the scattering process,

1151
00:58:12,530 --> 00:58:15,500
we want to be able to deduce
something about the potential.

1152
00:58:15,500 --> 00:58:26,190
So for example, we
can deduce the energy

1153
00:58:26,190 --> 00:58:30,920
by looking at the
position of the barriers,

1154
00:58:30,920 --> 00:58:33,380
and we can disentangle
the position of the energy

1155
00:58:33,380 --> 00:58:37,350
and the depth and the width
by looking at the phase shift,

1156
00:58:37,350 --> 00:58:38,680
by looking at the time delay.

1157
00:58:38,680 --> 00:58:40,940
So we can deduce all the
parameters of our potential

1158
00:58:40,940 --> 00:58:43,830
by looking at the resonance
points and the phase shift.

1159
00:58:43,830 --> 00:58:46,665
I want to do this more
generally for general potential.

1160
00:58:46,665 --> 00:58:48,040
And to set that
up, we need to be

1161
00:58:48,040 --> 00:58:50,150
a bit more general
than we've been.

1162
00:58:54,060 --> 00:58:56,730
So in general, if we
solve this potential,

1163
00:58:56,730 --> 00:59:02,270
as we talked about before, we
have A and B, e to the ikx,

1164
00:59:02,270 --> 00:59:05,850
e to the minus ikx out
here, and out here, we

1165
00:59:05,850 --> 00:59:11,278
have C e to the ikx and
D e to the minus ikx.

1166
00:59:14,235 --> 00:59:16,110
And I'm not going to
ask what happens inside.

1167
00:59:19,377 --> 00:59:20,960
Now we can do, as
discussed, two kinds

1168
00:59:20,960 --> 00:59:22,043
of scattering experiments.

1169
00:59:22,043 --> 00:59:24,220
We can send things in from
the left, in which case

1170
00:59:24,220 --> 00:59:27,100
A is nonzero, and then things
can either transmit or reflect,

1171
00:59:27,100 --> 00:59:29,755
but nothing's going to come
in from infinity, so D is 0.

1172
00:59:29,755 --> 00:59:31,130
Or we can do the
same in reverse.

1173
00:59:31,130 --> 00:59:33,254
Send things in from here,
and that corresponds to A

1174
00:59:33,254 --> 00:59:36,630
is 0, nothing coming in
this way but d0 and 0.

1175
00:59:36,630 --> 00:59:39,900
And more generally, we can
ask the following question.

1176
00:59:39,900 --> 00:59:42,950
Suppose I send some amount
of stuff in from the left

1177
00:59:42,950 --> 00:59:46,010
and I send some amount of stuff
in from the right, D. Then

1178
00:59:46,010 --> 00:59:48,650
that will tell me how much stuff
will be going out to the right

1179
00:59:48,650 --> 00:59:52,150
and how much stuff will
be going out to the left.

1180
00:59:52,150 --> 00:59:54,040
If you tell me how
much is coming in,

1181
00:59:54,040 --> 00:59:59,440
I will tell you how much
is coming out, B, C.

1182
00:59:59,440 --> 01:00:02,630
So if you could
solve this problem,

1183
01:00:02,630 --> 01:00:05,480
the answer is just some
paralinear relations

1184
01:00:05,480 --> 01:00:09,080
between these, and we can write
this as a matrix, which I will

1185
01:00:09,080 --> 01:00:15,030
S11, S12, S21, S22.

1186
01:00:15,030 --> 01:00:17,650
What this matrix is doing is
it takes the amplitude you're

1187
01:00:17,650 --> 01:00:19,120
sending in from
the left to right

1188
01:00:19,120 --> 01:00:21,344
and tells you the amplitude
coming out to the left

1189
01:00:21,344 --> 01:00:22,010
or to the right.

1190
01:00:22,010 --> 01:00:22,833
Yes?

1191
01:00:22,833 --> 01:00:25,870
AUDIENCE: How do we know
this relation is linear?

1192
01:00:25,870 --> 01:00:28,850
PROFESSOR: If you double the
amount of stuff coming in,

1193
01:00:28,850 --> 01:00:31,030
then you must double the
amount of stuff going out

1194
01:00:31,030 --> 01:00:32,420
or probability is not conserved.

1195
01:00:35,070 --> 01:00:37,120
Also, we've derived
these relations.

1196
01:00:37,120 --> 01:00:38,730
You know how the relations work.

1197
01:00:38,730 --> 01:00:42,769
The relations work by satisfying
a series of linear equations

1198
01:00:42,769 --> 01:00:44,310
between the various
coefficients such

1199
01:00:44,310 --> 01:00:46,855
that you have continuity
and differentiability at all

1200
01:00:46,855 --> 01:00:48,634
the matching points.

1201
01:00:48,634 --> 01:00:50,050
But the crucial
thing of linearity

1202
01:00:50,050 --> 01:00:54,034
is probability is conserved
and time evolution is linear.

1203
01:00:54,034 --> 01:00:54,700
Other questions?

1204
01:00:58,540 --> 01:01:01,100
So this is just some stupid
matrix, and we call it,

1205
01:01:01,100 --> 01:01:05,505
not surprisingly, the S
matrix in all of its majesty.

1206
01:01:09,510 --> 01:01:10,810
The basic idea is this.

1207
01:01:10,810 --> 01:01:13,685
For scattering problems, if
someone tells you the S matrix,

1208
01:01:13,685 --> 01:01:15,310
and in particular,
if someone tells you

1209
01:01:15,310 --> 01:01:18,892
how all the coefficients of
the S matrix vary with energy,

1210
01:01:18,892 --> 01:01:21,100
then you've completely solved
any scattering problem.

1211
01:01:21,100 --> 01:01:22,950
You tell me what A and D are.

1212
01:01:22,950 --> 01:01:23,450
Great.

1213
01:01:23,450 --> 01:01:26,071
I'll tell you exactly what
B and C are mode by mode,

1214
01:01:26,071 --> 01:01:27,570
and I can do this
for superposition.

1215
01:01:27,570 --> 01:01:30,377
So this allows you to completely
solve any scattering problem

1216
01:01:30,377 --> 01:01:31,960
in quantum mechanics
once you know it.

1217
01:01:31,960 --> 01:01:35,176
So it suffices to know S to
solve all scattering problems.

1218
01:01:38,270 --> 01:01:40,520
I want to now spend just a
little bit of time thinking

1219
01:01:40,520 --> 01:01:43,310
about what properties the
S matrix and its components

1220
01:01:43,310 --> 01:01:45,580
must satisfy.

1221
01:01:45,580 --> 01:01:47,210
What properties
must the S matrix

1222
01:01:47,210 --> 01:01:51,400
satisfy in order to
jibe well with the rest

1223
01:01:51,400 --> 01:01:54,610
of the rules of
quantum mechanics?

1224
01:01:54,610 --> 01:01:56,760
I'm not going to study
any particular system.

1225
01:01:56,760 --> 01:01:59,480
I just want to ask
general questions.

1226
01:01:59,480 --> 01:02:01,510
So the first thing
that must be true

1227
01:02:01,510 --> 01:02:04,285
is that stuff doesn't disappear.

1228
01:02:11,950 --> 01:02:16,120
Disapparate is probably
the appropriate.

1229
01:02:16,120 --> 01:02:18,870
Stuff doesn't leak
out of the world,

1230
01:02:18,870 --> 01:02:20,490
so whatever goes
in must come out.

1231
01:02:20,490 --> 01:02:23,760
What that means is norm of
A squared plus norm of D

1232
01:02:23,760 --> 01:02:26,970
squared, which is the
probability density in

1233
01:02:26,970 --> 01:02:29,030
and probability
density out, must

1234
01:02:29,030 --> 01:02:32,000
be equal to B squared
plus C squared.

1235
01:02:37,160 --> 01:02:39,422
Everyone agree with that?

1236
01:02:39,422 --> 01:02:42,340
AUDIENCE: Why can't stuff
stay in the potential?

1237
01:02:42,340 --> 01:02:44,960
PROFESSOR: Why can't stuff
stay in the potential?

1238
01:02:44,960 --> 01:02:46,350
That's a good.

1239
01:03:00,750 --> 01:03:03,260
So if we're looking at
fixed energy eigenstates,

1240
01:03:03,260 --> 01:03:05,070
we know that we're in
a stationary state,

1241
01:03:05,070 --> 01:03:09,200
so whatever the amplitude
going into the middle is,

1242
01:03:09,200 --> 01:03:12,187
it must also be coming
out of the middle.

1243
01:03:12,187 --> 01:03:13,770
There's another way
to say this, which

1244
01:03:13,770 --> 01:03:16,520
is let's think about it not
in terms of individual energy

1245
01:03:16,520 --> 01:03:17,020
eigenstates.

1246
01:03:17,020 --> 01:03:18,936
Let's think about it in
terms of wave packets.

1247
01:03:18,936 --> 01:03:20,590
So if we take a
wave packet of stuff

1248
01:03:20,590 --> 01:03:23,400
and we send in that wave packet,
it has some momentum, right?

1249
01:03:26,280 --> 01:03:28,190
This is going to get
delicate and technical.

1250
01:03:28,190 --> 01:03:31,040
Let me just stick with
the first statement, which

1251
01:03:31,040 --> 01:03:33,350
is that if stuff
is going in, then

1252
01:03:33,350 --> 01:03:36,140
it has to also be
coming out by the fact

1253
01:03:36,140 --> 01:03:38,150
that this is an
energy eigenstate.

1254
01:03:38,150 --> 01:03:39,650
The overall probability
distribution

1255
01:03:39,650 --> 01:03:40,810
is not changing in time.

1256
01:03:43,324 --> 01:03:44,990
If stuff went in and
it didn't come out,

1257
01:03:44,990 --> 01:03:45,770
that would mean
it's staying there.

1258
01:03:45,770 --> 01:03:46,700
That would mean
that the probability

1259
01:03:46,700 --> 01:03:48,060
density is changing in time.

1260
01:03:48,060 --> 01:03:50,060
That's not what happens
in an energy eigenstate.

1261
01:03:50,060 --> 01:03:53,631
The probability distribution
is time independent.

1262
01:03:53,631 --> 01:03:54,630
Everyone cool with that?

1263
01:04:02,380 --> 01:04:04,570
So let's think, though,
about what this is.

1264
01:04:04,570 --> 01:04:07,080
I can write this in
the following nice way.

1265
01:04:07,080 --> 01:04:12,440
I can write this as A
complex conjugate D complex

1266
01:04:12,440 --> 01:04:16,522
conjugate, AD.

1267
01:04:16,522 --> 01:04:17,980
And on the right
hand side, this is

1268
01:04:17,980 --> 01:04:23,525
equal to B complex conjugate,
C complex conjugate, BC.

1269
01:04:23,525 --> 01:04:25,400
I have done nothing
other than write this out

1270
01:04:25,400 --> 01:04:27,440
in some suggested form.

1271
01:04:27,440 --> 01:04:31,760
But B and C are equal
to the S matrix,

1272
01:04:31,760 --> 01:04:36,380
so BC is the S matrix times
A, and B complex conjugate,

1273
01:04:36,380 --> 01:04:39,550
C complex conjugate is the
transposed complex conjugate.

1274
01:04:39,550 --> 01:04:44,970
So this is equal to A complex
conjugate, D complex conjugate,

1275
01:04:44,970 --> 01:04:48,620
S transpose complex conjugate,
also known as adjoint,

1276
01:04:48,620 --> 01:04:51,370
and this is S on AD.

1277
01:04:55,010 --> 01:04:57,100
Yeah?

1278
01:04:57,100 --> 01:05:02,360
But this has to be equal
to this for any A and D.

1279
01:05:02,360 --> 01:05:06,750
So what must be true of
S dagger S as a matrix?

1280
01:05:06,750 --> 01:05:11,230
It's got to be the identity
as a matrix in order for this

1281
01:05:11,230 --> 01:05:14,450
to be true for all A and D. Ah.

1282
01:05:14,450 --> 01:05:15,170
That's cool.

1283
01:05:15,170 --> 01:05:16,500
Stuff doesn't disappear.

1284
01:05:16,500 --> 01:05:17,970
S is a unitary matrix.

1285
01:05:23,640 --> 01:05:24,980
So S is a unitary matrix.

1286
01:05:31,090 --> 01:05:33,290
Its inverse is its adjoint.

1287
01:05:43,110 --> 01:05:47,530
You'll study this in a little
more detail on the problem set.

1288
01:05:47,530 --> 01:05:49,190
You studied the
definition of unitary

1289
01:05:49,190 --> 01:05:52,316
on the last problem set.

1290
01:05:52,316 --> 01:05:53,690
So that's the
first thing about S

1291
01:05:53,690 --> 01:05:55,500
and it turns out to
be completely general.

1292
01:05:55,500 --> 01:05:57,990
Anytime, whether you're in one
dimension or two dimensions

1293
01:05:57,990 --> 01:06:01,550
or three, if you send stuff
in, it should not get stuck.

1294
01:06:01,550 --> 01:06:03,360
It should come out.

1295
01:06:03,360 --> 01:06:06,064
And when it does come
out, the statement

1296
01:06:06,064 --> 01:06:07,730
that it comes out for
energy eigenstates

1297
01:06:07,730 --> 01:06:10,850
is the statement that
S is a unitary matrix.

1298
01:06:10,850 --> 01:06:11,876
Questions on that.

1299
01:06:17,840 --> 01:06:29,760
So a consequence of that is
that the eigenvalues of S

1300
01:06:29,760 --> 01:06:32,510
are phases, pure phases.

1301
01:06:35,020 --> 01:06:38,800
So I can write S--
I'll write them

1302
01:06:38,800 --> 01:06:43,480
as S1 is equal to
e to the i of phi 1

1303
01:06:43,480 --> 01:06:47,824
and S2 is equal
to the i of phi 2.

1304
01:06:53,090 --> 01:06:56,970
So the statement that
S is a unitary matrix

1305
01:06:56,970 --> 01:07:00,362
leads to constraints
on the coefficients,

1306
01:07:00,362 --> 01:07:02,570
and you're going to derive
these on your problem set.

1307
01:07:02,570 --> 01:07:03,910
I'm just going to list them now.

1308
01:07:03,910 --> 01:07:07,960
The first is that
the magnitude of S11

1309
01:07:07,960 --> 01:07:11,840
is equal to the
magnitude of S22.

1310
01:07:11,840 --> 01:07:15,970
The magnitude of S12 is equal
to the magnitude of S21.

1311
01:07:24,200 --> 01:07:31,880
More importantly, S12 norm
squared plus S11 squared

1312
01:07:31,880 --> 01:07:33,990
is equal to 1.

1313
01:07:33,990 --> 01:07:43,670
And finally, S11, S12 complex
conjugate, plus S21, S22

1314
01:07:43,670 --> 01:07:47,010
complex conjugate is equal to 0.

1315
01:07:47,010 --> 01:07:51,820
So what do these mean?

1316
01:07:51,820 --> 01:07:53,453
What are these
conditions telling us?

1317
01:07:56,877 --> 01:07:58,460
They're telling us,
of course, they're

1318
01:07:58,460 --> 01:08:00,490
a consequence of
conservational probability,

1319
01:08:00,490 --> 01:08:01,950
but they have another meaning.

1320
01:08:01,950 --> 01:08:03,491
To get that other
meaning, let's look

1321
01:08:03,491 --> 01:08:07,230
at the definition of the
transmission amplitudes.

1322
01:08:07,230 --> 01:08:09,890
In particular, consider
the case that we send stuff

1323
01:08:09,890 --> 01:08:13,390
in from the left and
nothing in from the right.

1324
01:08:13,390 --> 01:08:16,830
That corresponds to D equals 0.

1325
01:08:16,830 --> 01:08:18,710
When D equals 0, what
does this tell us?

1326
01:08:18,710 --> 01:08:20,899
It tells us that B
is equal to-- and A

1327
01:08:20,899 --> 01:08:22,119
equals 1 for normalization.

1328
01:08:24,740 --> 01:08:30,810
B is equal to-- well, D is
equal to 0, so it's just S11 A,

1329
01:08:30,810 --> 01:08:35,029
so B over A is S11.

1330
01:08:35,029 --> 01:08:40,601
And similarly, C over A is S21.

1331
01:08:40,601 --> 01:08:42,100
But C over A is the
thing that we've

1332
01:08:42,100 --> 01:08:45,720
been calling the transmission
amplitude, little t,

1333
01:08:45,720 --> 01:08:48,109
and this is the reflection
amplitude, little r.

1334
01:08:48,109 --> 01:08:51,140
So this is the reflection
amplitude, if we send in stuff

1335
01:08:51,140 --> 01:08:53,479
and it bounces off,
reflects to the left,

1336
01:08:53,479 --> 01:08:55,869
and this is the transmission
amplitude for transmitting

1337
01:08:55,869 --> 01:08:56,410
to the right.

1338
01:08:58,529 --> 01:08:59,529
Everyone cool with that?

1339
01:09:02,279 --> 01:09:04,380
This is reflection
to the left, this

1340
01:09:04,380 --> 01:09:06,085
is transmission to the right.

1341
01:09:06,085 --> 01:09:08,210
By the same token, this is
going to be transmission

1342
01:09:08,210 --> 01:09:10,060
to the left and
reflection to the right.

1343
01:09:12,680 --> 01:09:14,840
So now let's look
at these conditions.

1344
01:09:14,840 --> 01:09:17,350
S11 is reflection and this
is going to be reflection.

1345
01:09:17,350 --> 01:09:19,350
This says that the
reflection to the left

1346
01:09:19,350 --> 01:09:22,875
is equal to the reflection
to the right in magnitude.

1347
01:09:25,910 --> 01:09:30,149
Before, what we saw was
for the simple step,

1348
01:09:30,149 --> 01:09:32,660
the reflection amplitude was
equal from left to right, not

1349
01:09:32,660 --> 01:09:34,430
just the magnitude,
but the actual value

1350
01:09:34,430 --> 01:09:36,100
was equal from the
left and the right.

1351
01:09:36,100 --> 01:09:38,350
It was a little bit of a
cheat because they were real.

1352
01:09:40,217 --> 01:09:41,800
And we saw that that
was a consequence

1353
01:09:41,800 --> 01:09:42,870
of just being the step.

1354
01:09:42,870 --> 01:09:43,970
We didn't know
anything more about it.

1355
01:09:43,970 --> 01:09:45,595
But now we see that
on general grounds,

1356
01:09:45,595 --> 01:09:47,309
on conservation of
probability grounds,

1357
01:09:47,309 --> 01:09:49,100
the magnitude of the
reflection to the left

1358
01:09:49,100 --> 01:09:52,510
and to the right for any
potential had better be equal.

1359
01:09:52,510 --> 01:09:57,490
And similarly, the magnitude of
the transmission for the left

1360
01:09:57,490 --> 01:10:03,120
and the transmission to the
right had better be equal,

1361
01:10:03,120 --> 01:10:06,540
all other things being-- if
you're sending in from the left

1362
01:10:06,540 --> 01:10:08,140
and then transmitting
to the left,

1363
01:10:08,140 --> 01:10:10,598
or sending in from the right
and transmitting to the right.

1364
01:10:10,598 --> 01:10:12,170
And what does this one tell us?

1365
01:10:12,170 --> 01:10:18,130
Well, S12 and S11, that tells
us that little t squared, which

1366
01:10:18,130 --> 01:10:23,890
is the total probability to
transmit, plus little r squared

1367
01:10:23,890 --> 01:10:27,040
is the total probability
to reflect, is equal to 1,

1368
01:10:27,040 --> 01:10:28,350
and we saw this last time, too.

1369
01:10:32,360 --> 01:10:35,630
This was the earlier definition
of nothing gets stuck.

1370
01:10:35,630 --> 01:10:39,771
And this one you'll study
on your problem set.

1371
01:10:39,771 --> 01:10:41,254
It's a little more subtle.

1372
01:10:45,440 --> 01:10:46,935
Questions?

1373
01:10:46,935 --> 01:10:47,435
Yeah?

1374
01:10:47,435 --> 01:10:50,345
AUDIENCE: Can you explain one
more time why is S unitary?

1375
01:10:50,345 --> 01:10:51,777
How did you get that?

1376
01:10:51,777 --> 01:10:54,110
PROFESSOR: So the way we got
S was unitary is first off,

1377
01:10:54,110 --> 01:10:55,740
this is just the
definition of S. S

1378
01:10:55,740 --> 01:10:57,800
is the matrix that,
for any energy,

1379
01:10:57,800 --> 01:11:00,500
relates A and D, the
ingoing amplitudes,

1380
01:11:00,500 --> 01:11:03,830
to the outgoing amplitudes B
and C. Just the definition.

1381
01:11:03,830 --> 01:11:06,640
Meanwhile, I claim that
stuff doesn't go away,

1382
01:11:06,640 --> 01:11:09,010
nothing gets stuck,
nothing disappears,

1383
01:11:09,010 --> 01:11:11,350
so the total probability
density of stuff going in

1384
01:11:11,350 --> 01:11:12,550
must be equal to the
total probability

1385
01:11:12,550 --> 01:11:13,675
density of stuff going out.

1386
01:11:15,826 --> 01:11:17,200
The probability
of stuff going in

1387
01:11:17,200 --> 01:11:19,970
can be expressed as this
row vector times this column

1388
01:11:19,970 --> 01:11:23,020
vector, and on the out, this
row vector times this column

1389
01:11:23,020 --> 01:11:25,560
vector, and then we use the
definition of the S matrix.

1390
01:11:25,560 --> 01:11:28,590
This column vector is equal
to S times this row vector.

1391
01:11:31,800 --> 01:11:34,090
BC is equal to S times AD.

1392
01:11:34,090 --> 01:11:36,470
And when we take the
transposed complex conjugate,

1393
01:11:36,470 --> 01:11:39,050
I get AD transposed
complex conjugate,

1394
01:11:39,050 --> 01:11:42,040
S transposed complex conjugate,
but that's the S adjoint.

1395
01:11:42,040 --> 01:11:46,730
But in order for this to be
true for any vectors A and D,

1396
01:11:46,730 --> 01:11:50,340
it must be that S dagger S
is unitary is the identity,

1397
01:11:50,340 --> 01:11:53,340
but that's the definition
of a unitary matrix.

1398
01:11:53,340 --> 01:11:55,270
Cool?

1399
01:11:55,270 --> 01:11:55,770
Others.

1400
01:11:59,240 --> 01:12:01,100
You're going to prove
a variety of things

1401
01:12:01,100 --> 01:12:07,370
on the problem set
about the scattering

1402
01:12:07,370 --> 01:12:08,930
matrix and its
coefficients, but I

1403
01:12:08,930 --> 01:12:12,600
want to show you two
properties of it.

1404
01:12:12,600 --> 01:12:17,630
The first is reasonably
tame, and it'll

1405
01:12:17,630 --> 01:12:21,135
make a little sharper the
step result we got earlier

1406
01:12:21,135 --> 01:12:23,260
that the reflection in both
directions off the step

1407
01:12:23,260 --> 01:12:25,170
potential was in fact
exactly the same.

1408
01:12:34,974 --> 01:12:36,890
Suppose our system is
time reversal invariant.

1409
01:12:40,390 --> 01:12:43,990
So if t goes to minus
t, nothing changes.

1410
01:12:43,990 --> 01:12:45,490
This would not be
true, for example,

1411
01:12:45,490 --> 01:12:47,365
if we had electric
currents in our system

1412
01:12:47,365 --> 01:12:50,797
because as we take t to minus
t, then the current reverses.

1413
01:12:50,797 --> 01:12:52,630
So if the current shows
up in the potential,

1414
01:12:52,630 --> 01:12:54,140
or if a magnetic
field due to a current

1415
01:12:54,140 --> 01:12:55,620
shows up in the
potential energy,

1416
01:12:55,620 --> 01:12:57,090
then as we change
t to minus t, we

1417
01:12:57,090 --> 01:12:58,590
change the direction
of the current,

1418
01:12:58,590 --> 01:13:01,250
we change the direction
of the magnetic field.

1419
01:13:01,250 --> 01:13:04,879
In simple systems where we
have time reversal invariance,

1420
01:13:04,879 --> 01:13:06,420
for example,
electrostatics, but not,

1421
01:13:06,420 --> 01:13:09,590
for example, magnetostatics,
suppose we have time reversal

1422
01:13:09,590 --> 01:13:13,560
invariance, then what you've
shown on a previous problem

1423
01:13:13,560 --> 01:13:19,060
set is that psi is a
solution, then psi star,

1424
01:13:19,060 --> 01:13:20,820
psi complex conjugate
is also a solution.

1425
01:13:23,920 --> 01:13:26,882
And using these, what you'll
see what you can show--

1426
01:13:26,882 --> 01:13:28,590
and I'm not going to
go through the steps

1427
01:13:28,590 --> 01:13:31,750
for this-- well, that's easy.

1428
01:13:31,750 --> 01:13:35,010
If we do the time reversal,
the wave function,

1429
01:13:35,010 --> 01:13:39,160
looking on the left or on the
right, so comparing these guys,

1430
01:13:39,160 --> 01:13:40,650
what changes?

1431
01:13:40,650 --> 01:13:43,244
Under time reversal,
we get a solution.

1432
01:13:43,244 --> 01:13:45,160
Given this solution, we
have another solution,

1433
01:13:45,160 --> 01:13:49,990
A star e to the minus ikx,
B star e to the plus ikx.

1434
01:13:49,990 --> 01:13:55,082
Minus star, star, plus.

1435
01:13:55,082 --> 01:13:56,540
So we can run
exactly the same game

1436
01:13:56,540 --> 01:13:58,168
but now with this amplitude.

1437
01:14:02,760 --> 01:14:05,680
And when you put the
conditions together,

1438
01:14:05,680 --> 01:14:08,850
which you'll do on the problem
set, another way to say this

1439
01:14:08,850 --> 01:14:12,220
is the same solution with k
to minus k and with A and B

1440
01:14:12,220 --> 01:14:13,780
replaced by their
complex conjugates

1441
01:14:13,780 --> 01:14:17,480
and C and D replaced by each
other's complex conjugates.

1442
01:14:17,480 --> 01:14:19,810
Then this implies
that it must be true

1443
01:14:19,810 --> 01:14:25,840
that A and D, which are now
the outgoing guys because we've

1444
01:14:25,840 --> 01:14:29,780
time reversed, is equal
to S-- and I'll write this

1445
01:14:29,780 --> 01:14:39,440
out explicitly-- S11, S12,
S21, S22, B star, and C star.

1446
01:14:42,530 --> 01:14:44,440
So these together give you that.

1447
01:14:48,950 --> 01:14:53,500
Therefore, S complex
conjugate S is equal to 1.

1448
01:14:57,460 --> 01:14:59,780
If S complex conjugate
S is equal to 1,

1449
01:14:59,780 --> 01:15:05,160
then S inverse is
equal to S transpose,

1450
01:15:05,160 --> 01:15:06,840
just putting this on the right.

1451
01:15:06,840 --> 01:15:12,027
So S transpose is equal to S
inverse is equal to S adjoint

1452
01:15:12,027 --> 01:15:13,110
because S is also unitary.

1453
01:15:15,910 --> 01:15:26,960
So time reversal invariance
implies, for example,

1454
01:15:26,960 --> 01:15:28,590
that S dagger is
equal to S star,

1455
01:15:28,590 --> 01:15:29,673
or S equal to S transpose.

1456
01:15:36,380 --> 01:15:37,880
This is what I
wanted to write here.

1457
01:15:37,880 --> 01:15:40,390
Gives us that S is
equal to S transpose.

1458
01:15:40,390 --> 01:15:46,110
And in particular, this tells
us that S21 is equal to S12.

1459
01:15:46,110 --> 01:15:48,950
The off diagonal
terms are equal,

1460
01:15:48,950 --> 01:15:52,650
not just in magnitude, which
was insured by unitary,

1461
01:15:52,650 --> 01:15:55,457
but if, in addition to
being a unitary system,

1462
01:15:55,457 --> 01:15:57,540
which of course, it should
be, if in addition it's

1463
01:15:57,540 --> 01:16:00,580
time reversal invariant, then we
see that the off diagonal terms

1464
01:16:00,580 --> 01:16:03,104
are equal not just in
magnitude but also in phase.

1465
01:16:03,104 --> 01:16:04,770
And as you know, the
phase is important.

1466
01:16:04,770 --> 01:16:06,480
The phase contains physics.

1467
01:16:06,480 --> 01:16:09,930
It tells you about time delays
and shifts in the scattering

1468
01:16:09,930 --> 01:16:10,576
process.

1469
01:16:10,576 --> 01:16:11,700
So the phases are the same.

1470
01:16:11,700 --> 01:16:13,200
That statement is
not a trivial one.

1471
01:16:13,200 --> 01:16:16,080
It contains physics.

1472
01:16:16,080 --> 01:16:18,960
So when the system is time
reversal invariant, the phases

1473
01:16:18,960 --> 01:16:20,890
as well as the
amplitudes are the same.

1474
01:16:20,890 --> 01:16:23,120
And you'll derive a series
of related conditions

1475
01:16:23,120 --> 01:16:27,482
or consequences for the S
matrix from various properties

1476
01:16:27,482 --> 01:16:28,940
of the system, for
example, parity,

1477
01:16:28,940 --> 01:16:31,670
if you could have a
symmetric potential.

1478
01:16:31,670 --> 01:16:33,375
But now in the last
few minutes, I just

1479
01:16:33,375 --> 01:16:35,920
want to tell you a
really lovely thing.

1480
01:16:35,920 --> 01:16:37,730
So it should be pretty
clear at this point

1481
01:16:37,730 --> 01:16:40,170
that all the information
about scattering

1482
01:16:40,170 --> 01:16:42,806
is contained in the S matrix and
its dependence on the energy.

1483
01:16:42,806 --> 01:16:44,680
If you know what the
incident amplitudes are,

1484
01:16:44,680 --> 01:16:47,236
you know what the
outgoing amplitudes are,

1485
01:16:47,236 --> 01:16:49,110
and that's cool because
you can measure this.

1486
01:16:49,110 --> 01:16:51,191
You can take a potential,
you can literally just

1487
01:16:51,191 --> 01:16:52,690
send in a beam of
particles, and you

1488
01:16:52,690 --> 01:16:54,970
can ask, how likely
are they to get out.

1489
01:16:54,970 --> 01:16:59,180
And more importantly, if I
build a wave packet, on average,

1490
01:16:59,180 --> 01:17:02,049
what's the time
delay or acceleration

1491
01:17:02,049 --> 01:17:03,340
of the transmitted wave packet?

1492
01:17:03,340 --> 01:17:04,890
And that way, I can
measure the phase as well.

1493
01:17:04,890 --> 01:17:06,630
I can measure both the
transmission probabilities

1494
01:17:06,630 --> 01:17:09,110
and the phases, or at least
the gradient of the phase

1495
01:17:09,110 --> 01:17:10,800
with energy.

1496
01:17:10,800 --> 01:17:11,300
Go ahead.

1497
01:17:11,300 --> 01:17:12,883
AUDIENCE: Is there
a special condition

1498
01:17:12,883 --> 01:17:16,330
that we can [? pose ?]
to see the resonance?

1499
01:17:16,330 --> 01:17:17,580
PROFESSOR: Excellent question.

1500
01:17:17,580 --> 01:17:20,569
Hold onto your
question for a second.

1501
01:17:20,569 --> 01:17:22,860
There's an enormous amount
of the physics of scattering

1502
01:17:22,860 --> 01:17:25,529
contained in the S matrix, and
you can measure the S matrix,

1503
01:17:25,529 --> 01:17:27,570
and you can measure its
dependence on the energy.

1504
01:17:27,570 --> 01:17:31,070
You can measure the coefficients
S12 and S22, their phases

1505
01:17:31,070 --> 01:17:33,780
and their amplitudes,
as a function of energy,

1506
01:17:33,780 --> 01:17:35,970
and you can plot them.

1507
01:17:35,970 --> 01:17:37,660
Here's what I want
to convince you of.

1508
01:17:37,660 --> 01:17:41,480
If you plot those and look
at how the functions behave

1509
01:17:41,480 --> 01:17:45,560
as functions of energy and ask,
how do those functions extend

1510
01:17:45,560 --> 01:17:48,680
to negative energy by
just drawing the line,

1511
01:17:48,680 --> 01:17:51,260
continuing the lines,
you can derive the energy

1512
01:17:51,260 --> 01:17:55,320
of any bound states
in the system, too.

1513
01:17:55,320 --> 01:17:59,950
Knowledge of the scattering is
enough to determine the bound

1514
01:17:59,950 --> 01:18:02,952
state energies of a system,
and let me show you that.

1515
01:18:02,952 --> 01:18:05,410
And this is one of the coolest
things in quantum mechanics.

1516
01:18:05,410 --> 01:18:07,760
Here's how this works.

1517
01:18:07,760 --> 01:18:09,740
We have, from the
definition of the S matrix,

1518
01:18:09,740 --> 01:18:13,930
that BC is equal to
the S matrix on AD,

1519
01:18:13,930 --> 01:18:15,790
where the wave
function-- let me just

1520
01:18:15,790 --> 01:18:20,930
put this back in the original
form-- is CD, AB, e to the ikx,

1521
01:18:20,930 --> 01:18:26,812
e to the minus ikx, and e to the
plus ikx, e to the minus ikx.

1522
01:18:26,812 --> 01:18:28,520
So that's the definition
of the S matrix.

1523
01:18:28,520 --> 01:18:31,730
The S matrix, at
a given energy e,

1524
01:18:31,730 --> 01:18:34,060
is a coefficient relation
matrix between the ingoing

1525
01:18:34,060 --> 01:18:36,820
and outgoing, or, more
to the point, A and D.

1526
01:18:36,820 --> 01:18:39,850
And in all of this, I've assumed
that the energy was positive,

1527
01:18:39,850 --> 01:18:42,840
that the k1 and k2
are positive and real.

1528
01:18:42,840 --> 01:18:44,340
But now let's ask
the question, what

1529
01:18:44,340 --> 01:18:46,215
would have happened if,
in the whole process,

1530
01:18:46,215 --> 01:18:53,490
I had taken the
energy less than 0?

1531
01:18:53,490 --> 01:18:57,280
If the energy were less
than 0, instead of k,

1532
01:18:57,280 --> 01:18:59,490
k would be replaced by i alpha.

1533
01:18:59,490 --> 01:19:01,240
Let's think about
what that does.

1534
01:19:01,240 --> 01:19:07,540
If k is i alpha, this is
e to the ik is minus alpha

1535
01:19:07,540 --> 01:19:11,350
and minus ik is plus alpha.

1536
01:19:11,350 --> 01:19:15,890
Similarly, ik times i,
that gives me a minus alpha

1537
01:19:15,890 --> 01:19:18,730
and this gives me a plus alpha.

1538
01:19:18,730 --> 01:19:19,770
Yeah?

1539
01:19:19,770 --> 01:19:21,800
So as equations, they're
the same equations

1540
01:19:21,800 --> 01:19:26,800
with k replaced by i alpha.

1541
01:19:26,800 --> 01:19:30,030
And now what must be
true for these states

1542
01:19:30,030 --> 01:19:32,870
to be normalizable?

1543
01:19:32,870 --> 01:19:34,890
What must be true,
for example, of A?

1544
01:19:38,070 --> 01:19:40,770
A must be 0 because at minus
infinity, there's divergence.

1545
01:19:40,770 --> 01:19:42,416
Not normalizable.

1546
01:19:42,416 --> 01:19:43,790
So in order to be
at bound state,

1547
01:19:43,790 --> 01:19:47,160
in order to have a physical
state, A must be 0.

1548
01:19:47,160 --> 01:19:49,416
What about D?

1549
01:19:49,416 --> 01:19:50,220
Same reason.

1550
01:19:50,220 --> 01:19:51,930
It's got to be 0 at
positive infinity.

1551
01:19:51,930 --> 01:19:54,470
These guys are convergent,
so C and B can be non-zero.

1552
01:19:54,470 --> 01:19:57,800
So now here's my question.

1553
01:19:57,800 --> 01:19:59,760
We know that these
relations must be true

1554
01:19:59,760 --> 01:20:04,690
because all these relations are
encoding is how a solution here

1555
01:20:04,690 --> 01:20:07,150
matches to a solution
here through a potential

1556
01:20:07,150 --> 01:20:10,140
in between with continuity of
the derivative and anything

1557
01:20:10,140 --> 01:20:11,900
else that's true of
that potential inside.

1558
01:20:11,900 --> 01:20:13,810
All that S is doing
from that point of view

1559
01:20:13,810 --> 01:20:15,610
is telling me how
these coefficients

1560
01:20:15,610 --> 01:20:17,150
match onto these coefficients.

1561
01:20:17,150 --> 01:20:18,480
Yes?

1562
01:20:18,480 --> 01:20:21,960
Now what, for a bound state--
if we have e less than 0,

1563
01:20:21,960 --> 01:20:23,510
what must be true?

1564
01:20:23,510 --> 01:20:30,200
It must be true that AD is equal
to 0, and in particular, 00.

1565
01:20:30,200 --> 01:20:31,310
So what are B and C?

1566
01:20:34,090 --> 01:20:37,000
Well, a matrix times
0 is equal to--

1567
01:20:37,000 --> 01:20:37,790
AUDIENCE: 0.

1568
01:20:37,790 --> 01:20:40,508
PROFESSOR: Unless?

1569
01:20:40,508 --> 01:20:41,867
AUDIENCE: [INAUDIBLE].

1570
01:20:41,867 --> 01:20:45,170
PROFESSOR: Unless the
matrix itself is diverging,

1571
01:20:45,170 --> 01:20:47,300
and then you have to be
more careful, but let's

1572
01:20:47,300 --> 01:20:49,340
be naive for the moment.

1573
01:20:49,340 --> 01:20:56,720
If A is 00, then in order
for B and C to be non-zero,

1574
01:20:56,720 --> 01:20:57,650
S must have a pole.

1575
01:21:01,580 --> 01:21:05,120
S must go like 1 over 0.

1576
01:21:05,120 --> 01:21:08,560
S must diverge at some
special value of the energy.

1577
01:21:08,560 --> 01:21:09,389
Well, that's easy.

1578
01:21:09,389 --> 01:21:11,930
That tells you that if you look
at any particular coefficient

1579
01:21:11,930 --> 01:21:15,440
in S, any of the
matrix elements of S,

1580
01:21:15,440 --> 01:21:17,940
the numerator can you whatever
you want, some finite number,

1581
01:21:17,940 --> 01:21:19,740
but the denominator
had better be?

1582
01:21:19,740 --> 01:21:20,600
AUDIENCE: 0.

1583
01:21:20,600 --> 01:21:21,590
PROFESSOR: 0.

1584
01:21:21,590 --> 01:21:23,830
So let's look at
the denominator.

1585
01:21:23,830 --> 01:21:30,466
If I compute S21-- actually,
let me do this over here.

1586
01:21:30,466 --> 01:21:32,890
No, it's all filled.

1587
01:21:32,890 --> 01:21:35,280
Let's do it over here.

1588
01:21:35,280 --> 01:21:39,060
If I look at S21 for
the potential well,

1589
01:21:39,060 --> 01:21:40,530
scattering off
the potential well

1590
01:21:40,530 --> 01:21:44,610
that we looked at at the
beginning of today's lecture,

1591
01:21:44,610 --> 01:21:46,840
this guy, and now
I'm going to look

1592
01:21:46,840 --> 01:21:51,420
at S21, one of the
coefficients of this guy,

1593
01:21:51,420 --> 01:21:55,990
also known as the scattering
amplitude t for the well.

1594
01:21:58,870 --> 01:22:05,020
This is equal to-- and it's a
godawful expression-- 2 k1 k2

1595
01:22:05,020 --> 01:22:12,210
e to the i k2 L over
2 k1 k2 cos of k

1596
01:22:12,210 --> 01:22:18,920
prime L minus i k squared plus
k1 squared plus k2 squared

1597
01:22:18,920 --> 01:22:24,910
times sine of k2 L. This
is some horrible thing.

1598
01:22:24,910 --> 01:22:30,220
But now I ask the condition,
when does this have a pole?

1599
01:22:30,220 --> 01:22:32,810
When the energy has
continued to be negative,

1600
01:22:32,810 --> 01:22:34,650
for what values does
this have a pole

1601
01:22:34,650 --> 01:22:36,410
or does the
denominator have a 0?

1602
01:22:36,410 --> 01:22:38,170
And the answer is
if you take this

1603
01:22:38,170 --> 01:22:39,780
and you massage
the equation, this

1604
01:22:39,780 --> 01:22:43,753
is equal to 0 a little bit, you
get the following expression,

1605
01:22:43,753 --> 01:22:53,870
k2 L upon 2 tangent of k2 L
upon 2 is equal to k1 L upon 2.

1606
01:22:53,870 --> 01:22:56,310
This is the condition
for the bound state

1607
01:22:56,310 --> 01:22:58,980
energies of the
square well, and we

1608
01:22:58,980 --> 01:23:03,080
computed it using knowledge
only of the scattering states.

1609
01:23:03,080 --> 01:23:05,320
If you took particles
and a square well

1610
01:23:05,320 --> 01:23:08,180
and roll the particles across
the square well potential

1611
01:23:08,180 --> 01:23:10,857
and measure it as a function
of energy, the scattering

1612
01:23:10,857 --> 01:23:12,440
amplitude, the
transmission amplitude,

1613
01:23:12,440 --> 01:23:15,720
and in particular S21, an
element of the S matrix,

1614
01:23:15,720 --> 01:23:17,910
and you plotted it as
a function of energy,

1615
01:23:17,910 --> 01:23:23,860
and then you approximated that
by a function of energy that

1616
01:23:23,860 --> 01:23:27,196
satisfies the basic properties
of unitarity, what you would

1617
01:23:27,196 --> 01:23:28,570
find is that when
you then extend

1618
01:23:28,570 --> 01:23:31,276
that function in
mathematica to minus

1619
01:23:31,276 --> 01:23:32,650
a particular value
of the energy,

1620
01:23:32,650 --> 01:23:36,000
the denominator
diverges at that energy.

1621
01:23:36,000 --> 01:23:38,912
You know that there
will be a bound state.

1622
01:23:38,912 --> 01:23:40,620
And so from scattering,
you've determined

1623
01:23:40,620 --> 01:23:42,236
the existence of a bound state.

1624
01:23:42,236 --> 01:23:44,610
This is how we find an awful
lot of the particles that we

1625
01:23:44,610 --> 01:23:49,050
actually deduce must
exist in the real world.

1626
01:23:49,050 --> 01:23:52,050
We'll pick up next time.

1627
01:23:52,050 --> 01:23:53,600
[APPLAUSE]