1
00:00:00,087 --> 00:00:00,670
PROFESSOR: OK.

2
00:00:00,670 --> 00:00:03,750
So what I'm going
to try to do now

3
00:00:03,750 --> 00:00:11,310
is set up again
this equation and do

4
00:00:11,310 --> 00:00:17,630
the analog of what
we're doing there

5
00:00:17,630 --> 00:00:25,940
and try to determine this
function fk in some nice way.

6
00:00:29,180 --> 00:00:29,695
All right.

7
00:00:33,690 --> 00:00:37,210
So let's think of this equation.

8
00:00:37,210 --> 00:00:38,970
I want to do it in pieces.

9
00:00:38,970 --> 00:00:50,410
So psi of r is going to
be equal to some formula.

10
00:00:50,410 --> 00:01:00,070
And then it has to be equal
to this right hand side.

11
00:01:00,070 --> 00:01:06,820
Let's write the e to the ikz
the way we've done it before.

12
00:01:06,820 --> 00:01:08,080
It's up there.

13
00:01:08,080 --> 00:01:09,910
So I'll write it here.

14
00:01:09,910 --> 00:01:15,520
Square root of 4 pi
over k square, sum of l

15
00:01:15,520 --> 00:01:22,090
equals 0 to infinity,
square root of 2l plus 1, i

16
00:01:22,090 --> 00:01:34,690
to the l yl 0 1 over 2 i e
to the ikr minus l pi over 2

17
00:01:34,690 --> 00:01:42,711
over r minus e the minus
ikr minus l pi over 2.

18
00:01:42,711 --> 00:01:43,210
Wow.

19
00:01:43,210 --> 00:01:47,320
It's tiring this r.

20
00:01:47,320 --> 00:01:55,990
Plus f of k of theta
e to the ikr over r.

21
00:02:07,883 --> 00:02:08,382
OK.

22
00:02:11,546 --> 00:02:13,360
So what do we have here?

23
00:02:13,360 --> 00:02:17,360
We've written the right
hand side of this equation.

24
00:02:17,360 --> 00:02:18,420
I copied it.

25
00:02:18,420 --> 00:02:24,765
I have not done anything except
taking r much greater than a.

26
00:02:28,150 --> 00:02:32,680
Because otherwise in the
plane wave into the ikz,

27
00:02:32,680 --> 00:02:35,770
I could not have expanded
the Bessel functions

28
00:02:35,770 --> 00:02:38,620
unless I took r greater than a.

29
00:02:38,620 --> 00:02:43,750
But that's good, because
we now have our waves.

30
00:02:43,750 --> 00:02:45,020
OK.

31
00:02:45,020 --> 00:02:47,600
We have our waves there.

32
00:02:47,600 --> 00:02:52,780
Now, look at this
right hand side.

33
00:02:52,780 --> 00:02:58,780
Where is the incoming wave
in this right hand side?

34
00:02:58,780 --> 00:03:00,720
The incoming wave is here.

35
00:03:05,020 --> 00:03:07,980
That's the only term
that is incoming,

36
00:03:07,980 --> 00:03:10,700
because this is
an outgoing wave,

37
00:03:10,700 --> 00:03:14,510
and this is an outgoing wave.

38
00:03:14,510 --> 00:03:19,960
So if I want to write
the left hand side,

39
00:03:19,960 --> 00:03:24,790
the incoming wave of
the left hand side

40
00:03:24,790 --> 00:03:27,000
has to be equal to this wave.

41
00:03:29,990 --> 00:03:33,650
And of course, the outgoing
wave of the left hand side

42
00:03:33,650 --> 00:03:37,280
will also have to be equal
to whatever is outgoing here,

43
00:03:37,280 --> 00:03:41,180
but the incoming must be this.

44
00:03:41,180 --> 00:03:43,640
So I'm going to write
this left hand side

45
00:03:43,640 --> 00:03:52,220
and already use this and put
4 pi over k square sum of l

46
00:03:52,220 --> 00:03:58,600
equals 0 to infinity 2l
plus 1 i to the l y l

47
00:03:58,600 --> 00:04:09,730
0 1 over 2 pi, big parentheses,
and one outgoing wave

48
00:04:09,730 --> 00:04:21,550
and one incoming wave minus
ikr minus l pi over 2 over r.

49
00:04:21,550 --> 00:04:26,380
And here, I don't
know what to put,

50
00:04:26,380 --> 00:04:29,710
but I've put already
there on the left hand

51
00:04:29,710 --> 00:04:34,540
side of this
equation for psi of r

52
00:04:34,540 --> 00:04:40,510
for the full solution, a wave
that matches the right hand

53
00:04:40,510 --> 00:04:46,930
side, because it has
the same incoming wave.

54
00:04:59,170 --> 00:05:03,910
And now, I'm going to use
some physical intuition

55
00:05:03,910 --> 00:05:08,240
to guess what we'll have
to put on this part.

56
00:05:08,240 --> 00:05:13,610
This is the step that requires
a little imagination, not too

57
00:05:13,610 --> 00:05:18,510
much, because we already
did something similar here.

58
00:05:18,510 --> 00:05:30,300
So what's happening
here and here is

59
00:05:30,300 --> 00:05:36,040
intuition I think you should
keep after weeks of this course

60
00:05:36,040 --> 00:05:38,390
when it's all forgotten,
there's some intuition

61
00:05:38,390 --> 00:05:40,880
that you should keep.

62
00:05:40,880 --> 00:05:46,520
And it's about this scattering
happening for each partial wave

63
00:05:46,520 --> 00:05:47,440
independent.

64
00:05:47,440 --> 00:05:49,325
Yes.

65
00:05:49,325 --> 00:05:50,780
AUDIENCE: 2i.

66
00:05:50,780 --> 00:05:53,080
PROFESSOR: 2i, yes.

67
00:05:53,080 --> 00:05:54,574
No, 2 pi there.

68
00:05:58,711 --> 00:05:59,210
Yes.

69
00:06:03,800 --> 00:06:05,180
Thank you.

70
00:06:05,180 --> 00:06:09,080
So here it is.

71
00:06:09,080 --> 00:06:15,320
This is a solution, and we've
got the intuition already.

72
00:06:15,320 --> 00:06:19,770
I will justify this later,
of course, very precisely.

73
00:06:19,770 --> 00:06:21,830
But I think that
this one you need

74
00:06:21,830 --> 00:06:24,920
to have a little bit
of an intuition of what

75
00:06:24,920 --> 00:06:26,540
you should do.

76
00:06:26,540 --> 00:06:30,650
And first, we said
each l works separately

77
00:06:30,650 --> 00:06:36,060
to create a solution of
the Schrodinger equation.

78
00:06:36,060 --> 00:06:40,050
That's superposition,
and it's [INAUDIBLE]..

79
00:06:40,050 --> 00:06:43,200
Each l is working separately.

80
00:06:43,200 --> 00:06:47,250
Each l is like a
scattering problem.

81
00:06:47,250 --> 00:06:54,140
Each l has a wave that comes
in and a wave that comes out,

82
00:06:54,140 --> 00:07:02,240
because these things, j and n,
have waves, and they have an

83
00:07:02,240 --> 00:07:03,320
in and out.

84
00:07:03,320 --> 00:07:08,080
So these have some in
wave and some out wave.

85
00:07:08,080 --> 00:07:12,210
And if each wave
works separately,

86
00:07:12,210 --> 00:07:18,470
it has an in wave and then out
wave, in a scattering problem,

87
00:07:18,470 --> 00:07:22,070
these waves must have
the same amplitude,

88
00:07:22,070 --> 00:07:25,940
because otherwise they wouldn't
have the same probability

89
00:07:25,940 --> 00:07:29,960
current, and probability
would get stuck.

90
00:07:29,960 --> 00:07:37,070
So this must be an outgoing
wave having this same amplitude

91
00:07:37,070 --> 00:07:37,965
as this wave.

92
00:07:40,790 --> 00:07:43,490
And by the argument
we have here,

93
00:07:43,490 --> 00:07:46,910
it just differs by a phase.

94
00:07:46,910 --> 00:08:00,650
So we'll put here e to the ikr
minus l pi over 2 plus 2i delta

95
00:08:00,650 --> 00:08:05,180
l over r.

96
00:08:05,180 --> 00:08:10,250
So that this wave, spherical
wave, that it's outgoing,

97
00:08:10,250 --> 00:08:13,460
it has the same
amplitude as this one,

98
00:08:13,460 --> 00:08:15,600
and cannot be the same.

99
00:08:15,600 --> 00:08:19,160
The only difference
can be a phase shift,

100
00:08:19,160 --> 00:08:20,670
and that's the phase shift.

101
00:08:23,750 --> 00:08:30,490
So your picture is scattering
in three dimensions.

102
00:08:30,490 --> 00:08:34,400
Looks like, OK, you
threw in a plane wave

103
00:08:34,400 --> 00:08:39,669
and out came a
spherical wave out.

104
00:08:39,669 --> 00:08:43,030
The other picture that is more
consistent with the way you

105
00:08:43,030 --> 00:08:48,220
solve it is that you have an
infinite set of partial waves

106
00:08:48,220 --> 00:08:52,735
for different l's, each
one scattering, the l

107
00:08:52,735 --> 00:08:57,460
equals 0, the equal 1,
the l equal 2, all of them

108
00:08:57,460 --> 00:09:00,320
scattering.

109
00:09:00,320 --> 00:09:02,770
So this corresponds
to an [? ansatz ?]

110
00:09:02,770 --> 00:09:07,960
in terms of phase
shifts, and now you

111
00:09:07,960 --> 00:09:11,230
can say you've
parameterized your ignorance

112
00:09:11,230 --> 00:09:13,000
in a physical way.

113
00:09:13,000 --> 00:09:17,290
You've discovered that all that
characterizes the scattering

114
00:09:17,290 --> 00:09:20,740
is, as it was in one
dimension, a phase shift.

115
00:09:20,740 --> 00:09:24,130
In one dimension, there
was a single phase shift,

116
00:09:24,130 --> 00:09:29,170
because you didn't have
all these general solutions

117
00:09:29,170 --> 00:09:31,060
that you had in
three dimensions.

118
00:09:31,060 --> 00:09:34,410
Your energy eigenstates
were momentum eigenstates

119
00:09:34,410 --> 00:09:37,510
there were
non-degenerate really.

120
00:09:37,510 --> 00:09:41,560
There was just a couple of
momentum eigenstates, wave in

121
00:09:41,560 --> 00:09:42,640
and waves out.

122
00:09:42,640 --> 00:09:44,980
Here is infinitely degenerate.

123
00:09:44,980 --> 00:09:46,940
There's spherical stuff.

124
00:09:46,940 --> 00:09:51,890
So there's a phase shift
for each value of l.

125
00:09:51,890 --> 00:09:55,550
So we've parameterized the
physics of the scattering

126
00:09:55,550 --> 00:09:59,360
problem in terms of phase
shift, and now, it's

127
00:09:59,360 --> 00:10:03,320
interesting to try to figure
out what is this quantity

128
00:10:03,320 --> 00:10:06,320
after all in terms
of the phase shifts.

129
00:10:06,320 --> 00:10:08,540
It's already here.

130
00:10:08,540 --> 00:10:10,680
We just have to solve it.

131
00:10:10,680 --> 00:10:16,010
So from this equation, I now
can say that this term cancels

132
00:10:16,010 --> 00:10:21,530
with this term, and now,
I can solve for this term

133
00:10:21,530 --> 00:10:26,870
f of theta e to the ikr by
collecting these other two

134
00:10:26,870 --> 00:10:27,665
terms together.

135
00:10:32,880 --> 00:10:41,240
And therefore, fk of theta e
to the ikr over r is equal,

136
00:10:41,240 --> 00:10:45,870
and it's exactly the same thing
I did here, cancel here, pass,

137
00:10:45,870 --> 00:10:49,380
and the mathematics is going
to be completely analogous,

138
00:10:49,380 --> 00:10:52,835
except that they have to
carry all that sum there.

139
00:10:57,570 --> 00:10:58,780
No big deal.

140
00:10:58,780 --> 00:11:00,210
So what is it?

141
00:11:00,210 --> 00:11:05,040
Square root of 4 pi
over k, sum from l

142
00:11:05,040 --> 00:11:12,820
equals 0 to infinity, 2l
plus 1, i to the l y l0.

143
00:11:16,340 --> 00:11:17,765
1 over 2i.

144
00:11:23,390 --> 00:11:27,230
OK, 1 over 2i.

145
00:11:27,230 --> 00:11:29,390
OK, I'll do it this way.

146
00:11:29,390 --> 00:11:39,290
e to the 2i delta l minus 1 e
to the ikr e to the minus il pi

147
00:11:39,290 --> 00:11:40,370
over 2.

148
00:11:40,370 --> 00:11:42,620
I think I got everything there.

149
00:11:42,620 --> 00:11:46,580
I put the first term
on the right hand side,

150
00:11:46,580 --> 00:11:48,410
I moved it to the
left hand side.

151
00:11:48,410 --> 00:11:51,960
The coefficient
was all the same.

152
00:11:51,960 --> 00:11:55,520
This was the coefficient they
both had the same coefficient.

153
00:11:55,520 --> 00:11:59,990
Then I just have to subtract
these two exponentials over r.

154
00:11:59,990 --> 00:12:04,040
So I did forget the r.

155
00:12:04,040 --> 00:12:06,500
So I just subtract
the two exponentials.

156
00:12:06,500 --> 00:12:13,380
Both exponentials have the e
to the ikr minus l pi over 2.

157
00:12:13,380 --> 00:12:17,390
And the difference is that the
first exponential on the top

158
00:12:17,390 --> 00:12:21,527
has the extra e to the
2i delta l minus 1,

159
00:12:21,527 --> 00:12:22,610
and the other one doesn't.

160
00:12:28,200 --> 00:12:29,635
So what do we get here?

161
00:12:33,740 --> 00:12:44,190
This part is e to the
i delta sine delta,

162
00:12:44,190 --> 00:12:55,030
and this part is e to
the minus i pi over 2

163
00:12:55,030 --> 00:13:03,150
to the power l, which
is minus i to the l,

164
00:13:03,150 --> 00:13:12,570
and i to the l times minus i
to the l is happily just 1.

165
00:13:12,570 --> 00:13:16,620
i times minus i is 1,
and 1 to the l is 1.

166
00:13:16,620 --> 00:13:25,740
So this term and
this term cancel.

167
00:13:25,740 --> 00:13:29,190
So finally, and I can
cancel happily they

168
00:13:29,190 --> 00:13:31,170
r dependence is all the same.

169
00:13:31,170 --> 00:13:36,510
I can cancel this r
dependence, this r dependence.

170
00:13:36,510 --> 00:13:41,820
And finally, we've
got fk of theta

171
00:13:41,820 --> 00:13:48,210
equals square root of
4 pi over k sum from l

172
00:13:48,210 --> 00:13:57,150
equals 0 to infinity square
root 2l plus 1 y L0 of theta e

173
00:13:57,150 --> 00:14:03,120
to the i delta l sine delta l.

174
00:14:10,370 --> 00:14:15,590
So that said, that's
our formula for fk

175
00:14:15,590 --> 00:14:17,240
in terms of the phase shift.

176
00:14:25,790 --> 00:14:26,980
So what have we achieved?

177
00:14:26,980 --> 00:14:30,720
We want f of k, because that
gives us the cross section.

178
00:14:30,720 --> 00:14:35,110
What we have figured out is
that the calculation of fk

179
00:14:35,110 --> 00:14:37,970
really requires knowing
the phase shift.

180
00:14:37,970 --> 00:14:43,720
And the phase shifts are defined
by that formula over there,

181
00:14:43,720 --> 00:14:49,240
where we have estimated
how one wave is connected

182
00:14:49,240 --> 00:14:52,450
to the other one, the
incoming and the outgoing

183
00:14:52,450 --> 00:14:57,220
for a given fixed f, for a
given partial wave, how they're

184
00:14:57,220 --> 00:15:00,780
offset by this phase shift.