1
00:00:00,500 --> 00:00:03,385
PROFESSOR: That brings us to
claim number four, which is
2
00:00:03,385 --> 00:00:07,480
perhaps the most important one.
3
00:00:07,480 --> 00:00:18,960
I may have said it already.b
The eigenfunctions of Q
4
00:00:18,960 --> 00:00:41,870
form a set of basis functions,
and then any reasonable psi
5
00:00:41,870 --> 00:00:53,010
can be written as
a superposition
6
00:00:53,010 --> 00:00:56,030
of Q eigenfunctions.
7
00:01:01,670 --> 00:01:05,523
OK, so let's just
make sense of this.
8
00:01:05,523 --> 00:01:09,170
Because not only, I
think we understand
9
00:01:09,170 --> 00:01:14,660
what this means, but let's
write it out mathematically.
10
00:01:14,660 --> 00:01:26,250
So the statement is any psi
of x, or this physical state,
11
00:01:26,250 --> 00:01:29,470
can be written as a
superposition of all
12
00:01:29,470 --> 00:01:34,000
these eigenfunctions So there
are numbers, alpha 1 psi
13
00:01:34,000 --> 00:01:40,160
1 of x plus alpha 2 psi 2 of x.
14
00:01:40,160 --> 00:01:43,780
Those are the expansion
coefficients with alphas.
15
00:01:43,780 --> 00:01:52,610
And in summary, we say from
sum over i, alpha i psi i of x.
16
00:01:52,610 --> 00:01:56,750
So the idea is that
those alpha i's exist
17
00:01:56,750 --> 00:01:59,990
and you can write them.
18
00:01:59,990 --> 00:02:02,780
So any wave function
that you have,
19
00:02:02,780 --> 00:02:05,810
you can write it
in a superposition
20
00:02:05,810 --> 00:02:11,050
of those eigenfunctions
of the Hermitian operator.
21
00:02:11,050 --> 00:02:16,460
And there are two
things to say here.
22
00:02:16,460 --> 00:02:20,080
One is that, how would you
calculate those alpha i's?
23
00:02:23,300 --> 00:02:26,190
Well, actually, if you
assume this equation,
24
00:02:26,190 --> 00:02:28,990
the calculation of
alpha i's is simple,
25
00:02:28,990 --> 00:02:32,650
because of this property.
26
00:02:32,650 --> 00:02:35,370
You're supposed to know
the eigenfunctions.
27
00:02:35,370 --> 00:02:39,180
You must have done the work to
calculate the eigenfunctions.
28
00:02:39,180 --> 00:02:41,920
So here is what you can do.
29
00:02:41,920 --> 00:02:44,950
You can do the
following integral.
30
00:02:44,950 --> 00:02:52,980
You can do this one, psi i psi.
31
00:02:52,980 --> 00:02:55,290
Let's calculate this thing.
32
00:02:55,290 --> 00:02:57,330
Remember what this is.
33
00:02:57,330 --> 00:03:04,255
This is an integral,
dx, of psi i star.
34
00:03:04,255 --> 00:03:06,560
That's psi.
35
00:03:06,560 --> 00:03:14,420
And psi is the sum over
j of alpha j psi j.
36
00:03:14,420 --> 00:03:15,870
You can use any letter.
37
00:03:15,870 --> 00:03:20,670
I used i for the sum, but
since I put that psi i,
38
00:03:20,670 --> 00:03:25,230
I would make a great
confusion if I used another i.
39
00:03:25,230 --> 00:03:30,010
So I should use j there.
40
00:03:30,010 --> 00:03:31,970
And what is this?
41
00:03:31,970 --> 00:03:33,915
Well, you're integrating
the part of this.
42
00:03:33,915 --> 00:03:34,840
That's a sum.
43
00:03:34,840 --> 00:03:36,700
So the sum can go out.
44
00:03:36,700 --> 00:03:46,940
It's the sum over j alpha j
integral of psi i star psi j d.
45
00:03:46,940 --> 00:03:49,610
And what is this delta ij?
46
00:03:49,610 --> 00:03:52,030
That is our nice orthonormality.
47
00:03:52,030 --> 00:04:00,210
So this is sum over
j alpha j, delta i j.
48
00:04:00,210 --> 00:04:03,110
Now, this is kind
of a simple sum.
49
00:04:03,110 --> 00:04:04,590
You can always be done.
50
00:04:04,590 --> 00:04:07,980
You should just think a second.
51
00:04:07,980 --> 00:04:12,180
You're summing over
j, and i is fixed.
52
00:04:12,180 --> 00:04:16,620
The only case when this
gives something is when j,
53
00:04:16,620 --> 00:04:19,089
and you're summing
over, is equal to i,
54
00:04:19,089 --> 00:04:21,779
which is a fixed number.
55
00:04:21,779 --> 00:04:24,540
Therefore, the only
thing that survives
56
00:04:24,540 --> 00:04:27,200
is j equals to i, so this is 1.
57
00:04:27,200 --> 00:04:30,450
And therefore, this is alpha i.
58
00:04:30,450 --> 00:04:33,910
So we did succeed
in calculating this,
59
00:04:33,910 --> 00:04:42,760
and in fact, alpha i is equal
to this integral of psi i
60
00:04:42,760 --> 00:04:44,310
with psi.
61
00:04:44,310 --> 00:04:47,030
So how do you
compute it now for i?
62
00:04:47,030 --> 00:04:48,390
You must do an integral.
63
00:04:48,390 --> 00:04:49,340
Of what?
64
00:04:49,340 --> 00:04:52,820
Of psi i star times
your wave function.
65
00:04:52,820 --> 00:04:55,830
So in this common interval.
66
00:04:55,830 --> 00:05:00,795
So the alpha i's are
given by these numbers.
67
00:05:00,795 --> 00:05:04,707
This would prove.
68
00:05:04,707 --> 00:05:09,710
The other thing
that you can check
69
00:05:09,710 --> 00:05:18,890
is if the wave function
squared dx is equal to 1.
70
00:05:22,940 --> 00:05:30,090
What does it imply
for the alpha i's?
71
00:05:30,090 --> 00:05:32,680
You see, the wave
function is normalized,
72
00:05:32,680 --> 00:05:36,902
but it's not a function of alpha
1, alpha 2, alpha 3, alpha 4,
73
00:05:36,902 --> 00:05:37,610
all these things.
74
00:05:37,610 --> 00:05:39,980
So I must calculate this.
75
00:05:39,980 --> 00:05:42,970
And now let's do it,
quickly, but do it.
76
00:05:42,970 --> 00:05:53,170
Sum over i, alpha i, psi i star,
sum over j, alpha j, psi j.
77
00:05:53,170 --> 00:05:56,662
See, that's the integral
of these things squared dx.
78
00:06:00,160 --> 00:06:01,612
I'm sorry.
79
00:06:01,612 --> 00:06:06,202
I went wrong here.
80
00:06:06,202 --> 00:06:08,760
The star is there.
81
00:06:08,760 --> 00:06:13,390
The first psi, starred,
the second psi.
82
00:06:13,390 --> 00:06:15,030
Now I got it right.
83
00:06:15,030 --> 00:06:24,070
Now, I take out the sums i, sum
over j, alpha i star alpha j,
84
00:06:24,070 --> 00:06:30,350
integral dx psi i star psi j.
85
00:06:30,350 --> 00:06:35,640
This is delta i j, therefore
j becomes equal to i,
86
00:06:35,640 --> 00:06:41,850
and you get sum over i
of alpha i star alpha
87
00:06:41,850 --> 00:06:48,100
i, which is the sum over i
of, then alpha i squared.
88
00:06:48,100 --> 00:06:49,810
OK.
89
00:06:49,810 --> 00:06:51,630
So that's what it says.
90
00:06:51,630 --> 00:06:52,140
Look.
91
00:06:52,140 --> 00:06:56,170
This is something that should
be internalized as well.
92
00:06:56,170 --> 00:07:01,690
The sum over i of the alpha
i squared is equal to 1.
93
00:07:01,690 --> 00:07:05,255
Whenever you have a
superposition of wave
94
00:07:05,255 --> 00:07:09,960
functions, and the whole
thing is normalized,
95
00:07:09,960 --> 00:07:15,640
and your wave functions
are orthonormal,
96
00:07:15,640 --> 00:07:16,875
then it's very simple.
97
00:07:16,875 --> 00:07:22,050
The normalization is computed
by doing the sums of squares
98
00:07:22,050 --> 00:07:23,760
of each coefficient.
99
00:07:23,760 --> 00:07:31,570
The mixings don't exist
because there's no mixes here.
100
00:07:31,570 --> 00:07:33,280
So everything is separate.
101
00:07:33,280 --> 00:07:34,700
Everything is unmixed.
102
00:07:34,700 --> 00:07:36,070
Everything is nice.
103
00:07:38,960 --> 00:07:40,030
So there you go.
104
00:07:40,030 --> 00:07:48,090
This is how you expand any
state in the collection
105
00:07:48,090 --> 00:07:51,970
of eigenfunctions of
any Hermitian operator
106
00:07:51,970 --> 00:07:55,071
that you are looking at.
107
00:07:55,071 --> 00:07:55,570
OK.
108
00:07:55,570 --> 00:07:57,960
So finally, we get it.
109
00:07:57,960 --> 00:08:02,790
We've done all the
work necessary to state
110
00:08:02,790 --> 00:08:04,840
the measurement possibility.
111
00:08:04,840 --> 00:08:09,710
How do we find what we measure?
112
00:08:09,710 --> 00:08:10,910
So here it is.
113
00:08:18,260 --> 00:08:19,730
Measurement Postulate.
114
00:08:32,510 --> 00:08:33,980
So here's the issue.
115
00:08:33,980 --> 00:08:35,715
We want to measure.
116
00:08:35,715 --> 00:08:39,890
I'm going to say
these things in words.
117
00:08:39,890 --> 00:08:43,590
You want to measure the
operator, q, of your state.
118
00:08:43,590 --> 00:08:47,120
The operator might be the
momentum, might be the energy,
119
00:08:47,120 --> 00:08:50,280
might be the angular momentum,
could be kinetic energy,
120
00:08:50,280 --> 00:08:51,830
could be potential energy.
121
00:08:51,830 --> 00:08:54,620
Any Hermitian operator.
122
00:08:54,620 --> 00:08:58,700
You want to measure
it in your state.
123
00:08:58,700 --> 00:09:02,450
The first thing that
the postulate will say
124
00:09:02,450 --> 00:09:07,310
is that you will, in general,
obtain just one number
125
00:09:07,310 --> 00:09:10,220
each time you do a
measurement, but that number
126
00:09:10,220 --> 00:09:15,020
is one of the eigenvalues
of this operator.
127
00:09:15,020 --> 00:09:18,800
So the set of
possible measurements,
128
00:09:18,800 --> 00:09:23,030
possible outcomes,
better say, is the set
129
00:09:23,030 --> 00:09:25,440
of eigenvalues of the operator.
130
00:09:25,440 --> 00:09:28,380
Those are the only
numbers you can get.
131
00:09:28,380 --> 00:09:31,430
But you can get them with
different probabilities.
132
00:09:31,430 --> 00:09:35,610
And for that, you
must use this plane.
133
00:09:35,610 --> 00:09:41,090
And you must, in a
sense, rewrite your state
134
00:09:41,090 --> 00:09:45,894
as a superposition of the
eigenfunctions, those alphas.
135
00:09:45,894 --> 00:09:51,090
And the probability
to measure q1
136
00:09:51,090 --> 00:09:53,617
is the probability
that you end up
137
00:09:53,617 --> 00:09:55,840
of this part of
the superposition,
138
00:09:55,840 --> 00:10:00,646
and it will be given by
alpha 1 squared, [INAUDIBLE].
139
00:10:00,646 --> 00:10:03,500
The probability
to measure q will
140
00:10:03,500 --> 00:10:08,390
be given by alpha 2 squared
and all of these numbers.
141
00:10:08,390 --> 00:10:15,490
So, and finally, that
after the measurement,
142
00:10:15,490 --> 00:10:17,060
another funny thing happens.
143
00:10:17,060 --> 00:10:23,630
The state that was this whole
sum collapses to that state
144
00:10:23,630 --> 00:10:25,290
that you obtained.
145
00:10:25,290 --> 00:10:31,280
So if you obtained q1, well, the
whole thing collapses to psi 1.
146
00:10:31,280 --> 00:10:32,840
After you've done
the measurement,
147
00:10:32,840 --> 00:10:36,260
the state of the
system becomes psi 1.
148
00:10:36,260 --> 00:10:38,630
So this is the spirit
of what happens.
149
00:10:38,630 --> 00:10:42,590
Let me write it out.
150
00:10:42,590 --> 00:10:57,390
If we measure Q
in the state psi,
151
00:10:57,390 --> 00:11:09,770
the possible values
obtained are q1, q2.
152
00:11:12,922 --> 00:11:26,070
The probability, p
i, to measure q i
153
00:11:26,070 --> 00:11:31,980
is p i equals alpha i squared.
154
00:11:31,980 --> 00:11:37,310
And remember what this
alpha i we calculated it.
155
00:11:37,310 --> 00:11:47,030
This overlap of psi
i with psi squared.
156
00:11:52,880 --> 00:11:58,370
And finally, after finding--
157
00:12:02,125 --> 00:12:09,970
after, let's write
it, the outcome, q i,
158
00:12:09,970 --> 00:12:23,020
the state of the
system becomes psi
159
00:12:23,020 --> 00:12:28,070
of x is equal to psi i of x.
160
00:12:28,070 --> 00:12:34,541
And this is a collapse
of the wave function.
161
00:12:42,720 --> 00:12:46,370
And it also means that after
you've done the measurement
162
00:12:46,370 --> 00:12:51,720
and you did obtain the value
of q i, you stay with psi i,
163
00:12:51,720 --> 00:12:54,872
if you measure it again, you
would keep obtaining q i.
164
00:13:02,420 --> 00:13:06,190
Why did it all become possible?
165
00:13:06,190 --> 00:13:10,190
It all became possible
because Hermitian operators
166
00:13:10,190 --> 00:13:14,060
are rich enough to
allow you to write
167
00:13:14,060 --> 00:13:17,030
any state as a superposition.
168
00:13:17,030 --> 00:13:21,200
And therefore, if you
want to measure momentum,
169
00:13:21,200 --> 00:13:23,510
you must find all the
eigenfunctions of momentum
170
00:13:23,510 --> 00:13:27,830
and rewrite your state as a
superposition of momentum.
171
00:13:27,830 --> 00:13:29,880
You want to do energy?
172
00:13:29,880 --> 00:13:31,900
Well, you must
rewrite your state
173
00:13:31,900 --> 00:13:34,370
as a superposition of
energy eigenstates,
174
00:13:34,370 --> 00:13:35,700
and then you can measure.
175
00:13:35,700 --> 00:13:37,970
Want to measure
angular momentum?
176
00:13:37,970 --> 00:13:41,000
Find the eigenstates
of angular momentum,
177
00:13:41,000 --> 00:13:45,440
use the theorem to rewrite your
whole state in different ways.
178
00:13:45,440 --> 00:13:48,820
And this is something we
said in the first lecture
179
00:13:48,820 --> 00:13:53,600
of this course, that any
vector in a vector space
180
00:13:53,600 --> 00:13:56,510
can be written in
infinitely many ways
181
00:13:56,510 --> 00:13:59,660
as different
superpositions of vectors.
182
00:13:59,660 --> 00:14:02,530
We wrote the arrow
and said, this vector
183
00:14:02,530 --> 00:14:06,772
is the sum of this and this,
and this plus this plus this,
184
00:14:06,772 --> 00:14:09,830
and this plus this plus this.
185
00:14:09,830 --> 00:14:13,040
And yes, you need
all that flexibility.
186
00:14:13,040 --> 00:14:17,750
For any measurement,
you rewrite the vector
187
00:14:17,750 --> 00:14:21,410
as the sum of the
eigenvectors, and then you
188
00:14:21,410 --> 00:14:24,910
can tell what are
your predictions.
189
00:14:24,910 --> 00:14:30,050
You need that flexibility that
any vector in a vector space
190
00:14:30,050 --> 00:14:32,660
can be written in
infinitely many ways
191
00:14:32,660 --> 00:14:37,320
as different linear
superpositions.
192
00:14:37,320 --> 00:14:41,060
So there's a couple
of things we can
193
00:14:41,060 --> 00:14:45,970
do to add intuition to this.
194
00:14:45,970 --> 00:14:48,690
I'll do, first, a
consistency check,
195
00:14:48,690 --> 00:14:51,230
and maybe I'll do
an example as well.
196
00:14:53,850 --> 00:14:57,934
And then we have to
define uncertainties,
197
00:14:57,934 --> 00:15:00,970
those of that phase.
198
00:15:00,970 --> 00:15:04,940
So any question about this
measurement postulate?
199
00:15:04,940 --> 00:15:07,530
Is there something
unclear about it?
200
00:15:10,590 --> 00:15:13,040
It's a very strange postulate.
201
00:15:13,040 --> 00:15:18,080
You see, it divides quantum
mechanics into two realms.
202
00:15:18,080 --> 00:15:20,840
There's the realm of the
Schrodinger equation,
203
00:15:20,840 --> 00:15:23,970
your wave function
evolves in time.
204
00:15:23,970 --> 00:15:27,190
And then there's a
realm of measurement.
205
00:15:27,190 --> 00:15:29,600
The Schroedinger
equation doesn't tell you
206
00:15:29,600 --> 00:15:31,760
what you're supposed
to do with measurement.
207
00:15:31,760 --> 00:15:35,540
But consistency with a
Schroedinger equations
208
00:15:35,540 --> 00:15:37,130
doesn't allow you many things.
209
00:15:37,130 --> 00:15:39,980
And this is apparently
the only thing we can do.
210
00:15:39,980 --> 00:15:45,380
And then we do a measurement,
but somehow, this psi of x
211
00:15:45,380 --> 00:15:49,472
collapses and becomes one of
the results of your measurement.
212
00:15:52,310 --> 00:15:55,160
People have wondered, if
the Schroedinger equation is
213
00:15:55,160 --> 00:15:58,400
all there is in the world,
why doesn't the result
214
00:15:58,400 --> 00:16:01,550
of the measurement come out
of the Schroedinger equation?
215
00:16:01,550 --> 00:16:05,150
Well, people think
very hard about it,
216
00:16:05,150 --> 00:16:10,460
and they come up with all
kinds of interesting things.
217
00:16:10,460 --> 00:16:13,590
Nevertheless, nothing
that comes out
218
00:16:13,590 --> 00:16:17,120
is sufficiently clear
and sufficiently useful
219
00:16:17,120 --> 00:16:19,970
to merit a discussion
at this moment.
220
00:16:19,970 --> 00:16:23,600
It's very interesting, and
it's subject of research,
221
00:16:23,600 --> 00:16:28,340
but nobody has found a flaw
with this way of stating things.
222
00:16:28,340 --> 00:16:31,190
And it's the simplest
way of stating things.
223
00:16:31,190 --> 00:16:35,415
And therefore, the measurement
is an extra assumption,
224
00:16:35,415 --> 00:16:36,940
an extra postulate.
225
00:16:36,940 --> 00:16:39,210
That's how a measurement works.
226
00:16:39,210 --> 00:16:42,290
And after you measure,
you leave the system,
227
00:16:42,290 --> 00:16:45,740
the Schroedinger equation
takes over and keeps evolving.
228
00:16:45,740 --> 00:16:48,410
You measure again,
something happens,
229
00:16:48,410 --> 00:16:51,360
there's some answer
that gets realized.
230
00:16:51,360 --> 00:16:56,230
Some answers are not
realized, and it so continues.