1
00:00:00,500 --> 00:00:04,120
PROFESSOR: That's how
it looks, a resonance.

2
00:00:04,120 --> 00:00:08,751
You can see it basically
in the phase shift.

3
00:00:08,751 --> 00:00:16,590
And great increase of the
phase of almost minutely pi

4
00:00:16,590 --> 00:00:20,632
over a very small
change of energy.

5
00:00:20,632 --> 00:00:26,200
And it should [INAUDIBLE]
with a very big [INAUDIBLE]..

6
00:00:26,200 --> 00:00:28,952
So this is how it looks.

7
00:00:28,952 --> 00:00:33,816
And I want to now
proceed, after if there

8
00:00:33,816 --> 00:00:36,370
are some questions,
of how do we search

9
00:00:36,370 --> 00:00:41,750
for residences a little
more mathematically rather

10
00:00:41,750 --> 00:00:43,360
than plotting them.

11
00:00:43,360 --> 00:00:46,010
How could I write an
equation for a resonance.

12
00:00:46,010 --> 00:00:48,887
Cannot say, oh, the
phase changes fast.

13
00:00:48,887 --> 00:00:54,090
Well, that's not a very
nice way of saying it.

14
00:00:54,090 --> 00:00:54,910
It's good.

15
00:00:54,910 --> 00:00:55,990
It's intuitive.

16
00:00:55,990 --> 00:00:59,110
But we should be
able to do better.

17
00:00:59,110 --> 00:01:01,230
So how do I find resonances?

18
00:01:01,230 --> 00:01:04,633
So let's model
resonances a little bit.

19
00:01:04,633 --> 00:01:06,565
How do we find resonances?

20
00:01:16,240 --> 00:01:19,165
So let's model this behavior.

21
00:01:19,165 --> 00:01:23,290
By that is writing
a formula that

22
00:01:23,290 --> 00:01:27,450
is simple enough that seems
to capture what's happening.

23
00:01:27,450 --> 00:01:29,670
And that formula's
going to inspire

24
00:01:29,670 --> 00:01:34,840
us to think of resonances
perhaps a little more clearly.

25
00:01:34,840 --> 00:01:42,480
So suppose you have a
resonance near k equal alpha.

26
00:01:42,480 --> 00:01:46,140
I claim the following
formula would be a good way

27
00:01:46,140 --> 00:01:47,490
to represent the resonance?

28
00:01:47,490 --> 00:01:56,060
We would say that tan
delta is equal to beta

29
00:01:56,060 --> 00:01:58,950
over alpha minus k.

30
00:02:02,846 --> 00:02:05,438
Or-- yeah, we would say that.

31
00:02:05,438 --> 00:02:16,680
[INAUDIBLE] Or if you wish,
delta is tan minus 1 of beta

32
00:02:16,680 --> 00:02:19,332
over alpha minus k.

33
00:02:19,332 --> 00:02:21,860
Why is that reasonable?

34
00:02:21,860 --> 00:02:28,950
It's a little surprising,
but not that surprising.

35
00:02:28,950 --> 00:02:36,730
You see that-- delta is
equal to minus pi over 2.

36
00:02:36,730 --> 00:02:39,630
The tangent of delta
goes to infinity.

37
00:02:39,630 --> 00:02:43,562
So there's something
going on here in which you

38
00:02:43,562 --> 00:02:44,960
have this property.

39
00:02:48,330 --> 00:02:50,860
So let's plot this.

40
00:02:50,860 --> 00:02:55,956
So let's plot beta
over alpha minus k.

41
00:02:55,956 --> 00:03:00,400
You need a clock
to understand this.

42
00:03:00,400 --> 00:03:03,700
So this is k, and we're
plotting this quantity.

43
00:03:03,700 --> 00:03:06,586
Well, it's going to go
crazy at k equal alpha.

44
00:03:06,586 --> 00:03:08,320
That we know.

45
00:03:08,320 --> 00:03:11,090
When k is less than
alpha, I'm going

46
00:03:11,090 --> 00:03:16,090
to assume that alpha
and beta are positive.

47
00:03:16,090 --> 00:03:17,996
They both have units of k.

48
00:03:24,420 --> 00:03:29,886
And when k is less than
alpha-- we begin here--

49
00:03:29,886 --> 00:03:33,220
then this denominator
is positive,

50
00:03:33,220 --> 00:03:38,000
the numerator is positive,
ratio is positive.

51
00:03:38,000 --> 00:03:39,570
It's small, maybe.

52
00:03:39,570 --> 00:03:42,407
And then suddenly,
when k reaches alpha

53
00:03:42,407 --> 00:03:45,365
it goes to infinity.

54
00:03:45,365 --> 00:03:51,540
So it's going to be like that.

55
00:03:51,540 --> 00:03:57,460
Now, it actually
is true that when

56
00:03:57,460 --> 00:04:03,586
k differs by alpha by
beta, it reaches value 1.

57
00:04:03,586 --> 00:04:08,056
So here is alpha minus beta.

58
00:04:08,056 --> 00:04:10,790
That point it reaches value 1.

59
00:04:13,440 --> 00:04:19,000
So if I want this
thing to be very sharp,

60
00:04:19,000 --> 00:04:23,540
I need beta to be small
so that it's little

61
00:04:23,540 --> 00:04:28,040
until it reaches beta within--

62
00:04:28,040 --> 00:04:32,040
distance beta within alpha,
and then it shoots up.

63
00:04:32,040 --> 00:04:40,570
So I want beta to be
small for sharp behavior.

64
00:04:48,650 --> 00:04:54,354
On the other hand here,
it goes the other way.

65
00:04:54,354 --> 00:04:57,560
It goes from minus
infinity back to 0,

66
00:04:57,560 --> 00:05:05,110
and has value minus
1 at alpha plus beta.

67
00:05:05,110 --> 00:05:10,420
So within minus beta, and
beta off of the center

68
00:05:10,420 --> 00:05:13,015
alpha, most of
the things happen.

69
00:05:16,980 --> 00:05:20,820
If we plot now the
tangent of this,

70
00:05:20,820 --> 00:05:32,610
or the arctangent of this, tan
minus 1 of beta, alpha minus k,

71
00:05:32,610 --> 00:05:37,530
well, if the tangent of
an angle is very little,

72
00:05:37,530 --> 00:05:40,250
the angle can be taken
to be very little.

73
00:05:40,250 --> 00:05:45,080
At this point, it
will reach pi over 2,

74
00:05:45,080 --> 00:05:49,340
so the angle is little,
will go to pi over 2,

75
00:05:49,340 --> 00:05:55,471
and then quickly becomes
larger than pi over 2,

76
00:05:55,471 --> 00:05:57,780
you're thinking tangents.

77
00:05:57,780 --> 00:06:02,610
So the tangent is going up,
is blowing up at pi over 2.

78
00:06:02,610 --> 00:06:05,420
Then continuously, it
goes to minus pi over 2,

79
00:06:05,420 --> 00:06:09,350
and then continuously goes
to 0 so it reaches pi.

80
00:06:14,840 --> 00:06:19,080
So this is the
behavior of delta.

81
00:06:19,080 --> 00:06:22,950
Delta is this tan minus
1 of beta over that.

82
00:06:22,950 --> 00:06:25,630
And delta is doing
the right thing.

83
00:06:25,630 --> 00:06:32,200
It's doing this
kind of behavior.

84
00:06:32,200 --> 00:06:33,230
There is a shift.

85
00:06:33,230 --> 00:06:37,090
I could add a constant
here to produce this shift,

86
00:06:37,090 --> 00:06:40,410
but it's not important
at this moment.

87
00:06:40,410 --> 00:06:42,320
The resonance is
doing this thing,

88
00:06:42,320 --> 00:06:44,980
up to a total shift
of pi that doesn't

89
00:06:44,980 --> 00:06:46,892
change the tangent of an angle.

90
00:06:49,860 --> 00:06:54,480
So this is one way
of modeling what's

91
00:06:54,480 --> 00:06:58,320
happening to the phase
shift near our resonance.

92
00:06:58,320 --> 00:07:00,400
So let's explore
it a little more.

93
00:07:03,450 --> 00:07:06,690
I can do a couple
of calculations.

94
00:07:06,690 --> 00:07:15,130
For example, I can compute
what is d delta dk at k

95
00:07:15,130 --> 00:07:16,970
equals alpha.

96
00:07:16,970 --> 00:07:19,240
That's should be
a nice quantity.

97
00:07:19,240 --> 00:07:21,261
Is a derivative of the face.

98
00:07:28,135 --> 00:07:31,250
At the resonance,
at the position

99
00:07:31,250 --> 00:07:33,090
alpha of the resonance.

100
00:07:33,090 --> 00:07:37,563
So here's delta, here
is k, and there's

101
00:07:37,563 --> 00:07:39,660
the derivative of this k.

102
00:07:39,660 --> 00:07:42,600
And how should it be?

103
00:07:42,600 --> 00:07:49,000
Well, basically, the
phase changes by amount pi

104
00:07:49,000 --> 00:07:52,760
over a distance beta or 2 beta.

105
00:07:52,760 --> 00:07:57,450
So this must be a
number divided by beta.

106
00:07:57,450 --> 00:08:01,080
You can calculate this
derivative from this equation.

107
00:08:01,080 --> 00:08:03,550
It's a nice exercise.

108
00:08:03,550 --> 00:08:07,472
It's actually just 1 over beta.

109
00:08:07,472 --> 00:08:13,130
That's a result. 1 over beta.

110
00:08:13,130 --> 00:08:17,210
The other quantity that
is nice to understand

111
00:08:17,210 --> 00:08:20,120
is how does this
scattering amplitude

112
00:08:20,120 --> 00:08:23,430
behave near the resonance.

113
00:08:23,430 --> 00:08:27,640
So what is the
value of as squared?

114
00:08:27,640 --> 00:08:32,409
Oh, that's the absolute
value of psi s squared,

115
00:08:32,409 --> 00:08:35,179
which is sine squared delta.

116
00:08:35,179 --> 00:08:37,160
That's the same
thing as As squared.

117
00:08:40,970 --> 00:08:44,944
Well, you know what is
the tangent of delta?

118
00:08:44,944 --> 00:08:49,890
A little trigonometric play
should be able to do it,

119
00:08:49,890 --> 00:08:53,500
and can give you the
sin squared delta.

120
00:08:53,500 --> 00:08:56,545
And here is the answer.

121
00:08:56,545 --> 00:09:01,950
It's beta squared over
beta squared plus alpha

122
00:09:01,950 --> 00:09:04,112
minus k squared.

123
00:09:04,112 --> 00:09:07,970
Kind of a nice,
almost bell shape.

124
00:09:07,970 --> 00:09:10,860
Of course, it's polynomial,
but it looks a little

125
00:09:10,860 --> 00:09:18,200
like just a nice symmetric
shape around alpha equal k.

126
00:09:18,200 --> 00:09:23,480
Now, this division is so famous
it has been given a name.

127
00:09:23,480 --> 00:09:27,100
It's called the
Breit-Wigner distribution.

128
00:09:27,100 --> 00:09:28,980
Breit-Wigner.

129
00:09:28,980 --> 00:09:43,100
But it's described as the
Breit-Wigner distribution,

130
00:09:43,100 --> 00:09:45,865
and it's usually referring
to terms of energy.

131
00:09:48,535 --> 00:09:50,280
Of energy, not momentum.

132
00:09:55,070 --> 00:09:59,840
So-- and it's--

133
00:09:59,840 --> 00:10:03,920
what should happen to
scattering amplitude in general

134
00:10:03,920 --> 00:10:06,710
when you have a resonance.

135
00:10:06,710 --> 00:10:12,950
So the way to do
this calculation now

136
00:10:12,950 --> 00:10:17,300
is to say, well, what
is alpha minus k?

137
00:10:17,300 --> 00:10:22,595
Let's try to relate it to the
energy minus the energy at k

138
00:10:22,595 --> 00:10:24,810
equal alpha.

139
00:10:24,810 --> 00:10:28,300
Well, this is h
squared k squared

140
00:10:28,300 --> 00:10:37,810
over 2 m minus h squared
alpha squared over 2 m,

141
00:10:37,810 --> 00:10:44,420
which is h squared over 2 m,
k squared minus alpha squared.

142
00:10:48,660 --> 00:10:55,250
On the other hand, I have
here alpha minus k squared.

143
00:10:55,250 --> 00:11:00,190
I don't have k squared
minus alpha squared.

144
00:11:00,190 --> 00:11:05,200
So-- approximations.

145
00:11:05,200 --> 00:11:12,120
if the resonance is narrow
enough, if beta is small,

146
00:11:12,120 --> 00:11:14,480
let's do an approximation.

147
00:11:14,480 --> 00:11:16,350
We do h squared over--

148
00:11:16,350 --> 00:11:17,970
everybody knows
this approximation,

149
00:11:17,970 --> 00:11:19,920
shouldn't be afraid of doing it.

150
00:11:19,920 --> 00:11:25,405
It's alpha-- how
could I write it--

151
00:11:25,405 --> 00:11:30,295
k minus alpha
times k plus alpha.

152
00:11:34,015 --> 00:11:39,400
And the approximation is that
all the interesting thing

153
00:11:39,400 --> 00:11:42,575
comes from the difference
between k and alpha,

154
00:11:42,575 --> 00:11:45,300
how close k is to alpha.

155
00:11:45,300 --> 00:11:49,925
So when k is close to
alpha, all the dependence

156
00:11:49,925 --> 00:11:51,770
is going to be here.

157
00:11:51,770 --> 00:11:56,880
This is going to be about 2
alpha when k is near alpha.

158
00:11:56,880 --> 00:11:59,600
And if it's a little
more than that,

159
00:11:59,600 --> 00:12:02,220
it doesn't matter,
because it's [INAUDIBLE]..

160
00:12:02,220 --> 00:12:06,710
So this could be
approximated to 2 alpha,

161
00:12:06,710 --> 00:12:14,050
and therefore this becomes h
squared alpha over m times k

162
00:12:14,050 --> 00:12:15,030
minus alpha.

163
00:12:19,450 --> 00:12:20,645
So that's a little help.

164
00:12:23,300 --> 00:12:30,570
Then size of s squared,
doing a little more

165
00:12:30,570 --> 00:12:33,080
of algebra with the
constants there.

166
00:12:33,080 --> 00:12:37,540
Probably you want to
do it with two lines.

167
00:12:37,540 --> 00:12:38,560
It's tricky.

168
00:12:38,560 --> 00:12:40,720
It's really simple.

169
00:12:40,720 --> 00:12:42,820
It's always written
in this form--

170
00:12:42,820 --> 00:12:51,540
1 over 4 gamma squared over
e minus e alpha squared

171
00:12:51,540 --> 00:12:57,010
plus 1 over four gamma
squared, and that's

172
00:12:57,010 --> 00:13:00,780
the so-called
Breit-Wigner distribution.

173
00:13:00,780 --> 00:13:07,670
And gamma is a
funny constant here.

174
00:13:07,670 --> 00:13:09,630
We'll try to
understand it better.

175
00:13:09,630 --> 00:13:14,050
2 alpha beta h squared over m.

176
00:13:17,600 --> 00:13:21,410
It has to be something that
depends on alpha and beta,

177
00:13:21,410 --> 00:13:23,960
because after all,
we weren't modeling

178
00:13:23,960 --> 00:13:25,820
the resonance with alpha beta.

179
00:13:28,850 --> 00:13:33,000
So this curve is very famous.

180
00:13:33,000 --> 00:13:40,224
That's the distribution of
the scattering amplitude

181
00:13:40,224 --> 00:13:43,010
over energies whenever
you have a resonance.

182
00:13:43,010 --> 00:13:45,130
So we should plot it.

183
00:13:45,130 --> 00:13:48,200
You have an e alpha.

184
00:13:48,200 --> 00:13:52,730
You have an e alpha
plus gamma over 2

185
00:13:52,730 --> 00:13:57,650
and e alpha minus gamma over 2.

186
00:13:57,650 --> 00:14:04,220
But the energy minus e alpha
is equal to the gamma over 2,

187
00:14:04,220 --> 00:14:08,830
you get the gamma squared over
4 so the total amplitude goes

188
00:14:08,830 --> 00:14:13,960
down to 1/2 of the
usual amplitudes.

189
00:14:13,960 --> 00:14:18,970
When the energy is equal
to e alpha, you get 1.

190
00:14:18,970 --> 00:14:23,894
1 for psi s squared.

191
00:14:23,894 --> 00:14:30,150
But when the energy differs
from e alpha by gamma over 2,

192
00:14:30,150 --> 00:14:32,600
you get half.

193
00:14:32,600 --> 00:14:42,160
So-- actually, I'm not
sure of the deflection

194
00:14:42,160 --> 00:14:43,860
point, where it is.

195
00:14:43,860 --> 00:14:45,224
Probably not there.

196
00:14:45,224 --> 00:14:47,684
Or is it there?

197
00:14:47,684 --> 00:14:50,174
I don't know.

198
00:14:50,174 --> 00:14:53,440
I drew it as if it is there.

199
00:14:53,440 --> 00:15:09,730
So that's the distribution, and
the width over here is gamma.

200
00:15:09,730 --> 00:15:14,780
So gamma is called the
width at half power,

201
00:15:14,780 --> 00:15:16,546
or at half intensity.

202
00:15:19,650 --> 00:15:21,780
Yeah.

203
00:15:21,780 --> 00:15:32,790
Width-- the half width
of the distribution.