1
00:00:00,000 --> 00:00:00,950
PROFESSOR: Great.

2
00:00:00,950 --> 00:00:03,800
So I will begin
with phase shifts

3
00:00:03,800 --> 00:00:09,920
and do the introduction of how
to make sure we can really--

4
00:00:09,920 --> 00:00:14,450
so this is the
important part of this.

5
00:00:14,450 --> 00:00:18,050
Just like when we added the
reflected and transmitted wave

6
00:00:18,050 --> 00:00:20,570
we could find the
solution I'm going

7
00:00:20,570 --> 00:00:25,760
to try to explain
why with this things

8
00:00:25,760 --> 00:00:27,740
we can find
solutions in general.

9
00:00:30,360 --> 00:00:35,840
So this is the subject
of partial waves,

10
00:00:35,840 --> 00:00:43,130
and it's a nice subject,
a little technical.

11
00:00:43,130 --> 00:00:47,480
There might seem to be
a lot of formulas here,

12
00:00:47,480 --> 00:00:54,590
but the ideas are relatively
simple once one keeps in mind

13
00:00:54,590 --> 00:00:57,020
the one dimensional analogies.

14
00:00:57,020 --> 00:01:00,950
The one dimensional analogies
are very valuable here,

15
00:01:00,950 --> 00:01:05,269
and we will
emphasize them a lot.

16
00:01:05,269 --> 00:01:10,640
So we will discuss partial
waves and face shifts.

17
00:01:32,540 --> 00:01:36,635
So it's time to simplify
this matters a little bit.

18
00:01:40,310 --> 00:01:43,550
And to do that I
will assume from now

19
00:01:43,550 --> 00:01:55,830
on that the potential is central
so v of r is equal to v of r.

20
00:01:55,830 --> 00:02:00,330
That will simplify the
azimuthal dependence.

21
00:02:00,330 --> 00:02:03,510
There will be no
azimuthal dependencies.

22
00:02:03,510 --> 00:02:06,210
You see, the thing is
spherical is symmetric,

23
00:02:06,210 --> 00:02:10,680
but still you're coming from
a particular direction, the z.

24
00:02:10,680 --> 00:02:14,520
So you can expect now that
the scatter wave depends

25
00:02:14,520 --> 00:02:18,540
on the angle of the
particle with respect

26
00:02:18,540 --> 00:02:20,610
to z because it's
spherically symmetrical.

27
00:02:20,610 --> 00:02:23,790
But it shouldn't depend
on five, the angle five,

28
00:02:23,790 --> 00:02:27,160
should just depend on theta.

29
00:02:27,160 --> 00:02:34,710
So expect f of theta.

30
00:02:40,646 --> 00:02:45,430
Now, a free particle
is something

31
00:02:45,430 --> 00:02:50,380
we all know how to
solve, e to the ikx.

32
00:02:50,380 --> 00:02:55,390
Why do we bother with the
free particle in so many ways?

33
00:02:55,390 --> 00:02:58,900
Because free particle
is very important.

34
00:02:58,900 --> 00:03:01,810
Part of the solution
is free particles.

35
00:03:01,810 --> 00:03:06,470
To some degree far away it
is free particles as well.

36
00:03:06,470 --> 00:03:08,800
And we need to
understand free particles

37
00:03:08,800 --> 00:03:11,540
in spherical coordinates.

38
00:03:11,540 --> 00:03:19,360
So it's something we've done
in 805 and sometimes in 804,

39
00:03:19,360 --> 00:03:22,570
and we look at the
radial equation which

40
00:03:22,570 --> 00:03:25,330
is associated to
spherical coordinates

41
00:03:25,330 --> 00:03:26,770
for a free particle.

42
00:03:26,770 --> 00:03:33,400
So we'll consider free
particle and we'd say, well,

43
00:03:33,400 --> 00:03:36,040
that's very simple
but it's not all that

44
00:03:36,040 --> 00:03:39,250
simple in spherical
coordinates, and you'd say, OK,

45
00:03:39,250 --> 00:03:41,530
if it's not simple, it's
spherical coordinates,

46
00:03:41,530 --> 00:03:43,020
why do we bother?

47
00:03:43,020 --> 00:03:45,250
We bother because
scattering is happening

48
00:03:45,250 --> 00:03:46,450
in spherical coordinates.

49
00:03:46,450 --> 00:03:51,040
So we can't escape having
to do the free particle

50
00:03:51,040 --> 00:03:52,140
in spherical coordinates.

51
00:03:52,140 --> 00:03:54,670
It is something you have to do.

52
00:03:54,670 --> 00:03:57,550
So what are solutions in
spherical coordinates?

53
00:03:57,550 --> 00:04:00,940
We'll have solution SI of r.

54
00:04:00,940 --> 00:04:03,280
Remember the language
with coordinates

55
00:04:03,280 --> 00:04:11,710
was a U of r divided by
r and of Ylm of omega.

56
00:04:11,710 --> 00:04:17,540
That was a typical solution, a
single solution of the showing

57
00:04:17,540 --> 00:04:20,550
our equation will--

58
00:04:20,550 --> 00:04:26,080
the U only depends on l, the
m disappears, so this is r.

59
00:04:26,080 --> 00:04:30,250
This r's are r's without the
vector because you're already

60
00:04:30,250 --> 00:04:35,150
talking about the
radial equations,

61
00:04:35,150 --> 00:04:37,690
and depend on the
energy and depend

62
00:04:37,690 --> 00:04:43,040
on the value of the
l quantum number.

63
00:04:43,040 --> 00:04:44,920
So what is the
Schrodinger equation?

64
00:04:44,920 --> 00:04:50,060
The radial equation is
minus h squared over 2m,

65
00:04:50,060 --> 00:04:54,220
the second the r squared plus.

66
00:04:54,220 --> 00:05:00,660
h squared over 2m l times
l plus 1 over r squared.

67
00:05:00,660 --> 00:05:04,750
Remember the potential
centrifugal barrier

68
00:05:04,750 --> 00:05:08,950
in the effective potential,
then you would have v of r here,

69
00:05:08,950 --> 00:05:13,960
but it's free particle,
so v of r is equal to 0.

70
00:05:13,960 --> 00:05:20,305
So if nothing else,
U of El of little r

71
00:05:20,305 --> 00:05:22,820
is equal to the
energy, which is h

72
00:05:22,820 --> 00:05:29,290
squared k squared over 2m UEl.

73
00:05:29,290 --> 00:05:31,720
And that's a
parliamentary session

74
00:05:31,720 --> 00:05:40,040
of the energy in terms of
the k squared, like that.

75
00:05:43,090 --> 00:05:47,370
Well, there's lots of
h squared, k squared,

76
00:05:47,370 --> 00:05:51,150
and 2m's, so we can
get rid of them.

77
00:05:51,150 --> 00:05:53,310
Cancel the h squared over 2m.

78
00:05:53,310 --> 00:05:59,640
You get minus d second
dr squared plus l times l

79
00:05:59,640 --> 00:06:03,270
plus 1 over r.

80
00:06:03,270 --> 00:06:10,170
UEl is equal to k squared UEl.

81
00:06:15,780 --> 00:06:17,130
It's a nice equation.

82
00:06:17,130 --> 00:06:19,860
It's the equation
of the free particle

83
00:06:19,860 --> 00:06:23,770
in spherical coordinates.

84
00:06:23,770 --> 00:06:26,140
Now, this is like the
Schrodinger equation.

85
00:06:30,080 --> 00:06:39,110
And I think when you look at
that you could get puzzled

86
00:06:39,110 --> 00:06:43,280
whether or not the value
of k squared or the energy

87
00:06:43,280 --> 00:06:48,020
might end up being quantized.

88
00:06:48,020 --> 00:06:50,420
With the Schrodinger
equation many times

89
00:06:50,420 --> 00:06:55,170
quantized is the energy, but
here it shouldn't happen.

90
00:06:55,170 --> 00:06:56,420
This is a free particle.

91
00:06:56,420 --> 00:06:59,040
All values of k
should be allowed,

92
00:06:59,040 --> 00:07:01,970
so there should be
no quantization.

93
00:07:01,970 --> 00:07:03,800
This is an r squared here.

94
00:07:09,680 --> 00:07:12,380
You can see one reason,
at least analytically,

95
00:07:12,380 --> 00:07:17,750
that there is no quantization
is that you can define

96
00:07:17,750 --> 00:07:23,360
a new variable row
equal kr and then

97
00:07:23,360 --> 00:07:25,700
this whole differential
equation becomes

98
00:07:25,700 --> 00:07:33,200
minus the second the row
squared plus l times l

99
00:07:33,200 --> 00:07:37,030
plus 1 over row squared.

100
00:07:40,730 --> 00:07:43,180
Well, I can put the other
number in there as well,

101
00:07:43,180 --> 00:07:44,120
or should I not?

102
00:07:44,120 --> 00:07:46,780
No, it's not done here.

103
00:07:46,780 --> 00:07:58,880
UEl is equal to UEl, and the k
squared disappeared completely.

104
00:07:58,880 --> 00:08:02,120
That tells you that the case
will kind of get quantized.

105
00:08:02,120 --> 00:08:05,340
If there is a solution of
this differential equation

106
00:08:05,340 --> 00:08:09,500
it holds for all values of k.

107
00:08:09,500 --> 00:08:11,960
And these are going to
be like plane waves,

108
00:08:11,960 --> 00:08:14,180
and maybe that's
another reason you

109
00:08:14,180 --> 00:08:17,030
can think that k
doesn't get quantized

110
00:08:17,030 --> 00:08:21,500
because these solutions are
not normalizable anyway,

111
00:08:21,500 --> 00:08:25,160
so it shouldn't get quantized.

112
00:08:25,160 --> 00:08:34,049
So with this equation in here
we get the two main solutions.

113
00:08:34,049 --> 00:08:36,630
The solutions of this
differential equation

114
00:08:36,630 --> 00:08:41,299
are vessel functions,
spherical vessel functions.

115
00:08:41,299 --> 00:08:50,940
UEl is equal to a constant
Al times row times the vessel

116
00:08:50,940 --> 00:08:57,710
function lowercase j of row.

117
00:08:57,710 --> 00:08:59,840
There's a row times
that function.

118
00:08:59,840 --> 00:09:03,350
That's the way it shows up.

119
00:09:03,350 --> 00:09:04,770
It's kind of interesting.

120
00:09:04,770 --> 00:09:09,440
It's because in fact you
have to divide U by r,

121
00:09:09,440 --> 00:09:12,980
so that would mean
dividing U by row,

122
00:09:12,980 --> 00:09:16,100
and it means that the radial
function is just the vessel

123
00:09:16,100 --> 00:09:18,890
function without anything else.

124
00:09:18,890 --> 00:09:21,980
And then there's the
other vessel function,

125
00:09:21,980 --> 00:09:30,790
the n of l a row times
of n of l of row.

126
00:09:34,200 --> 00:09:39,110
So those are spherical
vessel functions.

127
00:09:39,110 --> 00:09:41,250
As you're familiar
from the notation

128
00:09:41,250 --> 00:09:44,490
that j is the one that
this healthy at row

129
00:09:44,490 --> 00:09:49,020
equals 0 doesn't diverge
the n is the solution

130
00:09:49,020 --> 00:09:52,440
that diverges at the origin.

131
00:09:52,440 --> 00:09:57,690
And both of them
behave nicely far away.

132
00:09:57,690 --> 00:10:11,850
So Jl of x goes like 1 over x
sine of x minus l pi over 2,

133
00:10:11,850 --> 00:10:17,250
and ADA l of x
behaves like minus 1

134
00:10:17,250 --> 00:10:25,060
over x cosine of x
minus l pi over 2.

135
00:10:25,060 --> 00:10:32,220
This is for x big, x
much greater than 1,

136
00:10:32,220 --> 00:10:33,195
you have this behavior.

137
00:10:40,760 --> 00:10:45,610
So these are our
solutions, and here is

138
00:10:45,610 --> 00:10:48,790
the thing that we have to do.

139
00:10:48,790 --> 00:10:56,520
We have to rewrite our solutions
in terms of spherical waves

140
00:10:56,520 --> 00:11:01,020
because this was the spherical
wave so we should even write

141
00:11:01,020 --> 00:11:04,060
this part as a spherical wave.

142
00:11:04,060 --> 00:11:08,160
And this is a very
interesting and in some way

143
00:11:08,160 --> 00:11:15,840
strange representation of E to
the ikz You have E to the ikz

144
00:11:15,840 --> 00:11:18,270
that you have an
intuition for it

145
00:11:18,270 --> 00:11:20,710
as a plane wave
in the z direction

146
00:11:20,710 --> 00:11:26,250
represent it as an infinite
sum of incoming and outgoing

147
00:11:26,250 --> 00:11:27,610
spherical waves.

148
00:11:27,610 --> 00:11:28,860
That's what's going to happen.

149
00:11:36,290 --> 00:11:39,515
So this is the last
thing we need do here.

150
00:11:48,820 --> 00:11:55,690
We have that e to the ikz
is a plane wave solution,

151
00:11:55,690 --> 00:11:58,510
so it's a solution
of a free particle,

152
00:11:58,510 --> 00:12:03,970
so I should be able to write the
superpositions of the solutions

153
00:12:03,970 --> 00:12:06,460
that we have found.

154
00:12:06,460 --> 00:12:16,370
So it should be a superposition
of solutions of this type.

155
00:12:16,370 --> 00:12:28,000
So it could be a sum of
coefficients al times,

156
00:12:28,000 --> 00:12:37,700
well, alm you think of
some a's times solutions.

157
00:12:37,700 --> 00:12:40,130
Remember, we're writing
a full solution,

158
00:12:40,130 --> 00:12:44,140
so a full solution
you divide by r.

159
00:12:44,140 --> 00:12:46,370
So you divide by this quantity.

160
00:12:46,370 --> 00:12:54,160
So you could have
an alm Jl of row

161
00:12:54,160 --> 00:13:01,270
plus Blm ATA l of row times Ylm.

162
00:13:05,250 --> 00:13:08,240
So this should be
a general solution,

163
00:13:08,240 --> 00:13:12,060
and that would be a sum
over l's and m's of all

164
00:13:12,060 --> 00:13:14,130
those quantities.

165
00:13:14,130 --> 00:13:17,400
But that's a lot more
than what you need.

166
00:13:17,400 --> 00:13:24,270
First, this does not
diverge near r equals 0.

167
00:13:24,270 --> 00:13:30,460
It has no divergence anywhere
and the ATAs or the n's, I

168
00:13:30,460 --> 00:13:35,530
think they're n such and not
ATAs, the n's diverge for row

169
00:13:35,530 --> 00:13:36,360
equal to 0.

170
00:13:36,360 --> 00:13:42,600
So none of this are necessary,
so I can erase those.

171
00:13:48,620 --> 00:13:50,600
l and m.

172
00:13:50,600 --> 00:13:51,920
But there is more.

173
00:13:51,920 --> 00:13:57,630
This function is
invariant and there

174
00:13:57,630 --> 00:14:00,120
are some beautiful rotations.

175
00:14:00,120 --> 00:14:04,950
If you have your axis
here, here's the z,

176
00:14:04,950 --> 00:14:08,130
and you have a point here and
you rotate that the value of z

177
00:14:08,130 --> 00:14:09,090
doesn't change.

178
00:14:09,090 --> 00:14:13,210
It's independent of
phi for a given theta,

179
00:14:13,210 --> 00:14:16,110
z just depends on
r of cosine theta.

180
00:14:16,110 --> 00:14:20,860
So there's no phi dependence
but all the Ylm's with m

181
00:14:20,860 --> 00:14:23,220
difference from 0
have phi dependent.

182
00:14:23,220 --> 00:14:26,040
So m cannot be here either.

183
00:14:26,040 --> 00:14:27,640
m must be 0.

184
00:14:27,640 --> 00:14:31,950
So you must be
down to sum over l,

185
00:14:31,950 --> 00:14:41,450
al some coefficient,
Jl of row, Yl0.

186
00:14:48,530 --> 00:14:53,960
And all of those would be
perfectly good plane wave

187
00:14:53,960 --> 00:14:55,820
solutions.

188
00:14:55,820 --> 00:14:58,820
Whatever numbers you
choose for the little al's,

189
00:14:58,820 --> 00:15:02,540
those are good solutions
because we've build them

190
00:15:02,540 --> 00:15:06,770
by taking linear combinations
of exact solutions

191
00:15:06,770 --> 00:15:08,960
of this equation.

192
00:15:08,960 --> 00:15:14,060
But to represent this
quantity the al's

193
00:15:14,060 --> 00:15:17,160
must take particular values.

194
00:15:17,160 --> 00:15:18,770
So what is that formula?

195
00:15:18,770 --> 00:15:24,140
That formula is quite
famous, and perhaps even you

196
00:15:24,140 --> 00:15:28,310
could discuss this
in recitation.

197
00:15:28,310 --> 00:15:36,770
e to the ikz, which is e
to the ikr cosine theta,

198
00:15:36,770 --> 00:15:40,490
is the sum 4 pi.

199
00:15:40,490 --> 00:15:43,070
Now you have to get all
the constants right.

200
00:15:43,070 --> 00:15:46,700
Square root of 4 pi,
sum from l equals 0

201
00:15:46,700 --> 00:15:52,430
to infinity, square
root of 2l plus 1.

202
00:15:52,430 --> 00:15:56,070
Coefficients are pretty funny.

203
00:15:56,070 --> 00:15:59,500
They get worse very fast.

204
00:15:59,500 --> 00:16:10,780
Now you have of i to the
I, i to the l, Yl0 of theta

205
00:16:10,780 --> 00:16:14,810
doesn't depend on phi, Jl of kr.

206
00:16:21,010 --> 00:16:22,880
This is the expansion
that we need.

207
00:16:22,880 --> 00:16:26,470
There's no way we can make
problems with this problem

208
00:16:26,470 --> 00:16:29,500
unless we have this expansion.

209
00:16:29,500 --> 00:16:34,240
But now if Y the intuition
that I was telling you

210
00:16:34,240 --> 00:16:37,270
of these waves
coming in and out,

211
00:16:37,270 --> 00:16:42,370
well, you have e to the
ikz, you sum an infinite sum

212
00:16:42,370 --> 00:16:44,620
over partial waves.

213
00:16:44,620 --> 00:16:48,400
A partial wave is a
different value of l.

214
00:16:48,400 --> 00:16:49,960
These are partial waves.

215
00:16:49,960 --> 00:16:54,730
As I was saying, any solution
is a sum of partial waves

216
00:16:54,730 --> 00:16:56,530
is a sum over l.

217
00:16:56,530 --> 00:16:58,120
And where are the waves?

218
00:16:58,120 --> 00:17:10,569
Well, the Jl of kr far away
is a sine, and the sine of x

219
00:17:10,569 --> 00:17:17,050
is an exponential ix minus
e to the minus ix over 2.

220
00:17:17,050 --> 00:17:21,250
So here you have
exponentials of e to the ikr

221
00:17:21,250 --> 00:17:24,849
and exponentials of
e to the minus ikr,

222
00:17:24,849 --> 00:17:31,060
which are waves that are
here like outgoing waves

223
00:17:31,060 --> 00:17:32,360
and incoming waves.

224
00:17:32,360 --> 00:17:38,080
So the E to the ikz's are sum of
ingoing and outgoing spherical

225
00:17:38,080 --> 00:17:38,740
waves.

226
00:17:38,740 --> 00:17:42,790
And that's an intuition that
we will exploit very clearly

227
00:17:42,790 --> 00:17:43,920
to solve this problem.

228
00:17:43,920 --> 00:17:47,580
So we will do that next.