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YEN-JIE LEE: OK, happy
to see you again.
9
00:00:27,840 --> 00:00:31,260
Welcome back to 8.03.
10
00:00:31,260 --> 00:00:34,290
Today, as you see
on the slide, we're
11
00:00:34,290 --> 00:00:37,710
going to continue the discussion
of dispersive medium--
12
00:00:37,710 --> 00:00:40,200
how the waves and
vibration should
13
00:00:40,200 --> 00:00:44,090
be sent through this medium.
14
00:00:44,090 --> 00:00:50,370
And also, we will learn about
uncertainty principle today.
15
00:00:50,370 --> 00:00:52,200
Kind of interesting.
16
00:00:52,200 --> 00:00:57,120
That is connected back here
to what we discuss here.
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00:00:57,120 --> 00:01:00,970
And finally, if we
have time, we'll
18
00:01:00,970 --> 00:01:05,570
move to two-dimensional system
and three-dimensional system
19
00:01:05,570 --> 00:01:09,500
to look at two-dimensional waves
and three-dimensional waves.
20
00:01:09,500 --> 00:01:13,420
OK, that's the plan for today.
21
00:01:13,420 --> 00:01:17,210
Just a quick review about
what we have learned so far.
22
00:01:17,210 --> 00:01:21,590
Last time, we
discussed about shaking
23
00:01:21,590 --> 00:01:24,870
one end of this dispersive
medium which is actually
24
00:01:24,870 --> 00:01:28,160
a string with stiffness.
25
00:01:28,160 --> 00:01:32,210
And basically you would see that
the strategy that we have been
26
00:01:32,210 --> 00:01:38,120
following is to do a Fourier
transform to actually decompose
27
00:01:38,120 --> 00:01:42,710
the motion of the hand,
which is actually holding
28
00:01:42,710 --> 00:01:49,580
one end of the string, and
then decompose that into wave
29
00:01:49,580 --> 00:01:53,030
population in frequency space.
30
00:01:53,030 --> 00:01:55,280
OK, so that's what
we have been doing.
31
00:01:55,280 --> 00:02:00,510
And then, we know based on
the property of this medium,
32
00:02:00,510 --> 00:02:04,790
the dispersion relation, which
is omega as a function of k,
33
00:02:04,790 --> 00:02:11,750
we can propagate waves
with different frequency
34
00:02:11,750 --> 00:02:13,340
at different speeds.
35
00:02:13,340 --> 00:02:15,650
Then we can see how
this system will
36
00:02:15,650 --> 00:02:17,970
evolve as a function of time.
37
00:02:17,970 --> 00:02:22,360
That's the whole
idea and the strategy
38
00:02:22,360 --> 00:02:28,420
we approach this
interesting problem.
39
00:02:28,420 --> 00:02:32,095
Last time, we also
introduced AM radio.
40
00:02:35,430 --> 00:02:39,550
As we discussed before, if
we have a very simple-minded
41
00:02:39,550 --> 00:02:41,800
strategy to just
send the pulse--
42
00:02:41,800 --> 00:02:44,260
which is containing
information-- directly
43
00:02:44,260 --> 00:02:47,170
through this medium,
due to the dispersion
44
00:02:47,170 --> 00:02:49,600
relation which we
have this medium,
45
00:02:49,600 --> 00:02:53,860
different component would be
traveling at different speed.
46
00:02:53,860 --> 00:02:57,710
Therefore, the
information is smeared out
47
00:02:57,710 --> 00:02:59,950
after it travels
through a long distance.
48
00:02:59,950 --> 00:03:00,760
OK?
49
00:03:00,760 --> 00:03:01,860
That's the problem.
50
00:03:01,860 --> 00:03:10,350
And then the solution was
to use this approach, which
51
00:03:10,350 --> 00:03:16,380
is amplitude modulation mixer.
52
00:03:16,380 --> 00:03:18,980
That's actually
how AM radio works.
53
00:03:18,980 --> 00:03:26,380
So basically, we have a slowly
oscillating message or signal
54
00:03:26,380 --> 00:03:29,910
like music or voice
which we want to send,
55
00:03:29,910 --> 00:03:35,250
and then as we multiply that
by a really fast oscillating
56
00:03:35,250 --> 00:03:37,570
cosine tan.
57
00:03:37,570 --> 00:03:40,380
If we do this, assuming
that omega of 0
58
00:03:40,380 --> 00:03:45,750
is actually much, much higher
or larger than the typical scale
59
00:03:45,750 --> 00:03:47,200
of your signal--
60
00:03:47,200 --> 00:03:48,790
which is omega s--
61
00:03:48,790 --> 00:03:51,330
then, what is going to
happen is the following.
62
00:03:51,330 --> 00:03:54,350
Up to all the calculations
we have done last time,
63
00:03:54,350 --> 00:03:59,440
we found that the
resulting wave which
64
00:03:59,440 --> 00:04:04,830
is the amplitude as a
function of time and space,
65
00:04:04,830 --> 00:04:08,770
you can see that this can
factorize into two components.
66
00:04:08,770 --> 00:04:12,160
The first component is
virtually the original signal
67
00:04:12,160 --> 00:04:13,750
you are trying to send.
68
00:04:13,750 --> 00:04:17,630
Since you're traveling at
the speed of group velocity,
69
00:04:17,630 --> 00:04:21,130
and finally, the right hand
side-- the second component--
70
00:04:21,130 --> 00:04:28,300
is actually the contribution,
the really small structure
71
00:04:28,300 --> 00:04:29,830
of these high
frequency oscillation.
72
00:04:29,830 --> 00:04:32,200
We call it carrier,
and the carrier
73
00:04:32,200 --> 00:04:36,740
is still traveling at
the of face velocity.
74
00:04:36,740 --> 00:04:39,560
That's how we actually
finally understand
75
00:04:39,560 --> 00:04:42,010
what is the meaning
of group velocity
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00:04:42,010 --> 00:04:47,050
and the face velocity
through this example.
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00:04:47,050 --> 00:04:50,950
What I am going to do
today is to guide you
78
00:04:50,950 --> 00:04:53,720
through another
example which will
79
00:04:53,720 --> 00:04:58,870
ensure we can learn some more
insight from this calculation.
80
00:04:58,870 --> 00:05:03,250
Today, we are going to have
another test of function,
81
00:05:03,250 --> 00:05:09,250
which actually I can do Fourier
transforms really easily.
82
00:05:09,250 --> 00:05:12,700
And this function I'm
trying to introduce here,
83
00:05:12,700 --> 00:05:16,300
I have this functional
form exponential
84
00:05:16,300 --> 00:05:22,000
minus gamma times
absolute value of t, OK?
85
00:05:22,000 --> 00:05:24,850
The reason why I choose
absolute value of t
86
00:05:24,850 --> 00:05:28,440
is because I would like to
make it symmetric around 0.
87
00:05:32,760 --> 00:05:36,340
I can now do the usual
Fourier transform
88
00:05:36,340 --> 00:05:38,680
and then to extract
the wave population.
89
00:05:38,680 --> 00:05:43,110
The function of angular
frequency, c omega.
90
00:05:43,110 --> 00:05:45,010
c as a function of omega.
91
00:05:45,010 --> 00:05:47,500
And according to
the formula here,
92
00:05:47,500 --> 00:05:50,560
which we introduced
last time, we
93
00:05:50,560 --> 00:05:52,750
can quickly write
it down like this.
94
00:05:52,750 --> 00:05:56,410
Basically you get 1 over 2
pi integration from minus
95
00:05:56,410 --> 00:06:02,080
to infinity to infinity
integrating over time.
96
00:06:02,080 --> 00:06:05,950
This is the original
function, f of t.
97
00:06:05,950 --> 00:06:09,140
And multiply that by
exponential I omega t.
98
00:06:09,140 --> 00:06:13,060
And that's the way
we extract c omega.
99
00:06:13,060 --> 00:06:14,180
OK?
100
00:06:14,180 --> 00:06:16,090
Since we have this
absolute value
101
00:06:16,090 --> 00:06:23,960
here, basically the trick
is to change the interval,
102
00:06:23,960 --> 00:06:26,440
split the interval
into two pieces.
103
00:06:26,440 --> 00:06:30,850
So, y is actually
the negative t part,
104
00:06:30,850 --> 00:06:35,440
therefore, you get the
exponential plus t here,
105
00:06:35,440 --> 00:06:39,220
and the other part is
from 0 to infinity.
106
00:06:39,220 --> 00:06:42,310
Then the absolute value
doesn't change side.
107
00:06:42,310 --> 00:06:44,050
You have the
original exponential
108
00:06:44,050 --> 00:06:47,110
minus gamma times t.
109
00:06:47,110 --> 00:06:50,230
Then you can go ahead
and do with integration,
110
00:06:50,230 --> 00:06:53,680
and you get the two turns
and you get the functional
111
00:06:53,680 --> 00:06:58,000
form, which is c
omega equal to gamma
112
00:06:58,000 --> 00:07:02,300
over pi times gamma
square plus omega square.
113
00:07:02,300 --> 00:07:03,160
OK?
114
00:07:03,160 --> 00:07:07,440
From this simple exercise,
they are interesting things
115
00:07:07,440 --> 00:07:10,380
which we can learn from here.
116
00:07:10,380 --> 00:07:21,160
If I go ahead and draw f
of t as a function of time,
117
00:07:21,160 --> 00:07:24,850
this is what you will get.
118
00:07:24,850 --> 00:07:32,060
Suppose I set gamma
to be equal to 0.1.
119
00:07:32,060 --> 00:07:33,950
And I would like to
visualize this function
120
00:07:33,950 --> 00:07:36,070
and that's what we did here.
121
00:07:36,070 --> 00:07:40,930
You can see that from
the left hand side here,
122
00:07:40,930 --> 00:07:44,440
is f of t as a function of time.
123
00:07:44,440 --> 00:07:48,980
And you can see that
this like exponential
124
00:07:48,980 --> 00:07:54,490
of t k but symmetric that
mirror at the t equal to 0.
125
00:07:54,490 --> 00:07:58,990
And with a small
gamma value I choose,
126
00:07:58,990 --> 00:08:01,300
that means this
exponential decay will
127
00:08:01,300 --> 00:08:08,090
be really slow, therefore, you
have a pretty wide distribution
128
00:08:08,090 --> 00:08:11,430
as a function of time.
129
00:08:11,430 --> 00:08:14,634
However, if you look at the
right hand side, what did
130
00:08:14,634 --> 00:08:16,050
I show you in the
right hand side?
131
00:08:16,050 --> 00:08:21,650
Right hand side is c omega,
c as a function of omega,
132
00:08:21,650 --> 00:08:25,480
it's the population
in frequency space.
133
00:08:25,480 --> 00:08:29,610
And you can see that, if I
plug in gamma equal to 0.1
134
00:08:29,610 --> 00:08:31,590
into that equation,
then you would
135
00:08:31,590 --> 00:08:34,140
get a distribution
which is actually
136
00:08:34,140 --> 00:08:39,409
pretty narrow, around 0.
137
00:08:39,409 --> 00:08:41,120
That's actually
quite interesting.
138
00:08:41,120 --> 00:08:46,460
And now, if I change gamma,
I increase the gamma slowly
139
00:08:46,460 --> 00:08:52,370
so it changes to 0.2, you see
aha, that's what I expect--
140
00:08:52,370 --> 00:08:56,780
the f function graphed
in the coordinate space
141
00:08:56,780 --> 00:08:58,800
becomes narrower.
142
00:08:58,800 --> 00:09:00,800
But, on the other
hand, you pay the price
143
00:09:00,800 --> 00:09:07,200
that the wave population in the
frequency space becomes wider.
144
00:09:07,200 --> 00:09:10,270
OK, the distribution
become wider.
145
00:09:10,270 --> 00:09:11,650
I can increase and increase.
146
00:09:11,650 --> 00:09:14,200
Now it's gamma equal to 0.5.
147
00:09:14,200 --> 00:09:16,370
Gamma equal to 1.
148
00:09:16,370 --> 00:09:19,450
And now I have a
rather large gamma.
149
00:09:19,450 --> 00:09:25,750
Now it says 2.0, and you can
see that as a function of gamma,
150
00:09:25,750 --> 00:09:32,650
if I set the gamma to be 5,
and you can see that the wave,
151
00:09:32,650 --> 00:09:38,350
or say the waves of the wave
in the coordinate space,
152
00:09:38,350 --> 00:09:40,450
becomes really small.
153
00:09:40,450 --> 00:09:45,820
But if you look at the
corresponding c function,
154
00:09:45,820 --> 00:09:49,570
you can see that waves
becomes really large.
155
00:09:49,570 --> 00:09:53,680
This seems to be telling
us something interesting.
156
00:09:53,680 --> 00:09:58,840
It seems to me that I could
not choose a gamma value which
157
00:09:58,840 --> 00:10:03,430
simultaneously make waves
in a coordinate space
158
00:10:03,430 --> 00:10:07,000
narrow and those wave
populations in the frequency
159
00:10:07,000 --> 00:10:10,360
space narrow at the same time.
160
00:10:10,360 --> 00:10:15,670
I cannot actually do that based
on this simple-minded exercise.
161
00:10:15,670 --> 00:10:17,950
And what you are going
to do in your p-set
162
00:10:17,950 --> 00:10:20,680
is to go through another
parameterization, which
163
00:10:20,680 --> 00:10:22,570
is a Gaussian distribution.
164
00:10:22,570 --> 00:10:26,710
And you will see
very similar, hope
165
00:10:26,710 --> 00:10:31,450
for these very similar
conclusion from your exercise.
166
00:10:31,450 --> 00:10:33,560
So what is going on?
167
00:10:33,560 --> 00:10:37,660
And how do we
interpret this result?
168
00:10:37,660 --> 00:10:40,120
And why is this result
actually related
169
00:10:40,120 --> 00:10:42,850
to uncertainty principle?
170
00:10:42,850 --> 00:10:45,460
That's the first part
of the lecture, which
171
00:10:45,460 --> 00:10:48,700
we are going to discuss today.
172
00:10:48,700 --> 00:10:57,810
We can demonstrate this in
fact by one example of f
173
00:10:57,810 --> 00:11:00,670
of t, which is showing here.
174
00:11:00,670 --> 00:11:04,930
And we go through and change
the waves of this distribution.
175
00:11:04,930 --> 00:11:07,450
Of course, we can
also try to show
176
00:11:07,450 --> 00:11:13,180
this in a much more precise
mathematical definition.
177
00:11:13,180 --> 00:11:15,480
That's what we are
going to do now.
178
00:11:18,080 --> 00:11:20,600
The first thing which
we would need to do
179
00:11:20,600 --> 00:11:24,800
is to define how to
quantify the waves
180
00:11:24,800 --> 00:11:33,170
of the distribution in frequency
space and in coordinate space.
181
00:11:33,170 --> 00:11:41,650
First, we define that the
intensity of the signal
182
00:11:41,650 --> 00:11:49,040
is proportional
to f of t squared.
183
00:11:49,040 --> 00:11:53,805
OK, that is the to estimate
the size of the intensity.
184
00:11:53,805 --> 00:11:57,360
It kind of makes sense
because, for example,
185
00:11:57,360 --> 00:12:00,500
the energy of the
electromagnetic wave
186
00:12:00,500 --> 00:12:05,150
is actually proportional to
the wave function squared.
187
00:12:05,150 --> 00:12:10,220
That's kind of reasonable
to choose this definition.
188
00:12:10,220 --> 00:12:14,000
And then, once we have that
the definition of intensity,
189
00:12:14,000 --> 00:12:19,240
then I can now calculate
the average of some operator
190
00:12:19,240 --> 00:12:20,610
function.
191
00:12:20,610 --> 00:12:23,900
For example, I can
calculate g of t
192
00:12:23,900 --> 00:12:27,210
is a average of the g function.
193
00:12:27,210 --> 00:12:30,170
And in this definition
of intensity,
194
00:12:30,170 --> 00:12:34,770
how to calculate the
average is to do integration
195
00:12:34,770 --> 00:12:40,700
over minus infinity
to infinity over t.
196
00:12:40,700 --> 00:12:43,220
And this g function
is put right there
197
00:12:43,220 --> 00:12:46,280
and all the components
are weighted
198
00:12:46,280 --> 00:12:52,180
by this intensity estimator,
which f of t squared.
199
00:12:55,070 --> 00:12:59,720
Of course, since we are actually
calculating the average,
200
00:12:59,720 --> 00:13:02,930
we need to take out the
sum of the intensity.
201
00:13:02,930 --> 00:13:05,000
So, the sum of all
the intensity is
202
00:13:05,000 --> 00:13:08,440
an integration from minus
infinity to infinity--
203
00:13:08,440 --> 00:13:13,820
dt f of t squared.
204
00:13:13,820 --> 00:13:18,690
With this definition, we
can calculate the average.
205
00:13:18,690 --> 00:13:28,020
OK and don't forget our goal is
to have an estimator estimate
206
00:13:28,020 --> 00:13:31,510
the waves of sum distribution.
207
00:13:31,510 --> 00:13:35,320
Therefore, you are probably
very familiar with that.
208
00:13:35,320 --> 00:13:37,960
We use standard deviation.
209
00:13:37,960 --> 00:13:41,560
So basically that's also usually
associated with the exam,
210
00:13:41,560 --> 00:13:46,900
but this time it's associated
with some physical quantity.
211
00:13:46,900 --> 00:13:53,740
what is the estimator
of spread of time?
212
00:13:56,660 --> 00:13:58,800
Right.
213
00:13:58,800 --> 00:14:02,460
We can actually make
use of this definition
214
00:14:02,460 --> 00:14:06,630
and I can write
this notation that
215
00:14:06,630 --> 00:14:12,330
t-squared to be a quantity
which is associated
216
00:14:12,330 --> 00:14:16,080
with the size of spread in time.
217
00:14:16,080 --> 00:14:24,866
And now as you define to be the
average of t minus average of t
218
00:14:24,866 --> 00:14:25,366
squared.
219
00:14:27,960 --> 00:14:30,510
Basically you calculate
that difference with respect
220
00:14:30,510 --> 00:14:36,920
to the mean value, square it,
and then do the two averages
221
00:14:36,920 --> 00:14:39,620
again.
222
00:14:39,620 --> 00:14:42,110
All right, everybody
is following?
223
00:14:42,110 --> 00:14:43,960
Any questions?
224
00:14:43,960 --> 00:14:45,020
OK.
225
00:14:45,020 --> 00:14:48,830
If I have this so-called
standard deviation or spread
226
00:14:48,830 --> 00:14:54,210
of time definition here, then
I can write it down explicitly,
227
00:14:54,210 --> 00:14:57,790
and this will become minus
infinity to infinity,
228
00:14:57,790 --> 00:15:07,370
to disintegration over t, and
I have t minus n value of t,
229
00:15:07,370 --> 00:15:10,670
half of t squared.
230
00:15:10,670 --> 00:15:13,700
And of course I would take
out that normalization, which
231
00:15:13,700 --> 00:15:24,830
is a minus infinity to
infinity dt f of t squared
232
00:15:24,830 --> 00:15:31,670
And I can also do a similar
exercise for the frequency
233
00:15:31,670 --> 00:15:32,660
space.
234
00:15:32,660 --> 00:15:38,990
Basically, I can
define the spread
235
00:15:38,990 --> 00:15:47,143
of the frequency spectrum.
236
00:15:50,600 --> 00:15:53,770
And that I define
it to be delta omega
237
00:15:53,770 --> 00:15:56,700
squared, and this
will be defined
238
00:15:56,700 --> 00:16:02,100
as the average of omega
minus the mean value of omega
239
00:16:02,100 --> 00:16:02,600
squared.
240
00:16:07,800 --> 00:16:13,230
And with this definition,
we have an estimator
241
00:16:13,230 --> 00:16:18,150
of this spread of time,
and we have an estimator
242
00:16:18,150 --> 00:16:20,610
of spread the frequency.
243
00:16:20,610 --> 00:16:24,360
The phenomenon which we see
from here, from this exercise,
244
00:16:24,360 --> 00:16:30,660
going from low gamma value to
a large gamma value is that it
245
00:16:30,660 --> 00:16:37,290
seems to us that the spread of
the time or in coordinate space
246
00:16:37,290 --> 00:16:40,920
and the spread of the
distribution in the frequency
247
00:16:40,920 --> 00:16:45,190
space cannot
simultaneously be small.
248
00:16:45,190 --> 00:16:46,170
OK.
249
00:16:46,170 --> 00:16:50,880
Therefore, based on this
mathematical definition,
250
00:16:50,880 --> 00:16:54,950
our goal is now to
show that we can
251
00:16:54,950 --> 00:17:02,826
prove that delta
omega times delta t
252
00:17:02,826 --> 00:17:06,599
will be larger or equal to 1/2.
253
00:17:06,599 --> 00:17:10,650
That is an interesting
consequence based
254
00:17:10,650 --> 00:17:13,710
on this definition of spread.
255
00:17:13,710 --> 00:17:16,290
We can actually achieve
the lecture today.
256
00:17:19,230 --> 00:17:20,550
That's our goal.
257
00:17:20,550 --> 00:17:25,800
And we are going to try
to achieve this goal.
258
00:17:25,800 --> 00:17:30,450
Before we go ahead and
prove this relation
259
00:17:30,450 --> 00:17:35,670
delta omega times delta t
greater or equal to 1/2,
260
00:17:35,670 --> 00:17:41,010
we also realize
that when we discuss
261
00:17:41,010 --> 00:17:45,780
this spread of the
frequencies spectrum,
262
00:17:45,780 --> 00:17:49,140
if I write it down
here, if I try
263
00:17:49,140 --> 00:17:53,310
to calculate the
average of omega,
264
00:17:53,310 --> 00:17:56,880
then what I'm going to do is
to do the integration from
265
00:17:56,880 --> 00:18:00,870
minus infinity to
infinity, d omega.
266
00:18:00,870 --> 00:18:03,390
because now I'm
trying to calculate
267
00:18:03,390 --> 00:18:06,090
the mean value of omega.
268
00:18:06,090 --> 00:18:09,810
I have the omega times c omega.
269
00:18:12,600 --> 00:18:15,390
And the exponential
is i omega t.
270
00:18:24,700 --> 00:18:31,820
If I go ahead and
evaluate this integral,
271
00:18:31,820 --> 00:18:37,025
I integrate over omega, and
I have omega times c omega
272
00:18:37,025 --> 00:18:42,820
times exponential i omega t.
273
00:18:42,820 --> 00:18:47,500
And you can see that this
omega can actually be extracted
274
00:18:47,500 --> 00:18:50,110
from this exponential function.
275
00:18:50,110 --> 00:18:55,760
If I do differentiation,
which is spread through time,
276
00:18:55,760 --> 00:18:57,970
then I can actually
extract one omega out
277
00:18:57,970 --> 00:19:00,470
of the initial function.
278
00:19:00,470 --> 00:19:03,580
Therefore, what I'm going
to get is this will be equal
279
00:19:03,580 --> 00:19:14,390
to i partial t minus infinity
to infinity d omega, c omega,
280
00:19:14,390 --> 00:19:18,640
exponential minus i omega t.
281
00:19:18,640 --> 00:19:20,800
So you can see that
this is the design
282
00:19:20,800 --> 00:19:24,760
if I do a partiality relative
to with respect to t,
283
00:19:24,760 --> 00:19:29,260
then I take minus i omega out
of this exponential function
284
00:19:29,260 --> 00:19:33,690
and this i will make
minus i become 1.
285
00:19:33,690 --> 00:19:39,490
Therefore, you can see that this
integral, which I construct,
286
00:19:39,490 --> 00:19:43,790
is equal to i
partial, partial t.
287
00:19:43,790 --> 00:19:45,790
This function.
288
00:19:45,790 --> 00:19:52,210
OK, and you can quickly
realize that we know what
289
00:19:52,210 --> 00:19:54,330
this integral is doing right.
290
00:19:54,330 --> 00:19:59,470
According to the form which
I just did here, f of t
291
00:19:59,470 --> 00:20:02,900
is equal to this integral
which I actually just highlight
292
00:20:02,900 --> 00:20:03,910
there.
293
00:20:03,910 --> 00:20:08,366
Therefore, this is just f of t.
294
00:20:10,870 --> 00:20:16,500
that's kind of interesting
because that would give me i
295
00:20:16,500 --> 00:20:20,715
partial, partial t, f of t.
296
00:20:25,770 --> 00:20:30,570
Basically you can see that I
don't need to deal with omega,
297
00:20:30,570 --> 00:20:35,100
I can actually do a partial
relative with respect to time,
298
00:20:35,100 --> 00:20:38,490
then I can take one omega
out of the function which
299
00:20:38,490 --> 00:20:41,736
I have constructed.
300
00:20:41,736 --> 00:20:45,440
Any questions?
301
00:20:45,440 --> 00:20:52,490
All right, now I can calculate
what will be the mean omega.
302
00:20:52,490 --> 00:20:53,750
What would be the mean omega?
303
00:20:53,750 --> 00:20:58,570
The mean omega, according
to this definition here,
304
00:20:58,570 --> 00:21:03,590
this is how we calculate
the mean of some quantity,
305
00:21:03,590 --> 00:21:09,360
mean omega will be equal to
minus infinity to infinity,
306
00:21:09,360 --> 00:21:18,990
tt f star, t i partial,
partial t, f of t.
307
00:21:18,990 --> 00:21:22,430
OK sorry that this is
kind of close to here.
308
00:21:27,180 --> 00:21:32,100
The original definition I should
put omega got here, right?
309
00:21:32,100 --> 00:21:34,770
But instead of
putting omega there,
310
00:21:34,770 --> 00:21:39,810
I used the trick that this
i times partial partial t
311
00:21:39,810 --> 00:21:43,440
can generate an omega for me.
312
00:21:43,440 --> 00:21:46,860
Therefore, instead of
putting omega explicitly
313
00:21:46,860 --> 00:21:50,940
into the integral, I
put i partial partial t
314
00:21:50,940 --> 00:21:53,880
into the integral, then
I get 1 omega out of it,
315
00:21:53,880 --> 00:21:57,970
and that's equivalent
to the calculation
316
00:21:57,970 --> 00:22:00,930
with g of t equal to omega.
317
00:22:00,930 --> 00:22:02,606
OK, everybody's following?
318
00:22:07,470 --> 00:22:10,430
Therefore, I of
course still need
319
00:22:10,430 --> 00:22:14,420
to normalize the calculation.
320
00:22:14,420 --> 00:22:20,100
This is the denominator, which
is minus infinity to infinity,
321
00:22:20,100 --> 00:22:26,450
integral over tt f of t squared.
322
00:22:26,450 --> 00:22:30,280
OK, you can see that instead
of using omega directly,
323
00:22:30,280 --> 00:22:36,500
the I used this trick to use
i partial partial t to extract
324
00:22:36,500 --> 00:22:41,040
1 omega and I can calculate
the mean value of omega.
325
00:22:46,930 --> 00:22:49,630
Therefore I can also
calculate explicitly
326
00:22:49,630 --> 00:22:54,590
what would be the delta omega
square based on the definition
327
00:22:54,590 --> 00:22:56,080
which I outlined before.
328
00:22:56,080 --> 00:23:01,600
This would be the average value
of omega minus mean omega.
329
00:23:01,600 --> 00:23:06,880
Mean omega is a number,
and if I'd write it down
330
00:23:06,880 --> 00:23:15,050
explicitly I get minus
infinity to infinity, tt, i
331
00:23:15,050 --> 00:23:22,206
partial partial t, minus
average value omega, f
332
00:23:22,206 --> 00:23:30,520
of t squared divided
by minus infinity
333
00:23:30,520 --> 00:23:39,530
to infinity disintegration
over dt f of t squared.
334
00:23:39,530 --> 00:23:41,930
The take home
message is that I'm
335
00:23:41,930 --> 00:23:46,430
using this trick to
replace all the omega
336
00:23:46,430 --> 00:23:48,860
by i partial partial t.
337
00:23:48,860 --> 00:23:52,060
Therefore, in my
formula, you will
338
00:23:52,060 --> 00:23:56,570
see that originally, this
is supposed to be omega
339
00:23:56,570 --> 00:23:59,480
and now we were
using that trick.
340
00:23:59,480 --> 00:24:02,720
Therefore, it can be written
as pi partial partial t.
341
00:24:02,720 --> 00:24:08,450
And you'll realize what
this is used for afterwards.
342
00:24:08,450 --> 00:24:11,960
All right, so those
are just preparation.
343
00:24:11,960 --> 00:24:14,960
What we have done
is that my goal
344
00:24:14,960 --> 00:24:19,460
is to show that delta
omega times delta t
345
00:24:19,460 --> 00:24:22,670
is greater than or equal to 1/2.
346
00:24:22,670 --> 00:24:25,980
OK, that's my goal and
I'm preparing for that.
347
00:24:25,980 --> 00:24:29,590
And I have that definition
of delta t and delta omega.
348
00:24:29,590 --> 00:24:30,090
Yes?
349
00:24:30,090 --> 00:24:35,660
AUDIENCE: What do
you think [INAUDIBLE]
350
00:24:35,660 --> 00:24:39,940
YEN-JIE LEE: Oh sorry,
there should be--
351
00:24:39,940 --> 00:24:41,570
it should be like this.
352
00:24:41,570 --> 00:24:44,630
So I am taking partial
partial t out of f.
353
00:24:44,630 --> 00:24:46,040
OK, sorry.
354
00:24:46,040 --> 00:24:48,230
Good question.
355
00:24:48,230 --> 00:24:51,020
Any other mistakes?
356
00:24:51,020 --> 00:24:52,250
Very good.
357
00:24:52,250 --> 00:24:53,170
Not yet?
358
00:24:53,170 --> 00:24:54,380
All right.
359
00:24:54,380 --> 00:24:58,470
So now you can see that I have
the definition in my hand,
360
00:24:58,470 --> 00:25:00,620
and I am almost
there to show you
361
00:25:00,620 --> 00:25:03,500
that delta omega
times delta t is going
362
00:25:03,500 --> 00:25:08,030
to be greater or equal to 1/2.
363
00:25:08,030 --> 00:25:13,160
And what I'm going
to do after this--
364
00:25:13,160 --> 00:25:17,460
maybe you will be
even more mad at me--
365
00:25:17,460 --> 00:25:21,170
is to use exactly
the same trick which
366
00:25:21,170 --> 00:25:25,130
would be used to show
Heisenberg's Uncertainty
367
00:25:25,130 --> 00:25:29,300
Principle in quantum mechanics.
368
00:25:29,300 --> 00:25:33,590
basically what I'm going to do
is to consider a function which
369
00:25:33,590 --> 00:25:34,910
is r of t.
370
00:25:39,805 --> 00:25:44,840
r as a function kappa and t.
371
00:25:44,840 --> 00:25:48,830
and the definition of this
r function is like this.
372
00:25:48,830 --> 00:26:04,060
I define this r function to be t
minus average t minus i kappa i
373
00:26:04,060 --> 00:26:07,310
partial partial t minus omega.
374
00:26:10,202 --> 00:26:11,648
f of t.
375
00:26:14,550 --> 00:26:21,150
If you don't know where is
this relationship coming from,
376
00:26:21,150 --> 00:26:24,780
don't be worried because
you don't really need to.
377
00:26:24,780 --> 00:26:31,230
This is just to guide us through
this mathematical calculation.
378
00:26:31,230 --> 00:26:34,685
But if you can see directly
how this will help,
379
00:26:34,685 --> 00:26:36,433
the maybe you are Heisenberg.
380
00:26:39,243 --> 00:26:39,743
Maybe.
381
00:26:39,743 --> 00:26:40,980
So that's very nice.
382
00:26:40,980 --> 00:26:41,580
It's a test.
383
00:26:45,030 --> 00:26:48,390
What I am going to do
is to employ this r
384
00:26:48,390 --> 00:26:51,780
as a function of kappa of t.
385
00:26:51,780 --> 00:26:55,960
And the 2 for our purpose
to show that the delta omega
386
00:26:55,960 --> 00:26:58,370
and delta t greater than 1/2.
387
00:26:58,370 --> 00:27:01,330
And first, to make
my life easier,
388
00:27:01,330 --> 00:27:05,490
I would define this
to be capital T,
389
00:27:05,490 --> 00:27:12,090
and I would define this
thing to be capital omega.
390
00:27:12,090 --> 00:27:15,120
So that my mathematical
expression doesn't explode.
391
00:27:19,680 --> 00:27:27,210
Now I can consider this
ratio function r of kappa.
392
00:27:27,210 --> 00:27:30,870
This is defined
as minus infinity
393
00:27:30,870 --> 00:27:40,070
to infinity integrating
over t r kappa t divided
394
00:27:40,070 --> 00:27:48,700
by minus infinity to
infinity dt f of t squared.
395
00:27:48,700 --> 00:27:56,650
This is r function which is the
ratio of the area of r function
396
00:27:56,650 --> 00:28:00,067
and the area of the f function.
397
00:28:00,067 --> 00:28:02,150
You may say that, professor,
this is really crazy.
398
00:28:02,150 --> 00:28:04,990
Today is telling about
all the crazy things,
399
00:28:04,990 --> 00:28:07,232
but that is because
I would like to let
400
00:28:07,232 --> 00:28:11,200
you know that we are going to
see a very interesting result.
401
00:28:11,200 --> 00:28:14,050
So that's why I'm doing this.
402
00:28:14,050 --> 00:28:18,580
And if I construct
this r function,
403
00:28:18,580 --> 00:28:22,150
this r function will have
an interesting property.
404
00:28:22,150 --> 00:28:24,080
What is the
interesting property?
405
00:28:24,080 --> 00:28:26,710
I entered an integral
over something
406
00:28:26,710 --> 00:28:32,080
squared in the numerator
and the denominator.
407
00:28:32,080 --> 00:28:37,590
Now it means, what would be
the value of this r function?
408
00:28:37,590 --> 00:28:42,290
The r function would
be always positive.
409
00:28:42,290 --> 00:28:43,430
Right?
410
00:28:43,430 --> 00:28:45,920
Because this is a square,
this is a square, therefore,
411
00:28:45,920 --> 00:28:48,890
r is going to be positive.
412
00:28:48,890 --> 00:28:55,780
That means r is going to be
greater than or equal to 0.
413
00:28:55,780 --> 00:28:58,320
That's why we have
this r function.
414
00:28:58,320 --> 00:29:02,470
And the miracle will happen
because if I go ahead
415
00:29:02,470 --> 00:29:05,020
and calculate this r--
416
00:29:05,020 --> 00:29:08,350
before I calculate this
capital R function-- what's
417
00:29:08,350 --> 00:29:14,380
the function of kappa, I need to
actually deal with this small r
418
00:29:14,380 --> 00:29:18,250
as a function of
kappa and t squared.
419
00:29:18,250 --> 00:29:24,885
If I extract this component
and then calculate that,
420
00:29:24,885 --> 00:29:30,190
r kappa t squared.
421
00:29:30,190 --> 00:29:34,150
What I am going to do is
to use this expression r
422
00:29:34,150 --> 00:29:46,150
is equal to t, capital T minus
i kappa omega times f of t.
423
00:29:46,150 --> 00:29:48,940
So that my life would be easier.
424
00:29:48,940 --> 00:29:57,690
Then basically you get
t minus i kappa omega f.
425
00:29:57,690 --> 00:30:00,750
And then you need tje
complex conjugate.
426
00:30:00,750 --> 00:30:13,510
Basically, you get T cross
i kappa omega star f star.
427
00:30:13,510 --> 00:30:18,000
You can have T star,
but T is a real number.
428
00:30:18,000 --> 00:30:20,095
Therefore, it
doesn't do anything.
429
00:30:23,200 --> 00:30:26,884
Then, I can now go ahead
and collect all the terms.
430
00:30:26,884 --> 00:30:28,550
Then the first terms
which I can collect
431
00:30:28,550 --> 00:30:32,450
is everything
related to T times f.
432
00:30:32,450 --> 00:30:37,710
Then basically you
get the T f squared.
433
00:30:37,710 --> 00:30:43,920
That is coming from this
T times f times T times f.
434
00:30:46,680 --> 00:30:50,870
This term times this
term times this term.
435
00:30:50,870 --> 00:30:53,640
to give you the first term.
436
00:30:53,640 --> 00:30:55,600
And you also you can
connect another term
437
00:30:55,600 --> 00:31:00,740
which omega f squared.
438
00:31:00,740 --> 00:31:01,580
Right.
439
00:31:01,580 --> 00:31:09,601
Basically, you can
find that contribution.
440
00:31:17,590 --> 00:31:22,360
Use should have a kappa
square in front of it.
441
00:31:22,360 --> 00:31:24,160
Any questions so far?
442
00:31:24,160 --> 00:31:29,740
Basically, I collect the
terms related to omega times f
443
00:31:29,740 --> 00:31:31,780
and put it here.
444
00:31:31,780 --> 00:31:33,610
Finally, you have
the third term,
445
00:31:33,610 --> 00:31:52,020
which is i kappa T f omega star
f star minus omega f T f star.
446
00:31:52,020 --> 00:31:56,700
Basically, this small
r function squared
447
00:31:56,700 --> 00:32:01,270
can be written in
this functional form.
448
00:32:01,270 --> 00:32:04,150
We are almost there.
449
00:32:04,150 --> 00:32:06,840
What I'm going to discuss
first is that now I
450
00:32:06,840 --> 00:32:10,180
have these three terms.
451
00:32:10,180 --> 00:32:14,860
Number one, number
two, and number three.
452
00:32:17,380 --> 00:32:20,080
I can now attack
number three first.
453
00:32:23,300 --> 00:32:33,750
Number three, I'm going to
get i kappa Tf minus i partial
454
00:32:33,750 --> 00:32:42,100
partial t minus omega f star.
455
00:32:42,100 --> 00:32:47,710
Basically what I'm doing
is to take this omega here.
456
00:32:47,710 --> 00:32:50,170
This is omega star.
457
00:32:50,170 --> 00:32:53,110
And then use that
definition, write down
458
00:32:53,110 --> 00:32:55,420
the expression for omega--
459
00:32:55,420 --> 00:32:57,550
typical omega-- explicitly.
460
00:32:57,550 --> 00:33:02,660
Since I am writing omega star,
therefore, you get a minus i
461
00:33:02,660 --> 00:33:07,400
partial partial t minus
average omega out of it.
462
00:33:07,400 --> 00:33:10,360
That's why here you
have this expression
463
00:33:10,360 --> 00:33:13,615
and then multiple it by f, which
is the original expression.
464
00:33:17,420 --> 00:33:21,680
I also write this omega
capital Omega explicitly.
465
00:33:21,680 --> 00:33:31,290
I partial partial t
minus average Omega.
466
00:33:31,290 --> 00:33:34,080
f t f star.
467
00:33:37,760 --> 00:33:41,340
And you can immediately
realize that--
468
00:33:41,340 --> 00:33:47,160
OK, this whole thing is
multiplied by i times kappa.
469
00:33:47,160 --> 00:33:51,040
You can immediately recognize
that this term actually
470
00:33:51,040 --> 00:33:53,240
canceled because they are--
471
00:33:53,240 --> 00:33:56,300
actually they are
literally the same.
472
00:33:56,300 --> 00:33:59,950
And then what is
actually left over
473
00:33:59,950 --> 00:34:03,220
is the two terms,
which is in the middle.
474
00:34:03,220 --> 00:34:06,040
So basically, you are
going to get now I
475
00:34:06,040 --> 00:34:11,330
can multiply i and
cancel this minus i.
476
00:34:11,330 --> 00:34:17,065
Basically what you
get is kappa time
477
00:34:17,065 --> 00:34:21,461
T equals-- both terms have a
T, so I can extract this T out
478
00:34:21,461 --> 00:34:21,960
of it.
479
00:34:25,150 --> 00:34:35,449
f partial f star partial T cross
partial f partial T f star.
480
00:34:40,000 --> 00:34:43,199
After all those works,
you can see that this one
481
00:34:43,199 --> 00:34:45,600
looks pretty nice.
482
00:34:45,600 --> 00:34:46,429
This says what?
483
00:34:49,080 --> 00:34:54,980
This is not bad at all after
all those calculations basically
484
00:34:54,980 --> 00:35:02,880
these will be equal to kappa
T partial partial t f f star.
485
00:35:07,860 --> 00:35:10,930
Everybody's following or
everybody already lost?
486
00:35:14,280 --> 00:35:17,700
We are almost there.
487
00:35:17,700 --> 00:35:18,690
All right.
488
00:35:18,690 --> 00:35:20,170
Now, we have these three.
489
00:35:20,170 --> 00:35:21,990
Three originally is a beast.
490
00:35:21,990 --> 00:35:26,520
Looks really horrible and after
I write it down explicitly,
491
00:35:26,520 --> 00:35:28,700
it looks OK, not perfect.
492
00:35:28,700 --> 00:35:29,458
Yes?
493
00:35:29,458 --> 00:35:32,117
AUDIENCE: [INAUDIBLE]
494
00:35:32,117 --> 00:35:34,325
YEN-JIE LEE: The complex
conjugate of the f function.
495
00:35:36,960 --> 00:35:38,130
All right.
496
00:35:38,130 --> 00:35:44,070
Now I can put one, two, and
three into this integral.
497
00:35:44,070 --> 00:35:46,410
Then we are done.
498
00:35:46,410 --> 00:35:51,510
Now let's put numbers 3
into the integral first.
499
00:35:51,510 --> 00:35:56,652
I do a minus infinity to
infinity, number three, dt.
500
00:35:59,460 --> 00:36:02,130
What is going to happen?
501
00:36:02,130 --> 00:36:07,470
This will give you minus
infinity to infinity kappa T
502
00:36:07,470 --> 00:36:12,030
partial partial t f f star.
503
00:36:15,050 --> 00:36:18,810
And I can use integration by
parts so what I'm going to get
504
00:36:18,810 --> 00:36:28,620
is kappa T f f star
evaluating minus infinity
505
00:36:28,620 --> 00:36:35,870
and then plus infinity
minus kappa minus
506
00:36:35,870 --> 00:36:41,936
infinity to infinity
f square partial t
507
00:36:41,936 --> 00:36:46,030
partial capital T partial t d t.
508
00:36:48,690 --> 00:36:50,070
Let's look at this.
509
00:36:50,070 --> 00:36:55,440
Basically, what I'm doing is to
put in the numbers written back
510
00:36:55,440 --> 00:37:00,990
into this integral and then
use integration by parts.
511
00:37:00,990 --> 00:37:05,190
Basically you can see that
this is what you would expect.
512
00:37:08,170 --> 00:37:12,820
The interesting thing
is that this function
513
00:37:12,820 --> 00:37:18,070
is evaluated at crossing at
infinity and minus infinity.
514
00:37:18,070 --> 00:37:24,740
If you assume that your
f function is localized--
515
00:37:24,740 --> 00:37:30,110
it's confined in some
specific range of time,
516
00:37:30,110 --> 00:37:35,460
instead of spreading out
over the whole universe.
517
00:37:35,460 --> 00:37:41,230
That means this term
will be equal to 0
518
00:37:41,230 --> 00:37:45,710
because it's evaluated at
plus infinity time and minus
519
00:37:45,710 --> 00:37:48,890
infinity time.
520
00:37:48,890 --> 00:37:55,500
If the f function is localized,
then at the boundary of time,
521
00:37:55,500 --> 00:37:57,520
you are going to get 0.
522
00:37:57,520 --> 00:37:58,660
This term disappears.
523
00:37:58,660 --> 00:37:59,200
Very good.
524
00:37:59,200 --> 00:38:01,510
We've solved one problem.
525
00:38:01,510 --> 00:38:07,140
And this looks horrible, but
partial capital T, partial t,
526
00:38:07,140 --> 00:38:08,860
what is capital T?
527
00:38:08,860 --> 00:38:11,770
T is small t minus average of t.
528
00:38:11,770 --> 00:38:17,290
Average of t is a
number and t is just t.
529
00:38:17,290 --> 00:38:23,740
Therefore, partial t
partial small d is just 1.
530
00:38:23,740 --> 00:38:26,050
You can see that
there are hopes,
531
00:38:26,050 --> 00:38:29,290
things are becoming
simpler and simpler.
532
00:38:29,290 --> 00:38:33,430
Therefore, what I'm
going to get is this--
533
00:38:33,430 --> 00:38:41,940
minus kappa minus infinity
to infinity t t f squared.
534
00:38:44,650 --> 00:38:56,400
And then if you divide this by
this term, you can see that 3--
535
00:38:56,400 --> 00:39:02,110
number 3 term-- will give you
a contribution of minus kappa.
536
00:39:02,110 --> 00:39:03,870
That's all.
537
00:39:03,870 --> 00:39:07,260
Because once you plug
this integral back
538
00:39:07,260 --> 00:39:10,460
into this function, the
third term contribution
539
00:39:10,460 --> 00:39:13,140
gives you minus kappa.
540
00:39:13,140 --> 00:39:19,140
That's a very good news because
it's actually pretty simple.
541
00:39:19,140 --> 00:39:20,280
Any questions?
542
00:39:26,418 --> 00:39:33,840
AUDIENCE: [INAUDIBLE]
543
00:39:33,840 --> 00:39:35,972
YEN-JIE LEE: Oh,
you mean this one?
544
00:39:35,972 --> 00:39:37,418
AUDIENCE: No.
545
00:39:37,418 --> 00:39:38,382
YEN-JIE LEE: This one?
546
00:39:38,382 --> 00:39:40,310
AUDIENCE: To the left.
547
00:39:40,310 --> 00:39:41,274
YEN-JIE LEE: Oh, yeah.
548
00:39:41,274 --> 00:39:41,857
You are right.
549
00:39:41,857 --> 00:39:43,087
I missed a dt.
550
00:39:43,087 --> 00:39:43,920
Thank you very much.
551
00:39:43,920 --> 00:39:44,750
Very good.
552
00:39:44,750 --> 00:39:45,660
Yeah.
553
00:39:45,660 --> 00:39:49,140
Basically what I'm trying to
do is plug in the expression
554
00:39:49,140 --> 00:39:50,960
here into the integral.
555
00:39:54,886 --> 00:39:56,260
You can see that
the contribution
556
00:39:56,260 --> 00:39:59,040
from the third term that
number 2 is rather simple.
557
00:39:59,040 --> 00:40:04,030
It's just minus kappa.
558
00:40:04,030 --> 00:40:06,134
Let's also take a look
at the computation
559
00:40:06,134 --> 00:40:08,520
from the first and the second.
560
00:40:08,520 --> 00:40:17,130
Wife Number one, will give you
minus infinity to infinity t
561
00:40:17,130 --> 00:40:27,016
minus average of t
squared f of t squared dt.
562
00:40:27,016 --> 00:40:30,886
And this is divided
by minus infinity
563
00:40:30,886 --> 00:40:36,460
to infinity dt, f of t.
564
00:40:36,460 --> 00:40:41,852
This is not crazy at all
because this just the definition
565
00:40:41,852 --> 00:40:44,636
of delta t squared.
566
00:40:44,636 --> 00:40:48,785
Just a reminder that the
definition of delta t squared
567
00:40:48,785 --> 00:40:51,290
is written here.
568
00:40:51,290 --> 00:40:56,880
Therefore, this is
just delta t squared--
569
00:40:56,880 --> 00:40:59,740
the first term, which
looks really strange there,
570
00:40:59,740 --> 00:41:05,604
but in reality, it's
actually very simple.
571
00:41:05,604 --> 00:41:07,532
Let's look at the second term.
572
00:41:07,532 --> 00:41:14,430
This is kappa squared minus
infinity to infinity i
573
00:41:14,430 --> 00:41:22,828
partial partial t minus
average of omega f of t.
574
00:41:22,828 --> 00:41:24,820
And then square that.
575
00:41:24,820 --> 00:41:34,300
Divide it by minus infinity to
infinity dt, f of t squared.
576
00:41:34,300 --> 00:41:39,708
And that will give you kappa
squared delta omega squared.
577
00:41:44,000 --> 00:41:48,916
Basically, our conclusion
that this r function
578
00:41:48,916 --> 00:41:51,380
is a function of kappa.
579
00:41:51,380 --> 00:41:57,911
Essentially equal to the first
terms here delta t squared,
580
00:41:57,911 --> 00:42:03,315
the second term is plus kappa
squared of delta omega squared
581
00:42:03,315 --> 00:42:03,900
.
582
00:42:03,900 --> 00:42:07,440
And finally, the
third term is there.
583
00:42:07,440 --> 00:42:08,140
Minus kappa.
584
00:42:14,160 --> 00:42:19,020
And this would be
greater or equal to 0.
585
00:42:19,020 --> 00:42:22,000
Because what I am
doing is just summing
586
00:42:22,000 --> 00:42:26,820
all those positive functions.
587
00:42:26,820 --> 00:42:28,580
Then, take the rest.
588
00:42:28,580 --> 00:42:29,730
.
589
00:42:29,730 --> 00:42:33,208
Any questions?
590
00:42:33,208 --> 00:42:35,698
AUDIENCE: Why does the
integral from negative infinity
591
00:42:35,698 --> 00:42:38,805
to infinity dt f squared equal?
592
00:42:41,720 --> 00:42:43,740
YEN-JIE LEE: This one?
593
00:42:43,740 --> 00:42:44,240
This one?
594
00:42:47,040 --> 00:42:49,430
This is equal to zero, right?
595
00:42:49,430 --> 00:42:50,070
Oh, here?
596
00:42:50,070 --> 00:42:52,320
AUDIENCE: Yeah.
597
00:42:52,320 --> 00:42:53,392
Why does that--
598
00:42:53,392 --> 00:42:54,350
YEN-JIE LEE: Oh, I see.
599
00:42:54,350 --> 00:42:56,030
I see your point.
600
00:42:56,030 --> 00:42:59,330
This is an integrated minus
infinity to infinity number
601
00:42:59,330 --> 00:43:00,690
3 dt.
602
00:43:00,690 --> 00:43:02,210
It's the contribution here.
603
00:43:02,210 --> 00:43:06,710
Then, if I take a ratio between
this term and that term,
604
00:43:06,710 --> 00:43:10,050
then this is canceled
by the denominator.
605
00:43:10,050 --> 00:43:12,667
Therefore, what is actually
left over is minus kappa.
606
00:43:12,667 --> 00:43:14,750
AUDIENCE: OK.
607
00:43:14,750 --> 00:43:18,110
YEN-JIE LEE: This 3, the
contribution of 3 in green
608
00:43:18,110 --> 00:43:22,820
is already taking the ratio
when I evaluate the capital R
609
00:43:22,820 --> 00:43:24,870
function.
610
00:43:24,870 --> 00:43:27,530
Good question.
611
00:43:27,530 --> 00:43:30,800
Now you can see
that you can safely
612
00:43:30,800 --> 00:43:33,680
ignore what I have said so far.
613
00:43:33,680 --> 00:43:34,940
Everything you can ignore.
614
00:43:34,940 --> 00:43:38,120
Those are just
mathematics tricks.
615
00:43:38,120 --> 00:43:42,580
But what is very important is
that now I have this relation--
616
00:43:42,580 --> 00:43:45,940
delta t squared plus kappa
square plus delta omega
617
00:43:45,940 --> 00:43:48,800
squared minus k.
618
00:43:48,800 --> 00:43:51,380
This is a function of k.
619
00:43:51,380 --> 00:43:53,040
And I can actually minimize it.
620
00:43:57,760 --> 00:44:04,030
I can minimize R if I
carefully choose a kappa value.
621
00:44:04,030 --> 00:44:09,490
This kappa equal
to kappa mean value
622
00:44:09,490 --> 00:44:14,110
which makes the minimize
the R function is
623
00:44:14,110 --> 00:44:18,640
equal to 1/2 delta omega
squared, which I would not
624
00:44:18,640 --> 00:44:22,420
go over this calculation because
this is just a minimization
625
00:44:22,420 --> 00:44:22,920
problem.
626
00:44:25,450 --> 00:44:29,860
That means if plug that
in, what I'm getting
627
00:44:29,860 --> 00:44:39,865
is R kappa min will be equal
to delta T squared minus 1
628
00:44:39,865 --> 00:44:43,290
over 4 delta omega squared.
629
00:44:43,290 --> 00:44:45,594
That is greater or equal to.
630
00:44:45,594 --> 00:44:46,094
0.
631
00:44:52,110 --> 00:44:55,380
We arrive there.
632
00:44:55,380 --> 00:45:00,240
If I multiple both
sides by 4 delta omega
633
00:45:00,240 --> 00:45:09,140
squared you get delta t
squared delta omega squared
634
00:45:09,140 --> 00:45:13,780
greater or equal to 1 over 4.
635
00:45:13,780 --> 00:45:16,090
If you take the
square root of that,
636
00:45:16,090 --> 00:45:22,220
the you get delta t delta
omega greater or equal to 1/2.
637
00:45:22,220 --> 00:45:28,390
That's actually what we
started to try to prove right?
638
00:45:28,390 --> 00:45:30,850
You can see that
after all those works
639
00:45:30,850 --> 00:45:35,290
a lot of complicated
mathematic calculations,
640
00:45:35,290 --> 00:45:40,180
you can see that we
make no assumption,
641
00:45:40,180 --> 00:45:45,460
we are just using the
definition of the spread of time
642
00:45:45,460 --> 00:45:48,520
and the spread of frequency.
643
00:45:48,520 --> 00:45:52,600
We follow that definition and
the use of mathematical trick
644
00:45:52,600 --> 00:45:57,370
which we used to prove
Heisenberg's Uncertainty
645
00:45:57,370 --> 00:46:01,210
Principle and we arrive there.
646
00:46:01,210 --> 00:46:07,000
This means that this is
an intrinsic property
647
00:46:07,000 --> 00:46:09,160
of wave function.
648
00:46:09,160 --> 00:46:12,630
Intrinsic property means that
it's a mathematic property
649
00:46:12,630 --> 00:46:13,540
of wave function.
650
00:46:16,860 --> 00:46:22,310
What do I mean by this equation,
which we finally did right?
651
00:46:26,470 --> 00:46:28,360
After all those
hard work, we have
652
00:46:28,360 --> 00:46:31,160
to enjoy what we
have learned right
653
00:46:31,160 --> 00:46:34,330
from all of those crazy things.
654
00:46:34,330 --> 00:46:36,550
What do we learn?
655
00:46:36,550 --> 00:46:38,000
Look at this function.
656
00:46:38,000 --> 00:46:42,770
Delta t times delta omega,
greater or equal to 1/2.
657
00:46:42,770 --> 00:46:49,450
That means if I construct
a function, which
658
00:46:49,450 --> 00:46:53,410
is how I oscillate the
stream as a function of time,
659
00:46:53,410 --> 00:47:00,020
if I construct a really
narrow one to this very fast
660
00:47:00,020 --> 00:47:02,200
and then I stop--
661
00:47:02,200 --> 00:47:05,470
very narrow-- then you will
have a very small delta t.
662
00:47:08,660 --> 00:47:10,010
Now it sounds really nice.
663
00:47:10,010 --> 00:47:12,380
I produce a delta
function, delta t,
664
00:47:12,380 --> 00:47:20,540
but the delta omega space
is going to be a mess.
665
00:47:20,540 --> 00:47:24,640
It's going to be a super wide
distribution because delta t is
666
00:47:24,640 --> 00:47:26,510
really very, very small.
667
00:47:26,510 --> 00:47:28,880
That means you have
to compensate that
668
00:47:28,880 --> 00:47:34,410
by a rather large delta omega
because if you multiple delta t
669
00:47:34,410 --> 00:47:38,190
times delta omega, that is going
to be great or equal to 1/2.
670
00:47:41,540 --> 00:47:43,910
And is the consequence
of this, for example,
671
00:47:43,910 --> 00:47:49,340
for the discussion of AM radio.
672
00:47:49,340 --> 00:47:58,140
If I have an AM radio with
bandwidth delta omega.
673
00:47:58,140 --> 00:48:09,530
This is 2 pi delta nu and that
is something like 3 times 10
674
00:48:09,530 --> 00:48:14,790
to the 4 Hz.
675
00:48:14,790 --> 00:48:17,460
If I have some kind
of bandwidth which
676
00:48:17,460 --> 00:48:20,820
is actually roughly this value.
677
00:48:20,820 --> 00:48:24,090
I can now immediately
calculate what
678
00:48:24,090 --> 00:48:26,130
will be the resulting delta t.
679
00:48:26,130 --> 00:48:33,230
The resulting delta t
will be a few times 10
680
00:48:33,230 --> 00:48:40,190
to the minus 5 seconds
based on this equation.
681
00:48:42,700 --> 00:48:58,470
This means that if I'm trying
to send two signals in sequence
682
00:48:58,470 --> 00:49:02,030
through this AM radio.
683
00:49:02,030 --> 00:49:05,940
that mean if the delta t--
684
00:49:05,940 --> 00:49:08,880
the time difference
between the first
685
00:49:08,880 --> 00:49:11,040
and the second information--
686
00:49:11,040 --> 00:49:16,710
if the time difference
is large, if that delta-t
687
00:49:16,710 --> 00:49:23,000
between these two much, much
larger than 10 to the minus
688
00:49:23,000 --> 00:49:24,090
to the minus 5 seconds.
689
00:49:26,670 --> 00:49:31,270
Then I can actually easily
separate these two signals.
690
00:49:34,310 --> 00:49:38,060
On the other hand, if
I send then really,
691
00:49:38,060 --> 00:49:41,442
the two signal really
close to each other,
692
00:49:41,442 --> 00:49:45,780
if it looks like this,
then the receiver,
693
00:49:45,780 --> 00:49:48,350
the ones who will
receive the signal,
694
00:49:48,350 --> 00:49:51,410
will not be able to
separate, if this is just
695
00:49:51,410 --> 00:49:56,690
one signal or two signals,
or one pulse or two pulse
696
00:49:56,690 --> 00:49:58,160
which you are trying to send.
697
00:50:01,220 --> 00:50:03,840
Any questions so far?
698
00:50:03,840 --> 00:50:05,640
So you can see that
we can actually
699
00:50:05,640 --> 00:50:10,020
quantify what will be the
limitation in the resolution,
700
00:50:10,020 --> 00:50:15,340
tiny resolution, due to the
limitation of bandwidth delta
701
00:50:15,340 --> 00:50:15,840
omega.
702
00:50:18,570 --> 00:50:21,390
Before we take a
break, I would like
703
00:50:21,390 --> 00:50:24,730
to make a connection
to quantum physics.
704
00:50:24,730 --> 00:50:28,500
So if I look at this delta t
times delta omega greater than
705
00:50:28,500 --> 00:50:34,380
or equal to 1 over 2, this
expression, I can rewrite it.
706
00:50:34,380 --> 00:50:43,140
I can multiply t by velocity
v. And I get v times velocity
707
00:50:43,140 --> 00:50:47,820
and I can have
omega divided by v.
708
00:50:47,820 --> 00:50:51,180
And this would be better
or equal to 1 over 2.
709
00:50:51,180 --> 00:50:54,300
So I just multiply
v and divide by v,
710
00:50:54,300 --> 00:50:57,400
then actually you can solve.
711
00:50:57,400 --> 00:51:03,530
And that means this
will become delta x.
712
00:51:03,530 --> 00:51:08,570
And that, the second
term, will become delta k.
713
00:51:08,570 --> 00:51:12,140
And that would be greater
or equal to 1 over 2.
714
00:51:14,650 --> 00:51:24,240
In the quantum physics, momentum
is equal to h bar times k.
715
00:51:28,780 --> 00:51:34,372
Momentum will be equal
to h bar times k.
716
00:51:34,372 --> 00:51:39,160
And h bar is actually
the Planck constant.
717
00:51:39,160 --> 00:51:46,700
So that, actually you will
see that a few times in L4.
718
00:51:46,700 --> 00:51:47,650
OK.
719
00:51:47,650 --> 00:51:53,090
So if I have p equal
to h bar times k,
720
00:51:53,090 --> 00:52:00,520
that means I have delta x times
delta p greater or equal to h
721
00:52:00,520 --> 00:52:02,020
bar over 2.
722
00:52:04,690 --> 00:52:13,140
That is exactly the uncertainty
principle, which was actually
723
00:52:13,140 --> 00:52:16,170
introduced by Heisenberg.
724
00:52:16,170 --> 00:52:17,990
And what is actually
the meaning of this?
725
00:52:17,990 --> 00:52:23,070
So if we describe
all those particles
726
00:52:23,070 --> 00:52:30,070
we see by quantum
mechanical waves,
727
00:52:30,070 --> 00:52:36,240
if I have momentum p, now it
corresponds to a wave function,
728
00:52:36,240 --> 00:52:38,530
with wave number k.
729
00:52:38,530 --> 00:52:44,260
And the constant, which is
associated with p and the k
730
00:52:44,260 --> 00:52:48,040
is the Planck constant.
731
00:52:48,040 --> 00:52:54,930
So this means that if I
measure one particle really,
732
00:52:54,930 --> 00:53:00,260
really precisely
in a position, due
733
00:53:00,260 --> 00:53:04,380
to the nature of
wave function that
734
00:53:04,380 --> 00:53:09,130
means I will not have
a lot of information
735
00:53:09,130 --> 00:53:12,730
about the momentum
of that particle.
736
00:53:12,730 --> 00:53:17,580
And where this uncertainty
principle is coming from,
737
00:53:17,580 --> 00:53:24,850
it's coming from purely the
mathematics related to waves.
738
00:53:24,850 --> 00:53:27,520
As you can see there,
there's really nothing
739
00:53:27,520 --> 00:53:31,300
to do with quantum so far.
740
00:53:31,300 --> 00:53:34,690
Quantum I'm saying
actually only goes in
741
00:53:34,690 --> 00:53:39,110
after we prove the uncertainty
principle, delta omega
742
00:53:39,110 --> 00:53:40,510
times delta t.
743
00:53:40,510 --> 00:53:45,820
You can cannot have a very
precise frequency and a very
744
00:53:45,820 --> 00:53:50,020
precise position in a
coordinated space over time
745
00:53:50,020 --> 00:53:51,990
at the same time.
746
00:53:51,990 --> 00:53:55,230
And that actually has
direct consequence.
747
00:53:55,230 --> 00:53:59,660
That means if you are considered
in quantum mechanics, that
748
00:53:59,660 --> 00:54:02,760
is essentially the limitation
which will be posted,
749
00:54:02,760 --> 00:54:05,730
the uncertainty principle.
750
00:54:05,730 --> 00:54:07,960
So we will take a
five minute break.
751
00:54:07,960 --> 00:54:14,210
And we come back and we take a
look at 2-3 dimensional waves.
752
00:54:14,210 --> 00:54:16,932
And let me know if you
have any questions.
753
00:54:26,260 --> 00:54:27,670
So welcome, back everybody.
754
00:54:27,670 --> 00:54:31,930
So before we actually moved
to 2-3 dimensional waves,
755
00:54:31,930 --> 00:54:34,960
we will discuss a very
interesting topic,
756
00:54:34,960 --> 00:54:36,910
which is related
to the dispersion
757
00:54:36,910 --> 00:54:42,520
relation of the light actually.
758
00:54:42,520 --> 00:54:47,270
So if you use
spatial relativity,
759
00:54:47,270 --> 00:54:54,400
basically you can relate energy
to momentum and the mass.
760
00:54:54,400 --> 00:54:59,360
So E square will be equal to a p
square c square plus m square c
761
00:54:59,360 --> 00:55:01,180
to the 4.
762
00:55:01,180 --> 00:55:08,080
And you actually interpret
light as a photon,
763
00:55:08,080 --> 00:55:11,410
then basically E is actually
equal-- to the energy
764
00:55:11,410 --> 00:55:15,400
of the photon will be
equal to h bar times omega.
765
00:55:15,400 --> 00:55:18,640
So we are actually really
going really forward a bit.
766
00:55:18,640 --> 00:55:20,140
Because maybe some
of you actually
767
00:55:20,140 --> 00:55:21,610
haven't seen this before.
768
00:55:21,610 --> 00:55:25,030
But if you just believe
what I have said,
769
00:55:25,030 --> 00:55:28,090
basically you can
actually divide everything
770
00:55:28,090 --> 00:55:31,030
from the first formula, which
is the spatial relativity
771
00:55:31,030 --> 00:55:33,420
formula, by h bar square.
772
00:55:33,420 --> 00:55:35,830
Then you will be able
to derive and arrive
773
00:55:35,830 --> 00:55:39,490
the second formula, which
is omega square equal to c
774
00:55:39,490 --> 00:55:43,540
square k square
plus omega 0 square.
775
00:55:43,540 --> 00:55:45,250
And the omega 0 is
actually defined
776
00:55:45,250 --> 00:55:50,020
as mc square over h bar,
just for simplicity.
777
00:55:50,020 --> 00:55:54,200
So if we look at this
equation, this is essentially
778
00:55:54,200 --> 00:55:56,920
a dispersion relation.
779
00:55:56,920 --> 00:55:59,570
Now you have seen
this so many times.
780
00:55:59,570 --> 00:56:04,300
And this omega square equal to
c square k square plus omega 0
781
00:56:04,300 --> 00:56:06,940
square, this formula is
actually reminding you
782
00:56:06,940 --> 00:56:10,330
that this is actually
a dispersion relation.
783
00:56:10,330 --> 00:56:17,820
So what I mean by a
photon having mass here?
784
00:56:17,820 --> 00:56:22,920
That means the m term in this
special relativity formula
785
00:56:22,920 --> 00:56:25,540
is not 0.
786
00:56:25,540 --> 00:56:29,820
Therefore omega 0
will be non-zero.
787
00:56:29,820 --> 00:56:31,380
What is going to happen?
788
00:56:31,380 --> 00:56:35,850
That means the space
of velocity of light
789
00:56:35,850 --> 00:56:37,770
is going to be different.
790
00:56:37,770 --> 00:56:42,990
It depends on what
value of k you choose.
791
00:56:42,990 --> 00:56:44,820
That's kind of interesting.
792
00:56:44,820 --> 00:56:50,400
Because that means light
with different frequency
793
00:56:50,400 --> 00:56:51,990
or different
wavelengths is going
794
00:56:51,990 --> 00:56:56,460
to be traveling through the
vacuum at different speeds,
795
00:56:56,460 --> 00:56:57,210
if that's true.
796
00:56:59,860 --> 00:57:02,220
Everybody get it?
797
00:57:02,220 --> 00:57:03,240
Very good.
798
00:57:03,240 --> 00:57:05,880
So how do we actually test this?
799
00:57:05,880 --> 00:57:11,040
So that means I need a light
source, which are very,
800
00:57:11,040 --> 00:57:13,880
very far away from earth.
801
00:57:13,880 --> 00:57:17,540
Then I would like to
measure the delta t
802
00:57:17,540 --> 00:57:22,430
as a function of frequency,
for example, and analyzing.
803
00:57:22,430 --> 00:57:25,820
So how do we do that?
804
00:57:25,820 --> 00:57:30,590
So this is actually
possible if you actually
805
00:57:30,590 --> 00:57:35,562
use a natural light source,
which is the pulsar.
806
00:57:35,562 --> 00:57:36,770
So what is actually a pulsar?
807
00:57:36,770 --> 00:57:38,840
So what we actually
use, essentially a
808
00:57:38,840 --> 00:57:40,970
millisecond pulsar.
809
00:57:40,970 --> 00:57:46,320
So those are actually coming
from rapidly rotating neutron
810
00:57:46,320 --> 00:57:50,590
stars, and that those
rotating neutron stars will
811
00:57:50,590 --> 00:57:55,010
emit pulses of radiation
like x-ray and radio waves,
812
00:57:55,010 --> 00:57:56,354
at regular intervals.
813
00:57:56,354 --> 00:57:57,770
Because it's
essentially rotating,
814
00:57:57,770 --> 00:58:01,710
rotating, rotating
again and again.
815
00:58:01,710 --> 00:58:06,560
Based on this movie, basically
what it's showing here
816
00:58:06,560 --> 00:58:11,000
is a very old neutron star.
817
00:58:11,000 --> 00:58:13,430
It's actually in
a binary system.
818
00:58:13,430 --> 00:58:16,520
And this neutron star
can absorb the material
819
00:58:16,520 --> 00:58:18,500
from the other partner.
820
00:58:18,500 --> 00:58:20,230
So that actually is--
821
00:58:20,230 --> 00:58:23,400
the rotation speed
actually increased.
822
00:58:23,400 --> 00:58:29,300
And finally at the speed
of a millisecond per turn.
823
00:58:29,300 --> 00:58:30,980
So this actually
really happened.
824
00:58:30,980 --> 00:58:33,230
And we can actually
observe this.
825
00:58:33,230 --> 00:58:38,010
And if we are lucky,
the earth is essentially
826
00:58:38,010 --> 00:58:42,410
somehow in a spatial direction
such that the emitting radio
827
00:58:42,410 --> 00:58:46,520
wave actually pointing
from the pulsar to earth,
828
00:58:46,520 --> 00:58:51,800
then I can see the pulsar,
the amplitude of the light
829
00:58:51,800 --> 00:58:56,120
from pulsar essentially changing
rapidly as a function of time.
830
00:58:56,120 --> 00:58:59,740
And another very good news is
that typically those pulsars
831
00:58:59,740 --> 00:59:01,250
are really far away.
832
00:59:01,250 --> 00:59:07,850
For example, in this
example, pulsar B1937+21,
833
00:59:07,850 --> 00:59:12,840
this is essentially a pulsar
with rotation period of just
834
00:59:12,840 --> 00:59:15,830
1.6 milliseconds.
835
00:59:15,830 --> 00:59:19,250
And this is actually
something which is really
836
00:59:19,250 --> 00:59:22,310
happening really far away from
the Earth, which essentially
837
00:59:22,310 --> 00:59:25,250
is 16,000 light years away.
838
00:59:25,250 --> 00:59:27,520
And that we can
actually observe this.
839
00:59:27,520 --> 00:59:30,380
This is actually pretty
close to Sagitta,
840
00:59:30,380 --> 00:59:35,200
and you can actually
see this pulsar.
841
00:59:35,200 --> 00:59:38,510
And how does that
actually associate
842
00:59:38,510 --> 00:59:41,300
with the original
question we were posting?
843
00:59:41,300 --> 00:59:44,610
The original question
is, does the light
844
00:59:44,610 --> 00:59:49,052
with different frequency
travel at different speed.
845
00:59:49,052 --> 00:59:50,760
And this is essentially
a very nice tool.
846
00:59:50,760 --> 00:59:51,260
Right?
847
00:59:51,260 --> 00:59:53,610
Because it is emitting
the radio wave.
848
00:59:53,610 --> 00:59:57,520
And now I can just measure the
spectra as a function of time.
849
00:59:57,520 --> 01:00:01,280
And I will be able
to see if we actually
850
01:00:01,280 --> 01:00:04,190
can observe different speed.
851
01:00:04,190 --> 01:00:08,930
Because we know the rotation
in the world, and et cetera.
852
01:00:08,930 --> 01:00:15,127
And it also emits a wide
spectra of the frequency,
853
01:00:15,127 --> 01:00:15,960
the light frequency.
854
01:00:15,960 --> 01:00:18,380
Therefore, I can use
this as a light source
855
01:00:18,380 --> 01:00:23,760
far, far away from the Earth,
to see what will happen.
856
01:00:23,760 --> 01:00:27,300
So somebody actually
did this measurement,
857
01:00:27,300 --> 01:00:28,680
and this is that
what they found.
858
01:00:31,640 --> 01:00:34,740
They found a non-zero omega 0.
859
01:00:34,740 --> 01:00:37,580
A non-zero omega 0 was found.
860
01:00:37,580 --> 01:00:42,500
So that means the mass
will be 1.3 times 10
861
01:00:42,500 --> 01:00:47,360
to the minus 49 gram.
862
01:00:47,360 --> 01:00:48,830
That sounds really small.
863
01:00:48,830 --> 01:00:51,590
But it's not small at all.
864
01:00:51,590 --> 01:00:54,630
That's actually destroying the
whole understanding of light.
865
01:00:57,160 --> 01:00:58,866
What is going on?
866
01:00:58,866 --> 01:00:59,740
So we are in trouble.
867
01:01:02,950 --> 01:01:06,370
So after all this
discussion, et cetera,
868
01:01:06,370 --> 01:01:08,260
and also other
measurements which
869
01:01:08,260 --> 01:01:11,730
are sensitive to photon
mass, they actually
870
01:01:11,730 --> 01:01:14,392
threw out this
possible contribution.
871
01:01:14,392 --> 01:01:17,290
This is essentially is
just simply too large
872
01:01:17,290 --> 01:01:21,210
based on, for example,
measurement of magnetic field
873
01:01:21,210 --> 01:01:22,960
in the galaxy, et cetera.
874
01:01:22,960 --> 01:01:24,350
It doesn't really work.
875
01:01:24,350 --> 01:01:26,860
So what essentially
is really happening?
876
01:01:26,860 --> 01:01:34,150
The explanation is that the path
from the pulsar to the earth
877
01:01:34,150 --> 01:01:36,860
it's really not vacuum.
878
01:01:36,860 --> 01:01:39,250
There are a lot of--
879
01:01:39,250 --> 01:01:44,920
not a lot, but we have very
few or very dilute electrons,
880
01:01:44,920 --> 01:01:48,670
very diluted free electrons
all over the place.
881
01:01:48,670 --> 01:01:52,750
And that will change the
frequency and the speed
882
01:01:52,750 --> 01:01:55,210
of light slightly.
883
01:01:55,210 --> 01:01:57,830
Therefore you observe
the interesting--
884
01:01:57,830 --> 01:01:59,180
observe the effect.
885
01:01:59,180 --> 01:02:01,300
And we are going to
actually also talk
886
01:02:01,300 --> 01:02:04,330
about how the material
actually changes
887
01:02:04,330 --> 01:02:07,030
the behavior of the
electromagnetic wave
888
01:02:07,030 --> 01:02:09,010
in the coming lectures.
889
01:02:09,010 --> 01:02:11,820
I hope you find
this interesting.
890
01:02:11,820 --> 01:02:15,000
Any questions?
891
01:02:15,000 --> 01:02:16,540
All right.
892
01:02:16,540 --> 01:02:19,560
So we are going to move on.
893
01:02:19,560 --> 01:02:22,310
So far what we have
been discussing
894
01:02:22,310 --> 01:02:26,280
is always 1-dimensional waves.
895
01:02:26,280 --> 01:02:33,150
So for example, a string,
and also the sound
896
01:02:33,150 --> 01:02:34,480
save in a tube, et cetera.
897
01:02:34,480 --> 01:02:38,590
We always discuss things
which are in one dimension.
898
01:02:38,590 --> 01:02:42,190
But we are actually not
one dimensional animal.
899
01:02:42,190 --> 01:02:46,080
We are 3-dimensional And
of course, for example,
900
01:02:46,080 --> 01:02:49,490
these objects the
surface is 2-dimensional
901
01:02:49,490 --> 01:02:50,990
So there are many,
many things which
902
01:02:50,990 --> 01:02:53,740
are more than one dimension.
903
01:02:53,740 --> 01:02:56,080
So can-- the question
that I'm trying
904
01:02:56,080 --> 01:03:01,240
to ask is, can we actually
understand this kind of object,
905
01:03:01,240 --> 01:03:07,000
and how actually to
understand those objects
906
01:03:07,000 --> 01:03:10,330
and how do we actually
derive the normal amounts,
907
01:03:10,330 --> 01:03:12,730
and how do we
actually write down
908
01:03:12,730 --> 01:03:16,090
the general solution, which
describes a 2-dimensional
909
01:03:16,090 --> 01:03:17,610
or a 3-dimensional wave.
910
01:03:17,610 --> 01:03:22,510
That's actually the next topic
which I would like to discuss.
911
01:03:22,510 --> 01:03:28,480
So that's actually gets
started with a plate like this.
912
01:03:28,480 --> 01:03:35,770
So basically that plate is
actually a 2-dimensional.
913
01:03:35,770 --> 01:03:42,940
And assuming that this plate is
infinitely long, for a moment,
914
01:03:42,940 --> 01:03:45,820
very, very long.
915
01:03:45,820 --> 01:03:47,300
So what does that mean?
916
01:03:47,300 --> 01:03:54,460
This means that if I define
my x and y-coordinate, which
917
01:03:54,460 --> 01:04:01,000
is actually used to describe
the position of a specific point
918
01:04:01,000 --> 01:04:03,925
on this plate,
then basically you
919
01:04:03,925 --> 01:04:08,540
will see that they are
beautiful symmetries, which
920
01:04:08,540 --> 01:04:13,600
you can actually identify
from this simple example.
921
01:04:13,600 --> 01:04:17,290
What is actually the symmetry
which we can identify?
922
01:04:17,290 --> 01:04:20,650
Can anybody help me with that?
923
01:04:20,650 --> 01:04:21,760
AUDIENCE: x and y.
924
01:04:21,760 --> 01:04:22,510
YEN-JIE LEE: Yeah.
925
01:04:22,510 --> 01:04:25,300
So yeah, x and y
are symmetric, yes.
926
01:04:25,300 --> 01:04:28,600
And the other function of x,
what kind of symmetry to you
927
01:04:28,600 --> 01:04:29,716
have?
928
01:04:29,716 --> 01:04:30,720
AUDIENCE: Reflection.
929
01:04:30,720 --> 01:04:31,470
YEN-JIE LEE: Yeah.
930
01:04:31,470 --> 01:04:34,570
Also reflection,
and what I'm looking
931
01:04:34,570 --> 01:04:39,330
for is if I change x and
change y, what kind of symmetry
932
01:04:39,330 --> 01:04:40,095
do you have?
933
01:04:40,095 --> 01:04:41,214
AUDIENCE: Translation.
934
01:04:41,214 --> 01:04:42,630
YEN-JIE LEE:
Translation symmetry.
935
01:04:42,630 --> 01:04:44,720
Well, all of you are correct.
936
01:04:44,720 --> 01:04:46,940
But what I am trying
to focus on now
937
01:04:46,940 --> 01:04:48,615
is the translation symmetry.
938
01:04:51,150 --> 01:04:54,800
So if I use translation
symmetry, what I'm going to get
939
01:04:54,800 --> 01:04:58,280
is that I can already
know the functional
940
01:04:58,280 --> 01:05:01,410
form of the normal mode.
941
01:05:01,410 --> 01:05:05,000
Because essentially if
it's translation symmetric,
942
01:05:05,000 --> 01:05:07,940
as a function of x, it's
translation symmetric
943
01:05:07,940 --> 01:05:09,770
as a function of y.
944
01:05:09,770 --> 01:05:17,700
Then I can say is in the x
direction will be proportional
945
01:05:17,700 --> 01:05:20,360
to exponential iKxX.
946
01:05:24,030 --> 01:05:29,640
K underscore x is essentially
the wave number associated
947
01:05:29,640 --> 01:05:32,539
with the wave in
the x direction.
948
01:05:32,539 --> 01:05:34,080
So that's essentially
one consequence
949
01:05:34,080 --> 01:05:37,950
which we actually learned from
the discussion of symmetry.
950
01:05:37,950 --> 01:05:42,300
And in the y direction,
I can conclude also
951
01:05:42,300 --> 01:05:51,690
that the normal mode will be
proportional to exponential iKy
952
01:05:51,690 --> 01:05:57,970
times Y. Therefore,
I already know
953
01:05:57,970 --> 01:06:01,180
what will be the function
form of the normal mode
954
01:06:01,180 --> 01:06:05,410
of this highly symmetric system.
955
01:06:05,410 --> 01:06:06,150
What is that?
956
01:06:06,150 --> 01:06:14,680
The psi xy will be equal
to A times exponential iKx
957
01:06:14,680 --> 01:06:18,360
times X, exponential iKyY.
958
01:06:22,000 --> 01:06:24,710
So you can see that.
959
01:06:24,710 --> 01:06:28,320
And also I need to
take the real part.
960
01:06:28,320 --> 01:06:32,650
Something like this will
be possible in normal mode.
961
01:06:32,650 --> 01:06:36,650
Therefore without going
into detail basically,
962
01:06:36,650 --> 01:06:41,060
we will see that the
expected behavior of psi
963
01:06:41,060 --> 01:06:46,930
as a function of x and
y will be something
964
01:06:46,930 --> 01:06:56,420
like a sine Kx times
x, sin Ky times y.
965
01:06:56,420 --> 01:07:00,995
So that's actually the
kind of normal mode, which
966
01:07:00,995 --> 01:07:05,780
we will expect based on
the argument of translation
967
01:07:05,780 --> 01:07:07,880
symmetry.
968
01:07:07,880 --> 01:07:13,340
And of course if I now go back
from infinitely long system
969
01:07:13,340 --> 01:07:17,870
to a finite system, then you
can use the boundary condition
970
01:07:17,870 --> 01:07:21,730
to determine what
would be the K value,
971
01:07:21,730 --> 01:07:28,780
Kx value, and allow the Kx value
and allow the Ky value using
972
01:07:28,780 --> 01:07:30,710
boundary conditions.
973
01:07:30,710 --> 01:07:34,460
So actually without
doing any calculation,
974
01:07:34,460 --> 01:07:39,470
we can already find
that, so now if I
975
01:07:39,470 --> 01:07:43,820
have a plate with finite
size, basically you
976
01:07:43,820 --> 01:07:49,310
expect that I can have some
kind of normal mode, which this
977
01:07:49,310 --> 01:07:52,910
is the amplitude, a
projection in the x direction,
978
01:07:52,910 --> 01:07:55,000
it can be a sine function.
979
01:07:55,000 --> 01:08:00,830
And that it can become 0
at the left-hand side edge
980
01:08:00,830 --> 01:08:02,540
and the right-hand side edge.
981
01:08:02,540 --> 01:08:04,610
And in the y
direction it has to be
982
01:08:04,610 --> 01:08:11,780
also some kind of sine
wave as a function of y.
983
01:08:11,780 --> 01:08:14,870
And of course it goes
to 0 at the edge.
984
01:08:14,870 --> 01:08:18,620
Because if those are
actually the fixed boundary,
985
01:08:18,620 --> 01:08:20,420
for example.
986
01:08:20,420 --> 01:08:22,930
And if those are actually
not fixed boundary,
987
01:08:22,930 --> 01:08:24,859
then you expect that--
988
01:08:24,859 --> 01:08:27,649
like open-end solution.
989
01:08:27,649 --> 01:08:32,970
So you expect that
the distribution
990
01:08:32,970 --> 01:08:38,460
will be more like
a cosine function
991
01:08:38,460 --> 01:08:42,100
for the first normal mode.
992
01:08:42,100 --> 01:08:44,109
And if you look at
this, the structure
993
01:08:44,109 --> 01:08:47,890
of this kind of solution,
it looks really complicated.
994
01:08:47,890 --> 01:08:51,500
Because you have x direction
and you also have y direction.
995
01:08:51,500 --> 01:08:55,100
Both of them are
actually sine functions.
996
01:08:55,100 --> 01:08:59,300
And how do we actually visualize
this kind of sine function?
997
01:08:59,300 --> 01:09:03,790
And here is a demonstration,
which I have prepared.
998
01:09:03,790 --> 01:09:09,399
It's really a
2-dimensional plate.
999
01:09:09,399 --> 01:09:13,630
And as you can see
that under this plate,
1000
01:09:13,630 --> 01:09:18,040
I have a loudspeaker which
actually produces a sound wave
1001
01:09:18,040 --> 01:09:23,850
to try to excite one
of the normal mode.
1002
01:09:23,850 --> 01:09:29,910
And the one I am going to do is
to turn on this loud speaker.
1003
01:09:29,910 --> 01:09:31,600
You can hear the sound.
1004
01:09:31,600 --> 01:09:35,705
And I would like to
see the normal mode.
1005
01:09:35,705 --> 01:09:39,100
But it's very hard to see
that, without doing anything.
1006
01:09:39,100 --> 01:09:42,520
Because it's vibrating, but
its so fast that it is really
1007
01:09:42,520 --> 01:09:44,410
very difficult to see it.
1008
01:09:44,410 --> 01:09:49,359
So what I am going to do is to
pour some sand on the surface,
1009
01:09:49,359 --> 01:09:51,220
and see what is going to happen.
1010
01:09:51,220 --> 01:09:57,720
And if we look at this,
I am putting sand on it.
1011
01:09:57,720 --> 01:10:00,250
And you can see that, there
is something happening.
1012
01:10:02,950 --> 01:10:09,052
If I change the frequency to one
of the normal mode frequencies,
1013
01:10:09,052 --> 01:10:14,480
you can see that now we are
reaching some kind of resonance
1014
01:10:14,480 --> 01:10:18,770
and exciting one
of the normal mode.
1015
01:10:18,770 --> 01:10:21,890
And you can see that
the sand actually it
1016
01:10:21,890 --> 01:10:25,440
doesn't like to stay
on some of the plate.
1017
01:10:25,440 --> 01:10:28,050
Because it's
vibrating like crazy
1018
01:10:28,050 --> 01:10:31,380
and it's not very
comfortable to sit there.
1019
01:10:31,380 --> 01:10:35,670
So the sand, where will
the sand actually sit?
1020
01:10:35,670 --> 01:10:42,226
They will set at the place where
you don't have any vibration.
1021
01:10:42,226 --> 01:10:43,600
Because what we
are talking here,
1022
01:10:43,600 --> 01:10:45,970
is essentially some
kind of sine wave times
1023
01:10:45,970 --> 01:10:49,250
sine wave or cosine
wave times cosine wave.
1024
01:10:49,250 --> 01:10:53,320
That means there will
be nodes on the plate.
1025
01:10:53,320 --> 01:10:56,940
And those are
2-dimensional nodes.
1026
01:10:56,940 --> 01:11:00,680
In the 1-dimensional case,
we are talking about nodes,
1027
01:11:00,680 --> 01:11:04,800
it's actually the place where
you have zero amplitude.
1028
01:11:04,800 --> 01:11:07,980
And now I have
cosine times cosine.
1029
01:11:07,980 --> 01:11:11,990
Therefore, there will be a
complicated pattern appearing
1030
01:11:11,990 --> 01:11:16,290
which is essentially the place
the plate is not actually
1031
01:11:16,290 --> 01:11:18,560
moving at all as a
function of time.
1032
01:11:18,560 --> 01:11:21,270
And you can see that now
I can actually excite one
1033
01:11:21,270 --> 01:11:22,380
with the normal node.
1034
01:11:22,380 --> 01:11:24,890
And you can see a really
beautiful pattern.
1035
01:11:24,890 --> 01:11:30,280
And allow me to do this
and increase the frequency.
1036
01:11:30,280 --> 01:11:35,790
So that if we see if I can
excite another normal mode.
1037
01:11:35,790 --> 01:11:37,000
Look at what is happening.
1038
01:11:37,000 --> 01:11:44,820
So now you see that the number
of lines actually increased.
1039
01:11:44,820 --> 01:11:52,800
So this is actually
so-called Chladni figures.
1040
01:11:52,800 --> 01:11:55,950
Basically those
figures are actually
1041
01:11:55,950 --> 01:12:03,260
produced by this trying to
excite one of the normal mode.
1042
01:12:03,260 --> 01:12:08,690
And basically the sand will be
collected in the nodal lines.
1043
01:12:08,690 --> 01:12:15,310
And you can see that this higher
frequency input sound wave.
1044
01:12:15,310 --> 01:12:18,890
You can excite the higher
frequency in normal mode.
1045
01:12:18,890 --> 01:12:24,370
And of course I can continue to
increase and see what happens.
1046
01:12:24,370 --> 01:12:30,955
Now I'm increasing the frequency
even higher and higher.
1047
01:12:30,955 --> 01:12:36,894
You can see that now the
sound is actually rather loud.
1048
01:12:36,894 --> 01:12:39,890
And I am actually
putting more sand.
1049
01:12:39,890 --> 01:12:42,720
You can see that there are
more and more patterns.
1050
01:12:42,720 --> 01:12:45,450
Because now I am
increasing the frequency,
1051
01:12:45,450 --> 01:12:48,810
so that actually the higher
frequency normal modes
1052
01:12:48,810 --> 01:12:50,040
are excited.
1053
01:12:50,040 --> 01:12:55,190
And you will expect more nodes
for higher frequency ones.
1054
01:12:55,190 --> 01:13:00,220
And now I can even
go even higher
1055
01:13:00,220 --> 01:13:03,785
to see if I find success.
1056
01:13:03,785 --> 01:13:08,075
It's not easy now.
1057
01:13:08,075 --> 01:13:08,575
Look.
1058
01:13:12,900 --> 01:13:15,450
Probably this is a
very good way to design
1059
01:13:15,450 --> 01:13:16,680
the pattern of your t-shirt.
1060
01:13:19,510 --> 01:13:20,010
OK.
1061
01:13:20,010 --> 01:13:23,100
So how do we actually
understand all those patterns?
1062
01:13:23,100 --> 01:13:26,250
And we have already started.
1063
01:13:26,250 --> 01:13:29,260
This is actually something
related to cosine and sine
1064
01:13:29,260 --> 01:13:30,900
multiplied to each other.
1065
01:13:30,900 --> 01:13:32,880
And the next time
we are going to do
1066
01:13:32,880 --> 01:13:36,900
a more detailed calculation
and show you a few more demos
1067
01:13:36,900 --> 01:13:38,730
and see what we
can actually learn
1068
01:13:38,730 --> 01:13:41,310
from the 2-dimensional case.
1069
01:13:41,310 --> 01:13:42,430
Thank you very much.
1070
01:13:42,430 --> 01:13:44,109
I hope you enjoyed
the lecture today.
1071
01:13:44,109 --> 01:13:45,900
And if you have any
questions, let me know.
1072
01:13:48,940 --> 01:13:50,800
And you can actually
come forward and play
1073
01:13:50,800 --> 01:13:53,590
with those demos if you want.