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PROFESSOR: All right.

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00:00:27,890 --> 00:00:30,340
Hello.

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00:00:30,340 --> 00:00:35,360
So today, as people
slowly trickle in,

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00:00:35,360 --> 00:00:39,430
so today we're going to talk
about scattering states.

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00:00:39,430 --> 00:00:42,170
So we've spent a lot of time
talking about bound states,

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00:00:42,170 --> 00:00:44,140
states corresponding
to particles that

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00:00:44,140 --> 00:00:48,765
are localized in a region and
that can't get off to infinity.

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00:00:48,765 --> 00:00:50,890
And in particular, when we
distinguish bound states

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00:00:50,890 --> 00:00:52,431
from scattering
states, we're usually

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00:00:52,431 --> 00:00:54,440
talking about
energy eigenstates.

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00:00:54,440 --> 00:00:56,480
They're eigenstates
that are strictly

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00:00:56,480 --> 00:00:58,190
localized in some region.

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00:00:58,190 --> 00:01:00,380
For example in the infinite
well, all of the states

21
00:01:00,380 --> 00:01:03,580
are strictly localized
to be inside the well.

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00:01:03,580 --> 00:01:06,260
In the finite well,
as we saw last time,

23
00:01:06,260 --> 00:01:07,780
there are a finite
number of states,

24
00:01:07,780 --> 00:01:10,880
of energy eigenstates which
are bound inside the well.

25
00:01:10,880 --> 00:01:12,570
But there are also
states that aren't

26
00:01:12,570 --> 00:01:14,320
bound inside the well,
as hopefully you'll

27
00:01:14,320 --> 00:01:16,785
see in recitation,
and as we've discussed

28
00:01:16,785 --> 00:01:18,660
from qualitative structure
of wave functions,

29
00:01:18,660 --> 00:01:21,580
but under the eigenfunction.

30
00:01:21,580 --> 00:01:24,432
So why would we
particularly care?

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00:01:24,432 --> 00:01:25,890
We also did the
harmonic oscillator

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00:01:25,890 --> 00:01:27,756
where everything
was nice and bound.

33
00:01:27,756 --> 00:01:29,130
So why would we
particularly care

34
00:01:29,130 --> 00:01:31,338
about states that aren't
bound for scattering states?

35
00:01:31,338 --> 00:01:34,932
And the obvious answer
is, I am not bound.

36
00:01:34,932 --> 00:01:36,640
Most things in the
world are either bound

37
00:01:36,640 --> 00:01:37,670
or they are scattering states.

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00:01:37,670 --> 00:01:39,961
By scattering state just mean
things that can get away.

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00:01:39,961 --> 00:01:41,250
Things that can go away.

40
00:01:41,250 --> 00:01:51,640
So it's easy to
under emphasize, it's

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00:01:51,640 --> 00:01:56,614
easy to miss the importance
of scattering experiments.

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00:01:56,614 --> 00:01:58,530
It's easy to say look,
a scattering experiment

43
00:01:58,530 --> 00:02:00,540
is where you take
some fixed target

44
00:02:00,540 --> 00:02:02,290
and you throw
something at it and you

45
00:02:02,290 --> 00:02:03,831
look at how things
bounce off and you

46
00:02:03,831 --> 00:02:05,719
try to do something
about this object

47
00:02:05,719 --> 00:02:06,760
by helping it bounce off.

48
00:02:06,760 --> 00:02:11,810
For example, one of the
great experiments of our day

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00:02:11,810 --> 00:02:18,470
is the LEC, and it's a
gigantic ring, it's huge,

50
00:02:18,470 --> 00:02:21,580
at the foot of the Jura
Mountains outside of Geneva.

51
00:02:21,580 --> 00:02:24,310
And those are people and
that's the detector-- that's

52
00:02:24,310 --> 00:02:26,050
one of the detectors,
I think, at CMS.

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00:02:26,050 --> 00:02:28,397
And what are you doing
in this experiment?

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00:02:28,397 --> 00:02:30,230
In this experiment
you're taking two protons

55
00:02:30,230 --> 00:02:34,770
and you're accelerating
them ridiculously fast,

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00:02:34,770 --> 00:02:37,501
bounded only by the speed
light, but just barely.

57
00:02:37,501 --> 00:02:39,750
Until they have a tremendous
amount of kinetic energy,

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00:02:39,750 --> 00:02:40,850
and you're colliding
them into each other.

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00:02:40,850 --> 00:02:43,150
And the idea is if you
collide them into each other

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00:02:43,150 --> 00:02:45,390
and you watch how the
shrapnel comes flying off,

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00:02:45,390 --> 00:02:47,170
you can deduce
what must have been

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00:02:47,170 --> 00:02:48,711
going on when they
actually collided.

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00:02:48,711 --> 00:02:50,850
You could do something
about the structure.

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00:02:50,850 --> 00:02:52,725
How are protons built
up out of corks.

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00:02:52,725 --> 00:02:54,350
And it's easy to
think of a scattering

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experiment as some
bizarre thing that

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happens in particle
detectors under mountains.

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00:03:00,620 --> 00:03:05,000
But I am currently engaged
in a scattering experiment.

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00:03:05,000 --> 00:03:07,560
The scattering experiments
I'm currently engaged in

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00:03:07,560 --> 00:03:10,960
is light is bouncing from
light sources in the room,

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00:03:10,960 --> 00:03:13,260
off of very bits of each of you.

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00:03:13,260 --> 00:03:17,280
And some of it, and probably,
scatters directly into my eye.

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00:03:17,280 --> 00:03:18,100
That is shocking.

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00:03:18,100 --> 00:03:21,520
And through the scattering
process and deducing

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what I can from the
statistics of the photons

76
00:03:24,346 --> 00:03:25,970
that bounce into my
eye, I could deduce

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00:03:25,970 --> 00:03:28,450
that you're sitting
there and not over there,

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which is pretty awesome.

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So we actually get
a tremendous amount

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of the information about
the world around us

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00:03:33,820 --> 00:03:35,440
from scattering processes.

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These are not esoteric,
weird things that happen.

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Scattering is how you
interact with the world.

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00:03:41,290 --> 00:03:43,010
So when we do
scattering problems,

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00:03:43,010 --> 00:03:45,490
it's not because we're thinking
about particle colliders.

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00:03:45,490 --> 00:03:46,970
It's because at the end of
the day I want to understand--

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00:03:46,970 --> 00:03:47,260
AUDIENCE: [SNEEZE]

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00:03:47,260 --> 00:03:48,885
PROFESSOR: --how
surfaces-- bless you--

89
00:03:48,885 --> 00:03:50,080
how surfaces reflect.

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00:03:50,080 --> 00:03:52,690
I want to understand why when
you look in the mirror light

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00:03:52,690 --> 00:03:53,814
bounces back.

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00:03:53,814 --> 00:03:55,480
I want to understand
how you see things.

93
00:03:55,480 --> 00:03:57,146
So this is a much
broader-- scattering's

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00:03:57,146 --> 00:04:00,556
a much broader idea than
just the sort of legacy

95
00:04:00,556 --> 00:04:02,780
of how we study
particle physics.

96
00:04:02,780 --> 00:04:05,300
Scattering is what
we do constantly.

97
00:04:05,300 --> 00:04:07,880
And we're going to come back
to the computer in a bit.

98
00:04:07,880 --> 00:04:09,260
I'm going to close this for now.

99
00:04:09,260 --> 00:04:09,760
Yes, good.

100
00:04:12,940 --> 00:04:18,399
So today we're going to begin a
series of lectures in which we

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00:04:18,399 --> 00:04:20,760
study scattering
processes in one

102
00:04:20,760 --> 00:04:22,915
dimension in various
different settings.

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00:04:22,915 --> 00:04:24,290
So in particular
that means we're

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00:04:24,290 --> 00:04:28,200
going to be studying
quantum particles being sent

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00:04:28,200 --> 00:04:32,360
in from some distance,
incident on some potential,

106
00:04:32,360 --> 00:04:33,400
incident on some object.

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00:04:33,400 --> 00:04:35,650
And we're going to ask how
likely are they to continue

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00:04:35,650 --> 00:04:37,733
going through, how likely
are they to bounce back.

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00:04:40,730 --> 00:04:44,650
And so we're going to start
with an easy potential,

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00:04:44,650 --> 00:04:47,250
an easy object off which
to scatter something, which

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00:04:47,250 --> 00:04:49,970
is the absence of an
object, the free particle.

112
00:04:52,650 --> 00:04:57,050
And I want to think about
how the system evolves.

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00:04:57,050 --> 00:04:59,800
So for the free particle where
we just have something of mass,

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00:04:59,800 --> 00:05:03,380
m, and its potential
is constant,

115
00:05:03,380 --> 00:05:06,030
we know how to solve the--

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00:05:06,030 --> 00:05:06,530
Question?

117
00:05:06,530 --> 00:05:07,470
Yeah.

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00:05:07,470 --> 00:05:10,290
AUDIENCE: This may
be a little bit dumb,

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00:05:10,290 --> 00:05:14,324
but given that these
states, [INAUDIBLE]?

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00:05:18,040 --> 00:05:19,360
PROFESSOR: Fantastic question.

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00:05:19,360 --> 00:05:21,026
I'm going to come
back in just a second.

122
00:05:21,026 --> 00:05:24,110
It's a very good question.

123
00:05:24,110 --> 00:05:27,120
So we're interested
in energy eigenstates,

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00:05:27,120 --> 00:05:28,610
and we know the
energy eigenstates

125
00:05:28,610 --> 00:05:31,275
for the free particle-- we've
written these down many times--

126
00:05:31,275 --> 00:05:33,150
and I'm going to write
it in the phonic form.

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00:05:33,150 --> 00:05:35,220
Suppose we have a free
particle in an energy

128
00:05:35,220 --> 00:05:37,480
eigenstate with
energy e, then we

129
00:05:37,480 --> 00:05:40,200
know it's a superposition
of clean waves,

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00:05:40,200 --> 00:05:44,010
e to the i kx plus
b e to minus ikx.

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00:05:46,850 --> 00:05:49,820
And the normalization
is encoded in a and b,

132
00:05:49,820 --> 00:05:51,981
but, of course, these
are normalizable states.

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00:05:51,981 --> 00:05:53,480
We typically normalize
them with a 1

134
00:05:53,480 --> 00:05:55,760
over root 2 pi for delta
function normalizability.

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00:05:57,934 --> 00:06:00,350
Now, in order for this to be
a solution to the Schrodinger

136
00:06:00,350 --> 00:06:04,100
equation, we need the
frequency or the energy e

137
00:06:04,100 --> 00:06:08,410
is equal to h-bar squared
k squared upon 2m,

138
00:06:08,410 --> 00:06:10,660
and omega is this upon h-bar.

139
00:06:10,660 --> 00:06:12,690
And knowing that,
we can immediately

140
00:06:12,690 --> 00:06:13,730
write down the solution.

141
00:06:13,730 --> 00:06:14,910
Because these are
energy eigenstates,

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00:06:14,910 --> 00:06:17,451
we can immediately write down
the solution of the Schrodinger

143
00:06:17,451 --> 00:06:21,290
equation, which begins at
time t 0 in this state.

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00:06:29,035 --> 00:06:31,410
So there's the solution of
the time dependent Schrodinger

145
00:06:31,410 --> 00:06:31,909
equation.

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00:06:31,909 --> 00:06:35,080
So we've now completely solved
this in full generality, which

147
00:06:35,080 --> 00:06:38,510
is not so shocking since
it's a free particle.

148
00:06:38,510 --> 00:06:42,560
But let's think about what
these two possible states are.

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00:06:42,560 --> 00:06:44,720
What do these mean?

150
00:06:44,720 --> 00:06:48,380
So the first thing to
say is this component

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00:06:48,380 --> 00:06:50,879
of our superposition
is a wave, and I'm

152
00:06:50,879 --> 00:06:52,420
going to call it a
right moving wave.

153
00:06:52,420 --> 00:06:54,253
And why am I calling
it a right moving wave?

154
00:06:54,253 --> 00:06:57,300
If you draw the real part of
this, some moment in time,

155
00:06:57,300 --> 00:06:59,040
it's got some crests.

156
00:06:59,040 --> 00:07:01,660
And if you draw on it a
short bit of time later,

157
00:07:01,660 --> 00:07:03,900
I want to ask how
the wave has moved.

158
00:07:03,900 --> 00:07:06,000
And, for example,
a peak in the wave

159
00:07:06,000 --> 00:07:09,400
will be when kx
minus omega t is 0.

160
00:07:09,400 --> 00:07:15,780
If kx minus omega t is 0,
then kx is equal to omega t,

161
00:07:15,780 --> 00:07:20,010
or x is equal to omega over kt.

162
00:07:20,010 --> 00:07:22,810
And assuming as we have
that k is positive,

163
00:07:22,810 --> 00:07:25,290
then this says that
as time increases

164
00:07:25,290 --> 00:07:27,420
the position increases.

165
00:07:27,420 --> 00:07:29,580
And this one, by
exactly the same logic,

166
00:07:29,580 --> 00:07:33,095
x is equal to minus
omega over kt.

167
00:07:37,510 --> 00:07:43,576
So we'll call this a right
going or right moving,

168
00:07:43,576 --> 00:07:47,610
and we'll call this
a left moving wave.

169
00:07:47,610 --> 00:07:50,080
So the general solution is a
superposition, an arbitrary

170
00:07:50,080 --> 00:07:52,270
superposition, of a left
moving and a right moving wave.

171
00:07:52,270 --> 00:07:54,603
This should be familiar from
803 in your study of waves.

172
00:07:57,410 --> 00:08:00,910
But as was pointed
out just a moment ago,

173
00:08:00,910 --> 00:08:03,830
we've got a problem.

174
00:08:03,830 --> 00:08:06,550
Phi e or phi e is
not normalizable.

175
00:08:11,610 --> 00:08:13,690
And this is obvious
from if we-- let's

176
00:08:13,690 --> 00:08:15,470
have b equal 0 for simplicity.

177
00:08:15,470 --> 00:08:18,850
This is a pure phase, and the
norm squared of a pure phase

178
00:08:18,850 --> 00:08:20,270
is 1.

179
00:08:20,270 --> 00:08:22,275
And so the probability
density is constant.

180
00:08:22,275 --> 00:08:25,260
It's 1 from minus
infinity to infinity.

181
00:08:25,260 --> 00:08:28,740
So the probability of finding
it at any given point is equal.

182
00:08:28,740 --> 00:08:31,215
So assuming this is on
the real line, which

183
00:08:31,215 --> 00:08:33,220
is what we mean by saying
it's a free particle,

184
00:08:33,220 --> 00:08:35,640
there's no constant you
can multiply that by

185
00:08:35,640 --> 00:08:36,640
to make it normalizable.

186
00:08:36,640 --> 00:08:38,139
This is not a
normalizable function.

187
00:08:40,500 --> 00:08:43,289
So more precisely,
what we usually do

188
00:08:43,289 --> 00:08:50,610
is we take our states phi
sub k equal 1 over root 2 pi,

189
00:08:50,610 --> 00:08:53,160
e to the i kx.

190
00:08:53,160 --> 00:09:02,330
Such that phi k, phi k
prime is equal to delta.

191
00:09:02,330 --> 00:09:04,720
Sorry-- that's not
what I wanted to write.

192
00:09:04,720 --> 00:09:10,350
Such that-- well, fine.

193
00:09:10,350 --> 00:09:15,970
Phi k, phi k prime is equal
to delta of k minus k prime.

194
00:09:19,650 --> 00:09:20,932
This is as good as we can do.

195
00:09:20,932 --> 00:09:22,640
This is good as we
get to normalizing it.

196
00:09:22,640 --> 00:09:23,681
It's not normalized to 1.

197
00:09:23,681 --> 00:09:25,440
It's normalized 1k
is equal to k prime,

198
00:09:25,440 --> 00:09:27,939
to the value of the delta
function, which is ill-defined.

199
00:09:27,939 --> 00:09:30,230
So we've dealt with this
before, we've talked about it.

200
00:09:30,230 --> 00:09:31,702
So how do we deal with it?

201
00:09:31,702 --> 00:09:33,160
Well, we've just
learned, actually,

202
00:09:33,160 --> 00:09:34,320
a very important thing.

203
00:09:34,320 --> 00:09:35,940
This question,
very good question,

204
00:09:35,940 --> 00:09:37,648
led to a very important
observation which

205
00:09:37,648 --> 00:09:41,470
is that can a particle, can
a quantum mechanical particle

206
00:09:41,470 --> 00:09:45,050
be placed, meaningfully
said to be placed,

207
00:09:45,050 --> 00:09:50,290
in an energy eigenstate
which is bound?

208
00:09:50,290 --> 00:09:52,560
Can you put a particle
in the ground state

209
00:09:52,560 --> 00:09:54,010
of a harmonic oscillator?

210
00:09:54,010 --> 00:09:54,930
AUDIENCE: [INAUDIBLE]?

211
00:09:54,930 --> 00:09:55,640
PROFESSOR: Sure.

212
00:09:55,640 --> 00:09:56,140
That's fine.

213
00:09:56,140 --> 00:09:57,680
Can you put the
particle in the k

214
00:09:57,680 --> 00:09:59,285
equals 7 state of
a free particle?

215
00:09:59,285 --> 00:10:00,397
AUDIENCE: No.

216
00:10:00,397 --> 00:10:00,980
PROFESSOR: No.

217
00:10:00,980 --> 00:10:05,290
So these scattering
states, you can never

218
00:10:05,290 --> 00:10:08,010
truly put your particle
truly in a scattering state.

219
00:10:08,010 --> 00:10:10,694
It's always going to be
some approximate plane wave.

220
00:10:10,694 --> 00:10:12,110
Scattering states
are always going

221
00:10:12,110 --> 00:10:15,110
to be some plane
wave asymptotically.

222
00:10:15,110 --> 00:10:17,670
So we can't put the particles
directly in a plane wave.

223
00:10:17,670 --> 00:10:19,090
What can we do, however?

224
00:10:19,090 --> 00:10:20,052
AUDIENCE: [INAUDIBLE]

225
00:10:20,052 --> 00:10:20,760
PROFESSOR: Right.

226
00:10:20,760 --> 00:10:23,150
We can build a wave packet.

227
00:10:23,150 --> 00:10:26,070
We can use these as
a basis for states,

228
00:10:26,070 --> 00:10:28,820
which are normalizable
and reasonably localized.

229
00:10:28,820 --> 00:10:30,890
So we need to deal
with wave packets.

230
00:10:30,890 --> 00:10:33,130
So today is going to be,
the beginning of today

231
00:10:33,130 --> 00:10:36,034
is going to be dealing with
the evolution of a wave packet.

232
00:10:36,034 --> 00:10:37,700
And in particular
we'll start by looking

233
00:10:37,700 --> 00:10:41,210
at the evolution in time
of a well localized wave

234
00:10:41,210 --> 00:10:43,380
packet in the
potential corresponding

235
00:10:43,380 --> 00:10:44,900
to a free particle.

236
00:10:44,900 --> 00:10:46,650
And from that we're
going to learn already

237
00:10:46,650 --> 00:10:48,066
some interesting
things, and we'll

238
00:10:48,066 --> 00:10:50,510
use that intuition for the
next several lectures as well.

239
00:10:50,510 --> 00:10:53,680
Questions before we get going?

240
00:10:53,680 --> 00:10:54,195
All right.

241
00:10:57,080 --> 00:10:58,850
So here's going to
be my first example.

242
00:11:03,570 --> 00:11:09,080
So consider for
the free particle

243
00:11:09,080 --> 00:11:20,970
or consider a free particle
in a minimum uncertainty wave

244
00:11:20,970 --> 00:11:23,356
packet.

245
00:11:23,356 --> 00:11:25,230
So what is a minimum
uncertainty wave packet.

246
00:11:25,230 --> 00:11:25,680
We've talked--

247
00:11:25,680 --> 00:11:26,190
AUDIENCE: A Gaussian.

248
00:11:26,190 --> 00:11:27,390
PROFESSOR: A Gaussian, exactly.

249
00:11:27,390 --> 00:11:29,556
So the minimum uncertainly
wave packet's a Gaussian.

250
00:11:29,556 --> 00:11:33,410
Suppose that I take my particle
and I place it at time 0,

251
00:11:33,410 --> 00:11:35,494
x0 in a Gaussian.

252
00:11:35,494 --> 00:11:37,410
I'm going to properly
normalize this Gaussian.

253
00:11:37,410 --> 00:11:39,630
I hate these factors
of pi, but whatever.

254
00:11:39,630 --> 00:11:47,690
a root pi, e to the minus
x squared over 2a squared.

255
00:11:47,690 --> 00:11:52,500
So this is the wave function
at time 0, I declare.

256
00:11:52,500 --> 00:11:54,400
I will just prepare the
system in this state.

257
00:11:54,400 --> 00:11:55,816
And what I'm
interested in knowing

258
00:11:55,816 --> 00:11:58,180
is how does the
system evolve in time?

259
00:11:58,180 --> 00:12:01,080
What is x psi of x and t?

260
00:12:06,990 --> 00:12:09,005
So how do we solve this problem?

261
00:12:09,005 --> 00:12:11,380
Well, we could just plug it
into the Schrodinger equation

262
00:12:11,380 --> 00:12:13,588
directly and just by brute
force try to crank it out.

263
00:12:13,588 --> 00:12:17,210
But the easier thing to do
is to use the same technique

264
00:12:17,210 --> 00:12:19,136
we've used all the way along.

265
00:12:19,136 --> 00:12:20,760
We are going to take
the wave function,

266
00:12:20,760 --> 00:12:23,080
known wave function,
known dependence on x,

267
00:12:23,080 --> 00:12:25,900
expand it as a superposition
of energy eigenstates.

268
00:12:25,900 --> 00:12:29,220
Each energy eigenstates
evolved in a known fashion

269
00:12:29,220 --> 00:12:31,460
will evolve those
energy eigenstates

270
00:12:31,460 --> 00:12:35,180
in that superposition, and
do the sum again, re-sum it,

271
00:12:35,180 --> 00:12:38,430
to get the evolution as
a function of position.

272
00:12:38,430 --> 00:12:38,930
Cool?

273
00:12:38,930 --> 00:12:41,610
Yeah.

274
00:12:41,610 --> 00:12:52,938
AUDIENCE: So given that
the x-basis and the t-bases

275
00:12:52,938 --> 00:12:55,926
are uncountable, [INAUDIBLE]?

276
00:12:59,754 --> 00:13:00,420
PROFESSOR: Good.

277
00:13:00,420 --> 00:13:01,211
Here's the theorem.

278
00:13:01,211 --> 00:13:03,790
So the question is roughly how
do you know you can do that?

279
00:13:03,790 --> 00:13:05,665
How do you know you can
expand in the energy.

280
00:13:05,665 --> 00:13:08,320
And we have to go back to the
spectral theorem which tells us

281
00:13:08,320 --> 00:13:14,210
for any observable, the
corresponding operator

282
00:13:14,210 --> 00:13:17,639
has a basis of eigenfunctions.

283
00:13:17,639 --> 00:13:19,430
So this is a theorem
that I haven't proven,

284
00:13:19,430 --> 00:13:20,800
but that I'm telling you.

285
00:13:20,800 --> 00:13:23,525
And that we will
use over and over

286
00:13:23,525 --> 00:13:24,900
and over again,
spectral theorem.

287
00:13:24,900 --> 00:13:28,540
But what it tells you is that
any operator has an eigenbasis.

288
00:13:28,540 --> 00:13:31,650
So if we find any good operator
corresponding to an observable

289
00:13:31,650 --> 00:13:33,022
has a good eigenbasis.

290
00:13:33,022 --> 00:13:35,480
That means that if we find the
eigenfunctions of the energy

291
00:13:35,480 --> 00:13:38,770
operator, any state
can be expanded

292
00:13:38,770 --> 00:13:41,067
as a superposition
in that state.

293
00:13:41,067 --> 00:13:43,150
And the question of
accountable versus uncountable

294
00:13:43,150 --> 00:13:44,970
is a slightly subtle one.

295
00:13:44,970 --> 00:13:47,250
For example, how many points
are there on the circle?

296
00:13:47,250 --> 00:13:48,670
Well, there's an
uncountable number.

297
00:13:48,670 --> 00:13:50,586
How many momentum modes
are there on a circle?

298
00:13:50,586 --> 00:13:51,920
Well, that's countable.

299
00:13:51,920 --> 00:13:52,730
How does that work?

300
00:13:52,730 --> 00:13:54,480
How can you describe
that both in position

301
00:13:54,480 --> 00:13:55,512
and in momentum space.

302
00:13:55,512 --> 00:13:57,970
Ask me afterwards and we'll
talk about that in more detail.

303
00:13:57,970 --> 00:14:00,490
But don't get too hung up on
countable versus uncountable.

304
00:14:00,490 --> 00:14:01,600
It's not that big a difference.

305
00:14:01,600 --> 00:14:03,391
You're just summing--
it's just whether you

306
00:14:03,391 --> 00:14:05,840
sum over a continuous
or discrete thing.

307
00:14:05,840 --> 00:14:10,320
I'm being glib, but it's useful
to be glib at that level.

308
00:14:10,320 --> 00:14:13,330
Other questions before we-- OK.

309
00:14:13,330 --> 00:14:15,340
Good.

310
00:14:15,340 --> 00:14:16,223
Yeah.

311
00:14:16,223 --> 00:14:18,806
AUDIENCE: So the reason we are
going to use energy eigenstates

312
00:14:18,806 --> 00:14:22,750
is just because [INAUDIBLE]?

313
00:14:22,750 --> 00:14:23,500
PROFESSOR: Almost.

314
00:14:23,500 --> 00:14:26,094
So the question is are we
using the energy eigenstates

315
00:14:26,094 --> 00:14:27,510
because at the end
of the day it's

316
00:14:27,510 --> 00:14:29,594
going to boil down to
doing a Fourier transform,

317
00:14:29,594 --> 00:14:31,760
because the energy eigenstates
are just plain waves,

318
00:14:31,760 --> 00:14:32,870
e to the ix?

319
00:14:32,870 --> 00:14:36,285
And that will be true, but
that's not the reason we do it.

320
00:14:36,285 --> 00:14:37,160
It's a good question.

321
00:14:37,160 --> 00:14:38,326
So let me disentangle these.

322
00:14:38,326 --> 00:14:39,720
Well, I guess I can use this.

323
00:14:39,720 --> 00:14:42,510
The point is that if I
know that si-- for any wave

324
00:14:42,510 --> 00:14:45,180
function in any system, if
I know that phi of x at 0

325
00:14:45,180 --> 00:14:48,520
is some function, and I
know the energy eigenstates

326
00:14:48,520 --> 00:14:53,380
of that system, then I can
write this as sum over n, of cn,

327
00:14:53,380 --> 00:14:57,300
phi n of x for some
set of coefficient cn.

328
00:14:57,300 --> 00:14:58,930
So that's the spectral theorem.

329
00:14:58,930 --> 00:15:00,880
Pick an operator
here, the energy.

330
00:15:00,880 --> 00:15:02,200
Consider its eigenfunctions.

331
00:15:02,200 --> 00:15:04,260
They form a basis and any
state can be expanded.

332
00:15:04,260 --> 00:15:07,287
But once we've expanded
in the energy eigenbasis,

333
00:15:07,287 --> 00:15:08,870
specifically for the
energy eigenbasis

334
00:15:08,870 --> 00:15:12,310
where e phi n is
en phi n, then we

335
00:15:12,310 --> 00:15:14,060
know how these guys
evolve in time.

336
00:15:14,060 --> 00:15:19,080
They evolve by an e to the
minus i en t upon h-bar.

337
00:15:19,080 --> 00:15:21,250
And we also know that
the Schrodinger equation

338
00:15:21,250 --> 00:15:23,420
is a linear equation, so
we can superpose solutions

339
00:15:23,420 --> 00:15:24,590
and get a new solution.

340
00:15:24,590 --> 00:15:27,860
So now, this is the expression
for the full solution

341
00:15:27,860 --> 00:15:28,820
at time t.

342
00:15:28,820 --> 00:15:31,730
That's why we're using
energy eigenstates.

343
00:15:31,730 --> 00:15:33,515
In this simple case
of the free particle,

344
00:15:33,515 --> 00:15:37,660
it is a felicitous fact that
the energy eigenstates are also

345
00:15:37,660 --> 00:15:38,944
Fourier modes.

346
00:15:38,944 --> 00:15:40,360
But that won't be
true in general.

347
00:15:40,360 --> 00:15:44,340
For example, in the
harmonic oscillator system,

348
00:15:44,340 --> 00:15:46,090
the harmonic oscillator
energy eigenstates

349
00:15:46,090 --> 00:15:49,440
are Gaussians times
some special functions.

350
00:15:49,440 --> 00:15:50,830
And we know how to compute them.

351
00:15:50,830 --> 00:15:50,970
That's great.

352
00:15:50,970 --> 00:15:52,190
We use a raising,
lowering operators.

353
00:15:52,190 --> 00:15:52,689
It's nice.

354
00:15:52,689 --> 00:15:54,280
But they're not plane waves.

355
00:15:54,280 --> 00:15:56,730
Nonetheless, despite
not being Fourier modes,

356
00:15:56,730 --> 00:15:59,500
we can always expand
an arbitrary function

357
00:15:59,500 --> 00:16:01,950
in the energy eigenbasis.

358
00:16:01,950 --> 00:16:05,640
And once we've done so,
writing down the time evolution

359
00:16:05,640 --> 00:16:06,687
is straightforward.

360
00:16:06,687 --> 00:16:07,770
That answer your question?

361
00:16:07,770 --> 00:16:08,260
AUDIENCE: Yeah.

362
00:16:08,260 --> 00:16:09,090
PROFESSOR: Great.

363
00:16:09,090 --> 00:16:10,230
Anything else?

364
00:16:10,230 --> 00:16:10,730
OK.

365
00:16:10,730 --> 00:16:11,980
So let's do it for this guy.

366
00:16:14,752 --> 00:16:16,210
So what are the
energy eigenstates.

367
00:16:16,210 --> 00:16:17,480
They're these guys,
and I'm just going

368
00:16:17,480 --> 00:16:19,280
to write it as e to the
i kx without assuming

369
00:16:19,280 --> 00:16:20,110
that k is positive.

370
00:16:20,110 --> 00:16:21,960
So k could be
positive or negative,

371
00:16:21,960 --> 00:16:25,680
and energy is e to the
h-bar squared k squared.

372
00:16:25,680 --> 00:16:28,634
And so we can write this as--
we can write the wave function

373
00:16:28,634 --> 00:16:30,050
in this fashion,
but we could also

374
00:16:30,050 --> 00:16:32,710
write it in Fourier
transformed form,

375
00:16:32,710 --> 00:16:37,350
which is integral
dk upon root 2 pi

376
00:16:37,350 --> 00:16:38,700
for minus infinity to infinity.

377
00:16:38,700 --> 00:16:40,200
I'm mostly not going
to write the bounds.

378
00:16:40,200 --> 00:16:42,450
They're always going to be
minus infinity to infinity.

379
00:16:42,450 --> 00:16:50,720
Of dk 1 over root
2 pi e to the i kx.

380
00:16:50,720 --> 00:16:57,527
The plane wave times the Fourier
transform, psi tilde of k.

381
00:16:57,527 --> 00:16:59,860
So here all I'm doing is
defining for you the psi tilde.

382
00:17:03,290 --> 00:17:05,740
But you actually
computed this, so this

383
00:17:05,740 --> 00:17:06,740
is the Fourier transfer.

384
00:17:06,740 --> 00:17:10,220
You actually computed this on,
I think, the second problem set.

385
00:17:10,220 --> 00:17:18,920
So this is equal to the
integral dk over root 2 pi

386
00:17:18,920 --> 00:17:19,530
e to the i kx.

387
00:17:22,058 --> 00:17:23,849
And I'm going to get
the coefficients-- I'm

388
00:17:23,849 --> 00:17:27,115
going to be careful
about the coefficients a,

389
00:17:27,115 --> 00:17:31,963
so root a over 4 [INAUDIBLE] pi.

390
00:17:36,130 --> 00:17:42,340
e to the minus k squared
a squared over 2.

391
00:17:42,340 --> 00:17:44,060
So here, this is
just to remind us

392
00:17:44,060 --> 00:17:45,775
if we have a
Gaussian of width a,

393
00:17:45,775 --> 00:17:49,780
the Fourier transform is
Gaussian of width 1 upon a.

394
00:17:49,780 --> 00:17:53,930
So that's momentum and position
uncertainty in Fourier space.

395
00:17:53,930 --> 00:17:56,085
And you did this
in problem set two.

396
00:17:56,085 --> 00:17:58,460
So this is an alternate way
of writing this wave function

397
00:17:58,460 --> 00:17:59,570
through its Fourier transform.

398
00:17:59,570 --> 00:18:00,570
Everyone cool with that?

399
00:18:05,334 --> 00:18:07,000
But the nice thing
about this is that we

400
00:18:07,000 --> 00:18:09,460
know how to time evolve
this wave function.

401
00:18:09,460 --> 00:18:18,100
This tells us that that psi
of xt is equal to integral--

402
00:18:18,100 --> 00:18:21,840
and I'm going to get these
coefficients all pulled out--

403
00:18:21,840 --> 00:18:29,620
integral of root a over root pi.

404
00:18:29,620 --> 00:18:34,040
Integral dk upon root 2 pi.

405
00:18:34,040 --> 00:18:38,980
e to the i kx minus omega t.

406
00:18:38,980 --> 00:18:41,140
And remember, Omega
it depends on k,

407
00:18:41,140 --> 00:18:42,650
because it's h-bar
squared k squared

408
00:18:42,650 --> 00:18:44,900
upon 2m divided by h-bar.

409
00:18:44,900 --> 00:18:47,200
Times our Gaussian,
e to the minus

410
00:18:47,200 --> 00:18:50,215
k squared a squared upon 2.

411
00:18:54,080 --> 00:18:55,330
So everyone cool with that?

412
00:18:55,330 --> 00:18:57,200
So all I've done is
I've taken this line,

413
00:18:57,200 --> 00:18:59,630
I pulled out the
constant, and I've

414
00:18:59,630 --> 00:19:04,810
added the time evolution
of the Fourier mode.

415
00:19:04,810 --> 00:19:07,389
So now we have this integral
to give us the full,

416
00:19:07,389 --> 00:19:09,180
OK, so now we just have
to do the integral.

417
00:19:09,180 --> 00:19:10,520
And this is not so hard to do.

418
00:19:10,520 --> 00:19:12,800
But what makes it totally
tractable is that if I just

419
00:19:12,800 --> 00:19:14,757
do this with some
function of omega of k--

420
00:19:14,757 --> 00:19:17,090
I mean that's complicated,
know how to do that integral.

421
00:19:17,090 --> 00:19:21,300
But I happen to know that e
is h-bar squared k squared

422
00:19:21,300 --> 00:19:22,850
upon 2m, which
means that omega is

423
00:19:22,850 --> 00:19:26,980
equal to h-bar k
squared upon 2m.

424
00:19:26,980 --> 00:19:35,401
So I can rewrite this as
minus h-bar k squared upon 2m.

425
00:19:35,401 --> 00:19:35,900
Yes?

426
00:19:35,900 --> 00:19:37,649
AUDIENCE: Where is the
fourth [INAUDIBLE]?

427
00:19:39,769 --> 00:19:41,810
PROFESSOR: That's a square
root of a square root.

428
00:19:41,810 --> 00:19:42,475
AUDIENCE: Oh, I didn't see it.

429
00:19:42,475 --> 00:19:42,930
My bad.

430
00:19:42,930 --> 00:19:43,310
PROFESSOR: No, no.

431
00:19:43,310 --> 00:19:43,810
That's OK.

432
00:19:43,810 --> 00:19:47,640
It's a horrible,
horrible factor.

433
00:19:47,640 --> 00:19:50,522
So what do we take
away from this?

434
00:19:50,522 --> 00:19:52,230
Well, this can be
written in a nice form.

435
00:19:52,230 --> 00:19:53,560
Note that here we
have a k squared,

436
00:19:53,560 --> 00:19:55,351
here we have a k squared,
here we have a k.

437
00:19:55,351 --> 00:19:58,130
This is an integral over k,
so this is still Gaussian.

438
00:19:58,130 --> 00:20:01,400
It's still the exponential
of a quadratic function of k.

439
00:20:01,400 --> 00:20:03,857
So we can use our formulas
for Gaussian integrals

440
00:20:03,857 --> 00:20:04,690
to do this integral.

441
00:20:04,690 --> 00:20:06,273
So to make that a
little more obvious,

442
00:20:06,273 --> 00:20:08,694
let's simplify the form of this.

443
00:20:08,694 --> 00:20:10,110
So the form of
this is again going

444
00:20:10,110 --> 00:20:13,690
to be square root of
a over square root pi.

445
00:20:13,690 --> 00:20:18,530
Integral dk upon root 2 pi.

446
00:20:18,530 --> 00:20:19,699
e to the i kx.

447
00:20:19,699 --> 00:20:21,990
And I'm going to take this
term and this term and group

448
00:20:21,990 --> 00:20:22,900
them together because
they both have

449
00:20:22,900 --> 00:20:24,350
a k squared in the exponential.

450
00:20:24,350 --> 00:20:28,540
e to the minus k
squared upon 2 times--

451
00:20:28,540 --> 00:20:34,510
now instead of just a squared,
we have a squared plus i h-bar

452
00:20:34,510 --> 00:20:38,290
upon 2m t-- there's
a typo in my notes.

453
00:20:38,290 --> 00:20:40,890
Crap.

454
00:20:40,890 --> 00:20:41,390
Good.

455
00:20:41,390 --> 00:20:43,150
So before we do
anything else, let's

456
00:20:43,150 --> 00:20:44,620
just check dimensional analysis.

457
00:20:44,620 --> 00:20:46,287
This has units of one
of length squared.

458
00:20:46,287 --> 00:20:48,286
So this had better have
units of length squared.

459
00:20:48,286 --> 00:20:49,680
That's a length squared, good.

460
00:20:49,680 --> 00:20:51,000
This, is this a length squared?

461
00:20:51,000 --> 00:20:53,720
So that's momentum
times length, which

462
00:20:53,720 --> 00:20:56,520
is mass times length over time.

463
00:20:56,520 --> 00:20:59,700
Divide by mass, multiplied by
time, so that's length squared.

464
00:20:59,700 --> 00:21:00,255
Good.

465
00:21:00,255 --> 00:21:01,255
So our units make sense.

466
00:21:01,255 --> 00:21:03,150
AUDIENCE: [INAUDIBLE].

467
00:21:03,150 --> 00:21:04,630
PROFESSOR: Sorry?

468
00:21:04,630 --> 00:21:05,800
AUDIENCE: [INAUDIBLE].

469
00:21:05,800 --> 00:21:07,512
PROFESSOR: This, too, we need.

470
00:21:07,512 --> 00:21:08,554
AUDIENCE: In terms of 2m.

471
00:21:08,554 --> 00:21:09,553
PROFESSOR: Oh, this one.

472
00:21:09,553 --> 00:21:10,150
Good.

473
00:21:10,150 --> 00:21:11,170
Thank you.

474
00:21:11,170 --> 00:21:11,670
Yes.

475
00:21:11,670 --> 00:21:12,169
Thanks.

476
00:21:16,320 --> 00:21:18,460
So this is again a Gaussian.

477
00:21:18,460 --> 00:21:20,850
And look, before we
knew we have a Gaussian,

478
00:21:20,850 --> 00:21:23,380
the Fourier transform
is just this Gaussian.

479
00:21:23,380 --> 00:21:26,310
The only difference is it's
now a Gaussian with width.

480
00:21:26,310 --> 00:21:28,237
Not a, but a complex number.

481
00:21:28,237 --> 00:21:30,320
But if you go through the
analysis of the Gaussian

482
00:21:30,320 --> 00:21:32,190
integral, that's perfectly fine.

483
00:21:32,190 --> 00:21:33,730
It's not a problem at all.

484
00:21:33,730 --> 00:21:38,700
The effective width is this.

485
00:21:38,700 --> 00:21:40,450
So what does that tell
you about psi of t?

486
00:21:40,450 --> 00:21:44,420
So this tells you
that psi of x and t

487
00:21:44,420 --> 00:21:49,800
is equal to root a over root pi.

488
00:21:49,800 --> 00:21:50,840
And now the width.

489
00:21:50,840 --> 00:21:53,130
So we have the 1 over root a.

490
00:21:53,130 --> 00:21:55,490
But now the effective width
is this guy, a squared

491
00:21:55,490 --> 00:22:04,452
plus-- so it's square root of
a squared plus i h-bar upon mt.

492
00:22:04,452 --> 00:22:13,660
e to the minus x squared
over 4a squared plus-- oops,

493
00:22:13,660 --> 00:22:15,590
where did my 4-- there's
a 2 that-- there's

494
00:22:15,590 --> 00:22:17,025
a spurious 2 somewhere.

495
00:22:17,025 --> 00:22:19,260
I'm not sure where
that 2 came from.

496
00:22:19,260 --> 00:22:20,930
There should be a 2 there.

497
00:22:20,930 --> 00:22:24,360
a squared plus i h-bar upon mt.

498
00:22:39,160 --> 00:22:43,130
So at this point it's not
totally transparent to me

499
00:22:43,130 --> 00:22:45,450
what exactly this wave
function is telling me,

500
00:22:45,450 --> 00:22:48,486
because there's still a phase
downstairs in the width.

501
00:22:48,486 --> 00:22:50,110
So to get a little
more intuition let's

502
00:22:50,110 --> 00:22:51,430
look at something
that's purely real.

503
00:22:51,430 --> 00:22:53,513
Let's look at the probability
to find the particle

504
00:22:53,513 --> 00:22:55,072
at position x at time t.

505
00:22:55,072 --> 00:22:56,780
That's just the norm
squared of this guy.

506
00:22:56,780 --> 00:22:58,696
And if you go through a
little bit of algebra,

507
00:22:58,696 --> 00:23:02,610
this is equal to 1
over root pi times--

508
00:23:02,610 --> 00:23:09,170
I'm going to write out
three factors-- times 1

509
00:23:09,170 --> 00:23:18,850
over square root of-- yes,
good-- a squared plus h-bar

510
00:23:18,850 --> 00:23:27,280
upon 2ma squared
t squared times e

511
00:23:27,280 --> 00:23:35,480
to the minus x squared upon
2a squared plus h-bar squared

512
00:23:35,480 --> 00:23:37,800
over 2ma squared.

513
00:23:37,800 --> 00:23:43,080
Oops-- h-bar over 2ma squared t.

514
00:23:43,080 --> 00:23:45,760
So what do we get?

515
00:23:45,760 --> 00:23:49,610
The probability distribution
is again a Gaussian.

516
00:23:49,610 --> 00:23:51,480
But at any given
moment in time--

517
00:23:51,480 --> 00:23:53,830
this should be t squared--
at any given moment in time

518
00:23:53,830 --> 00:23:55,380
the width is changing.

519
00:23:55,380 --> 00:23:57,710
The width of this Gaussian
is changing in time.

520
00:23:57,710 --> 00:23:59,550
Notice this is
purely real again.

521
00:23:59,550 --> 00:24:02,720
And meanwhile, the
amplitude is also changing.

522
00:24:02,720 --> 00:24:05,260
And notice that it's changing
in the following way.

523
00:24:05,260 --> 00:24:09,720
At time 0, this is the
least it can possibly be.

524
00:24:09,720 --> 00:24:11,420
At time 0 this
denominator is the least

525
00:24:11,420 --> 00:24:13,045
it can possibly be
because that's gone.

526
00:24:13,045 --> 00:24:14,800
At any positive
time, the denominator

527
00:24:14,800 --> 00:24:17,550
is larger so the
probability has dropped off.

528
00:24:17,550 --> 00:24:21,470
So the probability at any
given point is decreasing.

529
00:24:21,470 --> 00:24:25,460
Except for we also
have this Gaussian

530
00:24:25,460 --> 00:24:28,580
whose width is increasing
in time, quadratically.

531
00:24:28,580 --> 00:24:30,450
It's getting wider
and wider and wider.

532
00:24:32,934 --> 00:24:34,850
And you can check-- so
let's check dimensional

533
00:24:34,850 --> 00:24:35,440
analysis again.

534
00:24:35,440 --> 00:24:37,148
This should have units
of length squared,

535
00:24:37,148 --> 00:24:39,160
and so this is
momentum times length

536
00:24:39,160 --> 00:24:43,110
divided by length divided by
mass-- that's just a velocity.

537
00:24:43,110 --> 00:24:45,050
So this is the velocity
squared times squared.

538
00:24:45,050 --> 00:24:45,550
That's good.

539
00:24:45,550 --> 00:24:47,110
So this has the correct units.

540
00:24:47,110 --> 00:24:48,960
So it's spreading
with the velocity

541
00:24:48,960 --> 00:24:51,800
v is equal to h-bar upon 2ma.

542
00:24:55,920 --> 00:24:58,130
And what was a? a was
the width at times

543
00:24:58,130 --> 00:24:59,670
0. a was the minimum width.

544
00:25:04,660 --> 00:25:09,550
So graphically, let's roll this
out-- what is this telling us?

545
00:25:13,720 --> 00:25:16,320
And incidentally, a
quick check, if this

546
00:25:16,320 --> 00:25:18,500
is supposed to be the
wave function at time t,

547
00:25:18,500 --> 00:25:21,250
it had better reproduce the
correct answer at time 0.

548
00:25:21,250 --> 00:25:24,010
So what is this wave
function at time 0?

549
00:25:24,010 --> 00:25:28,180
We lose that, we lose that, and
we recover the original wave

550
00:25:28,180 --> 00:25:30,290
function.

551
00:25:30,290 --> 00:25:31,010
So that's good.

552
00:25:39,399 --> 00:25:40,690
So graphically what's going on?

553
00:25:40,690 --> 00:25:43,740
So let's plot p of x and t.

554
00:25:43,740 --> 00:25:50,770
At time 0 we have a Gaussian
and it's centered at 0,

555
00:25:50,770 --> 00:25:54,890
and it's got a width,
which is just a.

556
00:25:54,890 --> 00:25:58,090
And this is at time t equals 0.

557
00:25:58,090 --> 00:26:04,640
At a subsequent time,
it's again centered

558
00:26:04,640 --> 00:26:07,810
at x equals 0--
modulating my bad art--

559
00:26:07,810 --> 00:26:13,310
but it's got a width, which
is a squared square root

560
00:26:13,310 --> 00:26:22,120
of a squared plus h-bar
upon 2ma squared, t squared.

561
00:26:26,250 --> 00:26:28,620
So it's become
shorter and wider such

562
00:26:28,620 --> 00:26:30,500
that it's still
properly normalized.

563
00:26:33,420 --> 00:26:35,910
And at much later
time it will be

564
00:26:35,910 --> 00:26:38,580
a much lower and much
broader Gaussian.

565
00:26:38,580 --> 00:26:42,880
Again, properly normalized, and
again centered at x equals 0.

566
00:26:42,880 --> 00:26:44,506
Is that cool?

567
00:26:44,506 --> 00:26:45,880
So what this says
is if you start

568
00:26:45,880 --> 00:26:49,520
with your well-localized
wave packet and you let go,

569
00:26:49,520 --> 00:26:50,070
it disperses.

570
00:26:52,625 --> 00:26:53,500
Does that make sense?

571
00:26:56,759 --> 00:26:57,550
Is that reasonable?

572
00:27:00,840 --> 00:27:01,800
How about this.

573
00:27:01,800 --> 00:27:04,630
What happens if you ran the
system backwards in time?

574
00:27:04,630 --> 00:27:09,090
If it's dispersing
and getting wider,

575
00:27:09,090 --> 00:27:10,780
then intuitively
you might expect

576
00:27:10,780 --> 00:27:15,201
that it'll get more sharp, if
you integrated it back in time.

577
00:27:15,201 --> 00:27:17,200
But, in fact, what happens
if we take t negative

578
00:27:17,200 --> 00:27:17,890
in this analysis?

579
00:27:17,890 --> 00:27:19,604
We didn't assume it
was positive, anyway.

580
00:27:19,604 --> 00:27:20,530
AUDIENCE: [INAUDIBLE].

581
00:27:20,530 --> 00:27:22,250
PROFESSOR: It disperses again.

582
00:27:22,250 --> 00:27:24,110
It comes from being a
very dispersed wave,

583
00:27:24,110 --> 00:27:26,193
to whap, to being very
well [INAUDIBLE], and whap,

584
00:27:26,193 --> 00:27:27,310
it disperses out again.

585
00:27:27,310 --> 00:27:32,080
And notice that the sharper the
wave function was localized,

586
00:27:32,080 --> 00:27:35,630
the smaller a was in the first
place, the faster it disperses.

587
00:27:35,630 --> 00:27:37,130
This has 1 upon a.

588
00:27:37,130 --> 00:27:38,780
It has an a downstairs.

589
00:27:38,780 --> 00:27:40,760
So the more sharp
it was at time 0,

590
00:27:40,760 --> 00:27:42,760
the faster it disperses away.

591
00:27:42,760 --> 00:27:45,320
This is what our
analysis predicts.

592
00:27:47,920 --> 00:27:50,635
So let me give you two
questions to think about.

593
00:27:50,635 --> 00:27:52,260
I'm not going to
answer them right now.

594
00:27:52,260 --> 00:27:55,522
I want you to think about
them-- not right-- well,

595
00:27:55,522 --> 00:27:57,480
think about them now,
but also think about them

596
00:27:57,480 --> 00:27:58,670
more broadly as you
do the problem set

597
00:27:58,670 --> 00:28:01,400
and as you're reading through
the reading for the next week.

598
00:28:01,400 --> 00:28:06,390
So the width in this example
was least at time t equals 0.

599
00:28:06,390 --> 00:28:07,617
Why?

600
00:28:07,617 --> 00:28:09,700
How could you build a wave
function, how could you

601
00:28:09,700 --> 00:28:13,630
take this result and build a
wave function whose width was

602
00:28:13,630 --> 00:28:16,150
minimum at some other
time, say, time t0.

603
00:28:18,872 --> 00:28:19,830
So that's one question.

604
00:28:19,830 --> 00:28:21,450
Second question.

605
00:28:21,450 --> 00:28:24,480
This wave function
is sitting still.

606
00:28:24,480 --> 00:28:26,725
How do you make it move?

607
00:28:26,725 --> 00:28:29,100
And either the second or the
third problem on the problem

608
00:28:29,100 --> 00:28:33,520
set is an introduction
to that question.

609
00:28:33,520 --> 00:28:36,380
So here's a challenge
to everyone.

610
00:28:36,380 --> 00:28:39,620
Construct a
well-localized wave packet

611
00:28:39,620 --> 00:28:42,360
that's moving, in the sense that
it has well-defined momentum

612
00:28:42,360 --> 00:28:44,742
expectation value.

613
00:28:44,742 --> 00:28:46,950
And again, hint, look at
your second or third problem

614
00:28:46,950 --> 00:28:48,690
on the problem set.

615
00:28:48,690 --> 00:28:51,090
And then repeat
this entire analysis

616
00:28:51,090 --> 00:28:56,940
and check how the probability
distribution evolves in time.

617
00:28:56,940 --> 00:28:58,990
And in particular,
what I'd like to do

618
00:28:58,990 --> 00:29:03,260
is verify that the probability
distribution disperses,

619
00:29:03,260 --> 00:29:05,920
but simultaneously
moves in the direction

620
00:29:05,920 --> 00:29:11,220
corresponding to the initial
momentum with that momentum.

621
00:29:11,220 --> 00:29:13,240
The center of the
wave packet moves

622
00:29:13,240 --> 00:29:16,562
according to the momentum that
you gave it in the first place.

623
00:29:16,562 --> 00:29:18,770
It's a free particle so
should the expectation value,

624
00:29:18,770 --> 00:29:22,460
the momentum change over time?

625
00:29:22,460 --> 00:29:23,640
No.

626
00:29:23,640 --> 00:29:24,910
And indeed it won't.

627
00:29:24,910 --> 00:29:27,910
So that's a challenge for you.

628
00:29:27,910 --> 00:29:30,345
OK, questions at this point?

629
00:29:30,345 --> 00:29:30,970
AUDIENCE: Yeah.

630
00:29:30,970 --> 00:29:34,770
So this wave or this
Gaussian's not moving,

631
00:29:34,770 --> 00:29:37,645
but it's centered over
0 at time 0 or x0?

632
00:29:37,645 --> 00:29:39,520
PROFESSOR: Sorry, this
is at x is equal to 0.

633
00:29:39,520 --> 00:29:40,061
AUDIENCE: OK.

634
00:29:40,061 --> 00:29:42,660
So is that-- so all
those [INAUDIBLE]

635
00:29:42,660 --> 00:29:44,034
are at x equals 0 over here.

636
00:29:44,034 --> 00:29:45,950
Shouldn't the amplitude
increase and not the--

637
00:29:45,950 --> 00:29:46,240
PROFESSOR: Good.

638
00:29:46,240 --> 00:29:47,448
So what I'm plotting-- sorry.

639
00:29:47,448 --> 00:29:49,570
So I'm plotting p of
x and t as a function

640
00:29:49,570 --> 00:29:51,160
of x for different times.

641
00:29:51,160 --> 00:29:52,650
So this is time 0.

642
00:29:52,650 --> 00:29:56,250
This was time t equals
something not 0, equals 1.

643
00:29:56,250 --> 00:30:00,320
And this is some time
t equal to large.

644
00:30:00,320 --> 00:30:01,668
Right.

645
00:30:01,668 --> 00:30:04,163
AUDIENCE: And so but
is it if x is always 0,

646
00:30:04,163 --> 00:30:08,155
then shouldn't the
[INAUDIBLE] not change

647
00:30:08,155 --> 00:30:10,224
because it would
equal [INAUDIBLE]?

648
00:30:10,224 --> 00:30:10,890
PROFESSOR: Good.

649
00:30:10,890 --> 00:30:13,790
So if x is 0, then the
Gaussian contribution

650
00:30:13,790 --> 00:30:15,030
is always giving me 1.

651
00:30:15,030 --> 00:30:17,245
But there's still
this amplitude.

652
00:30:17,245 --> 00:30:18,745
And the amplitude
is the denominator

653
00:30:18,745 --> 00:30:20,370
is getting larger
and larger with time.

654
00:30:20,370 --> 00:30:23,440
So, indeed, the amplitude
should be falling off.

655
00:30:23,440 --> 00:30:24,291
Is that--

656
00:30:24,291 --> 00:30:26,040
AUDIENCE: Yeah, but
will the width change?

657
00:30:26,040 --> 00:30:27,160
PROFESSOR: Yeah, absolutely.

658
00:30:27,160 --> 00:30:29,160
So what the width is, the
width is saying, look,

659
00:30:29,160 --> 00:30:32,520
how rapidly as we increase x,
how rapidly does this fall off

660
00:30:32,520 --> 00:30:33,650
as a Gaussian.

661
00:30:33,650 --> 00:30:37,130
And so if time is increasing,
if time is very, very

662
00:30:37,130 --> 00:30:38,980
large, than this
denominator's huge.

663
00:30:38,980 --> 00:30:42,010
So in order for this Gaussian
to suppress the wave function,

664
00:30:42,010 --> 00:30:43,520
x has to be very, very large.

665
00:30:43,520 --> 00:30:46,810
AUDIENCE: So we're assuming
[INAUDIBLE] x hasn't changed.

666
00:30:46,810 --> 00:30:47,700
PROFESSOR: Ah, good.

667
00:30:47,700 --> 00:30:50,520
Remember, so this x is not
the position of the particle.

668
00:30:50,520 --> 00:30:54,020
x is the position at which we're
evaluating the probability.

669
00:30:54,020 --> 00:30:56,022
So what we're plotting
is we're plotting-- good.

670
00:30:56,022 --> 00:30:57,980
Yeah, this is an easy
thing to get confused by.

671
00:30:57,980 --> 00:30:59,354
What is this
quantity telling us?

672
00:30:59,354 --> 00:31:02,000
This quantity is telling us the
probability, the probability

673
00:31:02,000 --> 00:31:04,710
[INAUDIBLE, that we find
the particle at position

674
00:31:04,710 --> 00:31:05,980
x at time t.

675
00:31:05,980 --> 00:31:08,109
So the x is like, look,
where do you want to look?

676
00:31:08,109 --> 00:31:09,900
You tell me the x and
I tell you how likely

677
00:31:09,900 --> 00:31:11,100
it is to be found there.

678
00:31:11,100 --> 00:31:13,844
So what this is telling
us is that the probability

679
00:31:13,844 --> 00:31:16,010
distribution to find the
particles start out sharply

680
00:31:16,010 --> 00:31:18,384
peaked, but then it becomes
more more and more dispersed.

681
00:31:18,384 --> 00:31:20,140
So at the initial
time, how likely

682
00:31:20,140 --> 00:31:22,980
am I to find the
particle, say, here?

683
00:31:22,980 --> 00:31:23,900
Not at all.

684
00:31:23,900 --> 00:31:26,700
But at a very late time, who
knows, it could be there.

685
00:31:26,700 --> 00:31:29,160
I mean my confidence
is very, very limited

686
00:31:29,160 --> 00:31:31,400
because the probability
distribution's very wide.

687
00:31:31,400 --> 00:31:32,210
Did that answer your question.

688
00:31:32,210 --> 00:31:32,523
AUDIENCE: Yes.

689
00:31:32,523 --> 00:31:32,680
PROFESSOR: Great.

690
00:31:32,680 --> 00:31:33,568
Yup?

691
00:31:33,568 --> 00:31:36,496
AUDIENCE: [INAUDIBLE] what
happens after [INAUDIBLE].

692
00:31:39,430 --> 00:31:40,730
PROFESSOR: Excellent question.

693
00:31:40,730 --> 00:31:41,896
That's a fantastic question.

694
00:31:41,896 --> 00:31:44,170
So we've talked about
measurement of position.

695
00:31:44,170 --> 00:31:45,878
We've talked about
measure-- So after you

696
00:31:45,878 --> 00:31:47,330
measure an
observable, the system

697
00:31:47,330 --> 00:31:50,800
is left in an eigenstate of
the observable corresponding

698
00:31:50,800 --> 00:31:52,360
to the observed eigenvalue.

699
00:31:52,360 --> 00:31:55,542
So suppose I have some potential
or I have some particle

700
00:31:55,542 --> 00:31:56,500
and it's moving around.

701
00:31:56,500 --> 00:31:59,270
It's in some complicated wave
function, and some complicated

702
00:31:59,270 --> 00:31:59,780
state.

703
00:31:59,780 --> 00:32:01,766
And then you measure
its position to be here.

704
00:32:01,766 --> 00:32:03,140
But my measurement
isn't perfect.

705
00:32:03,140 --> 00:32:05,181
I know it's here with some
reasonable confidence,

706
00:32:05,181 --> 00:32:06,980
with some reasonable width.

707
00:32:06,980 --> 00:32:08,862
What's going to be
the state thereafter?

708
00:32:08,862 --> 00:32:10,820
Well, it's going to be
in a state corresponding

709
00:32:10,820 --> 00:32:13,690
to being more or less here
with some uncertainty.

710
00:32:13,690 --> 00:32:15,230
Oh look, there's that state.

711
00:32:15,230 --> 00:32:16,957
And what happened subsequently?

712
00:32:16,957 --> 00:32:19,290
What happened subsequently
is my probability immediately

713
00:32:19,290 --> 00:32:21,085
decreases.

714
00:32:21,085 --> 00:32:22,710
And this is exactly
what we saw before.

715
00:32:22,710 --> 00:32:23,570
Thank you for asking
this question.

716
00:32:23,570 --> 00:32:24,450
It's a very good question.

717
00:32:24,450 --> 00:32:25,400
This is exactly
what we said before.

718
00:32:25,400 --> 00:32:27,720
If we can make an arbitrarily
precise measurement

719
00:32:27,720 --> 00:32:29,538
of the position, what
would be the width?

720
00:32:29,538 --> 00:32:30,596
AUDIENCE: [INAUDIBLE].

721
00:32:30,596 --> 00:32:31,262
PROFESSOR: Yeah.

722
00:32:31,262 --> 00:32:33,370
It'd be delta, so
with width would be 0.

723
00:32:33,370 --> 00:32:34,850
That little a would be 0.

724
00:32:34,850 --> 00:32:36,670
So how rapidly does it disperse?

725
00:32:36,670 --> 00:32:37,980
AUDIENCE: [INAUDIBLE].

726
00:32:37,980 --> 00:32:39,120
PROFESSOR: Pretty fast.

727
00:32:39,120 --> 00:32:40,050
And you have to
worry a little bit

728
00:32:40,050 --> 00:32:42,050
because that's getting
a little relativistic.

729
00:32:42,050 --> 00:32:42,949
Yeah.

730
00:32:42,949 --> 00:32:45,240
AUDIENCE: If you make a
measurement that's not perfect,

731
00:32:45,240 --> 00:32:47,250
how do you account
for that [INAUDIBLE]?

732
00:32:47,250 --> 00:32:47,916
PROFESSOR: Yeah.

733
00:32:47,916 --> 00:32:49,362
So OK.

734
00:32:49,362 --> 00:32:50,320
That's a good question.

735
00:32:50,320 --> 00:32:52,326
So you make a measurement,
it's not perfect,

736
00:32:52,326 --> 00:32:53,450
there was some uncertainty.

737
00:32:53,450 --> 00:32:55,530
And so let me rephrase
this question slightly.

738
00:32:55,530 --> 00:32:57,580
So here's another
version of this question.

739
00:32:57,580 --> 00:32:59,890
Suppose I make a
measurement of the system.

740
00:32:59,890 --> 00:33:02,480
If there are two possible
configurations, two

741
00:33:02,480 --> 00:33:04,010
possible states,
and they measure

742
00:33:04,010 --> 00:33:05,860
an eigenvalue corresponding
to one of them,

743
00:33:05,860 --> 00:33:07,937
then I know what state
the system is in.

744
00:33:07,937 --> 00:33:10,270
And then again, if I'm measuring
something like position

745
00:33:10,270 --> 00:33:12,061
where I'm inescapably
going to be measuring

746
00:33:12,061 --> 00:33:13,630
that position with
some uncertainty,

747
00:33:13,630 --> 00:33:15,320
what state is it
left in afterwards?

748
00:33:15,320 --> 00:33:17,590
Do I know what state
it's left in afterwards?

749
00:33:17,590 --> 00:33:19,400
No, I-- I know
approximately what state.

750
00:33:19,400 --> 00:33:20,858
It's going to be
a state corr-- OK,

751
00:33:20,858 --> 00:33:24,350
so then I have to have some
model for the probability

752
00:33:24,350 --> 00:33:26,010
distribution of
being in a state.

753
00:33:26,010 --> 00:33:27,536
But usually what
we'll do is we'll

754
00:33:27,536 --> 00:33:28,660
say we'll approximate that.

755
00:33:28,660 --> 00:33:30,058
We'll model that unknown.

756
00:33:30,058 --> 00:33:31,516
We don't know
exactly what state it

757
00:33:31,516 --> 00:33:33,210
is because we don't know exactly
what position we measured.

758
00:33:33,210 --> 00:33:35,220
Well, model that by
saying, look, by the law

759
00:33:35,220 --> 00:33:38,722
of large numbers it's going to
be roughly a Gaussian centered

760
00:33:38,722 --> 00:33:40,430
around the position
with the width, which

761
00:33:40,430 --> 00:33:42,536
is our expected uncertainty.

762
00:33:42,536 --> 00:33:44,910
Now, it depends on exactly
what measurement you're doing.

763
00:33:44,910 --> 00:33:46,701
Sometimes that's not
the right thing to do,

764
00:33:46,701 --> 00:33:48,900
but that's sort of a
maximally naive thing.

765
00:33:48,900 --> 00:33:50,900
This is an interesting
but complicated question,

766
00:33:50,900 --> 00:33:52,869
so come ask me in office hours.

767
00:33:52,869 --> 00:33:54,801
Yeah.

768
00:33:54,801 --> 00:33:56,733
AUDIENCE: So if this
probability isn't moving,

769
00:33:56,733 --> 00:33:59,160
what is that velocity we
[INAUDIBLE] over there?

770
00:33:59,160 --> 00:33:59,760
PROFESSOR: Yeah, exactly.

771
00:33:59,760 --> 00:34:00,690
So what is this velocity?

772
00:34:00,690 --> 00:34:02,356
So this velocity,
what is it telling us?

773
00:34:02,356 --> 00:34:05,500
It's telling us how rapidly
the wave spreads out.

774
00:34:05,500 --> 00:34:08,070
So does that indicate
anything moving?

775
00:34:08,070 --> 00:34:08,570
No.

776
00:34:08,570 --> 00:34:11,822
What's the expected value
of the position at any time?

777
00:34:11,822 --> 00:34:12,570
AUDIENCE: 0.

778
00:34:12,570 --> 00:34:14,139
PROFESSOR: 0, because
this is an even function.

779
00:34:14,139 --> 00:34:14,780
What's the position?

780
00:34:14,780 --> 00:34:16,480
Well, it's just as likely
to be here as here.

781
00:34:16,480 --> 00:34:18,440
It's just that the
probability of being here

782
00:34:18,440 --> 00:34:19,864
gets greater and greater
over time for a while

783
00:34:19,864 --> 00:34:21,613
and then it falls off
because the Gaussian

784
00:34:21,613 --> 00:34:23,000
is spreading and falling.

785
00:34:23,000 --> 00:34:23,500
Yeah.

786
00:34:23,500 --> 00:34:24,110
One last question.

787
00:34:24,110 --> 00:34:24,489
Yup.

788
00:34:24,489 --> 00:34:25,691
AUDIENCE: Can you tell
me a little bit more

789
00:34:25,691 --> 00:34:27,650
about negative time is
the same as positive time?

790
00:34:27,650 --> 00:34:28,239
PROFESSOR: Yeah.

791
00:34:28,239 --> 00:34:30,090
So the question is can
you talk a little bit more

792
00:34:30,090 --> 00:34:31,845
about why the behavior
as time increases

793
00:34:31,845 --> 00:34:36,010
is the same as the behavior
as we go backwards in time.

794
00:34:36,010 --> 00:34:38,230
And I'm not sure how
best to answer that,

795
00:34:38,230 --> 00:34:40,040
but let me give it a go.

796
00:34:40,040 --> 00:34:43,451
So the first thing is to
say look at our solution

797
00:34:43,451 --> 00:34:45,159
and look at the
probability distribution.

798
00:34:45,159 --> 00:34:47,909
The probability distribution
is completely even in time.

799
00:34:47,909 --> 00:34:49,511
It depends only on t squared.

800
00:34:49,511 --> 00:34:50,969
So if I ask what
happens at time t,

801
00:34:50,969 --> 00:34:53,130
it's the same as what
happened at time minus t.

802
00:34:53,130 --> 00:34:56,239
That's interesting in a
slightly unusual situation.

803
00:34:56,239 --> 00:34:57,780
Let's look back at
the wave function.

804
00:34:57,780 --> 00:34:59,774
Is that true of
the wave function?

805
00:34:59,774 --> 00:35:00,690
AUDIENCE: [INAUDIBLE].

806
00:35:00,690 --> 00:35:01,273
PROFESSOR: No.

807
00:35:01,273 --> 00:35:04,000
Because the phase
and the complex parts

808
00:35:04,000 --> 00:35:08,670
depend on-- OK, so the
amplitude doesn't depend on

809
00:35:08,670 --> 00:35:09,930
whether it's t or minus t.

810
00:35:09,930 --> 00:35:11,000
But the phase does.

811
00:35:11,000 --> 00:35:12,610
And that's a useful clue.

812
00:35:12,610 --> 00:35:13,730
Here's a better way.

813
00:35:13,730 --> 00:35:16,526
What was the weird thing that
made this t reversal invariant?

814
00:35:16,526 --> 00:35:18,150
There's another weird
property of this,

815
00:35:18,150 --> 00:35:19,150
which is that it's
centered at 0.

816
00:35:19,150 --> 00:35:20,080
It's sitting still.

817
00:35:20,080 --> 00:35:24,280
If we had given it momentum that
would also not have been true.

818
00:35:24,280 --> 00:35:28,310
So the best answer I
can give you, though,

819
00:35:28,310 --> 00:35:29,920
requires knowing
what happens when

820
00:35:29,920 --> 00:35:31,360
you have a finite momentum.

821
00:35:31,360 --> 00:35:33,440
So let me just sketch quickly.

822
00:35:33,440 --> 00:35:36,020
So when we have a momentum,
how do we take this wave packet

823
00:35:36,020 --> 00:35:40,000
and give it momentum k naught?

824
00:35:40,000 --> 00:35:42,180
So this is problem three
on your problem set,

825
00:35:42,180 --> 00:35:44,545
but I will answer it for you.

826
00:35:44,545 --> 00:35:47,770
You'd multiply by phase
e to the i k naught x.

827
00:35:52,270 --> 00:35:54,380
And the expectation
value will now

828
00:35:54,380 --> 00:36:00,080
shift to be h-bar k
naught for momentum.

829
00:36:00,080 --> 00:36:01,629
So what's that
going to do for us?

830
00:36:01,629 --> 00:36:03,170
Well, as we do the
Fourier transform,

831
00:36:03,170 --> 00:36:06,044
that's going to shift
k to k minus k naught.

832
00:36:06,044 --> 00:36:07,710
And so we can repeat
the entire analysis

833
00:36:07,710 --> 00:36:10,680
and get roughly k
and k naught here.

834
00:36:10,680 --> 00:36:16,290
The important thing is
that we end up with phases.

835
00:36:16,290 --> 00:36:20,360
So the way to read
off time 0 is going

836
00:36:20,360 --> 00:36:22,220
to be-- well, many
answers to this.

837
00:36:22,220 --> 00:36:25,560
As time increases or decreases,
we'll see from this phase

838
00:36:25,560 --> 00:36:28,417
that the entire wave packet
is going to continue moving,

839
00:36:28,417 --> 00:36:30,000
it's going to continue
walking across.

840
00:36:30,000 --> 00:36:32,920
And that breaks the
t minus t invariance.

841
00:36:32,920 --> 00:36:36,232
Because at minus t it
would have been over here.

842
00:36:36,232 --> 00:36:37,940
And then think about
that first question.

843
00:36:37,940 --> 00:36:40,500
How would you make it
become minimum wave packet

844
00:36:40,500 --> 00:36:43,032
at some time which
is not time equals 0?

845
00:36:43,032 --> 00:36:44,740
So when you have
answers to both of those

846
00:36:44,740 --> 00:36:45,790
you'll have the answer
to your question.

847
00:36:45,790 --> 00:36:46,145
Yeah.

848
00:36:46,145 --> 00:36:47,561
AUDIENCE: What
does it really mean

849
00:36:47,561 --> 00:36:49,888
for us to say that we have
a probability distribution

850
00:36:49,888 --> 00:36:53,255
and measure something
like negative time?

851
00:36:53,255 --> 00:36:57,462
Because it seems like we did
something at time equals 0,

852
00:36:57,462 --> 00:36:59,630
and we say the
probability that we

853
00:36:59,630 --> 00:37:01,400
would have found it
[INAUDIBLE] is here.

854
00:37:01,400 --> 00:37:01,755
PROFESSOR: Right.

855
00:37:01,755 --> 00:37:04,230
AUDIENCE: It seems that you
have to be aware [INAUDIBLE].

856
00:37:04,230 --> 00:37:04,896
PROFESSOR: Good.

857
00:37:04,896 --> 00:37:07,367
So that's not exactly what the
probability of distribution

858
00:37:07,367 --> 00:37:07,950
is telling us.

859
00:37:07,950 --> 00:37:09,470
So what the probability
and distribution is telling

860
00:37:09,470 --> 00:37:11,880
you is not what happens
when you make a measurement.

861
00:37:11,880 --> 00:37:14,530
It's given this state
what would happen?

862
00:37:14,530 --> 00:37:16,780
And so when you ask about
the probability distribution

863
00:37:16,780 --> 00:37:18,670
at time minus 7,
what you're saying

864
00:37:18,670 --> 00:37:21,100
is given the wave
function now, what must

865
00:37:21,100 --> 00:37:23,679
the wave function have
been a time earlier,

866
00:37:23,679 --> 00:37:25,720
such that Schrodinger
evolution, not measurement,

867
00:37:25,720 --> 00:37:28,260
but Shrodinger evolution
gives you this.

868
00:37:28,260 --> 00:37:30,850
So what that's telling you is
someone earlier-- had someone

869
00:37:30,850 --> 00:37:32,720
earlier been around
to do that experiment,

870
00:37:32,720 --> 00:37:34,012
what value would they have got?

871
00:37:34,012 --> 00:37:35,678
AUDIENCE: But wouldn't
that have changed

872
00:37:35,678 --> 00:37:36,860
the wave function of that?

873
00:37:36,860 --> 00:37:39,710
PROFESSOR: Absolutely, if
they had done the measurement.

874
00:37:39,710 --> 00:37:41,260
There's a difference between
knowing what the probability

875
00:37:41,260 --> 00:37:43,700
distribution would be if you
were to do a measurement,

876
00:37:43,700 --> 00:37:45,400
and actually doing a measurement
and changing the wave function.

877
00:37:45,400 --> 00:37:47,164
AUDIENCE: So for
the negative time,

878
00:37:47,164 --> 00:37:49,580
is it the earlier statement
about what the wave function--

879
00:37:49,580 --> 00:37:50,770
PROFESSOR: It's merely
saying about what the wave

880
00:37:50,770 --> 00:37:51,640
function was at an earlier time.

881
00:37:51,640 --> 00:37:53,666
AUDIENCE: It doesn't
really have measurements.

882
00:37:53,666 --> 00:37:55,790
PROFESSOR: No one's done
any measurements, exactly.

883
00:37:55,790 --> 00:37:57,170
Writing down the
probability distribution

884
00:37:57,170 --> 00:37:58,390
does not do a measurement.

885
00:37:58,390 --> 00:38:00,121
Exactly.

886
00:38:00,121 --> 00:38:00,870
One last question.

887
00:38:00,870 --> 00:38:01,910
Sorry, I really-- Yup.

888
00:38:01,910 --> 00:38:04,230
AUDIENCE: Why define it
negative 2, and then prepares

889
00:38:04,230 --> 00:38:05,619
the states and
its width times 0?

890
00:38:05,619 --> 00:38:07,410
PROFESSOR: Well, you
could ask the question

891
00:38:07,410 --> 00:38:08,285
slightly differently.

892
00:38:08,285 --> 00:38:12,010
If you had wanted to prepare
the state at time minus 7 such

893
00:38:12,010 --> 00:38:13,710
that you got this
state at time 0--

894
00:38:13,710 --> 00:38:14,390
AUDIENCE: OK.

895
00:38:14,390 --> 00:38:16,399
PROFESSOR: This is
the inverse of asking

896
00:38:16,399 --> 00:38:17,940
given that the state
is prepared now,

897
00:38:17,940 --> 00:38:20,150
what is it going
to be at time 7.

898
00:38:20,150 --> 00:38:22,110
So they're equally
reasonable questions.

899
00:38:22,110 --> 00:38:23,800
What should you
have done last week

900
00:38:23,800 --> 00:38:28,170
so that you got your
problem set turned in today?

901
00:38:28,170 --> 00:38:31,010
There was no problem
set due today.

902
00:38:31,010 --> 00:38:38,070
OK, so with that said, we've
done an analytic analysis--

903
00:38:38,070 --> 00:38:42,720
that was ridiculous-- we've done
a quick computation that showed

904
00:38:42,720 --> 00:38:49,640
analytically what happens
to a Gaussian wave packet.

905
00:39:06,370 --> 00:39:08,695
We've done an analysis
that showed us

906
00:39:08,695 --> 00:39:12,280
how the wave function
evolved over time.

907
00:39:12,280 --> 00:39:15,600
What I'd like to do now is use
some of the PhET simulations

908
00:39:15,600 --> 00:39:20,420
to see this effect, and also
to predict some more effects.

909
00:39:20,420 --> 00:39:22,170
So what I want you to
think about this as,

910
00:39:22,170 --> 00:39:24,330
so this is the PhET quantum
tunneling wave packet

911
00:39:24,330 --> 00:39:24,870
simulation.

912
00:39:24,870 --> 00:39:26,910
If you haven't played with
these, you totally should.

913
00:39:26,910 --> 00:39:27,710
They're fantastic.

914
00:39:27,710 --> 00:39:30,330
They're a great way of
developing some intuition.

915
00:39:30,330 --> 00:39:31,770
All props to the Colorado group.

916
00:39:31,770 --> 00:39:33,170
They're excellent.

917
00:39:33,170 --> 00:39:34,920
So what I'm going
to do is I'm going

918
00:39:34,920 --> 00:39:36,610
to run through a
series of experiments

919
00:39:36,610 --> 00:39:38,880
that we could, in principle,
do on a table top,

920
00:39:38,880 --> 00:39:41,050
although it would be
fabulously difficult.

921
00:39:41,050 --> 00:39:42,920
But the basic physics is simple.

922
00:39:42,920 --> 00:39:46,260
Instead I'm going to run them
on the computer on the table top

923
00:39:46,260 --> 00:39:48,882
because it's easier and cheaper.

924
00:39:48,882 --> 00:39:50,840
But I want you to think
of these as experiments

925
00:39:50,840 --> 00:39:52,260
that we're actually running.

926
00:39:52,260 --> 00:39:54,510
So here, what these diagram
show, for those of you who

927
00:39:54,510 --> 00:39:55,643
haven't played
with this, is this

928
00:39:55,643 --> 00:39:57,940
is the potential that I'm
going to be working with.

929
00:39:57,940 --> 00:39:59,860
The green, this
position, tells me

930
00:39:59,860 --> 00:40:03,180
the actual value of the
energy of my wave packet.

931
00:40:03,180 --> 00:40:05,890
And the width tells me how
broad that wave packet is.

932
00:40:05,890 --> 00:40:08,474
And roughly speaking,
how green it is

933
00:40:08,474 --> 00:40:10,890
tells me how much support I
have on that particular energy

934
00:40:10,890 --> 00:40:12,100
eigenstate.

935
00:40:12,100 --> 00:40:14,645
And I can change that by tuning
the initial width of the wave

936
00:40:14,645 --> 00:40:16,280
packet.

937
00:40:16,280 --> 00:40:18,790
So if I make width
very narrow, I

938
00:40:18,790 --> 00:40:20,900
need lots and lots of
different energy eigenstates

939
00:40:20,900 --> 00:40:22,870
to make an narrow
well-localized wave packet.

940
00:40:22,870 --> 00:40:25,830
And if I allow the wave
packet to be quite wide,

941
00:40:25,830 --> 00:40:27,840
then I don't need as
many energy eigenstates.

942
00:40:27,840 --> 00:40:30,390
So let me make that a
little more obvious.

943
00:40:30,390 --> 00:40:32,690
So fewer energy
eigenstates are needed,

944
00:40:32,690 --> 00:40:34,240
so we have a thinner
band in energy,

945
00:40:34,240 --> 00:40:35,698
more energy
eigenstates are needed,

946
00:40:35,698 --> 00:40:36,920
so I have more wide band.

947
00:40:36,920 --> 00:40:38,120
Is everyone cool with that?

948
00:40:38,120 --> 00:40:41,860
And what this program does is it
just integrates the Schrodinger

949
00:40:41,860 --> 00:40:45,420
equation-- well,
yeah, well it just

950
00:40:45,420 --> 00:40:47,530
integrates the
Schrodinger equation.

951
00:40:47,530 --> 00:40:50,860
So what I want to do is
I want to quickly-- oops.

952
00:40:56,130 --> 00:40:58,630
So this is the wave
function, its absolute value.

953
00:40:58,630 --> 00:41:02,666
And this is the probability
density, the norm squared.

954
00:41:02,666 --> 00:41:03,790
So let's see what happened.

955
00:41:03,790 --> 00:41:05,710
So what is this initial state?

956
00:41:05,710 --> 00:41:07,240
This initial
state-- in fact, I'm

957
00:41:07,240 --> 00:41:11,470
going to re-load
the configuration.

958
00:41:11,470 --> 00:41:14,070
So this initial
state corresponds

959
00:41:14,070 --> 00:41:18,860
to being more or less here
with some uncertainty.

960
00:41:18,860 --> 00:41:20,930
So that's our psi of x0.

961
00:41:20,930 --> 00:41:22,394
And let's see how this evolves.

962
00:41:22,394 --> 00:41:24,310
So first, before you
actually have it evolved,

963
00:41:24,310 --> 00:41:27,270
what do you think
is going to happen?

964
00:41:27,270 --> 00:41:32,660
And remember, it's got some
initial momentum as well.

965
00:41:32,660 --> 00:41:33,270
OK.

966
00:41:33,270 --> 00:41:36,000
So there it is evolving and
it's continuing to evolve,

967
00:41:36,000 --> 00:41:37,720
and it's really quite boring.

968
00:41:37,720 --> 00:41:39,730
This is a free particle.

969
00:41:39,730 --> 00:41:40,950
Wow, that was dull.

970
00:41:40,950 --> 00:41:42,550
OK, so let's try that again.

971
00:41:42,550 --> 00:41:44,355
Let's look also at
the real part and look

972
00:41:44,355 --> 00:41:47,452
at what the phase of the
wave function's doing.

973
00:41:47,452 --> 00:41:49,410
So here there's something
really cool going on.

974
00:41:49,410 --> 00:41:51,074
It might not be obvious.

975
00:41:51,074 --> 00:41:51,990
But look back at this.

976
00:41:51,990 --> 00:41:53,730
Look at how rapidly
the wave packet moves

977
00:41:53,730 --> 00:41:56,480
and how rapidly the phase moves.

978
00:41:56,480 --> 00:41:58,244
Which one's moving faster?

979
00:41:58,244 --> 00:41:59,160
AUDIENCE: [INAUDIBLE].

980
00:41:59,160 --> 00:42:00,610
PROFESSOR: Yeah,
the wave packet.

981
00:42:00,610 --> 00:42:02,082
And the wave
packet, it's kind of

982
00:42:02,082 --> 00:42:04,450
hard to tell exactly how
much faster it's moving.

983
00:42:04,450 --> 00:42:08,660
But we can get that from this.

984
00:42:08,660 --> 00:42:13,060
What's the velocity of a
single plane wave or the phase

985
00:42:13,060 --> 00:42:14,300
velocity?

986
00:42:14,300 --> 00:42:16,840
So the phase
velocity from problem

987
00:42:16,840 --> 00:42:23,590
set one is h-bar k squared upon
2m-- let me write this as omega

988
00:42:23,590 --> 00:42:29,999
upon k, which is just
equal to h-bar k upon 2m.

989
00:42:29,999 --> 00:42:31,790
But that's bad because
that's the momentum.

990
00:42:31,790 --> 00:42:35,690
Divide the momentum by the mass,
that's the classical velocity.

991
00:42:35,690 --> 00:42:38,824
So this says that the
phase of any plane wave

992
00:42:38,824 --> 00:42:40,615
is moving with half
the classical velocity.

993
00:42:44,460 --> 00:42:45,150
That's weird.

994
00:42:45,150 --> 00:42:46,691
On the other hand,
the group velocity

995
00:42:46,691 --> 00:42:48,860
of a wave packet, of a
normalizable wave packet

996
00:42:48,860 --> 00:42:51,460
is do mega dk.

997
00:42:51,460 --> 00:42:53,810
And do mega dk will pull it
down an extra factor of 2.

998
00:42:53,810 --> 00:42:56,442
This Is equal to h-bar k upon m.

999
00:42:56,442 --> 00:42:58,365
This is the classical velocity.

1000
00:42:58,365 --> 00:43:00,740
So this is the difference
between the phase and the group

1001
00:43:00,740 --> 00:43:05,160
velocity, and that's exactly
what we're seeing here.

1002
00:43:05,160 --> 00:43:06,660
So the first thing
you notice is you

1003
00:43:06,660 --> 00:43:08,980
see the difference between the
phase and the group velocity.

1004
00:43:08,980 --> 00:43:09,760
So that's good.

1005
00:43:09,760 --> 00:43:11,300
That should be reassuring.

1006
00:43:11,300 --> 00:43:11,930
But there's
something else that's

1007
00:43:11,930 --> 00:43:13,390
really quite nice about this.

1008
00:43:13,390 --> 00:43:18,347
Let's make the wave
packet really very narrow.

1009
00:43:18,347 --> 00:43:20,180
So here it's much more
narrow, and let's let

1010
00:43:20,180 --> 00:43:22,490
the system evolve.

1011
00:43:22,490 --> 00:43:24,490
And now you can see that
the probability density

1012
00:43:24,490 --> 00:43:26,760
is falling off quite rapidly.

1013
00:43:26,760 --> 00:43:29,360
Let's see that again.

1014
00:43:29,360 --> 00:43:30,886
Cool.

1015
00:43:30,886 --> 00:43:33,260
The probability distribution
is falling off very rapidly.

1016
00:43:33,260 --> 00:43:34,520
And there's also kind
of a cool thing that's

1017
00:43:34,520 --> 00:43:36,645
happening is you're seeing
an accumulation of phase

1018
00:43:36,645 --> 00:43:38,530
up here and a
diminution down here.

1019
00:43:38,530 --> 00:43:41,640
We can make that a
little more sharp.

1020
00:43:41,640 --> 00:43:44,840
Oops, sorry.

1021
00:43:44,840 --> 00:43:45,420
Yeah.

1022
00:43:45,420 --> 00:43:47,640
So that is a combination
of good things

1023
00:43:47,640 --> 00:43:49,520
and the code being
slightly badly behaved.

1024
00:43:51,984 --> 00:43:53,650
So the more narrow I
make that initial--

1025
00:43:53,650 --> 00:43:56,860
so let's-- so I made
that really narrow,

1026
00:43:56,860 --> 00:44:00,410
and how rapidly
does it fall off?

1027
00:44:00,410 --> 00:44:02,990
So that very narrow
wave packet by time 10

1028
00:44:02,990 --> 00:44:04,460
is like at a quarter.

1029
00:44:04,460 --> 00:44:10,460
But if we made it much wider,
that very narrow wave packet

1030
00:44:10,460 --> 00:44:14,364
started out much lower and it
stays roughly the same height.

1031
00:44:14,364 --> 00:44:16,030
It's just diminishing
very, very slowly.

1032
00:44:16,030 --> 00:44:19,140
And that's just the
1 over a effect.

1033
00:44:19,140 --> 00:44:21,690
So this is nice.

1034
00:44:21,690 --> 00:44:23,690
And I guess the way I'd
like to think about this

1035
00:44:23,690 --> 00:44:30,577
is this is confirming rather
nicely, our predictions.

1036
00:44:30,577 --> 00:44:32,660
So now I want to take a
slightly different system.

1037
00:44:32,660 --> 00:44:36,400
So this system is really
silly because we all

1038
00:44:36,400 --> 00:44:38,800
know what's going to happen.

1039
00:44:38,800 --> 00:44:44,319
Here's a hill, and if
this were the real world

1040
00:44:44,319 --> 00:44:46,360
I would think of this as
some potential hill that

1041
00:44:46,360 --> 00:44:48,902
has some nice finite fall off.

1042
00:44:48,902 --> 00:44:50,985
The sort of place you don't
want to go skiing off.

1043
00:44:50,985 --> 00:44:52,030
AUDIENCE: [INAUDIBLE].

1044
00:44:52,030 --> 00:44:54,030
PROFESSOR: Well, I don't.

1045
00:44:54,030 --> 00:44:57,050
So imagine you take
an object of mass m

1046
00:44:57,050 --> 00:45:00,990
and you let it slide along
in this potential, what

1047
00:45:00,990 --> 00:45:01,950
will happen?

1048
00:45:01,950 --> 00:45:03,852
It will move along
at some velocity.

1049
00:45:03,852 --> 00:45:06,060
It will get to this cliff,
it will fall off the cliff

1050
00:45:06,060 --> 00:45:08,870
and end up going much
faster at the bottom.

1051
00:45:08,870 --> 00:45:09,554
Yeah?

1052
00:45:09,554 --> 00:45:11,470
So if you kick it here,
it will get over there

1053
00:45:11,470 --> 00:45:13,640
and it will end up much faster.

1054
00:45:13,640 --> 00:45:20,460
What happens to
this wave packet.

1055
00:45:20,460 --> 00:45:22,210
So we're solving the
Schrodinger equation.

1056
00:45:22,210 --> 00:45:24,127
It's exactly the same
thing we've done before.

1057
00:45:24,127 --> 00:45:24,751
Let's solve it.

1058
00:45:24,751 --> 00:45:26,210
So here's our
initial wave packet,

1059
00:45:26,210 --> 00:45:27,730
nice and well-localized.

1060
00:45:27,730 --> 00:45:28,987
There's our wave function.

1061
00:45:28,987 --> 00:45:29,695
And what happens?

1062
00:45:38,104 --> 00:45:40,270
So at that point something
interesting has happened.

1063
00:45:40,270 --> 00:45:42,910
You see that for the most
part, most of the probability

1064
00:45:42,910 --> 00:45:45,574
is over here, but not all
the probability is over here.

1065
00:45:45,574 --> 00:45:46,990
There's still a
finite probability

1066
00:45:46,990 --> 00:45:51,150
that the particle stays
near the wall for a while.

1067
00:45:51,150 --> 00:45:54,300
Let's watch what happens to that
probability as time goes by.

1068
00:45:54,300 --> 00:45:56,220
And now you should
notice that it's

1069
00:45:56,220 --> 00:45:58,540
decayed into a
wave packet that's

1070
00:45:58,540 --> 00:46:00,440
over here, and then a
superposition of a wave

1071
00:46:00,440 --> 00:46:05,510
packet over here
moving to the left.

1072
00:46:05,510 --> 00:46:09,910
This wave has scattered off
the barrier going downhill.

1073
00:46:09,910 --> 00:46:15,170
It's mostly transmitted,
but some of it reflected.

1074
00:46:15,170 --> 00:46:16,240
Which is kind of spooky.

1075
00:46:16,240 --> 00:46:18,865
So let's watch that-- I'm going
to make this much more extreme.

1076
00:46:22,274 --> 00:46:24,190
So here's a much more
extreme version of this.

1077
00:46:24,190 --> 00:46:25,731
I'm now going to
make the energy very

1078
00:46:25,731 --> 00:46:28,477
close to the height
of the potential.

1079
00:46:28,477 --> 00:46:31,060
The energy is very, very close
to the height of the potential,

1080
00:46:31,060 --> 00:46:32,100
and the potential is far away.

1081
00:46:32,100 --> 00:46:34,516
We're still reasonably local--
I can make it a little less

1082
00:46:34,516 --> 00:46:38,135
localized just to really
crank up the suspense.

1083
00:46:40,710 --> 00:46:41,880
So watch what happens now.

1084
00:46:44,789 --> 00:46:46,580
So first off, what do
you expect to happen?

1085
00:46:46,580 --> 00:46:48,371
So we've made the energy
of the wave packet

1086
00:46:48,371 --> 00:46:50,440
just barely higher than
the energy of the hill.

1087
00:46:50,440 --> 00:46:52,815
So it's almost classically
disallowed from being up here.

1088
00:46:52,815 --> 00:46:54,439
What does that mean
about its velocity?

1089
00:46:54,439 --> 00:46:55,610
It's effective loss.

1090
00:46:55,610 --> 00:46:56,550
AUDIENCE: [INAUDIBLE].

1091
00:46:56,550 --> 00:46:56,810
PROFESSOR: Yeah.

1092
00:46:56,810 --> 00:46:58,790
The momentum, the expectation
value for momentum

1093
00:46:58,790 --> 00:47:00,000
should be very low
because it just

1094
00:47:00,000 --> 00:47:01,390
has a little bit
of spare energy.

1095
00:47:01,390 --> 00:47:04,260
So the expected value for the
momentum should be very low.

1096
00:47:04,260 --> 00:47:05,970
It should dribble, dribble,
dribble, dribble, dribble,

1097
00:47:05,970 --> 00:47:08,219
dribble, dribble till it
gets to the cliff, and then--

1098
00:47:08,219 --> 00:47:10,052
[WHISTLE]-- go flying off.

1099
00:47:10,052 --> 00:47:11,510
So let's see if
that's what we see.

1100
00:47:14,170 --> 00:47:20,750
So it's dribbling, but look
at the probability density

1101
00:47:20,750 --> 00:47:21,700
down here.

1102
00:47:21,700 --> 00:47:23,260
Oh, indeed, we do
see little waves,

1103
00:47:23,260 --> 00:47:25,320
and those waves are
moving off really fast.

1104
00:47:25,320 --> 00:47:29,470
But the amplitude is
exceedingly small.

1105
00:47:29,470 --> 00:47:32,320
Instead, what we're seeing is
a huge pile up of probability

1106
00:47:32,320 --> 00:47:37,180
just before the wall, similar
to what we saw a minute ago.

1107
00:47:37,180 --> 00:47:40,369
Except the probability
is much, much larger.

1108
00:47:40,369 --> 00:47:41,660
And what's going to happen now?

1109
00:47:44,873 --> 00:47:46,220
AUDIENCE: [INAUDIBLE].

1110
00:47:46,220 --> 00:47:47,220
PROFESSOR: It's leaking.

1111
00:47:47,220 --> 00:47:49,950
It's definitely leaking.

1112
00:47:49,950 --> 00:47:54,730
But an awful lot of
the probability density

1113
00:47:54,730 --> 00:47:57,340
is not going off the cliff.

1114
00:47:57,340 --> 00:47:59,650
This would be Thelma and
Louise cruising off the cliff

1115
00:47:59,650 --> 00:48:01,270
and then just not falling.

1116
00:48:01,270 --> 00:48:04,186
This is a very disconcerting--
you guys get to-- yeah.

1117
00:48:04,186 --> 00:48:06,600
[LAUGHTER]

1118
00:48:06,600 --> 00:48:08,820
So see what happens,
see what happened here.

1119
00:48:08,820 --> 00:48:12,480
We've got this nice big
amplitude to go across

1120
00:48:12,480 --> 00:48:14,502
or to reflect back,
to scatter back.

1121
00:48:14,502 --> 00:48:16,210
You really want to
fall down classically,

1122
00:48:16,210 --> 00:48:17,668
but quantum
mechanically you can't.

1123
00:48:17,668 --> 00:48:19,160
It's impossible.

1124
00:48:19,160 --> 00:48:22,185
So, in fact, the probability
they reflect back is 41,

1125
00:48:22,185 --> 00:48:23,810
and we're going to
see how to calculate

1126
00:48:23,810 --> 00:48:27,051
that in the next few lectures.

1127
00:48:27,051 --> 00:48:27,550
Yup.

1128
00:48:27,550 --> 00:48:31,237
AUDIENCE: So the transmission
was actually really high.

1129
00:48:31,237 --> 00:48:33,195
PROFESSOR: Transition
was reasonably high, yup.

1130
00:48:33,195 --> 00:48:35,003
AUDIENCE: And it was
really [INAUDIBLE].

1131
00:48:37,532 --> 00:48:38,990
PROFESSOR: It wasn't
really, sorry?

1132
00:48:38,990 --> 00:48:41,700
AUDIENCE: It wasn't really
apparent graphically.

1133
00:48:41,700 --> 00:48:43,200
PROFESSOR: Yeah,
it wasn't apparent.

1134
00:48:43,200 --> 00:48:44,908
The way it was apparent
was just the fact

1135
00:48:44,908 --> 00:48:47,380
that the amplitude that bounced
back was relatively small.

1136
00:48:47,380 --> 00:48:49,770
So what was going is any
little bit of probability that

1137
00:48:49,770 --> 00:48:52,100
fell down had a large
effective momentum.

1138
00:48:52,100 --> 00:48:54,690
Just ran off the
screen very rapidly.

1139
00:48:54,690 --> 00:48:57,980
So it's hard to see that in
the simulation, it's true.

1140
00:48:57,980 --> 00:49:02,030
We can make the-- we can
squeeze-- ah, there we go.

1141
00:49:02,030 --> 00:49:04,490
So here is-- that's
about as good

1142
00:49:04,490 --> 00:49:07,072
as I'm going to be able to get.

1143
00:49:07,072 --> 00:49:08,780
And it's just going
to be glacially slow.

1144
00:49:14,229 --> 00:49:16,270
But this is just going to
be preposterously slow.

1145
00:49:16,270 --> 00:49:18,430
I'm not even sure
there's much point

1146
00:49:18,430 --> 00:49:20,450
in-- But you can see
what's going to happen.

1147
00:49:20,450 --> 00:49:22,410
They're going to build up, we're
going to get a little leak off,

1148
00:49:22,410 --> 00:49:24,076
but for the most part,
the wave packet's

1149
00:49:24,076 --> 00:49:25,285
going to go back to the left.

1150
00:49:25,285 --> 00:49:27,575
Questions about this one
before we move on to the next?

1151
00:49:27,575 --> 00:49:28,242
Yeah.

1152
00:49:28,242 --> 00:49:29,158
AUDIENCE: [INAUDIBLE].

1153
00:49:32,182 --> 00:49:33,640
PROFESSOR: Are
the-- Yeah, exactly.

1154
00:49:33,640 --> 00:49:36,056
So r and t here are going to
be defined as the probability

1155
00:49:36,056 --> 00:49:37,860
that this wave packet
gets off to infinity

1156
00:49:37,860 --> 00:49:40,634
in either direction.

1157
00:49:40,634 --> 00:49:43,580
AUDIENCE: What happens if you
make the energy lower than--

1158
00:49:43,580 --> 00:49:45,121
PROFESSOR: Then you
can't have a wave

1159
00:49:45,121 --> 00:49:47,050
packet on the left
that's moving.

1160
00:49:47,050 --> 00:49:49,730
Because remember that
the wave packet in order

1161
00:49:49,730 --> 00:49:53,956
to have an expected momentum
needs to be oscillating.

1162
00:49:53,956 --> 00:49:56,580
But if the energy is lower than
the potential, its exponential.

1163
00:49:56,580 --> 00:49:58,550
So the expectation value
for the momentum is 0.

1164
00:49:58,550 --> 00:50:00,080
So you can't have a particle
moving in from the left

1165
00:50:00,080 --> 00:50:02,590
if it's got energy less than
the height of the barrier.

1166
00:50:02,590 --> 00:50:04,091
Cool?

1167
00:50:04,091 --> 00:50:04,590
Yeah.

1168
00:50:04,590 --> 00:50:06,750
AUDIENCE: Sorry, what's
this green thing?

1169
00:50:06,750 --> 00:50:09,208
PROFESSOR: The green thing is
the energy of the wave packet

1170
00:50:09,208 --> 00:50:10,620
that I'm sending in.

1171
00:50:10,620 --> 00:50:13,960
OK, so let's go to the next one.

1172
00:50:13,960 --> 00:50:16,280
So this is something
we'd like to understand.

1173
00:50:16,280 --> 00:50:17,780
The reason I'm
showing you this is I

1174
00:50:17,780 --> 00:50:21,580
want to explain this phenomena.

1175
00:50:21,580 --> 00:50:22,830
We already did the dispersion.

1176
00:50:22,830 --> 00:50:24,871
I want to explain this
phenomena that you reflect

1177
00:50:24,871 --> 00:50:27,680
when going downhill, which
is perhaps surprising.

1178
00:50:27,680 --> 00:50:29,680
And I want to ask how
efficiently do reflect off

1179
00:50:29,680 --> 00:50:31,450
any given barrier.

1180
00:50:31,450 --> 00:50:31,950
Yeah.

1181
00:50:31,950 --> 00:50:32,968
Question.

1182
00:50:32,968 --> 00:50:35,551
AUDIENCE: Can you describe the
[INAUDIBLE] physical experiment

1183
00:50:35,551 --> 00:50:37,429
that would show
that done, though?

1184
00:50:37,429 --> 00:50:38,720
PROFESSOR: Oh yeah, absolutely.

1185
00:50:38,720 --> 00:50:42,700
So suppose I have a
little capacitor plate

1186
00:50:42,700 --> 00:50:44,870
and there's a potential
difference across it

1187
00:50:44,870 --> 00:50:48,724
and a hole so a particle
could shoot threw it.

1188
00:50:48,724 --> 00:50:50,890
I mean it doesn't have to
be an insanely small hole,

1189
00:50:50,890 --> 00:50:51,390
but a hole.

1190
00:50:51,390 --> 00:50:53,232
So that if a particle
goes from here

1191
00:50:53,232 --> 00:50:54,940
to the other side of
the capacitor plate,

1192
00:50:54,940 --> 00:50:57,800
it will have accelerated across
the potential difference.

1193
00:50:57,800 --> 00:51:00,740
So if you think about the
effective potential energy,

1194
00:51:00,740 --> 00:51:02,490
the potential is
decreasing linearly.

1195
00:51:02,490 --> 00:51:03,970
It's a constant electric field.

1196
00:51:03,970 --> 00:51:07,710
So over this short domain
between the capacitor plates,

1197
00:51:07,710 --> 00:51:12,632
we have effectively a
linear potential energy.

1198
00:51:12,632 --> 00:51:15,090
So I have a particle that I
send with very little momentum,

1199
00:51:15,090 --> 00:51:16,180
but it carries some charge.

1200
00:51:16,180 --> 00:51:17,410
It gets on capacitor
plates, and if it

1201
00:51:17,410 --> 00:51:18,784
goes through the
capacitor plate,

1202
00:51:18,784 --> 00:51:20,520
it ends with a lot
more energy relative

1203
00:51:20,520 --> 00:51:22,300
to the potential energy.

1204
00:51:22,300 --> 00:51:23,220
That cool?

1205
00:51:23,220 --> 00:51:25,400
So a little capacitor
plate with a hole in it

1206
00:51:25,400 --> 00:51:28,100
is a beautiful example
of this system.

1207
00:51:28,100 --> 00:51:28,600
Good?

1208
00:51:28,600 --> 00:51:30,391
Now, making it infinitely
sharp, well, that

1209
00:51:30,391 --> 00:51:31,500
would require-- you know.

1210
00:51:31,500 --> 00:51:34,960
But making it very
sharp is no problem.

1211
00:51:34,960 --> 00:51:37,920
So here's the inverse
of what we just did,

1212
00:51:37,920 --> 00:51:42,850
which is sending a
particle into a barrier.

1213
00:51:42,850 --> 00:51:46,140
And so there you see it does
some quite awesome things.

1214
00:51:46,140 --> 00:51:48,320
And I want to pause
it right here.

1215
00:51:48,320 --> 00:51:49,550
This is the energy band.

1216
00:51:49,550 --> 00:51:51,510
So the green
represents-- it says

1217
00:51:51,510 --> 00:51:54,489
that at any given energy
inside this green band,

1218
00:51:54,489 --> 00:51:56,780
there's some contribution
from the corresponding energy

1219
00:51:56,780 --> 00:51:59,730
eigenstate to our wave packet.

1220
00:51:59,730 --> 00:52:02,200
There isn't any
contribution from states

1221
00:52:02,200 --> 00:52:04,680
up here where there's
no green, which

1222
00:52:04,680 --> 00:52:06,732
means that all the
contributions come

1223
00:52:06,732 --> 00:52:08,190
from energy
eigenstates with energy

1224
00:52:08,190 --> 00:52:09,960
below the height of the barrier.

1225
00:52:09,960 --> 00:52:14,080
None of them have enough
energy to cross the barrier.

1226
00:52:14,080 --> 00:52:17,390
And what we see when we
send in our wave packet

1227
00:52:17,390 --> 00:52:20,180
is some complicated mucking
around near the barrier,

1228
00:52:20,180 --> 00:52:23,610
but in particular,
right there, there's

1229
00:52:23,610 --> 00:52:26,100
a finite non-zero
probability that you're

1230
00:52:26,100 --> 00:52:28,810
found in the classically
disallowed region.

1231
00:52:28,810 --> 00:52:31,030
In the region where you
didn't have enough energy

1232
00:52:31,030 --> 00:52:32,670
to get there.

1233
00:52:32,670 --> 00:52:33,680
We see right there.

1234
00:52:33,680 --> 00:52:35,770
And meanwhile, there's
also a huge pile up here.

1235
00:52:35,770 --> 00:52:36,620
This is a hard wall.

1236
00:52:36,620 --> 00:52:38,320
Normally classically
from a hard wall,

1237
00:52:38,320 --> 00:52:39,980
you roll along, you
hit the hard wall,

1238
00:52:39,980 --> 00:52:42,740
and you bounce off
instantaneously

1239
00:52:42,740 --> 00:52:43,960
with the same velocity.

1240
00:52:43,960 --> 00:52:46,750
But here what we're
saying is no, the wave

1241
00:52:46,750 --> 00:52:48,940
doesn't just exactly
bounce off instantaneously.

1242
00:52:48,940 --> 00:52:51,760
We get this complicated piling
up and the interference effect.

1243
00:52:51,760 --> 00:52:54,590
So what's going on there?

1244
00:52:54,590 --> 00:52:56,482
Well, it's precisely
an interference effect.

1245
00:52:56,482 --> 00:52:57,690
So look at the wave function.

1246
00:52:57,690 --> 00:52:59,900
The wave function is a much
easier thing to look at.

1247
00:52:59,900 --> 00:53:02,610
So the red is the amplitude.

1248
00:53:02,610 --> 00:53:03,930
The red is the real part.

1249
00:53:03,930 --> 00:53:05,380
The black is the phase.

1250
00:53:05,380 --> 00:53:08,060
So the real part is actually
behaving quite reasonably.

1251
00:53:08,060 --> 00:53:10,179
It looks like it's just
bouncing right off.

1252
00:53:10,179 --> 00:53:12,470
But the physical thing is
the probability distribution.

1253
00:53:12,470 --> 00:53:15,260
The probability distribution
has interference terms

1254
00:53:15,260 --> 00:53:17,030
from the various
different contributions

1255
00:53:17,030 --> 00:53:18,844
to the wave function.

1256
00:53:18,844 --> 00:53:20,760
So at the collision those
interference effects

1257
00:53:20,760 --> 00:53:22,062
are very important.

1258
00:53:22,062 --> 00:53:23,770
And at late times
notice what's happened.

1259
00:53:23,770 --> 00:53:25,700
At late times the
probability distribution

1260
00:53:25,700 --> 00:53:28,979
all went right back off.

1261
00:53:28,979 --> 00:53:31,020
So that's something else
we'd like to understand.

1262
00:53:31,020 --> 00:53:34,522
Penetration into the
barrier from a classically

1263
00:53:34,522 --> 00:53:35,480
disallowed wave packet.

1264
00:53:35,480 --> 00:53:35,980
Yup.

1265
00:53:35,980 --> 00:53:40,124
AUDIENCE: [INAUDIBLE] the wave
packet is also expanding again?

1266
00:53:40,124 --> 00:53:40,790
PROFESSOR: Yeah.

1267
00:53:40,790 --> 00:53:41,831
The wave packet, exactly.

1268
00:53:41,831 --> 00:53:44,102
So the wave packet is
always going to expand.

1269
00:53:44,102 --> 00:53:46,310
And the reason is we're
taking our wave packet, which

1270
00:53:46,310 --> 00:53:48,560
is a reasonably well-localized
approximately Gaussian wave

1271
00:53:48,560 --> 00:53:49,100
packet.

1272
00:53:49,100 --> 00:53:51,340
So whatever else is
going on, while it's

1273
00:53:51,340 --> 00:53:53,465
in the part of the potential
where the potential is

1274
00:53:53,465 --> 00:53:56,030
constant, it's also
just dispersing.

1275
00:53:56,030 --> 00:53:57,240
And so we can see that here.

1276
00:53:57,240 --> 00:53:58,530
Let's make that more obvious.

1277
00:53:58,530 --> 00:54:03,642
Let's make the initial wave
packet much more narrow.

1278
00:54:03,642 --> 00:54:05,350
So if the wave packet's
much more narrow,

1279
00:54:05,350 --> 00:54:06,724
we're going to
watch it disperse,

1280
00:54:06,724 --> 00:54:09,950
and let's move the
wall over here.

1281
00:54:09,950 --> 00:54:12,630
So long before it hits the wall
it's going to disperse, right?

1282
00:54:12,630 --> 00:54:13,630
That's what we'd expect.

1283
00:54:13,630 --> 00:54:15,380
And lo, watch it disperse.

1284
00:54:15,380 --> 00:54:17,370
OK, now gets to the wall.

1285
00:54:17,370 --> 00:54:22,275
And it's sort of a mess.

1286
00:54:22,275 --> 00:54:24,400
Now, you see that there's
contributions to the wave

1287
00:54:24,400 --> 00:54:25,630
function over here,
there's support

1288
00:54:25,630 --> 00:54:27,260
for the wave function
moving off to the right.

1289
00:54:27,260 --> 00:54:29,634
But now notice that we have
support on energy eigenstates

1290
00:54:29,634 --> 00:54:31,999
with energy above the barrier.

1291
00:54:31,999 --> 00:54:34,040
So, indeed, it was possible
for some contribution

1292
00:54:34,040 --> 00:54:34,998
to go off to the right.

1293
00:54:38,430 --> 00:54:41,890
So one more of these guys.

1294
00:54:41,890 --> 00:54:42,970
I guess two more.

1295
00:54:42,970 --> 00:54:43,730
These are fun.

1296
00:54:43,730 --> 00:54:47,320
So here's a hard wall.

1297
00:54:47,320 --> 00:54:51,230
Let's make the-- So
here's a hard wall.

1298
00:54:51,230 --> 00:54:53,870
But it's a finite width barrier.

1299
00:54:53,870 --> 00:54:55,970
What happens?

1300
00:54:55,970 --> 00:54:59,530
Well, this is basically the
same as what we just saw.

1301
00:54:59,530 --> 00:55:01,230
We collide up against
that first wall,

1302
00:55:01,230 --> 00:55:02,782
and since there's
an exponential fall

1303
00:55:02,782 --> 00:55:04,565
off it doesn't really
matter that there's

1304
00:55:04,565 --> 00:55:06,440
this other wall where
it falls down over here

1305
00:55:06,440 --> 00:55:08,856
because the wave function is
just exponentially suppressed

1306
00:55:08,856 --> 00:55:11,070
in the classically
disallowed region.

1307
00:55:11,070 --> 00:55:12,110
So let's see that again.

1308
00:55:12,110 --> 00:55:13,990
It's always going to be
exponentially suppressed

1309
00:55:13,990 --> 00:55:15,531
in the classically
disallowed region.

1310
00:55:15,531 --> 00:55:18,930
As you see, there's that
big exponential suppression.

1311
00:55:18,930 --> 00:55:20,720
So there's some
finite particle here,

1312
00:55:20,720 --> 00:55:23,850
but you're really unlikely to
find it out here exponentially.

1313
00:55:23,850 --> 00:55:28,320
On the other hand, if we
make the barrier-- so let's

1314
00:55:28,320 --> 00:55:29,850
think about this barrier.

1315
00:55:29,850 --> 00:55:34,440
Classically, this is just
as good as a thick barrier.

1316
00:55:34,440 --> 00:55:36,770
You can't get passed this wall.

1317
00:55:36,770 --> 00:55:40,210
But quantum mechanically
what happens?

1318
00:55:40,210 --> 00:55:43,270
You go right through.

1319
00:55:43,270 --> 00:55:43,770
Just--

1320
00:55:43,770 --> 00:55:44,080
AUDIENCE: [INAUDIBLE].

1321
00:55:44,080 --> 00:55:45,038
PROFESSOR: --right on--

1322
00:55:45,038 --> 00:55:51,720
[LAUGHTER]

1323
00:55:51,720 --> 00:55:53,657
Yeah, uh-huh.

1324
00:55:53,657 --> 00:55:54,240
Right through.

1325
00:55:54,240 --> 00:55:56,530
It barely even changes shape.

1326
00:55:56,530 --> 00:55:57,830
It barely even changes shape.

1327
00:55:57,830 --> 00:55:58,955
Just goes right on through.

1328
00:55:58,955 --> 00:56:01,371
Now, there's some probability
that you go off to the left,

1329
00:56:01,371 --> 00:56:02,372
that you bounce off.

1330
00:56:02,372 --> 00:56:05,610
[LAUGHTER]

1331
00:56:05,610 --> 00:56:06,430
I share your pain.

1332
00:56:06,430 --> 00:56:09,210
I totally do.

1333
00:56:09,210 --> 00:56:14,394
So I would like to do another
experiment which demonstrates

1334
00:56:14,394 --> 00:56:17,060
the difference between classical
and quantum mechanical physics.

1335
00:56:19,744 --> 00:56:20,244
Right.

1336
00:56:20,244 --> 00:56:23,310
[LAUGHTER]

1337
00:56:23,310 --> 00:56:27,060
[APPLAUSE]

1338
00:56:27,060 --> 00:56:28,610
So we have some
explaining to do.

1339
00:56:28,610 --> 00:56:30,741
And this is going to turn
out to be-- surprisingly,

1340
00:56:30,741 --> 00:56:32,240
this is not a hard
thing to explain.

1341
00:56:32,240 --> 00:56:32,780
It's going to be real easy.

1342
00:56:32,780 --> 00:56:34,155
Just like the
dispersion, this is

1343
00:56:34,155 --> 00:56:37,060
going to be not a
hard thing to explain.

1344
00:56:37,060 --> 00:56:39,712
But the next one and this
is-- oh yeah, question.

1345
00:56:39,712 --> 00:56:42,319
AUDIENCE: Does the height
of the spike matter?

1346
00:56:42,319 --> 00:56:43,610
PROFESSOR: It does, absolutely.

1347
00:56:43,610 --> 00:56:46,640
So the height of the spike and
the width of the spike matter.

1348
00:56:46,640 --> 00:56:48,050
So let's make it a little wider.

1349
00:56:48,050 --> 00:56:49,841
And what we're going
to see now is oh, it's

1350
00:56:49,841 --> 00:56:54,045
a little less likely
to go through.

1351
00:56:54,045 --> 00:56:55,570
If we make it just
a tiny bit wider

1352
00:56:55,570 --> 00:56:57,695
we'll see that it's much
less likely to go through.

1353
00:56:57,695 --> 00:56:59,320
We'll see much stronger
reflected wave.

1354
00:56:59,320 --> 00:57:03,312
So both the height and width
are going to turn out to matter.

1355
00:57:03,312 --> 00:57:05,770
So here you can see that there's
an appreciable reflection,

1356
00:57:05,770 --> 00:57:06,590
which there wasn't previously.

1357
00:57:06,590 --> 00:57:08,173
Let's make it just
a little bit wider.

1358
00:57:15,670 --> 00:57:18,270
And now it's an awful lot
closer to half and half.

1359
00:57:18,270 --> 00:57:20,020
See the bottom two lumps?

1360
00:57:20,020 --> 00:57:22,325
Let's make it just
ever so slightly wider.

1361
00:57:25,844 --> 00:57:27,260
And again, looking
down here we're

1362
00:57:27,260 --> 00:57:28,613
going to have a transmitted
bit, but we're also

1363
00:57:28,613 --> 00:57:29,550
going to have a
reflected-- now it

1364
00:57:29,550 --> 00:57:32,010
looks like the reflected bit is
even a little bit larger maybe.

1365
00:57:32,010 --> 00:57:34,551
And we can actually compute the
reflection transmission here.

1366
00:57:34,551 --> 00:57:38,336
72% gets reflected and
28% get transmitted.

1367
00:57:38,336 --> 00:57:40,960
I really strongly encourage you
to play with these simulations.

1368
00:57:40,960 --> 00:57:42,584
They're both fun and
very illuminating.

1369
00:57:42,584 --> 00:57:43,175
Yeah.

1370
00:57:43,175 --> 00:57:45,905
AUDIENCE: [INAUDIBLE]
just the higher it got,

1371
00:57:45,905 --> 00:57:47,318
the less got transmitted?

1372
00:57:47,318 --> 00:57:47,943
PROFESSOR: Yes.

1373
00:57:47,943 --> 00:57:50,046
AUDIENCE: So if this were
a delta function would

1374
00:57:50,046 --> 00:57:50,990
[INAUDIBLE]?

1375
00:57:50,990 --> 00:57:53,350
PROFESSOR: Well,
what's the width?

1376
00:57:53,350 --> 00:57:54,130
AUDIENCE: 0.

1377
00:57:54,130 --> 00:57:57,290
PROFESSOR: So what
happens when we

1378
00:57:57,290 --> 00:58:00,657
make the well less and
less-- or more and more thin?

1379
00:58:00,657 --> 00:58:01,490
More, yeah, through.

1380
00:58:01,490 --> 00:58:02,990
So we've got a competition
between the heights

1381
00:58:02,990 --> 00:58:03,800
and the width.

1382
00:58:03,800 --> 00:58:05,250
So one of the problems
on your problem set,

1383
00:58:05,250 --> 00:58:07,166
either this week or next
week, I can't recall,

1384
00:58:07,166 --> 00:58:09,020
will be for the delta
function barrier,

1385
00:58:09,020 --> 00:58:10,562
compute the transmission
probability.

1386
00:58:10,562 --> 00:58:13,228
And, in fact, we're going to ask
you to compute the transmission

1387
00:58:13,228 --> 00:58:15,140
probability through
two delta functions.

1388
00:58:15,140 --> 00:58:16,450
And, in fact, that's this week.

1389
00:58:16,450 --> 00:58:18,230
And then next week
we're going to show you

1390
00:58:18,230 --> 00:58:20,245
a sneaky way of using
something called the S

1391
00:58:20,245 --> 00:58:23,650
matrix to construct bound
states for those guys.

1392
00:58:23,650 --> 00:58:26,510
Anyway, OK.

1393
00:58:26,510 --> 00:58:31,909
So next, the last
simulation today.

1394
00:58:31,909 --> 00:58:33,700
So this is the inverse
of what we just did.

1395
00:58:33,700 --> 00:58:34,865
Instead of having a
potential barrier,

1396
00:58:34,865 --> 00:58:35,950
we have a potential well.

1397
00:58:38,577 --> 00:58:39,910
So what do you expect to happen?

1398
00:58:42,690 --> 00:58:45,310
Well, there's our wave
packet, and it comes along,

1399
00:58:45,310 --> 00:58:48,084
and all heck breaks
loose inside.

1400
00:58:48,084 --> 00:58:50,000
We get some excitations
and it tunnels across.

1401
00:58:50,000 --> 00:58:51,940
But we see there's all
this sort of wiggling

1402
00:58:51,940 --> 00:58:54,670
around inside the wave function.

1403
00:58:54,670 --> 00:58:56,820
And we get some support
going off to the right,

1404
00:58:56,820 --> 00:58:59,120
some support going
off to the left.

1405
00:58:59,120 --> 00:59:00,670
But nothing sticks
around inside,

1406
00:59:00,670 --> 00:59:03,540
which it looks like
there is, but it's

1407
00:59:03,540 --> 00:59:06,700
going to slowly decay away as
it goes off to the boundary.

1408
00:59:06,700 --> 00:59:08,360
These are scattering states.

1409
00:59:08,360 --> 00:59:10,280
So one way to think about
what's going on here

1410
00:59:10,280 --> 00:59:13,190
is it first scatters off,
it scatters downhill,

1411
00:59:13,190 --> 00:59:15,622
and then it scatters uphill.

1412
00:59:15,622 --> 00:59:17,080
But there's a very
funny thing that

1413
00:59:17,080 --> 00:59:19,700
happens when you can scatter
uphill and scatter downhill.

1414
00:59:19,700 --> 00:59:21,940
As we saw, any time you
have scattering uphill

1415
00:59:21,940 --> 00:59:25,302
and there's some probability--
that you reflected

1416
00:59:25,302 --> 00:59:26,760
some probability
that you transmit.

1417
00:59:26,760 --> 00:59:27,990
And we scatter uphill.

1418
00:59:27,990 --> 00:59:29,573
There's some probability
you transmit,

1419
00:59:29,573 --> 00:59:31,940
and some probability
that you reflect.

1420
00:59:31,940 --> 00:59:35,540
But if you have both an
uphill and a downhill

1421
00:59:35,540 --> 00:59:39,290
when you have a well,
something amazing happens.

1422
00:59:39,290 --> 00:59:41,120
Consider this well.

1423
00:59:45,400 --> 00:59:47,660
This didn't work very well.

1424
00:59:47,660 --> 00:59:49,601
Let's take a much larger one.

1425
00:59:49,601 --> 00:59:50,100
Uh-hmm!

1426
00:59:53,250 --> 00:59:57,790
So at some point, I
think it was last year,

1427
00:59:57,790 --> 01:00:01,030
some student's cell phone
went off, and I was like oh,

1428
01:00:01,030 --> 01:00:02,290
come on, dude.

1429
01:00:02,290 --> 01:00:03,360
You can't do that.

1430
01:00:03,360 --> 01:00:06,520
And like five minutes later
my cell phone went off.

1431
01:00:06,520 --> 01:00:09,070
[LAUGHTER]

1432
01:00:09,070 --> 01:00:12,230
So I feel your pain, but please
don't let that happen again.

1433
01:00:16,120 --> 01:00:16,830
So here we go.

1434
01:00:16,830 --> 01:00:20,366
So this is a very similar
system-- well, the same system,

1435
01:00:20,366 --> 01:00:21,740
but with a different
wave packet,

1436
01:00:21,740 --> 01:00:23,031
and a slightly different width.

1437
01:00:25,850 --> 01:00:27,950
And now what happens?

1438
01:00:27,950 --> 01:00:32,250
Well, all this wave packet
just sort of goes on,

1439
01:00:32,250 --> 01:00:35,580
but how much probability is
going back off to the left?

1440
01:00:35,580 --> 01:00:36,890
Very little.

1441
01:00:36,890 --> 01:00:38,360
In fact, we can
really work this.

1442
01:00:41,770 --> 01:00:45,320
Sorry, I need this, otherwise
I'm very bad at this.

1443
01:01:03,306 --> 01:01:05,680
There's some probability that
reflects in the first well.

1444
01:01:05,680 --> 01:01:07,780
There's some probability that
reflects in the second well.

1445
01:01:07,780 --> 01:01:09,154
But the probability
that reflects

1446
01:01:09,154 --> 01:01:13,360
from the combination
of the two, 0.

1447
01:01:13,360 --> 01:01:14,220
How can that be?

1448
01:01:17,800 --> 01:01:21,450
So this is just like the
boxes at the very beginning.

1449
01:01:21,450 --> 01:01:22,820
There's some probability.

1450
01:01:22,820 --> 01:01:26,370
So to go in this system-- so
let's go back to the beginning.

1451
01:01:26,370 --> 01:01:30,280
There's some amplitude
to go from the left side

1452
01:01:30,280 --> 01:01:32,510
to the right-hand side by
going-- so how would you

1453
01:01:32,510 --> 01:01:34,620
go from the left-hand side
to the right-hand side?

1454
01:01:34,620 --> 01:01:39,520
You transmit and
then transmit, right?

1455
01:01:39,520 --> 01:01:42,500
So it's the transmission
amplitude times transmission

1456
01:01:42,500 --> 01:01:44,050
amplitude gives you the product.

1457
01:01:44,050 --> 01:01:46,200
But is that the only
thing that could happen?

1458
01:01:46,200 --> 01:01:48,434
What are other possible
things that could happen?

1459
01:01:48,434 --> 01:01:49,350
AUDIENCE: [INAUDIBLE].

1460
01:01:49,350 --> 01:01:50,016
PROFESSOR: Yeah.

1461
01:01:50,016 --> 01:01:53,330
You could transmit,
reflect, reflect, transmit.

1462
01:01:53,330 --> 01:01:55,270
You could also transmit,
reflect, reflect,

1463
01:01:55,270 --> 01:01:57,555
reflect, reflect, transmit.

1464
01:01:57,555 --> 01:01:59,680
And you could do that an
arbitrary number of times.

1465
01:01:59,680 --> 01:02:01,360
And each of those
contributions is

1466
01:02:01,360 --> 01:02:03,220
a contribution to the amplitude.

1467
01:02:03,220 --> 01:02:05,270
And when you compute the
probability of transmit,

1468
01:02:05,270 --> 01:02:07,520
you don't take some
of the probabilities.

1469
01:02:07,520 --> 01:02:10,440
You take the square of
the sum of the amplitudes.

1470
01:02:10,440 --> 01:02:12,010
And what we will
find when we-- I

1471
01:02:12,010 --> 01:02:13,926
guess we'll do this next
time when we actually

1472
01:02:13,926 --> 01:02:16,690
do this calculation-- is
that that interference

1473
01:02:16,690 --> 01:02:19,130
effect from taking the square
of the sum of the amplitudes

1474
01:02:19,130 --> 01:02:21,880
from each of the
possible bouncings

1475
01:02:21,880 --> 01:02:23,680
can interfere
destructively and it

1476
01:02:23,680 --> 01:02:25,200
can interfere constructively.

1477
01:02:25,200 --> 01:02:28,345
We can find points when the
reflection is extremely low.

1478
01:02:28,345 --> 01:02:32,524
And we can find points where
the transmission is perfect.

1479
01:02:32,524 --> 01:02:33,426
Yeah.

1480
01:02:33,426 --> 01:02:35,230
AUDIENCE: So two questions.

1481
01:02:35,230 --> 01:02:39,850
One, the computer says that the
reflection coefficient is 0.

1482
01:02:39,850 --> 01:02:42,882
Why did it look like it there
was a wave packet going off

1483
01:02:42,882 --> 01:02:43,810
back to the left?

1484
01:02:43,810 --> 01:02:44,610
PROFESSOR: Excellent question.

1485
01:02:44,610 --> 01:02:46,160
And, indeed, there
is a little bit here.

1486
01:02:46,160 --> 01:02:47,540
So now I have to tell you
a little bit about how

1487
01:02:47,540 --> 01:02:49,647
the computer is casting
this reflection amplitude.

1488
01:02:49,647 --> 01:02:51,230
The way it's doing
that is it's saying

1489
01:02:51,230 --> 01:02:53,970
suppose I have a state
which has a definite energy

1490
01:02:53,970 --> 01:02:56,319
at the center of
this distribution.

1491
01:02:56,319 --> 01:02:58,110
Then the corresponding
reflection amplitude

1492
01:02:58,110 --> 01:02:58,749
would be 0.

1493
01:02:58,749 --> 01:03:00,290
But, in fact, I have
a superposition.

1494
01:03:00,290 --> 01:03:02,980
I've taken a
superposition of them.

1495
01:03:02,980 --> 01:03:05,810
And so the contributions from
slightly different energies

1496
01:03:05,810 --> 01:03:08,980
are giving me some
small reflection.

1497
01:03:08,980 --> 01:03:10,547
AUDIENCE: And also,
why do you have

1498
01:03:10,547 --> 01:03:14,344
to fine-tune the
width of that as well?

1499
01:03:14,344 --> 01:03:15,010
PROFESSOR: Yeah.

1500
01:03:15,010 --> 01:03:16,600
That's a really-- let me
rephrase the question.

1501
01:03:16,600 --> 01:03:17,650
The question is why do
you have to fine-tune

1502
01:03:17,650 --> 01:03:18,290
the width of that well?

1503
01:03:18,290 --> 01:03:20,164
Let me ask the question
slightly differently.

1504
01:03:20,164 --> 01:03:22,090
How does the
reflection probability

1505
01:03:22,090 --> 01:03:23,660
depend on the width of the well?

1506
01:03:23,660 --> 01:03:27,181
What's going on at these
special values of the energy

1507
01:03:27,181 --> 01:03:28,930
or special values of
the width of the well

1508
01:03:28,930 --> 01:03:30,500
when the transmission
is perfect?

1509
01:03:30,500 --> 01:03:31,940
Which I will call a resonance.

1510
01:03:31,940 --> 01:03:34,220
Why is that happening?

1511
01:03:34,220 --> 01:03:35,367
And we'll see that.

1512
01:03:35,367 --> 01:03:37,200
So that's a question I
want you guys to ask.

1513
01:03:37,200 --> 01:03:39,820
So I'm done now with
the experiments.

1514
01:03:39,820 --> 01:03:47,234
You guys should play with
these on your own time

1515
01:03:47,234 --> 01:03:48,150
to get some intuition.

1516
01:03:48,150 --> 01:03:48,645
Yeah.

1517
01:03:48,645 --> 01:03:50,311
AUDIENCE: Isn't the
[INAUDIBLE] you just

1518
01:03:50,311 --> 01:03:52,740
showed actually a classical
[INAUDIBLE] objects?

1519
01:03:52,740 --> 01:03:55,340
PROFESSOR: So excellent.

1520
01:03:55,340 --> 01:03:57,065
So here's what
I'm going to tell.

1521
01:03:57,065 --> 01:03:58,440
I'm going to tell
you two things.

1522
01:03:58,440 --> 01:04:01,760
First off, if I take a
classical particle and I

1523
01:04:01,760 --> 01:04:05,047
put it through these potentials,
for example, this potential.

1524
01:04:05,047 --> 01:04:07,630
If I take a classical particle
and I put it in this potential,

1525
01:04:07,630 --> 01:04:11,039
it's got an energy here,
does it ever reflect back?

1526
01:04:11,039 --> 01:04:11,580
AUDIENCE: No.

1527
01:04:11,580 --> 01:04:12,580
PROFESSOR: No.

1528
01:04:12,580 --> 01:04:14,840
Always, always it
rolls and continues on.

1529
01:04:14,840 --> 01:04:15,920
Always.

1530
01:04:15,920 --> 01:04:17,970
So if I take a quantum
particle and I do this,

1531
01:04:17,970 --> 01:04:18,967
will it reflect back?

1532
01:04:18,967 --> 01:04:19,800
AUDIENCE: Sometimes.

1533
01:04:19,800 --> 01:04:21,280
PROFESSOR: Apparently sometimes.

1534
01:04:21,280 --> 01:04:22,550
Now, how do I encode that?

1535
01:04:22,550 --> 01:04:27,660
I encode that in studying the
solutions to the wave equation

1536
01:04:27,660 --> 01:04:31,850
dt minus-- sorry--
i h-bar dt psi

1537
01:04:31,850 --> 01:04:36,796
is equal to minus
h-bar squared upon 2m,

1538
01:04:36,796 --> 01:04:39,670
dt squared psi plus v psi.

1539
01:04:42,310 --> 01:04:45,770
So this property of the quantum
particle that it reflects

1540
01:04:45,770 --> 01:04:49,060
is a property of solutions
to this equation.

1541
01:04:49,060 --> 01:04:52,880
When I interpret the amplitude
squared as a probability.

1542
01:04:52,880 --> 01:04:55,030
But this equation
doesn't just govern--

1543
01:04:55,030 --> 01:04:57,710
this isn't the only place that
this equation's ever shown up.

1544
01:04:57,710 --> 01:04:59,550
An equation at least
very similar to this

1545
01:04:59,550 --> 01:05:01,640
shows up in studying
the propagation of waves

1546
01:05:01,640 --> 01:05:03,040
on a string, for example.

1547
01:05:03,040 --> 01:05:05,730
Where the potential is played
by the role of the tension

1548
01:05:05,730 --> 01:05:07,324
or the density of the string.

1549
01:05:07,324 --> 01:05:09,240
So as you have a wave
coming along the string,

1550
01:05:09,240 --> 01:05:10,850
as I'm sure you did
in 803, you have

1551
01:05:10,850 --> 01:05:12,290
a wave coming along a string.

1552
01:05:12,290 --> 01:05:15,340
And then the thickness or
density of the string changes,

1553
01:05:15,340 --> 01:05:17,370
becomes thicker or
thinner, then sometimes

1554
01:05:17,370 --> 01:05:19,203
you have reflections
off that interference--

1555
01:05:19,203 --> 01:05:21,270
from impedance mismatch
is one way to phrase it.

1556
01:05:21,270 --> 01:05:22,936
And as you move across
to the other side

1557
01:05:22,936 --> 01:05:24,100
you get reflections again.

1558
01:05:24,100 --> 01:05:25,840
And the thing that
matters is not

1559
01:05:25,840 --> 01:05:27,780
the amplitude of the--
one of the things

1560
01:05:27,780 --> 01:05:28,840
that matters is
not the amplitude,

1561
01:05:28,840 --> 01:05:29,798
but it's the intensity.

1562
01:05:29,798 --> 01:05:31,850
And the intensity's a square.

1563
01:05:31,850 --> 01:05:33,830
And so you get
interference effects.

1564
01:05:33,830 --> 01:05:37,000
So, indeed, this phenomena
of reflection and multiple

1565
01:05:37,000 --> 01:05:39,260
reflection shows up in
certain classical systems,

1566
01:05:39,260 --> 01:05:41,490
but it shows up not in
classical particle systems.

1567
01:05:41,490 --> 01:05:45,170
It shows up in classical wave
systems, like waves on a chain

1568
01:05:45,170 --> 01:05:46,830
or waves in a rope.

1569
01:05:46,830 --> 01:05:50,620
And they're governed by
effectively the same equations,

1570
01:05:50,620 --> 01:05:52,690
not exactly, but effectively
the same equations,

1571
01:05:52,690 --> 01:05:56,040
as the Schrodinger evolution
governs the wave function.

1572
01:05:56,040 --> 01:05:58,670
But the difference is that
those classical fields,

1573
01:05:58,670 --> 01:06:02,600
those classical continuous
objects have waves on them,

1574
01:06:02,600 --> 01:06:04,630
but those waves
are actual waves.

1575
01:06:04,630 --> 01:06:05,670
You see the rope.

1576
01:06:05,670 --> 01:06:07,260
The rope is everywhere.

1577
01:06:07,260 --> 01:06:09,430
In the case of the quantum
mechanical description

1578
01:06:09,430 --> 01:06:13,500
of a particle, the
particle could be anywhere,

1579
01:06:13,500 --> 01:06:14,460
but it is a particle.

1580
01:06:14,460 --> 01:06:15,668
It is a chunk, it is a thing.

1581
01:06:15,668 --> 01:06:18,870
And this wave is a probability
wave in our knowledge

1582
01:06:18,870 --> 01:06:20,500
or our command of the system.

1583
01:06:23,360 --> 01:06:24,940
Other questions?

1584
01:06:24,940 --> 01:06:26,210
OK.

1585
01:06:26,210 --> 01:06:33,246
So let's do the first example
of this kind of system.

1586
01:06:33,246 --> 01:06:35,620
So let's do the very first
example we talked about, apart

1587
01:06:35,620 --> 01:06:38,435
from the free particle,
which is the potential step.

1588
01:06:41,035 --> 01:06:42,535
First example is
the potential step.

1589
01:06:47,386 --> 01:06:48,760
And what we want
to do is we want

1590
01:06:48,760 --> 01:06:51,530
to find the eigenfunctions
of the following potential.

1591
01:06:51,530 --> 01:06:54,556
Constant, barrier, constant.

1592
01:06:54,556 --> 01:06:56,180
And I'm going to call
the height here V

1593
01:06:56,180 --> 01:06:57,429
naught, and the height here 0.

1594
01:06:57,429 --> 01:06:59,450
And I'll call this
position X equals 0.

1595
01:07:02,130 --> 01:07:06,416
And I want to send
in a wave-- think

1596
01:07:06,416 --> 01:07:09,040
about the physics of sending in
a particle that has an energy E

1597
01:07:09,040 --> 01:07:09,539
naught.

1598
01:07:14,871 --> 01:07:16,620
Now to think about the
dynamics, before we

1599
01:07:16,620 --> 01:07:20,644
think about time evolution
of a localized wave packet,

1600
01:07:20,644 --> 01:07:22,060
as we saw in the
free particle, it

1601
01:07:22,060 --> 01:07:24,080
behooves us to find
the energy eigenstates,

1602
01:07:24,080 --> 01:07:26,390
then we can deduce the
evolution of the wave packet

1603
01:07:26,390 --> 01:07:30,359
by doing a re-summation by using
the superposition principle.

1604
01:07:30,359 --> 01:07:32,150
So let's first find
the energy eigenstates.

1605
01:07:32,150 --> 01:07:33,900
And, in fact, this
will turn out to encode

1606
01:07:33,900 --> 01:07:35,290
all the information we need.

1607
01:07:35,290 --> 01:07:38,690
So we want to find phi e of x.

1608
01:07:38,690 --> 01:07:43,232
What is the energy eigenfunction
with energy E, let's say.

1609
01:07:43,232 --> 01:07:44,690
And we know the
answer on this side

1610
01:07:44,690 --> 01:07:45,770
and we know the
answer on this side

1611
01:07:45,770 --> 01:07:47,100
because it's just constant.

1612
01:07:47,100 --> 01:07:50,320
That's a free particle and we
know how to write the solution.

1613
01:07:50,320 --> 01:07:53,630
So we can write this
as this is equal to.

1614
01:07:53,630 --> 01:07:56,460
On the left-hand side it's
just a sum of plane waves.

1615
01:07:56,460 --> 01:08:01,150
It's exactly of that form.
a e to the i kx plus b

1616
01:08:01,150 --> 01:08:07,800
e to the minus i kx on the left.

1617
01:08:07,800 --> 01:08:14,500
And on the right we have c e
to the minus i alpha x-- sorry.

1618
01:08:14,500 --> 01:08:17,620
On this side it's disallowed,
classically disallowed.

1619
01:08:17,620 --> 01:08:19,670
So this is e to
the minus-- sorry,

1620
01:08:19,670 --> 01:08:26,340
do you-- plus alpha x
plus d e to minus alpha x.

1621
01:08:29,729 --> 01:08:38,229
Where h-bar squared k squared
upon 2m is equal to e.

1622
01:08:38,229 --> 01:08:42,729
And h-bar squared
alpha squared upon 2m

1623
01:08:42,729 --> 01:08:46,660
is equal to v naught minus
e, the positive quantity.

1624
01:08:51,560 --> 01:09:02,560
So we need to satisfy
our various continuity

1625
01:09:02,560 --> 01:09:04,702
and normalizability conditions.

1626
01:09:04,702 --> 01:09:07,160
And in particular, what had
the wave function better do out

1627
01:09:07,160 --> 01:09:07,733
this way?

1628
01:09:07,733 --> 01:09:08,649
AUDIENCE: [INAUDIBLE].

1629
01:09:08,649 --> 01:09:09,870
PROFESSOR: It had
better not divert--

1630
01:09:09,870 --> 01:09:11,910
are we going to be able to build
normalizable wave packets--

1631
01:09:11,910 --> 01:09:14,300
or are we going to be able
to find normalizable wave

1632
01:09:14,300 --> 01:09:16,279
functions of this form?

1633
01:09:16,279 --> 01:09:16,779
No.

1634
01:09:16,779 --> 01:09:18,810
Because there are always going
to plane waves on the left.

1635
01:09:18,810 --> 01:09:20,609
So the best we'll be able
to do is find delta function

1636
01:09:20,609 --> 01:09:22,390
normalizable energy eigenstates.

1637
01:09:22,390 --> 01:09:23,931
That's not such a
big deal because we

1638
01:09:23,931 --> 01:09:27,370
can build wave packets,
as we discussed before.

1639
01:09:27,370 --> 01:09:28,979
But on the other
hand, it's one thing

1640
01:09:28,979 --> 01:09:30,880
to be delta function
normalizable going

1641
01:09:30,880 --> 01:09:33,990
to just a wave, it's
another thing to diverge.

1642
01:09:33,990 --> 01:09:36,649
So if we want to build
something that's normalizable up

1643
01:09:36,649 --> 01:09:39,060
to a delta function
normalization condition,

1644
01:09:39,060 --> 01:09:41,520
this had better vanish.

1645
01:09:41,520 --> 01:09:42,020
Yeah?

1646
01:09:47,120 --> 01:09:50,569
But on top of that, we
need that the wave function

1647
01:09:50,569 --> 01:09:55,370
is continuous, so at x equals
we need that phi and phi

1648
01:09:55,370 --> 01:09:56,340
prime are continuous.

1649
01:09:59,870 --> 01:10:02,360
So that turns out to be
an easy set of equations.

1650
01:10:02,360 --> 01:10:07,980
So for phi, this says that
a plus b is equal to d.

1651
01:10:07,980 --> 01:10:13,210
And for phi prime,
this says that ik a.

1652
01:10:13,210 --> 01:10:17,020
And then the next one is going
to give me minus ik a minus b

1653
01:10:17,020 --> 01:10:20,820
is equal to, on the right-hand
side, minus alpha c.

1654
01:10:20,820 --> 01:10:25,090
Because the exponential's
all value to 0.

1655
01:10:25,090 --> 01:10:29,450
So we can invert
these to get that d

1656
01:10:29,450 --> 01:10:32,979
is equal to 2-- that's weird.

1657
01:10:32,979 --> 01:10:34,520
AUDIENCE: You have
a d in the bottom.

1658
01:10:34,520 --> 01:10:35,870
There's [INAUDIBLE].

1659
01:10:35,870 --> 01:10:38,490
PROFESSOR: Oh,
sorry. d. d. d. d. d.

1660
01:10:38,490 --> 01:10:39,000
Thank you.

1661
01:10:39,000 --> 01:10:42,190
To invert those you get 2k
over k plus i alpha, I think?

1662
01:10:42,190 --> 01:10:44,070
Yes.

1663
01:10:44,070 --> 01:10:49,210
And b is equal to k minus i
alpha over k plus i alpha.

1664
01:10:53,920 --> 01:10:55,530
And so you can
plug these back in

1665
01:10:55,530 --> 01:10:57,760
and get the explicit form
of the wave function.

1666
01:10:57,760 --> 01:10:59,840
Now, what condition
must the energy satisfy

1667
01:10:59,840 --> 01:11:02,410
order that this is a
solution that's continuous

1668
01:11:02,410 --> 01:11:07,289
and derivative is continuous?

1669
01:11:07,289 --> 01:11:09,955
Last time we found that in order
to satisfy for the finite well,

1670
01:11:09,955 --> 01:11:13,360
the continuity normalizability
conditions, only certain values

1671
01:11:13,360 --> 01:11:14,730
of the energy were allowed.

1672
01:11:14,730 --> 01:11:18,422
Here we've imposed
normalizability and continuity.

1673
01:11:18,422 --> 01:11:19,880
What's the condition
on the energy?

1674
01:11:24,249 --> 01:11:25,290
This is a trick question.

1675
01:11:25,290 --> 01:11:27,280
There isn't any.

1676
01:11:27,280 --> 01:11:29,130
For any energy we
could find a solution.

1677
01:11:29,130 --> 01:11:31,830
There was no consistency
condition amongst these.

1678
01:11:31,830 --> 01:11:32,710
Any energy.

1679
01:11:32,710 --> 01:11:35,194
So are the energy eigenvalues
continuous or discrete?

1680
01:11:35,194 --> 01:11:36,110
AUDIENCE: [INAUDIBLE].

1681
01:11:36,110 --> 01:11:37,026
PROFESSOR: Continuous.

1682
01:11:37,026 --> 01:11:39,290
Anything above 0 energy
is allowed for the system,

1683
01:11:39,290 --> 01:11:39,956
it's continuous.

1684
01:11:42,420 --> 01:11:44,350
So what does this wave
function look like?

1685
01:11:44,350 --> 01:11:46,600
Oh, I really should have
done this on the clean board.

1686
01:11:46,600 --> 01:11:48,225
What does this wave
function look like?

1687
01:11:48,225 --> 01:11:56,470
Well, it's oscillating out
here, and it's decaying in here,

1688
01:11:56,470 --> 01:11:57,220
and it's smooth.

1689
01:12:02,940 --> 01:12:05,420
So what's the meaning of this?

1690
01:12:05,420 --> 01:12:07,014
What's the physical meaning?

1691
01:12:07,014 --> 01:12:08,930
Well, here's the way I
want to interpret this.

1692
01:12:08,930 --> 01:12:10,475
How does this system
evolve in time?

1693
01:12:13,819 --> 01:12:15,610
How does this wave
function evolve in time?

1694
01:12:15,610 --> 01:12:18,050
So this is psi of x.

1695
01:12:18,050 --> 01:12:19,350
What's psi of x and t?

1696
01:12:23,160 --> 01:12:25,250
Well, it just gets hit
by an e to the minus

1697
01:12:25,250 --> 01:12:33,510
i omega t minus omega t plus
omega t minus i omega t.

1698
01:12:36,480 --> 01:12:38,729
Everyone cool with that?

1699
01:12:38,729 --> 01:12:40,270
So all I did is I
just said that this

1700
01:12:40,270 --> 01:12:42,630
is an energy eigenfunction
with energy e,

1701
01:12:42,630 --> 01:12:46,000
and frequency omega is
equal to e upon h-bar.

1702
01:12:46,000 --> 01:12:48,500
And saying that this is an
energy eigenstate tells you

1703
01:12:48,500 --> 01:12:51,000
that under time evolution,
it evolves by rotation

1704
01:12:51,000 --> 01:12:52,972
by an overall phase, e
to the minus i omega t.

1705
01:12:52,972 --> 01:12:54,930
And then I just multiplied
the whole thing by e

1706
01:12:54,930 --> 01:12:58,860
to the minus i omega
t and distributed it.

1707
01:12:58,860 --> 01:13:01,070
So here's the minus
i omega t minus i

1708
01:13:01,070 --> 01:13:05,370
omega t minus i
omega t in the phase.

1709
01:13:05,370 --> 01:13:07,755
So doing that, though, gives
us a simple interpretation.

1710
01:13:10,380 --> 01:13:15,000
Compared to our free particle,
e to the i kx minus omega t,

1711
01:13:15,000 --> 01:13:18,020
it's a wave moving to the
right with velocity omega

1712
01:13:18,020 --> 01:13:19,420
over k, the phase velocity.

1713
01:13:19,420 --> 01:13:21,535
This is a right-moving
contribution.

1714
01:13:21,535 --> 01:13:22,175
So plus.

1715
01:13:25,500 --> 01:13:29,760
And this is a
left-moving contribution.

1716
01:13:29,760 --> 01:13:30,760
Everyone cool with that?

1717
01:13:30,760 --> 01:13:32,140
So on the left-hand
side, what we

1718
01:13:32,140 --> 01:13:33,979
have is a superposition
of a wave moving

1719
01:13:33,979 --> 01:13:35,770
to the right and a wave
moving to the left.

1720
01:13:38,070 --> 01:13:38,570
Yeah?

1721
01:13:41,140 --> 01:13:44,970
On the right, what do we have?

1722
01:13:44,970 --> 01:13:52,060
We have an exponentially falling
function whose phase rotates.

1723
01:13:52,060 --> 01:13:54,001
Is this a traveling wave?

1724
01:13:54,001 --> 01:13:54,500
No.

1725
01:13:54,500 --> 01:13:55,660
It doesn't have any crests.

1726
01:13:55,660 --> 01:13:57,410
It just has an overall
phase that rotates.

1727
01:14:00,730 --> 01:14:02,700
And now here's one key thing.

1728
01:14:02,700 --> 01:14:04,880
If we look at this, what's b?

1729
01:14:04,880 --> 01:14:07,610
We did the
calculation of v. v is

1730
01:14:07,610 --> 01:14:09,660
k minus i alpha
over k plus i alpha.

1731
01:14:09,660 --> 01:14:11,060
What's the norm
square root of b?

1732
01:14:17,824 --> 01:14:19,240
Well, the norm
square root of b is

1733
01:14:19,240 --> 01:14:21,320
going to be multiply this
by its complex conjugate,

1734
01:14:21,320 --> 01:14:22,945
multiply this by its
complex conjugate.

1735
01:14:22,945 --> 01:14:25,640
But their each other's complex
conjugates will cancel.

1736
01:14:25,640 --> 01:14:27,390
The norm squared is 1.

1737
01:14:27,390 --> 01:14:28,480
So pure phase.

1738
01:14:28,480 --> 01:14:31,950
So this tells you
b is a pure phase.

1739
01:14:31,950 --> 01:14:33,120
So now look back at this.

1740
01:14:33,120 --> 01:14:37,837
If b is a pure phase--
sorry-- if b is a pure phase,

1741
01:14:37,837 --> 01:14:39,920
than this left-moving piece
has the same amplitude

1742
01:14:39,920 --> 01:14:41,450
as the right-moving piece.

1743
01:14:41,450 --> 01:14:42,770
This is a standing wave.

1744
01:14:45,400 --> 01:14:47,799
All it's doing is it's
rotating by an overall phase.

1745
01:14:47,799 --> 01:14:49,340
But it's a standing
wave because it's

1746
01:14:49,340 --> 01:14:51,480
a superposition of a wave
moving this way and a wave

1747
01:14:51,480 --> 01:14:53,146
moving that way with
the same amplitude.

1748
01:14:53,146 --> 01:14:54,215
Just slightly shifted.

1749
01:14:54,215 --> 01:14:55,840
So the fact that
there's a little shift

1750
01:14:55,840 --> 01:14:57,840
tells you that it's norm
squared is not constant.

1751
01:14:57,840 --> 01:14:58,673
It has a small wave.

1752
01:15:01,420 --> 01:15:03,990
So what we see is we
get a standing wave that

1753
01:15:03,990 --> 01:15:05,740
matches nicely onto a
decaying exponential

1754
01:15:05,740 --> 01:15:09,820
in this classically
disallowed region.

1755
01:15:09,820 --> 01:15:12,780
So what that tells you
is what's the probability

1756
01:15:12,780 --> 01:15:14,280
to get arbitrarily
far to the right?

1757
01:15:16,910 --> 01:15:17,480
0.

1758
01:15:17,480 --> 01:15:18,730
It's exponentially suppressed.

1759
01:15:18,730 --> 01:15:21,480
What's the probability that your
found some point on the left?

1760
01:15:21,480 --> 01:15:22,980
Well, it's the norm
squared of that,

1761
01:15:22,980 --> 01:15:25,470
which is some standing wave.

1762
01:15:25,470 --> 01:15:28,624
So the reflection of how
good is this is a mirror.

1763
01:15:28,624 --> 01:15:29,540
AUDIENCE: [INAUDIBLE].

1764
01:15:29,540 --> 01:15:30,915
PROFESSOR: It's
a perfect mirror.

1765
01:15:30,915 --> 01:15:32,750
Well, it's not-- Now
we have to quibble

1766
01:15:32,750 --> 01:15:34,379
about what you mean
by perfect mirror.

1767
01:15:34,379 --> 01:15:35,420
It reflects with a phase.

1768
01:15:38,120 --> 01:15:39,710
It reflects with a
phase beta, and I'm

1769
01:15:39,710 --> 01:15:42,810
going to call that phase-- I
want to pick conventions here

1770
01:15:42,810 --> 01:15:44,240
that are consistent throughout.

1771
01:15:44,240 --> 01:15:48,380
Let's call that e to the i phi.

1772
01:15:48,380 --> 01:15:51,424
So it reflects with
a phase phi, which

1773
01:15:51,424 --> 01:15:53,090
is to say that
[INAUDIBLE] squared is 1.

1774
01:15:59,790 --> 01:16:03,580
And what happens-- I
don't want to do this.

1775
01:16:08,400 --> 01:16:11,130
So in what situation
would you expect

1776
01:16:11,130 --> 01:16:15,440
this to be a truly perfect
mirror, this potential?

1777
01:16:15,440 --> 01:16:17,603
When do you expect reflection
to be truly perfect?

1778
01:16:17,603 --> 01:16:18,530
AUDIENCE: [INAUDIBLE].

1779
01:16:18,530 --> 01:16:19,170
PROFESSOR: Yeah, exactly.

1780
01:16:19,170 --> 01:16:20,961
So if the height here
were infinitely high,

1781
01:16:20,961 --> 01:16:22,140
then there's no probability.

1782
01:16:22,140 --> 01:16:23,181
You can't leak in at all.

1783
01:16:23,181 --> 01:16:24,560
There's no exponential tail.

1784
01:16:24,560 --> 01:16:25,590
It's just 0.

1785
01:16:25,590 --> 01:16:29,626
So in the case that v0 is much
greater than e, what do we get?

1786
01:16:29,626 --> 01:16:31,362
We get that v0 is
much greater than e.

1787
01:16:31,362 --> 01:16:35,200
That means alpha's gigantic,
and k is very, very small.

1788
01:16:35,200 --> 01:16:38,870
If alpha is gigantic and k is
very small, negligibly small,

1789
01:16:38,870 --> 01:16:42,020
than this just becomes
minus i alpha over i alpha.

1790
01:16:42,020 --> 01:16:44,640
This just becomes
minus 1 in that limit.

1791
01:16:44,640 --> 01:16:47,200
So in the limit that the wall
is really a truly hard wall,

1792
01:16:47,200 --> 01:16:48,850
this phase becomes minus 1.

1793
01:16:48,850 --> 01:16:51,886
What happens to a wave when it
bounces off a perfect mirror?

1794
01:16:51,886 --> 01:16:52,860
AUDIENCE: [INAUDIBLE].

1795
01:16:52,860 --> 01:16:55,370
PROFESSOR: Its phase
is inverted, exactly.

1796
01:16:55,370 --> 01:16:57,580
Here we see that the
phase shift is pi.

1797
01:16:57,580 --> 01:17:01,640
You get a minus 1 precisely when
the barrier is infinitely high.

1798
01:17:01,640 --> 01:17:03,760
When the barrier's
a finite height,

1799
01:17:03,760 --> 01:17:06,525
classically it would be
a perfect mirror still.

1800
01:17:06,525 --> 01:17:08,900
But quantum mechanically it's
no longer a perfect mirror.

1801
01:17:08,900 --> 01:17:11,100
There's a phase
shift which indicates

1802
01:17:11,100 --> 01:17:14,740
that the wave is extending a
little bit into the material.

1803
01:17:14,740 --> 01:17:16,160
The wave is extending
a little bit

1804
01:17:16,160 --> 01:17:18,290
into the classically
disallowed region.

1805
01:17:18,290 --> 01:17:21,310
This phase shift, called
the scattering phase shift,

1806
01:17:21,310 --> 01:17:22,810
is going to play a
huge role for us.

1807
01:17:22,810 --> 01:17:24,410
It encodes an enormous
amount of the physics,

1808
01:17:24,410 --> 01:17:25,930
as we'll see over
the next few weeks.

1809
01:17:25,930 --> 01:17:27,660
At the end of the semester
it'll be important for us

1810
01:17:27,660 --> 01:17:29,450
in our discussion
of bands and solids.

1811
01:17:33,050 --> 01:17:40,500
So at this point, though, we've
got a challenge ahead of us.

1812
01:17:40,500 --> 01:17:42,442
I'm going to be using
phrases like this

1813
01:17:42,442 --> 01:17:44,650
is the part of the wave
function moving to the right,

1814
01:17:44,650 --> 01:17:46,760
and that's the part of the wave
function moving to the left.

1815
01:17:46,760 --> 01:17:47,930
I'm going to say
the wave function is

1816
01:17:47,930 --> 01:17:49,740
a superposition of
those two things.

1817
01:17:49,740 --> 01:17:52,220
But I need a more
precise version of stuff

1818
01:17:52,220 --> 01:17:54,130
is going to the right.

1819
01:17:54,130 --> 01:17:58,650
And this is where the
probability density,

1820
01:17:58,650 --> 01:18:01,870
phi squared, and the probability
current, which you guys

1821
01:18:01,870 --> 01:18:10,640
have constructed on the problem
set, which is h-bar upon 2mi.

1822
01:18:13,350 --> 01:18:23,300
Psi star dx i minus
psi dx psi star.

1823
01:18:23,300 --> 01:18:26,370
These guys satisfy the
conservation equation

1824
01:18:26,370 --> 01:18:28,630
which is d RHO dt.

1825
01:18:28,630 --> 01:18:30,220
The time rate of
change of the density

1826
01:18:30,220 --> 01:18:34,580
at some particular
point at time t

1827
01:18:34,580 --> 01:18:37,260
is equal to minus the gradient,
derivative with respect

1828
01:18:37,260 --> 01:18:41,350
to x, of j.

1829
01:18:41,350 --> 01:18:42,725
And I should really
call this jx.

1830
01:18:47,260 --> 01:18:49,610
So the current in
the x direction.

1831
01:18:49,610 --> 01:18:56,110
So this is the
conservation equation,

1832
01:18:56,110 --> 01:18:59,170
just like for the
conservation of charge,

1833
01:18:59,170 --> 01:19:01,780
and you guys should have
all effectively derived

1834
01:19:01,780 --> 01:19:02,780
this on the problem set.

1835
01:19:02,780 --> 01:19:03,610
Yeah.

1836
01:19:03,610 --> 01:19:06,430
AUDIENCE: Is the i under
or above [INAUDIBLE]?

1837
01:19:06,430 --> 01:19:10,950
PROFESSOR: Is the i-- oh,
sorry, the i is below.

1838
01:19:10,950 --> 01:19:12,450
That's just my
horrible handwriting.

1839
01:19:16,320 --> 01:19:21,817
So what I want to
define very quickly

1840
01:19:21,817 --> 01:19:24,400
is I want to be able to come up
with an unambiguous definition

1841
01:19:24,400 --> 01:19:26,270
of how much stuff is
going left and how much

1842
01:19:26,270 --> 01:19:27,682
stuff is going right.

1843
01:19:27,682 --> 01:19:29,140
And the way I'm
going to do that is

1844
01:19:29,140 --> 01:19:33,350
I'm going to say that the
wave function in general--

1845
01:19:33,350 --> 01:19:38,370
I'll do that here-- when we have
a wave function at some point.

1846
01:19:38,370 --> 01:19:40,260
It can be approximated
if the potential is

1847
01:19:40,260 --> 01:19:41,975
roughly constant at that point.

1848
01:19:41,975 --> 01:19:43,930
It can be approximated
in the following way.

1849
01:19:43,930 --> 01:19:46,650
The wave function
is going to be psi

1850
01:19:46,650 --> 01:19:50,610
is equal to psi
incident plus psi

1851
01:19:50,610 --> 01:19:57,580
transmitted--
yeah, psi reflected

1852
01:19:57,580 --> 01:19:59,530
on the left of my barrier.

1853
01:19:59,530 --> 01:20:02,970
And psi transmitted
on the right.

1854
01:20:02,970 --> 01:20:06,089
Where psi incident is
that right-moving part.

1855
01:20:06,089 --> 01:20:07,880
The part that's moving
towards the barrier.

1856
01:20:07,880 --> 01:20:10,730
Psi reflected is the left-moving
part that's reflected.

1857
01:20:10,730 --> 01:20:12,602
Psi transmitted is
the part that is

1858
01:20:12,602 --> 01:20:14,310
on the right-hand
side, the entire thing.

1859
01:20:14,310 --> 01:20:16,768
So here the idea is I'm sending
in something from the left.

1860
01:20:16,768 --> 01:20:20,134
It can either reflect
or it can transmit.

1861
01:20:20,134 --> 01:20:22,300
So this is just my notation
for this for these guys.

1862
01:20:25,470 --> 01:20:27,950
And so what I wanted
to measure of how much

1863
01:20:27,950 --> 01:20:30,810
stuff is going left, how
much stuff is going right.

1864
01:20:30,810 --> 01:20:34,660
And the way to do that is to
use the probability current.

1865
01:20:34,660 --> 01:20:40,240
And to say that associated to
the incident term in the wave

1866
01:20:40,240 --> 01:20:42,840
function is an
incident current, which

1867
01:20:42,840 --> 01:20:47,659
is h-bar k upon
m from a squared.

1868
01:20:47,659 --> 01:20:49,950
So if we take this wave
function, take that expression,

1869
01:20:49,950 --> 01:20:52,850
plug it into j, this
is what you get.

1870
01:20:52,850 --> 01:20:56,490
If you take the reflected
part b to the minus i kx

1871
01:20:56,490 --> 01:21:00,980
minus omega d, and you plug it
into the current expression,

1872
01:21:00,980 --> 01:21:04,000
you get the part of-- sorry,
I should call this capital

1873
01:21:04,000 --> 01:21:06,370
J-- you get the part of
the current corresponding

1874
01:21:06,370 --> 01:21:12,410
to the reflected term, which
is h-bar k with a minus k

1875
01:21:12,410 --> 01:21:16,330
upon m v squared.

1876
01:21:16,330 --> 01:21:18,384
And we'll get that
J transmitted is 0.

1877
01:21:18,384 --> 01:21:20,300
Something that you proved
on your problem set.

1878
01:21:20,300 --> 01:21:24,770
If the wave function is real,
it must-- the current to 0.

1879
01:21:24,770 --> 01:21:27,000
So let's think about what
this is telling us quickly.

1880
01:21:27,000 --> 01:21:27,800
So what is this [INAUDIBLE]?

1881
01:21:27,800 --> 01:21:28,830
So what is a current?

1882
01:21:28,830 --> 01:21:33,250
A current is the amount of
stuff moving per unit time.

1883
01:21:33,250 --> 01:21:35,294
So charge times the velocity.

1884
01:21:35,294 --> 01:21:36,960
So what's the stuff
we're interested in?

1885
01:21:36,960 --> 01:21:38,830
It's the probability density.

1886
01:21:38,830 --> 01:21:39,580
There's a squared.

1887
01:21:39,580 --> 01:21:40,204
Norm a squared.

1888
01:21:40,204 --> 01:21:42,010
That's the
probability amplitude,

1889
01:21:42,010 --> 01:21:46,750
or probability density of
the right-moving piece.

1890
01:21:46,750 --> 01:21:48,390
Just consider it in isolation.

1891
01:21:48,390 --> 01:21:49,809
So that's the
probability density.

1892
01:21:49,809 --> 01:21:50,850
It's the amount of stuff.

1893
01:21:50,850 --> 01:21:55,310
And this is the momentum of
that wave, that's e to the i kx.

1894
01:21:55,310 --> 01:21:55,990
There's h-bar k.

1895
01:21:55,990 --> 01:21:58,160
That's the momentum
divided by the mass, which

1896
01:21:58,160 --> 01:21:59,625
is the classical velocity.

1897
01:21:59,625 --> 01:22:01,250
The amount of stuff
times the velocity.

1898
01:22:01,250 --> 01:22:02,266
That a current.

1899
01:22:02,266 --> 01:22:05,220
It's the current with
which probability

1900
01:22:05,220 --> 01:22:07,960
is flowing across a point.

1901
01:22:07,960 --> 01:22:11,250
Similarly, Jr, that's the amount
of stuff, and the velocity

1902
01:22:11,250 --> 01:22:13,832
is minus h-bar k upon m.

1903
01:22:13,832 --> 01:22:15,290
So again, the
current is the amount

1904
01:22:15,290 --> 01:22:16,605
of stuff times the velocity.

1905
01:22:19,190 --> 01:22:21,790
I'm going to define-- this
is just the definition,

1906
01:22:21,790 --> 01:22:23,910
it's the reasonable
choice of definition--

1907
01:22:23,910 --> 01:22:27,960
the transmission probability
is the ratio, the norm squared

1908
01:22:27,960 --> 01:22:31,494
of J transmitted
over J incident.

1909
01:22:31,494 --> 01:22:33,660
And here's why this is the
reasonable thing-- oops--

1910
01:22:33,660 --> 01:22:34,400
incident.

1911
01:22:34,400 --> 01:22:36,730
Here's why this is the
reasonable quantity.

1912
01:22:36,730 --> 01:22:41,382
This is saying how much current
is moving in from the left?

1913
01:22:41,382 --> 01:22:43,590
How much stuff is moving in
from-- And how much stuff

1914
01:22:43,590 --> 01:22:48,160
is moving off on the
right, to the right?

1915
01:22:48,160 --> 01:22:50,920
So what is this in this case?

1916
01:22:50,920 --> 01:22:54,890
0, although this is
a general definition.

1917
01:22:54,890 --> 01:22:57,404
And similarly, the reflection
you can either write as 1

1918
01:22:57,404 --> 01:22:59,820
minus the transmission, because
either you transmit or you

1919
01:22:59,820 --> 01:23:03,950
reflect, the probability
must sum to 1.

1920
01:23:03,950 --> 01:23:07,980
Or you can write r as
the current reflected.

1921
01:23:07,980 --> 01:23:11,230
What fraction of the incident
current is, in fact, reflected?

1922
01:23:11,230 --> 01:23:12,490
And here this is 1.

1923
01:23:12,490 --> 01:23:15,351
Because beta-- b is a phase,
and the norm squared of a phase

1924
01:23:15,351 --> 01:23:15,850
is 1.

1925
01:23:19,510 --> 01:23:21,490
So on your problems
set, you'll be

1926
01:23:21,490 --> 01:23:24,380
going through the computation
of various reflection

1927
01:23:24,380 --> 01:23:28,730
and transmission amplitudes
for this potential.

1928
01:23:28,730 --> 01:23:32,630
And we'll pick up on this with
the barrier uphill next week.

1929
01:23:32,630 --> 01:23:34,870
Have a good spring break, guys.