1
00:00:00,499 --> 00:00:09,180
PROFESSOR: So let's try
to do return to seeing,

2
00:00:09,180 --> 00:00:11,040
OK, we solved equation.

3
00:00:11,040 --> 00:00:12,500
We seem to be OK.

4
00:00:12,500 --> 00:00:15,260
What did we really approximate?

5
00:00:15,260 --> 00:00:19,220
We didn't approximate
saying h bar goes to 0.

6
00:00:19,220 --> 00:00:24,700
We did a more serious
physical approximation.

7
00:00:24,700 --> 00:00:31,340
And let's try to see
what we really did.

8
00:00:31,340 --> 00:00:41,040
So I think the whole clue is
in this top equation there.

9
00:00:41,040 --> 00:00:43,500
You have the first term
and the second term.

10
00:00:43,500 --> 00:00:46,770
And our claim is
that the second term

11
00:00:46,770 --> 00:00:52,270
is smaller than the first
term with h bar there.

12
00:00:52,270 --> 00:00:57,880
So for example-- now of
course, in this solution,

13
00:00:57,880 --> 00:01:02,520
the first term is identically
0 and the second term,

14
00:01:02,520 --> 00:01:06,960
the coefficient of h
bar, is identically 0.

15
00:01:06,960 --> 00:01:15,480
But we can look at one of those,
for example, and say that--

16
00:01:15,480 --> 00:01:17,940
so the validity of
the approximation.

17
00:01:17,940 --> 00:01:20,970
Validity of the approximation.

18
00:01:23,610 --> 00:01:26,460
It's pretty useful to do this.

19
00:01:26,460 --> 00:01:36,550
So we say, for example
there, that term, h bar

20
00:01:36,550 --> 00:01:42,100
s0 double prime that
enters into the order h bar

21
00:01:42,100 --> 00:01:46,930
part of the equation,
the absolute value of it

22
00:01:46,930 --> 00:01:52,390
must be much smaller than
a typical term s0 prime,

23
00:01:52,390 --> 00:01:59,050
for example, s0 prime
squared, in the first term.

24
00:02:03,610 --> 00:02:07,270
So each term-- so
basically, I'm saying

25
00:02:07,270 --> 00:02:13,600
each term in the first
bracket must be much larger

26
00:02:13,600 --> 00:02:17,290
than each term in
the second bracket.

27
00:02:17,290 --> 00:02:20,260
And you could have picked
any ones because they're

28
00:02:20,260 --> 00:02:22,480
all equal, after all.

29
00:02:22,480 --> 00:02:24,970
So let's see if
that is the case.

30
00:02:24,970 --> 00:02:32,050
So recall that s0 prime
from there is really

31
00:02:32,050 --> 00:02:34,870
p of x plus minus p of x.

32
00:02:34,870 --> 00:02:41,860
So what do we have here? h bar
and s0 double prime is dp vx.

33
00:02:45,390 --> 00:02:49,620
Must be much smaller
than p of x squared.

34
00:02:54,550 --> 00:02:57,510
Now it's a matter of playing
with these things a little bit

35
00:02:57,510 --> 00:03:03,840
until you find some way that
the equality tells you story.

36
00:03:03,840 --> 00:03:06,870
And the way I'll
do it is by saying

37
00:03:06,870 --> 00:03:13,740
that this is h bar 1
over p squared of x

38
00:03:13,740 --> 00:03:20,940
dp vx is much smaller than 1.

39
00:03:20,940 --> 00:03:26,970
And here I'll write
this as h bar d--

40
00:03:26,970 --> 00:03:28,395
no h bar.

41
00:03:28,395 --> 00:03:31,920
ddx of h bar over p.

42
00:03:36,370 --> 00:03:37,590
Look what I did.

43
00:03:37,590 --> 00:03:44,410
ddx of 1 over p is 1 over
p squared dp vx and the h

44
00:03:44,410 --> 00:03:45,460
bar I put it there.

45
00:03:48,360 --> 00:03:49,350
Here we go.

46
00:03:49,350 --> 00:03:52,770
What does this say?

47
00:03:52,770 --> 00:03:57,360
This is the local De
Broglie wave length.

48
00:03:57,360 --> 00:04:00,930
This is saying that
ddx of the local

49
00:04:00,930 --> 00:04:07,240
the De Broglie wave length
must be much smaller than 1.

50
00:04:07,240 --> 00:04:11,870
A nice result, your local
De Broglie wavelength

51
00:04:11,870 --> 00:04:15,160
must have a small derivative.

52
00:04:15,160 --> 00:04:18,180
So this is the
physics translation

53
00:04:18,180 --> 00:04:22,000
of the semi-classical
approximation. h bar going to 0

54
00:04:22,000 --> 00:04:26,820
is a mathematical device,
but this is physical.

55
00:04:26,820 --> 00:04:30,270
This is telling you
what should happen.

56
00:04:30,270 --> 00:04:35,940
Most of us look at that
and see an easier way

57
00:04:35,940 --> 00:04:39,040
to understand that equation.

58
00:04:39,040 --> 00:04:40,410
v lambda vx.

59
00:04:40,410 --> 00:04:41,940
It has the right units.

60
00:04:41,940 --> 00:04:43,530
Lambda has units of length.

61
00:04:43,530 --> 00:04:44,850
x has units of length.

62
00:04:44,850 --> 00:04:47,640
So that derivative
must have no units.

63
00:04:47,640 --> 00:04:49,960
And if it's supposed
to be small,

64
00:04:49,960 --> 00:04:52,980
it should be small
compared to 1.

65
00:04:52,980 --> 00:04:56,610
So this is the
conventional inequality.

66
00:04:56,610 --> 00:04:59,400
Most of us would prefer,
maybe, to write it

67
00:04:59,400 --> 00:05:07,650
like this-- lambda d lambda dx
is much smaller than lambda.

68
00:05:07,650 --> 00:05:10,860
And I think this is a
little clearer because this

69
00:05:10,860 --> 00:05:17,670
is how much the De Broglie
wave length changes

70
00:05:17,670 --> 00:05:21,570
over a distance equal to
the De Broglie wavelength.

71
00:05:21,570 --> 00:05:26,130
So you have a De Broglie wave
length and the next De Broglie

72
00:05:26,130 --> 00:05:26,890
wave length.

73
00:05:26,890 --> 00:05:28,590
How much did it change?

74
00:05:28,590 --> 00:05:32,610
That must be small compared
to the De Broglie wavelength.

75
00:05:32,610 --> 00:05:34,765
So the change of
the De Broglie wave

76
00:05:34,765 --> 00:05:37,110
length after you move one
De Broglie wave length

77
00:05:37,110 --> 00:05:40,110
must be smaller than the
De Broglie wave length.

78
00:05:40,110 --> 00:05:42,300
I don't know if you like it.

79
00:05:42,300 --> 00:05:46,060
Otherwise, you
can take this one.

80
00:05:46,060 --> 00:05:51,960
I'll do another one, another
version of the inequalities.

81
00:05:51,960 --> 00:05:53,860
And you can play with
those inequalities.

82
00:05:53,860 --> 00:05:59,080
It kind of takes a while
until you convince yourself

83
00:05:59,080 --> 00:06:01,390
that you're not
missing anything.

84
00:06:01,390 --> 00:06:07,240
Think of p squared equal
to m e minus v of x.

85
00:06:09,770 --> 00:06:13,100
Take a derivative, vvx.

86
00:06:13,100 --> 00:06:17,730
So I'll have p, p prime.

87
00:06:17,730 --> 00:06:19,100
This is 2 pp prime.

88
00:06:19,100 --> 00:06:23,660
But with this 2, I'm
going to cancel it.

89
00:06:23,660 --> 00:06:28,670
At some point, of course, we're
taking all kinds of factors

90
00:06:28,670 --> 00:06:30,780
of 2 and ignoring them.

91
00:06:30,780 --> 00:06:34,580
Remember that true De
Broglie wave length is h over

92
00:06:34,580 --> 00:06:36,620
p not h bar over p.

93
00:06:39,530 --> 00:06:44,120
Factors-- by the time you go
to this inequality, two pis

94
00:06:44,120 --> 00:06:45,400
are gone.

95
00:06:45,400 --> 00:06:46,530
m.

96
00:06:46,530 --> 00:06:48,230
We're differentiating
with respect

97
00:06:48,230 --> 00:06:50,540
to x and taking absolute value.

98
00:06:50,540 --> 00:06:53,960
So we'll write it like this.

99
00:06:53,960 --> 00:07:04,250
Or vvvx vx equals
1 over m pp prime.

100
00:07:04,250 --> 00:07:07,610
That is so far exact.

101
00:07:07,610 --> 00:07:10,665
Let me multiply by a lambda.

102
00:07:14,880 --> 00:07:16,185
So I'll have a lambda.

103
00:07:22,250 --> 00:07:44,200
And a lambda, vv dx,
is equal to lambda.

104
00:07:44,200 --> 00:07:44,830
OK.

105
00:07:44,830 --> 00:07:48,010
Lambda is h bar over p.

106
00:07:48,010 --> 00:07:54,635
So I can cancel one of these p's
and get h bar over m p prime.

107
00:07:57,610 --> 00:07:59,770
OK.

108
00:07:59,770 --> 00:08:01,540
H bar over m p prime.

109
00:08:01,540 --> 00:08:04,880
Now, look at this equation.

110
00:08:04,880 --> 00:08:05,380
I'm sorry.

111
00:08:05,380 --> 00:08:08,000
We'll play with this a little--

112
00:08:08,000 --> 00:08:09,330
like, trial and error.

113
00:08:09,330 --> 00:08:12,440
You're trying to move
around your inequality.

114
00:08:12,440 --> 00:08:15,220
So here we have
something-- h bar

115
00:08:15,220 --> 00:08:18,310
dp dx is much less than that.

116
00:08:18,310 --> 00:08:28,830
So this term is because
of this inequality

117
00:08:28,830 --> 00:08:34,100
is much smaller than
p squared over m.

118
00:08:34,100 --> 00:08:37,610
And now we have something nice.

119
00:08:37,610 --> 00:08:40,100
I'll write it here.

120
00:08:40,100 --> 00:08:50,330
Lambda of x dv dx is much
smaller than p squared over 2m.

121
00:08:53,930 --> 00:08:56,300
That's another nice one.

122
00:08:56,300 --> 00:08:58,135
I think this one is the--

123
00:09:01,250 --> 00:09:07,730
and this says that the potential
must be slowly varying for this

124
00:09:07,730 --> 00:09:13,610
to be true because the change
in the potential over at the De

125
00:09:13,610 --> 00:09:16,780
Broglie wavelength--
dv dx times lambda of x

126
00:09:16,780 --> 00:09:20,705
is an estimate for the change
of the potential over the De

127
00:09:20,705 --> 00:09:24,290
Broglie wavelength-- is much
smaller than the kinetic energy

128
00:09:24,290 --> 00:09:26,320
of the particle.

129
00:09:26,320 --> 00:09:30,740
So that's, again, another
thing that makes sense.

130
00:09:30,740 --> 00:09:34,490
It's kind of nice.

131
00:09:34,490 --> 00:09:40,070
So this is the wave at
the end of the day, this h

132
00:09:40,070 --> 00:09:42,920
bar going to zero
approximation has

133
00:09:42,920 --> 00:09:45,570
become a physical statement.

134
00:09:45,570 --> 00:09:50,720
It is a statement of
quantities varying

135
00:09:50,720 --> 00:09:55,610
slowly because after all, that's
what motivated the expansion

136
00:09:55,610 --> 00:09:57,570
from the beginning.

137
00:09:57,570 --> 00:10:01,490
So let's see if we ever
get in trouble with this.

138
00:10:01,490 --> 00:10:06,530
So we're trying to solve
physical problems of particles

139
00:10:06,530 --> 00:10:07,670
and potentials.

140
00:10:07,670 --> 00:10:11,990
And most of the times
we're interested in bound

141
00:10:11,990 --> 00:10:16,540
states or energy eigenstates,
at least the simple energy

142
00:10:16,540 --> 00:10:18,600
eigenstates are bound states.

143
00:10:18,600 --> 00:10:22,870
So here is a situation.

144
00:10:22,870 --> 00:10:25,780
We have a sketch of a situation.

145
00:10:28,690 --> 00:10:30,430
We have a v of x--

146
00:10:30,430 --> 00:10:32,320
this is x.

147
00:10:32,320 --> 00:10:34,750
This is a.

148
00:10:34,750 --> 00:10:38,260
v of x-- and some energy, e.

149
00:10:47,130 --> 00:10:50,610
And let's assume we're
looking close enough

150
00:10:50,610 --> 00:10:55,740
to the point x equals a so that
the v of x, however it curves,

151
00:10:55,740 --> 00:10:58,320
at that point is
roughly straight.

152
00:11:01,190 --> 00:11:05,120
That's a reasonable thing to do.

153
00:11:05,120 --> 00:11:12,200
So we'll model v of x minus e--

154
00:11:12,200 --> 00:11:17,210
v of x minus e-- as being
linear near x equals a.

155
00:11:17,210 --> 00:11:20,450
So this is g times
x minus a, where

156
00:11:20,450 --> 00:11:23,870
g is some positive
constant, which

157
00:11:23,870 --> 00:11:26,590
is the slope at this point.

158
00:11:30,990 --> 00:11:42,510
So look at x less than a.

159
00:11:42,510 --> 00:11:45,630
At this point, you are
in the loud region.

160
00:11:45,630 --> 00:11:51,190
You're in the region to
the left of the point a

161
00:11:51,190 --> 00:11:53,750
where your energy is
bigger than the potential.

162
00:11:53,750 --> 00:11:55,420
And that's perfectly allowed.

163
00:12:01,440 --> 00:12:08,280
So here, e minus v of x,
which is the negative of that,

164
00:12:08,280 --> 00:12:12,990
would be g a minus x.

165
00:12:12,990 --> 00:12:27,950
And p is square root of 2 m e
minus v of x, so g a minus x.

166
00:12:27,950 --> 00:12:28,780
That's p of x.

167
00:12:33,590 --> 00:12:37,890
So that's your position
dependent momentum.

168
00:12:37,890 --> 00:12:42,740
It's going to go
to 0 at that point.

169
00:12:42,740 --> 00:12:50,900
And lambda, which
is h bar over p,

170
00:12:50,900 --> 00:12:55,610
is h bar over square
root of 2 mg 1

171
00:12:55,610 --> 00:12:59,620
over square root of a minus x.

172
00:12:59,620 --> 00:13:02,300
So take the derivative--

173
00:13:02,300 --> 00:13:04,960
d lambda d x--

174
00:13:04,960 --> 00:13:08,710
take the absolute value of it.

175
00:13:08,710 --> 00:13:20,570
That's h bar over square root of
2 mg times 1/2 1 over a minus x

176
00:13:20,570 --> 00:13:22,110
to the three halves.

177
00:13:22,110 --> 00:13:24,050
We're differentiating
with respect to x.

178
00:13:27,220 --> 00:13:33,370
And now you see the trouble,
if you had not seen it

179
00:13:33,370 --> 00:13:40,840
before, the validity of the
semi-classical approximation

180
00:13:40,840 --> 00:13:48,610
is taken and requires d lambda
dx to be much smaller than 1.

181
00:13:48,610 --> 00:13:57,340
And as you approach x equals
to a, this grows without bound.

182
00:13:57,340 --> 00:14:00,880
It just becomes
bigger and bigger.

183
00:14:00,880 --> 00:14:05,800
You can choose g to be large and
you can choose m to be large.

184
00:14:05,800 --> 00:14:09,880
But still, you get closer and
closer, you eventually fail.

185
00:14:09,880 --> 00:14:14,890
This thing goes to
infinity as x goes to a

186
00:14:14,890 --> 00:14:16,870
and grows without limit.

187
00:14:16,870 --> 00:14:20,050
And the semi-classical
approximation crashes.

188
00:14:23,190 --> 00:14:30,370
You know, I would imagine
that many people got this far

189
00:14:30,370 --> 00:14:33,550
with this in my classical
approximation of writing

190
00:14:33,550 --> 00:14:35,140
this and that.

191
00:14:35,140 --> 00:14:41,210
But this is a
tremendous obstacle.

192
00:14:41,210 --> 00:14:42,890
Why is it an obstacle?

193
00:14:42,890 --> 00:14:45,680
Why can't we just
forget about that region

194
00:14:45,680 --> 00:14:49,830
because most of the times you
dealing with bound states.

195
00:14:49,830 --> 00:14:56,120
So you will have a very
slowly varying potential here.

196
00:14:56,120 --> 00:14:59,650
But if you want to
find bound states,

197
00:14:59,650 --> 00:15:02,920
you need the fact that there
is a forbidden region where

198
00:15:02,920 --> 00:15:06,130
the wave function
destroys itself.

199
00:15:06,130 --> 00:15:09,970
So whatever you can solve
for the wave function

200
00:15:09,970 --> 00:15:12,250
is slowly varying here.

201
00:15:12,250 --> 00:15:16,990
It's not enough because you
need to know how it decays.

202
00:15:16,990 --> 00:15:21,490
And therefore, you need
to face this corners

203
00:15:21,490 --> 00:15:25,900
where the semi-classical
approximation fails.

204
00:15:25,900 --> 00:15:31,330
Our problem here is that we know
how to write a solution here.

205
00:15:31,330 --> 00:15:34,330
We probably know how to
write the solution here.

206
00:15:34,330 --> 00:15:37,210
Those are these ones.

207
00:15:37,210 --> 00:15:40,930
But we have no idea
how to write them here.

208
00:15:40,930 --> 00:15:44,320
So we cannot connect
the two solutions.

209
00:15:44,320 --> 00:15:45,970
It's a serious difficulty.

210
00:15:49,320 --> 00:15:51,510
So people work hard on that.

211
00:15:51,510 --> 00:15:54,540
And I think that
is the breakthrough

212
00:15:54,540 --> 00:16:00,450
of the construction of
this WKBP [INAUDIBLE]..

213
00:16:00,450 --> 00:16:04,560
What they did is they
solved the equation

214
00:16:04,560 --> 00:16:09,450
exactly in this region,
assuming a linear potential.

215
00:16:09,450 --> 00:16:11,130
They solved it exactly.

216
00:16:11,130 --> 00:16:16,710
And then those functions, the
[INAUDIBLE] functions, show up.

217
00:16:16,710 --> 00:16:19,920
And you know how the
[INAUDIBLE] functions behave.

218
00:16:19,920 --> 00:16:21,930
So they solve it here.

219
00:16:21,930 --> 00:16:26,010
They related it to the solution
to the right, related it

220
00:16:26,010 --> 00:16:27,690
to the solution on the left.

221
00:16:27,690 --> 00:16:30,240
And in that way,
even though we don't

222
00:16:30,240 --> 00:16:33,270
have to write the
solution in this region,

223
00:16:33,270 --> 00:16:36,960
we know how a
solution on the middle

224
00:16:36,960 --> 00:16:39,240
connects to a
solution on the right.

225
00:16:39,240 --> 00:16:43,470
This is the subject of the
connection formulas in WKB.

226
00:16:43,470 --> 00:16:45,990
We will discuss that
next time, or we'll

227
00:16:45,990 --> 00:16:49,560
go through some of that analysis
because it's interesting

228
00:16:49,560 --> 00:16:52,380
and fairly non-trivial.

229
00:16:52,380 --> 00:16:56,410
But I will mention one of the
connection formulas and use it.

230
00:16:56,410 --> 00:17:00,710
That's the rest of what
we're going to do today.