1
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PROFESSOR: So what
are we trying to do?

2
00:00:03,010 --> 00:00:06,020
We're going to try to
write a matter wave.

3
00:00:08,860 --> 00:00:17,129
We have a particle with
energy e and momentum p.

4
00:00:17,129 --> 00:00:19,720
e is equal to h bar omega.

5
00:00:19,720 --> 00:00:22,660
So you can get the
omega of the wave.

6
00:00:22,660 --> 00:00:25,930
And p is equal to h bar k.

7
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You can get the k of the wave.

8
00:00:27,790 --> 00:00:33,910
So de Broglie has told you
that's the way to do it.

9
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That's the p and the k.

10
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But what is the wave?

11
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Really need the phase to--
how does the wave look like?

12
00:00:42,510 --> 00:00:46,380
So the thing is that I'm
going to do an argument based

13
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on superposition and very basic
ideas of probability to get--

14
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to find the shape of the wave.

15
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And look at this possibility.

16
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Suppose we have plane waves--

17
00:01:03,580 --> 00:01:12,550
plane waves in the
x plus direction.

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A particle that is moving
in the plus x direction.

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No need to be more general yet.

20
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So what could the wave be?

21
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Well, the wave could be
sine of kx minus omega t.

22
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Maybe that's the
de Broglie wave.

23
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Or maybe the de Broglie wave
is cosine of kx minus omega t.

24
00:01:45,380 --> 00:01:48,910
But maybe it's
neither one of them.

25
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Maybe it is an e to
ikx minus i omega t.

26
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These things move to the right.

27
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The minus sign is there.

28
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So with an always an e
to the minus i omega t.

29
00:02:13,340 --> 00:02:17,070
Or maybe it's the
other way around.

30
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It's e to the minus
ikx plus i omega t.

31
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So always an e to the i omega t.

32
00:02:28,550 --> 00:02:33,440
And then you have to change
the sine of the first term

33
00:02:33,440 --> 00:02:36,290
in order to get a wave
that is moving that way.

34
00:02:39,440 --> 00:02:42,780
And now you say, how am I ever
going to know which one is it?

35
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Maybe it's all of them, a
couple of them, none of them.

36
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That's we're going
to try to understand.

37
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So the argument is going to be
based on superposition and just

38
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the rough idea that
somehow this has

39
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to do with the existence
of particles having a wave.

40
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And it's very strange.

41
00:03:17,670 --> 00:03:19,650
In some sense, it's
very surprising.

42
00:03:19,650 --> 00:03:22,510
To me, it was very
surprising, this argument,

43
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when I first saw it.

44
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Because it almost seems
that there's no way

45
00:03:27,534 --> 00:03:28,950
you're going to
be able to decide.

46
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These are all waves, so
what difference can it make?

47
00:03:34,120 --> 00:03:35,375
But you can decide.

48
00:03:39,540 --> 00:03:42,060
So my first argument
is going to be,

49
00:03:42,060 --> 00:03:46,440
it's all going to be
based on superposition.

50
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Use superposition-- --position.

51
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Plus a vague notion
of probability--

52
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--bility.

53
00:04:05,230 --> 00:04:10,550
So I'm going to try to
produce with these waves

54
00:04:10,550 --> 00:04:17,040
a state of a particle that has
equal probability to be moving

55
00:04:17,040 --> 00:04:19,424
to the right or to the left.

56
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I'm going to try to
build a wave that

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has equal probability
of doing this thing.

58
00:04:28,710 --> 00:04:37,260
So in case 1, I would have to
put a sine of kx minus omega t.

59
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That's your wave that
is moving to the right.

60
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I have to change one sine here.

61
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Plus sine of kx.

62
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Say, plus omega t.

63
00:04:59,510 --> 00:05:05,020
And that would be a wave
that moves to the right.

64
00:05:05,020 --> 00:05:07,950
Just clearly, this is the
wave that moves to the left.

65
00:05:07,950 --> 00:05:12,130
And roughly speaking, by
having equal coefficients here,

66
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I get the sense
that this would be

67
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the only way I could produce a
wave that has equal probability

68
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to move to the left
and a particle that

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00:05:21,649 --> 00:05:22,440
moves to the right.

70
00:05:24,990 --> 00:05:30,680
On the other hand,
if I expand this

71
00:05:30,680 --> 00:05:38,050
you get twice sine
of kx cosine omega t.

72
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The fact is that this
is not acceptable.

73
00:05:48,970 --> 00:05:50,620
Why it's not acceptable?

74
00:05:50,620 --> 00:06:05,080
Because this wave function
vanishes for all x at t

75
00:06:05,080 --> 00:06:14,530
omega t equal to pi over
2, 3pi over 2, 5pi over 2.

76
00:06:14,530 --> 00:06:19,150
At all those times, the
wave is identically 0.

77
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The particle has disappeared.

78
00:06:21,110 --> 00:06:24,200
No probability of a particle.

79
00:06:24,200 --> 00:06:25,850
That's pretty bad.

80
00:06:25,850 --> 00:06:29,230
That can't be right.

81
00:06:29,230 --> 00:06:32,290
And suddenly, you've proven
something very surprising.

82
00:06:32,290 --> 00:06:36,260
This sort of wave just
can't be a matter particle.

83
00:06:36,260 --> 00:06:39,964
Again, in the way we're trying
to think of probabilities.

84
00:06:41,740 --> 00:06:45,730
Same argument for
2 for same reason.

85
00:06:45,730 --> 00:06:51,240
2-- So this is no good.

86
00:06:51,240 --> 00:06:51,930
No good.

87
00:06:55,150 --> 00:06:59,770
The wave function cannot
vanish everywhere at any time.

88
00:06:59,770 --> 00:07:01,915
If it vanished everywhere,
you have no particle.

89
00:07:01,915 --> 00:07:04,700
You have nothing.

90
00:07:04,700 --> 00:07:06,960
With 2, you can
do the same thing.

91
00:07:06,960 --> 00:07:09,310
You have a cosine
plus another cosine.

92
00:07:15,750 --> 00:07:18,790
Cosine omega t minus--

93
00:07:18,790 --> 00:07:27,260
kx minus omega t plus
cosine of kx plus omega t.

94
00:07:27,260 --> 00:07:34,040
That would be 2 cosine
kx cosine omega t.

95
00:07:34,040 --> 00:07:35,110
It has the same problems.

96
00:07:37,690 --> 00:07:40,660
Let's do case number 3.

97
00:07:40,660 --> 00:07:45,010
Case number 3 is based
on the philosophy

98
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that the wave that we have--
e to the ikx minus i omega t

99
00:07:55,000 --> 00:08:00,250
always has an e to the
minus i omega t as a phase.

100
00:08:00,250 --> 00:08:03,370
So to get a wave that moves
in the opposite direction,

101
00:08:03,370 --> 00:08:09,220
we have to do minus
ikx minus i omega t.

102
00:08:09,220 --> 00:08:12,790
Because I cannot
change that phase.

103
00:08:12,790 --> 00:08:16,070
Always this [INAUDIBLE].

104
00:08:16,070 --> 00:08:23,090
Now, in this case, we can
factor the time dependence.

105
00:08:23,090 --> 00:08:28,496
You have e to the ikx
e to the minus ikx

106
00:08:28,496 --> 00:08:31,488
e to the minus i omega t.

107
00:08:34,179 --> 00:08:41,230
And be left with 2 cosine
kx e to the minus i omega t.

108
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But that's not bad.

109
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This way function never
vanishes all over space.

110
00:08:53,620 --> 00:08:58,760
Because this is now a phase, and
this phase is always non-zero.

111
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The e to the minus i
omega t is never 0.

112
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The exponential of
something is never 0,

113
00:09:07,970 --> 00:09:13,390
unless that something
is real and negative.

114
00:09:13,390 --> 00:09:15,530
And a phase is never 0.

115
00:09:15,530 --> 00:09:21,530
So this function never
vanishes for all x--

116
00:09:21,530 --> 00:09:27,110
vanishes for all x.

117
00:09:27,110 --> 00:09:34,230
So it can vanish at
some point for all time.

118
00:09:34,230 --> 00:09:37,160
But those would be points where
you don't find the particle.

119
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The function is nonzero
everywhere else.

120
00:09:40,170 --> 00:09:42,520
So this is good.

121
00:09:42,520 --> 00:09:46,060
Suddenly, this wave,
for some reason,

122
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is much better behaved than
these things for superposition.

123
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Let's do the other
wave, the wave number 4.

124
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And wave number 4 is
also not problematic.

125
00:10:06,830 --> 00:10:14,020
So case 4, you would do
an e to the minus ikx e

126
00:10:14,020 --> 00:10:23,200
to the i omega t plus an e to
the ikx e to the i omega t.

127
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Always the same exponential.

128
00:10:25,330 --> 00:10:32,370
This is simply 2 cosine
of kx e to the i omega t.

129
00:10:32,370 --> 00:10:34,790
And it's also good.

130
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At least didn't get in trouble.

131
00:10:36,550 --> 00:10:39,340
We cannot prove it is
good at this point.

132
00:10:39,340 --> 00:10:42,950
We can only prove that you
are not getting in trouble.

133
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We are not capable of producing
a contradiction, so far.

134
00:10:48,490 --> 00:10:52,120
So actually, 3 and 4 are good.

135
00:10:52,120 --> 00:10:57,460
And the obvious question
that would come now

136
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is whether you can
use both of them

137
00:11:00,850 --> 00:11:06,790
or either one at the same time.

138
00:11:06,790 --> 00:11:13,160
So the next claim is that both
cannot be true at the same

139
00:11:13,160 --> 00:11:15,360
time.

140
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You cannot use both of
them at the same time.

141
00:11:19,000 --> 00:11:23,710
So suppose 3 and 4 are good.

142
00:11:23,710 --> 00:11:27,470
Both 3 and 4--

143
00:11:27,470 --> 00:11:32,150
and 4 are both good--

144
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both right, even.

145
00:11:35,600 --> 00:11:40,670
Then remember that
superimposing a state to itself

146
00:11:40,670 --> 00:11:43,490
doesn't change the state.

147
00:11:43,490 --> 00:11:46,260
So you can superimpose 3 and 4--

148
00:11:46,260 --> 00:11:50,870
e to the ikx minus i omega t.

149
00:11:50,870 --> 00:11:54,010
That's 3.

150
00:11:54,010 --> 00:11:58,090
You can add to it 4, which
is e to the minus ikx--

151
00:12:00,280 --> 00:12:01,825
minus omega t.

152
00:12:01,825 --> 00:12:04,240
I factor a sine.

153
00:12:04,240 --> 00:12:04,975
And that's 4.

154
00:12:07,930 --> 00:12:12,070
And that should still
represent this same particle

155
00:12:12,070 --> 00:12:14,080
moving to the right.

156
00:12:14,080 --> 00:12:22,060
But this thing is twice
cosine of kx minus omega t.

157
00:12:22,060 --> 00:12:24,880
So it would mean
that this represents

158
00:12:24,880 --> 00:12:27,620
a particle moving to the right.

159
00:12:27,620 --> 00:12:30,170
And we already know
that if this represents

160
00:12:30,170 --> 00:12:34,140
a particle moving to the
right, you get in trouble.

161
00:12:34,140 --> 00:12:38,330
So now, we have to
make a decision.

162
00:12:38,330 --> 00:12:40,230
We have to choose one of them.

163
00:12:40,230 --> 00:12:43,460
And it's a matter of convention
to choose one of them,

164
00:12:43,460 --> 00:12:46,490
but happily, everybody
has chosen the same one.

165
00:12:49,430 --> 00:12:54,962
So we are led, finally,
to our matter wave.

166
00:12:54,962 --> 00:12:56,170
We're going to make a choice.

167
00:13:01,310 --> 00:13:03,370
And here is the choice.

168
00:13:03,370 --> 00:13:12,350
Psi of x and t equal to
the ikx minus i omega t.

169
00:13:12,350 --> 00:13:15,800
The energy part will
always have a minus sign.

170
00:13:20,450 --> 00:13:35,750
Is the mother wave
or wave function

171
00:13:35,750 --> 00:13:50,200
for a particle with p
equal hk and e equal h bar

172
00:13:50,200 --> 00:13:56,350
omega according to de Broglie.

173
00:13:56,350 --> 00:13:59,650
You want to do 3
dimensions, no problem.

174
00:13:59,650 --> 00:14:07,520
You put e to the i k vector,
x vector, minus i omega t.

175
00:14:07,520 --> 00:14:12,790
On p, in this case,
is h bar k vector.

176
00:14:12,790 --> 00:14:16,970
So it's a plane wave
in 3 dimensions.

177
00:14:16,970 --> 00:14:20,600
So that's the beginning
of quantum mechanics.

178
00:14:20,600 --> 00:14:24,460
You have finally found
the wave corresponding

179
00:14:24,460 --> 00:14:26,530
to a matter particle.

180
00:14:26,530 --> 00:14:28,900
And it will be a
deductive process

181
00:14:28,900 --> 00:14:32,140
to figure out what
equation it satisfies,

182
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which will lead us to
the Schrodinger equation.