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PROFESSOR: Anything lingering
and disturbing or bewildering?
9
00:00:28,750 --> 00:00:29,420
No?
10
00:00:29,420 --> 00:00:30,570
Nothing?
11
00:00:30,570 --> 00:00:32,430
All right.
12
00:00:32,430 --> 00:00:43,285
OK, so the story so far is
basically three postulates.
13
00:00:46,140 --> 00:00:58,260
The first is that the
configuration of a particle
14
00:00:58,260 --> 00:01:06,410
is given by, or described
by, a wave function psi of x.
15
00:01:10,150 --> 00:01:11,310
Yeah?
16
00:01:11,310 --> 00:01:15,120
So in particular, just to
flesh this out a little more,
17
00:01:15,120 --> 00:01:17,650
if we were in 3D, for
example-- which we're not.
18
00:01:17,650 --> 00:01:19,490
We're currently in
our one dimensional
19
00:01:19,490 --> 00:01:20,710
tripped out tricycles.
20
00:01:20,710 --> 00:01:23,150
In 3D, the wave function
would be a function
21
00:01:23,150 --> 00:01:26,010
of all three
positions x, y and z.
22
00:01:29,860 --> 00:01:36,180
If we had two particles,
our wave function
23
00:01:36,180 --> 00:01:39,360
would be a function of the
position of each particle.
24
00:01:39,360 --> 00:01:42,790
x1, x2, and so on.
25
00:01:42,790 --> 00:01:46,566
So we'll go through lots of
details and examples later on.
26
00:01:46,566 --> 00:01:47,982
But for the most
part, we're going
27
00:01:47,982 --> 00:01:49,731
to be sticking with
single particle in one
28
00:01:49,731 --> 00:01:51,650
dimension for the
next few weeks.
29
00:01:51,650 --> 00:01:54,290
Now again, I want to emphasize
this is our first pass
30
00:01:54,290 --> 00:01:56,680
through our definition
of quantum mechanics.
31
00:01:56,680 --> 00:01:59,720
Once we use the language and
the machinery a little bit,
32
00:01:59,720 --> 00:02:04,460
we're going to develop a more
general, more coherent set
33
00:02:04,460 --> 00:02:06,850
of rules or definition
of quantum mechanics.
34
00:02:06,850 --> 00:02:09,210
But this is our first pass.
35
00:02:09,210 --> 00:02:12,950
Two, the meaning of
the wave function
36
00:02:12,950 --> 00:02:18,440
is that the norm squared psi of
x, norm squared, it's complex,
37
00:02:18,440 --> 00:02:29,450
dx is the probability of
finding the particle- There's
38
00:02:29,450 --> 00:02:30,320
an n in their.
39
00:02:30,320 --> 00:02:38,900
Finding the particle-- in the
region between x and x plus dx.
40
00:02:38,900 --> 00:02:40,690
So psi squared
itself, norm squared,
41
00:02:40,690 --> 00:02:44,430
is the probability density.
42
00:02:44,430 --> 00:02:46,550
OK?
43
00:02:46,550 --> 00:02:52,980
And third, the
superposition principle.
44
00:02:52,980 --> 00:02:55,130
If there are two
possible configurations
45
00:02:55,130 --> 00:02:58,250
the system can be in,
which in quantum mechanics
46
00:02:58,250 --> 00:03:00,570
means two different
wave functions that
47
00:03:00,570 --> 00:03:05,394
could describe the system
given psi 1 and psi 2, two
48
00:03:05,394 --> 00:03:06,810
wave functions
that could describe
49
00:03:06,810 --> 00:03:09,190
two different configurations
of the system.
50
00:03:09,190 --> 00:03:12,190
For example, the particles here
or the particles over here.
51
00:03:12,190 --> 00:03:21,360
It's also possible to find
the system in a superposition
52
00:03:21,360 --> 00:03:25,120
of those two psi is
equal to some arbitrary
53
00:03:25,120 --> 00:03:34,780
linear combination alpha
psi 1 plus beta psi 2 of x.
54
00:03:34,780 --> 00:03:35,280
OK?
55
00:03:41,490 --> 00:03:43,720
So some things to note--
so questions about those
56
00:03:43,720 --> 00:03:46,320
before we move on?
57
00:03:46,320 --> 00:03:49,030
No questions?
58
00:03:49,030 --> 00:03:50,330
Nothing?
59
00:03:50,330 --> 00:03:52,450
Really?
60
00:03:52,450 --> 00:03:55,105
You're going to make he
threaten you with something.
61
00:03:55,105 --> 00:03:56,230
I know there are questions.
62
00:03:56,230 --> 00:03:57,480
This is not trivial stuff.
63
00:04:00,560 --> 00:04:02,440
OK.
64
00:04:02,440 --> 00:04:06,409
So some things to note.
65
00:04:06,409 --> 00:04:07,825
The first is we
want to normalize.
66
00:04:11,455 --> 00:04:13,080
We will generally
normalize and require
67
00:04:13,080 --> 00:04:16,640
that the integral over
all possible positions
68
00:04:16,640 --> 00:04:19,839
of the probability density
psi of x norm squared
69
00:04:19,839 --> 00:04:20,839
is equal to 1.
70
00:04:20,839 --> 00:04:23,830
This is just saying that the
total probability that we find
71
00:04:23,830 --> 00:04:27,179
the particle somewhere
had better be one.
72
00:04:27,179 --> 00:04:28,970
This is like saying if
I know a particle is
73
00:04:28,970 --> 00:04:30,470
in one of two
boxes, because I've
74
00:04:30,470 --> 00:04:31,950
put a particle in
one of the boxes.
75
00:04:31,950 --> 00:04:33,510
I just don't remember which one.
76
00:04:33,510 --> 00:04:35,690
Then the probability that
it's in the first box
77
00:04:35,690 --> 00:04:37,523
plus probability that
it's in the second box
78
00:04:37,523 --> 00:04:38,720
must be 100% or one.
79
00:04:38,720 --> 00:04:42,020
If it's less, then the particle
has simply disappeared.
80
00:04:42,020 --> 00:04:45,140
And basic rule, things
don't just disappear.
81
00:04:45,140 --> 00:04:46,665
So probability
should be normalized.
82
00:04:49,430 --> 00:04:50,890
And this is our prescription.
83
00:04:50,890 --> 00:04:57,740
So a second thing to note is
that all reasonable, or non
84
00:04:57,740 --> 00:05:05,210
stupid, functions
psi of x are equally
85
00:05:05,210 --> 00:05:06,770
reasonable as wave functions.
86
00:05:16,510 --> 00:05:17,280
OK?
87
00:05:17,280 --> 00:05:21,300
So this is a very
reasonable function.
88
00:05:21,300 --> 00:05:22,420
It's nice and smooth.
89
00:05:22,420 --> 00:05:24,310
It converges to 0 infinity.
90
00:05:24,310 --> 00:05:27,140
It's got all the nice
properties you might want.
91
00:05:27,140 --> 00:05:30,005
This is also a
reasonable function.
92
00:05:30,005 --> 00:05:32,100
It's a little annoying,
but there it is.
93
00:05:32,100 --> 00:05:36,240
And they're both perfectly
reasonable as wave functions.
94
00:05:36,240 --> 00:05:38,940
This on the other
hand, not so much.
95
00:05:38,940 --> 00:05:39,740
So for two reasons.
96
00:05:39,740 --> 00:05:40,990
First off, it's discontinuous.
97
00:05:40,990 --> 00:05:43,390
And as you're going to
show in your problem set,
98
00:05:43,390 --> 00:05:45,462
discontinuities are very
bad for wave functions.
99
00:05:45,462 --> 00:05:47,420
So we need our wave
functions to be continuous.
100
00:05:47,420 --> 00:05:49,740
The second is over some
domain it's multi valued.
101
00:05:49,740 --> 00:05:51,230
There are two different
values of the function.
102
00:05:51,230 --> 00:05:52,780
That's also bad, because
what's the probability?
103
00:05:52,780 --> 00:05:54,500
It's the norm squared,
but if it two values,
104
00:05:54,500 --> 00:05:56,530
two values for the probability,
that doesn't make any sense.
105
00:05:56,530 --> 00:05:58,071
What's the probability
that I'm going
106
00:05:58,071 --> 00:06:00,150
to fall over in 10 seconds?
107
00:06:00,150 --> 00:06:05,000
Well, it's small, but it's not
actually equal to 1% or 3%.
108
00:06:05,000 --> 00:06:07,600
It's one of those.
109
00:06:07,600 --> 00:06:10,290
Hopefully is much
lower than that.
110
00:06:10,290 --> 00:06:14,640
So all reasonable
functions are equally
111
00:06:14,640 --> 00:06:17,270
reasonable as wave functions.
112
00:06:17,270 --> 00:06:19,710
And in particular, what
that means is all states
113
00:06:19,710 --> 00:06:21,520
corresponding to
reasonable wave functions
114
00:06:21,520 --> 00:06:25,890
are equally reasonable
as physical states.
115
00:06:25,890 --> 00:06:30,200
There's no primacy in wave
functions or in states.
116
00:06:30,200 --> 00:06:36,430
However, with that said,
some wave functions
117
00:06:36,430 --> 00:06:38,520
are more equal than others.
118
00:06:38,520 --> 00:06:40,120
OK?
119
00:06:40,120 --> 00:06:42,966
And this is important,
and coming up
120
00:06:42,966 --> 00:06:44,340
with a good
definition of this is
121
00:06:44,340 --> 00:06:45,696
going to be an important
challenge for us
122
00:06:45,696 --> 00:06:47,200
in the next couple of lectures.
123
00:06:47,200 --> 00:06:49,346
So in particular,
this wave function
124
00:06:49,346 --> 00:06:50,720
has a nice simple
interpretation.
125
00:06:50,720 --> 00:06:52,560
If I tell you this
is psi of x, then
126
00:06:52,560 --> 00:06:55,750
what can you tell me about the
particle whose wave function is
127
00:06:55,750 --> 00:06:56,560
the psi of x?
128
00:06:58,240 --> 00:06:59,490
What can you tell me about it?
129
00:06:59,490 --> 00:07:00,603
What do you know?
130
00:07:00,603 --> 00:07:02,700
AUDIENCE: [INAUDIBLE].
131
00:07:02,700 --> 00:07:04,330
PROFESSOR: It's here, right?
132
00:07:04,330 --> 00:07:05,166
It's not over here.
133
00:07:05,166 --> 00:07:06,290
Probability is basically 0.
134
00:07:06,290 --> 00:07:07,770
Probability is large.
135
00:07:07,770 --> 00:07:10,480
It's pretty much here with
this great confidence.
136
00:07:10,480 --> 00:07:12,940
What about this guy?
137
00:07:12,940 --> 00:07:15,190
Less informative, right?
138
00:07:15,190 --> 00:07:17,540
It's less obvious what this
wave function is telling me.
139
00:07:17,540 --> 00:07:19,623
So some wave functions are
more equal in the sense
140
00:07:19,623 --> 00:07:21,030
that they have-- i.e.
141
00:07:21,030 --> 00:07:22,405
they have simple
interpretations.
142
00:07:29,220 --> 00:07:35,650
So for example,
this wave function
143
00:07:35,650 --> 00:07:38,970
continuing on infinitely, this
wave function doesn't tell me
144
00:07:38,970 --> 00:07:41,116
where the particle is,
but what does it tell me?
145
00:07:41,116 --> 00:07:42,030
AUDIENCE: Momentum.
146
00:07:42,030 --> 00:07:43,180
PROFESSOR: The
momentum, exactly.
147
00:07:43,180 --> 00:07:45,000
So this is giving me
information about the momentum
148
00:07:45,000 --> 00:07:47,375
of the particle because it
has a well defined wavelength.
149
00:07:47,375 --> 00:07:50,810
So this one, I would also say
is more equal than this one.
150
00:07:50,810 --> 00:07:52,790
They're both perfectly
physical, but this one
151
00:07:52,790 --> 00:07:54,480
has a simple interpretation.
152
00:07:54,480 --> 00:07:57,670
And that's going to
be important for us.
153
00:07:57,670 --> 00:08:11,310
Related to that is that any
reasonable function psi of x
154
00:08:11,310 --> 00:08:24,980
can be expressed as a
superposition of more
155
00:08:24,980 --> 00:08:36,169
equal wave functions,
or more precisely easily
156
00:08:36,169 --> 00:08:46,056
interpretable wave functions.
157
00:08:46,056 --> 00:08:47,930
We saw this last time
in the Fourier theorem.
158
00:08:47,930 --> 00:08:50,764
The Fourier theorem said look,
take any wave function-- take
159
00:08:50,764 --> 00:08:52,430
any function, but I'm
going to interpret
160
00:08:52,430 --> 00:08:53,570
in the language of
quantum mechanics.
161
00:08:53,570 --> 00:08:56,069
Take any wave function which
is given by some complex valued
162
00:08:56,069 --> 00:08:57,460
function, and it
can be expressed
163
00:08:57,460 --> 00:09:00,530
as a superposition
of plane waves.
164
00:09:00,530 --> 00:09:07,810
1 over 2pi in our normalization
integral dk psi tilde of k,
165
00:09:07,810 --> 00:09:09,736
but this is a set
of coefficients.
166
00:09:09,736 --> 00:09:10,797
e to the ikx.
167
00:09:10,797 --> 00:09:11,880
So what are we doing here?
168
00:09:11,880 --> 00:09:14,251
We're saying pick a value of k.
169
00:09:14,251 --> 00:09:15,750
There's a number
associated with it,
170
00:09:15,750 --> 00:09:18,814
which is going to be an
a magnitude and a phase.
171
00:09:18,814 --> 00:09:20,230
And that's the
magnitude and phase
172
00:09:20,230 --> 00:09:22,380
of a plane wave, e to the ikx.
173
00:09:22,380 --> 00:09:25,980
Now remember that
e to the ikx is
174
00:09:25,980 --> 00:09:33,007
equal to cos kx plus i sin kx.
175
00:09:33,007 --> 00:09:35,090
Which you should all know,
but just to remind you.
176
00:09:35,090 --> 00:09:36,850
This is a periodic function.
177
00:09:36,850 --> 00:09:38,080
These are periodic functions.
178
00:09:38,080 --> 00:09:42,820
So this is a plane wave
with a definite wavelength,
179
00:09:42,820 --> 00:09:45,490
2pi upon k.
180
00:09:45,490 --> 00:09:47,990
So this is a more equal
wave function in the sense
181
00:09:47,990 --> 00:09:49,480
that it has a
definite wavelength.
182
00:09:49,480 --> 00:09:50,770
We know what its momentum is.
183
00:09:50,770 --> 00:09:52,840
Its momentum is h bar k.
184
00:09:52,840 --> 00:09:56,030
Any function, we're saying, can
be expressed as a superposition
185
00:09:56,030 --> 00:09:57,920
by summing over all
possible values of k,
186
00:09:57,920 --> 00:09:59,560
all possible
different wavelengths.
187
00:09:59,560 --> 00:10:03,030
Any function can be expressed
as a superposition of wave
188
00:10:03,030 --> 00:10:05,622
functions with a
definite momentum.
189
00:10:05,622 --> 00:10:07,300
That make sense?
190
00:10:07,300 --> 00:10:09,002
Fourier didn't think
about it that way,
191
00:10:09,002 --> 00:10:10,460
but from quantum
mechanics, this is
192
00:10:10,460 --> 00:10:11,960
the way we want
to think about it.
193
00:10:11,960 --> 00:10:12,950
It's just a true statement.
194
00:10:12,950 --> 00:10:13,991
It's a mathematical fact.
195
00:10:17,206 --> 00:10:18,080
Questions about that?
196
00:10:20,750 --> 00:10:24,820
Similarly, I claim that I can
expand the very same function,
197
00:10:24,820 --> 00:10:28,107
psi of x, as a
superposition of states,
198
00:10:28,107 --> 00:10:29,815
not with definite
momentum, but of states
199
00:10:29,815 --> 00:10:30,773
with definite position.
200
00:10:34,029 --> 00:10:35,820
So what's a state with
a definite position?
201
00:10:35,820 --> 00:10:36,929
AUDIENCE: Delta.
202
00:10:36,929 --> 00:10:38,470
PROFESSOR: A delta
function, exactly.
203
00:10:38,470 --> 00:10:40,510
So I claim that any
function psi of x
204
00:10:40,510 --> 00:10:45,569
can be expanded a
sum over all states
205
00:10:45,569 --> 00:10:46,610
with a definite position.
206
00:10:46,610 --> 00:10:49,470
So delta of-- well, what's a
state with a definite position?
207
00:10:49,470 --> 00:10:50,670
x0.
208
00:10:50,670 --> 00:10:53,810
Delta of x minus x0.
209
00:10:53,810 --> 00:10:54,420
OK?
210
00:10:54,420 --> 00:10:56,795
This goes bing when
x0 is equal to x.
211
00:10:56,795 --> 00:10:58,920
But I want a sum over all
possible delta functions.
212
00:10:58,920 --> 00:11:00,920
That means all
possible positions.
213
00:11:00,920 --> 00:11:04,930
That means all possible
values of x0, dx0.
214
00:11:04,930 --> 00:11:06,900
And I need some
coefficient function here.
215
00:11:06,900 --> 00:11:09,490
Well, the coefficient function
I'm going to call psi of x0.
216
00:11:13,210 --> 00:11:16,015
So is this true?
217
00:11:16,015 --> 00:11:17,640
Is it true that I
can take any function
218
00:11:17,640 --> 00:11:21,730
and expand it in a superposition
of delta functions?
219
00:11:21,730 --> 00:11:22,410
Absolutely.
220
00:11:22,410 --> 00:11:24,364
Because look at what
this equation does.
221
00:11:24,364 --> 00:11:26,030
Remember, delta
function is your friend.
222
00:11:26,030 --> 00:11:29,210
It's a map from integrals
to numbers or functions.
223
00:11:29,210 --> 00:11:31,280
So this integral, is
an integral over x0.
224
00:11:31,280 --> 00:11:33,000
Here we have a
delta of x minus x0.
225
00:11:33,000 --> 00:11:35,946
So this basically says
the value of this integral
226
00:11:35,946 --> 00:11:37,570
is what you get by
taking the integrand
227
00:11:37,570 --> 00:11:39,040
and replacing x by x0.
228
00:11:39,040 --> 00:11:41,250
Set x equals x0, that's
when delta equals 0.
229
00:11:41,250 --> 00:11:45,020
So this is equal to the argument
evaluated at x0 is equal to x.
230
00:11:45,020 --> 00:11:47,620
That's your psi of x.
231
00:11:47,620 --> 00:11:48,670
OK?
232
00:11:48,670 --> 00:11:52,310
Any arbitrarily ugly function
can be expressed either
233
00:11:52,310 --> 00:11:55,650
as a superposition of states
with definite momentum
234
00:11:55,650 --> 00:11:59,280
or a superposition of states
with definite position.
235
00:11:59,280 --> 00:12:00,022
OK?
236
00:12:00,022 --> 00:12:01,230
And this is going to be true.
237
00:12:01,230 --> 00:12:03,188
We're going to find this
is a general statement
238
00:12:03,188 --> 00:12:06,780
that any state can be expressed
as a superposition of states
239
00:12:06,780 --> 00:12:10,780
with a well defined
observable quantity
240
00:12:10,780 --> 00:12:12,457
for any observable
quantity you want.
241
00:12:12,457 --> 00:12:14,790
So let me give you just a
quick little bit of intuition.
242
00:12:14,790 --> 00:12:19,644
In 2D, this is a perfectly
good vector, right?
243
00:12:19,644 --> 00:12:21,310
Now here's a question
I want to ask you.
244
00:12:21,310 --> 00:12:22,310
Is that a superposition?
245
00:12:25,730 --> 00:12:26,230
Yeah.
246
00:12:26,230 --> 00:12:28,070
I mean every vector
can be written
247
00:12:28,070 --> 00:12:30,900
as the sum of other
vectors, right?
248
00:12:30,900 --> 00:12:34,052
And it can be done in an
infinite number of ways, right?
249
00:12:34,052 --> 00:12:35,510
So there's no such
thing as a state
250
00:12:35,510 --> 00:12:38,410
which is not a superposition.
251
00:12:38,410 --> 00:12:40,740
Every vector is a
superposition of other vectors.
252
00:12:40,740 --> 00:12:43,980
It's a sum of other vector.
253
00:12:43,980 --> 00:12:48,380
So in particular we often
find it useful to pick a basis
254
00:12:48,380 --> 00:12:50,992
and say look, I know what
I mean by the vector y,
255
00:12:50,992 --> 00:12:52,700
y hat is a unit vector
in this direction.
256
00:12:52,700 --> 00:12:54,325
I know what I mean
by the vector x hat.
257
00:12:54,325 --> 00:12:55,990
It's a unit vector
in this direction.
258
00:12:55,990 --> 00:12:59,630
And now I can ask, given that
these are my natural guys,
259
00:12:59,630 --> 00:13:03,660
the guys I want to attend
to, is this a superposition
260
00:13:03,660 --> 00:13:04,510
of x and y?
261
00:13:04,510 --> 00:13:06,585
Or is it just x or y?
262
00:13:06,585 --> 00:13:09,990
Well, that's a superposition.
263
00:13:09,990 --> 00:13:11,950
Whereas x hat itself is not.
264
00:13:11,950 --> 00:13:16,760
So this somehow is about finding
convenient choice of basis.
265
00:13:16,760 --> 00:13:18,920
But any given vector
can be expressed
266
00:13:18,920 --> 00:13:21,730
as a superposition of
some pair of basis vectors
267
00:13:21,730 --> 00:13:24,900
or a different pair
of basis vectors.
268
00:13:24,900 --> 00:13:28,060
There's nothing hallowed
about your choice of basis.
269
00:13:28,060 --> 00:13:30,392
There's no God given
basis for the universe.
270
00:13:30,392 --> 00:13:32,600
We look out in the universe
in the Hubble deep field,
271
00:13:32,600 --> 00:13:34,766
and you don't see somewhere
in the Hubble deep field
272
00:13:34,766 --> 00:13:37,470
an arrow going x, right?
273
00:13:37,470 --> 00:13:39,060
So there's no natural
choice of basis,
274
00:13:39,060 --> 00:13:41,090
but it's sometimes
convenient to pick a basis.
275
00:13:41,090 --> 00:13:42,590
This is the direction of
the surface of the earth.
276
00:13:42,590 --> 00:13:44,640
This is the direction
perpendicular to it.
277
00:13:44,640 --> 00:13:46,900
So sometimes
particular basis sets
278
00:13:46,900 --> 00:13:49,730
have particular meanings to us.
279
00:13:49,730 --> 00:13:50,821
That's true in vectors.
280
00:13:50,821 --> 00:13:51,820
This is along the earth.
281
00:13:51,820 --> 00:13:52,890
This is perpendicular to it.
282
00:13:52,890 --> 00:13:54,260
This would be slightly strange.
283
00:13:54,260 --> 00:13:56,450
Maybe if you're leaning.
284
00:13:56,450 --> 00:13:58,910
And similarly, this
is an expansion
285
00:13:58,910 --> 00:14:01,790
of a function as a
sum, as a superposition
286
00:14:01,790 --> 00:14:03,140
of other functions.
287
00:14:03,140 --> 00:14:06,030
And you could have done this
in any good space of functions.
288
00:14:06,030 --> 00:14:07,291
We'll talk about that more.
289
00:14:07,291 --> 00:14:08,790
These are particularly
natural ones.
290
00:14:08,790 --> 00:14:09,581
They're more equal.
291
00:14:09,581 --> 00:14:12,590
These are ones with different
definite values of position,
292
00:14:12,590 --> 00:14:14,270
different definite
values of momentum.
293
00:14:14,270 --> 00:14:15,520
Everyone cool?
294
00:14:15,520 --> 00:14:17,936
Quickly what's the momentum
associated to the plane wave e
295
00:14:17,936 --> 00:14:18,724
to the ikx?
296
00:14:18,724 --> 00:14:19,640
AUDIENCE: [INAUDIBLE].
297
00:14:22,220 --> 00:14:23,100
PROFESSOR: h bar k.
298
00:14:23,100 --> 00:14:23,600
Good.
299
00:14:33,180 --> 00:14:35,140
So now I want to
just quickly run over
300
00:14:35,140 --> 00:14:37,850
some concept questions for you.
301
00:14:37,850 --> 00:14:39,390
So whip out your clickers.
302
00:14:39,390 --> 00:14:41,030
OK, we'll do this verbally.
303
00:14:49,052 --> 00:14:50,385
All right, let's try this again.
304
00:14:50,385 --> 00:14:53,200
So how would you interpret
this wave function?
305
00:14:53,200 --> 00:14:54,452
AUDIENCE: e.
306
00:14:54,452 --> 00:14:55,160
PROFESSOR: Solid.
307
00:14:58,580 --> 00:15:00,080
How do you know
whether the particle
308
00:15:00,080 --> 00:15:02,332
is big or small by looking
at the wave function?
309
00:15:02,332 --> 00:15:03,248
AUDIENCE: [INAUDIBLE].
310
00:15:05,640 --> 00:15:06,780
PROFESSOR: All right.
311
00:15:06,780 --> 00:15:09,306
Two particles described by
a plane wave of the form e
312
00:15:09,306 --> 00:15:10,820
to the ikx.
313
00:15:10,820 --> 00:15:13,979
Particle one is a smaller
wavelength than particle two.
314
00:15:13,979 --> 00:15:15,520
Which particle has
a larger momentum?
315
00:15:15,520 --> 00:15:17,803
Think about it, but
don't say it out loud.
316
00:15:22,230 --> 00:15:24,647
And this sort of defeats the
purpose of the clicker thing,
317
00:15:24,647 --> 00:15:27,146
because now I'm supposed to be
able to know without you guys
318
00:15:27,146 --> 00:15:28,040
saying anything.
319
00:15:28,040 --> 00:15:29,549
So instead of
saying it out loud,
320
00:15:29,549 --> 00:15:30,840
here's what I'd like you to do.
321
00:15:30,840 --> 00:15:33,350
Talk to the person
next to you and discuss
322
00:15:33,350 --> 00:15:37,271
which one has the larger
323
00:15:37,271 --> 00:15:38,104
AUDIENCE: [CHATTER].
324
00:16:00,550 --> 00:16:02,000
All right.
325
00:16:02,000 --> 00:16:07,350
Cool, so which one has
the larger momentum?
326
00:16:07,350 --> 00:16:09,460
AUDIENCE: A.
327
00:16:09,460 --> 00:16:10,485
PROFESSOR: How come?
328
00:16:10,485 --> 00:16:12,760
[INTERPOSING VOICES]
329
00:16:12,760 --> 00:16:15,540
PROFESSOR: RIght,
smaller wavelength.
330
00:16:15,540 --> 00:16:18,262
P equals h bar k.
331
00:16:18,262 --> 00:16:22,330
k equals 2pi over lambda.
332
00:16:22,330 --> 00:16:23,880
Solid?
333
00:16:23,880 --> 00:16:25,869
Smaller wavelength,
higher momentum.
334
00:16:25,869 --> 00:16:27,660
If it has higher
momentum, what do you just
335
00:16:27,660 --> 00:16:30,642
intuitively expect to
know about its energy?
336
00:16:30,642 --> 00:16:31,940
It's probably higher.
337
00:16:31,940 --> 00:16:33,310
Are you positive about that?
338
00:16:33,310 --> 00:16:35,810
No, you need to know how the
energy depends on the momentum,
339
00:16:35,810 --> 00:16:38,130
but it's probably higher.
340
00:16:38,130 --> 00:16:39,990
So this is an
important little lesson
341
00:16:39,990 --> 00:16:41,990
that you probably all
know from optics and maybe
342
00:16:41,990 --> 00:16:43,030
from core mechanics.
343
00:16:43,030 --> 00:16:45,670
Shorter wavelength
thing, higher energy.
344
00:16:45,670 --> 00:16:47,010
Higher momentum for sure.
345
00:16:47,010 --> 00:16:50,550
Usually higher energy as well.
346
00:16:50,550 --> 00:16:53,190
Very useful rule of
thumb to keep in mind.
347
00:16:53,190 --> 00:16:54,490
Indeed, it's particle one.
348
00:16:54,490 --> 00:16:55,730
OK next one.
349
00:16:55,730 --> 00:16:59,490
Compared to the wave function
psi of x, it's Fourier
350
00:16:59,490 --> 00:17:02,660
transform, psi tilde of x
contains more information,
351
00:17:02,660 --> 00:17:06,740
or less, or the
same, or something.
352
00:17:06,740 --> 00:17:08,569
Don't say it out loud.
353
00:17:08,569 --> 00:17:12,579
OK, so how many people
know the answer?
354
00:17:12,579 --> 00:17:13,079
Awesome.
355
00:17:13,079 --> 00:17:16,380
And how many people
are not sure.
356
00:17:16,380 --> 00:17:17,869
OK, good.
357
00:17:17,869 --> 00:17:22,053
So talk to the person next to
you and convince them briefly.
358
00:17:43,100 --> 00:17:43,640
All right.
359
00:17:48,800 --> 00:17:49,590
So let's vote.
360
00:17:49,590 --> 00:17:52,980
A, more information.
361
00:17:52,980 --> 00:17:55,350
B, less information.
362
00:17:55,350 --> 00:17:57,690
C, same.
363
00:17:57,690 --> 00:17:59,586
OK, good you got it.
364
00:17:59,586 --> 00:18:00,710
So these are not hard ones.
365
00:18:04,600 --> 00:18:08,530
This function, which is
a sine wave of length l,
366
00:18:08,530 --> 00:18:10,080
0 outside of that region.
367
00:18:10,080 --> 00:18:12,340
Which is closer to true?
368
00:18:12,340 --> 00:18:15,370
f has a single well defined
wavelength for the most part?
369
00:18:15,370 --> 00:18:16,280
It's closer to true.
370
00:18:16,280 --> 00:18:17,570
This doesn't have to be exact.
371
00:18:17,570 --> 00:18:19,440
f has a single well
defined wavelengths.
372
00:18:19,440 --> 00:18:21,996
Or f is made up of a wide
range of wavelengths?
373
00:18:26,350 --> 00:18:28,382
Think it to yourself.
374
00:18:28,382 --> 00:18:29,590
Ponder that one for a minute.
375
00:18:40,380 --> 00:18:42,430
OK, now before we
get talking about it.
376
00:18:42,430 --> 00:18:44,152
Hold on, hold on, hold on.
377
00:18:44,152 --> 00:18:45,610
Since we don't have
clickers, but I
378
00:18:45,610 --> 00:18:47,410
want to pull off
the same effect,
379
00:18:47,410 --> 00:18:50,170
and we can do this,
because it's binary here.
380
00:18:50,170 --> 00:18:53,780
I want everyone close your eyes.
381
00:18:53,780 --> 00:18:56,020
Just close your eyes,
just for a moment.
382
00:18:56,020 --> 00:18:56,777
Yeah.
383
00:18:56,777 --> 00:18:58,610
Or close the eyes of
the person next to you.
384
00:18:58,610 --> 00:18:59,720
That's fine.
385
00:18:59,720 --> 00:19:02,610
And now and I want you to vote.
386
00:19:02,610 --> 00:19:05,200
A is f has a single
well defined wavelength.
387
00:19:05,200 --> 00:19:07,830
B is f has a wide
range of wavelengths.
388
00:19:07,830 --> 00:19:10,840
So how many people think
A, a single wavelength?
389
00:19:10,840 --> 00:19:12,540
OK.
390
00:19:12,540 --> 00:19:14,556
Lower your hands, good.
391
00:19:14,556 --> 00:19:18,000
And how many people think B,
a wide range of wavelengths?
392
00:19:18,000 --> 00:19:18,550
Awesome.
393
00:19:18,550 --> 00:19:20,340
So this is exactly what happens
when we actually use clickers.
394
00:19:20,340 --> 00:19:21,470
It's 50/50.
395
00:19:21,470 --> 00:19:24,040
So now you guys need to talk
to the person next to you
396
00:19:24,040 --> 00:19:26,225
and convince each
other of the truth.
397
00:19:26,225 --> 00:19:27,058
AUDIENCE: [CHATTER].
398
00:20:23,840 --> 00:20:30,719
All right, so the volume sort of
tones down as people, I think,
399
00:20:30,719 --> 00:20:31,510
come to resolution.
400
00:20:31,510 --> 00:20:33,020
Close your eyes again.
401
00:20:33,020 --> 00:20:34,700
Once more into the
breach, my friends.
402
00:20:34,700 --> 00:20:37,980
So close your eyes, and
now let's vote again.
403
00:20:37,980 --> 00:20:41,530
f of x has a single,
well defined wavelength.
404
00:20:41,530 --> 00:20:45,370
And now f of x is made up
of a range of wavelengths?
405
00:20:45,370 --> 00:20:46,090
OK.
406
00:20:46,090 --> 00:20:48,940
There's a dramatic
shift in the field to B,
407
00:20:48,940 --> 00:20:51,600
it has a wide range
of wavelengths, not
408
00:20:51,600 --> 00:20:52,920
a single wavelength.
409
00:20:52,920 --> 00:20:55,350
And that is, in fact,
the correct answer.
410
00:20:55,350 --> 00:20:58,010
OK, so learning happens.
411
00:20:58,010 --> 00:21:00,810
That was an empirical test.
412
00:21:00,810 --> 00:21:02,700
So does anyone want
to defend this view
413
00:21:02,700 --> 00:21:04,900
that f is made of a wide
range of wavelengths?
414
00:21:04,900 --> 00:21:06,190
Sure, bring it.
415
00:21:06,190 --> 00:21:08,650
AUDIENCE: So, the sine
wave is an infinite,
416
00:21:08,650 --> 00:21:11,217
and it cancels out
past minus l over 2
417
00:21:11,217 --> 00:21:12,800
and positive l over
2, which means you
418
00:21:12,800 --> 00:21:13,830
need to add a bunch
of wavelengths
419
00:21:13,830 --> 00:21:15,464
to actually cancel it out there.
420
00:21:15,464 --> 00:21:16,630
PROFESSOR: Awesome, exactly.
421
00:21:16,630 --> 00:21:17,240
Exactly.
422
00:21:17,240 --> 00:21:19,470
If you only had the thing
of a single wavelength,
423
00:21:19,470 --> 00:21:21,720
it would continue with a single
wavelength all the way out.
424
00:21:21,720 --> 00:21:23,386
In fact, there's a
nice way to say this.
425
00:21:23,386 --> 00:21:28,441
When you have a sine wave, what
can you say about it's-- we
426
00:21:28,441 --> 00:21:29,940
know that a sine
wave is continuous,
427
00:21:29,940 --> 00:21:32,580
and it's continuous
everywhere, right?
428
00:21:32,580 --> 00:21:34,220
It's also differentiable
everywhere.
429
00:21:34,220 --> 00:21:37,210
Its derivative is continuous
and differentiable everywhere,
430
00:21:37,210 --> 00:21:39,510
because it's a cosine, right?
431
00:21:39,510 --> 00:21:42,050
So if yo you take a
superposition of sines
432
00:21:42,050 --> 00:21:46,180
and cosines, do you ever
get a discontinuity?
433
00:21:46,180 --> 00:21:47,090
No.
434
00:21:47,090 --> 00:21:49,990
Do you ever get something whose
derivative is discontinuous?
435
00:21:49,990 --> 00:21:50,690
No.
436
00:21:50,690 --> 00:21:53,630
So how would you ever
reproduce a thing
437
00:21:53,630 --> 00:21:57,320
with a discontinuity
using sines and cosines?
438
00:21:57,320 --> 00:22:00,390
Well, you'd need some infinite
sum of sines and cosines
439
00:22:00,390 --> 00:22:03,400
where there's some technicality
about the infinite limit being
440
00:22:03,400 --> 00:22:04,900
singular, because
you can't do it
441
00:22:04,900 --> 00:22:07,200
a finite number of
sines and cosines.
442
00:22:07,200 --> 00:22:10,390
That function is continuous,
but its derivative
443
00:22:10,390 --> 00:22:12,240
is discontinuous.
444
00:22:12,240 --> 00:22:13,409
Yeah?
445
00:22:13,409 --> 00:22:15,450
So it's going to take an
infinite number of sines
446
00:22:15,450 --> 00:22:18,940
and cosines to reproduce
that little kink at the edge.
447
00:22:18,940 --> 00:22:19,658
Yeah?
448
00:22:19,658 --> 00:22:21,898
AUDIENCE: So a finite
number of sines and cosines
449
00:22:21,898 --> 00:22:26,176
doesn't mean finding-- or
an infinite number of sines
450
00:22:26,176 --> 00:22:28,384
and cosines doesn't mean
infinite [? regular ?] sines
451
00:22:28,384 --> 00:22:29,204
and cosines, right?
452
00:22:29,204 --> 00:22:33,430
Because over a finite
region [INAUDIBLE].
453
00:22:33,430 --> 00:22:35,864
PROFESSOR: That's true, but
you need arbitrarily-- so
454
00:22:35,864 --> 00:22:36,780
let's talk about that.
455
00:22:36,780 --> 00:22:38,804
That's an excellent question.
456
00:22:38,804 --> 00:22:39,970
That's a very good question.
457
00:22:39,970 --> 00:22:41,780
The question here
is look, there's
458
00:22:41,780 --> 00:22:43,696
two different things you
can be talking about.
459
00:22:43,696 --> 00:22:46,280
One is arbitrarily large and
arbitrarily short wavelengths,
460
00:22:46,280 --> 00:22:47,640
so an arbitrary
range of wavelengths.
461
00:22:47,640 --> 00:22:49,170
And the other is
an infinite number.
462
00:22:49,170 --> 00:22:50,520
But an infinite number
is silly, because there's
463
00:22:50,520 --> 00:22:51,932
a continuous variable here k.
464
00:22:51,932 --> 00:22:53,640
You got an infinite
number of wavelengths
465
00:22:53,640 --> 00:22:56,170
between one and 1.2, right?
466
00:22:56,170 --> 00:22:56,970
It's continuous.
467
00:22:56,970 --> 00:22:58,190
So which one do you mean?
468
00:22:58,190 --> 00:23:00,470
So let's go back
to this connection
469
00:23:00,470 --> 00:23:02,950
that we got a minute
ago from short distance
470
00:23:02,950 --> 00:23:04,080
and high momentum.
471
00:23:07,499 --> 00:23:09,790
That thing looks like it has
one particular wavelength.
472
00:23:09,790 --> 00:23:11,540
But I claim, in
order to reproduce
473
00:23:11,540 --> 00:23:14,980
that as a superposition of
states with definite momentum,
474
00:23:14,980 --> 00:23:17,714
I need arbitrarily
high wavelength.
475
00:23:17,714 --> 00:23:19,880
And why do I need arbitrarily
high wavelength modes?
476
00:23:19,880 --> 00:23:22,290
Why do we need to arbitrarily
high momentum modes?
477
00:23:22,290 --> 00:23:24,170
Well, it's because of this.
478
00:23:24,170 --> 00:23:24,950
We have a kink.
479
00:23:27,610 --> 00:23:32,330
And this feature, what's the
length scale of that feature?
480
00:23:32,330 --> 00:23:34,170
It's infinitesimally
small, which
481
00:23:34,170 --> 00:23:35,840
means I'm going to have to--
in order to reproduce that,
482
00:23:35,840 --> 00:23:37,465
in order to probe
it, I'm going to need
483
00:23:37,465 --> 00:23:40,130
a momentum that's
arbitrarily large.
484
00:23:40,130 --> 00:23:42,940
So it's really about the
range, not just the number.
485
00:23:42,940 --> 00:23:45,930
But you need arbitrarily
large momentum.
486
00:23:45,930 --> 00:23:50,890
To construct or detect an
arbitrarily small feature
487
00:23:50,890 --> 00:23:52,640
you need arbitrarily
large momentum modes.
488
00:23:52,640 --> 00:23:53,140
Yeah?
489
00:23:53,140 --> 00:23:56,390
AUDIENCE: Why do
you [INAUDIBLE]?
490
00:23:56,390 --> 00:23:58,330
Why don't you just
say, oh you need
491
00:23:58,330 --> 00:24:00,020
an arbitrary small wavelength?
492
00:24:00,020 --> 00:24:02,754
Why wouldn't you just
phrase that [INAUDIBLE]?
493
00:24:02,754 --> 00:24:04,420
PROFESSOR: I chose
to phrase it that way
494
00:24:04,420 --> 00:24:06,711
because I want an emphasize
and encourage-- I emphasize
495
00:24:06,711 --> 00:24:10,740
you to think and
encourage you to conflate
496
00:24:10,740 --> 00:24:13,140
short distance and
large momentum.
497
00:24:13,140 --> 00:24:16,764
I want the connection between
momentum and the length scale
498
00:24:16,764 --> 00:24:18,680
to be something that
becomes intuitive to you.
499
00:24:18,680 --> 00:24:20,440
So when I talk about
something with short features,
500
00:24:20,440 --> 00:24:22,940
I'm going to talk about it as
something with large momentum.
501
00:24:22,940 --> 00:24:25,960
And that's because in a
quantum mechanical system,
502
00:24:25,960 --> 00:24:27,990
something with
short wavelength is
503
00:24:27,990 --> 00:24:31,236
something that carries
large momentum.
504
00:24:31,236 --> 00:24:32,351
That cool?
505
00:24:32,351 --> 00:24:32,850
Great.
506
00:24:32,850 --> 00:24:33,930
Good question.
507
00:24:33,930 --> 00:24:36,270
AUDIENCE: So earlier you
said that any reasonable wave
508
00:24:36,270 --> 00:24:40,122
function, a possible
wave function,
509
00:24:40,122 --> 00:24:41,580
does that mean
they're not supposed
510
00:24:41,580 --> 00:24:43,100
to be Fourier transformable?
511
00:24:43,100 --> 00:24:45,640
PROFESSOR: That's
usually a condition.
512
00:24:45,640 --> 00:24:46,360
Yeah, exactly.
513
00:24:46,360 --> 00:24:47,600
We don't quite
phrase it that way.
514
00:24:47,600 --> 00:24:49,683
And in fact, there's a
problem on your problem set
515
00:24:49,683 --> 00:24:52,060
that will walk you
through what we will mean.
516
00:24:52,060 --> 00:24:53,700
What should be
true of the Fourier
517
00:24:53,700 --> 00:24:56,410
transform in order for this
to reasonably function.
518
00:24:56,410 --> 00:24:58,450
And among other things--
and your intuition
519
00:24:58,450 --> 00:25:00,710
here is exactly right--
among other things,
520
00:25:00,710 --> 00:25:03,190
being able to have a Fourier
transform where you don't have
521
00:25:03,190 --> 00:25:04,822
arbitrarily high
momentum modes is
522
00:25:04,822 --> 00:25:06,280
going to be an
important condition.
523
00:25:06,280 --> 00:25:09,460
That's going to turn to be
related to the derivative
524
00:25:09,460 --> 00:25:11,014
being continuous.
525
00:25:11,014 --> 00:25:12,180
That's a very good question.
526
00:25:12,180 --> 00:25:17,144
So that's the optional
problem 8 on problem set 2.
527
00:25:17,144 --> 00:25:17,810
Other questions?
528
00:25:22,340 --> 00:25:25,130
PROFESSOR: Cool, so that's
it for the clicker questions.
529
00:25:25,130 --> 00:25:26,490
Sorry for the technology fail.
530
00:25:29,400 --> 00:25:32,505
So I'm just going to
turn this off in disgust.
531
00:25:37,384 --> 00:25:38,425
That's really irritating.
532
00:25:41,580 --> 00:25:44,880
So today what I want
to start on is pick up
533
00:25:44,880 --> 00:25:47,070
on the discussion of the
uncertainty principle
534
00:25:47,070 --> 00:25:49,440
that we sort of
outlined previously.
535
00:25:49,440 --> 00:25:51,950
The fact that when we have a
wave function with reasonably
536
00:25:51,950 --> 00:25:53,460
well defined position
corresponding
537
00:25:53,460 --> 00:25:55,744
to a particle with reasonably
well defined position,
538
00:25:55,744 --> 00:25:58,160
it didn't have a reasonably
well defined momentum and vice
539
00:25:58,160 --> 00:25:58,850
versa.
540
00:25:58,850 --> 00:26:00,630
The certainty of
the momentum seems
541
00:26:00,630 --> 00:26:04,570
to imply lack of knowledge about
the position and vice versa.
542
00:26:04,570 --> 00:26:09,520
So in order to do that, we
need to define uncertainty.
543
00:26:09,520 --> 00:26:11,930
So I need to define for
you delta x and delta p.
544
00:26:14,470 --> 00:26:16,740
So first I just want
to run through what
545
00:26:16,740 --> 00:26:19,290
should be totally
remedial probability,
546
00:26:19,290 --> 00:26:23,570
but it's always useful
to just remember
547
00:26:23,570 --> 00:26:25,310
how these basic things work.
548
00:26:25,310 --> 00:26:28,620
So consider a set
of people in a room,
549
00:26:28,620 --> 00:26:32,370
and I want to plot the number
of people with a particular age
550
00:26:32,370 --> 00:26:36,240
as a function of the
age of possible ages.
551
00:26:36,240 --> 00:26:41,540
So let's say we have 16
people, and at 14 we have one,
552
00:26:41,540 --> 00:26:46,720
and at 15 we have 1,
and at 16 we have 3.
553
00:26:46,720 --> 00:26:50,170
And that's 16.
554
00:26:50,170 --> 00:26:53,705
And at 20 we have 2.
555
00:26:56,680 --> 00:26:58,715
And at 21 we have 4.
556
00:27:02,270 --> 00:27:03,935
And at 22 we have 5.
557
00:27:09,040 --> 00:27:10,070
And that's it.
558
00:27:10,070 --> 00:27:12,180
OK.
559
00:27:12,180 --> 00:27:23,020
So 1, 1, 3, 2, 4, 5.
560
00:27:23,020 --> 00:27:26,950
OK, so what's the
probability that any given
561
00:27:26,950 --> 00:27:29,750
person in this group of
16 has a particular age?
562
00:27:29,750 --> 00:27:30,647
I'll call it a.
563
00:27:33,340 --> 00:27:34,840
So how do we compute
the probability
564
00:27:34,840 --> 00:27:36,630
that they have age a?
565
00:27:36,630 --> 00:27:37,380
Well this is easy.
566
00:27:37,380 --> 00:27:40,460
It's the number that have
age a over the total number.
567
00:27:44,660 --> 00:27:47,140
So note an important thing,
an important side note,
568
00:27:47,140 --> 00:27:49,860
which is that the sum
over all possible ages
569
00:27:49,860 --> 00:27:52,955
of the probability that you
have age a is equal to 1,
570
00:27:52,955 --> 00:27:55,080
because it's just going to
be the sum of the number
571
00:27:55,080 --> 00:27:57,340
with a particular age over the
total number, which is just
572
00:27:57,340 --> 00:27:59,048
the sum of the number
with any given age.
573
00:28:04,540 --> 00:28:05,750
So here's some questions.
574
00:28:05,750 --> 00:28:09,264
So what's the most likely age?
575
00:28:09,264 --> 00:28:10,680
If you grabbed one
of these people
576
00:28:10,680 --> 00:28:12,495
from the room with
a giant Erector set,
577
00:28:12,495 --> 00:28:14,030
and pull out a person,
and let them dangle,
578
00:28:14,030 --> 00:28:15,613
and ask them what
their age is, what's
579
00:28:15,613 --> 00:28:17,288
the most likely they'll have?
580
00:28:17,288 --> 00:28:18,382
AUDIENCE: 22.
581
00:28:18,382 --> 00:28:18,965
PROFESSOR: 22.
582
00:28:21,211 --> 00:28:22,960
On the other hand,
what's the average age?
583
00:28:31,700 --> 00:28:35,420
Well, just by eyeball roughly
what do you think it is?
584
00:28:35,420 --> 00:28:37,030
So around 19 or 20.
585
00:28:37,030 --> 00:28:40,220
It turns out to
be 19.2 for this.
586
00:28:40,220 --> 00:28:40,840
OK.
587
00:28:40,840 --> 00:28:43,490
But if everyone had a little
sticker on their lapel
588
00:28:43,490 --> 00:28:47,770
that says I'm 14, 15, 16, 20,
21 or 22, how many people have
589
00:28:47,770 --> 00:28:49,910
the age 19.2?
590
00:28:49,910 --> 00:28:51,010
None, right?
591
00:28:51,010 --> 00:28:54,020
So a useful thing is
that the average need not
592
00:28:54,020 --> 00:28:55,940
be an observable value.
593
00:28:57,985 --> 00:28:59,610
This is going to come
back to haunt us.
594
00:28:59,610 --> 00:29:02,210
Oops, 19.4.
595
00:29:02,210 --> 00:29:03,550
That's what I got.
596
00:29:03,550 --> 00:29:11,300
So in particular how
did I get the average?
597
00:29:11,300 --> 00:29:13,050
I'm going to define
some notation.
598
00:29:13,050 --> 00:29:14,320
This notation is
going to stick with us
599
00:29:14,320 --> 00:29:15,736
for the rest of
quantum mechanics.
600
00:29:15,736 --> 00:29:18,800
The average age,
how do I compute it?
601
00:29:18,800 --> 00:29:21,330
So we all know this, but let
me just be explicit about it.
602
00:29:21,330 --> 00:29:25,240
It's the sum over
all possible ages
603
00:29:25,240 --> 00:29:29,610
of the number of
the number of people
604
00:29:29,610 --> 00:29:32,950
with that age times
the age divided
605
00:29:32,950 --> 00:29:34,800
by the total number of people.
606
00:29:37,100 --> 00:29:37,600
OK?
607
00:29:41,000 --> 00:29:46,892
So in this case, I'd go 14,14,
16, 16, 16, 20, 20, 21, 21,
608
00:29:46,892 --> 00:29:49,620
21 21, 22, 22, 22, 22, 22.
609
00:29:49,620 --> 00:29:51,662
And so that's all
I've written here.
610
00:29:51,662 --> 00:29:53,620
But notice that I can
write this in a nice way.
611
00:29:53,620 --> 00:29:56,462
This is equal to the sum
over all possible ages
612
00:29:56,462 --> 00:30:01,759
of a times the ratio of Na to N
with a ratio of Na to n total.
613
00:30:01,759 --> 00:30:03,800
That's just the probability
that any given person
614
00:30:03,800 --> 00:30:05,490
has a probability a.
615
00:30:05,490 --> 00:30:07,610
a times probability of a.
616
00:30:07,610 --> 00:30:11,680
So the expected value is the
sum over all possible values
617
00:30:11,680 --> 00:30:14,460
of the value times the
probability to get that value.
618
00:30:14,460 --> 00:30:15,840
Yeah?
619
00:30:15,840 --> 00:30:18,520
This is the same equation,
but I'm going to box it.
620
00:30:18,520 --> 00:30:21,820
It's a very useful relation.
621
00:30:21,820 --> 00:30:24,290
And so, again, does the
average have to be measurable?
622
00:30:24,290 --> 00:30:26,225
No, it certainly doesn't.
623
00:30:26,225 --> 00:30:29,310
And it usually isn't.
624
00:30:29,310 --> 00:30:33,590
So let's ask the same thing
for the square of ages.
625
00:30:33,590 --> 00:30:39,190
What is the average
of a squared?
626
00:30:39,190 --> 00:30:39,920
Square the ages.
627
00:30:39,920 --> 00:30:42,500
You might say, well, why
would I ever care about that?
628
00:30:42,500 --> 00:30:43,999
But let's just be
explicit about it.
629
00:30:43,999 --> 00:30:45,630
So following the
same logic here,
630
00:30:45,630 --> 00:30:48,460
the average of a squared,
the average value
631
00:30:48,460 --> 00:30:50,092
of the square of the
ages is, well, I'm
632
00:30:50,092 --> 00:30:51,550
going to do exactly
the same thing.
633
00:30:51,550 --> 00:30:53,060
It's just a squared, right?
634
00:30:53,060 --> 00:30:56,317
14 squared, 15 squared, 16
square, 16 squared, 16 squared.
635
00:30:56,317 --> 00:30:58,650
So this is going to give me
exactly the same expression.
636
00:30:58,650 --> 00:31:03,330
So over a of a squared
probability of measuring a.
637
00:31:07,140 --> 00:31:12,230
And more generally, the expected
value, or the average value
638
00:31:12,230 --> 00:31:14,802
of some function of
a is equal-- and this
639
00:31:14,802 --> 00:31:16,260
is something you
don't usually do--
640
00:31:16,260 --> 00:31:21,610
is equal to the sum over a of
f of a, the value of f given
641
00:31:21,610 --> 00:31:24,309
a particular value of a,
times the probability that you
642
00:31:24,309 --> 00:31:26,100
measure that value of
a in the first place.
643
00:31:29,320 --> 00:31:33,855
It's exactly the same
logic as averages.
644
00:31:37,340 --> 00:31:39,330
Right, cool.
645
00:31:39,330 --> 00:31:41,940
So here's a quick question.
646
00:31:41,940 --> 00:31:47,700
Is a squared equal to the
expected value of a squared?
647
00:31:47,700 --> 00:31:48,844
AUDIENCE: No.
648
00:31:48,844 --> 00:31:50,885
PROFESSOR: Right, in
general no, not necessarily.
649
00:31:56,810 --> 00:32:00,480
So for example,
the average value--
650
00:32:00,480 --> 00:32:02,780
suppose we have a Gaussian
centered at the origin.
651
00:32:02,780 --> 00:32:04,760
So here's a.
652
00:32:04,760 --> 00:32:07,076
Now a isn't age, but it's
something-- I don't know.
653
00:32:07,076 --> 00:32:09,570
You include infants or whatever.
654
00:32:09,570 --> 00:32:10,320
It's not age.
655
00:32:10,320 --> 00:32:13,340
Its happiness on a given day.
656
00:32:13,340 --> 00:32:17,920
So what's the average value?
657
00:32:17,920 --> 00:32:19,000
Meh.
658
00:32:19,000 --> 00:32:19,580
Right?
659
00:32:19,580 --> 00:32:21,910
Sort of vaguely neutral, right?
660
00:32:21,910 --> 00:32:24,080
But on the other hand,
if you take a squared,
661
00:32:24,080 --> 00:32:26,080
very few people have
a squared as zero.
662
00:32:26,080 --> 00:32:28,340
Most people have a
squared as not a 0 value.
663
00:32:28,340 --> 00:32:30,120
And most people are
sort of in the middle.
664
00:32:30,120 --> 00:32:34,130
Most people are sort of
hazy on what the day is.
665
00:32:34,130 --> 00:32:37,430
So in this case,
the expected value
666
00:32:37,430 --> 00:32:39,170
of a, or the average
value of a is 0.
667
00:32:39,170 --> 00:32:43,620
The average value of a
squared is not equal to 0.
668
00:32:43,620 --> 00:32:44,500
Yeah?
669
00:32:44,500 --> 00:32:46,791
And that's because the squared
has everything positive.
670
00:32:48,810 --> 00:32:51,560
So how do we characterize--
this gives us
671
00:32:51,560 --> 00:32:54,390
a useful tool for characterizing
the width of a distribution.
672
00:32:54,390 --> 00:32:56,890
So here we have a distribution
where its average value is 0,
673
00:32:56,890 --> 00:32:58,000
but its width is non-zero.
674
00:32:58,000 --> 00:33:00,810
And then the expectation
value of a squared,
675
00:33:00,810 --> 00:33:03,310
the expected value of
a squared, is non-zero.
676
00:33:03,310 --> 00:33:07,700
So how do we define the
width of a distribution?
677
00:33:07,700 --> 00:33:10,020
This is going to be
like our uncertainty.
678
00:33:10,020 --> 00:33:11,100
How happy are you today?
679
00:33:11,100 --> 00:33:11,910
Well, I'm not sure.
680
00:33:11,910 --> 00:33:12,880
How unsure are you?
681
00:33:12,880 --> 00:33:14,980
Well, that should give
us a precise measure.
682
00:33:14,980 --> 00:33:17,170
So let me define three things.
683
00:33:17,170 --> 00:33:19,060
First the deviation.
684
00:33:19,060 --> 00:33:22,770
So the deviation is going to be
a minus the average value of a.
685
00:33:22,770 --> 00:33:24,520
So this is just take
the actual value of a
686
00:33:24,520 --> 00:33:26,341
and subtract off the
average value of a.
687
00:33:26,341 --> 00:33:28,340
So we always get something
that's centered at 0.
688
00:33:32,087 --> 00:33:33,420
I'm going to write it like this.
689
00:33:36,240 --> 00:33:38,730
Note, by the way, just a
convenient thing to note.
690
00:33:38,730 --> 00:33:43,920
The average value of a
minus it's average value.
691
00:33:43,920 --> 00:33:46,564
Well, what's the
average value of 7?
692
00:33:46,564 --> 00:33:47,447
AUDIENCE: 7.
693
00:33:47,447 --> 00:33:48,280
PROFESSOR: OK, good.
694
00:33:48,280 --> 00:33:52,832
So that first term is
the average value of a.
695
00:33:52,832 --> 00:33:54,540
And that second term
is the average value
696
00:33:54,540 --> 00:33:58,090
of this number, which is
just this number minus a.
697
00:33:58,090 --> 00:33:59,931
So this is 0.
698
00:33:59,931 --> 00:34:00,430
Yeah?
699
00:34:04,092 --> 00:34:05,550
The average value
of a number is 0.
700
00:34:05,550 --> 00:34:06,966
The average value
of this variable
701
00:34:06,966 --> 00:34:10,469
is the average value of
that variable, but that's 0.
702
00:34:10,469 --> 00:34:12,750
So deviation is not a terribly
good thing on average,
703
00:34:12,750 --> 00:34:14,625
because on average the
deviation is always 0.
704
00:34:14,625 --> 00:34:17,602
That's what it means to
say this is the average.
705
00:34:17,602 --> 00:34:19,060
So the derivation
is saying how far
706
00:34:19,060 --> 00:34:21,560
is any particular
instance from the average.
707
00:34:21,560 --> 00:34:23,060
And if you average
those deviations,
708
00:34:23,060 --> 00:34:24,260
they always give you 0.
709
00:34:24,260 --> 00:34:25,909
So this is not a
very good measure
710
00:34:25,909 --> 00:34:28,630
of the actual width
of the system.
711
00:34:28,630 --> 00:34:31,559
But we can get a nice measure by
getting the deviation squared.
712
00:34:34,600 --> 00:34:36,880
And let's take the mean
of the derivation squared.
713
00:34:36,880 --> 00:34:40,829
So the mean of the derivation
squared, mean of a minus
714
00:34:40,829 --> 00:34:42,120
the average value of a squared.
715
00:34:45,179 --> 00:34:48,669
This is what I'm going to
call the standard deviation.
716
00:34:48,669 --> 00:34:50,460
Which is a little odd,
because really you'd
717
00:34:50,460 --> 00:34:52,418
want to call it the
standard deviation squared.
718
00:34:52,418 --> 00:34:53,886
But whatever.
719
00:34:53,886 --> 00:34:55,219
We use funny words.
720
00:34:59,450 --> 00:35:03,225
So now what does it mean if
the average value of a is 0?
721
00:35:03,225 --> 00:35:05,100
It means it's centered
at 0, but what does it
722
00:35:05,100 --> 00:35:07,830
mean if the standard
deviation of a is 0?
723
00:35:10,880 --> 00:35:15,610
So if the standard
deviation is 0,
724
00:35:15,610 --> 00:35:21,280
one then the distribution
has no width, right?
725
00:35:21,280 --> 00:35:23,450
Because if there was
any amplitude away
726
00:35:23,450 --> 00:35:25,600
from the average
value, then that
727
00:35:25,600 --> 00:35:27,960
would give a non-zero
strictly positive contribution
728
00:35:27,960 --> 00:35:31,280
to this average expectation,
and this wouldn't be 0 anymore.
729
00:35:31,280 --> 00:35:33,120
So standard deviation
is 0, as long
730
00:35:33,120 --> 00:35:35,578
as there's no width, which is
why the standard deviation is
731
00:35:35,578 --> 00:35:40,970
a good useful measure
of width or uncertainty.
732
00:35:40,970 --> 00:35:45,660
And just as a note, taking
this seriously and taking
733
00:35:45,660 --> 00:35:49,200
the square, so standard
deviation squared,
734
00:35:49,200 --> 00:35:52,320
this is equal to the
average value of a squared
735
00:35:52,320 --> 00:35:57,150
minus twice a times the average
value of a plus average value
736
00:35:57,150 --> 00:35:59,500
of a quantity squared.
737
00:35:59,500 --> 00:36:01,110
But if you do this
out, this is going
738
00:36:01,110 --> 00:36:05,770
to be equal to a squared
minus 2 average value
739
00:36:05,770 --> 00:36:07,260
of a average value of a.
740
00:36:07,260 --> 00:36:11,060
That's just minus
twice the average value
741
00:36:11,060 --> 00:36:14,180
of a quantity squared.
742
00:36:14,180 --> 00:36:16,360
And then plus average
value of a squared.
743
00:36:16,360 --> 00:36:19,230
So this is an alternate way of
writing the standard deviation.
744
00:36:19,230 --> 00:36:19,730
OK?
745
00:36:19,730 --> 00:36:23,170
So we can either write it in
this fashion or this fashion.
746
00:36:23,170 --> 00:36:29,075
And the notation for
this is delta a squared.
747
00:36:32,750 --> 00:36:33,580
OK?
748
00:36:33,580 --> 00:36:35,840
So when I talk about
an uncertainty, what
749
00:36:35,840 --> 00:36:38,520
I mean is, given
my distribution,
750
00:36:38,520 --> 00:36:40,094
I compute the
standard deviation.
751
00:36:40,094 --> 00:36:41,510
And the uncertainty
is going to be
752
00:36:41,510 --> 00:36:44,302
the square root of the
standard deviations squared.
753
00:36:44,302 --> 00:36:45,990
OK?
754
00:36:45,990 --> 00:36:48,880
So delta a, the words
I'm going to use for this
755
00:36:48,880 --> 00:36:59,430
is the uncertainty in a given
some probability distribution.
756
00:37:01,612 --> 00:37:03,820
Different probability
distributions are going to give
757
00:37:03,820 --> 00:37:05,020
me different delta a's.
758
00:37:09,240 --> 00:37:10,880
So one thing that's
sort of annoying
759
00:37:10,880 --> 00:37:12,706
is that when you
write delta a, there's
760
00:37:12,706 --> 00:37:14,330
nothing in the notation
that says which
761
00:37:14,330 --> 00:37:16,360
distribution you
were talking about.
762
00:37:16,360 --> 00:37:18,030
When you have multiple
distributions,
763
00:37:18,030 --> 00:37:21,582
or multiple possible probability
distributions, sometimes it's
764
00:37:21,582 --> 00:37:23,790
useful to just put given
the probability distribution
765
00:37:23,790 --> 00:37:25,670
p of a.
766
00:37:25,670 --> 00:37:28,170
This is not very often used,
but sometimes it's very helpful
767
00:37:28,170 --> 00:37:30,984
when you're doing calculations
just to keep track.
768
00:37:30,984 --> 00:37:34,010
Everyone cool with that?
769
00:37:34,010 --> 00:37:35,702
Yeah, questions?
770
00:37:35,702 --> 00:37:37,590
AUDIENCE: [INAUDIBLE]
delta a squared, right?
771
00:37:37,590 --> 00:37:38,980
PROFESSOR: Yeah, exactly.
772
00:37:38,980 --> 00:37:40,145
Of delta a squared.
773
00:37:40,145 --> 00:37:41,450
Yeah.
774
00:37:41,450 --> 00:37:43,550
Other questions?
775
00:37:43,550 --> 00:37:44,100
Yeah?
776
00:37:44,100 --> 00:37:45,890
AUDIENCE: So really it should
be parentheses [INAUDIBLE].
777
00:37:45,890 --> 00:37:47,620
PROFESSOR: Yeah, it's just
this is notation that's
778
00:37:47,620 --> 00:37:50,010
used typically, so I didn't
put the parentheses around
779
00:37:50,010 --> 00:37:53,140
precisely to alert you to the
stupidities of this notation.
780
00:37:59,580 --> 00:38:02,810
So any other questions?
781
00:38:02,810 --> 00:38:03,310
Good.
782
00:38:03,310 --> 00:38:06,970
OK, so let's just do the same
thing for continuous variables.
783
00:38:06,970 --> 00:38:08,610
Now for continuous variables.
784
00:38:14,516 --> 00:38:16,140
I'm just going to
write the expressions
785
00:38:16,140 --> 00:38:17,790
and just get them
out of the way.
786
00:38:17,790 --> 00:38:20,900
So the average value of
some x, given a probability
787
00:38:20,900 --> 00:38:23,220
distribution on x where x
is a continuous variable,
788
00:38:23,220 --> 00:38:24,952
is going to be equal
to the integral.
789
00:38:24,952 --> 00:38:26,910
Let's just say x is
defined from minus infinity
790
00:38:26,910 --> 00:38:31,500
to infinity, which is pretty
useful, or pretty typical.
791
00:38:31,500 --> 00:38:37,644
dx probability
distribution of x times x.
792
00:38:37,644 --> 00:38:38,560
I shouldn't use curvy.
793
00:38:38,560 --> 00:38:39,620
I should just use x.
794
00:38:42,150 --> 00:38:44,960
And similarly for
x squared, or more
795
00:38:44,960 --> 00:38:47,920
generally, for f of x,
the average value of f
796
00:38:47,920 --> 00:38:52,050
of x, or the expected value of
f of x given this probability
797
00:38:52,050 --> 00:38:54,900
distribution, is going to
be equal to the integral dx
798
00:38:54,900 --> 00:38:56,960
minus infinity to infinity.
799
00:38:56,960 --> 00:39:01,124
The probability distribution
of x times f of x.
800
00:39:01,124 --> 00:39:02,790
In direct analogy to
what we had before.
801
00:39:05,890 --> 00:39:08,850
So this is all just mathematics.
802
00:39:08,850 --> 00:39:11,620
And we define the
uncertainty in x
803
00:39:11,620 --> 00:39:15,940
is equal to the expectation
value of x squared
804
00:39:15,940 --> 00:39:20,155
minus the expected value
of x quantity squared.
805
00:39:23,560 --> 00:39:24,760
And this is delta x squared.
806
00:39:27,430 --> 00:39:30,210
If you see me dropping an
exponent or a factor of 2,
807
00:39:30,210 --> 00:39:33,110
please, please, please tell me.
808
00:39:33,110 --> 00:39:36,060
So thank you for that.
809
00:39:36,060 --> 00:39:39,720
All of that is just straight up
classical probability theory.
810
00:39:39,720 --> 00:39:42,750
And I just want to write this
in the notation of quantum
811
00:39:42,750 --> 00:39:43,660
mechanics.
812
00:39:43,660 --> 00:39:45,850
Given that the
system is in a state
813
00:39:45,850 --> 00:39:50,210
described by the wave function
psi of x, the average value,
814
00:39:50,210 --> 00:39:53,560
the expected value of x, the
typical value if you just
815
00:39:53,560 --> 00:39:56,160
observe the particle
at some moment,
816
00:39:56,160 --> 00:40:01,200
is equal to the integral over
all possible values of x.
817
00:40:01,200 --> 00:40:05,790
The probability distribution,
psi of x norm squared x.
818
00:40:08,480 --> 00:40:12,480
And similarly, for
any function of x,
819
00:40:12,480 --> 00:40:15,250
the expected value is going to
be equal to the integral dx.
820
00:40:15,250 --> 00:40:17,583
The probability distribution,
which is given by the norm
821
00:40:17,583 --> 00:40:23,710
squared of the wave function
times f of x minus infinity
822
00:40:23,710 --> 00:40:26,140
to infinity.
823
00:40:26,140 --> 00:40:30,010
And same definition
for uncertainty.
824
00:40:30,010 --> 00:40:33,070
And again, this notation
is really dangerous,
825
00:40:33,070 --> 00:40:37,010
because the expected value of
x depends on the probability
826
00:40:37,010 --> 00:40:37,790
distribution.
827
00:40:37,790 --> 00:40:39,985
In a physical system,
the expected value of x
828
00:40:39,985 --> 00:40:41,610
depends on what the
state of the system
829
00:40:41,610 --> 00:40:43,610
is, what the wave function
is, and this notation
830
00:40:43,610 --> 00:40:44,970
doesn't indicate that.
831
00:40:44,970 --> 00:40:47,450
So there are a couple of ways
to improve this notation.
832
00:40:47,450 --> 00:40:52,540
One of which is-- so this is,
again, a sort of side note.
833
00:40:52,540 --> 00:40:54,810
One way to improve
this notation x
834
00:40:54,810 --> 00:40:59,450
is to write the expected
value of x in the state psi,
835
00:40:59,450 --> 00:41:01,036
so you write psi as a subscript.
836
00:41:01,036 --> 00:41:02,910
Another notation that
will come back-- you'll
837
00:41:02,910 --> 00:41:05,040
see why this is a
useful notation later
838
00:41:05,040 --> 00:41:10,511
in the semester-- is
this notation, psi.
839
00:41:10,511 --> 00:41:12,510
And we will give meaning
to this notation later,
840
00:41:12,510 --> 00:41:13,610
but I just want to
alert you that it's
841
00:41:13,610 --> 00:41:15,690
used throughout books, and
it means the same thing
842
00:41:15,690 --> 00:41:17,814
as what we're talking about
the expected value of x
843
00:41:17,814 --> 00:41:19,880
given a particular state psi.
844
00:41:19,880 --> 00:41:20,640
OK?
845
00:41:20,640 --> 00:41:21,140
Yeah?
846
00:41:21,140 --> 00:41:23,480
AUDIENCE: To calculate the
expected value of momentum
847
00:41:23,480 --> 00:41:25,630
do you need to transform the--
848
00:41:25,630 --> 00:41:26,880
PROFESSOR: Excellent question.
849
00:41:26,880 --> 00:41:27,840
Excellent, excellent question.
850
00:41:27,840 --> 00:41:29,290
OK, so the question
is, how do we
851
00:41:29,290 --> 00:41:30,961
do the same thing for momentum?
852
00:41:30,961 --> 00:41:33,210
If you want to compute the
expected value of momentum,
853
00:41:33,210 --> 00:41:34,170
what do you have to do?
854
00:41:34,170 --> 00:41:36,790
Do you have to do some Fourier
transform to the wave function?
855
00:41:36,790 --> 00:41:39,830
So this is a
question that you're
856
00:41:39,830 --> 00:41:41,360
going to answer
on the problem set
857
00:41:41,360 --> 00:41:43,334
and that we made a
guess for last time.
858
00:41:43,334 --> 00:41:45,000
But quickly, let's
just think about what
859
00:41:45,000 --> 00:41:46,820
it's going to be
purely formally.
860
00:41:46,820 --> 00:41:50,540
Formally, if we want to know the
likely value of the momentum,
861
00:41:50,540 --> 00:41:53,210
the likely value the momentum,
it's a continuous variable.
862
00:41:53,210 --> 00:41:55,400
Just like any other
observable variable,
863
00:41:55,400 --> 00:41:58,700
we can write as the integral
over all possible values
864
00:41:58,700 --> 00:42:01,270
of momentum from,
let's say, it could
865
00:42:01,270 --> 00:42:03,980
be minus infinity to infinity.
866
00:42:03,980 --> 00:42:09,141
The probability of having that
momentum times momentum, right?
867
00:42:09,141 --> 00:42:10,140
Everyone cool with that?
868
00:42:10,140 --> 00:42:11,590
This is a tautology, right?
869
00:42:11,590 --> 00:42:14,930
This is what you
mean by probability.
870
00:42:14,930 --> 00:42:17,600
But we need to know if we have
a quantum mechanical system
871
00:42:17,600 --> 00:42:19,830
described by state
psi of x, how do
872
00:42:19,830 --> 00:42:23,066
we can get the probability
that you measure p?
873
00:42:23,066 --> 00:42:25,096
Do I want to do this now?
874
00:42:25,096 --> 00:42:27,430
Yeah, OK I do.
875
00:42:27,430 --> 00:42:30,780
And we need a guess.
876
00:42:30,780 --> 00:42:31,391
Question mark.
877
00:42:31,391 --> 00:42:33,140
We made a guess at the
end of last lecture
878
00:42:33,140 --> 00:42:34,765
that, in quantum
mechanics, this should
879
00:42:34,765 --> 00:42:41,820
be dp minus infinity to infinity
of the Fourier transform.
880
00:42:41,820 --> 00:42:46,700
Psi tilde of p up
to an h bar factor.
881
00:42:46,700 --> 00:42:53,160
Psi tilde of p, the Fourier
transform p norm squared.
882
00:42:53,160 --> 00:42:56,730
OK, so we're guessing that the
Fourier transform norm squared
883
00:42:56,730 --> 00:42:59,190
is equal to the probability
of measuring the associated
884
00:42:59,190 --> 00:43:00,850
momentum.
885
00:43:00,850 --> 00:43:03,370
So that's a guess.
886
00:43:03,370 --> 00:43:04,090
That's a guess.
887
00:43:04,090 --> 00:43:06,400
And so on your problem set
you're going to prove it.
888
00:43:06,400 --> 00:43:07,220
OK?
889
00:43:07,220 --> 00:43:08,940
So exactly the same
logic goes through.
890
00:43:08,940 --> 00:43:10,050
It's a very good
question, thanks.
891
00:43:10,050 --> 00:43:10,850
Other questions?
892
00:43:10,850 --> 00:43:12,698
Yeah?
893
00:43:12,698 --> 00:43:14,674
AUDIENCE: Is that p
the momentum itself?
894
00:43:14,674 --> 00:43:17,150
Or is that the probability?
895
00:43:17,150 --> 00:43:19,100
PROFESSOR: So this
is the probability
896
00:43:19,100 --> 00:43:20,990
of measuring momentum p.
897
00:43:20,990 --> 00:43:22,770
And that's the value p.
898
00:43:22,770 --> 00:43:23,980
We're summing over all p's.
899
00:43:23,980 --> 00:43:27,660
This is the probability,
and that's actually p.
900
00:43:27,660 --> 00:43:29,351
So the Fourier
transform is a function
901
00:43:29,351 --> 00:43:31,600
of the momentum in the same
way that the wave function
902
00:43:31,600 --> 00:43:34,380
is a function of
the position, right?
903
00:43:34,380 --> 00:43:36,080
So this is a function
of the momentum.
904
00:43:36,080 --> 00:43:39,917
It's norm squared
defines the probability.
905
00:43:39,917 --> 00:43:41,500
And then the p on
the right is this p,
906
00:43:41,500 --> 00:43:43,500
because we're computing
the expected value of p,
907
00:43:43,500 --> 00:43:44,640
or the average value of p.
908
00:43:44,640 --> 00:43:46,100
That make sense?
909
00:43:46,100 --> 00:43:47,530
Cool.
910
00:43:47,530 --> 00:43:48,426
Yeah?
911
00:43:48,426 --> 00:43:50,590
AUDIENCE: Are we then
multiplying by p squared
912
00:43:50,590 --> 00:43:52,027
if we're doing all p's?
913
00:43:52,027 --> 00:43:56,340
Because we have the dp times
p for each [INAUDIBLE].
914
00:43:56,340 --> 00:43:56,960
PROFESSOR: No.
915
00:43:56,960 --> 00:43:58,100
So that's a very good question.
916
00:43:58,100 --> 00:43:58,920
So let's go back.
917
00:43:58,920 --> 00:44:00,490
Very good question.
918
00:44:00,490 --> 00:44:02,470
Let me phrase it in
terms of position,
919
00:44:02,470 --> 00:44:03,430
because the same
question comes up.
920
00:44:03,430 --> 00:44:04,580
Thank you for asking that.
921
00:44:04,580 --> 00:44:05,250
Look at this.
922
00:44:05,250 --> 00:44:06,070
This is weird.
923
00:44:06,070 --> 00:44:08,570
I'm going to phrase this as a
dimensional analysis question.
924
00:44:08,570 --> 00:44:10,520
Tell me if this is the same
question as you're asking.
925
00:44:10,520 --> 00:44:12,310
This is a thing with
dimensions of what?
926
00:44:12,310 --> 00:44:13,542
Length, right?
927
00:44:13,542 --> 00:44:15,000
But over on the
right hand side, we
928
00:44:15,000 --> 00:44:19,305
have a length and a probability,
which is a number, and then
929
00:44:19,305 --> 00:44:19,930
another length.
930
00:44:19,930 --> 00:44:21,940
That looks like
x squared, right?
931
00:44:21,940 --> 00:44:23,570
So why are we getting something
with dimensions of length,
932
00:44:23,570 --> 00:44:25,610
not something with
dimensions of length squared?
933
00:44:25,610 --> 00:44:27,443
And the answer is this
is not a probability.
934
00:44:27,443 --> 00:44:29,970
It is a probability density.
935
00:44:29,970 --> 00:44:34,180
So it's got units of
probability per unit length.
936
00:44:34,180 --> 00:44:36,240
So this has dimensions
of one over length.
937
00:44:36,240 --> 00:44:39,380
So this quantity,
p of x dx, tells me
938
00:44:39,380 --> 00:44:41,930
the probability, which is a
pure number, no dimensions.
939
00:44:41,930 --> 00:44:44,690
The probability to find the
particle between x and x
940
00:44:44,690 --> 00:44:46,550
plus dx.
941
00:44:46,550 --> 00:44:47,110
Cool?
942
00:44:47,110 --> 00:44:50,100
So that was our
second postulate.
943
00:44:50,100 --> 00:44:52,455
Psi of x dx squared
is the probability
944
00:44:52,455 --> 00:44:54,640
of finding it in this domain.
945
00:44:54,640 --> 00:44:59,040
And so what we're doing is we're
summing over all such domains
946
00:44:59,040 --> 00:45:02,610
the probability times the value.
947
00:45:02,610 --> 00:45:03,490
Cool?
948
00:45:03,490 --> 00:45:05,730
So this is the difference
between discrete,
949
00:45:05,730 --> 00:45:09,115
where we didn't have these
probability densities,
950
00:45:09,115 --> 00:45:11,490
we just had numbers, pure
numbers and pure probabilities.
951
00:45:11,490 --> 00:45:14,776
Now we have probability
densities per unit whatever.
952
00:45:14,776 --> 00:45:15,405
Yeah?
953
00:45:15,405 --> 00:45:17,405
AUDIENCE: How do you
pronounce the last notation
954
00:45:17,405 --> 00:45:18,707
that you wrote?
955
00:45:18,707 --> 00:45:20,040
PROFESSOR: How do you pronounce?
956
00:45:20,040 --> 00:45:21,248
Good, that's a good question.
957
00:45:21,248 --> 00:45:23,500
The question is, how do
we pronounce these things.
958
00:45:23,500 --> 00:45:25,160
So this is called
the expected value
959
00:45:25,160 --> 00:45:28,190
of x, or the average value of
x, or most typically in quantum
960
00:45:28,190 --> 00:45:31,130
mechanics, the
expectation value of x.
961
00:45:31,130 --> 00:45:33,160
So you can call it
anything you want.
962
00:45:33,160 --> 00:45:34,830
This is the same thing.
963
00:45:34,830 --> 00:45:38,029
The psi is just to denote
that this is in the state psi.
964
00:45:38,029 --> 00:45:39,570
And it can be
pronounced in two ways.
965
00:45:39,570 --> 00:45:41,236
You can either say
the expectation value
966
00:45:41,236 --> 00:45:45,600
of x, or the expectation
of x in the state psi.
967
00:45:45,600 --> 00:45:48,610
And this would be
pronounced one of two ways.
968
00:45:48,610 --> 00:45:55,451
The expectation value of x in
the state psi, or psi x psi.
969
00:45:55,451 --> 00:45:55,950
Yeah.
970
00:45:55,950 --> 00:45:58,030
That's a very good question.
971
00:45:58,030 --> 00:45:59,240
But they mean the same thing.
972
00:45:59,240 --> 00:46:02,812
Now, I should emphasize that you
can have two ways of describing
973
00:46:02,812 --> 00:46:04,270
something that mean
the same thing,
974
00:46:04,270 --> 00:46:06,144
but they carry different
connotations, right?
975
00:46:08,500 --> 00:46:12,727
Like have a friend
who's a really nice guy.
976
00:46:12,727 --> 00:46:13,310
He's a mensch.
977
00:46:13,310 --> 00:46:14,310
He's a good guy.
978
00:46:14,310 --> 00:46:16,020
And so I could see
he's a nice guy,
979
00:46:16,020 --> 00:46:18,200
I could say he's
[? carinoso ?], and they
980
00:46:18,200 --> 00:46:21,490
mean different things
in different languages.
981
00:46:21,490 --> 00:46:24,500
It's the same idea, but they
have different flavors, right?
982
00:46:24,500 --> 00:46:27,310
So whatever your
native language is,
983
00:46:27,310 --> 00:46:29,082
you've got some analog of this.
984
00:46:29,082 --> 00:46:32,045
This means something in
a particular mathematical
985
00:46:32,045 --> 00:46:33,920
language for talking
about quantum mechanics.
986
00:46:33,920 --> 00:46:35,450
And this has a different flavor.
987
00:46:35,450 --> 00:46:36,896
It carries different
implications,
988
00:46:36,896 --> 00:46:38,270
and we'll see what
that is later.
989
00:46:38,270 --> 00:46:39,120
We haven't got there yet.
990
00:46:39,120 --> 00:46:39,620
Yeah?
991
00:46:39,620 --> 00:46:41,890
AUDIENCE: Why is there a
double notation of psi?
992
00:46:41,890 --> 00:46:44,460
PROFESSOR: Why is there
a double notation of psi?
993
00:46:44,460 --> 00:46:47,830
Yeah, we'll see later.
994
00:46:47,830 --> 00:46:50,530
Roughly speaking, it's because
in computing this expectation
995
00:46:50,530 --> 00:46:52,480
value, there's a psi squared.
996
00:46:52,480 --> 00:46:55,940
And so this is to
remind you of that.
997
00:46:55,940 --> 00:46:58,010
Other questions?
998
00:46:58,010 --> 00:47:00,060
Terminology is one of the
most annoying features
999
00:47:00,060 --> 00:47:00,640
of quantum mechanics.
1000
00:47:00,640 --> 00:47:01,139
Yeah?
1001
00:47:01,139 --> 00:47:04,435
AUDIENCE: So it seems like
this [INAUDIBLE] variance
1002
00:47:04,435 --> 00:47:07,301
is a really convenient
way of doing it.
1003
00:47:07,301 --> 00:47:08,800
How is it the
Heisenberg uncertainty
1004
00:47:08,800 --> 00:47:13,480
works exactly as it does for
this definition of variance.
1005
00:47:13,480 --> 00:47:15,105
PROFESSOR: That's a
very good question.
1006
00:47:17,285 --> 00:47:18,660
In order to answer
that question,
1007
00:47:18,660 --> 00:47:20,440
we need to actually work out
the Heisenberg uncertainty
1008
00:47:20,440 --> 00:47:20,940
relation.
1009
00:47:20,940 --> 00:47:25,182
So the question is, look, this
is some choice of uncertainty.
1010
00:47:25,182 --> 00:47:27,640
You could have chosen some
other definition of uncertainly.
1011
00:47:27,640 --> 00:47:29,590
We could have considered
the expectation
1012
00:47:29,590 --> 00:47:32,230
value of x to the fourth
minus x to the fourth
1013
00:47:32,230 --> 00:47:33,920
and taken the
fourth root of that.
1014
00:47:33,920 --> 00:47:36,100
So why this one?
1015
00:47:36,100 --> 00:47:38,460
And one answer is, indeed,
the uncertainty relation
1016
00:47:38,460 --> 00:47:40,050
works out quite nicely.
1017
00:47:40,050 --> 00:47:43,170
But then I think
important to say
1018
00:47:43,170 --> 00:47:45,837
here is that there are many ways
you could construct quantities.
1019
00:47:45,837 --> 00:47:47,378
This is a convenient
one, and we will
1020
00:47:47,378 --> 00:47:49,530
discover that it has nice
properties that we like.
1021
00:47:49,530 --> 00:47:53,662
There is no God given reason why
this had to be the right thing.
1022
00:47:53,662 --> 00:47:56,120
I can say more, but I don't
want to take the time to do it,
1023
00:47:56,120 --> 00:47:58,560
so ask in office hours.
1024
00:47:58,560 --> 00:48:01,180
OK, good.
1025
00:48:01,180 --> 00:48:02,680
The second part of
your question was
1026
00:48:02,680 --> 00:48:04,690
why does the Heisenberg
relation work out
1027
00:48:04,690 --> 00:48:05,750
nicely in terms of
these guys, and we
1028
00:48:05,750 --> 00:48:07,400
will study that in
extraordinary detail.
1029
00:48:07,400 --> 00:48:08,025
We'll see that.
1030
00:48:08,025 --> 00:48:11,240
So we're going to derive it
twice soon and then later.
1031
00:48:11,240 --> 00:48:12,690
The later version is better.
1032
00:48:12,690 --> 00:48:14,115
So let me work
out some examples.
1033
00:48:17,330 --> 00:48:21,380
Or actually, I'm going
to skip the examples
1034
00:48:21,380 --> 00:48:22,740
in the interest of time.
1035
00:48:22,740 --> 00:48:23,850
They're in the
notes, and so they'll
1036
00:48:23,850 --> 00:48:24,933
be posted on the web page.
1037
00:48:24,933 --> 00:48:28,341
By the way, the first 18
lectures of notes are posted.
1038
00:48:28,341 --> 00:48:29,590
I had a busy night last night.
1039
00:48:32,600 --> 00:48:34,547
So let's come back to
computing expectation
1040
00:48:34,547 --> 00:48:35,380
values for momentum.
1041
00:48:46,020 --> 00:48:48,000
So I want to go
back to this and ask
1042
00:48:48,000 --> 00:48:51,230
a silly-- I want to make some
progress towards deriving
1043
00:48:51,230 --> 00:48:51,910
this relation.
1044
00:48:51,910 --> 00:48:55,070
So I want to start over on
the definition of the expected
1045
00:48:55,070 --> 00:48:55,820
value of momentum.
1046
00:48:58,119 --> 00:49:00,660
And I'd like to do it directly
in terms of the wave function.
1047
00:49:00,660 --> 00:49:01,990
So how would we do this?
1048
00:49:01,990 --> 00:49:04,900
So one way of saying this is
what's the average value of p.
1049
00:49:04,900 --> 00:49:06,910
Well, I can phrase this
in terms of the wave
1050
00:49:06,910 --> 00:49:08,035
function the following way.
1051
00:49:08,035 --> 00:49:10,630
I'm going to sum over
all positions dx.
1052
00:49:10,630 --> 00:49:15,140
Expectation value of x
squared from minus infinity
1053
00:49:15,140 --> 00:49:16,620
to infinity.
1054
00:49:16,620 --> 00:49:19,065
And then the momentum
associated to the value x.
1055
00:49:23,287 --> 00:49:25,370
So it's tempting to write
something like this down
1056
00:49:25,370 --> 00:49:26,950
to think maybe
there's some p of x.
1057
00:49:29,476 --> 00:49:31,100
This is a tempting
thing to write down.
1058
00:49:33,700 --> 00:49:34,200
Can we?
1059
00:49:40,490 --> 00:49:46,490
Are we ever in a position
to say intelligently
1060
00:49:46,490 --> 00:49:50,010
that a particle--
that an electron
1061
00:49:50,010 --> 00:49:55,094
is both hard and white?
1062
00:49:55,094 --> 00:49:56,010
AUDIENCE: No.
1063
00:49:56,010 --> 00:49:58,060
PROFESSOR: No,
because being hard
1064
00:49:58,060 --> 00:50:01,676
is a superposition of being
black and white, right?
1065
00:50:01,676 --> 00:50:06,210
Are we ever in a position
to say that our particle has
1066
00:50:06,210 --> 00:50:08,850
a definite position
x and correspondingly
1067
00:50:08,850 --> 00:50:11,450
a definite momentum p.
1068
00:50:11,450 --> 00:50:12,930
It's not that we don't get too.
1069
00:50:12,930 --> 00:50:15,664
It's that it doesn't
make sense to do so.
1070
00:50:15,664 --> 00:50:17,330
In general, being in
a definite position
1071
00:50:17,330 --> 00:50:21,230
means being in a
superposition of having
1072
00:50:21,230 --> 00:50:24,520
different values for momentum.
1073
00:50:24,520 --> 00:50:27,100
And if you want a sharp
way of saying this,
1074
00:50:27,100 --> 00:50:29,065
look at these relations.
1075
00:50:29,065 --> 00:50:32,730
They claim that any
function can be expressed
1076
00:50:32,730 --> 00:50:37,040
as a superposition of states
with definite momentum, right?
1077
00:50:37,040 --> 00:50:41,880
Well, among other things a
state with definite position,
1078
00:50:41,880 --> 00:50:46,990
x0, can be written
as a superposition, 1
1079
00:50:46,990 --> 00:50:52,100
over 2pi integral dk.
1080
00:50:52,100 --> 00:50:56,659
I'll call this delta tilde of k.
1081
00:50:56,659 --> 00:50:57,200
e to the ikx.
1082
00:51:05,009 --> 00:51:07,050
If you haven't played with
delta functions before
1083
00:51:07,050 --> 00:51:09,655
and you haven't seen this,
then you will on the problem
1084
00:51:09,655 --> 00:51:11,230
set, because we
have a problem that
1085
00:51:11,230 --> 00:51:13,020
works through a
great many details.
1086
00:51:13,020 --> 00:51:17,840
But in particular, it's
clear that this is not--
1087
00:51:17,840 --> 00:51:20,672
this quantity can't be
a delta function of k,
1088
00:51:20,672 --> 00:51:22,880
because, if it were, this
would be just e to the ikx.
1089
00:51:22,880 --> 00:51:25,060
And that's definitely
not a delta function.
1090
00:51:25,060 --> 00:51:29,910
Meanwhile, what can you say
about the continuity structure
1091
00:51:29,910 --> 00:51:31,060
of a delta function.
1092
00:51:31,060 --> 00:51:33,300
Is it continuous?
1093
00:51:33,300 --> 00:51:33,960
No.
1094
00:51:33,960 --> 00:51:35,364
Its derivative isn't continuous.
1095
00:51:35,364 --> 00:51:36,280
Its second derivative.
1096
00:51:36,280 --> 00:51:38,363
None of its derivatives
are in any way continuous.
1097
00:51:38,363 --> 00:51:40,879
They're all absolutely
horrible, OK?
1098
00:51:40,879 --> 00:51:42,420
So how many momentum
modes am I going
1099
00:51:42,420 --> 00:51:44,919
to need to superimpose in order
to reproduce a function that
1100
00:51:44,919 --> 00:51:47,900
has this sort of structure?
1101
00:51:47,900 --> 00:51:48,790
An infinite number.
1102
00:51:48,790 --> 00:51:49,650
And it turns out
it's going to be
1103
00:51:49,650 --> 00:51:51,970
an infinite number with the
same amplitude, slightly
1104
00:51:51,970 --> 00:51:54,420
different phase, OK?
1105
00:51:54,420 --> 00:51:57,885
So you can never say
that you're in a state
1106
00:51:57,885 --> 00:51:59,760
with definite position
and definite momentum.
1107
00:51:59,760 --> 00:52:01,400
Being in a state with
definite position
1108
00:52:01,400 --> 00:52:07,210
means being in a superposition
of being in a superposition.
1109
00:52:07,210 --> 00:52:11,570
In fact, I'm just going
right down the answer here.
1110
00:52:11,570 --> 00:52:12,702
e to the ikx0.
1111
00:52:17,030 --> 00:52:21,060
Being in a state with
definite position
1112
00:52:21,060 --> 00:52:22,850
means being in a
superposition of states
1113
00:52:22,850 --> 00:52:26,420
with arbitrary momentum
and vice versa.
1114
00:52:26,420 --> 00:52:28,651
You cannot be in a state
with definite position,
1115
00:52:28,651 --> 00:52:29,400
definite momentum.
1116
00:52:29,400 --> 00:52:30,274
So this doesn't work.
1117
00:52:30,274 --> 00:52:33,280
So what we want is we
want some good definition.
1118
00:52:33,280 --> 00:52:34,780
So this does not work.
1119
00:52:34,780 --> 00:52:37,500
We want some good
definition of p
1120
00:52:37,500 --> 00:52:40,950
given that we're working
with a wave function which
1121
00:52:40,950 --> 00:52:43,500
is a function of x.
1122
00:52:43,500 --> 00:52:45,415
What is that good
definition of the momentum?
1123
00:52:51,310 --> 00:52:54,560
We have a couple of hints.
1124
00:52:54,560 --> 00:52:55,930
So hint the first.
1125
00:53:02,860 --> 00:53:04,810
So this is what we're after.
1126
00:53:04,810 --> 00:53:06,520
Hint the first is
that a wave-- we
1127
00:53:06,520 --> 00:53:09,950
know that given a wave with wave
number k, which is equal 2pi
1128
00:53:09,950 --> 00:53:15,470
over lambda, is associated,
according to de Broglie
1129
00:53:15,470 --> 00:53:19,460
and according to
Davisson-Germer experiments,
1130
00:53:19,460 --> 00:53:22,070
to a particle-- so
having a particle--
1131
00:53:22,070 --> 00:53:25,280
a wave, with wave number k or
wavelength lambda associated
1132
00:53:25,280 --> 00:53:29,590
particle with momentum
p is equal to h bar k.
1133
00:53:29,590 --> 00:53:31,670
Yeah?
1134
00:53:31,670 --> 00:53:34,540
But in particular, what is a
plane with wavelength lambda
1135
00:53:34,540 --> 00:53:36,570
or wave number k look like?
1136
00:53:36,570 --> 00:53:39,500
That's e to the iks.
1137
00:53:39,500 --> 00:53:41,890
And if I have a
wave, a plane wave
1138
00:53:41,890 --> 00:53:45,190
e to the iks, how do I
get h bar k out of it?
1139
00:53:52,650 --> 00:53:58,504
Note the following, the
derivative with respect to x.
1140
00:53:58,504 --> 00:53:59,920
Actually let me
do this down here.
1141
00:54:08,060 --> 00:54:12,910
Note that the derivative with
respect to x of e to the ikx
1142
00:54:12,910 --> 00:54:17,230
is equal to ik e to the ikx.
1143
00:54:24,030 --> 00:54:27,050
There's nothing up my sleeves.
1144
00:54:27,050 --> 00:54:30,140
So in particular, if
I want to get h bar k,
1145
00:54:30,140 --> 00:54:32,820
I can multiply by h
bar and divide by i.
1146
00:54:32,820 --> 00:54:35,510
Multiply by h bar, divide by
i, derivative with respect
1147
00:54:35,510 --> 00:54:39,140
to x e to the ikx.
1148
00:54:39,140 --> 00:54:42,490
And this is equal to
h bar k e to the ikx.
1149
00:54:45,160 --> 00:54:45,960
That's suggestive.
1150
00:54:49,200 --> 00:54:53,330
And I can write this
as p e to the ikx.
1151
00:55:00,620 --> 00:55:02,196
So let's quickly
check the units.
1152
00:55:05,170 --> 00:55:08,740
So first off, what are
the units of h bar?
1153
00:55:08,740 --> 00:55:10,130
Here's the super
easy to remember
1154
00:55:10,130 --> 00:55:13,450
the units of-- or
dimensions of h bar are.
1155
00:55:13,450 --> 00:55:18,550
Delta x delta p is h bar.
1156
00:55:18,550 --> 00:55:19,050
OK?
1157
00:55:19,050 --> 00:55:21,300
If you're ever in doubt,
if you just remember,
1158
00:55:21,300 --> 00:55:24,852
h bar has units of
momentum times length.
1159
00:55:24,852 --> 00:55:26,560
It's just the easiest
way to remember it.
1160
00:55:26,560 --> 00:55:27,893
You'll never forget it that way.
1161
00:55:27,893 --> 00:55:31,070
So if h bar has units of
momentum times length,
1162
00:55:31,070 --> 00:55:33,130
what are the units of k?
1163
00:55:33,130 --> 00:55:33,850
1 over length.
1164
00:55:33,850 --> 00:55:35,433
So does this
dimensionally make sense?
1165
00:55:35,433 --> 00:55:36,170
Yeah.
1166
00:55:36,170 --> 00:55:40,630
Momentum times length divided
by length number momentum.
1167
00:55:40,630 --> 00:55:41,130
Good.
1168
00:55:41,130 --> 00:55:42,781
So dimensionally we
haven't lied yet.
1169
00:55:46,000 --> 00:55:48,250
So this makes it tempting
to say something like, well,
1170
00:55:48,250 --> 00:55:51,530
hell h bar upon i
derivative with respect
1171
00:55:51,530 --> 00:55:57,425
to x is equal in some-- question
mark, quotation mark-- p.
1172
00:56:01,110 --> 00:56:03,510
Right?
1173
00:56:03,510 --> 00:56:06,250
So at this point it's just
tempting to say, look, trust
1174
00:56:06,250 --> 00:56:07,915
me, p is h bar upon idx.
1175
00:56:07,915 --> 00:56:09,290
But I don't know
about you, but I
1176
00:56:09,290 --> 00:56:11,910
find that deeply,
deeply unsatisfying.
1177
00:56:11,910 --> 00:56:14,540
So let me ask the question
slightly differently.
1178
00:56:14,540 --> 00:56:16,580
We've followed the
de Broglie relations,
1179
00:56:16,580 --> 00:56:18,586
and we've been led
to the idea that
1180
00:56:18,586 --> 00:56:19,960
using wave functions
that there's
1181
00:56:19,960 --> 00:56:22,400
some relationship
between the momentum,
1182
00:56:22,400 --> 00:56:24,850
the observable quantity that
you measure with sticks,
1183
00:56:24,850 --> 00:56:28,450
and meters, and stuff, and this
operator, this differential
1184
00:56:28,450 --> 00:56:32,170
operator, h bar upon on i
derivative with respect to x.
1185
00:56:32,170 --> 00:56:36,560
By the way, my notation for
dx is the partial derivative
1186
00:56:36,560 --> 00:56:38,652
with respect to x.
1187
00:56:38,652 --> 00:56:39,235
Just notation.
1188
00:56:42,350 --> 00:56:45,670
So if this is supposed
to be true in some sense,
1189
00:56:45,670 --> 00:56:49,590
what is momentum have
to do with a derivative?
1190
00:56:49,590 --> 00:56:51,090
Momentum is about
velocities, which
1191
00:56:51,090 --> 00:56:53,090
is like derivatives with
respect to time, right?
1192
00:56:53,090 --> 00:56:53,860
Times mass.
1193
00:56:53,860 --> 00:56:56,180
Mass times derivative with
respect to time, velocity.
1194
00:56:56,180 --> 00:56:57,960
So what does it have to do with
the derivative with respect
1195
00:56:57,960 --> 00:56:58,735
to position?
1196
00:57:02,480 --> 00:57:07,550
And this ties into the
most beautiful theorem
1197
00:57:07,550 --> 00:57:13,627
in classical mechanics, which
is the Noether's theorem, named
1198
00:57:13,627 --> 00:57:15,960
after the mathematician who
discovered it, Emmy Noether.
1199
00:57:19,070 --> 00:57:22,870
And just out of
curiosity, how many people
1200
00:57:22,870 --> 00:57:25,626
have seen Noether's
theorem in class.
1201
00:57:25,626 --> 00:57:26,970
Oh that's so sad.
1202
00:57:26,970 --> 00:57:27,640
That's a sin.
1203
00:57:27,640 --> 00:57:31,212
OK, so here's a statement
of Noether's theorem,
1204
00:57:31,212 --> 00:57:33,670
and it underlies an enormous
amount of classical mechanics,
1205
00:57:33,670 --> 00:57:34,920
but also of quantum mechanics.
1206
00:57:37,839 --> 00:57:39,630
Noether, incidentally,
was a mathematician.
1207
00:57:39,630 --> 00:57:41,899
There's a whole wonderful
story about Emmy Noether.
1208
00:57:41,899 --> 00:57:43,440
Ville went to her
and was like, look,
1209
00:57:43,440 --> 00:57:45,690
I'm trying to understand
the notion of energy.
1210
00:57:45,690 --> 00:57:48,130
And this guy down the
hall, Einstein, he
1211
00:57:48,130 --> 00:57:50,660
has a theory called general
relativity about curved space
1212
00:57:50,660 --> 00:57:52,650
times and how that has
something to do with gravity.
1213
00:57:52,650 --> 00:57:53,246
But it doesn't
make a lot of sense
1214
00:57:53,246 --> 00:57:54,870
to me, because I don't even
know how to define the energy.
1215
00:57:54,870 --> 00:57:56,620
So how do you define
momentum and energy
1216
00:57:56,620 --> 00:57:59,860
in this guy's crazy theory?
1217
00:57:59,860 --> 00:58:02,390
And so Noether, who
was a mathematician,
1218
00:58:02,390 --> 00:58:05,050
did all sorts of beautiful
stuff in algebra,
1219
00:58:05,050 --> 00:58:06,080
looked at the problem
and was like I don't even
1220
00:58:06,080 --> 00:58:07,450
know what it means in
classical mechanics.
1221
00:58:07,450 --> 00:58:08,835
So what is a mean in
classical mechanics?
1222
00:58:08,835 --> 00:58:09,970
So she went back to
classical mechanics
1223
00:58:09,970 --> 00:58:11,520
and, from first
principles, came up
1224
00:58:11,520 --> 00:58:13,680
with a good definition of
momentum, which turns out
1225
00:58:13,680 --> 00:58:16,190
to underlie the modern idea
of conserved quantities
1226
00:58:16,190 --> 00:58:17,250
and symmetries.
1227
00:58:17,250 --> 00:58:21,210
And it's had enormous
far reaching impact,
1228
00:58:21,210 --> 00:58:24,430
and say her name would praise.
1229
00:58:24,430 --> 00:58:28,780
So Noether tells us the
following statement,
1230
00:58:28,780 --> 00:58:39,230
to every symmetry-- and I
should say continuous symmetry--
1231
00:58:39,230 --> 00:58:46,060
to every symmetry is associated
a conserved quantity.
1232
00:58:52,590 --> 00:58:53,310
OK?
1233
00:58:53,310 --> 00:58:54,920
So in particular, what
do I mean by symmetry?
1234
00:58:54,920 --> 00:58:56,253
Well, for example, translations.
1235
00:58:56,253 --> 00:58:59,530
x goes to x plus some length l.
1236
00:58:59,530 --> 00:59:01,280
This could be done for
arbitrary length l.
1237
00:59:01,280 --> 00:59:03,571
So for example, translation
by this much or translation
1238
00:59:03,571 --> 00:59:04,150
by that much.
1239
00:59:04,150 --> 00:59:05,440
These are translations.
1240
00:59:05,440 --> 00:59:08,870
To every symmetry is associated
a conserved quantity.
1241
00:59:08,870 --> 00:59:11,165
What symmetry is
associated to translations?
1242
00:59:16,270 --> 00:59:18,142
Conservation of momentum, p dot.
1243
00:59:20,980 --> 00:59:27,910
Time translations, t
goes to t plus capital
1244
00:59:27,910 --> 00:59:30,790
T. What's a conserved
quantity associated
1245
00:59:30,790 --> 00:59:32,660
with time
translational symmetry?
1246
00:59:32,660 --> 00:59:35,230
Energy, which is
time independent.
1247
00:59:38,270 --> 00:59:40,290
And rotations.
1248
00:59:40,290 --> 00:59:41,280
Rotational symmetries.
1249
00:59:45,350 --> 00:59:51,680
x, as a vector, goes to
some rotation times x.
1250
00:59:54,540 --> 00:59:56,875
What's conserved by virtue
of rotational symmetry?
1251
00:59:56,875 --> 00:59:58,000
AUDIENCE: Angular momentum.
1252
00:59:58,000 --> 00:59:58,970
PROFESSOR: Angular momentum.
1253
00:59:58,970 --> 00:59:59,500
Rock on.
1254
01:00:03,760 --> 01:00:08,910
OK So quickly, I'm not going to
prove to you Noether's theorem.
1255
01:00:08,910 --> 01:00:12,800
It's one of the most beautiful
and important theorems
1256
01:00:12,800 --> 01:00:14,700
in physics, and you
should all study it.
1257
01:00:14,700 --> 01:00:16,200
But let me just
convince you quickly
1258
01:00:16,200 --> 01:00:18,002
that it's true in
classical mechanics.
1259
01:00:18,002 --> 01:00:20,210
And this was observed long
before Noether pointed out
1260
01:00:20,210 --> 01:00:23,080
why it was true in general.
1261
01:00:23,080 --> 01:00:25,930
What does it mean to have
transitional symmetry?
1262
01:00:25,930 --> 01:00:28,570
It means that, if I
do an experiment here
1263
01:00:28,570 --> 01:00:31,650
and I do it here, I get
exactly the same results.
1264
01:00:31,650 --> 01:00:33,820
I translate the system
and nothing changes.
1265
01:00:33,820 --> 01:00:34,320
Cool?
1266
01:00:34,320 --> 01:00:36,278
That's what I mean by
saying I have a symmetry.
1267
01:00:36,278 --> 01:00:38,540
You do this thing,
and nothing changes.
1268
01:00:38,540 --> 01:00:42,324
OK, so imagine I have a
particle, a classical particle,
1269
01:00:42,324 --> 01:00:43,740
and it's moving
in some potential.
1270
01:00:46,890 --> 01:00:49,564
This is u of x, right?
1271
01:00:49,564 --> 01:00:51,230
And we know what the
equations of motion
1272
01:00:51,230 --> 01:00:53,550
are in classical
mechanics from f
1273
01:00:53,550 --> 01:00:58,560
equals ma p dot is equal to
the force, which is minus
1274
01:00:58,560 --> 01:00:59,440
the gradient of u.
1275
01:00:59,440 --> 01:01:01,510
Minus the gradient of u.
1276
01:01:01,510 --> 01:01:02,010
Right?
1277
01:01:02,010 --> 01:01:05,680
That's f equals ma in
terms of the potential.
1278
01:01:05,680 --> 01:01:08,710
Now is the gradient of u 0?
1279
01:01:08,710 --> 01:01:10,175
No.
1280
01:01:10,175 --> 01:01:11,425
In this case, there's a force.
1281
01:01:13,671 --> 01:01:15,920
So if I do an experiment
here, do I get the same thing
1282
01:01:15,920 --> 01:01:17,583
as doing my experiment here?
1283
01:01:17,583 --> 01:01:18,430
AUDIENCE: No.
1284
01:01:18,430 --> 01:01:19,080
PROFESSOR: Certainly not.
1285
01:01:19,080 --> 01:01:21,163
The [? system ?] is not
translationally invariant.
1286
01:01:21,163 --> 01:01:23,210
The potential breaks that
translational symmetry.
1287
01:01:23,210 --> 01:01:26,490
What potential has
translational symmetry?
1288
01:01:26,490 --> 01:01:27,490
AUDIENCE: [INAUDIBLE].
1289
01:01:27,490 --> 01:01:28,870
PROFESSOR: Yeah, constant.
1290
01:01:28,870 --> 01:01:32,100
The only potential that has full
translational symmetry in one
1291
01:01:32,100 --> 01:01:35,970
dimension is translation
invariant, i.e.
1292
01:01:35,970 --> 01:01:37,500
constant.
1293
01:01:37,500 --> 01:01:38,650
OK?
1294
01:01:38,650 --> 01:01:40,180
What's the force?
1295
01:01:40,180 --> 01:01:40,680
AUDIENCE: 0.
1296
01:01:40,680 --> 01:01:41,221
PROFESSOR: 0.
1297
01:01:41,221 --> 01:01:41,890
0 gradient.
1298
01:01:41,890 --> 01:01:43,810
So what's p dot?
1299
01:01:43,810 --> 01:01:44,910
Yep.
1300
01:01:44,910 --> 01:01:45,720
Noether's theorem.
1301
01:01:45,720 --> 01:01:48,040
Solid.
1302
01:01:48,040 --> 01:01:49,270
OK.
1303
01:01:49,270 --> 01:01:51,360
Less trivial is
conservation of energy.
1304
01:01:51,360 --> 01:01:53,670
I claim and she claims--
and she's right--
1305
01:01:53,670 --> 01:01:56,770
that if the system has
the same dynamics at one
1306
01:01:56,770 --> 01:01:58,562
moment and a few moments
later and, indeed,
1307
01:01:58,562 --> 01:02:00,561
any amount of time later,
if the laws of physics
1308
01:02:00,561 --> 01:02:02,550
don't change in
time, then there must
1309
01:02:02,550 --> 01:02:05,260
be a conserved
quantity called energy.
1310
01:02:05,260 --> 01:02:08,740
There must be a
conserved quantity.
1311
01:02:08,740 --> 01:02:09,960
And that's Noether's theorem.
1312
01:02:09,960 --> 01:02:11,870
So this is the first
step, but this still
1313
01:02:11,870 --> 01:02:14,190
doesn't tell us what
momentum exactly
1314
01:02:14,190 --> 01:02:16,874
has to do with a derivative
with respect to space.
1315
01:02:16,874 --> 01:02:18,290
We see that there's
a relationship
1316
01:02:18,290 --> 01:02:22,380
between translations and
momentum conservation,
1317
01:02:22,380 --> 01:02:24,890
but what's the relationship?
1318
01:02:24,890 --> 01:02:26,270
So let's do this.
1319
01:02:26,270 --> 01:02:29,060
I'm going to define an
operation called translate by L.
1320
01:02:29,060 --> 01:02:30,520
And what translate
by L does is it
1321
01:02:30,520 --> 01:02:39,211
takes f of x and it maps
it to f of x minus L.
1322
01:02:39,211 --> 01:02:41,210
So this is a thing that
affects the translation.
1323
01:02:41,210 --> 01:02:43,880
And why do I say that's
a translation by L rather
1324
01:02:43,880 --> 01:02:47,490
than minus L. Well,
the point-- if you have
1325
01:02:47,490 --> 01:02:50,420
some function like this,
and it has a peak at 0,
1326
01:02:50,420 --> 01:02:57,630
then after the translation, the
peak is when x is equal to L.
1327
01:02:57,630 --> 01:02:58,130
OK?
1328
01:02:58,130 --> 01:02:59,940
So just to get the
signs straight.
1329
01:02:59,940 --> 01:03:02,660
So define this operation,
which takes a function of x
1330
01:03:02,660 --> 01:03:05,235
and translates it by L, but
leaves it otherwise identical.
1331
01:03:09,084 --> 01:03:10,500
So let's consider
how translations
1332
01:03:10,500 --> 01:03:13,160
behave on functions.
1333
01:03:13,160 --> 01:03:15,160
And this is really cute.
1334
01:03:15,160 --> 01:03:20,180
f of x minus L can be
written as a Taylor expansion
1335
01:03:20,180 --> 01:03:23,820
around the point x-- around
the point L equals 0.
1336
01:03:23,820 --> 01:03:26,250
So let's do Taylor
expansion for small L.
1337
01:03:26,250 --> 01:03:32,210
So this is equal to f of x
minus L derivative with respect
1338
01:03:32,210 --> 01:03:39,050
to x of f of x plus L squared
over 2 derivative squared,
1339
01:03:39,050 --> 01:03:42,780
two derivatives of x, f
of x plus dot, dot, dot.
1340
01:03:42,780 --> 01:03:43,280
Right?
1341
01:03:43,280 --> 01:03:46,306
I'm just Taylor expanding.
1342
01:03:46,306 --> 01:03:46,930
Nothing sneaky.
1343
01:03:49,510 --> 01:03:52,550
Let's add the next
term, actually.
1344
01:03:52,550 --> 01:03:54,820
Let me do this on
a whole new board.
1345
01:04:06,210 --> 01:04:08,865
All right, so we
have translate by L
1346
01:04:08,865 --> 01:04:15,130
on f of x is equal to f of x
minus L is equal to f of x.
1347
01:04:15,130 --> 01:04:17,530
Now Taylor expanding
minus L derivative
1348
01:04:17,530 --> 01:04:26,376
with respect to x of f
plus L squared over 2--
1349
01:04:26,376 --> 01:04:27,900
I'm not giving
myself enough space.
1350
01:04:27,900 --> 01:04:28,400
I'm sorry.
1351
01:04:30,990 --> 01:04:37,600
f of x minus L is equal to
f of x minus L with respect
1352
01:04:37,600 --> 01:04:46,150
to x of f of x plus L
squared over 2 to derivatives
1353
01:04:46,150 --> 01:04:51,430
of x f of x minus
L cubed over 6--
1354
01:04:51,430 --> 01:04:55,030
we're just Taylor expanding--
cubed with respect
1355
01:04:55,030 --> 01:05:00,800
to x of f of x and so on.
1356
01:05:00,800 --> 01:05:01,871
Yeah?
1357
01:05:01,871 --> 01:05:04,370
But I'm going to write this in
the following suggestive way.
1358
01:05:04,370 --> 01:05:13,610
This is equal to 1 times
f of x minus L derivative
1359
01:05:13,610 --> 01:05:17,730
with respect to x f
of x plus L squared
1360
01:05:17,730 --> 01:05:22,740
over 2 derivative with
respect to x squared times f
1361
01:05:22,740 --> 01:05:28,000
of x minus L cubed over 6
derivative cubed with respect
1362
01:05:28,000 --> 01:05:31,470
to x plus dot, dot, dot.
1363
01:05:31,470 --> 01:05:32,580
Everybody good with that?
1364
01:05:35,750 --> 01:05:38,740
But this is a series that
you should recognize,
1365
01:05:38,740 --> 01:05:41,700
a particular Taylor series
for a particular function.
1366
01:05:41,700 --> 01:05:44,754
It's a Taylor expansion for the
1367
01:05:44,754 --> 01:05:45,670
AUDIENCE: Exponential.
1368
01:05:45,670 --> 01:05:46,880
PROFESSOR: Exponential.
1369
01:05:46,880 --> 01:05:51,230
e to the minus L derivative
with respect to x f of x.
1370
01:05:59,689 --> 01:06:00,730
Which is kind of awesome.
1371
01:06:00,730 --> 01:06:02,220
So let's just check to make
sure that this makes sense
1372
01:06:02,220 --> 01:06:03,460
from dimensional grounds.
1373
01:06:03,460 --> 01:06:05,710
So that's a derivative with
respect to x as units of 1
1374
01:06:05,710 --> 01:06:06,120
over length.
1375
01:06:06,120 --> 01:06:07,870
That's a length, so
this is dimensionless,
1376
01:06:07,870 --> 01:06:09,370
so we can exponentiate it.
1377
01:06:09,370 --> 01:06:12,450
Now you might look at me and
say, look, this is silly.
1378
01:06:12,450 --> 01:06:14,680
You've taken an
operation like derivative
1379
01:06:14,680 --> 01:06:15,760
and exponentiated it.
1380
01:06:15,760 --> 01:06:17,631
What does that mean?
1381
01:06:17,631 --> 01:06:19,290
And that is what it means?
1382
01:06:19,290 --> 01:06:21,000
[LAUGHTER]
1383
01:06:21,000 --> 01:06:21,610
OK?
1384
01:06:21,610 --> 01:06:24,180
So we're going to do this all
the time in quantum mechanics.
1385
01:06:24,180 --> 01:06:26,452
We're going to do things
like exponentiate operations.
1386
01:06:26,452 --> 01:06:27,910
We'll talk about
it in more detail,
1387
01:06:27,910 --> 01:06:29,368
but we're always
going to define it
1388
01:06:29,368 --> 01:06:31,925
in this fashion as a
formal power series.
1389
01:06:31,925 --> 01:06:32,425
Questions?
1390
01:06:35,019 --> 01:06:36,560
AUDIENCE: Can you
transform operators
1391
01:06:36,560 --> 01:06:38,170
from one space to another?
1392
01:06:38,170 --> 01:06:39,680
PROFESSOR: Oh, you totally can.
1393
01:06:39,680 --> 01:06:40,930
But we'll come back to that.
1394
01:06:40,930 --> 01:06:43,400
We're going to talk about
operators next time.
1395
01:06:43,400 --> 01:06:48,750
OK, so here's where we are.
1396
01:06:48,750 --> 01:06:55,880
So from this what
is a derivative
1397
01:06:55,880 --> 01:06:56,950
with respect to x mean?
1398
01:06:56,950 --> 01:06:58,820
What does a derivative
with respect to x do?
1399
01:06:58,820 --> 01:07:00,280
Well a derivative
with respect to x
1400
01:07:00,280 --> 01:07:03,850
is something that generates
translations with respect
1401
01:07:03,850 --> 01:07:06,285
to x through a Taylor expansion.
1402
01:07:09,550 --> 01:07:12,240
If we have L be
arbitrarily small, right?
1403
01:07:12,240 --> 01:07:14,951
L is arbitrarily small.
1404
01:07:14,951 --> 01:07:17,200
What is the translation by
an arbitrarily small amount
1405
01:07:17,200 --> 01:07:18,140
of f of x?
1406
01:07:18,140 --> 01:07:19,610
Well, if L is
arbitrarily small, we
1407
01:07:19,610 --> 01:07:21,110
can drop all the
higher order terms,
1408
01:07:21,110 --> 01:07:24,542
and the change is just Ldx.
1409
01:07:24,542 --> 01:07:26,000
So the derivative
with respect to x
1410
01:07:26,000 --> 01:07:30,330
is telling us about
infinitesimal translations.
1411
01:07:30,330 --> 01:07:31,360
Cool?
1412
01:07:31,360 --> 01:07:33,490
The derivative with
respect to a position
1413
01:07:33,490 --> 01:07:35,520
is something that
tells you, or controls,
1414
01:07:35,520 --> 01:07:39,429
or generates infinitesimal
translations.
1415
01:07:39,429 --> 01:07:41,220
And if you exponentiate
it, you do it many,
1416
01:07:41,220 --> 01:07:43,350
many, many times in
a particular way,
1417
01:07:43,350 --> 01:07:46,410
you get a macroscopic
finite translation.
1418
01:07:46,410 --> 01:07:48,410
Cool?
1419
01:07:48,410 --> 01:07:50,820
So this gives us three things.
1420
01:07:50,820 --> 01:08:02,730
Translations in x are generated
by derivative with respect
1421
01:08:02,730 --> 01:08:03,230
to x.
1422
01:08:05,920 --> 01:08:14,410
But through Noether's
theorem translations,
1423
01:08:14,410 --> 01:08:20,390
in x are associated to
conservation of momentum.
1424
01:08:27,270 --> 01:08:30,720
So you shouldn't be so
shocked-- it's really not
1425
01:08:30,720 --> 01:08:44,050
totally shocking-- that in
quantum mechanics, where we're
1426
01:08:44,050 --> 01:08:48,500
very interested in the action
of things on functions,
1427
01:08:48,500 --> 01:08:51,319
not just in positions, but
on functions of position,
1428
01:08:51,319 --> 01:08:56,415
it shouldn't be totally shocking
that in quantum mechanics,
1429
01:08:56,415 --> 01:08:58,970
the derivative with
respect to x is related
1430
01:08:58,970 --> 01:09:00,810
to the momentum in
some particular way.
1431
01:09:03,950 --> 01:09:07,968
Similarly, translations
in t are going
1432
01:09:07,968 --> 01:09:09,384
to be generated
by what operation?
1433
01:09:12,380 --> 01:09:15,140
Derivative with respect to time.
1434
01:09:15,140 --> 01:09:18,520
So derivative with respect to
time from Noether's theorem
1435
01:09:18,520 --> 01:09:20,779
is associated with
conservation of energy.
1436
01:09:23,640 --> 01:09:25,810
That seems plausible.
1437
01:09:25,810 --> 01:09:27,689
Derivative with respect
to, I don't know,
1438
01:09:27,689 --> 01:09:31,779
an angle, a rotation.
1439
01:09:31,779 --> 01:09:34,260
That's going to be
associated with what?
1440
01:09:34,260 --> 01:09:36,390
Angular momentum?
1441
01:09:36,390 --> 01:09:38,080
But angular momentum
around the axis
1442
01:09:38,080 --> 01:09:39,240
for whom this is
the angle, so I'll
1443
01:09:39,240 --> 01:09:40,364
call that z for the moment.
1444
01:09:43,859 --> 01:09:48,140
And we're going to see these
pop up over and over again.
1445
01:09:48,140 --> 01:09:49,520
But here's the thing.
1446
01:09:52,819 --> 01:09:58,880
We started out with these
three principles today,
1447
01:09:58,880 --> 01:10:02,140
and we've let ourselves to
some sort of association
1448
01:10:02,140 --> 01:10:08,350
between the momentum and
the derivative like this.
1449
01:10:08,350 --> 01:10:08,850
OK?
1450
01:10:08,850 --> 01:10:10,558
And I've given you
some reason to believe
1451
01:10:10,558 --> 01:10:11,950
that this isn't totally insane.
1452
01:10:11,950 --> 01:10:13,440
Translations are
deeply connected
1453
01:10:13,440 --> 01:10:14,500
with conservation of momentum.
1454
01:10:14,500 --> 01:10:15,950
Transitional symmetry
is deeply connected
1455
01:10:15,950 --> 01:10:17,310
with conservation momentum.
1456
01:10:17,310 --> 01:10:18,770
And an infinitesimal
translation is
1457
01:10:18,770 --> 01:10:22,070
nothing but a derivative
with respect to position.
1458
01:10:22,070 --> 01:10:24,330
Those are deeply
linked concepts.
1459
01:10:24,330 --> 01:10:27,900
But I didn't derive anything.
1460
01:10:27,900 --> 01:10:29,570
I gave you no
derivation whatsoever
1461
01:10:29,570 --> 01:10:33,000
of the relationship between
d dx and the momentum.
1462
01:10:33,000 --> 01:10:36,170
Instead, I'm simply
going to declare it.
1463
01:10:36,170 --> 01:10:39,620
I'm going to declare that,
in quantum mechanics--
1464
01:10:39,620 --> 01:10:42,150
you cannot stop me--
in quantum mechanics,
1465
01:10:42,150 --> 01:10:48,300
p is represented by an
operator, it's represented
1466
01:10:48,300 --> 01:10:51,720
by the specific operator h bar
upon I derivative with respect
1467
01:10:51,720 --> 01:10:53,840
to x.
1468
01:10:53,840 --> 01:10:54,990
And this is a declaration.
1469
01:10:58,680 --> 01:11:00,890
OK?
1470
01:11:00,890 --> 01:11:02,700
It is simply a fact.
1471
01:11:02,700 --> 01:11:05,730
And when they say it's a fact,
I mean two things by that.
1472
01:11:05,730 --> 01:11:08,037
The first is it is a fact
that, in quantum mechanics,
1473
01:11:08,037 --> 01:11:10,120
momentum is represented
by derivative with respect
1474
01:11:10,120 --> 01:11:12,220
to x times h bar upon i.
1475
01:11:12,220 --> 01:11:15,639
Secondly, it is a fact that,
if you take this expression
1476
01:11:15,639 --> 01:11:17,930
and you work with the rest
of the postulates of quantum
1477
01:11:17,930 --> 01:11:19,346
mechanics, including
what's coming
1478
01:11:19,346 --> 01:11:22,730
next lecture about operators
and time evolution,
1479
01:11:22,730 --> 01:11:24,580
you reproduce the physics
of the real world.
1480
01:11:24,580 --> 01:11:26,030
You reproduce it beautifully.
1481
01:11:26,030 --> 01:11:28,540
You reproduce it so well that
no other models have even
1482
01:11:28,540 --> 01:11:32,000
ever vaguely come close to the
explanatory power of quantum
1483
01:11:32,000 --> 01:11:32,971
mechanics.
1484
01:11:32,971 --> 01:11:33,470
OK?
1485
01:11:33,470 --> 01:11:34,210
It is a fact.
1486
01:11:34,210 --> 01:11:36,710
It is not true in
some epistemic sense.
1487
01:11:36,710 --> 01:11:38,910
You can't sit back
and say, ah a priori
1488
01:11:38,910 --> 01:11:41,990
starting with the integers we
derive that p is equal to-- no,
1489
01:11:41,990 --> 01:11:43,050
it's a model.
1490
01:11:43,050 --> 01:11:44,509
But that's what physics does.
1491
01:11:44,509 --> 01:11:46,050
Physics doesn't tell
you what's true.
1492
01:11:46,050 --> 01:11:48,390
Physics doesn't tell
you what a priori
1493
01:11:48,390 --> 01:11:49,820
did the world have to look like.
1494
01:11:49,820 --> 01:11:52,412
Physics tells you
this is a good model,
1495
01:11:52,412 --> 01:11:54,370
and it works really well,
and it fits the data.
1496
01:11:54,370 --> 01:11:56,328
And to the degree that
it doesn't fit the data,
1497
01:11:56,328 --> 01:11:57,850
it's wrong.
1498
01:11:57,850 --> 01:11:58,414
OK?
1499
01:11:58,414 --> 01:11:59,705
This isn't something we derive.
1500
01:11:59,705 --> 01:12:01,250
This is something we declare.
1501
01:12:01,250 --> 01:12:03,940
We call it our model, and then
we use it to calculate stuff,
1502
01:12:03,940 --> 01:12:06,100
and we see if it
fits the real world.
1503
01:12:09,600 --> 01:12:12,041
Out, please, please leave.
1504
01:12:12,041 --> 01:12:12,540
Thank you.
1505
01:12:15,473 --> 01:12:24,730
[LAUGHTER]
1506
01:12:24,730 --> 01:12:25,750
I love MIT.
1507
01:12:25,750 --> 01:12:26,390
I really do.
1508
01:12:43,340 --> 01:12:45,280
So let me close
off at this point
1509
01:12:45,280 --> 01:12:48,450
with the following observation.
1510
01:12:48,450 --> 01:12:51,180
[LAUGHTER]
1511
01:12:51,180 --> 01:12:55,540
We live in a world
governed by probabilities.
1512
01:12:55,540 --> 01:12:57,910
There's a finite probability
that, at any given moment,
1513
01:12:57,910 --> 01:13:00,976
that two pirates might
walk into a room, OK?
1514
01:13:00,976 --> 01:13:01,475
[LAUGHTER]
1515
01:13:01,475 --> 01:13:03,340
You just never know.
1516
01:13:03,340 --> 01:13:08,350
[APPLAUSE]
1517
01:13:08,350 --> 01:13:13,240
But those probabilities can be
computed in quantum mechanics.
1518
01:13:13,240 --> 01:13:15,302
And they're computed
in the following ways.
1519
01:13:15,302 --> 01:13:16,760
They're computed
the following ways
1520
01:13:16,760 --> 01:13:18,420
as we'll study in great detail.
1521
01:13:18,420 --> 01:13:24,840
If I take a state, psi of x,
which is equal to e to the ikx,
1522
01:13:24,840 --> 01:13:28,000
this is a state that has
definite momentum h bar k.
1523
01:13:28,000 --> 01:13:28,990
Right?
1524
01:13:28,990 --> 01:13:29,720
We claimed this.
1525
01:13:29,720 --> 01:13:33,090
This was de Broglie
and Davisson-Germer.
1526
01:13:33,090 --> 01:13:35,450
Note the following,
take this operator
1527
01:13:35,450 --> 01:13:37,770
and act on this wave
function with this operator.
1528
01:13:37,770 --> 01:13:38,925
What do you get?
1529
01:13:38,925 --> 01:13:40,300
Well, we already
know, because we
1530
01:13:40,300 --> 01:13:42,760
constructed it to
have this property.
1531
01:13:42,760 --> 01:13:44,605
P hat on psi of
x-- and I'm going
1532
01:13:44,605 --> 01:13:46,230
to call this psi sub
k of x, because it
1533
01:13:46,230 --> 01:13:51,070
has a definite k-- is equal
to h bar k psi k of x.
1534
01:13:55,110 --> 01:13:58,455
A state with a definite
momentum has the property
1535
01:13:58,455 --> 01:14:00,580
that, when you hit it with
the operation associated
1536
01:14:00,580 --> 01:14:03,204
with momentum, you get back the
same function times a constant,
1537
01:14:03,204 --> 01:14:06,120
and that constant is
exactly the momentum we
1538
01:14:06,120 --> 01:14:09,486
ascribe to that plane wave.
1539
01:14:09,486 --> 01:14:10,730
Is that cool?
1540
01:14:10,730 --> 01:14:11,280
Yeah?
1541
01:14:11,280 --> 01:14:12,071
AUDIENCE: Question.
1542
01:14:12,071 --> 01:14:13,735
Just with notation,
what does the hat
1543
01:14:13,735 --> 01:14:14,744
above the p [INAUDIBLE]?
1544
01:14:14,744 --> 01:14:15,410
PROFESSOR: Good.
1545
01:14:15,410 --> 01:14:15,610
Excellent.
1546
01:14:15,610 --> 01:14:17,360
So the hat above the
P is to remind you
1547
01:14:17,360 --> 01:14:18,740
that P is on a number.
1548
01:14:18,740 --> 01:14:20,250
It's an operation.
1549
01:14:20,250 --> 01:14:23,370
It's a rule for
acting on functions.
1550
01:14:23,370 --> 01:14:25,670
We'll talk about that in
great detail next time.
1551
01:14:25,670 --> 01:14:27,560
But here's what I
want to emphasize.
1552
01:14:27,560 --> 01:14:30,792
This is a state which is equal
to all others in the sense
1553
01:14:30,792 --> 01:14:32,750
that it's a perfectly
reasonable wave function,
1554
01:14:32,750 --> 01:14:35,980
but it's more equal because it
has a simple interpretation.
1555
01:14:35,980 --> 01:14:37,350
Right?
1556
01:14:37,350 --> 01:14:39,904
The probability that I measure
the momentum to be h bar k
1557
01:14:39,904 --> 01:14:41,320
is one, and the
probability that I
1558
01:14:41,320 --> 01:14:45,010
measure it to be anything
else is 0, correct?
1559
01:14:45,010 --> 01:14:47,870
But I can always consider a
state which is a superposition.
1560
01:14:47,870 --> 01:14:54,426
Psi is equal to alpha, let's
just do 1 over 2 e to the ikx.
1561
01:14:54,426 --> 01:14:59,180
k1 x plus 1 over root
2 e to the minus ikx.
1562
01:15:09,070 --> 01:15:12,930
Is this state a state
with definite momentum?
1563
01:15:12,930 --> 01:15:15,170
If I act on this
state-- I'll call this i
1564
01:15:15,170 --> 01:15:18,590
sub s-- if I act on this state
with the momentum operator,
1565
01:15:18,590 --> 01:15:21,771
do I get back this
state times a constant?
1566
01:15:21,771 --> 01:15:22,270
No.
1567
01:15:22,270 --> 01:15:23,080
That's interesting.
1568
01:15:23,080 --> 01:15:24,659
And so it seems to
be that if we have
1569
01:15:24,659 --> 01:15:27,200
a state with definite momentum
and we act on it with momentum
1570
01:15:27,200 --> 01:15:28,810
operator, we get
back its momentum.
1571
01:15:28,810 --> 01:15:30,340
If we have a state
that's a superposition
1572
01:15:30,340 --> 01:15:33,006
of different momentum and we act
on it with a momentum operator,
1573
01:15:33,006 --> 01:15:35,330
this gives us h bar k 1,
this gives us h bar k2.
1574
01:15:35,330 --> 01:15:36,957
So it changes
which superposition
1575
01:15:36,957 --> 01:15:37,790
we're talking about.
1576
01:15:37,790 --> 01:15:41,330
We don't get back
our same state.
1577
01:15:41,330 --> 01:15:43,065
So the action of this
operator on a state
1578
01:15:43,065 --> 01:15:45,440
is going to tell us something
about whether the state has
1579
01:15:45,440 --> 01:15:48,570
definite value of the momentum.
1580
01:15:48,570 --> 01:15:50,490
And these coefficients
are going to turn out
1581
01:15:50,490 --> 01:15:52,910
to contain all the information
about the probability
1582
01:15:52,910 --> 01:15:53,660
of the system.
1583
01:15:53,660 --> 01:15:55,560
This is the
probability when norm
1584
01:15:55,560 --> 01:15:59,000
squared that will measure the
system to have momentum k1.
1585
01:15:59,000 --> 01:16:00,530
And this coefficient
norm squared
1586
01:16:00,530 --> 01:16:02,450
is going to tell us
the probability that we
1587
01:16:02,450 --> 01:16:05,750
have momentum k2.
1588
01:16:05,750 --> 01:16:10,730
So I think the
current wave function
1589
01:16:10,730 --> 01:16:21,020
is something like a
superposition of 1/10 psi
1590
01:16:21,020 --> 01:16:32,890
pirates plus 1 minus
is 1/100 square root.
1591
01:16:32,890 --> 01:16:35,760
To normalize it
properly psi no pirates.
1592
01:16:41,340 --> 01:16:43,670
And I'll leave you with
pondering this probability.
1593
01:16:43,670 --> 01:16:47,330
See you guys next time.
1594
01:16:47,330 --> 01:17:11,525
[APPLAUSE]
1595
01:17:11,525 --> 01:17:13,150
CHRISTOPHER SMITH:
We've come for Prof.
1596
01:17:13,150 --> 01:17:15,140
Allan Adams.
1597
01:17:15,140 --> 01:17:16,880
PROFESSOR: It is I.
1598
01:17:16,880 --> 01:17:21,110
CHRISTOPHER SMITH: When in
the chronicles of wasted time,
1599
01:17:21,110 --> 01:17:24,900
I see descriptions
of fairest rights,
1600
01:17:24,900 --> 01:17:29,650
and I see lovely
shows of lovely dames.
1601
01:17:29,650 --> 01:17:34,440
And descriptions of ladies
dead and lovely nights.
1602
01:17:34,440 --> 01:17:37,860
Then in the bosom of
fair loves depths.
1603
01:17:37,860 --> 01:17:43,730
Of eyes, of foot,
of eye, of brow.
1604
01:17:43,730 --> 01:17:48,090
I see the antique pens
do but express the beauty
1605
01:17:48,090 --> 01:17:50,770
that you master now.
1606
01:17:50,770 --> 01:17:55,550
So are all their praises but
prophecies of this, our time.
1607
01:17:55,550 --> 01:17:59,120
All you prefiguring.
1608
01:17:59,120 --> 01:18:02,942
But though they had
but diving eyes--
1609
01:18:02,942 --> 01:18:04,900
PROFESSOR: I was wrong
about the probabilities.
1610
01:18:04,900 --> 01:18:05,376
[LAUGHTER]
1611
01:18:05,376 --> 01:18:06,792
CHRISTOPHER SMITH:
But though they
1612
01:18:06,792 --> 01:18:09,120
had but diving eyes,
they had not skill
1613
01:18:09,120 --> 01:18:11,710
enough you're worth to sing.
1614
01:18:11,710 --> 01:18:15,020
For we which now behold
these present days
1615
01:18:15,020 --> 01:18:17,600
have eyes to behold.
1616
01:18:17,600 --> 01:18:21,440
[LAUGHTER]
1617
01:18:21,440 --> 01:18:23,180
But not tongues to praise.
1618
01:18:25,840 --> 01:18:28,200
[APPLAUSE]
1619
01:18:28,200 --> 01:18:29,144
It's not over.
1620
01:18:29,144 --> 01:18:31,990
You wait.
1621
01:18:31,990 --> 01:18:35,020
ARSHIA SURTI: Not marbled with
gilded monuments of princes
1622
01:18:35,020 --> 01:18:37,160
shall outlive this
powerful rhyme.
1623
01:18:37,160 --> 01:18:40,690
But you shall shine more
bright in these contents
1624
01:18:40,690 --> 01:18:43,370
that unswept stone
besmear its sluttish tide.
1625
01:18:43,370 --> 01:18:48,970
When wasteful war shall
statues overturn and broils
1626
01:18:48,970 --> 01:18:50,230
root out the work of masonry.
1627
01:18:50,230 --> 01:18:53,060
Nor Mars his sword.
1628
01:18:53,060 --> 01:18:57,390
Nor war's quick fire shall
burn the living record
1629
01:18:57,390 --> 01:18:59,640
of your memory.
1630
01:18:59,640 --> 01:19:04,810
Gainst death and all oblivious
enmity shall you pace forth.
1631
01:19:04,810 --> 01:19:06,570
Your praise shall
still find room,
1632
01:19:06,570 --> 01:19:08,233
even in the eyes
of all posterity.
1633
01:19:11,220 --> 01:19:15,910
So no judgment arise till
you yourself judgment arise.
1634
01:19:15,910 --> 01:19:18,060
You live in this and
dwell in lover's eyes.
1635
01:19:20,862 --> 01:19:23,200
[APPLAUSE]
1636
01:19:23,200 --> 01:19:26,220
CHRISTOPHER SMITH: Verily
happy Valentine's day upon you.
1637
01:19:26,220 --> 01:19:28,170
May your day be filled
with love and poetry.
1638
01:19:28,170 --> 01:19:31,448
Whatever state you're in,
we will always love you.
1639
01:19:31,448 --> 01:19:33,918
[LAUGHTER]
1640
01:19:33,918 --> 01:19:37,880
[APPLAUSE]
1641
01:19:37,880 --> 01:19:41,690
Signed, Jack Florian,
James [INAUDIBLE].
1642
01:19:41,690 --> 01:19:42,590
[LAUGHTER]
1643
01:19:42,590 --> 01:19:44,390
PROFESSOR: Thank you, sir.
1644
01:19:44,390 --> 01:19:45,582
Thank you.
1645
01:19:45,582 --> 01:19:46,790
CHRISTOPHER SMITH: Now we go.
1646
01:19:50,390 --> 01:19:51,940
[APPLAUSE]