1
00:00:00,499 --> 00:00:02,820
BARTON ZWIEBACH: Do
normal wave analysis
2
00:00:02,820 --> 00:00:09,820
to demonstrate that indeed these
things should not quite happen.
3
00:00:09,820 --> 00:00:24,610
So for that, so ordinary waves
and Galilean transformations.
4
00:00:34,920 --> 00:00:39,190
So when you have a wave,
as you've probably have
5
00:00:39,190 --> 00:00:42,760
seen many times before,
the key object in the wave
6
00:00:42,760 --> 00:00:46,810
is something called
the phaze of the wave.
7
00:00:46,810 --> 00:00:50,820
Phaze, the phaze.
8
00:00:50,820 --> 00:00:57,760
And it's controlled by this
quantity kx minus omega t.
9
00:00:57,760 --> 00:01:02,070
k being the wave number, omega
being the angular frequency
10
00:01:02,070 --> 00:01:03,960
and we spoke about.
11
00:01:03,960 --> 00:01:10,390
And the wave may be sine of that
phaze or cosine of that phaze
12
00:01:10,390 --> 00:01:16,930
or a linear combination of sines
and cosines, or E to this wave,
13
00:01:16,930 --> 00:01:21,580
any of those things
could be your wave.
14
00:01:21,580 --> 00:01:25,840
And whenever you have
such a wave, what we say
15
00:01:25,840 --> 00:01:31,370
is that the phaze of this
wave is a Galilean invariant.
16
00:01:34,910 --> 00:01:35,913
Invariant.
17
00:01:39,990 --> 00:01:45,620
What it means is that two
people looking at this wave,
18
00:01:45,620 --> 00:01:50,790
and they look at the
point on this wave,
19
00:01:50,790 --> 00:01:56,390
both people will agree on
the value of the phaze,
20
00:01:56,390 --> 00:02:00,290
because basically, the
reality of the wave
21
00:02:00,290 --> 00:02:05,270
is based on the phaze, and if
you have, for example, cosine
22
00:02:05,270 --> 00:02:09,800
of this phaze, the place
where this cosine is 0
23
00:02:09,800 --> 00:02:15,500
is some of the phaze, and if
the cosine is 0, the wave is 0,
24
00:02:15,500 --> 00:02:19,230
and everybody should agree that
the wave is 0 at that point.
25
00:02:19,230 --> 00:02:23,030
So if you have a place where the
wave has a maximum or a place
26
00:02:23,030 --> 00:02:26,960
where the wave is 0,
this is an ordinary wave,
27
00:02:26,960 --> 00:02:30,470
everybody would agree that at
that place you have a maximum
28
00:02:30,470 --> 00:02:32,570
and in that place you have a 0.
29
00:02:32,570 --> 00:02:39,390
So observers should agree
on the value of this phaze.
30
00:02:39,390 --> 00:02:42,140
It's going to be an invariant.
31
00:02:42,140 --> 00:02:47,060
And we can rewrite this phaze
in a perhaps more familiar way
32
00:02:47,060 --> 00:02:58,310
by factoring the k, and then
you have x minus omega over kt,
33
00:02:58,310 --> 00:03:03,710
and this is 2 pi over
lambda, x minus--
34
00:03:03,710 --> 00:03:11,880
this quantity is called
the velocity of the wave,
35
00:03:11,880 --> 00:03:13,820
and we'll write it this way.
36
00:03:17,840 --> 00:03:22,340
And I'll write in one last way--
37
00:03:22,340 --> 00:03:35,440
2 pi x over lambda minus
2 pi V over lambda t.
38
00:03:35,440 --> 00:03:43,390
And this quantity is omega
and this quantity is k.
39
00:03:49,150 --> 00:03:52,900
So this is our phaze.
40
00:03:52,900 --> 00:03:57,460
And we've said that it's
a Galilean invariant, so I
41
00:03:57,460 --> 00:04:00,540
will say that S should see--
42
00:04:00,540 --> 00:04:10,090
the observer S prime
should see the same phaze--
43
00:04:10,090 --> 00:04:19,690
phaze-- as S. So phi prime,
the phaze that S prime sees,
44
00:04:19,690 --> 00:04:25,720
must be equal to phi when
referring to the same point.
45
00:04:25,720 --> 00:04:37,495
When referring to the same
point at the same time.
46
00:04:49,180 --> 00:04:50,500
Let's write this.
47
00:04:50,500 --> 00:04:55,740
So phi prime should
be equal to phi.
48
00:04:55,740 --> 00:04:58,960
And phi, we've written there.
49
00:04:58,960 --> 00:05:03,130
2 pi over lambda x minus Vt.
50
00:05:06,230 --> 00:05:08,810
And this is so far
so good, but we
51
00:05:08,810 --> 00:05:12,800
want to write it in
terms of quantities
52
00:05:12,800 --> 00:05:14,750
that S prime measures.
53
00:05:14,750 --> 00:05:21,320
So this x should
be replaced by 2 pi
54
00:05:21,320 --> 00:05:35,330
over lambda x prime plus
Vt minus Vt like this.
55
00:05:35,330 --> 00:05:37,650
And I could even
do more if I wish.
56
00:05:37,650 --> 00:05:41,090
I could put t prime
here, because the t and t
57
00:05:41,090 --> 00:05:44,890
primes are the same.
58
00:05:44,890 --> 00:05:51,960
So phi prime, by the condition
that these phazes agree,
59
00:05:51,960 --> 00:05:56,260
it's given by this,
which is by the relation
60
00:05:56,260 --> 00:06:00,430
between the coordinates and
times of the two frames, just
61
00:06:00,430 --> 00:06:02,060
this quantity.
62
00:06:02,060 --> 00:06:12,020
So we can rewrite this as 2
pi over lambda x prime minus 2
63
00:06:12,020 --> 00:06:24,860
pi over lambda V 1 minus little
v over capital V t prime.
64
00:06:24,860 --> 00:06:27,500
I think I got the algebra right.
65
00:06:27,500 --> 00:06:31,640
2 pi over lambda, the sine--
66
00:06:31,640 --> 00:06:37,330
yes, I grouped those two
terms and rewrote in that way.
67
00:06:37,330 --> 00:06:39,310
So that is the phaze.
68
00:06:39,310 --> 00:06:42,160
And therefore, we
look at this phaze
69
00:06:42,160 --> 00:06:46,990
and see, oh, whenever
we have a wave,
70
00:06:46,990 --> 00:06:50,800
we can read the wave
number by looking
71
00:06:50,800 --> 00:06:54,940
at the factor
multiplying x, and we
72
00:06:54,940 --> 00:06:57,130
can read the
frequency by looking
73
00:06:57,130 --> 00:06:59,200
at the factor multiplying t.
74
00:06:59,200 --> 00:07:03,580
So you can do the same
thing in this case
75
00:07:03,580 --> 00:07:12,000
and read, therefore,
that omega prime,
76
00:07:12,000 --> 00:07:17,940
this whole quantity
is this, omega prime.
77
00:07:17,940 --> 00:07:21,940
And this is k prime,
because they can
78
00:07:21,940 --> 00:07:23,840
respond to the frame as prime.
79
00:07:23,840 --> 00:07:28,830
So omega prime is
equal to this 2 pi
80
00:07:28,830 --> 00:07:33,660
V over lambda, which
is omega, times 1
81
00:07:33,660 --> 00:07:46,090
minus V over V. And
k prime is equal to k
82
00:07:46,090 --> 00:07:50,790
or, what I wanted to show, that
lambda prime for a normal wave
83
00:07:50,790 --> 00:08:02,415
is equal to lambda for ordinary
wave moving in the medium.
84
00:08:05,460 --> 00:08:13,310
So at this moment, one
wonders, so what happened?
85
00:08:13,310 --> 00:08:14,540
What have we learned?
86
00:08:14,540 --> 00:08:19,160
Is that this wave function
is not like a sound wave.
87
00:08:19,160 --> 00:08:21,290
It's not like a water wave.
88
00:08:21,290 --> 00:08:24,200
We're doing everything
non-relativistic.
89
00:08:24,200 --> 00:08:29,070
But still, we're
seeing that you're not
90
00:08:29,070 --> 00:08:32,159
expected to have agreement.
91
00:08:32,159 --> 00:08:37,710
That is, if somebody
looks at one wave function
92
00:08:37,710 --> 00:08:40,799
and you look at the
same wave function,
93
00:08:40,799 --> 00:08:45,680
these two people will not
agree on the value of the wave
94
00:08:45,680 --> 00:08:47,740
function necessarily.
95
00:08:47,740 --> 00:08:56,070
So the things that we conclude--
so the conclusions are
96
00:08:56,070 --> 00:08:59,010
that waves are surprising.
97
00:08:59,010 --> 00:09:05,700
So size are not
directly measurable--
98
00:09:08,360 --> 00:09:15,780
measurable-- because if you had
a quantity for which you could
99
00:09:15,780 --> 00:09:19,830
measure, like a sound
wave or a water wave,
100
00:09:19,830 --> 00:09:21,660
and you could measure
aspects to it,
101
00:09:21,660 --> 00:09:24,630
they should agree between
different observables.
102
00:09:24,630 --> 00:09:26,940
So this is going to
be something that
103
00:09:26,940 --> 00:09:31,830
is not directly measurable--
not all of psi can be measured.
104
00:09:31,830 --> 00:09:35,210
Some of psi can be
measured, and you're already
105
00:09:35,210 --> 00:09:37,590
heard the hints of that.
106
00:09:37,590 --> 00:09:40,630
Because we said any
number that you multiply,
107
00:09:40,630 --> 00:09:43,230
you cannot measure, and in
the phase that you multiply,
108
00:09:43,230 --> 00:09:44,550
you cannot measure.
109
00:09:44,550 --> 00:09:47,410
So complex numbers
can't be measured,
110
00:09:47,410 --> 00:09:48,820
you measure real numbers.
111
00:09:48,820 --> 00:09:54,060
So at the end of the day, these
are not directly measurable,
112
00:09:54,060 --> 00:09:55,930
per se.
113
00:09:55,930 --> 00:10:05,950
The second thing is that
they're not Galilean invariant,
114
00:10:05,950 --> 00:10:09,130
and that sets the stage
to that problem 6.
115
00:10:09,130 --> 00:10:11,620
You see, the fact
that this phaze that
116
00:10:11,620 --> 00:10:15,280
controls these waves
is Galilean invariant
117
00:10:15,280 --> 00:10:19,390
led you to the quality
of the wavelengths,
118
00:10:19,390 --> 00:10:22,850
but these wavelengths
don't do that.
119
00:10:22,850 --> 00:10:26,520
The de Broglie wavelengths
don't transform
120
00:10:26,520 --> 00:10:31,800
as they would do for a
Galilean invariant wave.
121
00:10:31,800 --> 00:10:35,220
Therefore, this thing is
not Galilean invariant,
122
00:10:35,220 --> 00:10:37,890
and what does that mean?
123
00:10:37,890 --> 00:10:41,640
That if you have
two people and you
124
00:10:41,640 --> 00:10:46,840
ask, what is the value of the
wave function here at 103,
125
00:10:46,840 --> 00:10:49,960
the two observers might give
you a different complex number
126
00:10:49,960 --> 00:10:51,120
for the wave function.
127
00:10:51,120 --> 00:10:54,350
They will just not agree.
128
00:10:54,350 --> 00:10:56,480
Not all is lost,
because you will
129
00:10:56,480 --> 00:11:00,380
find how their measurements
can be compared.
130
00:11:00,380 --> 00:11:02,210
That will be the
task of the problem.
131
00:11:02,210 --> 00:11:06,350
How-- if you have a wave
function, how does your friend,
132
00:11:06,350 --> 00:11:08,960
that is moving
with some velocity,
133
00:11:08,960 --> 00:11:10,160
measure the wave function?
134
00:11:10,160 --> 00:11:12,750
What does this other
person measure?
135
00:11:12,750 --> 00:11:17,540
So the end result, if you have
a point here at some time t,
136
00:11:17,540 --> 00:11:22,190
the wave function
psi of x and t is not
137
00:11:22,190 --> 00:11:24,650
going to be the same as
the wave function measured
138
00:11:24,650 --> 00:11:27,930
by the prime
observer at x prime t
139
00:11:27,930 --> 00:11:34,510
prime, so this point is the
point x and t or x prime and t
140
00:11:34,510 --> 00:11:35,060
prime.
141
00:11:35,060 --> 00:11:38,437
These are two different
labels for the same point.
142
00:11:38,437 --> 00:11:40,020
You're talking about
the wave function
143
00:11:40,020 --> 00:11:42,950
at the same point
at the same time.
144
00:11:42,950 --> 00:11:44,400
You still don't agree.
145
00:11:44,400 --> 00:11:46,220
These two people will not agree.
146
00:11:46,220 --> 00:11:48,800
If they agreed,
this wave function
147
00:11:48,800 --> 00:11:51,710
would have a simpler
transformation law
148
00:11:51,710 --> 00:11:55,070
with a wavelength
that this can serve.
149
00:11:55,070 --> 00:12:03,210
So by simply discussing
the Galilean properties
150
00:12:03,210 --> 00:12:08,940
of this wave, we're led to know
that the de Broglie waves are
151
00:12:08,940 --> 00:12:14,070
not like normal matter
waves that propagate
152
00:12:14,070 --> 00:12:16,970
in a medium or simple.