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PROFESSOR: Today we're
going to just continue

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00:00:26,360 --> 00:00:28,040
what Allan Adams was doing.

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00:00:28,040 --> 00:00:32,340
He's away on a trip
in Europe, so he asked

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00:00:32,340 --> 00:00:33,900
me to give their lecture today.

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00:00:33,900 --> 00:00:39,650
And we'll just follow
what he told me to do.

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00:00:39,650 --> 00:00:44,280
He was sort of sad to give me
this lecture, because it's one

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00:00:44,280 --> 00:00:46,280
of the most interesting ones.

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00:00:46,280 --> 00:00:49,980
This is the one where you get
to see the Schrodinger equation.

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00:00:49,980 --> 00:00:56,660
But anyway, had to be a
way, so we'll do it here.

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00:00:56,660 --> 00:00:59,980
He also told me to take off
my shoes, but I won't do that.

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00:01:02,780 --> 00:01:05,760
So let's go ahead.

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00:01:05,760 --> 00:01:07,290
So what do we have so far.

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00:01:07,290 --> 00:01:12,910
It's going to be a list
of items that we have.

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00:01:12,910 --> 00:01:15,310
And what have we learned?

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00:01:15,310 --> 00:01:20,880
We know that
particles, or systems,

23
00:01:20,880 --> 00:01:29,180
are governed by wave functions,
described by wave functions.

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00:01:29,180 --> 00:01:33,710
Wave functions.

25
00:01:33,710 --> 00:01:38,520
And those are psi
of x at this moment.

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00:01:38,520 --> 00:01:44,060
And these are complex numbers,
belong to the complex numbers.

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00:01:44,060 --> 00:01:48,500
And they're continuous
and normalizable.

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00:01:48,500 --> 00:01:51,030
Their derivatives need
not be continuous,

29
00:01:51,030 --> 00:01:54,700
but the wave function
has to be continuous.

30
00:01:54,700 --> 00:01:57,770
It should be also
normalizable, and those

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00:01:57,770 --> 00:01:59,900
are two important
properties of it,

32
00:01:59,900 --> 00:02:02,223
continuous and normalizable.

33
00:02:07,910 --> 00:02:12,670
Second there's a probability
associated with this thing.

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00:02:12,670 --> 00:02:15,360
And this probability
is described

35
00:02:15,360 --> 00:02:20,200
by this special p of x.

36
00:02:20,200 --> 00:02:24,700
And given that x is a continuous
variable, you can say well,

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00:02:24,700 --> 00:02:28,560
what is the probability that the
particle is just at this point

38
00:02:28,560 --> 00:02:31,950
would be zero in general.

39
00:02:31,950 --> 00:02:34,670
You have to ask, typically,
what's the probability that I

40
00:02:34,670 --> 00:02:37,040
find it in a range.

41
00:02:37,040 --> 00:02:38,800
It's any continuous
variable that

42
00:02:38,800 --> 00:02:42,320
postulated to be given
by the square of the wave

43
00:02:42,320 --> 00:02:45,870
function and the x.

44
00:02:45,870 --> 00:02:55,276
Third there's superposition
of allowed states.

45
00:02:58,950 --> 00:03:01,720
So particles can be
in superpositions.

46
00:03:01,720 --> 00:03:05,330
So a wave function
that depends of x

47
00:03:05,330 --> 00:03:10,970
may quickly or generally be
given as a superposition of two

48
00:03:10,970 --> 00:03:11,746
wave functions.

49
00:03:16,260 --> 00:03:18,590
And this is seen in many ways.

50
00:03:18,590 --> 00:03:19,760
You have these boxes.

51
00:03:19,760 --> 00:03:21,420
A particle was a superposition.

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00:03:21,420 --> 00:03:24,370
A top, and side,
and hard, and soft.

53
00:03:24,370 --> 00:03:26,800
And photon superposition
of linearly

54
00:03:26,800 --> 00:03:29,960
polarized here or that way.

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00:03:29,960 --> 00:03:33,300
That's always
possible to explain.

56
00:03:33,300 --> 00:03:38,270
Now in addition to this,
to motivate the next one

57
00:03:38,270 --> 00:03:41,606
we talk about relations
between operators.

58
00:03:44,150 --> 00:03:50,820
There was an abstraction
going on in this course

59
00:03:50,820 --> 00:03:53,580
in the previous
lectures in which

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00:03:53,580 --> 00:03:56,260
the idea of the
momentum of a particle

61
00:03:56,260 --> 00:04:00,580
became that of an operator
on the wave function.

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00:04:03,800 --> 00:04:11,320
So as an aside,
operators momentum,

63
00:04:11,320 --> 00:04:14,050
we have the momentum
of a particle

64
00:04:14,050 --> 00:04:17,850
has been associated
with an operator h

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00:04:17,850 --> 00:04:21,730
bar over i [INAUDIBLE] x.

66
00:04:21,730 --> 00:04:25,250
Now two things here.

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00:04:25,250 --> 00:04:28,180
My taste, I like to
put h bar over i.

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00:04:28,180 --> 00:04:31,000
Allan likes to put I h bar.

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00:04:31,000 --> 00:04:35,700
That's exactly the same
thing, but no minus sign

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00:04:35,700 --> 00:04:37,640
is a good thing in my opinion.

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00:04:37,640 --> 00:04:41,150
So I avoid them when possible.

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00:04:41,150 --> 00:04:45,280
Now there's is d dx here
is a partial derivative,

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00:04:45,280 --> 00:04:50,130
and there seems to be no need
for partial derivatives here.

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00:04:50,130 --> 00:04:51,820
Why partial derivatives?

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00:04:51,820 --> 00:04:55,560
I only see functions of x.

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00:04:55,560 --> 00:04:56,240
Anybody?

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00:04:56,240 --> 00:04:56,930
Why?

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00:04:56,930 --> 00:04:57,590
Yes.

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00:04:57,590 --> 00:04:59,310
AUDIENCE: The complete
wave function also

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00:04:59,310 --> 00:05:01,905
depends on time, doesn't it?

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00:05:01,905 --> 00:05:03,280
PROFESSOR: Complete
wave function

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00:05:03,280 --> 00:05:04,680
depends on time as well.

83
00:05:04,680 --> 00:05:06,230
Yes, exactly.

84
00:05:06,230 --> 00:05:08,590
That's where we're
going to get to today.

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00:05:08,590 --> 00:05:12,740
This is the description of
this particle at some instant.

86
00:05:12,740 --> 00:05:16,480
So within [INAUDIBLE] time,
the time here is implicit.

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00:05:16,480 --> 00:05:20,570
It could be at some time,
now, later, some time,

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00:05:20,570 --> 00:05:22,090
but that's all we know.

89
00:05:22,090 --> 00:05:26,360
So in physics you're allowed
to ask the question, well,

90
00:05:26,360 --> 00:05:28,440
if I know the wave
function of this time

91
00:05:28,440 --> 00:05:31,290
and that seems to be what I
need to describe the physics,

92
00:05:31,290 --> 00:05:33,150
what will it be later?

93
00:05:33,150 --> 00:05:36,190
And later time will come
in, so therefore we'll

94
00:05:36,190 --> 00:05:40,950
stick to this
partial d dx in here.

95
00:05:40,950 --> 00:05:45,250
All right, so how
do we use that?

96
00:05:45,250 --> 00:05:50,980
We think of this operator as
acting on the wave functions

97
00:05:50,980 --> 00:05:54,430
to give you roughly the
momentum of the particle.

98
00:05:54,430 --> 00:05:58,540
And we've made it in
such a way that when

99
00:05:58,540 --> 00:06:02,670
we talk about the expectation
value of the momentum,

100
00:06:02,670 --> 00:06:06,290
the expected value of the
momentum of the particle,

101
00:06:06,290 --> 00:06:08,800
we compute the
following quantity.

102
00:06:08,800 --> 00:06:14,190
We compute the integral from
minus infinity to infinity dx.

103
00:06:14,190 --> 00:06:17,740
We put the conjugate of
the wave function here.

104
00:06:17,740 --> 00:06:23,730
And we put the operator,
h bar over i d dx acting

105
00:06:23,730 --> 00:06:25,640
on the wave function here.

106
00:06:28,350 --> 00:06:31,260
And that's supposed
to be sort of

107
00:06:31,260 --> 00:06:37,380
like saying that this evaluates
the momentum of the wave

108
00:06:37,380 --> 00:06:39,820
function.

109
00:06:39,820 --> 00:06:41,580
Why is that so?

110
00:06:41,580 --> 00:06:47,680
It is because if you're trying
to say, oh any wave function,

111
00:06:47,680 --> 00:06:51,300
a general wave
function need not be

112
00:06:51,300 --> 00:06:54,410
state in which the particle
has definite momentum.

113
00:06:54,410 --> 00:06:57,770
So I kind of just say the
momentum of a particle

114
00:06:57,770 --> 00:07:00,940
is the value of this
operator on the function.

115
00:07:00,940 --> 00:07:04,930
Because if I act with this
operator on the function,

116
00:07:04,930 --> 00:07:08,190
on the wave function,
it might not return me,

117
00:07:08,190 --> 00:07:09,310
the wave function.

118
00:07:09,310 --> 00:07:14,100
In fact in general, we've
seen that, for special wave

119
00:07:14,100 --> 00:07:21,220
functions, wave functions
of the form psi, a number,

120
00:07:21,220 --> 00:07:22,440
e to the ikx.

121
00:07:26,490 --> 00:07:34,900
Then p, let's think p hat
as the operator on psi.

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00:07:34,900 --> 00:07:40,180
Would be h bar
over i d dx on psi.

123
00:07:42,710 --> 00:07:43,660
Gives you what?

124
00:07:43,660 --> 00:07:48,760
Well, this h over I,
the h remains there.

125
00:07:48,760 --> 00:07:52,550
When you differentiate with
respect to x, the ik goes down,

126
00:07:52,550 --> 00:07:59,040
and the i cancels, so you get
hk times the same wave function.

127
00:07:59,040 --> 00:08:03,000
And for this, we think that
this wave function is a wave

128
00:08:03,000 --> 00:08:07,340
function with momentum hk.

129
00:08:07,340 --> 00:08:10,880
Because if you act with
the operator p on that wave

130
00:08:10,880 --> 00:08:14,260
function, it returns for you hk.

131
00:08:14,260 --> 00:08:23,750
So we think of this as
has p equal hk, h bar k.

132
00:08:23,750 --> 00:08:26,850
So the thing that
we want to do now

133
00:08:26,850 --> 00:08:29,830
is to make this a
little more general.

134
00:08:29,830 --> 00:08:31,731
This is just talking
about momentum,

135
00:08:31,731 --> 00:08:33,230
but in quantum
mechanics we're going

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00:08:33,230 --> 00:08:35,150
to have all kinds of operators.

137
00:08:35,150 --> 00:08:37,860
So we need to be more general.

138
00:08:37,860 --> 00:08:42,650
So this is going to be, as Allan
calls it, a math interlude.

139
00:08:50,800 --> 00:08:55,040
Based on the following question,
what this is an operator?

140
00:08:55,040 --> 00:09:00,050
And then the physics question,
what do measurable things

141
00:09:00,050 --> 00:09:01,950
have to do with our operators?

142
00:09:01,950 --> 00:09:12,750
So about operators aren't
measurable things, quantities.

143
00:09:16,980 --> 00:09:21,950
Now your view of operators
is going to evolve.

144
00:09:21,950 --> 00:09:23,450
It's going to evolve
in this course,

145
00:09:23,450 --> 00:09:25,310
It's going to evolve in 805.

146
00:09:25,310 --> 00:09:27,300
It probably will
continue to evolve.

147
00:09:27,300 --> 00:09:32,130
So we need to think
of what operators are.

148
00:09:32,130 --> 00:09:37,630
And there is a simple
way of thinking

149
00:09:37,630 --> 00:09:41,280
of operators that is going
to be the basis of much

150
00:09:41,280 --> 00:09:42,480
of the intuition.

151
00:09:42,480 --> 00:09:46,280
It's a mathematical way
of thinking of operators,

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00:09:46,280 --> 00:09:50,180
and what we'll sometimes
use it as a crutch.

153
00:09:50,180 --> 00:09:55,010
And the idea is that
basically operators

154
00:09:55,010 --> 00:10:00,390
are things that act on
objects and scramble them.

155
00:10:00,390 --> 00:10:02,860
So whenever you
have an operator,

156
00:10:02,860 --> 00:10:05,120
you really have
to figure out what

157
00:10:05,120 --> 00:10:07,550
are the objects
you're talking about.

158
00:10:07,550 --> 00:10:10,820
And then the operator
is some instruction

159
00:10:10,820 --> 00:10:14,290
on how to scramble it,
scramble this object.

160
00:10:14,290 --> 00:10:16,760
So for example, you
have an operator.

161
00:10:16,760 --> 00:10:20,560
You must see what it
does to any object here.

162
00:10:20,560 --> 00:10:24,530
The operator acts on the object
and gives you another object.

163
00:10:24,530 --> 00:10:28,390
So in a picture
you have all kinds

164
00:10:28,390 --> 00:10:37,020
of objects sets,
a set of objects.

165
00:10:37,020 --> 00:10:40,300
And the operators are things.

166
00:10:40,300 --> 00:10:42,680
You can have a
list of operators.

167
00:10:42,680 --> 00:10:45,780
And they come here and
move those objects,

168
00:10:45,780 --> 00:10:48,078
scramble them, do
something to them.

169
00:10:50,890 --> 00:10:53,960
And we should distinguish them,
because the objects are not

170
00:10:53,960 --> 00:10:57,610
operators, and the operators
are not the objects.

171
00:10:57,610 --> 00:11:03,070
So what is the simplest example
in mathematics of this thing

172
00:11:03,070 --> 00:11:08,850
is vectors and matrices.

173
00:11:08,850 --> 00:11:09,865
Simplest example.

174
00:11:15,090 --> 00:11:20,060
Objects are vectors.

175
00:11:20,060 --> 00:11:24,480
Operators are matrices.

176
00:11:24,480 --> 00:11:26,920
And how does that work?

177
00:11:26,920 --> 00:11:31,780
Well, you have a two
by two, the case where

178
00:11:31,780 --> 00:11:37,440
you have a set in which you have
vectors with two components.

179
00:11:37,440 --> 00:11:47,250
So example, a vector that
has two components v1 v2.

180
00:11:47,250 --> 00:11:53,170
And the matrices,
this is the object.

181
00:11:53,170 --> 00:11:57,060
And the operator is a matrix.

182
00:11:57,060 --> 00:12:06,390
a 11, a 12, a 21, a
22 as an operator.

183
00:12:06,390 --> 00:12:11,090
An m on a vector is a vector.

184
00:12:14,480 --> 00:12:17,870
If you are with a matrix
on a vector, this 2

185
00:12:17,870 --> 00:12:23,430
by 2 matrix on this common
vector, you get another vector.

186
00:12:23,430 --> 00:12:29,455
So that's this simplest example
of operators acting on objects.

187
00:12:31,980 --> 00:12:34,980
In our case, we're going to
talk about a more-- we're

188
00:12:34,980 --> 00:12:37,170
going to have to begin,
in quantum mechanics

189
00:12:37,170 --> 00:12:42,240
we're required to begin with a
more sophisticated one in which

190
00:12:42,240 --> 00:12:48,340
the objects are
going to be-- objects

191
00:12:48,340 --> 00:12:49,746
are going to be functions.

192
00:12:53,450 --> 00:13:02,165
In fact, I will write
complex functions of x.

193
00:13:06,220 --> 00:13:10,090
So let's see it, the
list of operators.

194
00:13:10,090 --> 00:13:12,680
And what do the operators do?

195
00:13:12,680 --> 00:13:18,010
The operators act
on the functions.

196
00:13:25,450 --> 00:13:28,330
So what is an operator?

197
00:13:28,330 --> 00:13:32,110
It's a rule on how
to take any function,

198
00:13:32,110 --> 00:13:34,710
and you must give a rule
on how to obtain from that

199
00:13:34,710 --> 00:13:37,940
function another function.

200
00:13:37,940 --> 00:13:40,500
So let's start with an examples.

201
00:13:40,500 --> 00:13:43,200
It's probably the
easiest thing to do.

202
00:13:43,200 --> 00:13:47,130
So an operator acts
on the functions.

203
00:13:47,130 --> 00:13:52,340
So an operator for
any function f of x

204
00:13:52,340 --> 00:13:54,033
will give you another function.

205
00:13:57,990 --> 00:14:01,760
Function of x.

206
00:14:04,600 --> 00:14:10,140
And here's operator o, we put
a hat sometimes for operators.

207
00:14:10,140 --> 00:14:21,750
So the simplest operator,
the operator one.

208
00:14:21,750 --> 00:14:27,220
We always, mathematicians, love
to begin with trivial examples.

209
00:14:27,220 --> 00:14:29,910
Illustrate anything
almost, and just kind of

210
00:14:29,910 --> 00:14:31,780
confuse you many times.

211
00:14:31,780 --> 00:14:35,170
But actually it's good
to get them of the way.

212
00:14:35,170 --> 00:14:38,710
So what is the operator one?

213
00:14:38,710 --> 00:14:41,350
One possibility it
takes any function

214
00:14:41,350 --> 00:14:44,660
and gives you the number one.

215
00:14:44,660 --> 00:14:45,850
Do you think that's it?

216
00:14:45,850 --> 00:14:48,950
Who thinks that's it?

217
00:14:48,950 --> 00:14:49,710
Nobody?

218
00:14:49,710 --> 00:14:50,470
Very good.

219
00:14:50,470 --> 00:14:53,000
that definitely is
not a good thing

220
00:14:53,000 --> 00:14:54,960
to do to give you
the number one.

221
00:14:54,960 --> 00:14:57,260
So this operator
does the following.

222
00:14:57,260 --> 00:14:58,480
I will write it like that.

223
00:14:58,480 --> 00:15:05,517
The operator one takes f
of x and gives you what?

224
00:15:05,517 --> 00:15:06,225
AUDIENCE: f of x.

225
00:15:06,225 --> 00:15:06,975
PROFESSOR: f of x.

226
00:15:06,975 --> 00:15:08,070
Correct.

227
00:15:08,070 --> 00:15:08,570
Good.

228
00:15:08,570 --> 00:15:12,440
So it's a very simple
operator, but it's an operator.

229
00:15:12,440 --> 00:15:15,640
It's like what matrix?

230
00:15:15,640 --> 00:15:17,980
The identity matrix.

231
00:15:17,980 --> 00:15:18,770
Very good.

232
00:15:18,770 --> 00:15:22,370
There could be a zero operator
that gives you nothing

233
00:15:22,370 --> 00:15:23,980
and would be the zero matrix.

234
00:15:23,980 --> 00:15:27,330
So let's write the more
interesting operator.

235
00:15:27,330 --> 00:15:30,164
The operator would d dx.

236
00:15:30,164 --> 00:15:30,955
That's interesting.

237
00:15:30,955 --> 00:15:35,240
The derivative can be
thought of as an operator

238
00:15:35,240 --> 00:15:38,670
because if you
start with f of x,

239
00:15:38,670 --> 00:15:43,390
it gives you another
function, d dx of f of x.

240
00:15:46,040 --> 00:15:48,830
And that's a rule to get
from one function to another.

241
00:15:48,830 --> 00:15:53,890
Therefore that's an operator,
qualifies as an operator.

242
00:15:53,890 --> 00:15:57,010
Another operator that
typically can confuse you

243
00:15:57,010 --> 00:16:01,560
is the operator x.

244
00:16:01,560 --> 00:16:02,850
x an operator?

245
00:16:02,850 --> 00:16:04,820
What does that mean?

246
00:16:04,820 --> 00:16:07,040
Well, you just
have to define it.

247
00:16:07,040 --> 00:16:10,500
At this moment, it
could mean many things.

248
00:16:10,500 --> 00:16:13,220
But you will see that
[INAUDIBLE] is the only thing

249
00:16:13,220 --> 00:16:15,820
that probably makes some sense.

250
00:16:15,820 --> 00:16:18,450
So what is the operator x?

251
00:16:18,450 --> 00:16:21,310
Well, it's the operator
that, given f of x,

252
00:16:21,310 --> 00:16:27,090
gives you the function
x times f of x.

253
00:16:27,090 --> 00:16:28,930
That's a reasonable thing to do.

254
00:16:28,930 --> 00:16:30,350
It's multiplying by x.

255
00:16:30,350 --> 00:16:32,710
It changes the function.

256
00:16:32,710 --> 00:16:35,590
You could define the operator
x squared that multiplies

257
00:16:35,590 --> 00:16:36,610
by x squared.

258
00:16:36,610 --> 00:16:41,090
And the only reasonable
thing is to multiply it by.

259
00:16:41,090 --> 00:16:43,330
You could divide
by it, and you may

260
00:16:43,330 --> 00:16:45,630
need to divide by it as well.

261
00:16:45,630 --> 00:16:47,460
And you could define
the operator 1

262
00:16:47,460 --> 00:16:51,650
over x gives you the
function times 1 over x.

263
00:16:51,650 --> 00:16:54,940
We will need that
sometime, but not now.

264
00:16:54,940 --> 00:16:59,410
Let's see another set of
operators where we give a name.

265
00:16:59,410 --> 00:17:01,380
It doesn't have a
name because it's not

266
00:17:01,380 --> 00:17:03,800
all that's useful in fact.

267
00:17:03,800 --> 00:17:05,640
But it's good to
illustrate things.

268
00:17:05,640 --> 00:17:11,810
They operator s
sub q for squared.

269
00:17:11,810 --> 00:17:15,710
S q for the first two
letters of the word square.

270
00:17:15,710 --> 00:17:19,910
That takes f of x
into f of x squared.

271
00:17:19,910 --> 00:17:21,069
That's another function.

272
00:17:21,069 --> 00:17:23,960
You could define more
functions like that.

273
00:17:23,960 --> 00:17:30,020
The operator p 42.

274
00:17:30,020 --> 00:17:33,295
That's another silly operator.

275
00:17:33,295 --> 00:17:35,960
Well certainly a lot
more silly than this one.

276
00:17:35,960 --> 00:17:37,330
That's not too bad.

277
00:17:37,330 --> 00:17:47,880
But the p 42 takes f of x And
gives you the number 42 times

278
00:17:47,880 --> 00:17:50,760
the constant function.

279
00:17:50,760 --> 00:17:52,300
So that's a function of x.

280
00:17:52,300 --> 00:17:54,045
It's trivial function of x.

281
00:17:57,860 --> 00:18:00,360
Now enough examples.

282
00:18:00,360 --> 00:18:01,660
So you get the idea.

283
00:18:01,660 --> 00:18:05,400
Operators act on functions
and give you functions.

284
00:18:05,400 --> 00:18:08,260
And we just need to
define them, and then we

285
00:18:08,260 --> 00:18:09,760
know what we're talking about.

286
00:18:09,760 --> 00:18:11,645
Yes?

287
00:18:11,645 --> 00:18:14,400
AUDIENCE: Is the Dirac
delta and operator?

288
00:18:14,400 --> 00:18:18,430
PROFESSOR: The Dirac
delta, well, you

289
00:18:18,430 --> 00:18:20,226
can think of it as an operator.

290
00:18:23,600 --> 00:18:26,650
So it all depends how
you define things.

291
00:18:26,650 --> 00:18:30,110
So how could I do find
the Dirac delta function

292
00:18:30,110 --> 00:18:32,140
to be an operator?

293
00:18:32,140 --> 00:18:34,590
So delta of x minus a.

294
00:18:34,590 --> 00:18:39,410
Can I call it the
operator delta hat of a?

295
00:18:39,410 --> 00:18:42,500
Well, I would have to tell
you what it does in functions.

296
00:18:42,500 --> 00:18:45,110
And probably I
would say delta had

297
00:18:45,110 --> 00:18:50,670
on a on a function of x is equal
to delta of x minus a times

298
00:18:50,670 --> 00:18:53,920
the function of x.

299
00:18:53,920 --> 00:18:55,820
And I'd say it's an operator.

300
00:18:55,820 --> 00:18:59,910
Now the question is, really,
is it a useful operator?

301
00:18:59,910 --> 00:19:04,350
And sometimes it will
be useful in fact.

302
00:19:04,350 --> 00:19:07,080
This is a more general
case of another operator

303
00:19:07,080 --> 00:19:08,330
that maybe I could define.

304
00:19:11,340 --> 00:19:17,960
o sub h of x is
the operator that

305
00:19:17,960 --> 00:19:23,825
takes f of x into h
of x times f of x.

306
00:19:27,330 --> 00:19:30,870
So that would be
another operator.

307
00:19:30,870 --> 00:19:34,900
Now there are operators
that are particularly nice,

308
00:19:34,900 --> 00:19:38,040
and there are the
so-called linear operators.

309
00:19:40,700 --> 00:19:42,685
So what is a linear operator?

310
00:19:45,610 --> 00:19:47,970
It's one that respects
superposition.

311
00:19:47,970 --> 00:19:54,590
So linear operator
respects superposition.

312
00:19:59,420 --> 00:20:03,780
So o hat is linear.

313
00:20:03,780 --> 00:20:08,980
o hat is a linear operator.

314
00:20:08,980 --> 00:20:16,420
If o hat on a times f
of x plus b times g of x

315
00:20:16,420 --> 00:20:19,310
does what you would
imagine it should do,

316
00:20:19,310 --> 00:20:22,790
it that's on the first, and
then it acts on the second.

317
00:20:22,790 --> 00:20:25,800
Acting on the first,
the number goes out

318
00:20:25,800 --> 00:20:28,620
and doesn't do anything,
say, on the number.

319
00:20:28,620 --> 00:20:29,510
It's linear.

320
00:20:29,510 --> 00:20:31,720
It's part of that idea.

321
00:20:31,720 --> 00:20:42,690
And it gives you o on f of
x plus b times o on g of x.

322
00:20:42,690 --> 00:20:48,005
So your operator may be linear,
or it may not be linear.

323
00:20:54,670 --> 00:20:56,890
And we have to just guess them.

324
00:20:56,890 --> 00:21:00,680
And you would imagine
that we can decide that,

325
00:21:00,680 --> 00:21:02,930
of the list of
operators that we have,

326
00:21:02,930 --> 00:21:12,910
let's see, one d dx-- how much?

327
00:21:12,910 --> 00:21:13,470
Which one?

328
00:21:13,470 --> 00:21:21,750
Sq, p 42, and o sub h of x.

329
00:21:25,770 --> 00:21:27,070
Let's see.

330
00:21:27,070 --> 00:21:31,130
Let's vote on each one
whether it's linear or not.

331
00:21:31,130 --> 00:21:35,190
A shouting match whether I
hear a stronger yes or no.

332
00:21:35,190 --> 00:21:35,750
OK?

333
00:21:35,750 --> 00:21:38,700
One is a linear operator, yes?

334
00:21:38,700 --> 00:21:39,340
AUDIENCE: Yes.

335
00:21:39,340 --> 00:21:40,860
PROFESSOR: No?

336
00:21:40,860 --> 00:21:42,350
Yes.

337
00:21:42,350 --> 00:21:43,680
All right.

338
00:21:43,680 --> 00:21:45,050
d dx linear.

339
00:21:45,050 --> 00:21:45,782
Yes?

340
00:21:45,782 --> 00:21:46,554
AUDIENCE: Yes.

341
00:21:46,554 --> 00:21:47,220
PROFESSOR: Good.

342
00:21:47,220 --> 00:21:48,660
That's strong enough.

343
00:21:48,660 --> 00:21:50,800
Don't need to hear
the other one.

344
00:21:50,800 --> 00:21:51,800
x hat.

345
00:21:51,800 --> 00:21:52,580
Linear operator?

346
00:21:52,580 --> 00:21:53,690
Yes or no?

347
00:21:53,690 --> 00:21:54,410
AUDIENCE: Yes.

348
00:21:54,410 --> 00:21:56,870
PROFESSOR: Yes, good.

349
00:21:56,870 --> 00:21:58,850
Squaring, linear operator?

350
00:21:58,850 --> 00:21:59,550
AUDIENCE: No.

351
00:21:59,550 --> 00:22:00,390
PROFESSOR: No.

352
00:22:00,390 --> 00:22:03,540
No way it could be
a linear operator.

353
00:22:03,540 --> 00:22:06,520
It just doesn't happen.

354
00:22:06,520 --> 00:22:14,330
If you have sq on f plus g,
it would be f plus g squared,

355
00:22:14,330 --> 00:22:18,470
which is f squared plus g
squared plus, unfortunately

356
00:22:18,470 --> 00:22:20,200
too, fg.

357
00:22:20,200 --> 00:22:28,600
And this thing ruins it, because
this is sq of f plus sq of g.

358
00:22:28,600 --> 00:22:30,210
It's even worse than that.

359
00:22:30,210 --> 00:22:37,680
You put sq on af,
by linearity it

360
00:22:37,680 --> 00:22:39,880
should be a times the operator.

361
00:22:39,880 --> 00:22:44,060
But when you square a times f,
you get a squared f squared.

362
00:22:44,060 --> 00:22:46,260
So you don't even
need two functions

363
00:22:46,260 --> 00:22:48,110
to see that it's not real.

364
00:22:48,110 --> 00:22:52,600
So definitely no.

365
00:22:52,600 --> 00:22:55,070
How about p 42?

366
00:22:55,070 --> 00:22:55,685
AUDIENCE: No.

367
00:22:55,685 --> 00:22:57,100
PROFESSOR: No, of course not.

368
00:22:57,100 --> 00:22:59,930
Because if you
add two functions,

369
00:22:59,930 --> 00:23:01,460
it still gives you 42.

370
00:23:01,460 --> 00:23:05,130
It doesn't get you 84, so no.

371
00:23:05,130 --> 00:23:07,840
How about oh of x?

372
00:23:07,840 --> 00:23:08,810
AUDIENCE: Yes.

373
00:23:08,810 --> 00:23:10,080
PROFESSOR: Yes, it does that.

374
00:23:10,080 --> 00:23:15,100
If you act with this operator on
a sum of functions distributive

375
00:23:15,100 --> 00:23:16,070
law, it works.

376
00:23:16,070 --> 00:23:18,210
So this is linear.

377
00:23:18,210 --> 00:23:18,710
Good.

378
00:23:18,710 --> 00:23:21,080
Linear operators
are important to us

379
00:23:21,080 --> 00:23:24,250
because we have
some superposition

380
00:23:24,250 --> 00:23:25,690
of allowed states.

381
00:23:25,690 --> 00:23:28,390
So if this is a state
and this is a state,

382
00:23:28,390 --> 00:23:29,910
this is also good state.

383
00:23:29,910 --> 00:23:35,120
So if we want superposition
to work well with our theory,

384
00:23:35,120 --> 00:23:36,860
we want linear operator.

385
00:23:36,860 --> 00:23:38,470
So that's good.

386
00:23:38,470 --> 00:23:41,730
So we have those
linear operators.

387
00:23:41,730 --> 00:23:45,420
And now operators have
another thing that

388
00:23:45,420 --> 00:23:48,980
makes them something special.

389
00:23:48,980 --> 00:23:56,380
It is the idea that there's
simpler object they can act on.

390
00:23:56,380 --> 00:23:59,920
We don't assume you've studied
linear algebra in this course,

391
00:23:59,920 --> 00:24:02,770
so whatever I'm
going to say, take it

392
00:24:02,770 --> 00:24:07,630
as motivation to learn some
linear algebra at some stage.

393
00:24:07,630 --> 00:24:10,830
You will be a little more
linear algebra in 805.

394
00:24:10,830 --> 00:24:13,970
But at this moment,
just basic idea.

395
00:24:13,970 --> 00:24:19,660
So whenever you have matrices,
one thing that people do

396
00:24:19,660 --> 00:24:23,450
is to see if there
are special vectors.

397
00:24:23,450 --> 00:24:27,170
Any arbitrary vector, when
you act within the matrix,

398
00:24:27,170 --> 00:24:30,060
is going to just jump and
go somewhere else, point

399
00:24:30,060 --> 00:24:31,920
in another direction.

400
00:24:31,920 --> 00:24:33,970
But there are some
special vectors

401
00:24:33,970 --> 00:24:38,040
that do act-- if you
have a given matrix m,

402
00:24:38,040 --> 00:24:43,560
there are some funny vectors
sometimes that acted by n

403
00:24:43,560 --> 00:24:45,390
remain the same direction.

404
00:24:45,390 --> 00:24:47,950
They may grow a little
or become smaller,

405
00:24:47,950 --> 00:24:50,020
but they remain
the same direction.

406
00:24:50,020 --> 00:24:52,340
These are called eigenvectors.

407
00:24:52,340 --> 00:24:55,000
And that constant
of proportionality,

408
00:24:55,000 --> 00:24:58,530
proportional to the action of
the operator on the vector,

409
00:24:58,530 --> 00:25:00,650
is called the eigenvalue.

410
00:25:00,650 --> 00:25:04,930
So these things
have generalizations

411
00:25:04,930 --> 00:25:06,290
for our operators.

412
00:25:06,290 --> 00:25:16,410
So operators can have special
functions, eigenfunctions.

413
00:25:23,630 --> 00:25:25,110
What are these eigenfunctions?

414
00:25:25,110 --> 00:25:28,640
So let's consider
that operator a hat.

415
00:25:28,640 --> 00:25:29,700
It's some operator.

416
00:25:29,700 --> 00:25:31,950
I don't know which
one of these, be we're

417
00:25:31,950 --> 00:25:33,710
going to talk about
linear operator.

418
00:25:33,710 --> 00:25:37,020
So linear operators
have eigenfunctions.

419
00:25:37,020 --> 00:25:38,560
A hat.

420
00:25:38,560 --> 00:25:41,040
So a hat.

421
00:25:41,040 --> 00:25:45,790
There may be functions that,
when you act with the operator,

422
00:25:45,790 --> 00:25:48,330
you sort of get the
same function up

423
00:25:48,330 --> 00:25:51,770
to possibly a constant a.

424
00:25:51,770 --> 00:25:55,250
So you may get a
times the function.

425
00:25:55,250 --> 00:25:58,690
And that's a pretty
unusual function,

426
00:25:58,690 --> 00:26:01,680
because, on most functions,
any given operator

427
00:26:01,680 --> 00:26:03,880
is going to make a mess
out of the function.

428
00:26:03,880 --> 00:26:05,560
But sometimes it does that.

429
00:26:05,560 --> 00:26:10,100
So to label them better
with respect to operator,

430
00:26:10,100 --> 00:26:13,280
I would put a subscript a,
which means that there's

431
00:26:13,280 --> 00:26:17,050
some special function
that has a parameter a,

432
00:26:17,050 --> 00:26:19,450
for which this
operator gives you

433
00:26:19,450 --> 00:26:21,870
a times that special function.

434
00:26:21,870 --> 00:26:24,670
And that special
function is called--

435
00:26:24,670 --> 00:26:29,526
this is the eigenfunction
and this is the eigenvalue.

436
00:26:36,830 --> 00:26:40,040
And that the
eigenvalue is a number.

437
00:26:40,040 --> 00:26:46,120
It's a complex number
c there over there.

438
00:26:46,120 --> 00:26:48,270
So these are special things.

439
00:26:48,270 --> 00:26:51,180
They don't necessarily
happen all the time to exist,

440
00:26:51,180 --> 00:26:54,500
but sometimes they do, and
then they're pretty useful.

441
00:26:54,500 --> 00:26:59,130
And we have one example of
them that is quite nice.

442
00:26:59,130 --> 00:27:12,650
For the operator a equal
p, we have eigenfunctions

443
00:27:12,650 --> 00:27:23,710
e to the ikx with eigenvalue hk.

444
00:27:23,710 --> 00:27:26,750
So this is the connection
to this whole thing.

445
00:27:26,750 --> 00:27:30,660
We wanted to make clear for
you that what you saw here,

446
00:27:30,660 --> 00:27:33,170
that this operator
acting on this function

447
00:27:33,170 --> 00:27:37,400
gives you something times this
function is a general fact

448
00:27:37,400 --> 00:27:39,580
about the operators.

449
00:27:39,580 --> 00:27:41,920
Operators have eigenfunctions.

450
00:27:41,920 --> 00:27:48,000
So eigenfunction e of x with
eigenvalue hk, because indeed

451
00:27:48,000 --> 00:27:54,000
p hat on this e to ikx,
as you see this h bar

452
00:27:54,000 --> 00:27:58,060
k times e to the ikx.

453
00:27:58,060 --> 00:28:00,160
So here you have p hat is the a.

454
00:28:00,160 --> 00:28:05,220
This is the function
labeled a would be like k.

455
00:28:05,220 --> 00:28:08,540
Here is something like k again.

456
00:28:08,540 --> 00:28:09,760
And here is this thing.

457
00:28:09,760 --> 00:28:14,140
But the main thing operator
on the function number

458
00:28:14,140 --> 00:28:15,990
times the function
is an eigenfunction.

459
00:28:15,990 --> 00:28:17,230
Yes?

460
00:28:17,230 --> 00:28:22,250
AUDIENCE: For a given operator,
is the eigenvalue [INAUDIBLE]?

461
00:28:22,250 --> 00:28:25,940
PROFESSOR: Well, for a given
operator good question.

462
00:28:25,940 --> 00:28:29,390
a is a list of values.

463
00:28:29,390 --> 00:28:32,580
So there may be many,
many, many eigenfunctions.

464
00:28:32,580 --> 00:28:35,410
Many cases infinitely
many eigenfunctions.

465
00:28:35,410 --> 00:28:40,730
In fact, here I can put
for k any number I want,

466
00:28:40,730 --> 00:28:42,500
and I get a different function.

467
00:28:42,500 --> 00:28:55,390
So a belongs to c and may take
many, or even infinite, values.

468
00:28:55,390 --> 00:29:01,650
If you remember for nice
matrices, n by n matrix

469
00:29:01,650 --> 00:29:05,950
may be a nice n by n matrix
because n eigenvectors

470
00:29:05,950 --> 00:29:09,730
and eigenvalues are
sometimes hard to generate,

471
00:29:09,730 --> 00:29:12,690
sometimes eigenvalues have
the same numbers and things

472
00:29:12,690 --> 00:29:13,330
like that.

473
00:29:17,510 --> 00:29:18,080
OK.

474
00:29:18,080 --> 00:29:24,680
Linearity is this some of two
eigenvectors and eigenvector.

475
00:29:24,680 --> 00:29:26,960
Yes?

476
00:29:26,960 --> 00:29:28,169
No?

477
00:29:28,169 --> 00:29:28,710
AUDIENCE: No.

478
00:29:28,710 --> 00:29:29,980
PROFESSOR: No, no.

479
00:29:29,980 --> 00:29:30,650
Correct.

480
00:29:30,650 --> 00:29:33,380
That's not necessarily true.

481
00:29:33,380 --> 00:29:39,400
If you have two
eigenvectors, they

482
00:29:39,400 --> 00:29:41,080
have different eigenvalues.

483
00:29:41,080 --> 00:29:44,580
So things don't work
out well necessarily.

484
00:29:44,580 --> 00:29:47,680
So an eigenvector plus
another eigenvector

485
00:29:47,680 --> 00:29:49,020
is not an eigenvector.

486
00:29:49,020 --> 00:29:56,920
So you have here, for
example, A f1 equals a1f1.

487
00:29:56,920 --> 00:30:07,340
And A f2 equal a2f2,
then a on f1 plus f2

488
00:30:07,340 --> 00:30:13,970
would be a1f1 plus
a2f2, and that's

489
00:30:13,970 --> 00:30:20,940
not equal to something
times f1 plus f2.

490
00:30:20,940 --> 00:30:24,000
It would have to be
something times f1 plus f2

491
00:30:24,000 --> 00:30:25,650
to be an eigenvector.

492
00:30:25,650 --> 00:30:29,420
So this is not necessarily
an eigenvector.

493
00:30:29,420 --> 00:30:33,290
And it doesn't help to put
a constant in front of here.

494
00:30:33,290 --> 00:30:34,320
Nothing helps.

495
00:30:34,320 --> 00:30:37,980
There's no way to
construct an eigenvector

496
00:30:37,980 --> 00:30:41,690
from two eigenvectors by
adding or subtracting.

497
00:30:41,690 --> 00:30:45,720
The size of the eigenvector
is not fixed either.

498
00:30:45,720 --> 00:30:56,516
If f is an eigenvector, then 3
times f is also an eigenvector.

499
00:30:59,700 --> 00:31:02,270
And we call it the
same eigenvector.

500
00:31:02,270 --> 00:31:05,700
Nobody would call it a
different eigenvector.

501
00:31:05,700 --> 00:31:07,860
It's really the same.

502
00:31:07,860 --> 00:31:10,710
OK, so how does that
relate to physics?

503
00:31:10,710 --> 00:31:12,700
Well, we've seen
it here already.

504
00:31:12,700 --> 00:31:16,640
that one operator that
we've learned to work with

505
00:31:16,640 --> 00:31:17,870
is the momentum operator.

506
00:31:17,870 --> 00:31:20,460
It has those eigenfunctions.

507
00:31:20,460 --> 00:31:25,370
So back to physics.

508
00:31:25,370 --> 00:31:27,190
We have other operators.

509
00:31:27,190 --> 00:31:29,370
Therefore we have
the P operator.

510
00:31:29,370 --> 00:31:30,660
That's good.

511
00:31:30,660 --> 00:31:32,970
We have the X operator.

512
00:31:32,970 --> 00:31:34,730
That's nice.

513
00:31:34,730 --> 00:31:36,850
It's multiplication by x.

514
00:31:36,850 --> 00:31:39,000
And why do we use it?

515
00:31:39,000 --> 00:31:41,560
Because sometimes you
have the energy operator.

516
00:31:46,430 --> 00:31:48,560
And what is the energy operator?

517
00:31:48,560 --> 00:31:51,640
The energy operator
is just the energy

518
00:31:51,640 --> 00:31:55,790
that you've always known, but
think of it as an operator.

519
00:31:55,790 --> 00:31:58,040
So how do we do that?

520
00:31:58,040 --> 00:32:01,750
Well, what is the energy
of a particle we've written

521
00:32:01,750 --> 00:32:07,300
p squared over 2m plus v of x.

522
00:32:07,300 --> 00:32:09,920
Well, that was the
energy of a particle,

523
00:32:09,920 --> 00:32:13,200
the momentum squared
over 2m plus v of x.

524
00:32:13,200 --> 00:32:17,470
So the energy operator
is hat here, hat there.

525
00:32:24,250 --> 00:32:28,460
And now it becomes an
interesting object.

526
00:32:28,460 --> 00:32:32,300
This energy operator
will be called E hat.

527
00:32:32,300 --> 00:32:34,190
It acts and functions.

528
00:32:34,190 --> 00:32:35,810
It is not a number.

529
00:32:35,810 --> 00:32:39,020
The energy is a number,
but the energy operator

530
00:32:39,020 --> 00:32:40,060
is not a number.

531
00:32:40,060 --> 00:32:43,220
Far from a number in fact.

532
00:32:43,220 --> 00:32:49,340
The energy operator is
minus h squared over 2m.

533
00:32:49,340 --> 00:32:52,340
d second the x squared.

534
00:32:52,340 --> 00:32:53,440
Why that?

535
00:32:53,440 --> 00:32:59,270
Well, because p was h bar
over i d dx as an operator.

536
00:32:59,270 --> 00:33:04,270
So this sort of arrow here,
it sort of the introduction.

537
00:33:04,270 --> 00:33:14,090
But after a while you just say
P hat is h bar over a i d dx.

538
00:33:14,090 --> 00:33:15,160
End of story.

539
00:33:15,160 --> 00:33:18,250
It's not like double arrow.

540
00:33:18,250 --> 00:33:19,670
It's just what it is.

541
00:33:19,670 --> 00:33:20,740
That operator.

542
00:33:20,740 --> 00:33:21,710
That's what we call it.

543
00:33:21,710 --> 00:33:26,390
So when we square it, the i
squares, the minus h squares,

544
00:33:26,390 --> 00:33:31,400
and d dx and d dx applied
twice is the second derivative.

545
00:33:31,400 --> 00:33:34,850
And here we get
v of X hat, which

546
00:33:34,850 --> 00:33:38,820
is your good potential, whatever
potential you're interested in,

547
00:33:38,820 --> 00:33:43,380
in which, whenever you see
an x, you put an X hat.

548
00:33:43,380 --> 00:33:45,310
And now this is an operator.

549
00:33:45,310 --> 00:33:50,000
So you see this is not a
number, not the function.

550
00:33:50,000 --> 00:33:52,020
It's just an operator.

551
00:33:52,020 --> 00:33:55,900
The operator has this
sort of operator v of x.

552
00:33:55,900 --> 00:34:00,590
Now what is this v of
x here as an operator?

553
00:34:00,590 --> 00:34:04,130
This is v of x as an operator
is just multiplication

554
00:34:04,130 --> 00:34:07,120
by the function v of x.

555
00:34:07,120 --> 00:34:12,120
You see, you have here that the
operator x is f of x like that.

556
00:34:12,120 --> 00:34:16,630
I could have written the
operator X hat to the n.

557
00:34:16,630 --> 00:34:18,380
What would it be?

558
00:34:18,380 --> 00:34:23,239
Well, if I add to
the function, this

559
00:34:23,239 --> 00:34:29,750
is a lot of X hats
acting on the function.

560
00:34:29,750 --> 00:34:31,440
Well, let the first one out.

561
00:34:31,440 --> 00:34:33,510
You let x times f of x.

562
00:34:33,510 --> 00:34:36,090
The second, that's
another x, another x.

563
00:34:36,090 --> 00:34:42,590
So this is just x to
the n times f of x.

564
00:34:42,590 --> 00:34:44,550
So lots of X hats.

565
00:34:44,550 --> 00:34:47,469
X hats To the
100th on a function

566
00:34:47,469 --> 00:34:49,889
is just X to the 100th
times a function.

567
00:34:49,889 --> 00:34:54,650
So v of x on a
function is just v

568
00:34:54,650 --> 00:34:56,520
of the number x on a function.

569
00:34:56,520 --> 00:35:00,240
It's just like this
operator, the O in which you

570
00:35:00,240 --> 00:35:03,550
multiply by a function.

571
00:35:03,550 --> 00:35:07,780
So please I hope this
is completely clear what

572
00:35:07,780 --> 00:35:10,500
this means as an operator.

573
00:35:10,500 --> 00:35:13,220
You take the wave function,
take two derivatives,

574
00:35:13,220 --> 00:35:18,790
and add the product of the wave
function times v of plane x.

575
00:35:18,790 --> 00:35:20,970
So I'll write it here maybe.

576
00:35:20,970 --> 00:35:22,040
So important.

577
00:35:22,040 --> 00:35:28,160
E hat and psi of x
is therefore minus h

578
00:35:28,160 --> 00:35:33,920
squared over 2m the [INAUDIBLE]
the x squared of psi of x

579
00:35:33,920 --> 00:35:38,122
plus just plain v
of x times psi of x.

580
00:35:42,157 --> 00:35:42,990
That's what it does.

581
00:35:42,990 --> 00:35:45,610
That's an operator.

582
00:35:45,610 --> 00:35:52,100
And for these
operators in general.

583
00:35:52,100 --> 00:35:53,830
Math interlude, is it over?

584
00:35:53,830 --> 00:35:55,560
Not quite.

585
00:35:55,560 --> 00:35:56,470
Wow.

586
00:35:56,470 --> 00:35:57,860
No, yes.

587
00:35:57,860 --> 00:36:00,196
Allan said at this
moment it's over,

588
00:36:00,196 --> 00:36:01,320
when you introduce it here.

589
00:36:04,760 --> 00:36:09,810
I'll say something more here,
but it's going to be over now.

590
00:36:09,810 --> 00:36:12,970
Our three continues
here then with four.

591
00:36:19,910 --> 00:36:37,563
Four, to each observable we
have an associated operator.

592
00:36:43,810 --> 00:36:53,420
So for momentum, we
have the operator P hat.

593
00:36:53,420 --> 00:37:00,460
And for position we
have the operator X hat.

594
00:37:00,460 --> 00:37:04,780
And for energy we have
the operator E hat.

595
00:37:08,150 --> 00:37:12,240
And these are
examples of operators.

596
00:37:12,240 --> 00:37:17,840
Example operator A hat
could be any of those.

597
00:37:17,840 --> 00:37:21,340
And there may be more
observables depending

598
00:37:21,340 --> 00:37:23,460
on the system you're working.

599
00:37:23,460 --> 00:37:25,790
If you have particles
in a line, there's

600
00:37:25,790 --> 00:37:28,719
not too many more
observables at this moment.

601
00:37:28,719 --> 00:37:30,135
If you have a
particle in general,

602
00:37:30,135 --> 00:37:32,840
you can have angular momentum.

603
00:37:32,840 --> 00:37:35,050
That's an interesting
observable.

604
00:37:35,050 --> 00:37:36,110
It can be others.

605
00:37:36,110 --> 00:37:41,240
So for any of those,
our definition

606
00:37:41,240 --> 00:37:44,360
is just like with
it for momentum.

607
00:37:44,360 --> 00:37:48,490
The expectation
value of the operator

608
00:37:48,490 --> 00:37:53,400
is computed by doing what
you did for momentum.

609
00:37:53,400 --> 00:37:56,210
You act with the operator
on the wave function

610
00:37:56,210 --> 00:38:02,950
here and multiply by the
compass conjugate function.

611
00:38:02,950 --> 00:38:06,150
And integrate just like
you did for momentum.

612
00:38:06,150 --> 00:38:11,080
This is going to be the
value that you expect.

613
00:38:11,080 --> 00:38:14,680
After many trials on
this wave function,

614
00:38:14,680 --> 00:38:19,040
you would expect the measured
value of this exhibit

615
00:38:19,040 --> 00:38:23,270
a distribution which its
expectation value, the mean,

616
00:38:23,270 --> 00:38:25,640
is given by this.

617
00:38:25,640 --> 00:38:29,430
Now there are other definitions.

618
00:38:29,430 --> 00:38:32,650
One definition that
already has been mentioned

619
00:38:32,650 --> 00:38:41,010
is that the uncertainty of
the operator on the state

620
00:38:41,010 --> 00:38:45,810
psi, the uncertainty,
is computed

621
00:38:45,810 --> 00:38:49,880
by taking the square
root of the expectation

622
00:38:49,880 --> 00:39:04,030
value of A squared minus
the expectation value of A,

623
00:39:04,030 --> 00:39:07,230
as a number, squared.

624
00:39:07,230 --> 00:39:10,810
Now the expectation value
of A squared, just simply

625
00:39:10,810 --> 00:39:12,830
here instead of A
you put A squared,

626
00:39:12,830 --> 00:39:14,830
so you've got A squared here.

627
00:39:14,830 --> 00:39:19,430
That unless the function is very
special, it's very different

628
00:39:19,430 --> 00:39:24,400
whole is bigger than the
expectation value of A squared.

629
00:39:24,400 --> 00:39:28,160
So this is a number, and
it's called the uncertainty.

630
00:39:28,160 --> 00:39:32,270
And that's the uncertainty
of the uncertainty principle.

631
00:39:32,270 --> 00:39:37,530
So for operators, we need to
have another observation that

632
00:39:37,530 --> 00:39:39,930
comes from matrices that
is going to be crucial

633
00:39:39,930 --> 00:39:45,580
for us is the observation that
operators don't necessarily

634
00:39:45,580 --> 00:39:47,510
commute.

635
00:39:47,510 --> 00:39:52,120
And we'll do the most
important example of that.

636
00:39:52,120 --> 00:39:56,470
So we'll try to see in
the operators associated

637
00:39:56,470 --> 00:40:00,530
with momentum and
position commute.

638
00:40:00,530 --> 00:40:04,050
And what we mean by commute
or don't communicate?

639
00:40:04,050 --> 00:40:07,420
Whether the order of
multiplication matters.

640
00:40:07,420 --> 00:40:10,820
Now we talked about
matrices at the beginning,

641
00:40:10,820 --> 00:40:14,800
and we said matrices act on
vectors to give you vectors.

642
00:40:14,800 --> 00:40:16,560
So do they commute?

643
00:40:16,560 --> 00:40:18,145
Well, matrices don't commute.

644
00:40:18,145 --> 00:40:21,260
The order matters for
matrices multiplication.

645
00:40:21,260 --> 00:40:25,500
So these operators we're
inventing here for physics,

646
00:40:25,500 --> 00:40:27,900
the order does matter as well.

647
00:40:27,900 --> 00:40:29,526
So commutation.

648
00:40:36,480 --> 00:40:42,210
So let's try to see if
we compute the operator p

649
00:40:42,210 --> 00:40:43,230
and x hat.

650
00:40:43,230 --> 00:40:49,360
Is it equal to the
operator x hat times p?

651
00:40:49,360 --> 00:40:51,220
This is a very good question.

652
00:40:51,220 --> 00:40:54,890
These are two operators
that we've defined.

653
00:40:54,890 --> 00:40:59,160
And we just want to know
if the order matters

654
00:40:59,160 --> 00:41:01,560
or if it doesn't matter.

655
00:41:01,560 --> 00:41:02,900
So how can I check it?

656
00:41:02,900 --> 00:41:06,260
I cannot just
check it like this,

657
00:41:06,260 --> 00:41:11,110
because operators are only clear
what they do is when they act

658
00:41:11,110 --> 00:41:12,540
on functions.

659
00:41:12,540 --> 00:41:15,220
So the only thing
that I can do is test

660
00:41:15,220 --> 00:41:19,430
if this thing acting on
functions give the same.

661
00:41:19,430 --> 00:41:23,710
So I'm going act with this
on the function f of x.

662
00:41:23,710 --> 00:41:26,960
And I'm going to have act with
this on the function f of x.

663
00:41:29,620 --> 00:41:33,040
Now what do I mean by
acting with p times

664
00:41:33,040 --> 00:41:35,010
x hat on the function f of x.

665
00:41:35,010 --> 00:41:39,060
This is by definition
you act first

666
00:41:39,060 --> 00:41:41,925
with the operator that
is next to the f and then

667
00:41:41,925 --> 00:41:42,550
with the other.

668
00:41:42,550 --> 00:41:48,285
So this is p hat on the
function x hat times f of x.

669
00:41:51,440 --> 00:41:57,050
So here I would have, this is
x hat on the function p hat

670
00:41:57,050 --> 00:41:58,700
f of x.

671
00:41:58,700 --> 00:42:04,160
See, if you have a series of
matrices, m1, m2, m3 acting

672
00:42:04,160 --> 00:42:06,330
on a vector, what do you mean?

673
00:42:06,330 --> 00:42:09,070
Act with this on the vector,
then with this on the vector,

674
00:42:09,070 --> 00:42:10,590
then with this.

675
00:42:10,590 --> 00:42:12,090
That's multiplication.

676
00:42:12,090 --> 00:42:14,370
So we're doing that.

677
00:42:14,370 --> 00:42:15,500
So let's evaluate.

678
00:42:15,500 --> 00:42:17,990
What is x operator on f of x?

679
00:42:17,990 --> 00:42:22,300
This is p hat on x times f of x.

680
00:42:25,510 --> 00:42:28,140
That's what the x operator
in the function is.

681
00:42:28,140 --> 00:42:30,570
Here, what this x hat?

682
00:42:30,570 --> 00:42:39,100
And now I have this, so I
have here h over i d dx of f.

683
00:42:39,100 --> 00:42:41,530
Let's go one more step here.

684
00:42:41,530 --> 00:42:50,460
This is h over i d ddx now
of this function, x f of x.

685
00:42:50,460 --> 00:42:55,440
And here I have just the x
function times this function.

686
00:42:55,440 --> 00:43:00,080
So h over i x df dx.

687
00:43:03,630 --> 00:43:05,840
Well, are these the same?

688
00:43:05,840 --> 00:43:11,750
No, because this d dx here is
not only acting on f like here.

689
00:43:11,750 --> 00:43:12,940
It's acting on the x.

690
00:43:12,940 --> 00:43:14,860
So this gives you two terms.

691
00:43:14,860 --> 00:43:18,100
One extra term on the
d dx acts on the x.

692
00:43:18,100 --> 00:43:20,390
And then one term
that is equal to this.

693
00:43:20,390 --> 00:43:22,060
So you don't get the same.

694
00:43:22,060 --> 00:43:28,340
So you get from here h over i
f of x, when you [INAUDIBLE]

695
00:43:28,340 --> 00:43:34,980
the x plus h over i x the df dx.

696
00:43:38,470 --> 00:43:39,790
So you don't get the same.

697
00:43:39,790 --> 00:43:43,050
So when I subtract
them, so when I

698
00:43:43,050 --> 00:43:54,280
do xp minus px acting on the
function f of x, what do I get?

699
00:43:54,280 --> 00:43:57,670
Well, I put them in this
order, x before the p.

700
00:43:57,670 --> 00:43:59,540
Doesn't matter
which one you take,

701
00:43:59,540 --> 00:44:01,280
but many people like this.

702
00:44:01,280 --> 00:44:05,150
Well, these terms cancel
and I get minus this thing.

703
00:44:05,150 --> 00:44:16,600
So I get minus h over i f
of x, or i h bar f of x.

704
00:44:16,600 --> 00:44:18,460
Wow.

705
00:44:18,460 --> 00:44:21,470
You got something very strange.

706
00:44:21,470 --> 00:44:26,740
The x times p minus p times
x gives you a number--

707
00:44:26,740 --> 00:44:30,900
an imaginary number, even
worse-- times f of x.

708
00:44:30,900 --> 00:44:34,630
So from this we
write the following.

709
00:44:34,630 --> 00:44:38,830
We say look, operators
are defined by the action

710
00:44:38,830 --> 00:44:42,970
and function, but for any
function, the only effect of xp

711
00:44:42,970 --> 00:44:51,600
minus px, which we call
the commutator of x with p.

712
00:44:51,600 --> 00:44:56,010
This definition, the bracket
of two things, of A B.

713
00:44:56,010 --> 00:45:00,830
Is defined to be A B minus B
A. It's called the commutator.

714
00:45:00,830 --> 00:45:06,280
x p is an operator acting
on f of x, gives you

715
00:45:06,280 --> 00:45:09,240
i h bar times f of x.

716
00:45:09,240 --> 00:45:13,530
So our kind of silly
operator that does nothing

717
00:45:13,530 --> 00:45:14,670
has appeared here.

718
00:45:14,670 --> 00:45:20,140
Because I could now
say that x hat with p

719
00:45:20,140 --> 00:45:24,070
is equal to i h bar
times the unit operator.

720
00:45:28,700 --> 00:45:30,790
Apart from the
Schrodinger equation,

721
00:45:30,790 --> 00:45:33,160
this is probably the
most important equation

722
00:45:33,160 --> 00:45:34,190
in quantum mechanics.

723
00:45:37,260 --> 00:45:41,960
It's the fact that x and b
are incompatible operators

724
00:45:41,960 --> 00:45:43,010
as you will see later.

725
00:45:43,010 --> 00:45:45,270
They don't commute.

726
00:45:45,270 --> 00:45:47,270
Their order matters.

727
00:45:47,270 --> 00:45:50,210
What's going to mean is
that when you measure one,

728
00:45:50,210 --> 00:45:52,280
you have difficulties
measuring the other.

729
00:45:52,280 --> 00:45:53,310
They interfere.

730
00:45:53,310 --> 00:45:55,560
They cannot be measured
simultaneously.

731
00:45:55,560 --> 00:45:58,950
All those things
are encapsulated

732
00:45:58,950 --> 00:46:02,530
in a very lovely
mathematical formula,

733
00:46:02,530 --> 00:46:06,355
which says that this is the
way these operators work.

734
00:46:09,560 --> 00:46:12,460
Any questions?

735
00:46:12,460 --> 00:46:14,410
Yes?

736
00:46:14,410 --> 00:46:17,140
AUDIENCE: When x-- the
commutator of x and p

737
00:46:17,140 --> 00:46:18,592
is itself an operator, right?

738
00:46:18,592 --> 00:46:19,300
PROFESSOR: RIght.

739
00:46:19,300 --> 00:46:22,190
AUDIENCE: So is that
what we're saying?

740
00:46:22,190 --> 00:46:27,370
When we had operators
before, we can't simply

741
00:46:27,370 --> 00:46:29,190
just cancel the f of x.

742
00:46:29,190 --> 00:46:31,520
I mean we're not really
canceling it, but it just

743
00:46:31,520 --> 00:46:35,820
because I h bar is the only
eigenvalue of the operator?

744
00:46:35,820 --> 00:46:40,240
PROFESSOR: Well, basically what
we've shown by this calculation

745
00:46:40,240 --> 00:46:43,910
is that this operator,
this combination

746
00:46:43,910 --> 00:46:46,890
is really the same as
the identity operator.

747
00:46:46,890 --> 00:46:50,480
That's all we've shown, that
some particular combination is

748
00:46:50,480 --> 00:46:52,530
the identity operator.

749
00:46:52,530 --> 00:46:56,200
Now this is very
deep, this equation.

750
00:46:56,200 --> 00:46:59,750
In fact, that's
the way Heisenberg

751
00:46:59,750 --> 00:47:02,800
invented quantum mechanics.

752
00:47:02,800 --> 00:47:05,610
He called it the
matrix mechanics,

753
00:47:05,610 --> 00:47:11,370
because he knew that operators
were related to matrices.

754
00:47:11,370 --> 00:47:14,420
It's a beautiful story how
he came up with this idea.

755
00:47:14,420 --> 00:47:16,660
It's very different from
what we're doing today

756
00:47:16,660 --> 00:47:19,340
that we're going to
follow Schrodinger today.

757
00:47:19,340 --> 00:47:25,850
But basically his analysis
led very quickly to this idea.

758
00:47:25,850 --> 00:47:28,030
And this is deep.

759
00:47:34,530 --> 00:47:35,590
Why is it deep?

760
00:47:38,350 --> 00:47:40,440
Depends who you ask.

761
00:47:40,440 --> 00:47:44,370
If you ask a mathematician,
they would probably tell you

762
00:47:44,370 --> 00:47:45,560
this equation is not deep.

763
00:47:45,560 --> 00:47:47,970
This is scary equation.

764
00:47:47,970 --> 00:47:50,730
And why is it scary?

765
00:47:50,730 --> 00:47:54,890
Because whenever a
mathematician see operators,

766
00:47:54,890 --> 00:47:58,630
they want to write matrices.

767
00:47:58,630 --> 00:48:02,120
So the mathematician, you
show him this equation,

768
00:48:02,120 --> 00:48:05,470
will say OK, Let me
try to figure out which

769
00:48:05,470 --> 00:48:08,010
matrices you're talking about.

770
00:48:08,010 --> 00:48:10,580
And this mathematician will
start doing calculations

771
00:48:10,580 --> 00:48:13,660
with two by two matrices,
and will say, no,

772
00:48:13,660 --> 00:48:15,850
I can't find two by
two matrices that

773
00:48:15,850 --> 00:48:18,550
behave like these operators.

774
00:48:18,550 --> 00:48:21,140
I can't find three by
three matrices either.

775
00:48:21,140 --> 00:48:22,570
And four by four.

776
00:48:22,570 --> 00:48:23,990
And five by five.

777
00:48:23,990 --> 00:48:26,400
And finds no matrix
really can do

778
00:48:26,400 --> 00:48:32,300
that, except if the matrix
is infinite dimensional.

779
00:48:32,300 --> 00:48:34,130
Infinite by infinite matrices.

780
00:48:34,130 --> 00:48:38,070
So that's why it's very
hard for a mathematician.

781
00:48:38,070 --> 00:48:40,070
This is the beginning
of quantum mechanics.

782
00:48:40,070 --> 00:48:43,960
This looks like a
trivial equation,

783
00:48:43,960 --> 00:48:47,090
and mathematicians
get scared by it.

784
00:48:47,090 --> 00:48:51,930
You show them for physicists
there will be angular momentum.

785
00:48:51,930 --> 00:48:56,010
The operators are like this,
and there's complicated

786
00:48:56,010 --> 00:48:59,720
into the [INAUDIBLE].

787
00:48:59,720 --> 00:49:02,170
The three components
of angular momentum

788
00:49:02,170 --> 00:49:04,340
have this commutation relation.

789
00:49:04,340 --> 00:49:07,100
And h bar here as well.

790
00:49:07,100 --> 00:49:08,000
Complicated.

791
00:49:08,000 --> 00:49:09,540
Three operators.

792
00:49:09,540 --> 00:49:10,930
They mix with each other.

793
00:49:10,930 --> 00:49:14,000
Show it to a mathematician,
he starts laughing at you.

794
00:49:14,000 --> 00:49:18,380
He says that best, the
simplest case, this is easy.

795
00:49:18,380 --> 00:49:19,390
This is complicated.

796
00:49:22,010 --> 00:49:23,550
It's very strange.

797
00:49:23,550 --> 00:49:26,420
But the reason this is
easier, the mathematician

798
00:49:26,420 --> 00:49:29,530
goes and, after five minutes,
comes to you with three

799
00:49:29,530 --> 00:49:33,450
by three matrices that
satisfies this relation.

800
00:49:33,450 --> 00:49:35,150
And here there weren't.

801
00:49:35,150 --> 00:49:38,250
And four by four that
satisfy, and five by five,

802
00:49:38,250 --> 00:49:40,310
and two by two, and all of them.

803
00:49:40,310 --> 00:49:43,010
We can calculate
all of them for you.

804
00:49:43,010 --> 00:49:45,770
But this one it's infinite
dimensional matrices,

805
00:49:45,770 --> 00:49:50,570
and it's very surprising, very
interesting, and very deep.

806
00:49:50,570 --> 00:49:55,300
All right, so we move
on a little bit more

807
00:49:55,300 --> 00:49:56,820
to the other observable.

808
00:49:56,820 --> 00:50:08,930
So after this, we have
more general observable.

809
00:50:08,930 --> 00:50:11,460
So let's talk a
little about them.

810
00:50:11,460 --> 00:50:15,290
That's another postulate
of quantum mechanics

811
00:50:15,290 --> 00:50:18,640
that continues with this
one, postulate number five.

812
00:50:18,640 --> 00:50:27,650
So once you measure,
upon measuring

813
00:50:27,650 --> 00:50:42,640
an observable A associated
with the operator A hat,

814
00:50:42,640 --> 00:50:43,605
two things happen.

815
00:50:46,130 --> 00:50:49,340
You measure this quantity
that could be momentum,

816
00:50:49,340 --> 00:50:55,360
could be energy, could
be position, you name it.

817
00:50:55,360 --> 00:51:02,560
The measured value
must be a number.

818
00:51:02,560 --> 00:51:13,460
It's one of the
eigenvalues of A hat.

819
00:51:13,460 --> 00:51:17,450
So actually those eigenvalues,
remember the definition

820
00:51:17,450 --> 00:51:18,920
of the eigenvalues.

821
00:51:18,920 --> 00:51:19,990
It's there.

822
00:51:19,990 --> 00:51:23,940
I said many, but
whenever you measure,

823
00:51:23,940 --> 00:51:27,750
the only possibilities
that you get this number.

824
00:51:27,750 --> 00:51:32,600
So you measure the momentum, you
must get this hk, for example.

825
00:51:32,600 --> 00:51:37,870
So observables, we have
an associated operator,

826
00:51:37,870 --> 00:51:40,480
and the measured values
are the eigenvalues.

827
00:51:43,630 --> 00:51:47,120
Now these eigenvalues, in
order to be observable,

828
00:51:47,120 --> 00:51:48,940
they should be a real numbers.

829
00:51:48,940 --> 00:51:51,160
And we said oh,
they can be complex.

830
00:51:51,160 --> 00:51:56,190
Well, we will limit
the kind of observables

831
00:51:56,190 --> 00:51:59,380
to things that have
real eigenvalues,

832
00:51:59,380 --> 00:52:03,830
and these are going to be called
later on permission operators.

833
00:52:03,830 --> 00:52:07,770
At this moment, the notes,
they don't mention them.

834
00:52:07,770 --> 00:52:11,240
You're going to
get them confused.

835
00:52:11,240 --> 00:52:17,650
So anyway, special operators
that have real eigenvalues.

836
00:52:17,650 --> 00:52:21,350
So we mentioned here they
will have to be a real.

837
00:52:21,350 --> 00:52:26,050
Have to be real.

838
00:52:26,050 --> 00:52:30,330
And then the second one, which
is an even stranger thing that

839
00:52:30,330 --> 00:52:34,000
happens is something you've
already seen in examples.

840
00:52:34,000 --> 00:52:37,360
After you measure, the
whole wave function

841
00:52:37,360 --> 00:52:42,970
goes into the state which is the
eigenfunction of the operator.

842
00:52:42,970 --> 00:53:00,950
So after measurement system
collapses into psi a.

843
00:53:04,060 --> 00:53:10,080
The measure value is one
over the eigenvalues a of A.

844
00:53:10,080 --> 00:53:14,020
And the system
collapses into psi a.

845
00:53:14,020 --> 00:53:21,780
So psi a is such that
A hat psi a is a psi a.

846
00:53:21,780 --> 00:53:26,960
So this is the eigenvector with
eigenvalue a that you measured.

847
00:53:26,960 --> 00:53:29,570
So after you
measure the momentum

848
00:53:29,570 --> 00:53:34,160
and you found that its h
bar k, the wave function

849
00:53:34,160 --> 00:53:37,140
is the wave function
of momentum h bar k.

850
00:53:37,140 --> 00:53:40,450
If at the beginning, it was
a superposition of many,

851
00:53:40,450 --> 00:53:44,140
as Fourier told you,
then after measuring,

852
00:53:44,140 --> 00:53:46,420
if you get one
component of momentum,

853
00:53:46,420 --> 00:53:48,670
that's all that is left
of the wave function.

854
00:53:48,670 --> 00:53:50,560
It collapses.

855
00:53:50,560 --> 00:53:53,950
This collapse is a
very strange thing,

856
00:53:53,950 --> 00:53:57,370
and is something about
quantum mechanics

857
00:53:57,370 --> 00:54:02,740
that people are a little
uncomfortable with,

858
00:54:02,740 --> 00:54:07,230
and try to understand better,
but surprisingly nobody

859
00:54:07,230 --> 00:54:14,170
has understood it better after
60 years of thinking about it.

860
00:54:14,170 --> 00:54:15,670
And it works very well.

861
00:54:15,670 --> 00:54:18,250
It's a very strange thing.

862
00:54:18,250 --> 00:54:22,320
Because for example, if you
have a wave function that

863
00:54:22,320 --> 00:54:28,240
says your particle can be
anywhere, after you measure it

864
00:54:28,240 --> 00:54:30,710
where it is, the
whole wave function

865
00:54:30,710 --> 00:54:35,240
becomes a delta function at
the position that you measure.

866
00:54:35,240 --> 00:54:38,640
So everything on
the wave function,

867
00:54:38,640 --> 00:54:41,540
when you do a
measurement, basically

868
00:54:41,540 --> 00:54:45,110
collapses as we'll see.

869
00:54:45,110 --> 00:54:49,650
Now for example,
let's do an example.

870
00:54:49,650 --> 00:54:50,150
Position.

871
00:54:54,420 --> 00:54:59,130
So you have a wave
function psi of x.

872
00:54:59,130 --> 00:55:09,810
You find measure and
find the particle at x0.

873
00:55:13,350 --> 00:55:16,830
Measure and you find
the particle at x0.

874
00:55:16,830 --> 00:55:21,930
So measure what?

875
00:55:21,930 --> 00:55:22,900
I should be clear.

876
00:55:22,900 --> 00:55:23,630
Measure position.

877
00:55:27,000 --> 00:55:28,580
So we said two things.

878
00:55:28,580 --> 00:55:32,860
The measured value is one
of the eigenvalues of a,

879
00:55:32,860 --> 00:55:34,690
and after measurement,
the system

880
00:55:34,690 --> 00:55:36,790
collapses to eigenfunctions.

881
00:55:36,790 --> 00:55:42,310
Now here we really need a
little of your intuition.

882
00:55:42,310 --> 00:55:47,010
Our position eigenstate is a
particle a localized at one

883
00:55:47,010 --> 00:55:47,510
place.

884
00:55:47,510 --> 00:55:49,850
What is the best
function associated

885
00:55:49,850 --> 00:55:51,430
to a position eigenstate?

886
00:55:51,430 --> 00:55:52,990
It's a delta function.

887
00:55:52,990 --> 00:55:57,440
The function that says it's at
some point and nowhere else.

888
00:55:57,440 --> 00:56:04,040
So eigenfunctions delta
of x minus x0, it's

889
00:56:04,040 --> 00:56:08,510
a function as a function of x.

890
00:56:08,510 --> 00:56:14,140
It peaks at x0, and
it's 0 everywhere else.

891
00:56:14,140 --> 00:56:18,330
And this is, when you
find a particle at x0,

892
00:56:18,330 --> 00:56:19,810
this is the wave function.

893
00:56:19,810 --> 00:56:24,830
The wave function must be
proportional to this quantity.

894
00:56:24,830 --> 00:56:27,595
Now you can't normalize
this wave function.

895
00:56:27,595 --> 00:56:30,665
It's a small complication, but
we shouldn't worry about it

896
00:56:30,665 --> 00:56:31,165
too much.

897
00:56:33,680 --> 00:56:37,760
Basically you really can't
localize a particle perfectly,

898
00:56:37,760 --> 00:56:40,890
so that's the little problem
with this wave function.

899
00:56:40,890 --> 00:56:43,540
You've studied how you can
represent delta functions

900
00:56:43,540 --> 00:56:47,130
as limits and probably
intuitively those limits

901
00:56:47,130 --> 00:56:48,740
are the best things.

902
00:56:48,740 --> 00:56:52,430
But this is the wave
function, so after you

903
00:56:52,430 --> 00:56:56,400
measure the system, you go into
an eigenstate of the operator.

904
00:56:56,400 --> 00:56:59,420
Is this an eigenstate
of the x operator?

905
00:56:59,420 --> 00:57:01,740
What a strange question.

906
00:57:01,740 --> 00:57:02,850
But it is.

907
00:57:02,850 --> 00:57:10,310
Look, if you put the x operator
on delta of x minus x zero,

908
00:57:10,310 --> 00:57:11,870
what is it supposed to do?

909
00:57:11,870 --> 00:57:14,620
It's supposed to multiply by x.

910
00:57:14,620 --> 00:57:21,160
So it's x times delta
of x minus x zero.

911
00:57:21,160 --> 00:57:23,950
If you had a little experience
with delta functions,

912
00:57:23,950 --> 00:57:27,270
you'd know that this
function is 0 everywhere,

913
00:57:27,270 --> 00:57:33,230
except when x is equal to x0,
so this x can be turned into x0.

914
00:57:33,230 --> 00:57:37,220
It just never at any other
place it contributes.

915
00:57:37,220 --> 00:57:44,490
This x really can be turned into
x0 times delta of x minus x0.

916
00:57:44,490 --> 00:57:47,680
Because delta functions are
really used to do integrals.

917
00:57:47,680 --> 00:57:51,160
And if you do the
integral of this function,

918
00:57:51,160 --> 00:57:53,860
you will see that it gives you
the same value as the integral

919
00:57:53,860 --> 00:57:55,770
of this function.

920
00:57:55,770 --> 00:57:57,230
So there you have it.

921
00:57:57,230 --> 00:58:00,290
The operator acting
on the eigenfunction

922
00:58:00,290 --> 00:58:01,790
is a number times this.

923
00:58:01,790 --> 00:58:04,970
So these are indeed
eigenfunctions

924
00:58:04,970 --> 00:58:07,390
of the x operator.

925
00:58:07,390 --> 00:58:11,610
And what you measured was an
eigenvalue of the x operator.

926
00:58:11,610 --> 00:58:16,780
Eigenvalue of x.

927
00:58:16,780 --> 00:58:20,970
And this is an
eigenfunction of x.

928
00:58:26,980 --> 00:58:34,570
So we can do the same
with the momentum.

929
00:58:34,570 --> 00:58:39,255
Eigenvalues and eigenfunctions,
we've seen them more properly.

930
00:58:42,000 --> 00:58:46,270
Now we'll go to the sixth
postulate, the last postulate

931
00:58:46,270 --> 00:58:55,220
that we'll want to talk about
is the one of general operators,

932
00:58:55,220 --> 00:58:59,970
and general eigenfunctions,
and what happens with them.

933
00:58:59,970 --> 00:59:10,590
So let's take now our operator
A and its functions that

934
00:59:10,590 --> 00:59:12,310
can be found.

935
00:59:12,310 --> 00:59:32,300
So six, given an observable A
hat and its eigenfunctions phi

936
00:59:32,300 --> 00:59:34,620
a of x.

937
00:59:34,620 --> 00:59:40,285
So an a runs over many
values, many values.

938
00:59:43,910 --> 00:59:47,730
OK so let's consider this case.

939
00:59:47,730 --> 00:59:51,310
Now eigenfunctions
of an operator

940
00:59:51,310 --> 00:59:54,610
are very interesting objects.

941
00:59:54,610 --> 01:00:00,290
You see the eigenfunctions of
momentum were of this form.

942
01:00:00,290 --> 01:00:04,610
And they allow you to
expand via the Fourier

943
01:00:04,610 --> 01:00:08,750
any wave function as super
positions of these things.

944
01:00:08,750 --> 01:00:12,140
Fourier told you you
can't expand any function

945
01:00:12,140 --> 01:00:14,940
in eigenfunctions
of the momentum,

946
01:00:14,940 --> 01:00:17,900
or the result is more general.

947
01:00:17,900 --> 01:00:21,790
For observables
in general you can

948
01:00:21,790 --> 01:00:24,660
expand functions,
arbitrary functions,

949
01:00:24,660 --> 01:00:27,630
in terms of the eigenfunctions.

950
01:00:27,630 --> 01:00:31,940
Now for that, remember,
an eigenfunction

951
01:00:31,940 --> 01:00:35,070
is not determined up to scale.

952
01:00:35,070 --> 01:00:39,030
You change multiplied by three,
it's an eigenfunction still.

953
01:00:39,030 --> 01:00:43,470
So people like to normalize
them nicely, the eigenfunctions.

954
01:00:43,470 --> 01:00:47,580
You construct them like this,
and you normalize them nicely.

955
01:00:47,580 --> 01:00:50,300
So how do you normalize them?

956
01:00:50,300 --> 01:00:59,150
Normalize them by saying
that the integral over x

957
01:00:59,150 --> 01:01:06,930
of psi a star of x psi b
of x is going to be what?

958
01:01:06,930 --> 01:01:12,560
OK, basically what you want
is that these eigenfunctions

959
01:01:12,560 --> 01:01:14,190
be basically orthogonal.

960
01:01:14,190 --> 01:01:17,020
Each one orthogonal to the next.

961
01:01:17,020 --> 01:01:20,360
So you want this to
be 0 unless these two

962
01:01:20,360 --> 01:01:22,260
different eigenfunctions
are different.

963
01:01:22,260 --> 01:01:24,840
And when they are
the same, you want

964
01:01:24,840 --> 01:01:26,820
them to be just
like wave functions,

965
01:01:26,820 --> 01:01:30,640
that their total integral of
psi squared is equal to 1.

966
01:01:30,640 --> 01:01:32,695
So what you put
here is delta ab.

967
01:01:35,380 --> 01:01:38,050
Now this is something
that you can always

968
01:01:38,050 --> 01:01:40,200
do with eigenfunctions.

969
01:01:40,200 --> 01:01:42,680
It's proven in
mathematics books.

970
01:01:42,680 --> 01:01:44,760
It's not all that
simple to prove,

971
01:01:44,760 --> 01:01:47,980
but this can always be done.

972
01:01:47,980 --> 01:01:53,740
And when we need examples,
we'll do it ourselves.

973
01:01:53,740 --> 01:01:58,370
So given an operator that
you have its eigenfunctions

974
01:01:58,370 --> 01:02:00,230
like that, two things happen.

975
01:02:00,230 --> 01:02:12,800
One can expand psi as psi of x.

976
01:02:12,800 --> 01:02:17,160
Any arbitrary wave
function as the sum.

977
01:02:17,160 --> 01:02:18,770
Or sometimes an integral.

978
01:02:18,770 --> 01:02:22,340
So some people like to write
this and put them an integral

979
01:02:22,340 --> 01:02:24,910
on top of that.

980
01:02:24,910 --> 01:02:26,990
You can write it
whichever way you want.

981
01:02:26,990 --> 01:02:28,150
It doesn't matter.

982
01:02:28,150 --> 01:02:33,850
Of coefficients times
the eigenfunctions.

983
01:02:33,850 --> 01:02:38,330
So just like any wave could be
a written, a Fourier coefficient

984
01:02:38,330 --> 01:02:40,330
[INAUDIBLE] Fourier function.

985
01:02:40,330 --> 01:02:47,180
Any state can be written a
superposition of these things.

986
01:02:47,180 --> 01:02:49,580
So that's one.

987
01:02:49,580 --> 01:03:10,940
And two, the probability of
measuring A hat and getting a.

988
01:03:10,940 --> 01:03:16,150
So a one of the particular
values that you can get.

989
01:03:16,150 --> 01:03:22,390
That probability is given by
the square of this coefficient.

990
01:03:24,930 --> 01:03:29,010
Ca squared.

991
01:03:29,010 --> 01:03:36,870
So this is P, the probability,
to measure in psi and to get a.

992
01:03:36,870 --> 01:03:40,150
I think, actually,
let's put an a0 there.

993
01:03:44,430 --> 01:03:49,360
So here we go.

994
01:03:49,360 --> 01:03:52,600
So it's a very
interesting thing.

995
01:03:52,600 --> 01:03:55,190
Basically you expand
the wave function

996
01:03:55,190 --> 01:03:57,950
in terms of these
eigenfunctions,

997
01:03:57,950 --> 01:04:02,070
and these coefficients
give you the probabilities

998
01:04:02,070 --> 01:04:05,130
of measuring these numbers.

999
01:04:05,130 --> 01:04:10,380
So we can illustrate that
again with the delta functions,

1000
01:04:10,380 --> 01:04:12,450
and we'll do it
quick, because we

1001
01:04:12,450 --> 01:04:17,000
get to the punchline of this
lecture with the Schrodinger

1002
01:04:17,000 --> 01:04:18,720
equation.

1003
01:04:18,720 --> 01:04:24,540
So what do we have?

1004
01:04:24,540 --> 01:04:29,920
Well, let me think of
the operator X example.

1005
01:04:29,920 --> 01:04:33,220
Operator X, the
eigenfunctions are

1006
01:04:33,220 --> 01:04:38,812
delta of x minus x0 for all x0.

1007
01:04:38,812 --> 01:04:40,020
These are the eigenfunctions.

1008
01:04:44,290 --> 01:04:47,270
And we'll write sort
of a trivial equation,

1009
01:04:47,270 --> 01:04:50,730
but it sort of illustrates
what's going on.

1010
01:04:50,730 --> 01:04:59,410
Psi of x as a superposition
over an integral over x0

1011
01:04:59,410 --> 01:05:02,930
of delta of x minus x0.

1012
01:05:06,260 --> 01:05:07,520
Psi of x0.

1013
01:05:10,520 --> 01:05:13,550
Delta of x minus x zero.

1014
01:05:20,280 --> 01:05:23,725
OK, first let's check
that this make sense.

1015
01:05:23,725 --> 01:05:26,500
Here we're integrating over x0.

1016
01:05:26,500 --> 01:05:28,290
x0 is the variable.

1017
01:05:28,290 --> 01:05:32,430
This thing shoots
and fires whenever

1018
01:05:32,430 --> 01:05:37,030
x is equal to x--
whenever x0 is equal to x.

1019
01:05:37,030 --> 01:05:38,890
Therefore the whole
result of that integral

1020
01:05:38,890 --> 01:05:42,590
is psi of x is a little
funny how it's written,

1021
01:05:42,590 --> 01:05:46,250
because you have x minus
0, which is the same

1022
01:05:46,250 --> 01:05:50,660
as delta of x0 minus x
is just the same thing.

1023
01:05:50,660 --> 01:05:54,690
And you integrate over x0,
and you get just psi of x.

1024
01:05:54,690 --> 01:05:57,490
But what have you achieved here?

1025
01:05:57,490 --> 01:06:03,410
You've achieved the analogue
of this equation in which these

1026
01:06:03,410 --> 01:06:08,660
are the psi a.

1027
01:06:08,660 --> 01:06:10,945
These are the coefficients Ca.

1028
01:06:14,700 --> 01:06:19,980
And this is the
sum, this integral.

1029
01:06:19,980 --> 01:06:21,420
So there you go.

1030
01:06:21,420 --> 01:06:26,000
Any wave function can be written
as the sum of coefficients

1031
01:06:26,000 --> 01:06:30,010
times the eigenfunctions
of the operator.

1032
01:06:30,010 --> 01:06:33,920
And what is the probability
to find the particle at x0?

1033
01:06:33,920 --> 01:06:35,340
Well, it's from here.

1034
01:06:35,340 --> 01:06:37,040
The coefficients, the a squared.

1035
01:06:37,040 --> 01:06:41,910
That's exactly
what we had before.

1036
01:06:41,910 --> 01:06:46,080
So this is getting
basically what we want.

1037
01:06:46,080 --> 01:06:48,440
So this brings us
to the final stage

1038
01:06:48,440 --> 01:06:53,430
of this lecture in which
we have to get the time

1039
01:06:53,430 --> 01:06:55,580
evolution finally.

1040
01:06:55,580 --> 01:07:01,180
So how does it happen?

1041
01:07:01,180 --> 01:07:04,270
Well it happens in a
very interesting way.

1042
01:07:04,270 --> 01:07:07,175
So maybe I'll call it
seven, Schrodinger equation.

1043
01:07:13,950 --> 01:07:21,910
So as with any fundamental
equation in physics,

1044
01:07:21,910 --> 01:07:27,530
there's experimental evidence
and suddenly, however, you

1045
01:07:27,530 --> 01:07:31,100
have to do a conceptual leap.

1046
01:07:31,100 --> 01:07:34,440
Experimental evidence doesn't
tell you the equation.

1047
01:07:34,440 --> 01:07:36,790
It suggests the equation.

1048
01:07:36,790 --> 01:07:40,500
And it tells you probably
what you're doing is right.

1049
01:07:40,500 --> 01:07:44,110
So what we're going to do now
is collect some of the evidence

1050
01:07:44,110 --> 01:07:49,220
we had and look at an
equation, and then just have

1051
01:07:49,220 --> 01:07:53,670
a flash of inspiration,
change something very little,

1052
01:07:53,670 --> 01:07:55,545
and suddenly that's the
Schrodinger equation.

1053
01:07:59,260 --> 01:08:02,240
Allan told me, in
fact, still sometimes

1054
01:08:02,240 --> 01:08:05,540
are disappointed that we
don't derive the Schrodinger

1055
01:08:05,540 --> 01:08:06,160
equation.

1056
01:08:06,160 --> 01:08:08,690
Now let's derive
it mathematically.

1057
01:08:08,690 --> 01:08:11,700
But you also don't derive
Newton's equations.

1058
01:08:11,700 --> 01:08:14,520
F equal ma.

1059
01:08:14,520 --> 01:08:16,800
You have an
inspiration, you get it.

1060
01:08:16,800 --> 01:08:17,979
Newton got it.

1061
01:08:17,979 --> 01:08:21,319
And then you use it and
you see it makes sense.

1062
01:08:21,319 --> 01:08:23,580
It has to be a
sensible equation,

1063
01:08:23,580 --> 01:08:26,069
and you can test very quickly
whether your equation is

1064
01:08:26,069 --> 01:08:27,020
sensible.

1065
01:08:27,020 --> 01:08:29,300
But you can't quite derive it.

1066
01:08:29,300 --> 01:08:33,020
In 805 we come a little closer
to deriving the Schrodinger

1067
01:08:33,020 --> 01:08:37,390
equation, which we say unitary
time evolution, something

1068
01:08:37,390 --> 01:08:39,819
that I haven't
explained what it is,

1069
01:08:39,819 --> 01:08:41,910
implies the
Schrodinger equation.

1070
01:08:41,910 --> 01:08:43,740
And that's a mathematical fact.

1071
01:08:43,740 --> 01:08:47,720
And you can begin unitary
time evolution, define it,

1072
01:08:47,720 --> 01:08:50,279
and you derive the
Schrodinger equation.

1073
01:08:50,279 --> 01:08:53,180
But that's just
saying that you've

1074
01:08:53,180 --> 01:08:55,840
substituted the Schrodinger
equation by saying there

1075
01:08:55,840 --> 01:08:57,950
is unitary time evolution.

1076
01:08:57,950 --> 01:09:00,689
The Schrodinger question
really comes from something

1077
01:09:00,689 --> 01:09:03,660
a little deeper than that.

1078
01:09:03,660 --> 01:09:06,279
Experimentally it comes
from something else.

1079
01:09:06,279 --> 01:09:08,790
So how does it come?

1080
01:09:08,790 --> 01:09:12,220
Well, you've studied some of
the history of this subject,

1081
01:09:12,220 --> 01:09:18,979
and you've seen that Planck
postulated quantized energies

1082
01:09:18,979 --> 01:09:24,660
in multiples of h bar omega.

1083
01:09:24,660 --> 01:09:26,770
And then came Einstein
and said look,

1084
01:09:26,770 --> 01:09:32,000
in fact, the energy of
a photon is h bar omega.

1085
01:09:32,000 --> 01:09:36,960
And the momentum of
the photon was h bar k.

1086
01:09:36,960 --> 01:09:44,510
So all these people, starting
with Planck and then Einstein,

1087
01:09:44,510 --> 01:09:47,290
understood what the photon is.

1088
01:09:47,290 --> 01:09:53,399
The quantum of
photons for photons,

1089
01:09:53,399 --> 01:09:58,890
you have E is equal h bar
omega, and the momentum

1090
01:09:58,890 --> 01:10:02,090
is equal to h bar k.

1091
01:10:02,090 --> 01:10:04,610
I write them as a vector,
because the momentum

1092
01:10:04,610 --> 01:10:09,160
is a vector, but we also write
them in this because p equal h

1093
01:10:09,160 --> 01:10:13,530
bar k, assuming you move
just in one direction.

1094
01:10:13,530 --> 01:10:15,280
And that's the way
it's been written.

1095
01:10:15,280 --> 01:10:22,710
So this is the result of much
work beginning by Planck,

1096
01:10:22,710 --> 01:10:26,110
Einstein, and Compton.

1097
01:10:29,230 --> 01:10:32,480
So you may recall
Einstein said in 1905

1098
01:10:32,480 --> 01:10:36,050
for that there seemed to be
this quantum of light that

1099
01:10:36,050 --> 01:10:38,350
carry energy h omega.

1100
01:10:38,350 --> 01:10:41,130
Planck didn't quite like that.

1101
01:10:41,130 --> 01:10:43,080
And people were not
all that convinced.

1102
01:10:43,080 --> 01:10:47,620
Experiments were done
by Millikan in 1915,

1103
01:10:47,620 --> 01:10:51,350
and people were still
not quite convinced.

1104
01:10:51,350 --> 01:10:54,640
And then came Compton and
did Compton scattering.

1105
01:10:54,640 --> 01:10:57,250
And then people said, yeah,
they seem to be particles.

1106
01:10:57,250 --> 01:10:58,790
No way out of that.

1107
01:10:58,790 --> 01:11:03,960
And they satisfy
such a relation.

1108
01:11:03,960 --> 01:11:10,200
Now there was
something about this

1109
01:11:10,200 --> 01:11:15,460
that was quite nice, that
these photons are associated

1110
01:11:15,460 --> 01:11:21,140
with waves, and that
was not too surprising,

1111
01:11:21,140 --> 01:11:28,960
because people understood that
electromagnetic waves are waves

1112
01:11:28,960 --> 01:11:30,700
that correspond to photons.

1113
01:11:30,700 --> 01:11:33,550
So you can also
see that this says

1114
01:11:33,550 --> 01:11:40,220
that E p is equal
to h bar omega k

1115
01:11:40,220 --> 01:11:42,950
as an equation between vectors.

1116
01:11:42,950 --> 01:11:48,000
You see the E is the first, and
the p is the second equation.

1117
01:11:48,000 --> 01:11:52,540
And this is actually a
relativistic equation.

1118
01:11:52,540 --> 01:11:55,310
It's a wonderful
relativistic equation,

1119
01:11:55,310 --> 01:11:59,910
because energy and momentum
form what is called

1120
01:11:59,910 --> 01:12:03,340
a relativity of four vector.

1121
01:12:03,340 --> 01:12:08,550
It's the four vector-- this is
a little aside on relativity--

1122
01:12:08,550 --> 01:12:11,310
four vector.

1123
01:12:11,310 --> 01:12:16,080
The index mew runs
from 0, 1, 2, 3.

1124
01:12:16,080 --> 01:12:20,760
Just like the x mews,
which are t and x.

1125
01:12:20,760 --> 01:12:25,790
Run from x0, which is t, x1,
which is x, x2 which is y,

1126
01:12:25,790 --> 01:12:28,290
three-- these are four vectors.

1127
01:12:28,290 --> 01:12:30,500
And this is a four vector.

1128
01:12:30,500 --> 01:12:31,760
This is a four vector.

1129
01:12:31,760 --> 01:12:34,720
This all seemed quite
pretty, and this

1130
01:12:34,720 --> 01:12:38,200
was associated to photons.

1131
01:12:38,200 --> 01:12:42,080
But then came De Broglie.

1132
01:12:42,080 --> 01:12:46,020
And De Broglie had
a very daring idea

1133
01:12:46,020 --> 01:12:49,370
that even though this
was written for photons,

1134
01:12:49,370 --> 01:12:53,150
it was true for particles
as well, for any particle.

1135
01:12:53,150 --> 01:13:01,805
De Broglie says good for
particles, all particles.

1136
01:13:04,770 --> 01:13:08,810
And these particles
are really waves.

1137
01:13:08,810 --> 01:13:13,810
So what if he write--
he wrote psi of x and t

1138
01:13:13,810 --> 01:13:17,530
is equal a wave associated
to a matter particle.

1139
01:13:17,530 --> 01:13:25,790
And it would be an e to
the i kx minus omega t.

1140
01:13:25,790 --> 01:13:26,505
That's a wave.

1141
01:13:29,220 --> 01:13:37,970
And you know that this wave
has momentum p equal h bar k.

1142
01:13:37,970 --> 01:13:42,690
If k is positive,
look at this sign.

1143
01:13:42,690 --> 01:13:46,080
If this sign is like
that, then k is positive.

1144
01:13:46,080 --> 01:13:49,640
This is a wave that is
moving to the right.

1145
01:13:49,640 --> 01:13:51,885
So p being hk.

1146
01:13:51,885 --> 01:13:55,080
If k is positive, p is positive,
is moving to the right,

1147
01:13:55,080 --> 01:14:03,890
this is a wave moving to the
right, and has this momentum.

1148
01:14:03,890 --> 01:14:07,160
So it should also
have an energy.

1149
01:14:07,160 --> 01:14:11,520
Compton said that this is
relativistic because this all

1150
01:14:11,520 --> 01:14:12,820
comes from photons.

1151
01:14:12,820 --> 01:14:19,920
So if the momentum is given
by that, and the energy

1152
01:14:19,920 --> 01:14:23,160
must also be given by
a similar relation.

1153
01:14:23,160 --> 01:14:26,520
In fact, he mostly
said, look, you

1154
01:14:26,520 --> 01:14:32,120
must have the energy being
equal to h bar omega.

1155
01:14:32,120 --> 01:14:37,960
The momentum, therefore,
would be equal to h bar k.

1156
01:14:37,960 --> 01:14:42,560
And I will sometimes
erase these things.

1157
01:14:42,560 --> 01:14:46,310
So what happens with this thing?

1158
01:14:46,310 --> 01:14:49,570
Well, momentum equal to hk.

1159
01:14:49,570 --> 01:14:56,120
We've already understood
this as momentum operator

1160
01:14:56,120 --> 01:14:59,655
being h bar over i d dx.

1161
01:15:04,510 --> 01:15:09,290
So this fact that these
two must go together and be

1162
01:15:09,290 --> 01:15:13,030
true for particles was
De Broglie's insight,

1163
01:15:13,030 --> 01:15:15,850
and the connection
to relativity.

1164
01:15:15,850 --> 01:15:17,790
Now here we have this.

1165
01:15:17,790 --> 01:15:22,820
So now we just have to try
to figure out what could we

1166
01:15:22,820 --> 01:15:25,060
do for the energy.

1167
01:15:25,060 --> 01:15:27,500
Could we have an
energy operator?

1168
01:15:27,500 --> 01:15:31,820
What would the energy
operator have to do?

1169
01:15:31,820 --> 01:15:34,730
Well, if the energy
operator is supposed

1170
01:15:34,730 --> 01:15:38,995
to give us h bar omega,
the only thing it could be

1171
01:15:38,995 --> 01:15:45,400
is that the energy
is i h bar d dt.

1172
01:15:45,400 --> 01:15:45,940
Why?

1173
01:15:45,940 --> 01:15:50,400
Because you go again
at the wave function.

1174
01:15:50,400 --> 01:15:55,740
And you think i h bar d
dt, and what do you get?

1175
01:15:55,740 --> 01:16:08,675
i h bar d dt on the wave
function is equal to i h bar.

1176
01:16:08,675 --> 01:16:14,420
You take the d dt,
you get minus i omega

1177
01:16:14,420 --> 01:16:16,450
times the whole wave function.

1178
01:16:16,450 --> 01:16:22,984
So this is equal h bar omega
times the wave function,

1179
01:16:22,984 --> 01:16:25,240
times the wave
function like that.

1180
01:16:25,240 --> 01:16:26,490
So here it is.

1181
01:16:26,490 --> 01:16:30,380
This is the operator
that realizes the energy,

1182
01:16:30,380 --> 01:16:34,870
just like this is the operator
that realizes the momentum.

1183
01:16:34,870 --> 01:16:40,760
You could say these are the
main relations that we have.

1184
01:16:40,760 --> 01:16:46,110
So if you have
this wave function,

1185
01:16:46,110 --> 01:16:55,480
it corresponds to a particle
with momentum hk and energy h

1186
01:16:55,480 --> 01:16:56,680
omega.

1187
01:16:56,680 --> 01:17:01,680
So now we write this.

1188
01:17:01,680 --> 01:17:15,670
So for this psi
that we have here,

1189
01:17:15,670 --> 01:17:22,030
h bar over i d dx
of psi of x and t

1190
01:17:22,030 --> 01:17:27,890
is equal the value of the
momentum times psi of x and t.

1191
01:17:27,890 --> 01:17:30,880
That is something we've seen.

1192
01:17:30,880 --> 01:17:32,810
But then there's a second one.

1193
01:17:32,810 --> 01:17:42,940
For this psi, we also that i
h bar d dt of psi of x and t

1194
01:17:42,940 --> 01:17:48,270
is equal to the energy of
that particle times x and t,

1195
01:17:48,270 --> 01:17:52,010
because the energy of that
particle is h bar omega.

1196
01:17:56,430 --> 01:18:02,770
And look, this is familiar.

1197
01:18:02,770 --> 01:18:08,820
And here the t plays no role,
but here the t plays a role.

1198
01:18:08,820 --> 01:18:14,670
And this is prescribing you
how a wave function of energy E

1199
01:18:14,670 --> 01:18:17,830
evolves in time.

1200
01:18:17,830 --> 01:18:19,260
So you're almost there.

1201
01:18:19,260 --> 01:18:22,930
You have something
very deep in here.

1202
01:18:22,930 --> 01:18:25,290
It's telling you if you
know the wave function

1203
01:18:25,290 --> 01:18:30,240
and it has energy E, this
is how it looks later.

1204
01:18:30,240 --> 01:18:33,620
You can take this derivative
and solve this differential

1205
01:18:33,620 --> 01:18:35,570
equation.

1206
01:18:35,570 --> 01:18:38,200
Now this differential
equation is kind of trivial

1207
01:18:38,200 --> 01:18:40,260
because E is a number here.

1208
01:18:45,330 --> 01:18:51,180
But if you know that you have
a particle with energy E,

1209
01:18:51,180 --> 01:18:54,800
that's how it evolves in time.

1210
01:18:54,800 --> 01:18:59,190
So came Schrodinger and
looked at this equation.

1211
01:19:02,300 --> 01:19:09,970
Psi of x and t equal
E psi of x and t.

1212
01:19:09,970 --> 01:19:14,240
This is true for any
particle that has energy E.

1213
01:19:14,240 --> 01:19:18,580
How can I make out of
this a full equation?

1214
01:19:18,580 --> 01:19:22,390
Because maybe I don't
know what is the energy E.

1215
01:19:22,390 --> 01:19:26,410
The energy E might be
anything in general.

1216
01:19:26,410 --> 01:19:27,490
What can I do?

1217
01:19:30,260 --> 01:19:32,090
Very simple.

1218
01:19:32,090 --> 01:19:34,710
One single replacement
in that equation.

1219
01:19:38,670 --> 01:19:40,790
Done.

1220
01:19:40,790 --> 01:19:41,490
It's over.

1221
01:19:41,490 --> 01:19:43,000
That's the Schrodinger equation.

1222
01:19:43,000 --> 01:19:48,308
It's the energy operator
that we introduced before.

1223
01:19:48,308 --> 01:19:48,807
Inspiration.

1224
01:19:53,150 --> 01:19:57,110
Change E to E hat.

1225
01:19:57,110 --> 01:19:58,920
This is the
Schrodinger equation.

1226
01:19:58,920 --> 01:20:02,570
Now what has really
happened here,

1227
01:20:02,570 --> 01:20:07,520
this equation that was trivial
for a wave function that

1228
01:20:07,520 --> 01:20:10,700
represented a particle
with energy E,

1229
01:20:10,700 --> 01:20:15,360
if this is the energy operator,
this is not so easy anymore.

1230
01:20:15,360 --> 01:20:18,970
Because remember, the energy
operator, for example,

1231
01:20:18,970 --> 01:20:26,590
was p squared over
2m plus v of x.

1232
01:20:26,590 --> 01:20:33,590
And this was minus h squared
over 2m d second dx squared

1233
01:20:33,590 --> 01:20:39,640
plus v of x acting
on wave functions.

1234
01:20:39,640 --> 01:20:44,040
So now you've got a really
interesting equation,

1235
01:20:44,040 --> 01:20:49,070
because you don't assume
that the energy is a number,

1236
01:20:49,070 --> 01:20:50,740
because you don't know it.

1237
01:20:50,740 --> 01:20:52,740
In general, if the
particle is moving

1238
01:20:52,740 --> 01:20:55,830
in a complicated
potential, you don't know

1239
01:20:55,830 --> 01:20:58,180
what are the possible energies.

1240
01:20:58,180 --> 01:21:02,830
But this is symbolically
what must be happening,

1241
01:21:02,830 --> 01:21:06,630
because if this particle
has a definite energy,

1242
01:21:06,630 --> 01:21:11,040
then this energy operator
gives you the energy acting

1243
01:21:11,040 --> 01:21:13,560
on the function,
and then you recover

1244
01:21:13,560 --> 01:21:18,160
what you know is true for a
particle of a given energy.

1245
01:21:18,160 --> 01:21:20,820
So in general, the
Schrodinger equation

1246
01:21:20,820 --> 01:21:22,810
is a complicated equation.

1247
01:21:22,810 --> 01:21:24,810
Let's write it now completely.

1248
01:21:24,810 --> 01:21:26,540
So this is the
Schrodinger equation.

1249
01:21:30,390 --> 01:21:32,590
And if we write
it completely, it

1250
01:21:32,590 --> 01:21:41,030
will read i h bar d psi dt is
equal to minus h bar squared

1251
01:21:41,030 --> 01:21:49,880
over 2m d second dx squared
of psi plus v of x times psi--

1252
01:21:49,880 --> 01:21:53,900
psi of x and t, psi of x and t.

1253
01:21:59,650 --> 01:22:02,090
So it's an equation, a
differential equation.

1254
01:22:02,090 --> 01:22:07,920
It's first order in time,
and second order in space.

1255
01:22:07,920 --> 01:22:13,210
So let me say three things
about this equation and finish.

1256
01:22:13,210 --> 01:22:17,790
First, it requires
complex numbers.

1257
01:22:17,790 --> 01:22:21,330
If psi would be real, everything
on the right hand side

1258
01:22:21,330 --> 01:22:22,150
would be real.

1259
01:22:22,150 --> 01:22:25,980
But with an i it would
spoil it, so complex numbers

1260
01:22:25,980 --> 01:22:28,300
have to be there.

1261
01:22:28,300 --> 01:22:31,290
Second, it's a linear equation.

1262
01:22:31,290 --> 01:22:33,310
It satisfies the proposition.

1263
01:22:33,310 --> 01:22:36,270
So if one wave function
satisfies the Schrodinger

1264
01:22:36,270 --> 01:22:38,860
equation, the sum
of wave functions,

1265
01:22:38,860 --> 01:22:42,380
and another wave function
does, the sum does.

1266
01:22:42,380 --> 01:22:44,490
Third, it's deterministic.

1267
01:22:44,490 --> 01:22:49,430
If you know psi at
x and time equals 0,

1268
01:22:49,430 --> 01:22:53,320
you can calculate psi
at any later time,

1269
01:22:53,320 --> 01:22:57,890
because this is a first order
differential equation in time.

1270
01:22:57,890 --> 01:23:00,030
This equation will
be the subject of all

1271
01:23:00,030 --> 01:23:02,300
what we'll do in this course.

1272
01:23:02,300 --> 01:23:03,940
So that's it for today.

1273
01:23:03,940 --> 01:23:05,620
Thank you.

1274
01:23:05,620 --> 01:23:08,370
[APPLAUSE]