1
00:00:00,500 --> 00:00:01,580
PROFESSOR: OK.

2
00:00:01,580 --> 00:00:03,180
That is the first step.

3
00:00:03,180 --> 00:00:06,170
But to make things
really clear, we

4
00:00:06,170 --> 00:00:11,960
have to do the second step that
it involves the Bi solution.

5
00:00:11,960 --> 00:00:19,280
So let me say something
about Bi as well.

6
00:00:26,630 --> 00:00:32,900
So Bi has similar
asymptotic expansions.

7
00:00:32,900 --> 00:00:37,970
A Bi solution, if psi, you
have an exact Bi solution.

8
00:00:41,610 --> 00:00:48,630
When psi on the left would
behave as minus 1 over pi,

9
00:00:48,630 --> 00:00:51,360
this is for u negative.

10
00:00:51,360 --> 00:01:02,790
1 over u to the 1/4 sine of 2/3
u to the 3/2 minus pi over 4.

11
00:01:02,790 --> 00:01:10,890
And psi on the right
of x would behave as 1

12
00:01:10,890 --> 00:01:16,060
over square root of pi,
1 over u to the 1/4, e

13
00:01:16,060 --> 00:01:21,290
to the 2/3, u over 3/2.

14
00:01:21,290 --> 00:01:21,870
OK.

15
00:01:21,870 --> 00:01:26,730
This is the asymptotic
behavior of B. So

16
00:01:26,730 --> 00:01:30,890
what's noteworthy about it?

17
00:01:30,890 --> 00:01:35,330
First, that it oscillates
for u negative.

18
00:01:35,330 --> 00:01:36,500
That's to the left.

19
00:01:36,500 --> 00:01:37,310
It oscillates.

20
00:01:37,310 --> 00:01:41,000
Instead of a cosine, a
sine, the same phase shift.

21
00:01:43,720 --> 00:01:50,410
And then that on the other side
for u positive, it blows up.

22
00:01:50,410 --> 00:01:52,630
So it uses the
other exponential.

23
00:01:52,630 --> 00:01:56,020
And this all is reasonable.

24
00:01:56,020 --> 00:01:59,890
And in fact, it allows
you to connect as well.

25
00:01:59,890 --> 00:02:11,550
So in this case, we would have
this term looks like this one.

26
00:02:17,240 --> 00:02:27,680
And the left one looks like--

27
00:02:35,570 --> 00:02:40,670
so indeed, if you
had the Bi solution,

28
00:02:40,670 --> 00:02:45,950
the matching requires
that the D term over there

29
00:02:45,950 --> 00:02:52,070
is matched to the B term
over here because that's how

30
00:02:52,070 --> 00:02:56,150
the Bi solution is connected.

31
00:02:56,150 --> 00:02:59,090
So we now need to
relate the B to the D.

32
00:02:59,090 --> 00:03:04,470
So I will write down
the relation here.

33
00:03:04,470 --> 00:03:07,380
So in this case,
there's no factor of 2,

34
00:03:07,380 --> 00:03:09,840
but there is a minus sign.

35
00:03:09,840 --> 00:03:13,820
So we have B must
be equal to minus D.

36
00:03:13,820 --> 00:03:19,250
So you can take B equals to
minus 1 and D equals to 1.

37
00:03:19,250 --> 00:03:32,690
And we get 1 over square root
of kfx sine x to A. Kfx prime dx

38
00:03:32,690 --> 00:03:42,500
prime minus pi over 4 connects
in principle to minus 1

39
00:03:42,500 --> 00:03:52,870
over square root of kappa of x
e to the integral from A to x,

40
00:03:52,870 --> 00:03:56,450
kappa of x prime dx prime.

41
00:03:56,450 --> 00:04:00,120
I think I have it all right.

42
00:04:00,120 --> 00:04:01,660
B is there.

43
00:04:01,660 --> 00:04:03,060
B is minus 1.

44
00:04:03,060 --> 00:04:05,700
It's over there.

45
00:04:05,700 --> 00:04:08,310
D is equal to 1.

46
00:04:08,310 --> 00:04:09,750
That's the D term.

47
00:04:09,750 --> 00:04:12,300
That's the other
connection condition.

48
00:04:18,290 --> 00:04:18,790
OK.

49
00:04:18,790 --> 00:04:22,180
So the technicalities are gone.

50
00:04:22,180 --> 00:04:26,980
But the concept now requires
a serious discussion.

51
00:04:29,510 --> 00:04:32,790
I'll give you a little
time to take it in.

52
00:04:36,740 --> 00:04:41,150
We've connected every functions.

53
00:04:41,150 --> 00:04:43,940
That was the first work
we did at the beginning

54
00:04:43,940 --> 00:04:44,970
of this lecture.

55
00:04:44,970 --> 00:04:49,190
We did, in the top blackboard
justify, this expansions.

56
00:04:49,190 --> 00:04:53,540
We discussed how the Bi function
would originate from a counter

57
00:04:53,540 --> 00:04:55,370
integral as well.

58
00:04:55,370 --> 00:05:00,270
And we didn't derive the
Bi asymptotic expansions,

59
00:05:00,270 --> 00:05:03,170
but they're similarly derived.

60
00:05:03,170 --> 00:05:05,655
And then we used those
to connect things.

61
00:05:08,420 --> 00:05:12,590
And now the question is whether
these equations can be used.

62
00:05:12,590 --> 00:05:16,400
If I know the solution
on the left is like that,

63
00:05:16,400 --> 00:05:19,970
can I say the solution on
the right is like that?

64
00:05:19,970 --> 00:05:22,550
Or if I know the solution
on the right is like that,

65
00:05:22,550 --> 00:05:25,280
can I say the equation
is on the left?

66
00:05:25,280 --> 00:05:26,910
And the same thing here.

67
00:05:26,910 --> 00:05:28,640
If I know the
solution on the left,

68
00:05:28,640 --> 00:05:31,070
do I know the
solution on the right?

69
00:05:31,070 --> 00:05:33,320
And if I know the
solution on the right,

70
00:05:33,320 --> 00:05:36,710
do I know the
solution on the left?

71
00:05:36,710 --> 00:05:39,680
Now it looks like yes,
that's what you derive.

72
00:05:39,680 --> 00:05:43,670
But remember, it's
not quite simple

73
00:05:43,670 --> 00:05:50,120
because our whole discussion
assume that OK, the potential

74
00:05:50,120 --> 00:05:52,040
is strictly linear.

75
00:05:52,040 --> 00:05:55,760
But in reality, the
potential starts to deviate.

76
00:05:55,760 --> 00:05:59,150
So these things
that were right here

77
00:05:59,150 --> 00:06:04,610
are not exact
asymptotic expansions.

78
00:06:04,610 --> 00:06:09,360
That is the exact dominant term
of the asymptotic expansions.

79
00:06:09,360 --> 00:06:12,710
But here there
may be extra terms

80
00:06:12,710 --> 00:06:15,950
because the solution
does not correspond

81
00:06:15,950 --> 00:06:18,410
to a potential is
exactly linear.

82
00:06:18,410 --> 00:06:21,140
And anyway, this is
not an exact solution

83
00:06:21,140 --> 00:06:24,120
of the differential equation.

84
00:06:24,120 --> 00:06:29,270
So there is a possibility
of ambiguities here

85
00:06:29,270 --> 00:06:36,060
that actually indicate
the following.

86
00:06:36,060 --> 00:06:40,820
I claim this connection can
only be done in this way.

87
00:06:44,220 --> 00:06:52,420
You can only take this equality
from the right to the left.

88
00:06:52,420 --> 00:06:59,250
That is, if you
know that you just

89
00:06:59,250 --> 00:07:03,510
have a decaying
exponential, you know

90
00:07:03,510 --> 00:07:07,060
it connects to this function.

91
00:07:07,060 --> 00:07:07,560
Why?

92
00:07:07,560 --> 00:07:10,440
Because if you know you
have a decaying exponential,

93
00:07:10,440 --> 00:07:13,530
there's 0 chance
there is a growing

94
00:07:13,530 --> 00:07:15,810
exponential on the right.

95
00:07:15,810 --> 00:07:19,350
You may have a barrier
that extends forever,

96
00:07:19,350 --> 00:07:22,960
and there's definitely only
a decay in exponential,

97
00:07:22,960 --> 00:07:26,010
however the barrier looks.

98
00:07:26,010 --> 00:07:28,440
If there's only a
decaying exponential,

99
00:07:28,440 --> 00:07:33,270
you know that this growing
exponential is just not there.

100
00:07:33,270 --> 00:07:35,610
The coefficient here is 0.

101
00:07:35,610 --> 00:07:38,640
So you don't have the
ambiguity that maybe you

102
00:07:38,640 --> 00:07:41,310
have a little bit of a
growing exponential that

103
00:07:41,310 --> 00:07:42,930
connects to this.

104
00:07:42,930 --> 00:07:49,240
So you can go from
here to there.

105
00:07:49,240 --> 00:07:54,780
On the other hand, you
cannot quite go from the left

106
00:07:54,780 --> 00:07:56,890
to the right.

107
00:07:56,890 --> 00:07:59,560
Why is that?

108
00:07:59,560 --> 00:08:02,420
The reason is the following.

109
00:08:02,420 --> 00:08:06,370
If you just know
this is on the left,

110
00:08:06,370 --> 00:08:11,500
you would say, OK, now I predict
a decaying exponential there.

111
00:08:11,500 --> 00:08:14,440
But your calculation
is not totally exact,

112
00:08:14,440 --> 00:08:19,300
so there could be an
infinitesimal wave

113
00:08:19,300 --> 00:08:22,840
of the other type on the
left, an infinitesimal wave

114
00:08:22,840 --> 00:08:24,730
of this kind.

115
00:08:24,730 --> 00:08:27,400
But that infinitesimal
wave of this kind

116
00:08:27,400 --> 00:08:32,950
goes into a growing exponential,
and a growing exponential

117
00:08:32,950 --> 00:08:38,350
eventually overtakes this
one and ruins your solution.

118
00:08:38,350 --> 00:08:46,870
So you cannot really use this
equation from right to left.

119
00:08:46,870 --> 00:08:51,190
The small possibility of
error on the left side,

120
00:08:51,190 --> 00:08:53,140
because you don't
have exact solutions,

121
00:08:53,140 --> 00:08:56,400
translates into an
ambiguity concerning

122
00:08:56,400 --> 00:08:59,230
a positively
growing exponential,

123
00:08:59,230 --> 00:09:00,850
and it would overtake this.

124
00:09:00,850 --> 00:09:03,700
So if you say oh, I
have this, therefore I

125
00:09:03,700 --> 00:09:05,860
have this decaying
exponential, this

126
00:09:05,860 --> 00:09:08,710
may be a very inaccurate
thing because maybe you

127
00:09:08,710 --> 00:09:11,350
have a little bit
of the sine that you

128
00:09:11,350 --> 00:09:14,290
didn't see because it was
much smaller than this term.

129
00:09:14,290 --> 00:09:17,350
But that would give you
a growing exponential.

130
00:09:17,350 --> 00:09:22,220
So you cannot use that
equation in that direction.

131
00:09:22,220 --> 00:09:27,830
Similarly here, you
have another direction.

132
00:09:27,830 --> 00:09:29,650
The direction is this one.

133
00:09:32,339 --> 00:09:32,880
It's crucial.

134
00:09:37,870 --> 00:09:43,360
You can go that way, but you
cannot go the other direction

135
00:09:43,360 --> 00:09:44,900
as well.

136
00:09:44,900 --> 00:09:48,610
And the reason is
kind of similar.

137
00:09:48,610 --> 00:09:51,240
Suppose you have
here, you say, OK,

138
00:09:51,240 --> 00:09:53,310
I have this growing exponential.

139
00:09:53,310 --> 00:09:57,010
And say somebody comes in, no,
there's a little bit of cosine

140
00:09:57,010 --> 00:09:58,090
here.

141
00:09:58,090 --> 00:10:00,220
And you say well, but
a little bit of cosine

142
00:10:00,220 --> 00:10:03,490
gives me a decaying
exponential that is irrelevant

143
00:10:03,490 --> 00:10:04,990
compared to this one.

144
00:10:04,990 --> 00:10:08,590
So you are safe in going
from this side to that side.

145
00:10:08,590 --> 00:10:11,050
You're going into
growing exponential.

146
00:10:11,050 --> 00:10:17,260
Any error on the left
will produce a small error

147
00:10:17,260 --> 00:10:19,410
on the right.

148
00:10:19,410 --> 00:10:23,840
Of the other hand, you
cannot go, honestly,

149
00:10:23,840 --> 00:10:31,430
from right to left because
if you had a decaying

150
00:10:31,430 --> 00:10:34,490
exponential-- this is
a growing exponential.

151
00:10:34,490 --> 00:10:37,910
If you had the
decaying exponential,

152
00:10:37,910 --> 00:10:39,380
you wouldn't see it here.

153
00:10:39,380 --> 00:10:40,950
This one takes over.

154
00:10:40,950 --> 00:10:46,310
But the decaying exponential
produces a nice cosine

155
00:10:46,310 --> 00:10:50,340
that is quite
comparable to this one.

156
00:10:50,340 --> 00:10:54,970
So a decaying exponential
that you don't see here

157
00:10:54,970 --> 00:11:00,050
produces on this side a term
that is comparable to this one.

158
00:11:00,050 --> 00:11:05,460
So you also cannot go from this
direction to that direction.

159
00:11:05,460 --> 00:11:10,250
So these are the
connection formulas

160
00:11:10,250 --> 00:11:12,620
and these are the directions.

161
00:11:12,620 --> 00:11:17,750
We've done this for this
configuration in which we have

162
00:11:17,750 --> 00:11:23,870
a turning point of this form
at the point A, in which you're

163
00:11:23,870 --> 00:11:29,060
allowed to the left and
forbidden to the right.

164
00:11:29,060 --> 00:11:37,990
For convenience, it's
useful to have a formula--

165
00:11:37,990 --> 00:11:44,400
let's go here-- where you have
the opposite direction in which

166
00:11:44,400 --> 00:11:50,250
you have a B here, and you
have allowed to the right

167
00:11:50,250 --> 00:11:52,710
and forbidden to the left.

168
00:11:52,710 --> 00:11:58,200
I will write those formulas
because they are many times

169
00:11:58,200 --> 00:12:00,750
used and you will
need them in general.

170
00:12:00,750 --> 00:12:04,402
And Griffiths and other
books don't have the patience

171
00:12:04,402 --> 00:12:05,610
to give you all the formulas.

172
00:12:08,560 --> 00:12:16,060
This is kappa of
x prime, vx prime.

173
00:12:16,060 --> 00:12:27,040
This is growing or
decaying exponential?

174
00:12:27,040 --> 00:12:30,700
This should be called
the growing exponential

175
00:12:30,700 --> 00:12:32,840
on the left.

176
00:12:32,840 --> 00:12:35,150
This is the forbidden region.

177
00:12:35,150 --> 00:12:39,420
As x becomes more and more
left, the interval is bigger.

178
00:12:39,420 --> 00:12:43,130
So this is a growing
exponential on the left.

179
00:12:43,130 --> 00:12:54,020
1 over k of x sine b to x,
k of x prime, vx prime minus

180
00:12:54,020 --> 00:12:56,450
pi over 4.

181
00:12:56,450 --> 00:12:58,850
And this one, again,
just like that one,

182
00:12:58,850 --> 00:13:04,600
you go into the
growing exponential.

183
00:13:04,600 --> 00:13:06,010
So that's that relation.

184
00:13:08,910 --> 00:13:13,720
And I kind of write these
things following this picture.

185
00:13:13,720 --> 00:13:17,820
So you're allowed to the right
and forbidden on the left side.

186
00:13:17,820 --> 00:13:19,530
The allowed way
function the right,

187
00:13:19,530 --> 00:13:21,375
the forbidden way
function to the left.

188
00:13:24,510 --> 00:13:29,790
1 over kappa of x,
exponential of minus x

189
00:13:29,790 --> 00:13:39,034
to B. Kappa of x prime
vx prime goes into--

190
00:13:39,034 --> 00:13:45,040
a decay in exponential you can
follow into the other region.

191
00:13:45,040 --> 00:13:48,580
And that's what we
do here, as before.

192
00:13:48,580 --> 00:13:52,060
The decaying exponential
is safe because if you

193
00:13:52,060 --> 00:13:54,850
know there's just a
decaying exponential,

194
00:13:54,850 --> 00:13:58,660
there's no possibility of
having the growing exponential.

195
00:13:58,660 --> 00:14:08,440
So these are your conditions
for that turning point,

196
00:14:08,440 --> 00:14:10,570
and that's how you use them.

197
00:14:10,570 --> 00:14:15,130
So one example that we will
not get to discuss now,

198
00:14:15,130 --> 00:14:20,620
but will be in the notes, is
tunneling across a barrier.

199
00:14:20,620 --> 00:14:24,250
And you will use the connection
conditions in this case

200
00:14:24,250 --> 00:14:31,150
here, and for this case, the
other ones that we've done.

201
00:14:31,150 --> 00:14:34,530
So you need for tunneling
both connection conditions.

202
00:14:34,530 --> 00:14:37,380
It's a very nice exercise.

203
00:14:37,380 --> 00:14:39,660
I recommend that
you play with it,

204
00:14:39,660 --> 00:14:42,370
and we'll put it in
the notes as well.

205
00:14:42,370 --> 00:14:45,980
So let's stop here.