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PROFESSOR: All right.

9
00:00:22,120 --> 00:00:24,140
So, we'll get started.

10
00:00:24,140 --> 00:00:27,030
And as I mentioned,
to some degree

11
00:00:27,030 --> 00:00:31,290
this is going to be review on
the setting of our notation

12
00:00:31,290 --> 00:00:33,100
and conventions clear.

13
00:00:33,100 --> 00:00:36,735
So, our first topic is
the Schrodinger equation.

14
00:00:47,560 --> 00:00:50,110
So this Schrodinger
equation is an equation

15
00:00:50,110 --> 00:00:52,340
that takes the following form.

16
00:00:52,340 --> 00:00:56,180
I h bar partial
derivative of this object

17
00:00:56,180 --> 00:01:03,340
called the wave function
that depends on x and t

18
00:01:03,340 --> 00:01:09,000
is equal to minus h squared
over 2m second derivative

19
00:01:09,000 --> 00:01:17,103
with respect to x plus v
of x and t Psi of x and t.

20
00:01:21,450 --> 00:01:22,910
And that's the full equation.

21
00:01:22,910 --> 00:01:24,430
That's the Schrodinger equation.

22
00:01:24,430 --> 00:01:28,460
Now actually, this is not
the Schrodinger equation

23
00:01:28,460 --> 00:01:32,060
in most generality, but it's
the Schrodinger equation

24
00:01:32,060 --> 00:01:35,310
for the case that you
have a potential that

25
00:01:35,310 --> 00:01:37,960
depends on x and t.

26
00:01:37,960 --> 00:01:41,760
For the case that we are doing
non-relativistic physics,

27
00:01:41,760 --> 00:01:46,560
because this thing you may
remember is p squared over 2m

28
00:01:46,560 --> 00:01:49,400
is the kinetic energy operator.

29
00:01:49,400 --> 00:01:53,380
So p squared over 2m
is non-relativistic.

30
00:01:53,380 --> 00:01:55,740
That's a non-relativistic
kinetic energy.

31
00:01:55,740 --> 00:01:57,870
So this is non-relativistic.

32
00:01:57,870 --> 00:02:01,200
Moreover, we have
just one x here.

33
00:02:01,200 --> 00:02:04,460
That means it's a
particle in one dimension.

34
00:02:04,460 --> 00:02:08,800
So we've done a few things,
but this is generally enough

35
00:02:08,800 --> 00:02:11,530
to illustrate our ideas.

36
00:02:11,530 --> 00:02:14,560
And the most
important thing that

37
00:02:14,560 --> 00:02:19,190
should be said at
this point is that Psi

38
00:02:19,190 --> 00:02:27,020
of x and t-- which is
the wave function--

39
00:02:27,020 --> 00:02:29,200
belongs to the complex numbers.

40
00:02:34,530 --> 00:02:35,980
It's a complex number.

41
00:02:35,980 --> 00:02:38,600
And that's by necessity.

42
00:02:38,600 --> 00:02:43,410
If Psi would be real, this
quantity-- the right hand

43
00:02:43,410 --> 00:02:44,760
side-- would be real.

44
00:02:44,760 --> 00:02:47,340
The potential is a real number.

45
00:02:47,340 --> 00:02:51,125
On the left hand side, on the
other hand, if Psi is real,

46
00:02:51,125 --> 00:02:54,630
its derivative would be real,
and this would be imaginary.

47
00:02:54,630 --> 00:02:59,620
So, it's just impossible to get
the solution of this equation

48
00:02:59,620 --> 00:03:02,790
if Psi is real.

49
00:03:02,790 --> 00:03:08,070
So, Psi complex is really
the fundamental thing

50
00:03:08,070 --> 00:03:11,300
that can be said about
this wave function.

51
00:03:11,300 --> 00:03:16,690
Now, you've used complex
numbers in physics all the time,

52
00:03:16,690 --> 00:03:20,210
and even in electromagnetism,
you use complex numbers.

53
00:03:20,210 --> 00:03:23,900
But you use them really
in an auxiliary way only.

54
00:03:23,900 --> 00:03:29,060
You didn't use them in an
absolutely necessary way.

55
00:03:29,060 --> 00:03:30,860
So, for example.

56
00:03:30,860 --> 00:03:41,630
In E&M, you had an electric
field, for example,

57
00:03:41,630 --> 00:03:44,620
for a circularly polarized wave.

58
00:03:48,920 --> 00:03:51,000
And you would write it as this.

59
00:03:55,330 --> 00:03:56,410
Let me put the z here.

60
00:04:00,046 --> 00:04:00,545
Zero.

61
00:04:09,690 --> 00:04:14,190
X hat plus y hat--
those are unit vectors.

62
00:04:14,190 --> 00:04:17,440
I is a complex number.

63
00:04:17,440 --> 00:04:19,420
It's the square root of minus 1.

64
00:04:19,420 --> 00:04:24,680
E to the IKZ minus omega t.

65
00:04:24,680 --> 00:04:27,610
You typically wrote
things like that,

66
00:04:27,610 --> 00:04:32,280
but, in fact, you
always meant real part.

67
00:04:35,650 --> 00:04:39,580
An electric field
is a real quantity.

68
00:04:39,580 --> 00:04:43,740
And the Maxwell's equations
are real equations.

69
00:04:43,740 --> 00:04:46,740
This is a circularly
polarized wave.

70
00:04:46,740 --> 00:04:49,390
And this whole
thing-- by the time

71
00:04:49,390 --> 00:04:53,280
you take the real part of
this, all these complex numbers

72
00:04:53,280 --> 00:04:55,300
play absolutely no role.

73
00:04:55,300 --> 00:05:00,370
It's just a neat way of writing
a complicated electric field

74
00:05:00,370 --> 00:05:05,160
in which the x component and the
y component are out of phase,

75
00:05:05,160 --> 00:05:08,370
and that you have a wave at
the same time propagating

76
00:05:08,370 --> 00:05:10,930
in the z direction.

77
00:05:10,930 --> 00:05:18,090
So this-- in the
here, E is real,

78
00:05:18,090 --> 00:05:22,780
and all i's are auxiliary.

79
00:05:28,350 --> 00:05:31,840
This is completely
different from the case

80
00:05:31,840 --> 00:05:33,590
of the Schrodinger equation.

81
00:05:33,590 --> 00:05:36,990
This i there is fundamental.

82
00:05:36,990 --> 00:05:40,090
The Psi is the
dynamical variable,

83
00:05:40,090 --> 00:05:42,460
and it has to be complex.

84
00:05:42,460 --> 00:05:45,910
So, we make a few remarks
about the Schrodinger equation

85
00:05:45,910 --> 00:05:48,610
to get started.

86
00:05:48,610 --> 00:05:56,490
First remark is that this
is first order differential

87
00:05:56,490 --> 00:05:58,750
equation in time.

88
00:06:05,230 --> 00:06:07,020
This has implications.

89
00:06:07,020 --> 00:06:09,930
Those two derivatives
are maybe--

90
00:06:09,930 --> 00:06:12,080
for some funny
Hamiltonians, you can

91
00:06:12,080 --> 00:06:15,070
have even more than two
derivatives or more complicated

92
00:06:15,070 --> 00:06:15,800
things.

93
00:06:15,800 --> 00:06:19,740
But definitely there's just
one derivative in time.

94
00:06:19,740 --> 00:06:22,887
So, what this means
is that if you

95
00:06:22,887 --> 00:06:26,960
know the wave function
all over space,

96
00:06:26,960 --> 00:06:28,940
you can calculate
what it's going

97
00:06:28,940 --> 00:06:31,260
to be a little time later.

98
00:06:31,260 --> 00:06:34,850
Because if you know
it all over space,

99
00:06:34,850 --> 00:06:37,770
you can calculate
this right hand side

100
00:06:37,770 --> 00:06:39,850
and know what is
the time derivative.

101
00:06:39,850 --> 00:06:41,500
And with the time
derivative, you

102
00:06:41,500 --> 00:06:44,240
can figure it out what
it's going to be later.

103
00:06:44,240 --> 00:06:47,720
A first order differential
equation in time

104
00:06:47,720 --> 00:06:51,020
is something that if you know
the quantity at one time,

105
00:06:51,020 --> 00:06:52,830
the differential
equation tells you

106
00:06:52,830 --> 00:06:54,830
what it's going to be later.

107
00:06:54,830 --> 00:06:59,200
So, that's really sufficient.

108
00:06:59,200 --> 00:07:11,680
Psi of x-- of all x's-- at some
time t naught determines Psi

109
00:07:11,680 --> 00:07:12,720
at all times.

110
00:07:17,480 --> 00:07:20,820
Second property,
fundamental property.

111
00:07:20,820 --> 00:07:22,155
The equation is linear.

112
00:07:28,130 --> 00:07:30,100
So, if you have
two solutions, you

113
00:07:30,100 --> 00:07:33,430
can form a third by
superimposing them,

114
00:07:33,430 --> 00:07:36,780
and you can superimpose them
with complex coefficients.

115
00:07:36,780 --> 00:07:41,110
So, if you have two
solutions, Psi 1 and Psi 2,

116
00:07:41,110 --> 00:07:50,480
then a 1 Psi 1 plus A2
Psi 2 is a solution.

117
00:07:50,480 --> 00:07:54,650
And here the a's belong
to the complex numbers.

118
00:07:54,650 --> 00:08:00,430
So A 1 and A 2 are
complex numbers.

119
00:08:04,850 --> 00:08:07,300
As far as complex
numbers are concerned,

120
00:08:07,300 --> 00:08:09,550
the first thing you
just need to know

121
00:08:09,550 --> 00:08:13,210
is the definition of the
length of a complex number.

122
00:08:13,210 --> 00:08:16,720
So, if you have z, a
typical name people

123
00:08:16,720 --> 00:08:21,210
use for a complex number,
having two components.

124
00:08:21,210 --> 00:08:26,225
A plus ib, where
a and b are real.

125
00:08:29,840 --> 00:08:33,049
There's the definition of
the complex conjugate, which

126
00:08:33,049 --> 00:08:37,049
is a minus ib, and
there's the definition

127
00:08:37,049 --> 00:08:39,840
of the length of the
complex number, which

128
00:08:39,840 --> 00:08:44,470
is square root of a
squared plus b squared,

129
00:08:44,470 --> 00:08:48,380
which is the square
root of z times z star.

130
00:08:54,150 --> 00:08:57,700
So, that's for your
complex number.

131
00:08:57,700 --> 00:09:03,790
So, the property that this makes
this into a physical theory

132
00:09:03,790 --> 00:09:06,860
and goes beyond math
is what you know

133
00:09:06,860 --> 00:09:11,140
is the interpretation of the
wave function as a probability.

134
00:09:11,140 --> 00:09:12,770
So, what do we construct?

135
00:09:12,770 --> 00:09:20,580
We construct p of x and
t, or sometimes called

136
00:09:20,580 --> 00:09:24,920
the row of x and t as a density.

137
00:09:24,920 --> 00:09:30,330
And it's defined
as Psi star of x t.

138
00:09:30,330 --> 00:09:35,100
Now, here the notation
means this Psi star--

139
00:09:35,100 --> 00:09:38,490
we'd put the star
here-- it really

140
00:09:38,490 --> 00:09:43,600
means Psi of x and
t complex conjugate.

141
00:09:43,600 --> 00:09:45,590
You complex conjugate
the wave function.

142
00:09:45,590 --> 00:09:46,400
And you get that.

143
00:09:46,400 --> 00:09:49,070
We'd put the star here,
and typically don't

144
00:09:49,070 --> 00:09:51,940
put the parentheses, unless
you have to complex conjugate

145
00:09:51,940 --> 00:09:54,040
something that's a
little ambiguous.

146
00:09:54,040 --> 00:09:59,960
So, Psi star of x and
t times Psi of x and t.

147
00:09:59,960 --> 00:10:02,495
And this is called the
probability density.

148
00:10:05,460 --> 00:10:08,140
Probability density.

149
00:10:11,880 --> 00:10:19,660
And the interpretation is
that if you take p of x and t

150
00:10:19,660 --> 00:10:25,950
and multiply by little dx,
this is the probability

151
00:10:25,950 --> 00:10:41,770
to find the particle in the
interval x comma x plus dx

152
00:10:41,770 --> 00:10:42,720
at time t.

153
00:10:47,300 --> 00:10:50,780
So, this is our
probability density.

154
00:10:50,780 --> 00:10:56,600
It's a way to make physics
out of the wave function.

155
00:10:56,600 --> 00:10:59,460
It's a postulate.

156
00:10:59,460 --> 00:11:02,620
And so the consequence
of this postulate,

157
00:11:02,620 --> 00:11:05,570
since we're describing
just one particle,

158
00:11:05,570 --> 00:11:09,920
is that we must have the
particle as somewhere.

159
00:11:09,920 --> 00:11:22,136
So, if we add the probabilities
that the particle is somewhere

160
00:11:22,136 --> 00:11:26,646
all over space, this
is the probability

161
00:11:26,646 --> 00:11:31,350
that the particle is in
this little dx we integrated

162
00:11:31,350 --> 00:11:34,570
that must be equal to 1.

163
00:11:34,570 --> 00:11:36,925
And this must hold
for all times.

164
00:11:40,780 --> 00:11:42,950
In terms of things
to notice here,

165
00:11:42,950 --> 00:11:47,650
maybe one thing you can
notice is the units of Psi.

166
00:11:47,650 --> 00:11:53,040
The units of Psi must be 1
over square root of length,

167
00:11:53,040 --> 00:11:55,860
because when we square it,
then we multiply it by length,

168
00:11:55,860 --> 00:11:58,605
we get one, which has no units.

169
00:12:03,710 --> 00:12:07,090
Key property of the
Schrodinger equation.

170
00:12:07,090 --> 00:12:12,200
We will revisit the Schrodinger
equation later and derive it,

171
00:12:12,200 --> 00:12:14,870
sort of the way [? De ?]
[? Rack ?] derives it in his

172
00:12:14,870 --> 00:12:16,070
textbook.

173
00:12:16,070 --> 00:12:19,970
As just a consequence of
unitary time evolution,

174
00:12:19,970 --> 00:12:23,020
it would be a very
neat derivation.

175
00:12:23,020 --> 00:12:25,070
It will give you a
feeling that you really

176
00:12:25,070 --> 00:12:28,040
understand something deep
about quantum mechanics.

177
00:12:28,040 --> 00:12:32,500
And it will be
true, that feeling.

178
00:12:32,500 --> 00:12:35,200
But here, we're going to
go the other way around.

179
00:12:35,200 --> 00:12:41,470
Just simply ask the
question-- suppose

180
00:12:41,470 --> 00:12:51,280
you have a wave function
such that the integral

181
00:12:51,280 --> 00:12:56,915
of this quantity at some
specific time is equal to one.

182
00:13:01,490 --> 00:13:07,230
Will this integral be
equal to one for all times,

183
00:13:07,230 --> 00:13:12,020
given that it is one
at some given time?

184
00:13:12,020 --> 00:13:15,170
Now, you say, well,
why do you ask that?

185
00:13:15,170 --> 00:13:18,890
I ask that because actually
this could be a problem.

186
00:13:18,890 --> 00:13:22,850
We've said that if you know the
wave function all over space

187
00:13:22,850 --> 00:13:25,210
at one time, it's
determined everywhere.

188
00:13:25,210 --> 00:13:27,140
So any time later.

189
00:13:27,140 --> 00:13:31,130
Therefore, if I know the wave
function at time equal zero

190
00:13:31,130 --> 00:13:35,850
is good-- time equal t zero--
is a good wave function,

191
00:13:35,850 --> 00:13:39,370
I might warranty that when I
saw the Schrodinger equation,

192
00:13:39,370 --> 00:13:43,400
the wave function will be
normalized, well, later?

193
00:13:43,400 --> 00:13:44,840
Yes, you are.

194
00:13:44,840 --> 00:13:48,820
And it's a simple or
interesting exercise

195
00:13:48,820 --> 00:13:52,570
that we'll call it
the quick calculation

196
00:13:52,570 --> 00:13:54,300
that I'll leave
it for you to do.

197
00:13:57,940 --> 00:14:14,040
Which is show that d dt of
this integral Psi of x and t

198
00:14:14,040 --> 00:14:17,900
squared dx is equal to zero.

199
00:14:17,900 --> 00:14:20,400
So, basically what
this is saying.

200
00:14:20,400 --> 00:14:24,680
You got one but, think of
this integral-- I'm sorry,

201
00:14:24,680 --> 00:14:29,920
I'm missing a dx here-- think
of this integral for all times.

202
00:14:29,920 --> 00:14:33,650
Now it could be a
function of time,

203
00:14:33,650 --> 00:14:36,310
because you put an
arbitrary time here.

204
00:14:36,310 --> 00:14:38,660
The integral might
depend on time.

205
00:14:38,660 --> 00:14:42,600
So, it's a good question to
think of that integral that

206
00:14:42,600 --> 00:14:45,730
may be a function of time
and take its derivative.

207
00:14:45,730 --> 00:14:49,410
If its derivative is
zero for all times,

208
00:14:49,410 --> 00:14:54,780
and that sometimes equal to
one, it will be one forever.

209
00:14:54,780 --> 00:14:58,880
So, you must show
that this is true.

210
00:14:58,880 --> 00:15:03,240
Now, this I think you've done
one way or another several ways

211
00:15:03,240 --> 00:15:05,270
maybe in 804.

212
00:15:05,270 --> 00:15:07,220
But I ask you to do it again.

213
00:15:07,220 --> 00:15:11,880
So this is left for you as a
way to warm up on this object.

214
00:15:11,880 --> 00:15:17,260
And you will see actually
that it's a little subtle.

215
00:15:17,260 --> 00:15:20,880
It's a little delicate,
because how is it going to go?

216
00:15:20,880 --> 00:15:25,430
You're going to go in and take
the derivative of Psi Psi star.

217
00:15:25,430 --> 00:15:27,219
You're going to take
the derivative of Psi

218
00:15:27,219 --> 00:15:29,260
and you're going to use
the Schrodinger equation.

219
00:15:29,260 --> 00:15:31,740
You're going to take the
derivative of Psi star,

220
00:15:31,740 --> 00:15:34,180
and you're going to use
the complex conjugate

221
00:15:34,180 --> 00:15:36,170
of the Schrodinger equation.

222
00:15:36,170 --> 00:15:37,720
It's going to be a little messy.

223
00:15:37,720 --> 00:15:41,230
But then you're going to
do integration by parts,

224
00:15:41,230 --> 00:15:44,010
and you're going to get zero,
but only if you throw away

225
00:15:44,010 --> 00:15:45,650
the terms at infinity.

226
00:15:45,650 --> 00:15:49,420
And what gives you the
right to throw them away?

227
00:15:49,420 --> 00:15:50,770
You will have to think.

228
00:15:50,770 --> 00:15:54,530
And the answer is that
you will throw them away

229
00:15:54,530 --> 00:15:57,360
if the wave function
goes to zero

230
00:15:57,360 --> 00:16:02,640
at infinity, which must do it.

231
00:16:02,640 --> 00:16:05,260
The wave function must
go to zero at infinity,

232
00:16:05,260 --> 00:16:07,700
because if it didn't
go to zero at infinity,

233
00:16:07,700 --> 00:16:10,170
it went to a
constant at infinity,

234
00:16:10,170 --> 00:16:15,140
it would pick up an
un-normalizable thing here.

235
00:16:15,140 --> 00:16:19,200
So the wave function definitely
has to go to zero at infinity.

236
00:16:19,200 --> 00:16:21,870
But that will also
not be quite enough

237
00:16:21,870 --> 00:16:24,600
if you're careful about
what you're doing.

238
00:16:24,600 --> 00:16:28,720
You will have to demand that the
derivative of the wave function

239
00:16:28,720 --> 00:16:30,550
doesn't blow up.

240
00:16:30,550 --> 00:16:34,840
It's not asking too much,
but it's asking something.

241
00:16:34,840 --> 00:16:37,620
A function could go
to zero, presumably,

242
00:16:37,620 --> 00:16:40,770
and its derivative at
the same time blow up,

243
00:16:40,770 --> 00:16:43,760
but it would be a very
pathological function.

244
00:16:43,760 --> 00:16:47,650
This will bring us to
something that we said.

245
00:16:47,650 --> 00:16:50,430
We're going to
try to be precise,

246
00:16:50,430 --> 00:16:53,340
but it's not so
easy to be precise.

247
00:16:53,340 --> 00:16:56,520
When you try to be
precise, you can

248
00:16:56,520 --> 00:16:59,040
exaggerate and go
precise to a point

249
00:16:59,040 --> 00:17:02,210
that you're paralyzed with
fear with every equation.

250
00:17:02,210 --> 00:17:04,859
We don't want to get that far.

251
00:17:04,859 --> 00:17:07,750
We want you to notice what
happens and just look at it

252
00:17:07,750 --> 00:17:09,200
and state what you need.

253
00:17:09,200 --> 00:17:11,770
Why can't we be precise?

254
00:17:11,770 --> 00:17:14,800
Because at the end of
the day, this equation

255
00:17:14,800 --> 00:17:18,170
is extraordinarily
complicated, and maybe crazy.

256
00:17:18,170 --> 00:17:20,640
The potential is crazy enough.

257
00:17:20,640 --> 00:17:25,970
So, functions-- mathematicians
can invent crazy functions,

258
00:17:25,970 --> 00:17:29,920
things like a function that is
one for every rational number

259
00:17:29,920 --> 00:17:32,410
and zero for every
rational number.

260
00:17:32,410 --> 00:17:37,360
Put that for a potential here,
and who knows what one gets.

261
00:17:37,360 --> 00:17:39,530
So, we're going to
take mild functions.

262
00:17:39,530 --> 00:17:42,400
We're not going to make
them a very complicated,

263
00:17:42,400 --> 00:17:45,700
and we're going to be stating
very soon what we need.

264
00:17:45,700 --> 00:17:47,780
So, what you need
for this to work

265
00:17:47,780 --> 00:17:49,850
is that the function
goes to zero

266
00:17:49,850 --> 00:17:51,570
and the relative goes to zero.

267
00:17:51,570 --> 00:17:52,606
Yes.

268
00:17:52,606 --> 00:17:55,730
AUDIENCE: The potential
has to be real always?

269
00:17:55,730 --> 00:17:58,070
PROFESSOR: The potential
is real at this moment.

270
00:17:58,070 --> 00:17:59,250
Yes.

271
00:17:59,250 --> 00:18:01,550
For the discussion
that we're doing here,

272
00:18:01,550 --> 00:18:04,688
v is also a real number.

273
00:18:07,616 --> 00:18:09,292
AUDIENCE: So it
can't be complex?

274
00:18:09,292 --> 00:18:10,000
PROFESSOR: Sorry?

275
00:18:10,000 --> 00:18:11,380
AUDIENCE: Can it be complex?

276
00:18:11,380 --> 00:18:13,570
PROFESSOR: It could be
in certain applications

277
00:18:13,570 --> 00:18:15,880
for particles in
electromagnetic fields.

278
00:18:15,880 --> 00:18:17,650
You can have
something that looks

279
00:18:17,650 --> 00:18:19,830
like a complex Hamiltonian.

280
00:18:19,830 --> 00:18:22,890
So we will not discuss that
in this couple of lectures,

281
00:18:22,890 --> 00:18:25,400
but maybe later.

282
00:18:25,400 --> 00:18:25,900
Yes.

283
00:18:25,900 --> 00:18:27,316
AUDIENCE: Are there
any conditions

284
00:18:27,316 --> 00:18:31,610
that the potential has
to be time-dependent?

285
00:18:31,610 --> 00:18:34,280
PROFESSOR: Well, at this
moment, I put it time dependent.

286
00:18:34,280 --> 00:18:36,820
Also, it complicated
potentials, but they're

287
00:18:36,820 --> 00:18:38,400
sometimes necessary.

288
00:18:38,400 --> 00:18:41,150
And we will discuss
some of them.

289
00:18:41,150 --> 00:18:43,670
We will have very simple
time dependencies.

290
00:18:43,670 --> 00:18:47,410
Otherwise, it's difficult
to solve this equation.

291
00:18:47,410 --> 00:18:49,420
But very soon-- in
about five minutes,

292
00:18:49,420 --> 00:18:52,200
I will say-- let's consider
time-independent things

293
00:18:52,200 --> 00:18:55,470
to review the things that are a
little more basic and important

294
00:18:55,470 --> 00:18:59,790
and that you should
definitely remember well.

295
00:18:59,790 --> 00:19:06,000
OK, so that's this part of
the Schrodinger equation.

296
00:19:06,000 --> 00:19:08,560
I want to remind you of
another concept called

297
00:19:08,560 --> 00:19:10,890
the current--
probability current.

298
00:19:13,890 --> 00:19:17,980
Probability current.

299
00:19:17,980 --> 00:19:18,840
What is it?

300
00:19:18,840 --> 00:19:22,420
It's a j of x and
t-- that you will

301
00:19:22,420 --> 00:19:26,570
review in the homework--
is given by h over m,

302
00:19:26,570 --> 00:19:32,490
the imaginary part of
Psi star d Psi over dx.

303
00:19:32,490 --> 00:19:34,500
So, it's a real quantity.

304
00:19:37,320 --> 00:19:41,820
And it's called a
probability current.

305
00:19:41,820 --> 00:19:47,520
And it goes together with
this probability density,

306
00:19:47,520 --> 00:19:56,480
this probability density
that we wrote over here.

307
00:19:56,480 --> 00:20:00,260
So it's the current
associated to that density.

308
00:20:06,870 --> 00:20:09,480
Let's think a second
what this means.

309
00:20:09,480 --> 00:20:14,180
In electromagnetism, you have
currents and charged densities.

310
00:20:14,180 --> 00:20:19,580
So in E&M, you have a current.

311
00:20:19,580 --> 00:20:22,790
It's a vector and
a charged density.

312
00:20:22,790 --> 00:20:25,500
Now, this current
could also be a vector.

313
00:20:25,500 --> 00:20:27,870
If you're working in
more than one dimension,

314
00:20:27,870 --> 00:20:29,790
it would be a vector.

315
00:20:29,790 --> 00:20:31,730
But if you have
electromagnetism,

316
00:20:31,730 --> 00:20:36,040
the most famous thing associated
to electromagnetism currents

317
00:20:36,040 --> 00:20:39,800
and charged densities is the
so-called conservation law.

318
00:20:44,860 --> 00:20:47,490
This differential
equations satisfied

319
00:20:47,490 --> 00:20:49,950
by the current and the density.

320
00:20:49,950 --> 00:20:54,770
Divergence of j plus d
Rho dt is equal to zero.

321
00:20:54,770 --> 00:20:56,983
That means charge conservation.

322
00:20:59,560 --> 00:21:01,550
You may or may
not remember that.

323
00:21:01,550 --> 00:21:05,410
If you don't, it's a good
time to review it in E&M

324
00:21:05,410 --> 00:21:08,750
and check on that,
discuss it in recitation.

325
00:21:08,750 --> 00:21:10,960
Think about it.

326
00:21:10,960 --> 00:21:14,990
This means charge conservation
as we usually understand,

327
00:21:14,990 --> 00:21:18,610
and the way to do it--
I'm saying just in words--

328
00:21:18,610 --> 00:21:20,700
is you think of
a volume, you can

329
00:21:20,700 --> 00:21:23,450
see how much charge
is inside, and you

330
00:21:23,450 --> 00:21:26,070
see that the rate of
change of the charge

331
00:21:26,070 --> 00:21:30,750
is proportional to the current
that is escaping the volume.

332
00:21:30,750 --> 00:21:34,680
Which is to say, charge is
never destroyed or created.

333
00:21:34,680 --> 00:21:37,720
It can escape a volume,
because the charges are moving,

334
00:21:37,720 --> 00:21:41,140
but if it doesn't escape, well,
the charge remains the same.

335
00:21:41,140 --> 00:21:42,920
So, this is charge conservation.

336
00:21:42,920 --> 00:21:45,150
And this is the same thing.

337
00:21:45,150 --> 00:21:47,720
So the divergence
of j in this case

338
00:21:47,720 --> 00:21:55,740
reduces to dj dx plus
d Rho dt equals zero.

339
00:22:00,530 --> 00:22:03,250
It has a very similar
interpretation.

340
00:22:03,250 --> 00:22:06,500
So, perhaps in
equations, it's easier

341
00:22:06,500 --> 00:22:09,290
to think of interpretation.

342
00:22:09,290 --> 00:22:14,250
Consider the real line
and the point a and b,

343
00:22:14,250 --> 00:22:16,550
with a less than b.

344
00:22:16,550 --> 00:22:21,190
And define the
probability pab of t

345
00:22:21,190 --> 00:22:31,940
of finding the particle in
this interval between a and b

346
00:22:31,940 --> 00:22:32,675
at any time.

347
00:22:37,580 --> 00:22:42,620
You should be able to
show-- and it's again

348
00:22:42,620 --> 00:22:44,330
another thing to review.

349
00:22:44,330 --> 00:22:45,370
This you can review.

350
00:22:49,120 --> 00:22:50,795
And this review as well.

351
00:22:54,630 --> 00:22:58,360
You will use this differential
equation, things like that,

352
00:22:58,360 --> 00:23:06,620
to show that dpab dt-- the
rate at which the probability

353
00:23:06,620 --> 00:23:10,300
that you find the particle
in this interval changes

354
00:23:10,300 --> 00:23:12,780
depends on what the
current is doing here

355
00:23:12,780 --> 00:23:14,820
and what the current
is doing here.

356
00:23:14,820 --> 00:23:27,861
So, it's actually given by j of
a and t minus j at b at time t.

357
00:23:30,630 --> 00:23:38,740
You can show, and
please try to show it.

358
00:23:38,740 --> 00:23:41,210
So, what does that mean?

359
00:23:41,210 --> 00:23:43,730
You can have the particle
here at any time.

360
00:23:43,730 --> 00:23:46,900
But if you want to know how
the probability changing,

361
00:23:46,900 --> 00:23:50,010
you must see how
it's leaking from a

362
00:23:50,010 --> 00:23:51,570
or how it's leaking from b.

363
00:23:51,570 --> 00:23:54,740
Now j's are defined,
by convention,

364
00:23:54,740 --> 00:23:56,310
positive to the right.

365
00:23:56,310 --> 00:23:59,620
So, if there's a current--
a bit of current at a,

366
00:23:59,620 --> 00:24:01,370
it increases the probability.

367
00:24:01,370 --> 00:24:05,850
This particle is sort of
moving into the interval.

368
00:24:05,850 --> 00:24:09,544
And here at b, there's a
positive current decreases

369
00:24:09,544 --> 00:24:10,210
the probability.

370
00:24:16,040 --> 00:24:21,500
Finally, for wave functions,
the last thing we say

371
00:24:21,500 --> 00:24:27,110
is that these wave functions
are-- you want them normalized,

372
00:24:27,110 --> 00:24:28,900
but we can work with
them and they're

373
00:24:28,900 --> 00:24:32,940
physically equivalent if they
differ just by a constant.

374
00:24:32,940 --> 00:24:44,080
So Psi 1 and Psi 2 are
said to be equivalent

375
00:24:44,080 --> 00:24:51,610
if Psi 1 of x and t is equal to
some complex constant of Psi 2

376
00:24:51,610 --> 00:24:52,850
of x and t.

377
00:24:56,650 --> 00:25:00,540
Now, you would say,
well, I don't like that.

378
00:25:00,540 --> 00:25:02,760
I like normalized
wave functions,

379
00:25:02,760 --> 00:25:05,490
and you could have
a point there.

380
00:25:05,490 --> 00:25:08,645
But even if these are
normalized functions,

381
00:25:08,645 --> 00:25:11,110
they could differ by a phase.

382
00:25:11,110 --> 00:25:13,620
And they would still be
physically equivalent.

383
00:25:13,620 --> 00:25:17,940
This part of the definition
of the theory-- the definition

384
00:25:17,940 --> 00:25:21,440
of the theory is that these
wave functions are really

385
00:25:21,440 --> 00:25:25,370
physically equivalent
and indistinguishable.

386
00:25:25,370 --> 00:25:29,580
And that puts a constraint on
the way we define observables.

387
00:25:29,580 --> 00:25:32,380
Any observable should
have this property

388
00:25:32,380 --> 00:25:36,900
that, whether we used this
wave function or the other,

389
00:25:36,900 --> 00:25:40,660
they give you the
same observables.

390
00:25:40,660 --> 00:25:44,700
So, if your wave
functions are normalized,

391
00:25:44,700 --> 00:25:48,540
this can be complex
constant of length one.

392
00:25:48,540 --> 00:25:52,280
Then one normalized implies
the other is normalized.

393
00:25:52,280 --> 00:25:55,880
If they're not normalized, you
can say, look, the only reason

394
00:25:55,880 --> 00:25:59,610
I'm not normalizing
it because I don't

395
00:25:59,610 --> 00:26:03,000
gain all that much by
normalizing it, in fact.

396
00:26:03,000 --> 00:26:06,190
I can do almost everything
without normalizing the wave

397
00:26:06,190 --> 00:26:06,720
function.

398
00:26:06,720 --> 00:26:08,680
So, why should I bother?

399
00:26:08,680 --> 00:26:13,490
And we'll explain that
also as well very soon.

400
00:26:13,490 --> 00:26:15,890
So, this is something
that this part

401
00:26:15,890 --> 00:26:19,160
of the physical interpretation
that we should keep.

402
00:26:19,160 --> 00:26:23,200
So, now we've reviewed
the Schrodinger equation.

403
00:26:23,200 --> 00:26:27,430
Next thing we want to say is
the most important solutions

404
00:26:27,430 --> 00:26:31,590
of the Schrodinger equations
are those energy Eigenstates,

405
00:26:31,590 --> 00:26:33,110
stationary states.

406
00:26:33,110 --> 00:26:37,520
And let's just go
through that subject

407
00:26:37,520 --> 00:26:39,130
and explain what it was.

408
00:26:39,130 --> 00:26:42,560
So, I'm going to
start erasing here.

409
00:26:47,040 --> 00:26:51,530
So we're going to look at--
whoops-- stationary solutions.

410
00:26:55,910 --> 00:26:59,320
Now, I've used this
week wave function

411
00:26:59,320 --> 00:27:07,250
with a capital
Psi for a purpose,

412
00:27:07,250 --> 00:27:10,110
because I want to distinguish
it from another Psi

413
00:27:10,110 --> 00:27:12,700
that we're going to
encounter very soon.

414
00:27:12,700 --> 00:27:15,640
So, stationary solutions.

415
00:27:15,640 --> 00:27:22,370
And we'll take it-- from now
assume v is time-independent.

416
00:27:28,360 --> 00:27:30,460
The case is
sufficiently important

417
00:27:30,460 --> 00:27:34,530
that we may as well do it.

418
00:27:34,530 --> 00:27:38,240
So, in this case, the
Schrodinger equation

419
00:27:38,240 --> 00:27:44,020
is written as I h bar
d Psi dt, and we'll

420
00:27:44,020 --> 00:27:49,910
write it with something
called an h hat acting on Psi.

421
00:27:49,910 --> 00:27:53,010
And h hat at this
point is nothing else

422
00:27:53,010 --> 00:27:58,560
than minus h squared over 2m
second derivative with respect

423
00:27:58,560 --> 00:28:01,880
to x plus v of x.

424
00:28:01,880 --> 00:28:06,730
We say that h hat is an operator
acting on the wave function

425
00:28:06,730 --> 00:28:10,740
Psi on the right.

426
00:28:10,740 --> 00:28:13,200
Operator acting on that--
what does that mean?

427
00:28:13,200 --> 00:28:18,440
Basically, when we say an
operator acts on some space,

428
00:28:18,440 --> 00:28:21,530
we mean that it takes
elements of that space

429
00:28:21,530 --> 00:28:23,790
and moves them
around in the space.

430
00:28:23,790 --> 00:28:29,240
So, you've got a wave function,
which is a complex number that

431
00:28:29,240 --> 00:28:32,190
depends on x and t
ultimately, and then you

432
00:28:32,190 --> 00:28:36,170
act with this thing, which
involves taking derivatives,

433
00:28:36,170 --> 00:28:39,380
multiplying by v
of x, and you still

434
00:28:39,380 --> 00:28:43,420
got some complex
function of x and t.

435
00:28:43,420 --> 00:28:47,140
So, this is called the
Hamiltonian operator,

436
00:28:47,140 --> 00:28:49,390
and it's written like that.

437
00:28:49,390 --> 00:28:51,770
This Hamiltonian operator
is time-independent.

438
00:28:59,300 --> 00:29:03,190
So, what is a stationary state?

439
00:29:03,190 --> 00:29:08,970
A stationary state-- the way
it's defined is as follows.

440
00:29:08,970 --> 00:29:28,370
A stationary state of energy
e-- which is a real number--

441
00:29:28,370 --> 00:29:35,020
is a Psi of x and t
of the following form.

442
00:29:35,020 --> 00:29:36,470
It's a simple form.

443
00:29:36,470 --> 00:29:41,910
It's a pure exponential in
time times a function that

444
00:29:41,910 --> 00:29:44,340
just depends on x.

445
00:29:44,340 --> 00:29:46,715
So, it's a pretty simple object.

446
00:29:49,980 --> 00:29:51,130
So what is it?

447
00:29:51,130 --> 00:29:56,230
We say that this is
a stationary state.

448
00:29:56,230 --> 00:30:03,250
e to the minus i Et
over H bar Psi of x.

449
00:30:03,250 --> 00:30:08,560
And this Psi is in purpose
different from this Psi.

450
00:30:08,560 --> 00:30:10,740
It doesn't have the
bar at the bottom,

451
00:30:10,740 --> 00:30:12,540
and that signals
to you that that's

452
00:30:12,540 --> 00:30:14,000
the time-independent one.

453
00:30:14,000 --> 00:30:20,270
So this also belongs
to the complex numbers,

454
00:30:20,270 --> 00:30:23,910
but doesn't depend on time.

455
00:30:23,910 --> 00:30:30,220
So, it's called stationary
because, as it turns out,

456
00:30:30,220 --> 00:30:35,160
when we will compute expectation
values of any observable

457
00:30:35,160 --> 00:30:38,800
on this state, in
this stationary state,

458
00:30:38,800 --> 00:30:41,110
it will be time-independent.

459
00:30:41,110 --> 00:30:43,240
In particular, you
know, one thing

460
00:30:43,240 --> 00:30:48,070
that observable is the
probability density.

461
00:30:48,070 --> 00:30:53,260
And when you look at that,
you have Psi star and Psi.

462
00:30:53,260 --> 00:30:58,040
Since E is real,
this phase cancels--

463
00:30:58,040 --> 00:31:00,980
this is really a face,
because E is real.

464
00:31:00,980 --> 00:31:05,270
Therefore, Psi star Psi, the
e cancels, and all the time

465
00:31:05,270 --> 00:31:09,760
dependence cancels
and goes away.

466
00:31:09,760 --> 00:31:14,300
Same thing here for the j.

467
00:31:14,300 --> 00:31:17,760
The x derivative over here
it doesn't do anything

468
00:31:17,760 --> 00:31:19,180
to that phase.

469
00:31:19,180 --> 00:31:24,170
Therefore, the phase e
to the i Et over H bar

470
00:31:24,170 --> 00:31:25,660
cancels from there two.

471
00:31:25,660 --> 00:31:29,120
And the current also
has no time dependence.

472
00:31:29,120 --> 00:31:33,685
So, this will be the case
for any operator that

473
00:31:33,685 --> 00:31:37,180
is called a
time-independent operator.

474
00:31:37,180 --> 00:31:41,150
It will have time-independent
expectation values.

475
00:31:41,150 --> 00:31:45,420
So you can ask anything about
some familiar operator-- energy

476
00:31:45,420 --> 00:31:49,740
operator, momentum operator,
angular momentum operator-- all

477
00:31:49,740 --> 00:31:52,170
the famous operators
of quantum mechanics,

478
00:31:52,170 --> 00:31:56,330
and it will have real
expectation values.

479
00:31:56,330 --> 00:32:02,150
So, as you, you're
supposed to now plug this

480
00:32:02,150 --> 00:32:04,380
into this equation.

481
00:32:04,380 --> 00:32:07,500
And it's a famous result.

482
00:32:07,500 --> 00:32:10,370
Let's just do it.

483
00:32:10,370 --> 00:32:12,400
Plug back into the top equation.

484
00:32:12,400 --> 00:32:14,990
So, we have I H bar.

485
00:32:14,990 --> 00:32:18,620
The DET will only
act on the phase,

486
00:32:18,620 --> 00:32:28,080
because the Psi has
no time-dependence.

487
00:32:28,080 --> 00:32:31,575
And on the other hand,
on the right hand side,

488
00:32:31,575 --> 00:32:36,570
the H has nothing
to do with time,

489
00:32:36,570 --> 00:32:40,460
and therefore it can slide
through the exponential

490
00:32:40,460 --> 00:32:43,220
until it hits Psi.

491
00:32:43,220 --> 00:32:47,590
So here we have H-- well,
I'll put the exponential

492
00:32:47,590 --> 00:32:52,935
in front-- H on little Psi.

493
00:32:57,330 --> 00:33:01,630
So, we multiply here,
and what do we get?

494
00:33:01,630 --> 00:33:03,510
Well, the H bars cancel.

495
00:33:03,510 --> 00:33:05,910
The i at minus i gives you one.

496
00:33:05,910 --> 00:33:08,750
You get that E in front.

497
00:33:08,750 --> 00:33:13,510
So you get E times
this phase Psi of x.

498
00:33:13,510 --> 00:33:15,200
And the phase is
supposed to be here,

499
00:33:15,200 --> 00:33:18,250
but I cancel it with
this phase as well.

500
00:33:18,250 --> 00:33:22,125
And I get on the
right hand side H Psi.

501
00:33:22,125 --> 00:33:26,230
I will put it as
a left hand Psi.

502
00:33:26,230 --> 00:33:29,805
And this is the time-independent
Schrodinger equation.

503
00:33:34,180 --> 00:33:40,390
So far this is really
a simple matter.

504
00:33:40,390 --> 00:33:43,240
We've written a solution
that will represent

505
00:33:43,240 --> 00:33:46,570
the stationary state,
but then this energy

506
00:33:46,570 --> 00:33:50,340
should be such that you
can solve this equation.

507
00:33:50,340 --> 00:33:52,220
And as you've
learned before, it's

508
00:33:52,220 --> 00:33:55,670
something not so easy
to solve that equation.

509
00:33:55,670 --> 00:34:03,330
So what do we want to
say about this equation?

510
00:34:03,330 --> 00:34:07,540
Well, we have a lot to
say, and a few things

511
00:34:07,540 --> 00:34:11,679
will be pointed out now
that are very important.

512
00:34:11,679 --> 00:34:15,860
So, we have a
differential equation now.

513
00:34:15,860 --> 00:34:21,150
This differential equation has
second derivatives with respect

514
00:34:21,150 --> 00:34:21,780
to x.

515
00:34:21,780 --> 00:34:23,940
Now it has no time derivatives.

516
00:34:23,940 --> 00:34:26,929
The time has been factored out.

517
00:34:26,929 --> 00:34:29,530
Time is not a problem anymore.

518
00:34:29,530 --> 00:34:32,230
This equation, in
fact, looks quite

519
00:34:32,230 --> 00:34:37,969
real in that it seems that
Psi could even be real here.

520
00:34:37,969 --> 00:34:40,139
And in fact, yes,
there's no problem

521
00:34:40,139 --> 00:34:42,409
with this Psi being real.

522
00:34:42,409 --> 00:34:47,730
The total Psi just can't
be real in general.

523
00:34:47,730 --> 00:34:51,370
But this one can be
a real, and we'll

524
00:34:51,370 --> 00:34:54,409
consider those cases as well.

525
00:34:54,409 --> 00:34:57,480
So, things that
we want to say is

526
00:34:57,480 --> 00:35:00,180
that this is a second
order differential

527
00:35:00,180 --> 00:35:01,950
equations in space.

528
00:35:01,950 --> 00:35:12,470
So second order differential
equation in space.

529
00:35:12,470 --> 00:35:13,905
You could write it here.

530
00:35:13,905 --> 00:35:16,940
The H operator has
partial derivatives,

531
00:35:16,940 --> 00:35:20,540
but this time time,
you might as well

532
00:35:20,540 --> 00:35:24,680
say that this is minus
h squared over 2m.

533
00:35:24,680 --> 00:35:32,800
The second Psi vx squared
plus v of x tines Psi of x.

534
00:35:32,800 --> 00:35:36,360
Because Psi only depends
on x, might as well

535
00:35:36,360 --> 00:35:38,060
write it as complete derivative.

536
00:35:38,060 --> 00:35:43,080
So, second order
differential equation.

537
00:35:43,080 --> 00:35:46,790
And therefore, the
strategy for this equation

538
00:35:46,790 --> 00:35:48,970
is a little out there in
relation to the Schrodinger

539
00:35:48,970 --> 00:35:49,470
equation.

540
00:35:49,470 --> 00:35:51,280
We said, in the
Schrodinger equation,

541
00:35:51,280 --> 00:35:55,370
we know the wave function
everywhere, you know it later.

542
00:35:55,370 --> 00:36:00,360
Here, if you know it at one
point-- the wave function--

543
00:36:00,360 --> 00:36:04,060
and you know the derivative
at that one point,

544
00:36:04,060 --> 00:36:06,240
you have it everywhere.

545
00:36:06,240 --> 00:36:07,080
Why is that?

546
00:36:07,080 --> 00:36:09,530
Because that's how you solve
a differential equation.

547
00:36:09,530 --> 00:36:12,480
If you know the wave
function and the derivative

548
00:36:12,480 --> 00:36:14,450
at the point, you go to
the equation and say,

549
00:36:14,450 --> 00:36:19,100
I know the wave function and
I know the first derivative,

550
00:36:19,100 --> 00:36:21,150
and I know the
second derivative.

551
00:36:21,150 --> 00:36:25,330
So, a little later I can know
what the first derivative is,

552
00:36:25,330 --> 00:36:27,830
and if I know what the first
derivative is a little later,

553
00:36:27,830 --> 00:36:31,060
I can then know what the wave
function is a little later,

554
00:36:31,060 --> 00:36:33,230
and you just integrate
it numerically.

555
00:36:33,230 --> 00:36:39,300
So, you just need to know the
wave function Psi of x zero

556
00:36:39,300 --> 00:36:52,000
and Psi prime at x zero
suffice for a solution

557
00:36:52,000 --> 00:36:57,550
when v is regular.

558
00:36:57,550 --> 00:37:02,580
But this v is not too
complicated, or too strange,

559
00:37:02,580 --> 00:37:04,910
because you can always
find exceptions.

560
00:37:04,910 --> 00:37:07,160
You have the square
well potential,

561
00:37:07,160 --> 00:37:09,770
and you say, oh, I know
the wave function is here

562
00:37:09,770 --> 00:37:11,515
and its derivative is zero.

563
00:37:11,515 --> 00:37:12,890
Does that determine
the solution?

564
00:37:12,890 --> 00:37:15,220
No, because it's infinite.

565
00:37:15,220 --> 00:37:19,050
There's no space here, really,
and you should work here.

566
00:37:19,050 --> 00:37:24,460
So, basically, unless v
is really pathological,

567
00:37:24,460 --> 00:37:28,720
Psi and Psi prime are enough
to solve for everything.

568
00:37:28,720 --> 00:37:31,600
And that actually means
something very important,

569
00:37:31,600 --> 00:37:37,330
that if Psi is equal to zero
at x zero is equal to zero,

570
00:37:37,330 --> 00:37:41,830
and Psi prime at x zero
is equal to zero, then

571
00:37:41,830 --> 00:37:46,110
under these regular conditions,
Psi of all x is zero.

572
00:37:49,110 --> 00:37:51,115
Because you have a
differential equation which

573
00:37:51,115 --> 00:37:54,500
the initial value is zero,
the Psi prime is zero.

574
00:37:54,500 --> 00:37:56,650
And you go through
the equation, you

575
00:37:56,650 --> 00:37:59,800
see that every solution
has to be zero.

576
00:37:59,800 --> 00:38:02,290
It's the only possibility here.

577
00:38:02,290 --> 00:38:05,340
So what happens now is the
following-- that you have

578
00:38:05,340 --> 00:38:09,200
a physical understanding
that your wave function,

579
00:38:09,200 --> 00:38:13,740
when it becomes zero-- it may
do it slowly that it's becoming

580
00:38:13,740 --> 00:38:18,450
zero, but never quite being
zero-- but if it's zero,

581
00:38:18,450 --> 00:38:22,420
it does it with Psi prime
different from zero,

582
00:38:22,420 --> 00:38:26,790
so the wave function
is not zero all over.

583
00:38:26,790 --> 00:38:31,320
So, this is a pretty important
fact that is useful many times

584
00:38:31,320 --> 00:38:36,540
when you try to understand
the nature of solutions.

585
00:38:36,540 --> 00:38:38,710
So what else do we have here?

586
00:38:38,710 --> 00:38:42,830
Well, we have energy
Eigenstates on the spectrum.

587
00:38:48,130 --> 00:38:51,840
So, what is an
energy Eigenstate?

588
00:38:51,840 --> 00:38:53,975
Well, it's a solution
of this equation.

589
00:38:56,800 --> 00:39:05,480
So a solution Psi--
a solution for Psi

590
00:39:05,480 --> 00:39:09,260
is an energy Eigenstate.

591
00:39:13,990 --> 00:39:25,035
Then, this set of values
of E is this spectrum.

592
00:39:31,500 --> 00:39:34,800
And these two values
of E-- if there's

593
00:39:34,800 --> 00:39:38,500
a value of E that has
more than one solution,

594
00:39:38,500 --> 00:39:41,520
we say the spectrum
is degenerate.

595
00:39:41,520 --> 00:40:05,460
So a degenerate spectrum is
more than one Psi for a given E.

596
00:40:05,460 --> 00:40:11,340
So, these are just definitions,
but they're used all the time.

597
00:40:11,340 --> 00:40:15,200
So, our energy Eigenstates
are the solutions of this.

598
00:40:15,200 --> 00:40:18,120
The funny thing
about this equation

599
00:40:18,120 --> 00:40:21,370
is that sometimes
the requirement

600
00:40:21,370 --> 00:40:24,540
that Psi be normalized
means that you can't always

601
00:40:24,540 --> 00:40:27,030
find the solution
for any value of E.

602
00:40:27,030 --> 00:40:29,240
So, only specific
values of Es are

603
00:40:29,240 --> 00:40:32,360
allowed-- you know that for
the harmonic oscillator,

604
00:40:32,360 --> 00:40:34,180
for example-- and
therefore there's

605
00:40:34,180 --> 00:40:37,990
something called the spectrum,
which is the allowed values.

606
00:40:37,990 --> 00:40:40,010
And many times you
have degeneracies,

607
00:40:40,010 --> 00:40:42,260
and that makes for very
interesting physics.

608
00:40:50,700 --> 00:40:53,980
Let's say a couple more things
about the nature of this wave

609
00:40:53,980 --> 00:40:54,760
function.

610
00:40:54,760 --> 00:41:00,820
So, what kind of
potentials do we allow?

611
00:41:00,820 --> 00:41:08,745
We will allow potentials
that can fail to be bounded.

612
00:41:11,660 --> 00:41:13,540
What do we allow?

613
00:41:13,540 --> 00:41:17,710
We allow failure of continuity.

614
00:41:23,300 --> 00:41:26,890
Certainly, we must allow
that in our potentials

615
00:41:26,890 --> 00:41:28,870
that we consider,
because you have

616
00:41:28,870 --> 00:41:30,860
even the finite square well.

617
00:41:30,860 --> 00:41:34,250
The potential is not continuous.

618
00:41:34,250 --> 00:41:37,555
You can allow as well
failure to be bounded.

619
00:41:42,940 --> 00:41:45,100
So, what is a typical example?

620
00:41:45,100 --> 00:41:50,190
The harmonic oscillator,
the x squared potential.

621
00:41:50,190 --> 00:41:51,150
It's not bounded.

622
00:41:51,150 --> 00:41:54,340
It goes to infinity.

623
00:41:54,340 --> 00:41:57,680
So, we can fail
to be continuous,

624
00:41:57,680 --> 00:42:01,210
but we can fail at one
point, another point,

625
00:42:01,210 --> 00:42:03,770
but we shouldn't fail at
infinitely many points,

626
00:42:03,770 --> 00:42:05,300
presumably.

627
00:42:05,300 --> 00:42:07,280
So, it's piecewise continuous.

628
00:42:07,280 --> 00:42:14,270
It can fail to be bounded, and
it can include delta functions.

629
00:42:19,340 --> 00:42:23,540
Which is pretty interesting,
because a lot of physics

630
00:42:23,540 --> 00:42:26,770
uses delta functions,
but a delta function

631
00:42:26,770 --> 00:42:28,400
is a complicated thing.

632
00:42:28,400 --> 00:42:32,060
We'll include delta
functions but not

633
00:42:32,060 --> 00:42:36,410
derivatives of them, nor powers.

634
00:42:39,450 --> 00:42:44,330
So we won't take anything more
strange than delta functions,

635
00:42:44,330 --> 00:42:45,895
collections of delta functions.

636
00:42:48,920 --> 00:42:54,720
So, this is really how delicate
your potentials will be.

637
00:42:54,720 --> 00:42:57,840
They will not be more
complicated than that.

638
00:42:57,840 --> 00:43:04,260
But for that, we
will assume, and it

639
00:43:04,260 --> 00:43:07,340
will be completely
consistent to require

640
00:43:07,340 --> 00:43:09,590
the following for
the wave functions.

641
00:43:09,590 --> 00:43:21,500
So Psi is continuous-- Psi of
x-- is continuous and bounded.

642
00:43:25,760 --> 00:43:35,580
And its derivative is bounded.

643
00:43:35,580 --> 00:43:37,470
Psi prime is bounded.

644
00:43:46,270 --> 00:43:48,270
AUDIENCE: What about Psi's
behavior at infinity?

645
00:43:48,270 --> 00:43:49,087
PROFESSOR: Sorry?

646
00:43:49,087 --> 00:43:50,712
AUDIENCE: What kind
of extra conditions

647
00:43:50,712 --> 00:43:54,200
do we have to impose of
Psi's behavior at infinity?

648
00:43:54,200 --> 00:43:59,810
PROFESSOR: Well, I will not
impose any condition that

649
00:43:59,810 --> 00:44:02,054
is further than that,
except the condition

650
00:44:02,054 --> 00:44:03,345
that they've been normalizable.

651
00:44:05,930 --> 00:44:10,800
And even that we will
be a little-- how would

652
00:44:10,800 --> 00:44:14,350
I say, not too
demanding on that.

653
00:44:14,350 --> 00:44:16,040
Because there will
be wave functions,

654
00:44:16,040 --> 00:44:19,190
like momentum Eigenstates
that can't be normalized.

655
00:44:19,190 --> 00:44:21,250
So, we'll leave it at that.

656
00:44:21,250 --> 00:44:24,680
I think probably this is
what you should really box,

657
00:44:24,680 --> 00:44:27,910
because for a
momentum Eigenstate,

658
00:44:27,910 --> 00:44:31,675
e to the ipx over h bar.

659
00:44:34,460 --> 00:44:36,360
This is a momentum Eigenstate.

660
00:44:36,360 --> 00:44:38,050
This is continuous.

661
00:44:38,050 --> 00:44:39,090
It's bounded.

662
00:44:39,090 --> 00:44:40,560
The derivative is bounded.

663
00:44:40,560 --> 00:44:43,460
It is not normalizable,
but it's so useful

664
00:44:43,460 --> 00:44:48,470
that we must include in the
list of things that we allow.

665
00:44:48,470 --> 00:44:51,260
So, bound states
and non-bound states

666
00:44:51,260 --> 00:44:52,950
are things that are
not normalizable.

667
00:44:52,950 --> 00:44:54,540
So, I don't put normalization.

668
00:44:54,540 --> 00:44:58,420
Now, if you put normalization,
then the wave function

669
00:44:58,420 --> 00:45:01,280
will go to zero at infinity.

670
00:45:01,280 --> 00:45:03,880
And that's all you
would want to impose.

671
00:45:03,880 --> 00:45:04,660
Nothing else.

672
00:45:04,660 --> 00:45:07,440
So, really in some
sense, this is it.

673
00:45:07,440 --> 00:45:09,072
You don't want more than that.

674
00:45:09,072 --> 00:45:11,030
AUDIENCE: Is normalization
sufficient to ensure

675
00:45:11,030 --> 00:45:13,192
the derivative also goes
to zero at infinity?

676
00:45:13,192 --> 00:45:13,574
PROFESSOR: Sorry?

677
00:45:13,574 --> 00:45:15,990
AUDIENCE: Is normalization
sufficient to ensure that the--

678
00:45:15,990 --> 00:45:17,220
PROFESSOR: Not that I know.

679
00:45:17,220 --> 00:45:18,190
I don't think so.

680
00:45:18,190 --> 00:45:19,940
AUDIENCE: Then why is
integration by price

681
00:45:19,940 --> 00:45:21,840
generically valid?

682
00:45:21,840 --> 00:45:24,590
PROFESSOR: It's probably
valid for restricted kinds

683
00:45:24,590 --> 00:45:26,470
of potentials.

684
00:45:26,470 --> 00:45:28,250
So you could not
prove it in general.

685
00:45:31,150 --> 00:45:33,680
So, you know,
there may be things

686
00:45:33,680 --> 00:45:36,800
that one can generalize and
be a little more general,

687
00:45:36,800 --> 00:45:38,860
but I'm trying to
be conservative.

688
00:45:38,860 --> 00:45:41,810
I know that for any
decent potential--

689
00:45:41,810 --> 00:45:45,630
and we definitely need
Psi prime bounded.

690
00:45:45,630 --> 00:45:50,000
And wave functions that
go to zero, the only ones

691
00:45:50,000 --> 00:45:52,730
I know that also have
Psi prime going to zero.

692
00:45:52,730 --> 00:45:55,840
But I don't think it's easy
to prove that's generic,

693
00:45:55,840 --> 00:45:59,000
unless you make
more assumptions.

694
00:45:59,000 --> 00:46:03,510
So, all right.

695
00:46:03,510 --> 00:46:05,810
So, this we'll have
for our wave functions,

696
00:46:05,810 --> 00:46:09,290
and now I want to say
a couple of things

697
00:46:09,290 --> 00:46:12,180
about properties
of the Eigenstates.

698
00:46:12,180 --> 00:46:16,370
Now, we will calculate
many of these Eigenstates,

699
00:46:16,370 --> 00:46:19,900
but we need to understand
some of the basic properties

700
00:46:19,900 --> 00:46:20,990
that they have.

701
00:46:20,990 --> 00:46:25,080
And there's really two
types of identities

702
00:46:25,080 --> 00:46:28,290
that I want you to be
very aware that they

703
00:46:28,290 --> 00:46:33,370
play some sort of dual role-- a
pretty interesting dual role--

704
00:46:33,370 --> 00:46:36,420
that has to do with
these wave functions.

705
00:46:36,420 --> 00:46:44,050
So, the Eigenstates
of-- Eigenstates of H

706
00:46:44,050 --> 00:46:46,440
hat-- these are the
energy Eigenstates.

707
00:46:48,970 --> 00:46:56,440
you can consider them
and make a list of them.

708
00:46:56,440 --> 00:47:02,690
So, you have an energy E zero
less than or equal an E 1, E 2.

709
00:47:02,690 --> 00:47:03,860
Just goes like that.

710
00:47:03,860 --> 00:47:07,450
And you have a Psi zero, Psi 1.

711
00:47:07,450 --> 00:47:09,080
All this wave functions.

712
00:47:09,080 --> 00:47:15,250
And then H hat Psi N
is equal to E N Psi

713
00:47:15,250 --> 00:47:18,596
N. You have a set of solutions.

714
00:47:22,090 --> 00:47:26,595
So, this is what will happen
if you have a good problem.

715
00:47:26,595 --> 00:47:32,470
A reasonable potential, and
nothing terribly strange

716
00:47:32,470 --> 00:47:33,550
going on.

717
00:47:33,550 --> 00:47:36,100
There would be a
lot of solutions,

718
00:47:36,100 --> 00:47:41,030
and they can be chosen
to be orthonormal.

719
00:47:41,030 --> 00:47:49,240
Now at first sight, it's a
funny term to use-- orthonormal.

720
00:47:49,240 --> 00:47:52,430
This is a term that
we use for vectors.

721
00:47:52,430 --> 00:47:54,930
Two vectors are
orthogonal, and we

722
00:47:54,930 --> 00:47:58,231
say they're orthonormal if they
have unit length, and things

723
00:47:58,231 --> 00:47:58,730
like that.

724
00:47:58,730 --> 00:48:05,540
But what do we mean the two
functions are orthonormal?

725
00:48:05,540 --> 00:48:09,140
Well, our function's vectors.

726
00:48:09,140 --> 00:48:13,030
Well, that's a little dubious.

727
00:48:13,030 --> 00:48:16,530
But the way we will think
in quantum mechanics

728
00:48:16,530 --> 00:48:18,820
is that, in some
sense, functions

729
00:48:18,820 --> 00:48:22,930
are vectors in an infinite
dimensional space.

730
00:48:22,930 --> 00:48:26,080
So, they're just vectors,
but not in three dimensions.

731
00:48:26,080 --> 00:48:26,980
Why?

732
00:48:26,980 --> 00:48:28,090
Think of it.

733
00:48:28,090 --> 00:48:31,630
If you have a function,
you have to give values--

734
00:48:31,630 --> 00:48:35,710
independent values-- at many
points-- Infinitely many.

735
00:48:35,710 --> 00:48:37,150
And if you give
all those values,

736
00:48:37,150 --> 00:48:38,150
you've got the function.

737
00:48:38,150 --> 00:48:40,647
If you have a vector, you
have to give components,

738
00:48:40,647 --> 00:48:41,730
and you've got the vector.

739
00:48:41,730 --> 00:48:44,180
So, in a sense, to
give a function,

740
00:48:44,180 --> 00:48:45,810
I have to give a lot of numbers.

741
00:48:45,810 --> 00:48:48,750
And I can say the
first vector is

742
00:48:48,750 --> 00:48:52,190
the value along the direction--
the first component is

743
00:48:52,190 --> 00:48:54,320
the value around zero.

744
00:48:54,320 --> 00:49:00,430
The second unit vector is the
value of about 0.01, 9.02,

745
00:49:00,430 --> 00:49:01,490
going on and on.

746
00:49:01,490 --> 00:49:03,240
And then list of
them, and you have

747
00:49:03,240 --> 00:49:06,240
a vector of infinite dimensions.

748
00:49:06,240 --> 00:49:09,480
You say, totally
useless. [LAUGHTER] No,

749
00:49:09,480 --> 00:49:11,290
it's not totally useless.

750
00:49:11,290 --> 00:49:13,810
Actually, if you
visualize that--

751
00:49:13,810 --> 00:49:16,440
and we'll do it
later more-- you will

752
00:49:16,440 --> 00:49:21,120
be able to understand many
formulas as natural extensions.

753
00:49:21,120 --> 00:49:27,250
So, what does it mean that these
two functions are orthonormal?

754
00:49:27,250 --> 00:49:31,490
Well, a dot product,
or orthonormality,

755
00:49:31,490 --> 00:49:33,715
is to say that the
dot product is zero.

756
00:49:33,715 --> 00:49:41,060
And the way we dot product
functions Psi m and Psi n of x

757
00:49:41,060 --> 00:49:46,690
is we take their values at
the same point with star one,

758
00:49:46,690 --> 00:49:47,455
and we integrate.

759
00:49:51,000 --> 00:49:55,830
And, if this is
equal to delta mn,

760
00:49:55,830 --> 00:49:57,915
we say the functions
are orthonormal.

761
00:50:05,810 --> 00:50:09,720
So, ortho, for
orthogonal, which says

762
00:50:09,720 --> 00:50:13,420
that if m is different from
n, the Kronecker delta,

763
00:50:13,420 --> 00:50:16,505
that symbol is equal to
one if the two labels

764
00:50:16,505 --> 00:50:19,560
are the same, or zero otherwise.

765
00:50:19,560 --> 00:50:21,580
If they're different,
you get zero.

766
00:50:21,580 --> 00:50:23,620
The inner product--
this left hand

767
00:50:23,620 --> 00:50:26,150
side is called the
inner product-- is zero.

768
00:50:26,150 --> 00:50:31,090
On the other hand, if they are
the same, if m is equal to n,

769
00:50:31,090 --> 00:50:33,650
it says that the
Psi squared is one.

770
00:50:33,650 --> 00:50:36,420
Kind of like a wave function
that is well normalized.

771
00:50:36,420 --> 00:50:40,560
So we say normal
for orthonormal.

772
00:50:40,560 --> 00:50:44,220
So these are orthonormal wave
functions, and that's good.

773
00:50:44,220 --> 00:50:46,540
This is called orthonormality.

774
00:50:46,540 --> 00:50:50,010
But then there is a
more subtle property,

775
00:50:50,010 --> 00:50:53,650
which is that this
set of functions

776
00:50:53,650 --> 00:50:58,770
is enough to expand any
function in this interval

777
00:50:58,770 --> 00:51:00,560
that you're doing your
quantum mechanics.

778
00:51:00,560 --> 00:51:03,245
So, if you have any
reasonable function,

779
00:51:03,245 --> 00:51:08,750
it can be written as a
superposition of these ones.

780
00:51:08,750 --> 00:51:12,320
So, this differential
equation furnishes for you

781
00:51:12,320 --> 00:51:15,280
a collection of functions
that are very useful.

782
00:51:15,280 --> 00:51:17,080
So this is orthonormality.

783
00:51:17,080 --> 00:51:20,550
This is also
completeness, which is

784
00:51:20,550 --> 00:51:26,780
to say that any
function can be written

785
00:51:26,780 --> 00:51:29,780
as a sum of of this function.

786
00:51:29,780 --> 00:51:31,280
So I will write it as this.

787
00:51:31,280 --> 00:51:34,740
Psi of x-- an
arbitrary Psi of x--

788
00:51:34,740 --> 00:51:39,630
can be written as
bm Psi n of x n

789
00:51:39,630 --> 00:51:43,945
equals zero to infinity,
where the bn's are complex.

790
00:51:50,950 --> 00:51:56,980
So, this is an assumption, but
it's a very solid assumption.

791
00:51:56,980 --> 00:51:59,900
When you study differential
equations of this type--

792
00:51:59,900 --> 00:52:01,790
Sturm-Liouville
problem-- this is

793
00:52:01,790 --> 00:52:04,680
one thing that
mathematicians prove for you,

794
00:52:04,680 --> 00:52:06,700
and it's not all that easy.

795
00:52:06,700 --> 00:52:08,570
But the collection
of wave functions

796
00:52:08,570 --> 00:52:10,140
is good in this sense.

797
00:52:10,140 --> 00:52:13,690
It provides you a
complete set of things

798
00:52:13,690 --> 00:52:17,080
that any function can be
written in terms of that.

799
00:52:17,080 --> 00:52:21,812
I'm not saying this satisfies
any particular equation.

800
00:52:21,812 --> 00:52:26,050
You see, this function satisfies
very particular equations--

801
00:52:26,050 --> 00:52:29,900
those equations-- but this
is an arbitrary function.

802
00:52:29,900 --> 00:52:32,470
And it can be written
as a sum of this.

803
00:52:32,470 --> 00:52:35,570
See, these equations
have different energies

804
00:52:35,570 --> 00:52:37,760
for different Psi's.

805
00:52:37,760 --> 00:52:42,380
This Psi here satisfies
no obvious equation.

806
00:52:42,380 --> 00:52:47,790
But here is a problem
that this is useful for.

807
00:52:47,790 --> 00:52:54,840
Suppose you're given a wave
function at, at the given time,

808
00:52:54,840 --> 00:52:57,290
you know what it looks like.

809
00:52:57,290 --> 00:53:00,865
So, here is your wave function.

810
00:53:04,190 --> 00:53:06,600
Psi.

811
00:53:06,600 --> 00:53:10,520
And you know that
Psi at x and time

812
00:53:10,520 --> 00:53:15,430
equals to zero happens to
be equal to this Psi of x

813
00:53:15,430 --> 00:53:16,590
that we wrote above.

814
00:53:16,590 --> 00:53:20,707
So, it's equal to bn Psi n of x.

815
00:53:26,680 --> 00:53:33,910
Well, if you know that, if you
can calculate this coefficient,

816
00:53:33,910 --> 00:53:36,610
the wave function of time
equals zero is known,

817
00:53:36,610 --> 00:53:39,470
say, and it was given
by this thing, which

818
00:53:39,470 --> 00:53:41,780
is then written in this form.

819
00:53:41,780 --> 00:53:44,420
If you can write
it in this form,

820
00:53:44,420 --> 00:53:47,630
you've solved the problem
of time evolution,

821
00:53:47,630 --> 00:53:53,190
because then Psi
of x at any time

822
00:53:53,190 --> 00:54:00,440
is just simply obtained by
evolving each component.

823
00:54:00,440 --> 00:54:10,910
Which is bn e to the minus
iEnt over h bar Psi n of x.

824
00:54:10,910 --> 00:54:12,530
So this is the important result.

825
00:54:16,870 --> 00:54:20,720
Now, look what has happened.

826
00:54:20,720 --> 00:54:23,570
We have replaced each term.

827
00:54:23,570 --> 00:54:25,420
We added this exponential.

828
00:54:25,420 --> 00:54:26,900
Why?

829
00:54:26,900 --> 00:54:31,240
Because then each one
of these is a solution

830
00:54:31,240 --> 00:54:34,880
of the full
Schrodinger equation.

831
00:54:34,880 --> 00:54:38,170
And therefore a superposition
with complex coefficients

832
00:54:38,170 --> 00:54:41,660
is still a solution of
the Schrodinger equation.

833
00:54:41,660 --> 00:54:44,280
Therefore, this thing
I've put by hand

834
00:54:44,280 --> 00:54:46,580
is, you would say it's ad hoc.

835
00:54:46,580 --> 00:54:48,440
No, it's not.

836
00:54:48,440 --> 00:54:51,030
We've put it by
hand, yes, but we've

837
00:54:51,030 --> 00:54:54,000
produced a solution of the
Schrodinger equation, which

838
00:54:54,000 --> 00:54:55,740
has another virtue.

839
00:54:55,740 --> 00:54:59,040
When t is equal to
zero, it becomes

840
00:54:59,040 --> 00:55:01,490
what you know the
wave function is.

841
00:55:01,490 --> 00:55:04,340
So, since this solves the
Schrodinger equation--

842
00:55:04,340 --> 00:55:06,810
time equals zero gives
you the right answer.

843
00:55:06,810 --> 00:55:08,890
And you remember that
the Schrodinger equation,

844
00:55:08,890 --> 00:55:10,830
if you know that
time equals zero,

845
00:55:10,830 --> 00:55:12,970
the term is a wave
function everywhere--

846
00:55:12,970 --> 00:55:15,100
this is the solution.

847
00:55:15,100 --> 00:55:16,860
It's not just a solution.

848
00:55:16,860 --> 00:55:19,780
It's the solution.

849
00:55:19,780 --> 00:55:23,250
So, you've solved this equation,
and it's a very nice thing.

850
00:55:23,250 --> 00:55:25,360
It all depends, of
course, on having

851
00:55:25,360 --> 00:55:27,660
found the coefficients bn.

852
00:55:27,660 --> 00:55:29,930
Because typically
at time equals zero,

853
00:55:29,930 --> 00:55:32,560
you may know what
the wave function is,

854
00:55:32,560 --> 00:55:34,440
but you may not
know how to write it

855
00:55:34,440 --> 00:55:37,780
in terms of these
coefficients bn.

856
00:55:37,780 --> 00:55:39,610
So, what do you do then?

857
00:55:39,610 --> 00:55:41,770
If you don't know
those coefficients,

858
00:55:41,770 --> 00:55:43,650
you can calculate them.

859
00:55:43,650 --> 00:55:46,550
How do you calculate them?

860
00:55:46,550 --> 00:55:50,250
Well, you use orthonormality.

861
00:55:50,250 --> 00:55:57,120
So you actually take this and
integrate against another Psi

862
00:55:57,120 --> 00:55:58,550
star.

863
00:55:58,550 --> 00:56:02,640
So you take a Psi star
sub m and integrate--

864
00:56:02,640 --> 00:56:04,340
multiply and integrate.

865
00:56:04,340 --> 00:56:08,860
And then the right hand side
will get the Kronecker delta

866
00:56:08,860 --> 00:56:10,760
that will pick out one term.

867
00:56:10,760 --> 00:56:14,830
So, I'm just saying in
words a two line calculation

868
00:56:14,830 --> 00:56:19,940
that you should do if you
don't see this as obvious.

869
00:56:19,940 --> 00:56:22,330
Because it's a
kind of calculation

870
00:56:22,330 --> 00:56:23,920
that you do a few times in life.

871
00:56:23,920 --> 00:56:28,360
Then it becomes obvious
and you never do it again.

872
00:56:28,360 --> 00:56:38,710
It's minus infinity to infinity
dx Psi m star of x Psi of x dx.

873
00:56:38,710 --> 00:56:42,250
So, bm is given
by this quantity,

874
00:56:42,250 --> 00:56:45,360
or bn is given by this quantity.

875
00:56:45,360 --> 00:56:50,440
You obtain it from here
plus orthonormality.

876
00:56:50,440 --> 00:56:52,510
So, once you have
this bn, you can

877
00:56:52,510 --> 00:56:57,210
do something that may-- if
you look at these things

878
00:56:57,210 --> 00:57:00,940
and say, well, I'm
bored, what should I do?

879
00:57:00,940 --> 00:57:02,070
I say, well, you have bm.

880
00:57:02,070 --> 00:57:03,380
Plug it back.

881
00:57:03,380 --> 00:57:06,740
What happens then?

882
00:57:06,740 --> 00:57:09,120
You say, why would
I plug it back?

883
00:57:09,120 --> 00:57:10,800
I don't need to plug it back.

884
00:57:10,800 --> 00:57:16,020
And that's true, but it's
not a crazy thing to do,

885
00:57:16,020 --> 00:57:20,540
because it somehow must
lead to some identity.

886
00:57:20,540 --> 00:57:23,380
Because you solve an equation
and then plug it back

887
00:57:23,380 --> 00:57:29,710
and try to see if
somehow it makes sense.

888
00:57:29,710 --> 00:57:33,120
So either it makes sense, or
you learned something new.

889
00:57:33,120 --> 00:57:37,580
So, we were supposed
to calculate the bn's.

890
00:57:37,580 --> 00:57:41,350
And now we have them, so
I can plug this back here.

891
00:57:41,350 --> 00:57:48,780
So what do I Get Psi of x now is
equal to the sum from n equals

892
00:57:48,780 --> 00:57:51,020
zero to infinity.

893
00:57:51,020 --> 00:57:58,815
bn-- but this bn is the integral
of Psi n star of x prime.

894
00:57:58,815 --> 00:58:01,500
I put here Psi of x prime.

895
00:58:01,500 --> 00:58:03,240
dx prime.

896
00:58:03,240 --> 00:58:05,840
I don't want to confuse
the x's with x prime,

897
00:58:05,840 --> 00:58:10,460
so I should put the x
primes all over here.

898
00:58:10,460 --> 00:58:12,580
Psi n of x.

899
00:58:15,430 --> 00:58:17,540
Well, can I do the integral?

900
00:58:17,540 --> 00:58:19,290
No.

901
00:58:19,290 --> 00:58:24,060
So, have I gained anything?

902
00:58:24,060 --> 00:58:27,760
Well, you've gained something
if you write it in a way

903
00:58:27,760 --> 00:58:35,910
that Psi is equal to
something times Psi.

904
00:58:35,910 --> 00:58:37,790
That doesn't look
all that simple,

905
00:58:37,790 --> 00:58:41,750
but we can at least organize it.

906
00:58:41,750 --> 00:58:46,170
Let's assume things
are convergent enough

907
00:58:46,170 --> 00:58:48,860
that you can change orders
of sums and integrals.

908
00:58:48,860 --> 00:58:52,840
That's an assumption
we always do.

909
00:58:52,840 --> 00:58:53,890
I'll write it like this.

910
00:58:53,890 --> 00:58:54,920
dx prime.

911
00:58:54,920 --> 00:58:59,570
And now I'll put the
sum here equals zero

912
00:58:59,570 --> 00:59:06,170
to infinity of Psi
n star of x prime.

913
00:59:06,170 --> 00:59:08,830
And I'll put the other
Psi here as well.

914
00:59:08,830 --> 00:59:13,390
The Psi n of x over here.

915
00:59:13,390 --> 00:59:18,400
I'll put the parentheses, and
finally the Psi of x prime

916
00:59:18,400 --> 00:59:18,900
here.

917
00:59:23,530 --> 00:59:27,030
So, now it's put in a nice way.

918
00:59:27,030 --> 00:59:28,840
And it's a nice way
because it allows

919
00:59:28,840 --> 00:59:33,070
you to learn something new
and interesting about this.

920
00:59:33,070 --> 00:59:35,460
And what is that?

921
00:59:35,460 --> 00:59:40,790
That this must be a
very peculiar function,

922
00:59:40,790 --> 00:59:45,670
such that integrated
against Psi gives you Psi.

923
00:59:45,670 --> 00:59:48,540
And what could it be?

924
00:59:48,540 --> 00:59:51,210
Well, this is of
the form, if you

925
00:59:51,210 --> 00:59:54,770
wish-- the x prime--
some function

926
00:59:54,770 --> 01:00:00,670
of x and x prime--
times Psi of x prime.

927
01:00:00,670 --> 01:00:02,850
So, this k is this thing.

928
01:00:08,540 --> 01:00:12,340
Well, you can try to
think what this is.

929
01:00:12,340 --> 01:00:14,230
If you put the delta
function here--

930
01:00:14,230 --> 01:00:16,020
which may be a little
bit of a cheat--

931
01:00:16,020 --> 01:00:18,540
you will figure out
the right answer.

932
01:00:18,540 --> 01:00:23,200
This must be a function
that sort of picks out

933
01:00:23,200 --> 01:00:26,710
the value of the function
at x by integrating.

934
01:00:26,710 --> 01:00:29,800
So it only cares
about the value at x.

935
01:00:29,800 --> 01:00:32,290
So, it must be a delta function.

936
01:00:32,290 --> 01:00:38,280
So, in fact, this
is a delta function,

937
01:00:38,280 --> 01:00:41,080
or should be a delta function.

938
01:00:41,080 --> 01:00:46,050
And therefore the
claim is that we now

939
01:00:46,050 --> 01:00:52,070
have a very curious identity
that looks as follows.

940
01:00:52,070 --> 01:00:57,750
It looks like n equal zero
to infinity, Psi n star

941
01:00:57,750 --> 01:01:03,460
of x prime Psi n
of x is actually

942
01:01:03,460 --> 01:01:06,385
delta of x minus x prime.

943
01:01:10,170 --> 01:01:11,850
So, this must be true.

944
01:01:11,850 --> 01:01:15,560
If what we said at
the beginning is true,

945
01:01:15,560 --> 01:01:21,660
that you can expand any function
in terms of the Eigenfunctions,

946
01:01:21,660 --> 01:01:25,630
then, well, that's not
such a trivial assumption.

947
01:01:25,630 --> 01:01:30,370
And therefore, it allows you
to prove something fairly

948
01:01:30,370 --> 01:01:34,050
surprising, that
this must be true,

949
01:01:34,050 --> 01:01:36,750
that this identity must be true.

950
01:01:36,750 --> 01:01:41,880
And I want you to realize
and compare and contrast

951
01:01:41,880 --> 01:01:44,450
with this identity here.

952
01:01:44,450 --> 01:01:47,400
One is completeness.

953
01:01:47,400 --> 01:01:48,333
One is orthonormality.

954
01:01:51,530 --> 01:01:55,490
There are two kinds
of sums going on here.

955
01:01:55,490 --> 01:02:05,250
Here is sum over space, and
you keep labels arbitrary--

956
01:02:05,250 --> 01:02:07,810
label indices arbitrary.

957
01:02:07,810 --> 01:02:09,430
So, sum over space.

958
01:02:09,430 --> 01:02:13,660
These functions depend
on space and on labels.

959
01:02:13,660 --> 01:02:16,140
Sum over space, and
keep the labels,

960
01:02:16,140 --> 01:02:20,710
and you get sort of a
unit matrix in this space,

961
01:02:20,710 --> 01:02:22,930
in the space of labels.

962
01:02:22,930 --> 01:02:30,540
Here, you keep the positions
arbitrary, but sum over labels.

963
01:02:30,540 --> 01:02:33,030
And now you get
like a unit matrix

964
01:02:33,030 --> 01:02:35,530
in the space of positions.

965
01:02:35,530 --> 01:02:38,450
Something is one--
but actually infinite,

966
01:02:38,450 --> 01:02:42,640
but you couldn't do better--
when x is equal to x prime.

967
01:02:42,640 --> 01:02:44,580
So, if you think
of it as a matrix,

968
01:02:44,580 --> 01:02:49,310
this function in x and x prime
is a very strange matrix,

969
01:02:49,310 --> 01:02:52,410
with two indices, x and x prime.

970
01:02:52,410 --> 01:02:55,050
And when x is different
from x prime, it's zero,

971
01:02:55,050 --> 01:02:57,250
but when x is equal
to x prime, it's one.

972
01:02:57,250 --> 01:02:59,510
But it has to be
a delta function,

973
01:02:59,510 --> 01:03:01,820
because continuous variables.

974
01:03:01,820 --> 01:03:03,480
But it's the same idea.

975
01:03:03,480 --> 01:03:08,090
So, actually if you think
of these two things, x and m

976
01:03:08,090 --> 01:03:15,010
as dual variables, this
is a matrix variable,

977
01:03:15,010 --> 01:03:19,700
and then you're sort of
keeping these two indices open

978
01:03:19,700 --> 01:03:21,660
and summing over
the other index.

979
01:03:21,660 --> 01:03:24,500
Multiplying in one way
you get a unit matrix.

980
01:03:24,500 --> 01:03:27,340
Here, you do the
other way around.

981
01:03:27,340 --> 01:03:30,250
You have a matrix in m and n.

982
01:03:30,250 --> 01:03:32,680
This is a more familiar
matrix, but then you

983
01:03:32,680 --> 01:03:35,100
sum over the other things.

984
01:03:35,100 --> 01:03:38,780
So, they're dual,
and two properties

985
01:03:38,780 --> 01:03:44,840
that look very different in the
way you express them in words.

986
01:03:44,840 --> 01:03:47,380
One is that they're orthonormal.

987
01:03:47,380 --> 01:03:49,350
The other is that
they're complete.

988
01:03:49,350 --> 01:03:51,280
And then suddenly
then the mathematics

989
01:03:51,280 --> 01:03:56,370
tells you there's a nice
duality between them.

990
01:03:56,370 --> 01:04:00,350
So, the last thing
I want to say today

991
01:04:00,350 --> 01:04:03,830
is about expectation
values, which

992
01:04:03,830 --> 01:04:11,150
is another concept we
have to review and recall.

993
01:04:11,150 --> 01:04:14,300
So let's give those ideas.

994
01:04:14,300 --> 01:04:22,265
So, if we have a
time-dependent operator-- no,

995
01:04:22,265 --> 01:04:25,350
a time independent-- we'll do
a time-independent operator,

996
01:04:25,350 --> 01:04:26,175
I'm sorry.

997
01:04:26,175 --> 01:04:31,280
Time-Independent operator.

998
01:04:37,050 --> 01:04:40,500
And this operator
will be called A hat.

999
01:04:40,500 --> 01:04:43,650
No time dependence
on the operator.

1000
01:04:43,650 --> 01:04:48,960
So, then we have the
expectation value

1001
01:04:48,960 --> 01:04:52,040
of this operator on
a normalized state.

1002
01:04:52,040 --> 01:04:53,590
So what does that mean?

1003
01:04:53,590 --> 01:05:01,330
The expectation value of this
operator on a state-- on a wave

1004
01:05:01,330 --> 01:05:03,100
function here.

1005
01:05:03,100 --> 01:05:05,880
Now, this wave function
is time-dependent.

1006
01:05:05,880 --> 01:05:10,880
So this expectation
value of this operator

1007
01:05:10,880 --> 01:05:14,830
is expected to be
a function of time.

1008
01:05:14,830 --> 01:05:17,180
And how is it defined?

1009
01:05:17,180 --> 01:05:20,610
It's defined by doing
the following integral.

1010
01:05:20,610 --> 01:05:23,870
Again, from minus
infinity to infinity,

1011
01:05:23,870 --> 01:05:31,220
dx Psi star of x and t,
and then the operator

1012
01:05:31,220 --> 01:05:34,780
A acting on Psi of x and t.

1013
01:05:40,980 --> 01:05:44,240
And Psi is supposed to
be a normalized state.

1014
01:05:53,780 --> 01:05:57,920
So, notice the notation here.

1015
01:05:57,920 --> 01:06:01,540
We put the Psi here because
of the expectation-- whenever

1016
01:06:01,540 --> 01:06:03,000
somebody asks you
the expectation

1017
01:06:03,000 --> 01:06:07,120
value for an operator, it
has to be on a given state.

1018
01:06:07,120 --> 01:06:08,500
So you put the state.

1019
01:06:08,500 --> 01:06:12,050
Then you realize that this is
a time-dependent wave function

1020
01:06:12,050 --> 01:06:15,400
typically, so it
could depend on time.

1021
01:06:15,400 --> 01:06:18,030
Now, we said about
stationary states

1022
01:06:18,030 --> 01:06:20,520
that if the state is
stationary, there's

1023
01:06:20,520 --> 01:06:22,840
a single time exponential here.

1024
01:06:22,840 --> 01:06:28,430
There's just one term, e to
the minus iEt over h bar.

1025
01:06:28,430 --> 01:06:32,270
And if A, of course, is a
time-independent operator,

1026
01:06:32,270 --> 01:06:34,150
you won't care about
the exponential.

1027
01:06:34,150 --> 01:06:36,600
You will cancel this
one, and there will not

1028
01:06:36,600 --> 01:06:39,110
be a time dependence there.

1029
01:06:39,110 --> 01:06:42,360
But if this state
is not stationary--

1030
01:06:42,360 --> 01:06:45,540
like most states
are not stationary--

1031
01:06:45,540 --> 01:06:47,400
remember it's very important.

1032
01:06:47,400 --> 01:06:50,160
If you have a stationary
state, and you superimpose

1033
01:06:50,160 --> 01:06:54,280
another stationary state,
the result is not stationary.

1034
01:06:54,280 --> 01:06:56,540
Stationary is a
single exponential.

1035
01:06:56,540 --> 01:06:59,330
More than one exponential
is not stationary.

1036
01:06:59,330 --> 01:07:03,260
So when you have this, you
could have time dependence.

1037
01:07:03,260 --> 01:07:05,020
So that's why we wrote it.

1038
01:07:05,020 --> 01:07:07,890
Whenever you have a state
that is not stationary,

1039
01:07:07,890 --> 01:07:11,150
there is time dependence.

1040
01:07:11,150 --> 01:07:15,290
Now, you could do
the following thing.

1041
01:07:15,290 --> 01:07:21,940
So here is a simple but
important calculation

1042
01:07:21,940 --> 01:07:24,580
that should be done.

1043
01:07:24,580 --> 01:07:28,930
And it's the expectation
value of H. So

1044
01:07:28,930 --> 01:07:34,170
what is the expectation value
of the Hamiltonian at time

1045
01:07:34,170 --> 01:07:40,300
t on this wave function Psi
that we've computed there?

1046
01:07:40,300 --> 01:07:43,290
So, we would have to
do that whole integral.

1047
01:07:43,290 --> 01:07:50,070
And in fact, I ask
you that you do it.

1048
01:07:50,070 --> 01:07:51,870
It's not too hard.

1049
01:07:51,870 --> 01:07:56,350
In fact, I will say
it's relatively simple.

1050
01:07:56,350 --> 01:08:02,360
And you have H on
Psi of x and t,

1051
01:08:02,360 --> 01:08:06,540
and then you must substitute
this Psi equal the sum of bn

1052
01:08:06,540 --> 01:08:07,525
Psi n.

1053
01:08:10,230 --> 01:08:13,250
And you have two sums.

1054
01:08:13,250 --> 01:08:17,189
And the H acting on each
side n-- you know what it is.

1055
01:08:17,189 --> 01:08:20,229
And then the two sums-- you
can do the integral using

1056
01:08:20,229 --> 01:08:22,229
orthonormality.

1057
01:08:22,229 --> 01:08:27,109
It's a relatively
standard calculation.

1058
01:08:27,109 --> 01:08:28,890
You should be able to do it.

1059
01:08:28,890 --> 01:08:31,560
If you find it hard, you
will see it, of course,

1060
01:08:31,560 --> 01:08:32,819
in the notes.

1061
01:08:32,819 --> 01:08:36,020
But it's the kind of thing
that I want you to review.

1062
01:08:36,020 --> 01:08:38,960
So, what is the answer here?

1063
01:08:38,960 --> 01:08:41,500
It's a famous answer.

1064
01:08:41,500 --> 01:08:47,229
It's bm squared En.

1065
01:08:47,229 --> 01:08:52,490
So, you get the expected
value of the energy.

1066
01:08:52,490 --> 01:08:57,350
It's a weighted average over all
of the stationary states that

1067
01:08:57,350 --> 01:09:00,069
are involved in this state
that you've been building.

1068
01:09:00,069 --> 01:09:04,140
So your state has a little
bit of Psi zero, Psi 1, Psi 2,

1069
01:09:04,140 --> 01:09:05,040
Psi 3.

1070
01:09:05,040 --> 01:09:10,270
And for each one, you square its
component and multiply by En.

1071
01:09:10,270 --> 01:09:13,490
And this is time-independent.

1072
01:09:13,490 --> 01:09:16,010
And you say, well,
you told me that only

1073
01:09:16,010 --> 01:09:19,260
for stationary states,
things are time-independent.

1074
01:09:19,260 --> 01:09:23,300
Yes, only for stationary
states, all operators

1075
01:09:23,300 --> 01:09:25,300
are time-independent,
but the Hamiltonian

1076
01:09:25,300 --> 01:09:28,029
is a very special operator.

1077
01:09:28,029 --> 01:09:30,420
It's an energy
operator, and this

1078
01:09:30,420 --> 01:09:32,330
is a time independent system.

1079
01:09:32,330 --> 01:09:34,729
It's not being
driven by something,

1080
01:09:34,729 --> 01:09:37,910
so you would expect the
energy to be conserved.

1081
01:09:37,910 --> 01:09:40,540
And this is pretty
much the statement

1082
01:09:40,540 --> 01:09:44,630
of conservation of energy,
the time-independence

1083
01:09:44,630 --> 01:09:46,520
of this thing.

1084
01:09:46,520 --> 01:09:51,120
My last remark is technical
about normalizations,

1085
01:09:51,120 --> 01:09:56,100
and it's something
you may find useful.

1086
01:09:56,100 --> 01:10:03,390
If you have a wave function
that is Psi, which is not

1087
01:10:03,390 --> 01:10:11,640
normalized, you may say,
OK, let's normalize it.

1088
01:10:11,640 --> 01:10:14,920
So, what is the
normalized wave function?

1089
01:10:14,920 --> 01:10:18,130
The normalized wave
function is Psi

1090
01:10:18,130 --> 01:10:25,340
divided by the square root of
the integral of Psi star Psi

1091
01:10:25,340 --> 01:10:25,840
dx.

1092
01:10:29,820 --> 01:10:33,920
You see, this is a number, and
you take the square root of it.

1093
01:10:33,920 --> 01:10:38,640
And this is the Psi of x and t.

1094
01:10:38,640 --> 01:10:42,945
If the Psi is not normalized,
this thing is normalized.

1095
01:10:47,410 --> 01:10:52,240
So, think of doing this here.

1096
01:10:52,240 --> 01:10:56,530
Suppose you don't
want to work too hard,

1097
01:10:56,530 --> 01:11:00,550
and you want to normalize
your wave function.

1098
01:11:00,550 --> 01:11:03,130
So, your Psi is not normalized.

1099
01:11:03,130 --> 01:11:06,730
Well, then this is
definitely normalized.

1100
01:11:06,730 --> 01:11:07,840
You should check that.

1101
01:11:07,840 --> 01:11:10,130
Square it, an integrate
it, and you'll see.

1102
01:11:10,130 --> 01:11:11,560
You'll get one.

1103
01:11:11,560 --> 01:11:17,200
But then I can then calculate
the expectation value of A

1104
01:11:17,200 --> 01:11:20,700
on that state,
and wherever I see

1105
01:11:20,700 --> 01:11:25,120
a Psi that should be normalized,
I put this whole thing.

1106
01:11:25,120 --> 01:11:26,870
So what do I end up with?

1107
01:11:26,870 --> 01:11:29,620
I end up with this
integral from infinity

1108
01:11:29,620 --> 01:11:41,270
to infinity dx Psi star A A
hat Psi divided by the integral

1109
01:11:41,270 --> 01:11:49,680
from minus infinity to
infinity of Psi star Psi dx.

1110
01:11:49,680 --> 01:11:53,460
If you don't want to normalize
a wave function, that's OK.

1111
01:11:53,460 --> 01:11:57,590
You can still calculate
its expectation value

1112
01:11:57,590 --> 01:12:01,250
by working with a
not-normalized wave function.

1113
01:12:01,250 --> 01:12:08,180
So in this definition,
Psi is not normalized,

1114
01:12:08,180 --> 01:12:11,690
but you still get
the right value.

1115
01:12:11,690 --> 01:12:13,850
OK, so that's it for today.

1116
01:12:13,850 --> 01:12:18,910
Next time we'll do properties
of the spectrum in one dimension

1117
01:12:18,910 --> 01:12:23,340
and begin something new called
the variational problem.

1118
01:12:23,340 --> 01:12:23,940
All right.

1119
01:12:23,940 --> 01:12:24,840
[APPLAUSE]

1120
01:12:24,840 --> 01:12:26,990
Thank you, thank you.