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PROFESSOR: OK, so
I want to start out

9
00:00:25,030 --> 00:00:27,420
by reviewing a few
things and putting

10
00:00:27,420 --> 00:00:29,620
some machinery together.

11
00:00:29,620 --> 00:00:31,560
Unfortunately, this
thing is sort of stuck.

12
00:00:31,560 --> 00:00:35,300
We're going to need a
later, so I don't know.

13
00:00:35,300 --> 00:00:37,700
I'll put it up for now.

14
00:00:37,700 --> 00:00:40,410
So first just a bit of notation.

15
00:00:40,410 --> 00:00:42,250
This symbol, you
should think of it

16
00:00:42,250 --> 00:00:44,520
like the dot product,
or the inner product.

17
00:00:44,520 --> 00:00:49,160
It's just saying that
bracket f g is the integral.

18
00:00:49,160 --> 00:00:50,450
It's a number that you get.

19
00:00:50,450 --> 00:00:52,780
So this is a number that
you get from the function

20
00:00:52,780 --> 00:00:55,820
f and the function g
by taking f, taking

21
00:00:55,820 --> 00:00:58,840
its complex conjugate,
multiplying it by g,

22
00:00:58,840 --> 00:01:00,710
and then integrating
overall positions.

23
00:01:00,710 --> 00:01:01,490
All right?

24
00:01:01,490 --> 00:01:02,740
So it's a way to get a number.

25
00:01:02,740 --> 00:01:05,310
And you should think about it
as the analog for functions

26
00:01:05,310 --> 00:01:07,530
of the dot product for vectors.

27
00:01:07,530 --> 00:01:11,490
It's a way to get a
number out of two vectors.

28
00:01:11,490 --> 00:01:15,860
And so, for example, with
vectors we could do v dot w,

29
00:01:15,860 --> 00:01:16,940
and this is some number.

30
00:01:16,940 --> 00:01:20,712
And it has a nice
property that v dot v,

31
00:01:20,712 --> 00:01:23,170
we can think it as v squared,
it's something like a length.

32
00:01:23,170 --> 00:01:25,480
It's strictly positive,
and it's something

33
00:01:25,480 --> 00:01:26,760
like the length of a vector.

34
00:01:26,760 --> 00:01:31,320
Similarly, if I take f and
take its bracket with f,

35
00:01:31,320 --> 00:01:34,205
this is equal to the
integral dx of f squared,

36
00:01:34,205 --> 00:01:36,580
and in particular, f could be
complex, so f norm squared.

37
00:01:36,580 --> 00:01:38,000
This is strictly non-negative.

38
00:01:38,000 --> 00:01:41,550
It could vanish, but it's
not negative at a point,

39
00:01:41,550 --> 00:01:42,740
hence the norm squared.

40
00:01:42,740 --> 00:01:46,990
So this will be zero
if and only if what?

41
00:01:46,990 --> 00:01:49,010
f is 0, f is the
0 function, right.

42
00:01:49,010 --> 00:01:51,852
So the same way that
if you take a vector,

43
00:01:51,852 --> 00:01:54,310
and you take its dot product
with itself, take it the norm,

44
00:01:54,310 --> 00:01:56,470
it's 0 if an only
if the vector is 0.

45
00:01:56,470 --> 00:01:59,270
So this beast satisfies a lot
of the properties of a dot

46
00:01:59,270 --> 00:01:59,920
product.

47
00:01:59,920 --> 00:02:01,980
You should think about
it as morally equivalent.

48
00:02:01,980 --> 00:02:04,190
We'll talk about that
in more detail later.

49
00:02:04,190 --> 00:02:06,670
Second, basic postulate
of quantum mechanics,

50
00:02:06,670 --> 00:02:08,972
to every observable is
associated an operator,

51
00:02:08,972 --> 00:02:11,180
and it's an operator acting
on the space of functions

52
00:02:11,180 --> 00:02:14,340
or on the space
of wave functions.

53
00:02:14,340 --> 00:02:16,249
And to every operator
corresponding

54
00:02:16,249 --> 00:02:17,790
to an observable in
quantum mechanics

55
00:02:17,790 --> 00:02:20,060
are associated a special
set of functions called

56
00:02:20,060 --> 00:02:22,640
the eigenfunctions, such
that when the operator acts

57
00:02:22,640 --> 00:02:24,640
on that function, it gives
you the same function

58
00:02:24,640 --> 00:02:26,560
back times a constant.

59
00:02:26,560 --> 00:02:29,100
What these functions
mean, physically,

60
00:02:29,100 --> 00:02:32,860
is they are the wave functions
describing configurations

61
00:02:32,860 --> 00:02:35,930
with a definite value of the
corresponding observable.

62
00:02:35,930 --> 00:02:38,240
If I'm in an eigenfunction
of position with

63
00:02:38,240 --> 00:02:41,610
eigenvalue x naught, awesome.

64
00:02:41,610 --> 00:02:44,740
Thank you, AV person, thank you.

65
00:02:44,740 --> 00:02:48,742
So if your system is described
by a wave function which

66
00:02:48,742 --> 00:02:50,200
is an eigenfunction
of the position

67
00:02:50,200 --> 00:02:52,470
operator with
eigenvalue x naught,

68
00:02:52,470 --> 00:02:55,084
that means you can be
confident that the system is

69
00:02:55,084 --> 00:02:56,500
in the configuration
corresponding

70
00:02:56,500 --> 00:02:59,111
to having a definite
position x naught.

71
00:02:59,111 --> 00:02:59,610
Right?

72
00:02:59,610 --> 00:03:02,360
It's not a superposition
of different positions.

73
00:03:02,360 --> 00:03:04,980
It is at x naught.

74
00:03:04,980 --> 00:03:07,567
Similarly, momentum,
momentum has eigenfunctions,

75
00:03:07,567 --> 00:03:08,900
and we know what these guys are.

76
00:03:08,900 --> 00:03:12,300
These are the exponentials,
e to the iKX's.

77
00:03:12,300 --> 00:03:14,890
They're the eigenfunctions, and
those are the wave functions

78
00:03:14,890 --> 00:03:18,130
describing states with
definite value of the momentum,

79
00:03:18,130 --> 00:03:19,780
of the associated observable.

80
00:03:19,780 --> 00:03:26,640
Energy as an operator, energy is
described by an operator, which

81
00:03:26,640 --> 00:03:34,340
has eigenfunctions which I'll
call phi sub n, with energy

82
00:03:34,340 --> 00:03:37,910
as E sub n, those
are the eigenvalues.

83
00:03:37,910 --> 00:03:40,920
And if I tell you that your
wave function is the state phi

84
00:03:40,920 --> 00:03:43,330
sub 2, what that tells
you is that the system has

85
00:03:43,330 --> 00:03:46,540
a definite energy, E
sub 2, corresponding

86
00:03:46,540 --> 00:03:48,883
to that eigenvalue.

87
00:03:48,883 --> 00:03:51,166
Cool?

88
00:03:51,166 --> 00:03:53,040
And this is true for
any physical observable.

89
00:03:53,040 --> 00:03:54,350
But these are sort
of the basic ones

90
00:03:54,350 --> 00:03:56,849
that we'll keep focusing on,
position, momentum, and energy,

91
00:03:56,849 --> 00:03:58,930
for the next while.

92
00:03:58,930 --> 00:04:01,360
Now a nice property about
these eigenfunctions

93
00:04:01,360 --> 00:04:04,740
is that for different
eigenvalues,

94
00:04:04,740 --> 00:04:07,220
the associated wave functions
are different functions.

95
00:04:07,220 --> 00:04:08,320
And what I mean by
saying they're different

96
00:04:08,320 --> 00:04:09,736
functions is that
they're actually

97
00:04:09,736 --> 00:04:12,540
orthogonal functions in the
sense of this dot product.

98
00:04:12,540 --> 00:04:16,050
If I have a state
corresponding to be at x 0,

99
00:04:16,050 --> 00:04:17,579
definite position
x 0, that means

100
00:04:17,579 --> 00:04:20,371
they're in eigenfunction of
position with eigenvalue x 0,

101
00:04:20,371 --> 00:04:21,829
and I have another
that corresponds

102
00:04:21,829 --> 00:04:24,630
to being at x1, an eigenfunction
of the position operator

103
00:04:24,630 --> 00:04:28,240
or the eigenvalue x1, then
these wave functions are

104
00:04:28,240 --> 00:04:29,920
orthogonal to each other.

105
00:04:29,920 --> 00:04:32,960
And we get 0 if x 0
is not equal to x1.

106
00:04:32,960 --> 00:04:35,250
Everyone cool with that?

107
00:04:35,250 --> 00:04:37,520
Now, meanwhile not only
are they orthogonal

108
00:04:37,520 --> 00:04:39,837
but they're normalized
in a particular way.

109
00:04:39,837 --> 00:04:41,670
The inner product gives
me a delta function,

110
00:04:41,670 --> 00:04:44,490
which goes beep once, so that
if I integrate against it

111
00:04:44,490 --> 00:04:45,319
I get a 1.

112
00:04:45,319 --> 00:04:46,360
Same thing with momentum.

113
00:04:46,360 --> 00:04:48,734
And you do this, this you're
checking on the problem set.

114
00:04:48,734 --> 00:04:50,930
I don't remember if it
was last one or this one.

115
00:04:50,930 --> 00:04:53,410
And for the energies,
energy 1, if I

116
00:04:53,410 --> 00:04:58,436
know the system is in state
energy 1, and let's say e sub n

117
00:04:58,436 --> 00:05:00,625
and e sub m, those
are different states

118
00:05:00,625 --> 00:05:02,250
if n and m are not
equal to each other.

119
00:05:02,250 --> 00:05:04,867
And this inner product
is 0 if n and m are not

120
00:05:04,867 --> 00:05:06,450
equal to each other
and 1 if they are.

121
00:05:06,450 --> 00:05:08,220
Their properly normalized.

122
00:05:08,220 --> 00:05:10,240
Everyone cool with that?

123
00:05:10,240 --> 00:05:11,769
Yeah.

124
00:05:11,769 --> 00:05:13,768
AUDIENCE: Is it possible
that two eigenfunctions

125
00:05:13,768 --> 00:05:15,514
have the same eigenvalue?

126
00:05:15,514 --> 00:05:16,430
PROFESSOR: Absolutely.

127
00:05:16,430 --> 00:05:18,650
It is absolutely possible
for two eigenfunctions

128
00:05:18,650 --> 00:05:19,996
to have the same eigenvalue.

129
00:05:19,996 --> 00:05:21,120
That is certainly possible.

130
00:05:21,120 --> 00:05:23,210
AUDIENCE: [INAUDIBLE]

131
00:05:23,210 --> 00:05:25,257
PROFESSOR: Yeah, good.

132
00:05:25,257 --> 00:05:26,840
Thank you, this is
a good technicality

133
00:05:26,840 --> 00:05:28,970
that I didn't want to get into,
but I'll go and get into it.

134
00:05:28,970 --> 00:05:30,053
It's a very good question.

135
00:05:30,053 --> 00:05:31,900
So the question is,
is it possible for two

136
00:05:31,900 --> 00:05:34,620
different eigenfunctions to
have the same eigenvalue.

137
00:05:34,620 --> 00:05:36,640
Could there be two states
with the same energy ,

138
00:05:36,640 --> 00:05:38,550
different states, same energy?

139
00:05:38,550 --> 00:05:40,080
Yeah, that's
absolutely possible.

140
00:05:40,080 --> 00:05:41,280
And we'll run into that.

141
00:05:41,280 --> 00:05:43,550
And there's nice
physics encoded in it.

142
00:05:43,550 --> 00:05:45,200
But let's think about
what that means.

143
00:05:45,200 --> 00:05:47,400
The subsequent question is
well, if that's the case,

144
00:05:47,400 --> 00:05:49,910
are they really
still orthogonal?

145
00:05:49,910 --> 00:05:51,450
And here's the crucial thing.

146
00:05:51,450 --> 00:05:55,190
The crucial thing is, let's
say I take one function,

147
00:05:55,190 --> 00:05:58,750
I'll call the function phi 1,
consider the function phi 1.

148
00:05:58,750 --> 00:06:02,330
And let it have energy E1,
so that E acting on phi 1

149
00:06:02,330 --> 00:06:05,020
is equal to E1 phi 1.

150
00:06:05,020 --> 00:06:08,600
And let there be
another function, phi 2,

151
00:06:08,600 --> 00:06:10,810
such that the energy
operator acting on phi 2

152
00:06:10,810 --> 00:06:14,050
is also equal to E1 phi 2.

153
00:06:14,050 --> 00:06:16,700
These are said to be degenerate.

154
00:06:16,700 --> 00:06:18,950
Degenerate doesn't mean you
go out and trash your car,

155
00:06:18,950 --> 00:06:22,510
degenerate that the
energies are the same.

156
00:06:22,510 --> 00:06:23,997
So what does this tell me?

157
00:06:23,997 --> 00:06:25,080
This tells me a cool fact.

158
00:06:25,080 --> 00:06:30,390
If I take a wave function phi,
and I will call this phi star,

159
00:06:30,390 --> 00:06:32,560
in honor of Shri
Kulkarni, so I've

160
00:06:32,560 --> 00:06:35,770
got this phi star, which is
a linear combination alpha

161
00:06:35,770 --> 00:06:39,270
phi 1 plus beta phi 2,
a linear combination

162
00:06:39,270 --> 00:06:42,700
of them, a superposition
of those two states.

163
00:06:42,700 --> 00:06:46,060
Is this also an
energy eigenfunction?

164
00:06:46,060 --> 00:06:52,630
Yeah, because if I act on phi
star with E, then it's linear,

165
00:06:52,630 --> 00:06:55,960
so E acting on phi star is
E acting on alpha phi 1,

166
00:06:55,960 --> 00:06:57,370
alpha's a constant,
doesn't care.

167
00:06:57,370 --> 00:06:59,090
Phi 1 gives me an E1.

168
00:06:59,090 --> 00:07:02,060
Similarly, E acting on
phi 2 gives me an E1.

169
00:07:02,060 --> 00:07:06,490
So if I act with E
on this guy, this

170
00:07:06,490 --> 00:07:09,229
is equal to, from both of these
I get an overall factor of E1.

171
00:07:09,229 --> 00:07:11,270
So notice that we get the
same vector back, times

172
00:07:11,270 --> 00:07:16,260
a constant, a common constant.

173
00:07:16,260 --> 00:07:18,237
So when we have
degenerate eigenfunctions,

174
00:07:18,237 --> 00:07:20,320
we can take arbitrary
linear combinations to them,

175
00:07:20,320 --> 00:07:22,700
get another degenerate
eigenfunction.

176
00:07:22,700 --> 00:07:23,522
Cool?

177
00:07:23,522 --> 00:07:25,230
So this is like,
imagine I have a vector,

178
00:07:25,230 --> 00:07:26,460
and I have another vector.

179
00:07:26,460 --> 00:07:28,380
And they share the property
that they're both eigenfunctions

180
00:07:28,380 --> 00:07:29,240
of some operator.

181
00:07:29,240 --> 00:07:30,948
That means any linear
combination of them

182
00:07:30,948 --> 00:07:32,284
is also, right?

183
00:07:32,284 --> 00:07:33,950
So there's a whole
vector space, there's

184
00:07:33,950 --> 00:07:36,850
a whole space of
possible functions

185
00:07:36,850 --> 00:07:38,704
that all have the
same eigenvalue.

186
00:07:38,704 --> 00:07:40,870
So now you say, well, look,
are these two orthogonal

187
00:07:40,870 --> 00:07:41,360
to each other?

188
00:07:41,360 --> 00:07:41,640
No.

189
00:07:41,640 --> 00:07:41,820
These two?

190
00:07:41,820 --> 00:07:42,240
No.

191
00:07:42,240 --> 00:07:43,140
But here's the thing.

192
00:07:43,140 --> 00:07:45,306
If you have a vector space,
if you have a the space,

193
00:07:45,306 --> 00:07:48,030
you can always find
orthogonal guys and a basis

194
00:07:48,030 --> 00:07:50,190
for that space, yes?

195
00:07:50,190 --> 00:07:53,470
So while it's not true that
the eigenfunctions are always

196
00:07:53,470 --> 00:07:56,460
orthogonal, it is true--

197
00:07:56,460 --> 00:07:59,920
we will not prove this, but we
will discuss the proof of it

198
00:07:59,920 --> 00:08:01,770
later by pulling the
mathematician out

199
00:08:01,770 --> 00:08:03,930
of the closet--

200
00:08:03,930 --> 00:08:06,370
the proof will say that
it is possible to find

201
00:08:06,370 --> 00:08:09,620
a set of eigenfunctions which
are orthogonal in precisely

202
00:08:09,620 --> 00:08:12,380
this fashion, even if
there are degeneracies.

203
00:08:12,380 --> 00:08:12,880
OK?

204
00:08:12,880 --> 00:08:14,960
That theorem is called
the spectral theorem.

205
00:08:14,960 --> 00:08:16,724
And we'll discuss it later.

206
00:08:16,724 --> 00:08:18,140
So it is always
possible to do so.

207
00:08:18,140 --> 00:08:20,460
But you must be alert that
there may be degeneracies.

208
00:08:20,460 --> 00:08:22,140
There aren't always
degeneracies.

209
00:08:22,140 --> 00:08:24,040
In fact, degeneracies
are very special.

210
00:08:24,040 --> 00:08:27,840
Why should two numbers
happen to be the same?

211
00:08:27,840 --> 00:08:29,840
Something has to be forcing
them to be the same.

212
00:08:29,840 --> 00:08:31,715
That's going to be an
important theme for us.

213
00:08:31,715 --> 00:08:32,966
But it certainly is possible.

214
00:08:32,966 --> 00:08:33,549
Good question.

215
00:08:33,549 --> 00:08:36,085
Other questions?

216
00:08:36,085 --> 00:08:36,585
Yeah.

217
00:08:36,585 --> 00:08:42,340
AUDIENCE: [INAUDIBLE]

218
00:08:42,340 --> 00:08:46,110
PROFESSOR: Yeah, so using
the triangular brackets--

219
00:08:46,110 --> 00:08:49,140
so there's another notation for
the same thing, which is f g,

220
00:08:49,140 --> 00:08:51,351
but this carries some
slightly different weight.

221
00:08:51,351 --> 00:08:53,600
It mean something slightly--
you'll see this in books,

222
00:08:53,600 --> 00:08:55,500
and this means something
very similar to this.

223
00:08:55,500 --> 00:08:56,680
But I'm not going to
use this notation.

224
00:08:56,680 --> 00:08:58,010
It's called Dirac notation.

225
00:08:58,010 --> 00:08:59,360
We'll talk about it
later in the semester,

226
00:08:59,360 --> 00:09:00,640
but we're not going to
talk about it just yet.

227
00:09:00,640 --> 00:09:02,490
But when you see
this, effectively it

228
00:09:02,490 --> 00:09:05,565
means the same thing as this.

229
00:09:05,565 --> 00:09:07,260
This is sort of like dialect.

230
00:09:07,260 --> 00:09:10,590
You know, it's like
French and Quebecois.

231
00:09:10,590 --> 00:09:12,290
Other questions?

232
00:09:12,290 --> 00:09:14,190
My wife's Canadian.

233
00:09:14,190 --> 00:09:15,750
Other questions?

234
00:09:15,750 --> 00:09:17,760
OK.

235
00:09:17,760 --> 00:09:19,200
So given this fact,
given the fact

236
00:09:19,200 --> 00:09:20,574
that we can
associate observables

237
00:09:20,574 --> 00:09:23,210
to operators, operators
come with special functions,

238
00:09:23,210 --> 00:09:25,460
the eigenfunctions, those
eigenfunctions corresponding

239
00:09:25,460 --> 00:09:27,410
to have a definite
value of the observable,

240
00:09:27,410 --> 00:09:29,320
and they're orthonormal.

241
00:09:29,320 --> 00:09:30,940
This tells us,
and this is really

242
00:09:30,940 --> 00:09:32,523
the statement of the
spectral theorem,

243
00:09:32,523 --> 00:09:36,340
that any function can be
expanded in a basis of states

244
00:09:36,340 --> 00:09:38,550
with definite values
of some observable.

245
00:09:38,550 --> 00:09:40,200
So for example,
consider position.

246
00:09:40,200 --> 00:09:41,720
I claim that any
wave function can

247
00:09:41,720 --> 00:09:44,660
be expanded as a
superposition of states

248
00:09:44,660 --> 00:09:46,190
with definite position.

249
00:09:46,190 --> 00:09:48,510
So here's an arbitrary
function, here's

250
00:09:48,510 --> 00:09:52,140
this set of states with definite
position, the delta functions.

251
00:09:52,140 --> 00:09:55,290
And I can write any
function as a superposition

252
00:09:55,290 --> 00:09:57,960
with some coefficients of
states with definite position,

253
00:09:57,960 --> 00:10:01,814
integrating over all
possible positions, x0.

254
00:10:01,814 --> 00:10:03,480
And this is also sort
of trivially true,

255
00:10:03,480 --> 00:10:04,910
because what's this integral?

256
00:10:04,910 --> 00:10:07,650
Well, it's an integral, dx0
over all possible positions

257
00:10:07,650 --> 00:10:09,170
of this delta function.

258
00:10:09,170 --> 00:10:12,090
But we're evaluating
at x, so this is 0

259
00:10:12,090 --> 00:10:13,550
unless x is equal to x0.

260
00:10:13,550 --> 00:10:15,125
So I can just put
in x instead of x0,

261
00:10:15,125 --> 00:10:16,670
and that gives me psi of x.

262
00:10:16,670 --> 00:10:18,760
Sort of tautological We
can do the same thing

263
00:10:18,760 --> 00:10:20,442
for momentum eigenfunctions.

264
00:10:20,442 --> 00:10:22,150
I claim that any
function can be expanded

265
00:10:22,150 --> 00:10:24,340
in a superposition of
momentum eigenfunctions, where

266
00:10:24,340 --> 00:10:27,160
I sum over all possible
values in the momentum

267
00:10:27,160 --> 00:10:30,120
with some weight.

268
00:10:30,120 --> 00:10:31,670
This psi tilde of
K is just telling

269
00:10:31,670 --> 00:10:34,480
me how much amplitude there
is at that wave number.

270
00:10:34,480 --> 00:10:35,120
Cool?

271
00:10:35,120 --> 00:10:37,660
But this is the Fourier theorem,
it's a Fourier expansion.

272
00:10:37,660 --> 00:10:40,864
So purely mathematically,
we know that this is true.

273
00:10:40,864 --> 00:10:42,530
But there's also the
physical statement.

274
00:10:42,530 --> 00:10:45,050
Any state can be expressed
as a superposition of states

275
00:10:45,050 --> 00:10:47,950
with definite momentum.

276
00:10:47,950 --> 00:10:50,300
There's a math in here, but
there's also physics in it.

277
00:10:50,300 --> 00:10:52,370
Finally, this is less obvious
from a mathematical point

278
00:10:52,370 --> 00:10:54,036
of view, because I
haven't even told you

279
00:10:54,036 --> 00:10:56,560
what energy is, any wave
function can be expanded

280
00:10:56,560 --> 00:10:58,360
in states with definite energy.

281
00:10:58,360 --> 00:11:02,110
So this is a state, my state
En, with definite energy,

282
00:11:02,110 --> 00:11:05,200
with some coefficient summed
over all possible values

283
00:11:05,200 --> 00:11:07,790
of the energy.

284
00:11:07,790 --> 00:11:11,750
Given any physical observable,
any physical observable,

285
00:11:11,750 --> 00:11:13,860
momentum, position,
angular momentum,

286
00:11:13,860 --> 00:11:16,769
whatever, given any
physical observable,

287
00:11:16,769 --> 00:11:18,310
a given wave function
can be expanded

288
00:11:18,310 --> 00:11:21,920
as some superposition of
having definite values of that.

289
00:11:21,920 --> 00:11:25,950
Will it in general have definite
values of the observable?

290
00:11:25,950 --> 00:11:28,500
Well a general state be
an energy eigenfunction?

291
00:11:28,500 --> 00:11:29,420
No.

292
00:11:29,420 --> 00:11:35,210
But any state is a superposition
of energy eigenfunctions.

293
00:11:35,210 --> 00:11:38,240
Will a random state
have definite position?

294
00:11:38,240 --> 00:11:38,870
Certainly not.

295
00:11:38,870 --> 00:11:41,010
You could have
this wave function.

296
00:11:41,010 --> 00:11:43,110
Superposition.

297
00:11:43,110 --> 00:11:45,210
Yeah.

298
00:11:45,210 --> 00:11:50,710
AUDIENCE: Why is the
energy special such

299
00:11:50,710 --> 00:11:54,680
that you can make an arbitrary
state with a countable number

300
00:11:54,680 --> 00:11:56,954
of energy eigenfunctions
rather than having

301
00:11:56,954 --> 00:11:58,120
to do a continuous spectrum?

302
00:11:58,120 --> 00:11:58,890
PROFESSOR: Excellent question.

303
00:11:58,890 --> 00:12:00,340
So I'm going to phrase
that slightly differently.

304
00:12:00,340 --> 00:12:01,280
It's an excellent
question, and we'll

305
00:12:01,280 --> 00:12:03,460
come to that at the
end of today's lecture.

306
00:12:03,460 --> 00:12:06,190
So the question is,
those are integrals, that

307
00:12:06,190 --> 00:12:08,115
is a sum over discrete things.

308
00:12:08,115 --> 00:12:08,615
Why?

309
00:12:08,615 --> 00:12:12,012
Why is the possible values
of the position continuous,

310
00:12:12,012 --> 00:12:14,470
possible values of momentum
continuous, and possible values

311
00:12:14,470 --> 00:12:16,690
of energy discrete?

312
00:12:16,690 --> 00:12:20,050
The answer to this
will become apparent

313
00:12:20,050 --> 00:12:22,152
over the course of your
next few problem sets.

314
00:12:22,152 --> 00:12:23,610
You have to do some
problems to get

315
00:12:23,610 --> 00:12:25,190
your fingers dirty to
really understand this.

316
00:12:25,190 --> 00:12:26,648
But here's the
statement, and we'll

317
00:12:26,648 --> 00:12:30,700
see the first version of this
at the end of today's lecture.

318
00:12:30,700 --> 00:12:33,420
Sometimes the allowed energies
of a system, the energy

319
00:12:33,420 --> 00:12:35,240
eigenvalues, are discrete.

320
00:12:35,240 --> 00:12:37,170
Sometimes they are continuous.

321
00:12:37,170 --> 00:12:41,056
They will be discrete when you
have bound states, states that

322
00:12:41,056 --> 00:12:42,930
are trapped in some
region and aren't allowed

323
00:12:42,930 --> 00:12:44,410
to get arbitrarily far away.

324
00:12:44,410 --> 00:12:48,160
They'll be continuous
when you have states that

325
00:12:48,160 --> 00:12:50,720
can get arbitrarily far away.

326
00:12:50,720 --> 00:12:54,090
Sometimes the momentum will be
allowed to be discrete values,

327
00:12:54,090 --> 00:12:56,930
sometimes it will be allowed
to be continuous values.

328
00:12:56,930 --> 00:12:59,416
And we'll see exactly
why subsequently.

329
00:12:59,416 --> 00:13:00,790
But the thing I
want to emphasize

330
00:13:00,790 --> 00:13:03,720
is that I'm writing this to
emphasize that it's possible

331
00:13:03,720 --> 00:13:05,920
that each of these can be
discrete or continuous.

332
00:13:05,920 --> 00:13:09,120
The important thing is that once
you pick your physical system,

333
00:13:09,120 --> 00:13:11,572
you ask what are the allowed
values of position, what

334
00:13:11,572 --> 00:13:13,030
are the allowed
values of momentum,

335
00:13:13,030 --> 00:13:15,940
and what are the allowed
values of energy.

336
00:13:15,940 --> 00:13:18,290
And then you sum over
all possible values.

337
00:13:18,290 --> 00:13:21,450
Now, in the examples we looked
at yesterday, or last lecture,

338
00:13:21,450 --> 00:13:23,070
the energy could
have been discrete,

339
00:13:23,070 --> 00:13:25,070
as in the case of
the infinite well,

340
00:13:25,070 --> 00:13:28,680
or continuous, as in the
case of the free particle.

341
00:13:28,680 --> 00:13:30,180
In the case of a
continuous particle

342
00:13:30,180 --> 00:13:31,810
this would have
been an integral.

343
00:13:31,810 --> 00:13:35,890
In the case of the system
such as a free particle, where

344
00:13:35,890 --> 00:13:37,890
the energy could take any
of a continuous number

345
00:13:37,890 --> 00:13:40,424
of possible values, this would
be a continuous integral.

346
00:13:40,424 --> 00:13:41,840
To deal with that,
I'm often going

347
00:13:41,840 --> 00:13:45,660
to use the notation, just
shorthand, integral sum.

348
00:13:45,660 --> 00:13:47,500
Which I know is a
horrible bastardization

349
00:13:47,500 --> 00:13:50,120
of all that's good and
just, but on the other hand,

350
00:13:50,120 --> 00:13:53,000
emphasizes the fact
that in some systems

351
00:13:53,000 --> 00:13:55,200
you will get continuous,
in some systems discrete,

352
00:13:55,200 --> 00:13:57,520
and sometimes you'll have
both continuous and discrete.

353
00:13:57,520 --> 00:14:00,380
For example, in
hydrogen, in hydrogen

354
00:14:00,380 --> 00:14:02,310
we'll find that there
are bound states

355
00:14:02,310 --> 00:14:04,830
where the electron is stuck
to the hydrogen nucleus,

356
00:14:04,830 --> 00:14:05,920
to the proton.

357
00:14:05,920 --> 00:14:08,510
And there are discrete
allowed energy levels

358
00:14:08,510 --> 00:14:10,170
for that configuration.

359
00:14:10,170 --> 00:14:12,640
However, once you
ionize the hydrogen,

360
00:14:12,640 --> 00:14:15,290
the electron can add
any energy you want.

361
00:14:15,290 --> 00:14:16,170
It's no longer bound.

362
00:14:16,170 --> 00:14:18,090
It can just get
arbitrarily far away.

363
00:14:18,090 --> 00:14:21,360
And there are an uncountable
infinity, a continuous set

364
00:14:21,360 --> 00:14:22,744
of possible states.

365
00:14:22,744 --> 00:14:24,410
So in that situation,
we'll find that we

366
00:14:24,410 --> 00:14:27,230
have both the discrete
and continuous series

367
00:14:27,230 --> 00:14:28,582
of possible states.

368
00:14:28,582 --> 00:14:29,082
Yeah.

369
00:14:29,082 --> 00:14:32,420
AUDIENCE: [INAUDIBLE]

370
00:14:32,420 --> 00:14:34,620
PROFESSOR: Yeah, sure,
if you work on a lattice.

371
00:14:34,620 --> 00:14:36,870
So for example, consider the
following quantum system.

372
00:14:36,870 --> 00:14:38,550
I have an undergraduate.

373
00:14:38,550 --> 00:14:41,760
And that undergraduate has
been placed in 1 of 12 boxes.

374
00:14:41,760 --> 00:14:42,450
OK?

375
00:14:42,450 --> 00:14:44,309
Now, what's the state
of the undergraduate?

376
00:14:44,309 --> 00:14:44,850
I don't know.

377
00:14:44,850 --> 00:14:46,410
Is it a definite position state?

378
00:14:46,410 --> 00:14:47,290
It might be.

379
00:14:47,290 --> 00:14:49,660
But probably it's a
superposition, an arbitrary

380
00:14:49,660 --> 00:14:51,360
superposition, right?

381
00:14:51,360 --> 00:14:54,912
Very impressive
undergraduates at MIT.

382
00:14:54,912 --> 00:14:55,745
OK, other questions.

383
00:14:58,355 --> 00:14:59,491
Yeah.

384
00:14:59,491 --> 00:15:00,990
AUDIENCE: Do these
three [INAUDIBLE]

385
00:15:00,990 --> 00:15:03,580
hold even if the probability
changes over time?

386
00:15:03,580 --> 00:15:04,170
PROFESSOR: Excellent question.

387
00:15:04,170 --> 00:15:05,090
We'll come back to that.

388
00:15:05,090 --> 00:15:06,631
Very good question,
leading question.

389
00:15:06,631 --> 00:15:09,340
OK, so we have this.

390
00:15:09,340 --> 00:15:12,130
The next thing is that
energy eigenfunctions satisfy

391
00:15:12,130 --> 00:15:15,540
some very special properties.

392
00:15:15,540 --> 00:15:17,340
And in particular,
energy eigenfunctions

393
00:15:17,340 --> 00:15:20,785
have the property from the
Schrodinger equation i h

394
00:15:20,785 --> 00:15:26,150
bar d t on psi of x and t is
equal to the energy operator

395
00:15:26,150 --> 00:15:28,680
acting on psi of x and t.

396
00:15:28,680 --> 00:15:30,150
This tells us that
if we have psi

397
00:15:30,150 --> 00:15:37,000
x 0 time t 0 is equal to phi
n of x, as we saw last time,

398
00:15:37,000 --> 00:15:41,080
then the wave function,
psi at x at time t

399
00:15:41,080 --> 00:15:43,100
is equal to phi n of x.

400
00:15:43,100 --> 00:15:46,200
And it only changes by an
overall phase, e to the minus i

401
00:15:46,200 --> 00:15:50,510
En t over h bar.

402
00:15:50,510 --> 00:15:52,760
And this ratio En
upon h bar will often

403
00:15:52,760 --> 00:15:56,130
be written omega n is
equal to En over h bar.

404
00:15:56,130 --> 00:15:59,252
This is just the
Dupre relations.

405
00:15:59,252 --> 00:16:01,340
Everyone cool with that?

406
00:16:01,340 --> 00:16:02,600
So are energy eigenfunctions--

407
00:16:07,020 --> 00:16:07,570
how to say.

408
00:16:07,570 --> 00:16:12,800
No wave function is more
morally good than another.

409
00:16:12,800 --> 00:16:14,727
But some are
particularly convenient.

410
00:16:14,727 --> 00:16:16,560
Energy eigenfunctions
have the nice property

411
00:16:16,560 --> 00:16:17,990
that while they're not
in a definite position

412
00:16:17,990 --> 00:16:20,156
and they don't necessarily
have a definite momentum,

413
00:16:20,156 --> 00:16:22,950
they do evolve over time in
a particularly simple way.

414
00:16:22,950 --> 00:16:25,690
And that and the
superposition principle

415
00:16:25,690 --> 00:16:27,250
allow me to write the following.

416
00:16:27,250 --> 00:16:34,400
If I know that this is my wave
function at psi at x at time 0,

417
00:16:34,400 --> 00:16:38,230
so let's say in all these cases,
this is psi of x at time 0,

418
00:16:38,230 --> 00:16:40,590
how does this state
evolve forward in time?

419
00:16:44,909 --> 00:16:45,950
It's kind of complicated.

420
00:16:45,950 --> 00:16:49,060
How does this description,
how does psi tilde of k

421
00:16:49,060 --> 00:16:50,715
evolve forward in time?

422
00:16:50,715 --> 00:16:51,840
Again, kind of complicated.

423
00:16:51,840 --> 00:16:54,497
But when expressed in terms
of the energy eigenstates,

424
00:16:54,497 --> 00:16:56,330
the answer to how it
evolves forward in time

425
00:16:56,330 --> 00:16:58,460
is very simple, because
I know that this

426
00:16:58,460 --> 00:17:01,440
is a superposition, a
linear combination of states

427
00:17:01,440 --> 00:17:02,600
with definite energy.

428
00:17:02,600 --> 00:17:05,079
States with definite
energy evolve with a phase.

429
00:17:05,079 --> 00:17:07,010
And the Schrodinger
equation is linear,

430
00:17:07,010 --> 00:17:08,700
so solutions of the
Schrodinger equation

431
00:17:08,700 --> 00:17:11,680
evolve to become solutions
of the Schrodinger equation.

432
00:17:11,680 --> 00:17:13,950
So how does this state
evolve forward in time?

433
00:17:13,950 --> 00:17:18,476
It evolves forward with a phase,
e to the minus i omega n t.

434
00:17:18,476 --> 00:17:21,560
One for every different
terms in this sum.

435
00:17:21,560 --> 00:17:23,020
Cool?

436
00:17:23,020 --> 00:17:25,089
So we are going to harp
on energy functions,

437
00:17:25,089 --> 00:17:28,300
not because they're more moral,
or more just, or more good,

438
00:17:28,300 --> 00:17:31,040
but because they're more
convenient for solving the time

439
00:17:31,040 --> 00:17:33,680
evolution problem in
quantum mechanics.

440
00:17:33,680 --> 00:17:37,020
So most of today is going
to be about this expansion

441
00:17:37,020 --> 00:17:40,470
and qualitative features
of energy eigenfunctions.

442
00:17:40,470 --> 00:17:42,690
Cool?

443
00:17:42,690 --> 00:17:43,510
OK.

444
00:17:43,510 --> 00:17:45,670
And just to close
that out, I just

445
00:17:45,670 --> 00:17:47,420
want to remind you of
a couple of examples

446
00:17:47,420 --> 00:17:50,022
that we did last time,
just get them on board.

447
00:17:50,022 --> 00:17:51,355
So the first is a free particle.

448
00:17:54,440 --> 00:17:57,220
So for free particle, we have
that our wave functions--

449
00:18:00,380 --> 00:18:04,125
well, actually let me
not write that down.

450
00:18:04,125 --> 00:18:06,000
Actually, let me skip
over the free particle,

451
00:18:06,000 --> 00:18:06,999
because it's so trivial.

452
00:18:06,999 --> 00:18:10,417
Let me just talk about
the infinite well.

453
00:18:10,417 --> 00:18:12,000
So the potential is
infinite out here,

454
00:18:12,000 --> 00:18:13,750
and it's 0 inside
the well, and it

455
00:18:13,750 --> 00:18:18,790
goes from 0 to L. This is
just my choice of notation.

456
00:18:18,790 --> 00:18:22,300
And the energy
operator, as usual,

457
00:18:22,300 --> 00:18:26,165
is p squared upon
2m plus u of x.

458
00:18:26,165 --> 00:18:27,790
You might say, where
did I derive this,

459
00:18:27,790 --> 00:18:29,000
and the answer is I
didn't derive this.

460
00:18:29,000 --> 00:18:30,030
I just wrote it down.

461
00:18:30,030 --> 00:18:32,690
It's like force in
Newton's equations.

462
00:18:32,690 --> 00:18:34,730
You just declare some
force and you ask,

463
00:18:34,730 --> 00:18:36,190
what system does is model.

464
00:18:36,190 --> 00:18:38,120
So here's my system.

465
00:18:38,120 --> 00:18:41,630
It has what looks like a
classical kind of energy,

466
00:18:41,630 --> 00:18:43,450
except these are all operators.

467
00:18:43,450 --> 00:18:48,120
And the potential here is this
guy, it's 0 between 0 and L,

468
00:18:48,120 --> 00:18:49,730
and it's infinite elsewhere.

469
00:18:49,730 --> 00:18:51,740
And as we saw last
time, the solutions

470
00:18:51,740 --> 00:18:53,480
to the energy
eigenvalue equation

471
00:18:53,480 --> 00:18:55,190
are particularly simple.

472
00:18:55,190 --> 00:18:59,500
Phi sub n of x is
equal to root properly

473
00:18:59,500 --> 00:19:04,620
normalized 2 upon
L sine of Kn x,

474
00:19:04,620 --> 00:19:08,790
where kn is equal to
n plus 1 pi, where

475
00:19:08,790 --> 00:19:11,670
n is an integer upon L.

476
00:19:11,670 --> 00:19:14,070
And these were chosen to
satisfy our boundary conditions,

477
00:19:14,070 --> 00:19:16,403
that the wave function must
vanish here, hence the sine,

478
00:19:16,403 --> 00:19:19,690
and K was chosen so that it
turned over and just hit 0

479
00:19:19,690 --> 00:19:25,210
as we got to L. And that gave us
that the allowed energies were

480
00:19:25,210 --> 00:19:29,050
discrete, because the En, which
you can get by just plugging

481
00:19:29,050 --> 00:19:31,200
into the energy
eigenvalue equation,

482
00:19:31,200 --> 00:19:34,860
was equal to h bar squared
Kn squared upon 2m.

483
00:19:37,550 --> 00:19:39,370
So this tells us a nice thing.

484
00:19:39,370 --> 00:19:41,430
First off, in this system,
if I take a particle

485
00:19:41,430 --> 00:19:43,940
and I throw it in here
in some arbitrary state

486
00:19:43,940 --> 00:19:46,070
so that at time t
equals zero the wave

487
00:19:46,070 --> 00:19:54,170
function x 0 is equal to
sum over n phi n of x Cn.

488
00:19:54,170 --> 00:19:56,440
OK?

489
00:19:56,440 --> 00:19:57,220
Can I do this?

490
00:19:57,220 --> 00:19:58,845
Can I just pick some
arbitrary function

491
00:19:58,845 --> 00:20:01,320
which is a superposition
of energy eigenstates?

492
00:20:01,320 --> 00:20:02,956
Sure, because any function is.

493
00:20:02,956 --> 00:20:04,955
Any function can be
described as a superposition

494
00:20:04,955 --> 00:20:06,560
of energy eigenfunctions.

495
00:20:06,560 --> 00:20:09,350
And if I use the
energy eigenfunctions,

496
00:20:09,350 --> 00:20:11,600
it will automatically satisfy
the boundary conditions.

497
00:20:11,600 --> 00:20:13,190
All good things will happen.

498
00:20:13,190 --> 00:20:15,170
So this is perfectly
fine initial condition.

499
00:20:15,170 --> 00:20:16,950
What is the system at time t?

500
00:20:20,986 --> 00:20:22,360
Yeah, we just pick
up the phases.

501
00:20:22,360 --> 00:20:24,100
And what phase is this guy?

502
00:20:24,100 --> 00:20:28,420
It's this, e to the
minus i omega n t.

503
00:20:28,420 --> 00:20:31,150
And when I write omega n, let
me be more explicit about that,

504
00:20:31,150 --> 00:20:33,820
that's En over h bar.

505
00:20:33,820 --> 00:20:38,340
So that's h bar Kn
squared upon 2m t.

506
00:20:41,420 --> 00:20:42,550
Cool?

507
00:20:42,550 --> 00:20:48,740
So there is our solution for
arbitrary initial conditions

508
00:20:48,740 --> 00:20:54,069
to the infinite square well
problem in quantum mechanics.

509
00:20:54,069 --> 00:20:56,610
And you're going to study this
in some detail on your problem

510
00:20:56,610 --> 00:20:58,950
set.

511
00:20:58,950 --> 00:21:01,920
But just to start with a
little bit of intuition,

512
00:21:01,920 --> 00:21:04,390
let's look at the wave
functions and the probability

513
00:21:04,390 --> 00:21:06,177
distributions for the
lowest lying states.

514
00:21:06,177 --> 00:21:07,760
So for example, let's
look at the wave

515
00:21:07,760 --> 00:21:11,360
function for the ground state,
what I will call psi sub 0.

516
00:21:11,360 --> 00:21:14,959
And this is from 0 to L.
And I put these bars here

517
00:21:14,959 --> 00:21:16,750
not because we're
looking at the potential.

518
00:21:16,750 --> 00:21:23,954
I'm going to be plotting the
real part of the wave function.

519
00:21:23,954 --> 00:21:26,450
But I put these walls
here just to emphasize

520
00:21:26,450 --> 00:21:29,617
that that's where the walls are,
at x equals 0 and x equals L.

521
00:21:29,617 --> 00:21:30,700
So what does it look like?

522
00:21:30,700 --> 00:21:33,100
Well, the first one is
going to sine of Kn x.

523
00:21:33,100 --> 00:21:34,010
n is 0.

524
00:21:34,010 --> 00:21:38,804
Kn is going to be pi upon L.
So that's again just this guy.

525
00:21:38,804 --> 00:21:40,470
Now, what's the
probability distribution

526
00:21:40,470 --> 00:21:43,734
associated with psi 0?

527
00:21:43,734 --> 00:21:45,025
Where do you find the particle?

528
00:21:47,920 --> 00:21:51,850
So we know that it's just
the norm squared of this wave

529
00:21:51,850 --> 00:21:56,580
function and the norm
squared is here at 0, it's 0

530
00:21:56,580 --> 00:21:58,040
and it rises
linearly, because sine

531
00:21:58,040 --> 00:21:59,580
is linear for small values.

532
00:21:59,580 --> 00:22:02,980
That makes this
quadratic, and a maximum,

533
00:22:02,980 --> 00:22:04,600
and then quadratic again.

534
00:22:04,600 --> 00:22:06,710
So there's our
probability distribution.

535
00:22:06,710 --> 00:22:08,014
Now, here's a funny thing.

536
00:22:08,014 --> 00:22:09,930
Imagine I take a particle,
classical particle,

537
00:22:09,930 --> 00:22:11,607
and I put it in a box.

538
00:22:11,607 --> 00:22:13,690
And you put it in a box,
and you tell it, OK, it's

539
00:22:13,690 --> 00:22:14,390
got some energy.

540
00:22:14,390 --> 00:22:16,210
So classically it's
got some momentum.

541
00:22:16,210 --> 00:22:17,834
So it's sort of
bouncing back and forth

542
00:22:17,834 --> 00:22:20,000
and just bounces off the
arbitrarily hard walls

543
00:22:20,000 --> 00:22:20,880
and moves around.

544
00:22:20,880 --> 00:22:22,880
Where are you most likely
to find that particle?

545
00:22:27,850 --> 00:22:29,530
Where does it spend
most of its time?

546
00:22:33,052 --> 00:22:35,010
It spends the same amount
of time at any point.

547
00:22:35,010 --> 00:22:36,410
It's moving at
constant velocity.

548
00:22:36,410 --> 00:22:39,247
It goes boo, boo,
boo, boo, right?

549
00:22:39,247 --> 00:22:40,830
So what's the
probability distribution

550
00:22:40,830 --> 00:22:44,020
for finding it at any
point inside, classically?

551
00:22:44,020 --> 00:22:45,411
Constant.

552
00:22:45,411 --> 00:22:47,660
Classically, the probability
distribution is constant.

553
00:22:47,660 --> 00:22:49,576
You're just as likely
to find it near the wall

554
00:22:49,576 --> 00:22:51,615
as not near the wall.

555
00:22:51,615 --> 00:22:53,740
However, quantum mechanically,
for the lowest lying

556
00:22:53,740 --> 00:22:55,090
state that is clearly not true.

557
00:22:55,090 --> 00:22:58,310
You're really likely to
find it near the wall.

558
00:22:58,310 --> 00:23:01,060
What's up with that?

559
00:23:01,060 --> 00:23:03,630
So that's a question that
I want to put in your head

560
00:23:03,630 --> 00:23:05,850
and have you think about.

561
00:23:05,850 --> 00:23:08,309
You're going to see a similar
effect arising over and over.

562
00:23:08,309 --> 00:23:09,891
And we're going to
see at the very end

563
00:23:09,891 --> 00:23:12,460
that that is directly related,
the fact that this goes to 0,

564
00:23:12,460 --> 00:23:15,370
is directly related,
and I'm not kidding,

565
00:23:15,370 --> 00:23:16,940
to the transparency of diamond.

566
00:23:22,385 --> 00:23:24,000
OK, I think it was pretty cool.

567
00:23:26,720 --> 00:23:27,470
They're expensive.

568
00:23:30,399 --> 00:23:31,940
It's also related
to the transparency

569
00:23:31,940 --> 00:23:34,148
of cubic zirconium, which
I guess is less impressive.

570
00:23:37,020 --> 00:23:39,090
So the first state,
again, let's look

571
00:23:39,090 --> 00:23:43,790
at the real part of psi 1,
the first excited state.

572
00:23:43,790 --> 00:23:46,240
Well, this is now a
sine with one extra--

573
00:23:46,240 --> 00:23:50,940
with a 2 here, 2 pi,
so it goes through 0.

574
00:23:50,940 --> 00:23:53,740
So the probability distribution
associated with psi 1,

575
00:23:53,740 --> 00:23:56,774
and I should say write this
as a function of x, looks

576
00:23:56,774 --> 00:23:58,190
like, well, again,
it's quadratic.

577
00:23:58,190 --> 00:24:01,310
But it has a 0
again in the middle.

578
00:24:01,310 --> 00:24:04,250
So it's going to look like--

579
00:24:04,250 --> 00:24:07,670
oops, my bad art defeats me.

580
00:24:07,670 --> 00:24:08,580
OK, there we go.

581
00:24:08,580 --> 00:24:10,010
So now it's even worse.

582
00:24:10,010 --> 00:24:12,100
Not only is unlikely
to be out here,

583
00:24:12,100 --> 00:24:14,183
it's also very unlikely
to be found in the middle.

584
00:24:14,183 --> 00:24:16,750
In fact, there is 0 probability
you'll find it in the middle.

585
00:24:16,750 --> 00:24:18,700
That's sort of surprising.

586
00:24:18,700 --> 00:24:21,290
But you can quickly
guess what happens as you

587
00:24:21,290 --> 00:24:23,770
go to very high energies.

588
00:24:23,770 --> 00:24:30,010
The real part of psi n let's
say 10,000, 10 to the 4,

589
00:24:30,010 --> 00:24:31,610
what is that going to look like?

590
00:24:31,610 --> 00:24:33,620
Well, this had no
0s, this had one 0,

591
00:24:33,620 --> 00:24:35,300
and every time you
increase n by 1,

592
00:24:35,300 --> 00:24:37,570
you're just going to add
one more 0 to the sign.

593
00:24:37,570 --> 00:24:39,570
That's an interesting
and suggestive fact.

594
00:24:39,570 --> 00:24:42,650
So if it's size of
10,000, how many nodes

595
00:24:42,650 --> 00:24:45,411
are there going to be in
the middle of the domain?

596
00:24:45,411 --> 00:24:45,910
10,000.

597
00:24:45,910 --> 00:24:48,690
And the amplitude is
going to be the same.

598
00:24:48,690 --> 00:24:51,220
I'm not to be able to do
this, but you get the idea.

599
00:24:51,220 --> 00:24:53,520
And now if I construct the
probability distribution,

600
00:24:53,520 --> 00:24:55,780
what's the probability
distribution going to be?

601
00:24:55,780 --> 00:25:02,030
Probability of the 10,000th
psi sub 10 to the 4 of x.

602
00:25:02,030 --> 00:25:05,830
Well, it's again going
to be strictly positive.

603
00:25:05,830 --> 00:25:10,040
And if you are not able to make
measurements on the scale of L

604
00:25:10,040 --> 00:25:14,340
upon 10,000, but just say like
L over 3, because you have

605
00:25:14,340 --> 00:25:17,331
a thumb and you don't have an
infinitely accurate meter, what

606
00:25:17,331 --> 00:25:17,830
do you see?

607
00:25:17,830 --> 00:25:20,384
You see effectively a constant
probability distribution.

608
00:25:20,384 --> 00:25:22,050
And actually, I
shouldn't draw it there.

609
00:25:22,050 --> 00:25:23,834
I should draw it
through the half,

610
00:25:23,834 --> 00:25:27,170
because sine squared over
2 averages to one half,

611
00:25:27,170 --> 00:25:30,500
or, sorry, sine squared averages
to one half over many periods.

612
00:25:30,500 --> 00:25:33,660
So what we see is that
the classical probability

613
00:25:33,660 --> 00:25:38,380
distribution constant
does arise when we look

614
00:25:38,380 --> 00:25:41,590
at very high energy states.

615
00:25:41,590 --> 00:25:42,880
Cool?

616
00:25:42,880 --> 00:25:46,084
But it is manifestly
not a good description.

617
00:25:46,084 --> 00:25:48,250
The classical description
is not a good description.

618
00:25:48,250 --> 00:25:51,070
Your intuition is
crappy at low energies,

619
00:25:51,070 --> 00:25:53,570
near the ground state, where
quantum effects are dominating,

620
00:25:53,570 --> 00:25:55,980
because indeed, classically
there was no minimum energy.

621
00:25:55,980 --> 00:25:57,930
Quantum effects have
to be dominating there.

622
00:25:57,930 --> 00:25:59,780
And here we see that even the
probability distribution's

623
00:25:59,780 --> 00:26:01,542
radically different
than our intuition.

624
00:26:01,542 --> 00:26:02,042
Yeah.

625
00:26:02,042 --> 00:26:12,650
AUDIENCE: [INAUDIBLE]

626
00:26:12,650 --> 00:26:13,900
PROFESSOR: Keep working on it.

627
00:26:13,900 --> 00:26:18,820
So I want you all to
think about what--

628
00:26:18,820 --> 00:26:21,250
you're not, I promise
you, unless you've already

629
00:26:21,250 --> 00:26:22,230
seen some quantum
mechanics, you're

630
00:26:22,230 --> 00:26:24,271
not going to be able to
answer this question now.

631
00:26:24,271 --> 00:26:27,010
But I want you to have it as
an uncomfortable little piece

632
00:26:27,010 --> 00:26:31,420
of sand in the back
of your oyster mind--

633
00:26:31,420 --> 00:26:38,020
no offense-- what
is causing that 0?

634
00:26:38,020 --> 00:26:39,300
Why are we getting 0?

635
00:26:39,300 --> 00:26:40,575
And I'll give you a hint.

636
00:26:40,575 --> 00:26:42,960
In quantum mechanics,
anytime something interesting

637
00:26:42,960 --> 00:26:45,985
happens it's because of
superposition and interference.

638
00:26:49,220 --> 00:26:50,850
All right.

639
00:26:50,850 --> 00:26:55,044
So with all that said, so any
questions now over this story

640
00:26:55,044 --> 00:26:56,585
about energy
eigenfunctions expanding

641
00:26:56,585 --> 00:27:01,136
in a basis, et cetera,
before we get moving?

642
00:27:01,136 --> 00:27:03,370
No, OK.

643
00:27:03,370 --> 00:27:05,880
In that case, get
out your clickers.

644
00:27:05,880 --> 00:27:07,620
We're going to test
your knowledge.

645
00:27:17,720 --> 00:27:20,836
Channel 41, for those of
you who have to adjust it.

646
00:27:20,836 --> 00:27:32,380
[CHATTER]

647
00:27:32,380 --> 00:27:32,880
Wow.

648
00:27:37,100 --> 00:27:38,100
That's kind of worrying.

649
00:27:42,980 --> 00:27:43,480
Aha.

650
00:27:58,400 --> 00:28:01,780
OK, ready?

651
00:28:01,780 --> 00:28:07,010
OK, channel 41, and here we go.

652
00:28:18,300 --> 00:28:19,789
So go ahead and start now.

653
00:28:19,789 --> 00:28:21,830
Sorry, there was a little
technical glitch there.

654
00:28:21,830 --> 00:28:24,640
So psi 1 and psi
2 are eigenstates.

655
00:28:24,640 --> 00:28:27,590
They're non-degenerate, meaning
the energies are different.

656
00:28:27,590 --> 00:28:30,020
Is a superposition psi 1 plus
psi 2 also an eigenstate?

657
00:28:43,398 --> 00:28:45,580
All right, four more seconds.

658
00:28:48,610 --> 00:28:49,560
All right.

659
00:28:49,560 --> 00:28:52,910
I want everyone to turn
to the person next to you

660
00:28:52,910 --> 00:28:54,050
and discuss this.

661
00:28:54,050 --> 00:28:56,538
You've got about 30 seconds
to discuss, or a minute.

662
00:28:56,538 --> 00:29:39,780
[CHATTER]

663
00:29:39,780 --> 00:29:41,940
All right.

664
00:29:41,940 --> 00:29:45,230
I want everyone, now that you've
got an answer, click again,

665
00:29:45,230 --> 00:29:47,300
put in your current best guess.

666
00:29:50,564 --> 00:29:51,230
Oh, wait, sorry.

667
00:29:51,230 --> 00:29:54,180
For some reason I have
to start over again.

668
00:29:54,180 --> 00:29:56,080
OK, now click.

669
00:30:02,690 --> 00:30:05,090
This is the best.

670
00:30:05,090 --> 00:30:08,180
I'm such a convert to clickers,
this is just fantastic.

671
00:30:08,180 --> 00:30:11,750
So you guys went from,
so roughly you all

672
00:30:11,750 --> 00:30:20,480
went from about 30, 60,
10, to what are we now?

673
00:30:20,480 --> 00:30:26,990
8, 82, and 10.

674
00:30:26,990 --> 00:30:29,230
So it sounds like you guys
are predicting answer b.

675
00:30:29,230 --> 00:30:30,200
And the answer is--

676
00:30:33,587 --> 00:30:34,420
I like the suspense.

677
00:30:34,420 --> 00:30:37,170
There we go.

678
00:30:37,170 --> 00:30:38,150
B, good.

679
00:30:38,150 --> 00:30:40,160
So here's a quick question.

680
00:30:42,890 --> 00:30:44,020
So why?

681
00:30:44,020 --> 00:30:49,910
And the reason why is that if
we have E on psi 1 plus psi 2,

682
00:30:49,910 --> 00:30:56,540
this is equal to E on psi 1 plus
E on psi 2, operator, operator,

683
00:30:56,540 --> 00:30:59,200
operator, but this
is equal to E 1

684
00:30:59,200 --> 00:31:05,380
psi 1 E 2 psi 2, which if
E1 and E2 are not equal,

685
00:31:05,380 --> 00:31:09,070
which is not equal to E
times psi 1 plus psi 2.

686
00:31:09,070 --> 00:31:10,350
Right?

687
00:31:10,350 --> 00:31:14,740
Not equal to E anything
times psi 1 plus psi 2.

688
00:31:14,740 --> 00:31:16,280
And it needs to
be, in order to be

689
00:31:16,280 --> 00:31:18,300
an eigenfunction, an
eigenfunction of the energy

690
00:31:18,300 --> 00:31:20,960
operator.

691
00:31:20,960 --> 00:31:22,538
Yeah.

692
00:31:22,538 --> 00:31:24,162
AUDIENCE: So I was
thinking about this,

693
00:31:24,162 --> 00:31:26,037
if this was kind of a
silly random case where

694
00:31:26,037 --> 00:31:27,140
one of the energies is 0.

695
00:31:27,140 --> 00:31:29,640
Does this only happen if you
have something that's infinite?

696
00:31:29,640 --> 00:31:31,360
PROFESSOR: Yeah, that's
a really good question.

697
00:31:31,360 --> 00:31:33,340
So first off, how do
you measure an energy?

698
00:31:36,240 --> 00:31:38,790
Do you ever measure an energy?

699
00:31:38,790 --> 00:31:41,365
Do you ever measure a
voltage, the actual value

700
00:31:41,365 --> 00:31:43,490
of the scalar potential,
the electromagnetic scalar

701
00:31:43,490 --> 00:31:43,670
potential?

702
00:31:43,670 --> 00:31:44,169
No.

703
00:31:44,169 --> 00:31:46,259
You measure a difference.

704
00:31:46,259 --> 00:31:47,550
Do you ever measure the energy?

705
00:31:47,550 --> 00:31:49,296
No, you measure a
difference in energy.

706
00:31:49,296 --> 00:31:51,670
So the absolute value of energy
is sort of a silly thing.

707
00:31:51,670 --> 00:31:54,440
But we always talk
about it as if it's not.

708
00:31:54,440 --> 00:31:55,680
We say, that's got energy 14.

709
00:31:55,680 --> 00:31:58,200
It's a little bit suspicious.

710
00:31:58,200 --> 00:32:00,480
So to answer your
question, there's

711
00:32:00,480 --> 00:32:02,290
nothing hallowed
about the number 0,

712
00:32:02,290 --> 00:32:04,282
although we will often
refer to zero energy

713
00:32:04,282 --> 00:32:05,490
with a very specific meaning.

714
00:32:05,490 --> 00:32:06,980
What we really mean
in that case is

715
00:32:06,980 --> 00:32:09,420
the value of the potential
energy at infinity.

716
00:32:09,420 --> 00:32:11,220
So when I say energy,
usually what I mean

717
00:32:11,220 --> 00:32:12,790
is relative to the
value at infinity.

718
00:32:12,790 --> 00:32:14,415
So then let me ask
your question again.

719
00:32:14,415 --> 00:32:16,510
Your question is it
possible to have energy 0?

720
00:32:16,510 --> 00:32:17,802
Absolutely, and we'll see that.

721
00:32:17,802 --> 00:32:20,093
And it's actually going to
be really interesting what's

722
00:32:20,093 --> 00:32:21,910
true of states with
energy 0 in that sense.

723
00:32:21,910 --> 00:32:23,451
Second part of your
question, though,

724
00:32:23,451 --> 00:32:25,690
is how does energy
being 0 fit into this?

725
00:32:25,690 --> 00:32:26,970
Well, does that save us?

726
00:32:26,970 --> 00:32:29,210
Suppose one of
the energies is 0.

727
00:32:29,210 --> 00:32:31,690
Then that says E on psi
1 plus psi 2 is equal to,

728
00:32:31,690 --> 00:32:34,790
let's say E2 is 0.

729
00:32:34,790 --> 00:32:35,950
Well, that term is gone.

730
00:32:35,950 --> 00:32:37,074
So there's just the one E1.

731
00:32:37,074 --> 00:32:38,657
Are we in energy eigenstate?

732
00:32:38,657 --> 00:32:40,240
No, because it's
still not of the form

733
00:32:40,240 --> 00:32:42,290
E times psi 1 plus psi 2.

734
00:32:42,290 --> 00:32:44,740
So it doesn't save us, but
it's an interesting question

735
00:32:44,740 --> 00:32:46,200
for the future.

736
00:32:46,200 --> 00:32:48,290
All right.

737
00:32:48,290 --> 00:32:50,935
Next question, four parts.

738
00:33:07,100 --> 00:33:10,250
So the question says x
and p commute to i h bar.

739
00:33:10,250 --> 00:33:11,470
We've shown this.

740
00:33:11,470 --> 00:33:14,580
Is p x equal to i h
bar, and is ip plus cx

741
00:33:14,580 --> 00:33:15,910
the same as cx plus ip?

742
00:33:29,920 --> 00:33:33,740
If you're really unsure you
can ask the person next to you,

743
00:33:33,740 --> 00:33:37,170
but you don't have to.

744
00:33:37,170 --> 00:33:41,290
OK, so this is looking good.

745
00:33:41,290 --> 00:33:44,060
Everyone have an answer in?

746
00:33:44,060 --> 00:33:44,560
No?

747
00:33:49,770 --> 00:33:55,310
Five, four, three,
two, one, OK, good.

748
00:33:55,310 --> 00:34:03,242
So the answer is C, which most
of you got, but not everyone.

749
00:34:03,242 --> 00:34:05,200
A bunch of you put D. So
let's talk through it.

750
00:34:05,200 --> 00:34:10,639
So remember what the definition
of the commutator is.

751
00:34:10,639 --> 00:34:15,219
x with p by definition
is equal to xp minus px.

752
00:34:15,219 --> 00:34:18,500
If we change the
order here, px is

753
00:34:18,500 --> 00:34:22,630
equal to minus
this, px minus xp.

754
00:34:22,630 --> 00:34:24,620
It's just the definition
of the commutator.

755
00:34:24,620 --> 00:34:27,800
So on the other hand, if you
add things, does 7 plus 6

756
00:34:27,800 --> 00:34:28,940
equal 6 plus 7?

757
00:34:28,940 --> 00:34:29,440
Yeah.

758
00:34:29,440 --> 00:34:31,116
Well, of course 6
times 7 is 7 times 6.

759
00:34:31,116 --> 00:34:32,699
So that's not a
terribly good analogy.

760
00:34:36,880 --> 00:34:39,880
Does the order of addition
of operators matter?

761
00:34:39,880 --> 00:34:42,540
No.

762
00:34:42,540 --> 00:34:44,639
Yeah.

763
00:34:44,639 --> 00:34:45,630
Yeah, exactly.

764
00:34:45,630 --> 00:34:46,130
Exactly.

765
00:34:46,130 --> 00:34:47,130
So it's slightly sneaky.

766
00:34:47,130 --> 00:34:49,840
OK, next question.

767
00:34:49,840 --> 00:34:51,170
OK, this one has five.

768
00:34:54,860 --> 00:34:56,670
f and g are both wave functions.

769
00:34:56,670 --> 00:34:58,090
c is a constant.

770
00:34:58,090 --> 00:35:03,150
Then if we take the inner
product c times f with g,

771
00:35:03,150 --> 00:35:05,160
this is equal to what?

772
00:35:21,000 --> 00:35:24,240
Three, two, one, OK.

773
00:35:24,240 --> 00:35:25,720
So the answer is--

774
00:35:25,720 --> 00:35:27,610
so this one definitely discuss.

775
00:35:27,610 --> 00:35:29,778
Discuss with the
person next to you.

776
00:35:29,778 --> 00:35:55,520
[CHATTER]

777
00:35:55,520 --> 00:35:56,280
All right.

778
00:36:01,440 --> 00:36:05,910
OK, go ahead and enter
your guess again,

779
00:36:05,910 --> 00:36:08,490
or your answer again,
let it not be a guess.

780
00:36:11,220 --> 00:36:12,855
OK, 10 seconds.

781
00:36:15,450 --> 00:36:15,950
Wow.

782
00:36:15,950 --> 00:36:16,630
OK, fantastic.

783
00:36:16,630 --> 00:36:17,630
That works like a champ.

784
00:36:17,630 --> 00:36:19,040
So what's the answer?

785
00:36:19,040 --> 00:36:20,970
Yes, complex conjugation.

786
00:36:20,970 --> 00:36:21,970
Don't screw that one up.

787
00:36:21,970 --> 00:36:24,110
It's very easy to forget,
but it matters a lot.

788
00:36:28,297 --> 00:36:29,380
Cursor keeps disappearing.

789
00:36:29,380 --> 00:36:32,360
OK, next one.

790
00:36:32,360 --> 00:36:40,120
A wave function
has been expressed

791
00:36:40,120 --> 00:36:41,570
as a sum of energy
eigenfunctions.

792
00:36:41,570 --> 00:36:45,445
Here I'm calling them mu rather
than phi, but same thing.

793
00:36:45,445 --> 00:36:47,070
Compared to the
original wave function,

794
00:36:47,070 --> 00:36:49,920
the set of coefficients, given
that we're using the energy

795
00:36:49,920 --> 00:36:52,730
basis, the set of coefficients
contains more or less

796
00:36:52,730 --> 00:36:55,390
the same information, or
it can't be determined.

797
00:37:00,650 --> 00:37:02,360
OK, five seconds.

798
00:37:06,990 --> 00:37:07,780
All right.

799
00:37:07,780 --> 00:37:11,660
And the answer is C, great.

800
00:37:11,660 --> 00:37:13,390
OK, next one.

801
00:37:16,429 --> 00:37:17,720
So right now we're normalizing.

802
00:37:22,540 --> 00:37:23,470
OK.

803
00:37:23,470 --> 00:37:25,820
All stationary states, or
all energy eigenstates,

804
00:37:25,820 --> 00:37:28,570
have the form that spatial
and time dependence

805
00:37:28,570 --> 00:37:30,760
is the spatial dependence,
the energy eigenfunction,

806
00:37:30,760 --> 00:37:35,110
times a phase, so that the norm
squared is time independent.

807
00:37:35,110 --> 00:37:37,290
Consider the sum of two
non-degenerate energy

808
00:37:37,290 --> 00:37:40,280
eigenstates psi 1 and psi 2.

809
00:37:40,280 --> 00:37:42,990
Non-degenerate means they
have different energy.

810
00:37:45,540 --> 00:37:47,140
Is the wave function stationary?

811
00:37:47,140 --> 00:37:48,840
Is the probability
distribution time

812
00:37:48,840 --> 00:37:50,420
independent or is
it time dependent?

813
00:38:00,600 --> 00:38:02,980
This one's not trivial.

814
00:38:02,980 --> 00:38:03,530
Oh, shoot.

815
00:38:03,530 --> 00:38:04,730
I forgot to get it started.

816
00:38:04,730 --> 00:38:06,399
Sorry.

817
00:38:06,399 --> 00:38:08,940
It's particularly non-trivial
if you can't enter your answer.

818
00:38:08,940 --> 00:38:09,210
Right.

819
00:38:09,210 --> 00:38:10,626
So go ahead and
enter your answer.

820
00:38:17,438 --> 00:38:19,420
Whoo, yeah.

821
00:38:19,420 --> 00:38:20,715
This one always kills people.

822
00:38:31,315 --> 00:38:32,190
No chatting just yet.

823
00:38:32,190 --> 00:38:33,565
Test yourself,
not your neighbor.

824
00:38:38,427 --> 00:38:40,010
It's fine to look
deep into your soul,

825
00:38:40,010 --> 00:38:41,980
but don't look deep into the
soul of the person sitting next

826
00:38:41,980 --> 00:38:42,480
to you.

827
00:38:46,660 --> 00:38:47,590
All right.

828
00:38:47,590 --> 00:38:51,840
So at this point, chat
with your neighbor.

829
00:38:51,840 --> 00:38:54,086
Let me just give
you some presage.

830
00:38:54,086 --> 00:38:57,720
The parallel strategy's probably
not so good, because about half

831
00:38:57,720 --> 00:39:00,273
of you got it right, and about
half of you got it wrong.

832
00:39:00,273 --> 00:40:07,070
[CHATTER]

833
00:40:07,070 --> 00:40:07,950
All right.

834
00:40:07,950 --> 00:40:09,375
Let's vote again.

835
00:40:11,980 --> 00:40:15,720
And hold on, starting now.

836
00:40:15,720 --> 00:40:16,480
OK, vote again.

837
00:40:16,480 --> 00:40:18,063
You've got 10 seconds
to enter a vote.

838
00:40:24,520 --> 00:40:25,710
Wow.

839
00:40:25,710 --> 00:40:28,110
OK, two seconds.

840
00:40:28,110 --> 00:40:29,020
Good.

841
00:40:29,020 --> 00:40:36,940
So the distribution on
this one went from 30, 50,

842
00:40:36,940 --> 00:40:47,290
20 initially, to now
it is 10, 80, and 10.

843
00:40:47,290 --> 00:40:51,030
Amazingly, you guys got worse.

844
00:40:51,030 --> 00:40:55,620
The answer is C. And I want
you to discuss with each other

845
00:40:55,620 --> 00:40:57,174
why it's C.

846
00:40:57,174 --> 00:41:27,840
[CHATTER]

847
00:41:27,840 --> 00:41:29,420
All right.

848
00:41:29,420 --> 00:41:32,060
OK.

849
00:41:32,060 --> 00:41:33,830
So let me talk you through it.

850
00:41:33,830 --> 00:41:36,230
So the wave function,
we've said psi of x and t

851
00:41:36,230 --> 00:41:45,865
is equal to phi 1 at x, e to
the minus i omega 1 t plus phi

852
00:41:45,865 --> 00:41:50,940
2 of x e to the
minus i omega 2 t.

853
00:41:50,940 --> 00:41:52,670
So great, we take
the norm squared.

854
00:41:52,670 --> 00:41:55,342
What's the probability to
find it at x at time t.

855
00:41:55,342 --> 00:41:56,800
The probability
density is the norm

856
00:41:56,800 --> 00:41:59,260
squared of this guy,
psi squared, which

857
00:41:59,260 --> 00:42:04,590
is equal to phi 1 complex
conjugate e to the plus i omega

858
00:42:04,590 --> 00:42:12,200
1 t plus phi 2 complex conjugate
e to the plus i omega 2t times

859
00:42:12,200 --> 00:42:16,210
the thing itself phi 1
of x e to the minus i

860
00:42:16,210 --> 00:42:23,320
omega 1 t plus phi 2 of x e to
the minus i omega 2t, right?

861
00:42:23,320 --> 00:42:25,870
So this has four terms.

862
00:42:25,870 --> 00:42:28,470
The first term is
psi 1 norm squared.

863
00:42:28,470 --> 00:42:30,650
The phases cancel, right?

864
00:42:30,650 --> 00:42:33,340
You're going to see this
happen a billion times in 804.

865
00:42:33,340 --> 00:42:35,850
The first term is going
to be phi 1 norm squared.

866
00:42:35,850 --> 00:42:37,970
There's another term, which
is phi 2 norm squared.

867
00:42:37,970 --> 00:42:40,345
Again the phases exactly
cancel, even the minus i omega 2

868
00:42:40,345 --> 00:42:42,080
t to the plus i omega 2 t.

869
00:42:42,080 --> 00:42:44,470
Plus phi 2 squared.

870
00:42:44,470 --> 00:42:47,280
But then there are two cross
terms, the interference terms.

871
00:42:47,280 --> 00:42:55,180
Plus phi 1 complex conjugate
phi 2 e to the i omega 1 t

872
00:42:55,180 --> 00:43:00,670
e to the plus i omega 1 t, i
omega 1 t, and e to the minus

873
00:43:00,670 --> 00:43:01,869
i omega 2t, minus omega 2.

874
00:43:01,869 --> 00:43:04,160
So we have a cross-term which
depends on the difference

875
00:43:04,160 --> 00:43:05,194
in frequencies.

876
00:43:05,194 --> 00:43:07,110
Frequencies are like
energies modulo on h-bar,

877
00:43:07,110 --> 00:43:09,240
so it's a difference
in energies.

878
00:43:09,240 --> 00:43:10,740
And then there's
another term, which

879
00:43:10,740 --> 00:43:12,670
is the complex
conjugate of this guy,

880
00:43:12,670 --> 00:43:15,950
phi 2 star times phi 1 phi
2 complex conjugate phi 1

881
00:43:15,950 --> 00:43:17,825
and the phases are also
the complex conjugate

882
00:43:17,825 --> 00:43:28,350
e to the minus i omega 1 minus
omega 2 t of x of x of x of x.

883
00:43:28,350 --> 00:43:31,880
So is there time dependence
in this, in principle?

884
00:43:31,880 --> 00:43:33,890
Absolutely, from the
interference terms.

885
00:43:33,890 --> 00:43:35,500
Were we not in
the superposition,

886
00:43:35,500 --> 00:43:37,470
we would not have
interference terms.

887
00:43:37,470 --> 00:43:40,330
Time dependence comes from
interference, when we expand

888
00:43:40,330 --> 00:43:41,600
in energy eigenfunctions.

889
00:43:41,600 --> 00:43:43,187
Cool?

890
00:43:43,187 --> 00:43:44,270
However, can these vanish?

891
00:43:44,270 --> 00:43:44,769
When?

892
00:43:48,450 --> 00:43:50,460
Sorry, say again?

893
00:43:50,460 --> 00:43:54,577
Great, so when omega 1
equals omega 2, what happens?

894
00:43:54,577 --> 00:43:55,660
Time dependence goes away.

895
00:43:55,660 --> 00:44:00,750
But omega 1 is e 1 over h bar,
omega 2 is e 2 over h bar,

896
00:44:00,750 --> 00:44:03,595
and we started out by saying
these are non-degenerate.

897
00:44:03,595 --> 00:44:05,970
So if they're non-degenerate,
the energies are different,

898
00:44:05,970 --> 00:44:08,361
the frequencies are different,
so that doesn't help us.

899
00:44:08,361 --> 00:44:09,860
How do we kill this
time dependence?

900
00:44:12,860 --> 00:44:13,399
Yes.

901
00:44:13,399 --> 00:44:15,190
If the two functions
aren't just orthogonal

902
00:44:15,190 --> 00:44:17,150
in a functional sense, but
if we have the following.

903
00:44:17,150 --> 00:44:18,410
Suppose phi 1 is like this.

904
00:44:18,410 --> 00:44:21,430
It's 0 everywhere except for
in some lump that's phi 1,

905
00:44:21,430 --> 00:44:25,230
and phi 2 is 0
everywhere except here.

906
00:44:25,230 --> 00:44:27,670
Then anywhere that phi 1
is non-zero, phi 2 is zero.

907
00:44:27,670 --> 00:44:30,790
And anywhere where phi 2
is non-zero, phi 1 is zero.

908
00:44:30,790 --> 00:44:33,940
So this can point-wise vanish.

909
00:44:33,940 --> 00:44:37,250
Do you expect this to
happen generically?

910
00:44:37,250 --> 00:44:39,850
Does it happen for the
energy eigenfunctions

911
00:44:39,850 --> 00:44:41,270
in the infinite square well?

912
00:44:44,710 --> 00:44:45,360
Sine waves?

913
00:44:47,981 --> 00:44:48,480
No.

914
00:44:48,480 --> 00:44:50,100
They have zero at
isolated points,

915
00:44:50,100 --> 00:44:53,019
but they're non-zero
generically.

916
00:44:53,019 --> 00:44:54,310
Yeah, so it doesn't work there.

917
00:44:54,310 --> 00:44:55,875
What about for
the free particle?

918
00:44:55,875 --> 00:44:57,250
Well, those are
just plain waves.

919
00:44:57,250 --> 00:44:58,680
Does that ever happen?

920
00:44:58,680 --> 00:44:59,610
No.

921
00:44:59,610 --> 00:45:02,540
OK, so this is an
incredibly special case.

922
00:45:02,540 --> 00:45:04,040
We'll actually see
it in one problem

923
00:45:04,040 --> 00:45:05,485
on a problem set later on.

924
00:45:05,485 --> 00:45:07,130
It's a very special case.

925
00:45:07,130 --> 00:45:09,935
So technically, the
answer is C. And I

926
00:45:09,935 --> 00:45:11,570
want you guys to
keep your minds open

927
00:45:11,570 --> 00:45:15,310
on these sorts of questions,
when does a spatial dependence

928
00:45:15,310 --> 00:45:17,370
matter and when are
there interference terms.

929
00:45:17,370 --> 00:45:18,260
Those are two
different questions,

930
00:45:18,260 --> 00:45:19,718
and I want you to
tease them apart.

931
00:45:19,718 --> 00:45:21,490
OK?

932
00:45:21,490 --> 00:45:22,590
Cool?

933
00:45:22,590 --> 00:45:23,784
Yeah?

934
00:45:23,784 --> 00:45:26,462
AUDIENCE: Is a valid
way to think about this

935
00:45:26,462 --> 00:45:34,594
to think that you're fixing
the initial [INAUDIBLE]

936
00:45:34,594 --> 00:45:36,760
PROFESSOR: That's a very
good way to think about it.

937
00:45:36,760 --> 00:45:39,150
That's exactly right.

938
00:45:39,150 --> 00:45:41,082
That's a very,
very good question.

939
00:45:41,082 --> 00:45:43,040
Let me say that subtly
differently, and tell me

940
00:45:43,040 --> 00:45:44,956
if this agrees with what
you were just saying.

941
00:45:44,956 --> 00:45:47,806
So I can look at
this wave function,

942
00:45:47,806 --> 00:45:49,930
and I already know that
the overall phase of a wave

943
00:45:49,930 --> 00:45:51,150
function doesn't matter.

944
00:45:51,150 --> 00:45:52,960
That's what it is to say a
stationary state is stationary.

945
00:45:52,960 --> 00:45:54,960
It's got an overall phase
that's the only thing,

946
00:45:54,960 --> 00:45:56,540
norm squared it goes away.

947
00:45:56,540 --> 00:46:01,950
So I can write this as e to
the minus i omega 1 t times phi

948
00:46:01,950 --> 00:46:08,750
1 of x plus phi 2 of x
e to the minus i omega

949
00:46:08,750 --> 00:46:12,506
2 minus omega 1 t.

950
00:46:12,506 --> 00:46:13,920
Is that what you mean?

951
00:46:13,920 --> 00:46:15,170
So that's one way to do this.

952
00:46:15,170 --> 00:46:16,503
We could also do something else.

953
00:46:16,503 --> 00:46:22,752
We could do e to the minus i
omega 1 plus omega 2 upon 2 t.

954
00:46:22,752 --> 00:46:24,210
And this is more,
I think, what you

955
00:46:24,210 --> 00:46:26,085
were thinking of, a sort
of average frequency

956
00:46:26,085 --> 00:46:28,070
and then a relative
frequency, and then

957
00:46:28,070 --> 00:46:31,050
the change in the frequencies
on these two terms.

958
00:46:31,050 --> 00:46:31,760
Absolutely.

959
00:46:31,760 --> 00:46:36,720
So you can organize
this in many, many ways.

960
00:46:36,720 --> 00:46:39,670
But your question gets at
a very important point,

961
00:46:39,670 --> 00:46:41,670
which is that the overall
phase doesn't matter.

962
00:46:41,670 --> 00:46:46,400
But relative phases in a
superposition do matter.

963
00:46:46,400 --> 00:46:48,727
So when does a phase
matter in a wave function?

964
00:46:48,727 --> 00:46:50,560
It does not matter if
it's an overall phase.

965
00:46:50,560 --> 00:46:53,170
But it does matter if it's a
relative phase between terms

966
00:46:53,170 --> 00:46:54,790
in a superposition.

967
00:46:54,790 --> 00:46:55,870
Cool?

968
00:46:55,870 --> 00:46:57,510
Very good question.

969
00:46:57,510 --> 00:47:00,330
Other questions?

970
00:47:00,330 --> 00:47:03,550
If not, then I have some.

971
00:47:03,550 --> 00:47:07,930
So, consider a system
which is in the state--

972
00:47:07,930 --> 00:47:10,227
so I should give you five--

973
00:47:10,227 --> 00:47:12,810
system is in a state which is a
linear combination of n equals

974
00:47:12,810 --> 00:47:14,280
1 and n equals 2 eigenstates.

975
00:47:16,956 --> 00:47:18,580
What's the probability
that measurement

976
00:47:18,580 --> 00:47:20,082
will give us energy E1?

977
00:47:20,082 --> 00:47:21,373
And it's in this superposition.

978
00:47:28,390 --> 00:47:29,215
OK, five seconds.

979
00:47:33,780 --> 00:47:34,660
OK, fantastic.

980
00:47:34,660 --> 00:47:36,450
What's the answer?

981
00:47:36,450 --> 00:47:37,497
Yes, C, great.

982
00:47:37,497 --> 00:47:38,580
OK, everyone got that one.

983
00:47:38,580 --> 00:47:42,100
So one's a slightly more
interesting question.

984
00:47:45,580 --> 00:47:55,210
Suppose I have an infinite
well with width L.

985
00:47:55,210 --> 00:47:57,150
How does the energy,
the ground state energy,

986
00:47:57,150 --> 00:48:01,175
compare to that of a
system with a wider well?

987
00:48:16,070 --> 00:48:21,335
So L versus a larger
L. OK, four seconds.

988
00:48:25,140 --> 00:48:28,940
OK, quickly discuss amongst
yourselves, like 10 seconds.

989
00:48:28,940 --> 00:48:45,160
[CHATTER]

990
00:48:45,160 --> 00:48:47,300
All right.

991
00:48:47,300 --> 00:48:48,185
Now click again.

992
00:48:51,493 --> 00:48:51,993
Yeah.

993
00:48:56,080 --> 00:48:57,520
All right.

994
00:48:57,520 --> 00:48:58,260
Five seconds.

995
00:48:58,260 --> 00:49:02,270
One, two, three,
four, five, great.

996
00:49:02,270 --> 00:49:08,470
OK, the answer is A. OK,
great, because the energy

997
00:49:08,470 --> 00:49:11,530
of the infinite well
goes like K squared.

998
00:49:11,530 --> 00:49:18,974
K goes like 1 over L. So the
energy is, if we make it wider,

999
00:49:18,974 --> 00:49:21,140
the energy if we make it
wider is going to be lower.

1000
00:49:27,260 --> 00:49:28,725
And last couple of questions.

1001
00:49:31,510 --> 00:49:34,900
OK, so t equals 0.

1002
00:49:34,900 --> 00:49:38,350
Could the wave function for an
electron in an infinite square

1003
00:49:38,350 --> 00:49:44,880
well of width a, rather than
L, be A sine squared of pi x

1004
00:49:44,880 --> 00:49:48,680
upon a, where A is suitably
chosen to be normalized?

1005
00:50:10,460 --> 00:50:13,870
All right, you've got
about five seconds left.

1006
00:50:17,550 --> 00:50:21,420
And OK, we are at chance.

1007
00:50:21,420 --> 00:50:24,710
We are at even
odds, and the answer

1008
00:50:24,710 --> 00:50:28,380
is not a superposition
of A and B,

1009
00:50:28,380 --> 00:50:31,130
so I encourage you to discuss
with the people around you.

1010
00:50:31,130 --> 00:50:34,567
[CHATTER]

1011
00:50:51,731 --> 00:50:52,230
Great.

1012
00:50:52,230 --> 00:50:54,400
What properties had it
better satisfy in order

1013
00:50:54,400 --> 00:50:56,774
to be a viable wave function?

1014
00:50:56,774 --> 00:50:58,440
What properties should
the wave function

1015
00:50:58,440 --> 00:51:00,391
have so that it's reasonable?

1016
00:51:00,391 --> 00:51:00,890
Yeah.

1017
00:51:00,890 --> 00:51:02,080
Is it zero at the ends?

1018
00:51:02,080 --> 00:51:03,001
Yeah.

1019
00:51:03,001 --> 00:51:03,500
Good.

1020
00:51:03,500 --> 00:51:04,841
Is it smooth?

1021
00:51:04,841 --> 00:51:07,100
Yeah.

1022
00:51:07,100 --> 00:51:07,830
Exactly.

1023
00:51:07,830 --> 00:51:09,691
And so you can write
it as a superposition.

1024
00:51:09,691 --> 00:51:10,190
Excellent.

1025
00:51:10,190 --> 00:51:11,692
So the answer is?

1026
00:51:11,692 --> 00:51:12,192
Yeah.

1027
00:51:17,516 --> 00:51:18,015
All right.

1028
00:51:20,780 --> 00:51:21,750
Vote again.

1029
00:51:25,170 --> 00:51:26,794
OK, I might have
missed a few people.

1030
00:51:26,794 --> 00:51:27,710
So go ahead and start.

1031
00:51:31,800 --> 00:51:33,185
OK, five more seconds.

1032
00:51:37,230 --> 00:51:37,950
All right.

1033
00:51:37,950 --> 00:51:43,820
So we went from 50-50 to 77-23.

1034
00:51:43,820 --> 00:51:44,820
That's pretty good.

1035
00:51:44,820 --> 00:51:47,260
What's the answer?

1036
00:51:47,260 --> 00:51:47,890
A. Why?

1037
00:51:51,710 --> 00:51:53,892
Is this an energy eigenstate?

1038
00:51:53,892 --> 00:51:55,160
No.

1039
00:51:55,160 --> 00:51:56,960
Does that matter?

1040
00:51:56,960 --> 00:51:57,620
No.

1041
00:51:57,620 --> 00:51:59,710
What properties had this
wave function better

1042
00:51:59,710 --> 00:52:05,460
satisfy to be a reasonable wave
function in this potential?

1043
00:52:05,460 --> 00:52:06,660
Say again?

1044
00:52:06,660 --> 00:52:08,020
It's got to vanish at the walls.

1045
00:52:08,020 --> 00:52:09,853
It's got to satisfy the
boundary conditions.

1046
00:52:09,853 --> 00:52:12,290
What else must be true
of this wave function?

1047
00:52:12,290 --> 00:52:13,010
Normalizable.

1048
00:52:13,010 --> 00:52:14,240
Is it normalizable?

1049
00:52:14,240 --> 00:52:14,880
Yeah.

1050
00:52:14,880 --> 00:52:16,400
What else?

1051
00:52:16,400 --> 00:52:16,900
Continuous.

1052
00:52:16,900 --> 00:52:18,524
It better not have
any discontinuities.

1053
00:52:18,524 --> 00:52:19,471
Is it continuous?

1054
00:52:19,471 --> 00:52:19,970
Great.

1055
00:52:19,970 --> 00:52:20,470
OK.

1056
00:52:20,470 --> 00:52:23,691
Is there any reason that this
is a stupid wave function?

1057
00:52:23,691 --> 00:52:24,190
No.

1058
00:52:24,190 --> 00:52:25,500
It's perfectly reasonable.

1059
00:52:25,500 --> 00:52:29,930
It's not an energy
eigenfunction, but--

1060
00:52:29,930 --> 00:52:31,330
Yeah, cool?

1061
00:52:31,330 --> 00:52:32,150
Yeah.

1062
00:52:32,150 --> 00:52:34,252
AUDIENCE: This is sort
of like a math question.

1063
00:52:34,252 --> 00:52:35,768
So to write that
at a superposition,

1064
00:52:35,768 --> 00:52:38,636
you have to write it like
basically a Fourier sign

1065
00:52:38,636 --> 00:52:39,592
series?

1066
00:52:39,592 --> 00:52:42,940
Isn't the [INAUDIBLE]
function even, though?

1067
00:52:42,940 --> 00:52:46,426
PROFESSOR: On this domain,
that and the sines are even.

1068
00:52:46,426 --> 00:52:48,925
So this is actually odd, but
we're only looking at it from 0

1069
00:52:48,925 --> 00:52:53,480
to L. So, I mean
that half of it.

1070
00:52:53,480 --> 00:52:56,080
The sines are odd, but we're
only looking at the first peak.

1071
00:52:56,080 --> 00:52:56,780
So you could just
as well have written

1072
00:52:56,780 --> 00:52:59,420
that as cosine of the
midpoint plus the distance

1073
00:52:59,420 --> 00:53:01,595
from the midpoint.

1074
00:53:01,595 --> 00:53:03,470
Actually, let me say
that again, because it's

1075
00:53:03,470 --> 00:53:05,511
a much better question I
just give it shrift for.

1076
00:53:10,050 --> 00:53:11,010
So here's the question.

1077
00:53:11,010 --> 00:53:15,359
The question is, look, so
sine is an odd function,

1078
00:53:15,359 --> 00:53:16,900
but sine squared is
an even function.

1079
00:53:16,900 --> 00:53:19,670
So how can you expand sine
squared, an even function,

1080
00:53:19,670 --> 00:53:21,780
in terms of sines,
an odd function?

1081
00:53:21,780 --> 00:53:23,860
But think about this physically.

1082
00:53:23,860 --> 00:53:30,570
Here's sine squared in our
domain, and here's sine.

1083
00:53:30,570 --> 00:53:31,810
Now what do you mean by even?

1084
00:53:31,810 --> 00:53:33,905
Usually by even we mean
reflection around zero.

1085
00:53:33,905 --> 00:53:35,780
But I could just as well
have said reflection

1086
00:53:35,780 --> 00:53:36,770
around the origin.

1087
00:53:36,770 --> 00:53:39,030
This potential is symmetric.

1088
00:53:39,030 --> 00:53:41,745
And the energy eigenfunctions
are symmetric about the origin.

1089
00:53:41,745 --> 00:53:44,120
They're not symmetric about
reflection around this point.

1090
00:53:44,120 --> 00:53:46,380
But they are symmetric about
reflection around this point.

1091
00:53:46,380 --> 00:53:48,210
That's a particularly
natural place to call it 0.

1092
00:53:48,210 --> 00:53:49,930
So I was calling them sine
because I was calling this 0,

1093
00:53:49,930 --> 00:53:51,304
but I could have
called it cosine

1094
00:53:51,304 --> 00:53:55,470
if I called this
0, for the same Kx.

1095
00:53:55,470 --> 00:53:58,570
And indeed, can we
expand this sine

1096
00:53:58,570 --> 00:54:01,570
squared function in terms
of a basis of these sines

1097
00:54:01,570 --> 00:54:02,610
on the domain 0 to L?

1098
00:54:02,610 --> 00:54:04,030
Absolutely.

1099
00:54:04,030 --> 00:54:07,220
Very good question.

1100
00:54:07,220 --> 00:54:09,355
And lastly, last
clicker question.

1101
00:54:12,700 --> 00:54:13,460
Oops.

1102
00:54:13,460 --> 00:54:16,030
Whatever.

1103
00:54:16,030 --> 00:54:17,210
OK.

1104
00:54:17,210 --> 00:54:20,600
At t equals 0, a particle is
described by the wave function

1105
00:54:20,600 --> 00:54:21,130
we just saw.

1106
00:54:23,461 --> 00:54:25,710
Which of the following is
true about the wave function

1107
00:54:25,710 --> 00:54:26,600
at subsequent times?

1108
00:54:37,650 --> 00:54:38,210
5 seconds.

1109
00:54:42,776 --> 00:54:43,730
Whew.

1110
00:54:43,730 --> 00:54:44,650
Oh, OK.

1111
00:54:44,650 --> 00:54:47,360
In the last few seconds we had
an explosive burst for A, B,

1112
00:54:47,360 --> 00:54:49,870
and C. So our
current distribution

1113
00:54:49,870 --> 00:54:54,960
is 8, 16, 10, and 67,
sounds like 67 is popular.

1114
00:54:54,960 --> 00:54:58,630
Discuss quickly, very quickly,
with the person next to you.

1115
00:54:58,630 --> 00:55:02,510
[CHATTER]

1116
00:55:13,670 --> 00:55:17,093
OK, and vote again.

1117
00:55:21,580 --> 00:55:22,980
OK, five seconds.

1118
00:55:22,980 --> 00:55:24,180
Get your last vote in.

1119
00:55:28,240 --> 00:55:29,300
All right.

1120
00:55:29,300 --> 00:55:33,472
And the answer is D. Yay.

1121
00:55:37,289 --> 00:55:38,580
So let's think about the logic.

1122
00:55:38,580 --> 00:55:39,950
Let's go through the logic here.

1123
00:55:39,950 --> 00:55:42,360
So as was pointed out by
a student up here earlier,

1124
00:55:42,360 --> 00:55:46,070
the wave function
sine squared of pi x

1125
00:55:46,070 --> 00:55:49,100
can be expanded in terms of
the energy eigenfunction.

1126
00:55:49,100 --> 00:55:51,150
Any reasonable function
can be expanded

1127
00:55:51,150 --> 00:55:53,800
in terms of a superposition
of definite energy states

1128
00:55:53,800 --> 00:55:55,600
of energy eigenfunctions.

1129
00:55:55,600 --> 00:55:58,490
So that means we can
write psi at some time

1130
00:55:58,490 --> 00:56:04,060
as a superposition Cn sine of n
pi x upon a e to the minus i e

1131
00:56:04,060 --> 00:56:09,450
n t upon h bar, since those are,
in fact, the eigenfunctions.

1132
00:56:09,450 --> 00:56:10,200
So we can do that.

1133
00:56:10,200 --> 00:56:11,877
Now, when we look at
the time evolution,

1134
00:56:11,877 --> 00:56:13,710
we know that each term
in that superposition

1135
00:56:13,710 --> 00:56:15,450
evolves with a phase.

1136
00:56:15,450 --> 00:56:19,220
The overall wave function
does not evolve with a phase.

1137
00:56:19,220 --> 00:56:21,360
It is not an energy eigenstate.

1138
00:56:21,360 --> 00:56:23,460
There are going to be
interference terms due

1139
00:56:23,460 --> 00:56:26,207
to the fact that
it's a superposition.

1140
00:56:26,207 --> 00:56:28,540
So its probability distribution
is not time-independent.

1141
00:56:28,540 --> 00:56:29,640
It is a superposition.

1142
00:56:29,640 --> 00:56:33,430
And so the wave function doesn't
rotate by an overall phase.

1143
00:56:33,430 --> 00:56:37,030
However, we can solve
the Schrodinger equation,

1144
00:56:37,030 --> 00:56:37,810
as we did before.

1145
00:56:37,810 --> 00:56:40,440
The wave function is
expanded at time 0

1146
00:56:40,440 --> 00:56:43,544
as the energy eigenfunctions
times some set of coefficients.

1147
00:56:43,544 --> 00:56:44,960
And the time
evolution corresponds

1148
00:56:44,960 --> 00:56:48,230
to adding two each independent
term in the superposition

1149
00:56:48,230 --> 00:56:52,120
the appropriate phase for
that energy eigenstate.

1150
00:56:52,120 --> 00:56:54,040
Cool?

1151
00:56:54,040 --> 00:56:54,540
All right.

1152
00:56:54,540 --> 00:57:03,220
So the answer is D. And that's
it for the clicker questions.

1153
00:57:03,220 --> 00:57:07,050
OK, so any questions on the
clicker questions so far?

1154
00:57:07,050 --> 00:57:09,050
OK, those are going to
be posted on the web site

1155
00:57:09,050 --> 00:57:12,190
so you can go over them.

1156
00:57:12,190 --> 00:57:14,250
And now back to
energy eigenfunctions.

1157
00:57:24,770 --> 00:57:27,730
So what I want to talk about
now is the qualitative behavior

1158
00:57:27,730 --> 00:57:29,140
of energy eigenfunctions.

1159
00:57:29,140 --> 00:57:31,460
Suppose I know I have
an energy eigenfunction.

1160
00:57:31,460 --> 00:57:33,955
What can I say generally
about its structure?

1161
00:57:38,870 --> 00:57:41,175
So let me ask the question,
qualitative behavior.

1162
00:57:47,270 --> 00:57:49,670
So suppose someone hands
you a potential U of x.

1163
00:57:49,670 --> 00:57:51,990
Someone hands you some
potential, U of x,

1164
00:57:51,990 --> 00:57:54,900
and says, look, I've
got this potential.

1165
00:57:54,900 --> 00:57:57,580
Maybe I'll draw it for you.

1166
00:57:57,580 --> 00:57:59,850
It's got some wiggles,
and then a big wiggle,

1167
00:57:59,850 --> 00:58:01,880
and then it's got a
big wiggle, and then--

1168
00:58:01,880 --> 00:58:03,195
do I want to do that?

1169
00:58:03,195 --> 00:58:04,275
Yeah, let's do that.

1170
00:58:04,275 --> 00:58:08,089
Then a big wiggle, and
something like this.

1171
00:58:08,089 --> 00:58:09,630
And someone shows
you this potential.

1172
00:58:09,630 --> 00:58:11,921
And they say, look, what are
the energy eigenfunctions?

1173
00:58:13,770 --> 00:58:17,510
Well, OK, free
particle was easy.

1174
00:58:17,510 --> 00:58:19,070
The infinite square
well was easy.

1175
00:58:19,070 --> 00:58:21,207
We could solve
that analytically.

1176
00:58:21,207 --> 00:58:23,290
The next involved solving
a differential equation.

1177
00:58:23,290 --> 00:58:25,360
So what differential equation
is this going to lead us to?

1178
00:58:25,360 --> 00:58:27,570
Well, we know that the
energy eigenvalue equation

1179
00:58:27,570 --> 00:58:32,070
is minus h bar
squared upon 2 m phi

1180
00:58:32,070 --> 00:58:38,570
prime prime of x plus U of x
phi x, so that's the energy

1181
00:58:38,570 --> 00:58:41,620
operator acting on
phi, is equal to,

1182
00:58:41,620 --> 00:58:47,949
saying that it's an energy
eigenfunction, phi sub E,

1183
00:58:47,949 --> 00:58:49,740
says that it's equal
to the energy operator

1184
00:58:49,740 --> 00:58:52,670
acting on this eigenfunction
is just a constant E phi sub

1185
00:58:52,670 --> 00:58:54,870
E of x.

1186
00:58:54,870 --> 00:58:56,680
And I'm going to work
at moment in time,

1187
00:58:56,680 --> 00:59:00,420
so we're going to drop all the
t dependence for the moment.

1188
00:59:00,420 --> 00:59:01,920
So this is the
differential equation

1189
00:59:01,920 --> 00:59:07,072
we need to solve where U of
x is this god-awful function.

1190
00:59:07,072 --> 00:59:08,780
Do you think it's very
likely that you're

1191
00:59:08,780 --> 00:59:11,670
going to be able to
solve this analytically?

1192
00:59:11,670 --> 00:59:13,700
Probably not.

1193
00:59:13,700 --> 00:59:18,761
However, some basic
ideas will help

1194
00:59:18,761 --> 00:59:21,010
you get an intuition for
what the wave function should

1195
00:59:21,010 --> 00:59:22,240
look like.

1196
00:59:22,240 --> 00:59:25,910
And I cannot overstate the
importance of being able

1197
00:59:25,910 --> 00:59:29,560
to eyeball a system and guess
the qualitative features of its

1198
00:59:29,560 --> 00:59:32,030
wave functions,
because that intuition,

1199
00:59:32,030 --> 00:59:34,590
that ability to estimate, is
going to contain an awful lot

1200
00:59:34,590 --> 00:59:35,340
of physics.

1201
00:59:35,340 --> 00:59:36,790
So let's try to extract it.

1202
00:59:36,790 --> 00:59:38,930
So in order to do
so, I want to start

1203
00:59:38,930 --> 00:59:40,945
by massaging this
equation into a form which

1204
00:59:40,945 --> 00:59:42,070
is particularly convenient.

1205
00:59:44,790 --> 00:59:47,910
So in particular, I'm going to
write this equation as phi sub

1206
00:59:47,910 --> 00:59:49,449
E prime prime.

1207
00:59:49,449 --> 00:59:51,740
So what I'm going to do is
I'm going to take this term,

1208
00:59:51,740 --> 00:59:53,615
I'm going to notice this
has two derivatives,

1209
00:59:53,615 --> 00:59:56,026
this has no derivatives,
this has no derivatives.

1210
00:59:56,026 --> 00:59:59,130
And I'm going to move this
term over here and combine

1211
00:59:59,130 --> 01:00:01,600
these terms into E
minus U of x, and I'm

1212
01:00:01,600 --> 01:00:05,320
going to divide each side by 2m
upon h bar squared with a minus

1213
01:00:05,320 --> 01:00:08,260
sign, giving me that
phi prime prime of E

1214
01:00:08,260 --> 01:00:14,420
of x upon phi E of x dividing
through by this phi E

1215
01:00:14,420 --> 01:00:18,250
is equal to minus 2m
over h bar squared.

1216
01:00:21,682 --> 01:00:23,140
And let's just get
our signs right.

1217
01:00:23,140 --> 01:00:24,560
We've got the minus
from here, so this

1218
01:00:24,560 --> 01:00:25,960
is going to be E minus U of x.

1219
01:00:33,931 --> 01:00:35,680
So you might look at
that and think, well,

1220
01:00:35,680 --> 01:00:40,010
why is that any better than
what I've just written down.

1221
01:00:40,010 --> 01:00:42,724
But what is the second
derivative of function?

1222
01:00:42,724 --> 01:00:44,890
It's telling you not its
slope, but it's telling you

1223
01:00:44,890 --> 01:00:46,550
how the slope changes.

1224
01:00:46,550 --> 01:00:49,588
It's telling about the
curvature of the function.

1225
01:00:49,588 --> 01:00:52,046
And what this is telling me is
something very, very useful.

1226
01:00:54,179 --> 01:00:55,970
So for example, let's
look at the function.

1227
01:00:55,970 --> 01:00:57,360
Let's assume that
the function is real,

1228
01:00:57,360 --> 01:00:58,901
although we know in
general it's not.

1229
01:00:58,901 --> 01:01:02,100
Let's assume that the function
is real for simplicity.

1230
01:01:02,100 --> 01:01:05,910
So we're going to plot
the real part of phi

1231
01:01:05,910 --> 01:01:07,210
in the vertical axis.

1232
01:01:07,210 --> 01:01:08,630
And this is x.

1233
01:01:08,630 --> 01:01:15,130
Suppose the real part of phi
is positive at some point.

1234
01:01:15,130 --> 01:01:17,070
Phi prime prime,
if it's positive,

1235
01:01:17,070 --> 01:01:19,297
tells us that not only
is the slope positive,

1236
01:01:19,297 --> 01:01:20,130
but it's increasing.

1237
01:01:20,130 --> 01:01:21,600
Or it doesn't tell us
anything about the slope,

1238
01:01:21,600 --> 01:01:23,080
but it tells us that whatever
the slope, it's increasing.

1239
01:01:23,080 --> 01:01:25,650
If it's negative, the slope is
increasing as we increase x.

1240
01:01:25,650 --> 01:01:28,210
If it's positive, it's
increasing as we increase x.

1241
01:01:28,210 --> 01:01:31,450
So it's telling us that the
wave function looks like this,

1242
01:01:31,450 --> 01:01:34,880
locally, something like that.

1243
01:01:34,880 --> 01:01:39,040
If phi is negative,
if phi is negative,

1244
01:01:39,040 --> 01:01:41,254
then if this
quantity is positive,

1245
01:01:41,254 --> 01:01:42,920
then phi prime prime
has to be negative.

1246
01:01:42,920 --> 01:01:45,150
But negative is curving down.

1247
01:01:50,210 --> 01:01:54,780
So if this quantity, which
I will call the curvature,

1248
01:01:54,780 --> 01:02:01,260
if this quantity is positive,
it curves away from the axis.

1249
01:02:01,260 --> 01:02:07,100
So this is phi prime prime
over phi greater than 0.

1250
01:02:07,100 --> 01:02:09,700
If this quantity is
positive, the function

1251
01:02:09,700 --> 01:02:11,050
curves away from the axis.

1252
01:02:11,050 --> 01:02:12,440
Cool?

1253
01:02:12,440 --> 01:02:16,270
If this quantity is negative,
phi prime prime upon

1254
01:02:16,270 --> 01:02:19,490
phi less than 0,
exactly the opposite.

1255
01:02:19,490 --> 01:02:20,920
This has to be negative.

1256
01:02:20,920 --> 01:02:23,480
If phi is positive, then phi
prime prime has to be negative.

1257
01:02:23,480 --> 01:02:24,907
It has to be curving down.

1258
01:02:27,710 --> 01:02:30,290
And similarly, if
phi is negative,

1259
01:02:30,290 --> 01:02:33,170
then phi prime prime
has to be positive,

1260
01:02:33,170 --> 01:02:34,500
and it has to curve up.

1261
01:02:34,500 --> 01:02:37,840
So if this quantity is positive,
if the curvature is positive,

1262
01:02:37,840 --> 01:02:39,440
it curves away from the axis.

1263
01:02:39,440 --> 01:02:41,950
If the curvature is negative,
if this quantity is negative,

1264
01:02:41,950 --> 01:02:44,120
it curves towards the axis.

1265
01:02:44,120 --> 01:02:47,590
So what does that tell you about
solutions when the curvature is

1266
01:02:47,590 --> 01:02:49,206
positive or negative?

1267
01:02:49,206 --> 01:02:50,330
It tells you the following.

1268
01:02:52,930 --> 01:02:54,980
It tells you that,
imagine we have

1269
01:02:54,980 --> 01:02:59,450
a function where phi prime
prime over phi is constant.

1270
01:02:59,450 --> 01:03:03,050
And in particular, let's
let phi prime prime over phi

1271
01:03:03,050 --> 01:03:06,760
be a constant,
which is positive.

1272
01:03:06,760 --> 01:03:10,671
And I'll call that positive
constant kappa squared.

1273
01:03:10,671 --> 01:03:12,170
And to emphasize
that it's positive,

1274
01:03:12,170 --> 01:03:14,174
I'm going to call
it kappa squared.

1275
01:03:14,174 --> 01:03:15,090
It's a positive thing.

1276
01:03:15,090 --> 01:03:16,980
It's a real number squared.

1277
01:03:16,980 --> 01:03:18,635
What does the
solution look like?

1278
01:03:22,510 --> 01:03:24,120
Well, this quantity is positive.

1279
01:03:24,120 --> 01:03:25,661
It's always going
to be curving away.

1280
01:03:25,661 --> 01:03:28,287
So we have solutions that
look like this or solutions

1281
01:03:28,287 --> 01:03:29,120
that look like this.

1282
01:03:29,120 --> 01:03:29,990
Can it ever be 0?

1283
01:03:32,907 --> 01:03:34,740
Yeah, sure, it could
be an inflection point.

1284
01:03:34,740 --> 01:03:36,698
So for example, here the
curvature is positive,

1285
01:03:36,698 --> 01:03:40,170
but at this point the curvature
has to switch to be like this.

1286
01:03:40,170 --> 01:03:41,960
What functions are of this form?

1287
01:03:44,752 --> 01:03:45,960
Let me give you another hint.

1288
01:03:45,960 --> 01:03:46,550
Here's one.

1289
01:03:46,550 --> 01:03:49,400
Is this curvature positive?

1290
01:03:49,400 --> 01:03:50,057
Yes.

1291
01:03:50,057 --> 01:03:50,890
What about this one?

1292
01:03:53,470 --> 01:03:53,970
Yup.

1293
01:03:53,970 --> 01:03:56,220
Those are all
positive curvature.

1294
01:03:56,220 --> 01:03:57,910
And these are exponentials.

1295
01:03:57,910 --> 01:03:59,930
And the solution to this
differential equation

1296
01:03:59,930 --> 01:04:06,670
is e to the plus kappa x
or e to the minus kappa x.

1297
01:04:06,670 --> 01:04:08,940
And an arbitrary
solution of this equation

1298
01:04:08,940 --> 01:04:12,130
is a superposition A e
to the kappa x plus B

1299
01:04:12,130 --> 01:04:14,346
e to the minus kappa x.

1300
01:04:14,346 --> 01:04:16,244
Everyone cool with that?

1301
01:04:16,244 --> 01:04:17,660
When this quantity
is positive, we

1302
01:04:17,660 --> 01:04:19,390
get growing and
collapsing exponentials.

1303
01:04:21,890 --> 01:04:22,390
Yeah?

1304
01:04:25,610 --> 01:04:29,990
On the other hand, if
phi prime prime over phi

1305
01:04:29,990 --> 01:04:34,010
is a negative number, i.e.
minus what I'll call k

1306
01:04:34,010 --> 01:04:44,010
squared, then the curvature
has to be negative.

1307
01:04:44,010 --> 01:04:47,460
And what functions have
everywhere negative curvature?

1308
01:04:47,460 --> 01:04:49,169
Sinusoidals.

1309
01:04:49,169 --> 01:04:49,669
Cool?

1310
01:04:53,540 --> 01:05:00,445
And the general solution
is A e to the i K x plus B

1311
01:05:00,445 --> 01:05:03,226
e to the minus i K x.

1312
01:05:03,226 --> 01:05:06,710
So that differential equation,
also known as sine and cosine.

1313
01:05:09,170 --> 01:05:09,670
Cool?

1314
01:05:12,540 --> 01:05:19,765
So putting that together
with our original function,

1315
01:05:19,765 --> 01:05:21,140
let's bring this up.

1316
01:05:24,110 --> 01:05:26,680
So we want to think about
the wave functions here.

1317
01:05:26,680 --> 01:05:29,014
But in order to think about
the energy eigenstates,

1318
01:05:29,014 --> 01:05:30,305
we need to decide on an energy.

1319
01:05:32,850 --> 01:05:35,794
We need to pick an
energy, because you

1320
01:05:35,794 --> 01:05:37,960
can't find the solution
without figuring the energy.

1321
01:05:37,960 --> 01:05:38,950
But notice something nice here.

1322
01:05:38,950 --> 01:05:40,260
So suppose the energy is e.

1323
01:05:40,260 --> 01:05:42,360
And let me just draw
E. This is a constant.

1324
01:05:42,360 --> 01:05:43,187
The energy is this.

1325
01:05:43,187 --> 01:05:45,520
So this is the value of E.
Here we're drawing potential.

1326
01:05:45,520 --> 01:05:48,489
But this is the value of the
energy, which is a constant.

1327
01:05:48,489 --> 01:05:49,280
It's just a number.

1328
01:05:53,680 --> 01:05:57,200
If you had a classical particle
moving in this potential,

1329
01:05:57,200 --> 01:05:58,760
what would happen?

1330
01:05:58,760 --> 01:05:59,650
It would roll around.

1331
01:05:59,650 --> 01:06:01,360
So for example, let's say
you gave it this energy

1332
01:06:01,360 --> 01:06:02,210
by putting it here.

1333
01:06:02,210 --> 01:06:04,168
And think of this as a
gravitational potential.

1334
01:06:04,168 --> 01:06:06,157
You put it here, you let
go, and it falls down.

1335
01:06:06,157 --> 01:06:07,990
And it'll keep rolling
until it gets up here

1336
01:06:07,990 --> 01:06:10,230
to the classical turning point.

1337
01:06:10,230 --> 01:06:12,080
And at that point,
its kinetic energy

1338
01:06:12,080 --> 01:06:13,710
must be 0, because
its potential energy

1339
01:06:13,710 --> 01:06:15,410
is its total energy,
at which point

1340
01:06:15,410 --> 01:06:17,670
it will turn around
and fall back.

1341
01:06:17,670 --> 01:06:18,359
Yes?

1342
01:06:18,359 --> 01:06:20,150
If you take your ball,
and you put it here,

1343
01:06:20,150 --> 01:06:21,525
and you let it
roll, does it ever

1344
01:06:21,525 --> 01:06:23,960
get here, to this position?

1345
01:06:23,960 --> 01:06:25,990
No, because it doesn't
have enough energy.

1346
01:06:25,990 --> 01:06:29,620
Classically, this is
a forbidden position.

1347
01:06:29,620 --> 01:06:31,790
So given an energy
and given a potential,

1348
01:06:31,790 --> 01:06:38,200
we can break the system up
into classically allowed zones

1349
01:06:38,200 --> 01:06:39,760
and classically forbidden zones.

1350
01:06:45,170 --> 01:06:47,120
Cool?

1351
01:06:47,120 --> 01:06:49,210
Now, in a classically
allowed zone,

1352
01:06:49,210 --> 01:06:53,670
the energy is greater
than the potential.

1353
01:06:53,670 --> 01:06:55,170
And in a classically
forbidden zone,

1354
01:06:55,170 --> 01:06:57,300
the energy is less
than the potential.

1355
01:07:00,530 --> 01:07:03,230
Everyone cool with that?

1356
01:07:03,230 --> 01:07:06,080
But this tells us
something really nice.

1357
01:07:06,080 --> 01:07:09,429
If the energy is greater
than the potential,

1358
01:07:09,429 --> 01:07:10,970
what do you know
about the curvature?

1359
01:07:15,050 --> 01:07:15,550
Yeah.

1360
01:07:15,550 --> 01:07:17,175
If we're in a
classically allowed zone,

1361
01:07:17,175 --> 01:07:20,150
so the energy is greater
than the potential,

1362
01:07:20,150 --> 01:07:22,860
then this quantity is positive,
there's a minus sign here,

1363
01:07:22,860 --> 01:07:23,920
so this is negative.

1364
01:07:23,920 --> 01:07:25,760
So the curvature is negative.

1365
01:07:28,920 --> 01:07:31,680
Remember, curvature
is negative means

1366
01:07:31,680 --> 01:07:35,000
that we curve towards the axis.

1367
01:07:35,000 --> 01:07:37,015
So in a classically
allowed region,

1368
01:07:37,015 --> 01:07:38,640
the wave function
should be sinusoidal.

1369
01:07:43,840 --> 01:07:46,050
What about in the classically
forbidden regions?

1370
01:07:46,050 --> 01:07:47,280
In the classically
forbidden regions,

1371
01:07:47,280 --> 01:07:48,890
the energy is less
than the potential.

1372
01:07:48,890 --> 01:07:50,848
That means in magnitude
this is less than this,

1373
01:07:50,848 --> 01:07:53,270
this is a negative
number, minus sign,

1374
01:07:53,270 --> 01:07:55,365
the curvature is
going to be minus

1375
01:07:55,365 --> 01:07:59,050
times a minus is a positive,
so the curvature's positive.

1376
01:08:02,260 --> 01:08:05,190
So the solutions are
either growing exponentials

1377
01:08:05,190 --> 01:08:10,000
or shrinking exponentials
or superpositions of them.

1378
01:08:10,000 --> 01:08:11,810
Everyone cool with that?

1379
01:08:11,810 --> 01:08:14,340
So let's think about
a simple example.

1380
01:08:14,340 --> 01:08:16,452
Let's work through this
in a simple example.

1381
01:08:16,452 --> 01:08:18,575
And let me give you a
little bit more board space.

1382
01:08:22,430 --> 01:08:26,122
Simple example would be a
potential that looks like this.

1383
01:08:26,122 --> 01:08:27,580
And let's just
suppose that we want

1384
01:08:27,580 --> 01:08:33,760
to find an energy eigenfunction
with energy that's E. Well,

1385
01:08:33,760 --> 01:08:36,250
this is a classically
allowed zone,

1386
01:08:36,250 --> 01:08:39,939
and these are the classically
forbidden regions.

1387
01:08:39,939 --> 01:08:43,370
Now I want to ask, what does
the wave function look like?

1388
01:08:43,370 --> 01:08:46,630
And I don't want to draw it
on top of the energy diagram,

1389
01:08:46,630 --> 01:08:49,037
because wave function
is not an energy.

1390
01:08:49,037 --> 01:08:50,620
Wave function is a
different quantity,

1391
01:08:50,620 --> 01:08:52,370
because it's got
different axes and I want

1392
01:08:52,370 --> 01:08:54,279
it drawn on a different plot.

1393
01:08:54,279 --> 01:08:57,960
So but as a function of x--

1394
01:08:57,960 --> 01:09:00,935
so just to get the
positions straight,

1395
01:09:00,935 --> 01:09:02,560
these are the bounds
of the classically

1396
01:09:02,560 --> 01:09:04,899
allowed and forbidden regions.

1397
01:09:04,899 --> 01:09:07,140
What do we expect?

1398
01:09:07,140 --> 01:09:09,830
Well, we expect that it's
going to be sinusoidal in here.

1399
01:09:13,160 --> 01:09:15,770
We expect that it's going
to be exponential growing

1400
01:09:15,770 --> 01:09:20,399
or converging out here, exp.

1401
01:09:20,399 --> 01:09:25,100
But one last important thing is
that not only is the curvature

1402
01:09:25,100 --> 01:09:28,270
negative in here in these
classically allowed regions,

1403
01:09:28,270 --> 01:09:30,080
but the magnitude
of the curvature,

1404
01:09:30,080 --> 01:09:32,870
how rapidly it's turning over,
how big that second derivative

1405
01:09:32,870 --> 01:09:35,040
is, depends on the
difference between the energy

1406
01:09:35,040 --> 01:09:35,865
and the potential.

1407
01:09:35,865 --> 01:09:38,240
The greater the difference,
the more rapid the curvature,

1408
01:09:38,240 --> 01:09:40,500
the more rapid the turning
over and fluctuation.

1409
01:09:40,500 --> 01:09:43,450
If the differences between the
potential and the true energy,

1410
01:09:43,450 --> 01:09:46,720
the total energy, is small, then
the curvature is very small.

1411
01:09:46,720 --> 01:09:49,990
So the derivative
changes very gradually.

1412
01:09:49,990 --> 01:09:51,660
What does that tell us?

1413
01:09:51,660 --> 01:09:54,194
That tells us that in here the
wave function is oscillating

1414
01:09:54,194 --> 01:09:56,110
rapidly, because the
curvature, the difference

1415
01:09:56,110 --> 01:09:58,320
between the energy and
the potential is large,

1416
01:09:58,320 --> 01:10:01,690
and so the wave function
is oscillating rapidly.

1417
01:10:01,690 --> 01:10:04,200
As we get out towards the
classical turning points,

1418
01:10:04,200 --> 01:10:07,640
the wave function will be
oscillating less rapidly.

1419
01:10:07,640 --> 01:10:09,865
The slope will be
changing more gradually.

1420
01:10:09,865 --> 01:10:11,810
And as a consequence,
two things happen.

1421
01:10:11,810 --> 01:10:13,768
Let me actually draw this
slightly differently.

1422
01:10:17,647 --> 01:10:19,230
So as a consequence
two things happen.

1423
01:10:19,230 --> 01:10:21,130
One is the wavelength
gets longer,

1424
01:10:21,130 --> 01:10:22,847
because the
curvature is smaller.

1425
01:10:22,847 --> 01:10:24,680
And the second is the
amplitude gets larger,

1426
01:10:24,680 --> 01:10:27,280
because it keeps on having
a positive slope for longer

1427
01:10:27,280 --> 01:10:31,520
and longer, and it takes
longer to curve back down.

1428
01:10:31,520 --> 01:10:34,260
So here we have
rapid oscillations.

1429
01:10:34,260 --> 01:10:41,430
And then the oscillations get
longer and longer wavelength,

1430
01:10:41,430 --> 01:10:43,892
until we get out to the
classical turning point.

1431
01:10:43,892 --> 01:10:45,225
And at this point, what happens?

1432
01:10:47,704 --> 01:10:49,120
Yeah, it's got to
be [INAUDIBLE]..

1433
01:10:49,120 --> 01:10:51,860
Now, here we have some sine,
and some superpositions

1434
01:10:51,860 --> 01:10:53,647
of sine and cosines,
exponentials.

1435
01:10:53,647 --> 01:10:55,730
And in particular, it
arrives here with some slope

1436
01:10:55,730 --> 01:10:57,490
and with some value.

1437
01:10:57,490 --> 01:11:00,140
We know this side we've
got to get exponentials.

1438
01:11:00,140 --> 01:11:02,330
And so this sum of sines
and cosines at this point

1439
01:11:02,330 --> 01:11:06,400
must match the sum
of exponentials.

1440
01:11:06,400 --> 01:11:08,282
How must it do so?

1441
01:11:08,282 --> 01:11:10,490
What must be true of the
wave function at this point?

1442
01:11:13,880 --> 01:11:16,090
Can it be discontinuous?

1443
01:11:16,090 --> 01:11:17,850
Can its derivative
be discontinuous?

1444
01:11:17,850 --> 01:11:18,350
No.

1445
01:11:18,350 --> 01:11:21,800
So the value and the
derivative must be continuous.

1446
01:11:21,800 --> 01:11:24,690
So that tells us precisely
which linear combination

1447
01:11:24,690 --> 01:11:29,089
of positive growing and
shrinking exponentials we get.

1448
01:11:29,089 --> 01:11:30,630
So we'll get some
linear combination,

1449
01:11:30,630 --> 01:11:32,360
which may do this for awhile.

1450
01:11:32,360 --> 01:11:33,860
But since it's got
some contribution

1451
01:11:33,860 --> 01:11:36,490
of positive exponential, it'll
just grow exponentially off

1452
01:11:36,490 --> 01:11:37,982
to infinity.

1453
01:11:37,982 --> 01:11:39,940
And as the energy gets
further and further away

1454
01:11:39,940 --> 01:11:42,780
from the potential, now in
their negative sine, what

1455
01:11:42,780 --> 01:11:45,444
happens to the rate of growth?

1456
01:11:45,444 --> 01:11:46,610
It gets more and more rapid.

1457
01:11:46,610 --> 01:11:48,540
So this just diverges
more and more rapidly.

1458
01:11:48,540 --> 01:11:51,554
Similarly, out here we
have to match the slope.

1459
01:11:51,554 --> 01:11:52,970
And we know that
the curvature has

1460
01:11:52,970 --> 01:11:55,530
to be now positive,
so it has to do this.

1461
01:11:59,120 --> 01:12:00,040
So two questions.

1462
01:12:00,040 --> 01:12:03,820
First off, is this sketch
of the wave function

1463
01:12:03,820 --> 01:12:05,560
a reasonable sketch,
given what we

1464
01:12:05,560 --> 01:12:08,570
know about curvature and this
potential of a wave function

1465
01:12:08,570 --> 01:12:09,280
with that energy?

1466
01:12:13,007 --> 01:12:14,840
Are there ways in which
it's a bad estimate?

1467
01:12:17,798 --> 01:12:19,590
AUDIENCE: [INAUDIBLE]

1468
01:12:19,590 --> 01:12:21,685
PROFESSOR: OK, excellent.

1469
01:12:21,685 --> 01:12:24,600
AUDIENCE: On the right side,
could it have crossed zero?

1470
01:12:24,600 --> 01:12:25,720
PROFESSOR: Absolutely, it
could have crossed zero.

1471
01:12:25,720 --> 01:12:26,920
So I may have drawn this badly.

1472
01:12:26,920 --> 01:12:28,060
It turned out it
was a little subtle.

1473
01:12:28,060 --> 01:12:28,740
It's not obvious.

1474
01:12:28,740 --> 01:12:30,340
Maybe it actually punched
all the way through zero,

1475
01:12:30,340 --> 01:12:31,660
and then it diverged
down negative.

1476
01:12:31,660 --> 01:12:32,785
That's absolutely positive.

1477
01:12:32,785 --> 01:12:35,075
So that was one of the
quibbles you could have.

1478
01:12:35,075 --> 01:12:36,450
Another quibble
you could have is

1479
01:12:36,450 --> 01:12:39,200
that it looks like I have
constant wavelength in here.

1480
01:12:39,200 --> 01:12:41,010
But the potential's
actually changing.

1481
01:12:41,010 --> 01:12:43,070
And what you should
chalk this up to,

1482
01:12:43,070 --> 01:12:46,299
if you'll pardon the pun, is
my artistic skills are limited.

1483
01:12:46,299 --> 01:12:48,340
So this is always going
to be sort of inescapable

1484
01:12:48,340 --> 01:12:50,505
when you qualitatively
draw something.

1485
01:12:50,505 --> 01:12:53,130
On a test, I'm not going to bag
you points on things like that.

1486
01:12:53,130 --> 01:12:54,584
That's what I want to emphasize.

1487
01:12:54,584 --> 01:12:56,250
But the second thing,
is there something

1488
01:12:56,250 --> 01:12:57,622
bad about this wave function?

1489
01:12:57,622 --> 01:12:58,830
Yes, you've already named it.

1490
01:12:58,830 --> 01:13:01,090
What's bad about
this wave function?

1491
01:13:01,090 --> 01:13:03,150
It's badly non-normalizable.

1492
01:13:03,150 --> 01:13:06,820
It diverges off to infinity
out here and out here.

1493
01:13:06,820 --> 01:13:10,110
What does that tell you?

1494
01:13:10,110 --> 01:13:10,860
It's not physical.

1495
01:13:10,860 --> 01:13:11,170
Good.

1496
01:13:11,170 --> 01:13:13,045
What else does it tell
you about this system?

1497
01:13:16,110 --> 01:13:18,220
Sorry?

1498
01:13:18,220 --> 01:13:18,870
Excellent.

1499
01:13:18,870 --> 01:13:21,670
Is this an allowable energy?

1500
01:13:21,670 --> 01:13:22,840
No.

1501
01:13:22,840 --> 01:13:25,130
If the wave function
has this energy,

1502
01:13:25,130 --> 01:13:28,340
it is impossible to
make it continuous,

1503
01:13:28,340 --> 01:13:30,010
assuming that I
drew it correctly,

1504
01:13:30,010 --> 01:13:31,530
and have it converge.

1505
01:13:31,530 --> 01:13:33,640
Is this wave function allowable?

1506
01:13:33,640 --> 01:13:36,180
No, because it does not satisfy
our boundary conditions.

1507
01:13:36,180 --> 01:13:38,763
Our boundary conditions are that
the wave function must vanish

1508
01:13:38,763 --> 01:13:41,710
out here and it must
vanish out here at infinity

1509
01:13:41,710 --> 01:13:43,785
in order to be normalizable.

1510
01:13:43,785 --> 01:13:44,410
Here we failed.

1511
01:13:47,590 --> 01:13:49,610
Now, you can imagine
that-- so let's decrease

1512
01:13:49,610 --> 01:13:50,610
the energy a little bit.

1513
01:13:50,610 --> 01:13:53,760
If we decrease the energy,
our trial energy just a little

1514
01:13:53,760 --> 01:13:55,200
tiny bit, what happens?

1515
01:13:55,200 --> 01:13:58,570
Well, that's going to decrease
the curvature in here.

1516
01:13:58,570 --> 01:14:01,260
We decrease, we bring the energy
in just a little tiny bit.

1517
01:14:01,260 --> 01:14:03,550
That means this is a
little bit smaller.

1518
01:14:03,550 --> 01:14:05,130
The potential stays the same.

1519
01:14:05,130 --> 01:14:06,760
So the curvature in
the allowed region

1520
01:14:06,760 --> 01:14:09,010
is just a little
tiny bit smaller.

1521
01:14:09,010 --> 01:14:10,600
And meanwhile,
the allowed region

1522
01:14:10,600 --> 01:14:12,649
has got just a
little bit thinner.

1523
01:14:12,649 --> 01:14:14,940
And what that will do is the
curvature's a little less,

1524
01:14:14,940 --> 01:14:17,500
the region's a little
less, so now we have--

1525
01:14:28,130 --> 01:14:30,885
Sorry, I get excited.

1526
01:14:30,885 --> 01:14:33,010
And if we tweak the energy,
what's going to happen?

1527
01:14:33,010 --> 01:14:35,176
Well, it's going to arrive
here a little bit sooner.

1528
01:14:38,220 --> 01:14:40,490
And let's imagine
something like this.

1529
01:14:40,490 --> 01:14:45,250
And if we chose the
energy just right,

1530
01:14:45,250 --> 01:14:48,120
we would get it to match
to a linear combination

1531
01:14:48,120 --> 01:14:50,560
of collapsing and
growing exponentials,

1532
01:14:50,560 --> 01:14:53,085
where the contribution from
the growing exponential

1533
01:14:53,085 --> 01:14:54,210
in this direction vanishes.

1534
01:14:57,180 --> 01:14:59,320
There's precisely one
value of the energy

1535
01:14:59,320 --> 01:15:01,992
that lets me do that with
this number of wiggles.

1536
01:15:01,992 --> 01:15:03,950
And so then it goes
through and does its thing.

1537
01:15:07,860 --> 01:15:11,570
And we need it to
happen on both sides.

1538
01:15:11,570 --> 01:15:14,510
Now if I take that solution,
so that it achieves convergence

1539
01:15:14,510 --> 01:15:18,740
out here, and it achieves
convergence out here,

1540
01:15:18,740 --> 01:15:21,630
and I take that energy and
I increase it by epsilon,

1541
01:15:21,630 --> 01:15:24,450
by just the tiniest little bit,
what will happen to this wave

1542
01:15:24,450 --> 01:15:26,057
function?

1543
01:15:26,057 --> 01:15:26,640
It'll diverge.

1544
01:15:26,640 --> 01:15:28,670
It will no longer
be normalizable.

1545
01:15:28,670 --> 01:15:31,070
When you have classically
forbidden regions,

1546
01:15:31,070 --> 01:15:36,010
are the allowed energies
continuous or discrete?

1547
01:15:36,010 --> 01:15:38,820
And that answers a question
from earlier in the class.

1548
01:15:38,820 --> 01:15:40,570
And it also is going
to be the beginning

1549
01:15:40,570 --> 01:15:43,153
of the answer to the question,
why is the spectrum of hydrogen

1550
01:15:43,153 --> 01:15:44,000
discrete.

1551
01:15:44,000 --> 01:15:45,960
See you next time.