1
00:00:01,730 --> 00:00:03,770
BARTON ZWIEBACH: We
were faced last time

2
00:00:03,770 --> 00:00:05,660
with a question
of interpretation

3
00:00:05,660 --> 00:00:08,760
of the Schrodinger
wave function.

4
00:00:08,760 --> 00:00:15,470
And so to recap the main
ideas that we were looking at,

5
00:00:15,470 --> 00:00:19,820
we derive this
Schrodinger equation,

6
00:00:19,820 --> 00:00:25,550
basically derived it
from simple ideas--

7
00:00:25,550 --> 00:00:31,010
having operators, energy
operator, momentum operator,

8
00:00:31,010 --> 00:00:36,680
and exploring how the de
Broglie wavelength associated

9
00:00:36,680 --> 00:00:40,970
to a particle would be a wave
that would solve the equation.

10
00:00:40,970 --> 00:00:43,455
And the equation was the
Schrodinger equation,

11
00:00:43,455 --> 00:00:46,250
a free Schrodinger
equation, and then

12
00:00:46,250 --> 00:00:48,110
we added the
potential to make it

13
00:00:48,110 --> 00:00:53,720
interacting and that way,
we motivated the Schrodinger

14
00:00:53,720 --> 00:01:10,250
equation and took this form
of x and t psi of x and t.

15
00:01:10,250 --> 00:01:15,400
And this is a dynamical equation
that governs the wave function.

16
00:01:15,400 --> 00:01:18,030
But the interpretation
that we've

17
00:01:18,030 --> 00:01:22,840
had for the wave function,
we discussed what Born said,

18
00:01:22,840 --> 00:01:26,670
was that it's related
to probabilities

19
00:01:26,670 --> 00:01:31,230
and psi squared
multiplied by a little dx

20
00:01:31,230 --> 00:01:33,690
would give you the
probability to find

21
00:01:33,690 --> 00:01:39,220
the particle in that little
dx at some particular time.

22
00:01:39,220 --> 00:01:50,430
So psi of x and t squared
dx would be the probability

23
00:01:50,430 --> 00:01:56,700
to find the particle at
that interval dx around x.

24
00:01:56,700 --> 00:02:00,000
And if you're describing the
physics of your Schrodinger

25
00:02:00,000 --> 00:02:02,940
equation is that of
a single particle,

26
00:02:02,940 --> 00:02:04,650
which is the case here--

27
00:02:04,650 --> 00:02:12,525
one coordinate, the coordinate
of the particle, this integral,

28
00:02:12,525 --> 00:02:17,550
if you integrate
this all over space,

29
00:02:17,550 --> 00:02:22,930
must be 1 for the
probability to make sense.

30
00:02:22,930 --> 00:02:28,420
So the total probability of
finding the particle must be 1,

31
00:02:28,420 --> 00:02:30,040
must be somewhere.

32
00:02:30,040 --> 00:02:32,200
If it's in one part
of another part

33
00:02:32,200 --> 00:02:35,230
or another part, this
probabilities-- for this

34
00:02:35,230 --> 00:02:37,075
to be a probability
distribution,

35
00:02:37,075 --> 00:02:41,470
it has to be well-normalized,
which means 1.

36
00:02:41,470 --> 00:02:50,350
And we said that this equation
was interesting but somewhat

37
00:02:50,350 --> 00:02:55,960
worrisome, because if the
normalization of the wave

38
00:02:55,960 --> 00:03:07,880
function satisfies, if this
holds for t equal t-nought

39
00:03:07,880 --> 00:03:13,490
then the Schrodinger equation,
if you know the wave function

40
00:03:13,490 --> 00:03:18,890
all over space for t equal
t-nought, which is what you

41
00:03:18,890 --> 00:03:23,450
would need to know in order
to check that this is working,

42
00:03:23,450 --> 00:03:28,550
a t equal t-nought, you take
the psi of x and t-nought,

43
00:03:28,550 --> 00:03:30,650
integrate it.

44
00:03:30,650 --> 00:03:36,900
But if you know psi of x
and t-nought for all x,

45
00:03:36,900 --> 00:03:39,200
then the Schrodinger
equation tells you

46
00:03:39,200 --> 00:03:43,290
what the wave function
is at a later time.

47
00:03:43,290 --> 00:03:46,320
Because it gives you the
time derivative of the wave

48
00:03:46,320 --> 00:03:49,580
function in terms of
data about the wave

49
00:03:49,580 --> 00:03:52,350
function all over space.

50
00:03:52,350 --> 00:03:55,470
So automatically, the
Schrodinger equation

51
00:03:55,470 --> 00:04:02,230
must make it true that this
will hold at later times.

52
00:04:02,230 --> 00:04:06,651
You cannot force the wave
function to satisfy this at all

53
00:04:06,651 --> 00:04:07,150
times.

54
00:04:07,150 --> 00:04:11,650
You can force it maybe
to satisfy at one time,

55
00:04:11,650 --> 00:04:16,290
but once it satisfies it at
this time, then it will evolve,

56
00:04:16,290 --> 00:04:20,470
and it better be that
at every time later,

57
00:04:20,470 --> 00:04:23,730
it still satisfies
this equation.

58
00:04:23,730 --> 00:04:28,640
So this is a very
important constraint.

59
00:04:28,640 --> 00:04:37,910
So we'll basically develop this
throughout the lecture today.

60
00:04:37,910 --> 00:04:43,610
We're going to make a big point
of this trying to explain why

61
00:04:43,610 --> 00:04:47,650
the conditions that we're going
to impose on the wave function

62
00:04:47,650 --> 00:04:51,770
are necessary; what it teaches
you about the Hamiltonian,

63
00:04:51,770 --> 00:04:55,970
we'll teach you that it's
a Hermitian operator;

64
00:04:55,970 --> 00:05:00,120
what do you learn
about probability--

65
00:05:00,120 --> 00:05:03,810
you will learn that there
is a probability current;

66
00:05:03,810 --> 00:05:09,720
and all kinds of things will
come out of taking seriously

67
00:05:09,720 --> 00:05:12,930
the interpretation
of this probability,

68
00:05:12,930 --> 00:05:18,210
the main point being that
we can be sure it behaves

69
00:05:18,210 --> 00:05:23,130
as a probability at one time,
but then for later times,

70
00:05:23,130 --> 00:05:26,670
the behaviors and probability
the Schrodinger equation must

71
00:05:26,670 --> 00:05:27,600
help--

72
00:05:27,600 --> 00:05:31,770
must somehow be part of
the reason this works out.

73
00:05:31,770 --> 00:05:35,750
So that's what we're
going to try to do.

74
00:05:35,750 --> 00:05:43,550
Now, when we write an equation
like this, and more explicitly,

75
00:05:43,550 --> 00:05:55,460
this means integral of psi star
of x and t, psi of x and t dx

76
00:05:55,460 --> 00:05:56,300
equal 1.

77
00:05:59,270 --> 00:06:02,690
You can imagine that not
all kind of functions

78
00:06:02,690 --> 00:06:05,660
will satisfy it.

79
00:06:05,660 --> 00:06:09,020
In particular, any wave
function, for example,

80
00:06:09,020 --> 00:06:13,250
that at infinity approaches
a constant will never

81
00:06:13,250 --> 00:06:16,700
satisfy this, because
if infinity, you

82
00:06:16,700 --> 00:06:20,300
approach a constant,
then the integral

83
00:06:20,300 --> 00:06:22,980
is going to be infinite.

84
00:06:22,980 --> 00:06:25,020
And it's just not
going to work out.

85
00:06:25,020 --> 00:06:31,920
So the wave function cannot
approach a finite number,

86
00:06:31,920 --> 00:06:35,350
a finite constant as
x goes to infinity.

87
00:06:35,350 --> 00:06:39,810
So in order for this to hold--

88
00:06:39,810 --> 00:07:04,190
order to guarantee this can
even hold, can conceivably hold,

89
00:07:04,190 --> 00:07:07,610
it will require a bit
of boundary conditions.

90
00:07:07,610 --> 00:07:12,290
And we'll say that
the limit as x

91
00:07:12,290 --> 00:07:16,770
goes to infinity
or minus infinity--

92
00:07:16,770 --> 00:07:22,790
plus/minus infinity of psi of
x and t will be equal to 0.

93
00:07:25,640 --> 00:07:28,210
It better be true.

94
00:07:28,210 --> 00:07:31,240
And we'll ask a little more.

95
00:07:31,240 --> 00:07:36,700
Now, you could say,
look, certainly

96
00:07:36,700 --> 00:07:39,610
the limit of this
function could not

97
00:07:39,610 --> 00:07:45,190
be in number, because it
would be non-zero number,

98
00:07:45,190 --> 00:07:47,270
the interval will diverge.

99
00:07:47,270 --> 00:07:49,840
But maybe there is no limit.

100
00:07:49,840 --> 00:07:54,160
The wave function is so crazy
that it can be integrated,

101
00:07:54,160 --> 00:07:57,250
but suddenly, it
has a little spike

102
00:07:57,250 --> 00:08:00,820
and it just doesn't
have a normal limit.

103
00:08:00,820 --> 00:08:05,170
That could conceivably
be the case.

104
00:08:05,170 --> 00:08:10,000
Nevertheless, it doesn't seem
to happen in any example that

105
00:08:10,000 --> 00:08:11,770
is of relevance.

106
00:08:11,770 --> 00:08:16,210
So we will assume that
the situations are not

107
00:08:16,210 --> 00:08:19,740
that crazy that this happened.

108
00:08:19,740 --> 00:08:23,860
So we'll take wave
functions that necessarily

109
00:08:23,860 --> 00:08:26,560
go to 0 at infinity.

110
00:08:26,560 --> 00:08:32,830
And that certainly is good.

111
00:08:32,830 --> 00:08:36,010
You cannot prove it's
a necessary condition,

112
00:08:36,010 --> 00:08:41,080
but if it holds, it
simplifies many, many things,

113
00:08:41,080 --> 00:08:46,300
and essentially, if the wave
function is good enough to have

114
00:08:46,300 --> 00:08:49,690
a limit, then the
limit must be 0.

115
00:08:49,690 --> 00:08:51,610
The other thing
that we will want

116
00:08:51,610 --> 00:09:04,710
is that d psi/dx, the limit as
x goes to plus/minus infinity

117
00:09:04,710 --> 00:09:05,380
is bounded.

118
00:09:09,220 --> 00:09:14,050
That is, yes, the limit may
exist and it may be a number,

119
00:09:14,050 --> 00:09:17,510
but it's not infinite.

120
00:09:17,510 --> 00:09:21,890
And In every example
that I know of--

121
00:09:21,890 --> 00:09:25,340
in fact, when this goes to
0, this goes to 0 as well--

122
00:09:25,340 --> 00:09:31,640
but this is basically all
you will ever need in order

123
00:09:31,640 --> 00:09:35,900
to make sense of the wave
functions and their integrals

124
00:09:35,900 --> 00:09:37,700
that we're going to be doing.

125
00:09:37,700 --> 00:09:40,670
Now you shouldn't
be too surprised

126
00:09:40,670 --> 00:09:45,080
that you need to say something
about this wave function

127
00:09:45,080 --> 00:09:46,760
in the analysis
that will follow,

128
00:09:46,760 --> 00:09:50,150
because the derivative--

129
00:09:50,150 --> 00:09:52,220
you have the function
and its derivative,

130
00:09:52,220 --> 00:09:56,190
because certainly, there
are two derivatives here.

131
00:09:56,190 --> 00:10:04,020
So when we manipulate
these quantities

132
00:10:04,020 --> 00:10:08,890
inside the integrals,
you will see very soon--

133
00:10:08,890 --> 00:10:15,280
single derivatives will show up
and we'll have to control them.

134
00:10:15,280 --> 00:10:16,870
So the only thing
that I'm saying

135
00:10:16,870 --> 00:10:20,920
is that when you see
a wave function that

136
00:10:20,920 --> 00:10:24,320
satisfies this
property, you know

137
00:10:24,320 --> 00:10:27,290
that unless the function
is extremely crazy,

138
00:10:27,290 --> 00:10:32,600
it's a function that goes
to 0 at plus/minus infinity.

139
00:10:32,600 --> 00:10:35,540
And it's the relative
pursuant it also goes to 0,

140
00:10:35,540 --> 00:10:42,570
but it will be enough to say
that it maybe goes to a number.

141
00:10:42,570 --> 00:10:47,820
Now there's another
possibility thing for confusion

142
00:10:47,820 --> 00:10:52,930
here with things that
we've been saying before.

143
00:10:52,930 --> 00:10:58,410
We've said before that the
physics of a wave function

144
00:10:58,410 --> 00:11:05,890
is not altered by multiplying
the wave function by a number.

145
00:11:05,890 --> 00:11:11,150
We said that psi added to
psi is the same state; psi

146
00:11:11,150 --> 00:11:14,820
is the same state as
square root of 2 psi--

147
00:11:14,820 --> 00:11:17,760
all this is the same
physics, but here it

148
00:11:17,760 --> 00:11:25,080
looks a little surprising if you
wish, because if I have a psi

149
00:11:25,080 --> 00:11:31,290
and I got this
already working out,

150
00:11:31,290 --> 00:11:37,540
if I multiply psi by square
root of 2, it will not hold.

151
00:11:37,540 --> 00:11:43,334
So there seems to be a little
maybe something with the words

152
00:11:43,334 --> 00:11:44,250
that we're been using.

153
00:11:44,250 --> 00:11:47,970
It's not exactly
right and I want

154
00:11:47,970 --> 00:11:52,050
to make sure there is no
room for confusion here,

155
00:11:52,050 --> 00:11:56,730
and it's the following fact.

156
00:11:56,730 --> 00:12:03,330
Here, this wave function
has been normalized.

157
00:12:03,330 --> 00:12:08,810
So there's two kinds of wave
functions that you can have--

158
00:12:08,810 --> 00:12:12,205
wave functions that can be
normalized and wave functions

159
00:12:12,205 --> 00:12:13,640
that cannot be normalized.

160
00:12:13,640 --> 00:12:18,940
Suppose somebody comes to you
and gives you a psi of x and t.

161
00:12:22,530 --> 00:12:25,240
Or let's assume that--

162
00:12:25,240 --> 00:12:27,320
I'll put x and t.

163
00:12:27,320 --> 00:12:29,270
No problem.

164
00:12:29,270 --> 00:12:36,270
Now suppose you go and
start doing this integral--

165
00:12:36,270 --> 00:12:42,230
integral of psi squared dx.

166
00:12:42,230 --> 00:12:44,990
And then you find that
it's not equal to 1

167
00:12:44,990 --> 00:12:49,840
but is equal to
some value N, which

168
00:12:49,840 --> 00:12:52,080
is different from 1 maybe.

169
00:12:55,690 --> 00:13:10,700
If this happens, we say that
psi is normalizable, which

170
00:13:10,700 --> 00:13:16,580
means it can be normalized.

171
00:13:16,580 --> 00:13:21,440
And using this idea that
changing the value--

172
00:13:21,440 --> 00:13:23,660
the coefficient
of the function--

173
00:13:23,660 --> 00:13:32,030
doesn't change too
much, we simply say, use

174
00:13:32,030 --> 00:13:42,280
instead psi prime, which
is equal to psi over

175
00:13:42,280 --> 00:13:50,860
square root of N. And look what
a nice property this psi prime

176
00:13:50,860 --> 00:13:52,180
has.

177
00:13:52,180 --> 00:14:04,430
If you integrate psi prime
squared, it would be equal--

178
00:14:04,430 --> 00:14:06,500
because you have psi
prime here is squared,

179
00:14:06,500 --> 00:14:13,300
it would be equal to the
integral of PSI squared divided

180
00:14:13,300 --> 00:14:15,730
by the number N--

181
00:14:15,730 --> 00:14:18,940
because there's two of them--

182
00:14:18,940 --> 00:14:25,680
dx, and the number goes out and
you have the integral of psi

183
00:14:25,680 --> 00:14:34,000
squared dx, but that integral
was exactly N, so that's 1.

184
00:14:34,000 --> 00:14:42,570
So if your wave function
has a finite integral

185
00:14:42,570 --> 00:14:51,610
in this sense, a number
that is less than infinity,

186
00:14:51,610 --> 00:14:56,430
then psi can be normalized.

187
00:14:56,430 --> 00:15:01,480
And if you're going to
work with probabilities,

188
00:15:01,480 --> 00:15:05,480
you should use instead
this wave function,

189
00:15:05,480 --> 00:15:11,050
which is the original wave
function divided by a number.

190
00:15:11,050 --> 00:15:15,280
So they realize
that, in some sense,

191
00:15:15,280 --> 00:15:18,390
you can delay all of
this and you can always

192
00:15:18,390 --> 00:15:23,010
work with wave functions
that are normalizable,

193
00:15:23,010 --> 00:15:27,000
but only when you're going to
calculate your probabilities.

194
00:15:27,000 --> 00:15:31,350
You can take the trouble
to actually normalize them

195
00:15:31,350 --> 00:15:35,280
and those are the ones
you use in these formulas.

196
00:15:35,280 --> 00:15:42,480
So the idea remains that we work
flexibly with wave functions

197
00:15:42,480 --> 00:15:48,300
and multiply them by numbers and
nothing changes as long as you

198
00:15:48,300 --> 00:15:52,860
realize that you cannot change
the fact that the wave function

199
00:15:52,860 --> 00:15:57,240
is normalizable by multiplying
it by any finite number,

200
00:15:57,240 --> 00:15:59,490
it will still be normalized.

201
00:15:59,490 --> 00:16:05,900
And if it's normalizable, it's
equivalent to a normalized wave

202
00:16:05,900 --> 00:16:06,730
function.

203
00:16:06,730 --> 00:16:09,940
So those two words
sound very similar,

204
00:16:09,940 --> 00:16:11,670
but they're a little different.

205
00:16:11,670 --> 00:16:17,250
One is normalizable, which
means it has an integral of psi

206
00:16:17,250 --> 00:16:20,730
squared finite, and
normalize is one

207
00:16:20,730 --> 00:16:25,450
that already has been
adjusted to do this

208
00:16:25,450 --> 00:16:31,490
and can be used to define
a probability distribution.

209
00:16:31,490 --> 00:16:32,030
OK.

210
00:16:32,030 --> 00:16:35,540
So that, in a way
of introduction

211
00:16:35,540 --> 00:16:39,160
to the problem
that we have to do,

212
00:16:39,160 --> 00:16:43,660
our serious problem
is indeed justifying

213
00:16:43,660 --> 00:16:48,820
that the time evolution doesn't
mess up the normalization

214
00:16:48,820 --> 00:16:51,480
and how does it do that?