1
00:00:00,500 --> 00:00:03,080
PROFESSOR: So here it
is, connections formula,

2
00:00:03,080 --> 00:00:03,890
connection.

3
00:00:14,480 --> 00:00:20,400
So we'll take a
situation as follows.

4
00:00:20,400 --> 00:00:25,490
Here is the point of x equals a.

5
00:00:25,490 --> 00:00:27,790
I'll put just a here.

6
00:00:27,790 --> 00:00:30,230
Here is the x-axis.

7
00:00:30,230 --> 00:00:34,550
And I will imagine that I
have a linear potential.

8
00:00:40,530 --> 00:00:43,290
So we have a linear
potential here.

9
00:00:43,290 --> 00:00:45,810
Why do we imagine
a linear potential?

10
00:00:45,810 --> 00:00:50,610
It's because the thing we
would really kind of want

11
00:00:50,610 --> 00:00:57,090
to think about is what is called
the turning point, energy,

12
00:00:57,090 --> 00:00:58,050
and a potential.

13
00:00:58,050 --> 00:00:59,820
There's a turning point.

14
00:00:59,820 --> 00:01:05,110
But it's clear that near
enough to the turning point,

15
00:01:05,110 --> 00:01:06,600
this is linear.

16
00:01:06,600 --> 00:01:09,720
And the problem for
the WKB approximation

17
00:01:09,720 --> 00:01:14,650
fails, as we discussed, is
precisely in this region.

18
00:01:14,650 --> 00:01:18,780
Now, I will make one
claim that this is really

19
00:01:18,780 --> 00:01:20,970
what's going to connect here.

20
00:01:20,970 --> 00:01:26,130
These two asymptotic expansions
that we've found here

21
00:01:26,130 --> 00:01:31,950
are in fact WKB approximations
of our calculations.

22
00:01:31,950 --> 00:01:39,620
So our whole procedure
here is going

23
00:01:39,620 --> 00:01:45,290
to be assuming that we have
a linear potential here,

24
00:01:45,290 --> 00:01:48,420
and I will take the linear
potential seriously.

25
00:01:48,420 --> 00:01:52,670
So I will imagine it
even goes forever.

26
00:01:52,670 --> 00:01:56,960
And what we're going
to do is find a way

27
00:01:56,960 --> 00:01:58,470
from the turning point.

28
00:01:58,470 --> 00:02:01,115
So the energy is
going to be here.

29
00:02:03,740 --> 00:02:06,474
Here is v of x.

30
00:02:06,474 --> 00:02:10,190
x v of x, the energy.

31
00:02:10,190 --> 00:02:16,850
And then we have this region,
which is a WKB region.

32
00:02:21,140 --> 00:02:27,030
And we have this region over
here that is also WKB region.

33
00:02:33,180 --> 00:02:36,900
And the region in the
middle is not a WKB region.

34
00:02:36,900 --> 00:02:42,290
It's a region where
all solutions fail,

35
00:02:42,290 --> 00:02:44,830
WKB solutions fail.

36
00:02:44,830 --> 00:02:48,580
They're not valid
in this region.

37
00:02:48,580 --> 00:02:57,370
So let's assume, we have here,
therefore, a potential that

38
00:02:57,370 --> 00:03:04,290
is linear, and, therefore
v of x minus the energy

39
00:03:04,290 --> 00:03:09,402
is going to be a
number times x minus a,

40
00:03:09,402 --> 00:03:13,200
where g is a positive constant.

41
00:03:13,200 --> 00:03:15,090
It has some units.

42
00:03:15,090 --> 00:03:23,040
v minus e is a linear function
that vanishes at x equals a.

43
00:03:23,040 --> 00:03:27,910
So this is a nice
description of the situation.

44
00:03:27,910 --> 00:03:37,390
And then you could
write a WKB solution.

45
00:03:37,390 --> 00:03:40,170
So what is your WKB solution?

46
00:03:40,170 --> 00:03:50,910
Let's write the WKB solution,
psi WKB on the right,

47
00:03:50,910 --> 00:03:56,130
and then we'll write a WKB
solution on the left, the right

48
00:03:56,130 --> 00:03:56,860
and left.

49
00:03:56,860 --> 00:03:59,040
So here is what we.

50
00:03:59,040 --> 00:04:04,260
Typical WKB solution
on the right.

51
00:04:04,260 --> 00:04:07,080
That is a forbidden region.

52
00:04:07,080 --> 00:04:10,860
Square root of kappa
of x, you would

53
00:04:10,860 --> 00:04:15,990
have a decrease in
exponential, e to the minus a

54
00:04:15,990 --> 00:04:23,130
to x kappa of x
prime, d x prime.

55
00:04:23,130 --> 00:04:31,350
And then you have b over square
root of kappa of x, e to the a

56
00:04:31,350 --> 00:04:37,650
to x kappa of x prime, dx prime.

57
00:04:41,100 --> 00:04:44,100
So this is what you've
been told is supposed

58
00:04:44,100 --> 00:04:46,950
to be your WKB approximation.

59
00:04:49,560 --> 00:04:52,350
Decay in exponential,
a pre-factor,

60
00:04:52,350 --> 00:04:54,960
growing exponential,
a pre-factor.

61
00:04:54,960 --> 00:04:58,830
And kappa of x is something that
you know from this potential.

62
00:04:58,830 --> 00:05:04,650
So this you would write for
any potential on the right

63
00:05:04,650 --> 00:05:07,590
or on the left as well.

64
00:05:07,590 --> 00:05:19,180
And here, your kappa of x
squared is 2m over h squared,

65
00:05:19,180 --> 00:05:21,480
v of x minus e.

66
00:05:26,520 --> 00:05:30,480
That's the definition
of kappa squared.

67
00:05:30,480 --> 00:05:35,100
You're in the region
where v is greater than e,

68
00:05:35,100 --> 00:05:44,400
and therefore, this is equal to
2m g over h squared x minus a.

69
00:05:47,970 --> 00:05:52,650
So our next step is
a little calculation.

70
00:05:52,650 --> 00:05:57,600
We should simplify all
these quantities here.

71
00:05:57,600 --> 00:06:01,110
We should evaluate them,
because we have a kappa.

72
00:06:01,110 --> 00:06:03,450
You have a linear function.

73
00:06:03,450 --> 00:06:05,010
We can do the integrals.

74
00:06:05,010 --> 00:06:07,140
We can put what kappa is.

75
00:06:07,140 --> 00:06:11,160
So everything can be done here.

76
00:06:11,160 --> 00:06:13,590
So if we do it, what do we get?

77
00:06:13,590 --> 00:06:14,600
So evaluate.

78
00:06:21,000 --> 00:06:23,070
We get the following.

79
00:06:23,070 --> 00:06:26,130
Now, I will introduce some
notation to write this.

80
00:06:26,130 --> 00:06:34,020
So first, I'll write
it, psi write of x WKB

81
00:06:34,020 --> 00:06:37,020
equals 1 over
square root of eta.

82
00:06:37,020 --> 00:06:42,930
That's a symbol I haven't
defined yet, u to the 1/4,

83
00:06:42,930 --> 00:06:50,820
e to the minus 2/3,
u to the 3/2 plus b

84
00:06:50,820 --> 00:06:58,290
square root of eta 1 over
u to the 1/4, e to the 2/3,

85
00:06:58,290 --> 00:06:59,280
u to the 3/2.

86
00:07:01,800 --> 00:07:03,300
OK, that's what you get.

87
00:07:03,300 --> 00:07:06,210
You'll recognize
the a and the b's.

88
00:07:06,210 --> 00:07:14,880
And here is what we need to
introduce a couple of variables

89
00:07:14,880 --> 00:07:18,150
to make sure we know
what every symbol is.

90
00:07:18,150 --> 00:07:19,800
And here, they are.

91
00:07:19,800 --> 00:07:26,190
u is eta x minus a.

92
00:07:26,190 --> 00:07:32,840
And eta is 2m g over
h squared to the 1/3.

93
00:07:38,640 --> 00:07:44,890
Just roughly k squared
is proportional to u.

94
00:07:44,890 --> 00:07:48,730
So k is proportional
to u to the 1/2.

95
00:07:48,730 --> 00:07:53,810
And therefore, square root of
k gives you the u to the 1/4.

96
00:07:53,810 --> 00:07:57,420
The integrals also
work that way.

97
00:07:57,420 --> 00:08:01,440
And you see already
here, things that

98
00:08:01,440 --> 00:08:05,600
look like the array function.

99
00:08:05,600 --> 00:08:09,790
And that will become
even clearer soon.

100
00:08:09,790 --> 00:08:20,990
So this is our solution for the
WKB expression on the right,

101
00:08:20,990 --> 00:08:25,500
but let's do also the WKB
expression on the left.

102
00:08:51,290 --> 00:08:54,540
The WKB expression
on the left would

103
00:08:54,540 --> 00:08:58,680
be another wave of this
kind, but with cosines

104
00:08:58,680 --> 00:09:01,260
or with exponentials
that we've done before.

105
00:09:01,260 --> 00:09:05,370
So what did it
used to look like?

106
00:09:05,370 --> 00:09:09,060
It used to look like
k of x and integrals

107
00:09:09,060 --> 00:09:19,210
of plus minus i integrals
of k of x prime dx prime.

108
00:09:19,210 --> 00:09:23,140
That was the way the WKB
solution looked like.

109
00:09:23,140 --> 00:09:33,580
I'm now trying to write this
expression for the WKB solution

110
00:09:33,580 --> 00:09:34,390
on the right.

111
00:09:34,390 --> 00:09:39,580
Now, I want to make a
remark that even though I'm

112
00:09:39,580 --> 00:09:46,150
using the linear
potential, in general,

113
00:09:46,150 --> 00:09:50,110
if your potential,
even eventually

114
00:09:50,110 --> 00:09:59,740
becomes something different
in this region, I'm good.

115
00:09:59,740 --> 00:10:04,660
So I'm moving away
from the bad point

116
00:10:04,660 --> 00:10:09,190
sufficiently so that the
WKB approximation is good.

117
00:10:09,190 --> 00:10:12,790
But I don't want to
move away so much, so

118
00:10:12,790 --> 00:10:16,480
that the potential fails
dramatically to be linear.

119
00:10:16,480 --> 00:10:24,290
I have to straddle
limited region in fact.

120
00:10:24,290 --> 00:10:25,610
I don't want to go too far.

121
00:10:28,590 --> 00:10:33,300
So these were your
solutions before in a region

122
00:10:33,300 --> 00:10:37,470
where you're in the
classically allowed region.

123
00:10:37,470 --> 00:10:42,210
Now, this form of the solutions
will not be practical for us.

124
00:10:42,210 --> 00:10:47,010
So I will write them, instead
with sines and cosines.

125
00:10:47,010 --> 00:10:49,185
It will be better
for us to write it

126
00:10:49,185 --> 00:10:50,360
with sines and cosines.

127
00:10:50,360 --> 00:10:54,940
So let me write it
with sines and cosines.

128
00:10:54,940 --> 00:11:08,520
Psi left now WKB of u or x.

129
00:11:08,520 --> 00:11:09,320
Sorry.

130
00:11:09,320 --> 00:11:14,160
It's going to be c over
square root of k of x,

131
00:11:14,160 --> 00:11:19,830
and I will write the cosine
of integral from x to a.

132
00:11:19,830 --> 00:11:22,210
You see, I'm on the left.

133
00:11:22,210 --> 00:11:25,740
So it's natural to write
integral from x to a.

134
00:11:25,740 --> 00:11:30,330
On the other hand, on the right
was natural to write integrals

135
00:11:30,330 --> 00:11:32,430
from a to x.

136
00:11:32,430 --> 00:11:36,760
And that's what we did,
cosine from x to a.

137
00:11:41,150 --> 00:11:48,900
We'll have here k of x
prime vx prime plus d

138
00:11:48,900 --> 00:11:58,660
over square root of k of x
sine of x to a k of x prime dx

139
00:11:58,660 --> 00:12:00,130
prime.

140
00:12:00,130 --> 00:12:04,050
And you could say
look, that's good.

141
00:12:04,050 --> 00:12:08,370
You just traded the exponentials
for sines and cosines,

142
00:12:08,370 --> 00:12:10,820
and that's fine.

143
00:12:14,220 --> 00:12:16,890
Still, for what we're
going to do later,

144
00:12:16,890 --> 00:12:20,790
it's convenient to write
our WKB approximations still

145
00:12:20,790 --> 00:12:23,010
in a little different way.

146
00:12:23,010 --> 00:12:28,350
I can add a phase here
and a phase there.

147
00:12:28,350 --> 00:12:31,230
And that still is the
most general solution,

148
00:12:31,230 --> 00:12:35,490
because they're both
sines and cosines.

149
00:12:35,490 --> 00:12:40,530
And unless I would put a
phase that differs pi over 2,

150
00:12:40,530 --> 00:12:43,230
and one becomes equal
to the other, no.

151
00:12:43,230 --> 00:12:49,350
I'm going to shift
both by pi over 4,

152
00:12:49,350 --> 00:12:53,520
and therefore, these are
still different functions,

153
00:12:53,520 --> 00:12:57,660
and it's convenient to do that.

154
00:12:57,660 --> 00:13:02,350
So this is our onset
for the WKB solution.

155
00:13:07,830 --> 00:13:09,720
We have to do the
same thing with it

156
00:13:09,720 --> 00:13:19,200
here in which we expand and
evaluate in terms of and eta

157
00:13:19,200 --> 00:13:20,380
in this quantity.

158
00:13:20,380 --> 00:13:22,570
So what do we get?

159
00:13:22,570 --> 00:13:31,770
We get c over square root of
eta u to the 1/4 cosine 2/3 u

160
00:13:31,770 --> 00:13:40,800
to the 3/2 minus pi over 4 plus
d over square root of eta u

161
00:13:40,800 --> 00:13:52,250
to the 1/4 sine of 2/3 u
to the 3/2 minus pi over 4.

162
00:13:52,250 --> 00:14:01,095
So this is our
psi left WKB of x.

163
00:14:07,571 --> 00:14:08,070
OK.

164
00:14:11,640 --> 00:14:13,600
We're in good shape.

165
00:14:13,600 --> 00:14:15,930
I'm going to raise this up.

166
00:14:22,300 --> 00:14:31,290
And now you could say, all
right, you've done the WKB.

167
00:14:31,290 --> 00:14:36,960
But on the other hand, how
about the exact solution?

168
00:14:36,960 --> 00:14:39,920
What is the exact
solution to this problem?

169
00:14:39,920 --> 00:14:42,620
You've written the
WKB approximation

170
00:14:42,620 --> 00:14:46,550
for the linear potential,
the WKB approximation

171
00:14:46,550 --> 00:14:48,600
for the linear potential.

172
00:14:48,600 --> 00:14:52,880
Now, what is the exact
solution for the problem?

173
00:14:52,880 --> 00:14:55,250
Well, your Schrodinger
equation is

174
00:14:55,250 --> 00:14:59,780
minus h squared over
2n psi double dot

175
00:14:59,780 --> 00:15:07,775
of x plus of x minus the
energy on psi is equal to 0.

176
00:15:11,920 --> 00:15:14,890
Well, we've written this
already a couple of times.

177
00:15:14,890 --> 00:15:19,690
So this is minus h
squared over 2m psi

178
00:15:19,690 --> 00:15:25,780
double dot plus g x
minus a psi equals 0.

179
00:15:25,780 --> 00:15:28,900
That is our v minus e.

180
00:15:33,940 --> 00:15:42,090
And in terms of the u variable,
the differential equation,

181
00:15:42,090 --> 00:15:49,275
of course, becomes the second
psi, the u squared equal u psi.

182
00:15:52,090 --> 00:15:56,830
That's why the area
function is relevant here.

183
00:15:56,830 --> 00:15:58,960
We get back to that equation.

184
00:15:58,960 --> 00:16:06,950
That is the equation for
the exact linear potential,

185
00:16:06,950 --> 00:16:11,810
and those would be
the exact solutions.

186
00:16:11,810 --> 00:16:18,530
So our exact solutions are psi
being some linear combination

187
00:16:18,530 --> 00:16:24,010
of the array function and
the other array function.

188
00:16:30,360 --> 00:16:36,810
And now let's do one case first
before we get into something

189
00:16:36,810 --> 00:16:39,840
that is more complicated.

190
00:16:39,840 --> 00:16:48,850
Let's take psi to
be the solution

191
00:16:48,850 --> 00:16:54,310
to be the array
function ai of u.

192
00:16:54,310 --> 00:16:55,520
That's your solution now.

193
00:16:58,780 --> 00:17:03,310
If that is your
solution, you would

194
00:17:03,310 --> 00:17:09,310
have this behavior,
the behavior we've

195
00:17:09,310 --> 00:17:11,105
noted here for this solution.

196
00:17:28,150 --> 00:17:32,320
And now you compare and
you look at what you got.

197
00:17:32,320 --> 00:17:36,110
You say, all right,
I have a solution,

198
00:17:36,110 --> 00:17:38,600
and I know how it
looks rigorously,

199
00:17:38,600 --> 00:17:41,840
how it looks to the
right and to the left.

200
00:17:41,840 --> 00:17:46,620
That's how it looks, and
this is the exact solution.

201
00:17:46,620 --> 00:17:52,040
This is the WKB solution, and I
know how it looks to the right

202
00:17:52,040 --> 00:17:54,770
and how it looks to the left.

203
00:17:54,770 --> 00:17:58,460
And I see the correspondence.

204
00:17:58,460 --> 00:18:08,410
We see this term in
yellow is a solution

205
00:18:08,410 --> 00:18:10,520
that this a decaying
exponential,

206
00:18:10,520 --> 00:18:11,590
and we see it here.

207
00:18:17,560 --> 00:18:25,520
And we have the red
term, the oscillations.

208
00:18:28,110 --> 00:18:32,430
And we see it also here.

209
00:18:38,190 --> 00:18:41,380
So here is that
connection condition.

210
00:18:41,380 --> 00:18:45,630
In fact, when you derive
this asymptotic expansions,

211
00:18:45,630 --> 00:18:49,080
you did the true
connection condition,

212
00:18:49,080 --> 00:18:53,850
because you connected a
single function from the left

213
00:18:53,850 --> 00:18:55,710
and from the right.

214
00:18:55,710 --> 00:18:57,130
Here it was to the right.

215
00:18:57,130 --> 00:18:58,630
Here it was to the left.

216
00:18:58,630 --> 00:19:03,010
And it's a connection formula
because it's the same function.

217
00:19:03,010 --> 00:19:09,030
So here, we see that if
we wrote the WKB solution,

218
00:19:09,030 --> 00:19:15,540
and we were aiming at
maybe this solution,

219
00:19:15,540 --> 00:19:24,210
we would have that this term
goes, connects with this term.

220
00:19:24,210 --> 00:19:28,710
And the coefficients
differ by 1/2.

221
00:19:28,710 --> 00:19:31,440
That 1/2 has no
other explanation

222
00:19:31,440 --> 00:19:34,470
than the asymptotic expansions.

223
00:19:34,470 --> 00:19:37,470
There's the square root
of pi, these things,

224
00:19:37,470 --> 00:19:40,750
and there's the two functions.

225
00:19:40,750 --> 00:19:45,550
So if you put the 1/2 is
there in the exponential,

226
00:19:45,550 --> 00:19:54,800
so you could put the 1/2
here, and 2a, if you wish.

227
00:19:54,800 --> 00:20:01,390
So you realize that in
order to have a connection

228
00:20:01,390 --> 00:20:05,275
2a would be equal to c.

229
00:20:08,140 --> 00:20:11,620
The two coefficients here
are related in that way.

230
00:20:11,620 --> 00:20:14,740
There's a number here
and a number here.

231
00:20:14,740 --> 00:20:18,790
It happens to be 2a
and c in this case.

232
00:20:18,790 --> 00:20:20,920
So 2a is equal to c.

233
00:20:20,920 --> 00:20:27,670
So if we choose a equals
1, we would have c equal 2.

234
00:20:27,670 --> 00:20:32,680
So here is a matching
condition therefore.

235
00:20:32,680 --> 00:20:39,070
We have that a WKB
solution that has

236
00:20:39,070 --> 00:20:46,240
the c factor with c equals 2, so
it would be 2 overs square root

237
00:20:46,240 --> 00:20:53,080
of k of x in here
times cosine of this x

238
00:20:53,080 --> 00:21:03,250
over a k of x prime, dx prime
minus pi over 4 is connected--

239
00:21:03,250 --> 00:21:05,170
we'll see more on that--

240
00:21:05,170 --> 00:21:09,970
with 1 over square
root of kappa of x,

241
00:21:09,970 --> 00:21:20,170
this solution with a equals
to 1, what we have here,

242
00:21:20,170 --> 00:21:27,040
e to the minus a over x
kappa of x prime dx prime.

243
00:21:30,500 --> 00:21:43,170
So that is the first
connection condition.

244
00:21:43,170 --> 00:21:46,010
We will discuss it further
in a couple of minutes,

245
00:21:46,010 --> 00:21:52,150
but this is how
you connect things.

246
00:21:52,150 --> 00:21:54,860
A single solution of the
differential equation

247
00:21:54,860 --> 00:21:58,150
looks like this term and
like that term on the left

248
00:21:58,150 --> 00:21:59,200
and on the right.

249
00:21:59,200 --> 00:22:02,260
Here is your WKB solution.

250
00:22:02,260 --> 00:22:06,280
This general solution
looks like this term,

251
00:22:06,280 --> 00:22:08,740
and it then should
match to something

252
00:22:08,740 --> 00:22:13,850
that looks like that if you
have the array function.

253
00:22:13,850 --> 00:22:20,240
So in general, for
not linear potentials,

254
00:22:20,240 --> 00:22:22,310
this is the term that
gives rise to that,

255
00:22:22,310 --> 00:22:26,180
so we take that
this term in general

256
00:22:26,180 --> 00:22:31,930
is matched to the top term
in the blackboard here.