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PROFESSOR: All right, so
for the next six lectures,

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00:00:28,980 --> 00:00:33,250
including today, we're
going to finish off

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00:00:33,250 --> 00:00:36,680
the course with the
application of everything

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00:00:36,680 --> 00:00:39,560
we've studied so far
to a couple of ideas.

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00:00:39,560 --> 00:00:42,820
The first being the existence
of solids-- why we have solids

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00:00:42,820 --> 00:00:45,290
and why we have conductivity
in solids, which

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00:00:45,290 --> 00:00:48,020
is basic properties
of materials.

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00:00:48,020 --> 00:00:51,340
In particular, the story I want
you guys to leave the course

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00:00:51,340 --> 00:00:54,860
with is an understanding of
why diamonds are transparent

17
00:00:54,860 --> 00:00:58,400
and why copper
isn't, which is sort

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00:00:58,400 --> 00:00:59,797
of a crude fact about the world.

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00:00:59,797 --> 00:01:01,630
But we can explain it
from first principles,

20
00:01:01,630 --> 00:01:03,840
which is pretty awesome.

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00:01:03,840 --> 00:01:07,550
The second is we're going to
come back to this idea of spin

22
00:01:07,550 --> 00:01:09,330
and of 1/2 integral
angular momentum.

23
00:01:09,330 --> 00:01:11,510
This intrinsic angular
momentum of the electron.

24
00:01:11,510 --> 00:01:14,339
And we're going to use that
to motivate a couple of ideas.

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00:01:14,339 --> 00:01:16,630
First off, we're going to
tie back to Bell's inequality

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00:01:16,630 --> 00:01:17,730
from the very beginning,
and then we're

27
00:01:17,730 --> 00:01:19,730
going to do a little bit
on quantum computation.

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00:01:19,730 --> 00:01:21,220
That will be the
last two lectures.

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00:01:21,220 --> 00:01:25,817
So before I get started
though with today, two things.

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00:01:25,817 --> 00:01:27,400
First, I'm going to
ask for questions.

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00:01:27,400 --> 00:01:30,430
But the second is I got
some really good questions

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00:01:30,430 --> 00:01:32,080
about hydrogen, so
I want to wrap up

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00:01:32,080 --> 00:01:35,570
with one last
comment on hydrogen

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00:01:35,570 --> 00:01:37,970
because it's entertaining.

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00:01:37,970 --> 00:01:40,762
But before I get started
on that, questions

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00:01:40,762 --> 00:01:41,970
about everything up till now?

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00:01:49,560 --> 00:01:50,425
Yeah.

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00:01:50,425 --> 00:01:52,365
AUDIENCE: So in the
[INAUDIBLE] there's

39
00:01:52,365 --> 00:01:55,980
someting called spirit orbit
couplings, another edition,

40
00:01:55,980 --> 00:02:00,180
the [INAUDIBLE], that
we haven't talked about,

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00:02:00,180 --> 00:02:02,538
in addition throughout
this egression.

42
00:02:02,538 --> 00:02:04,923
Is that something we can
do with our knowledge now?

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00:02:04,923 --> 00:02:06,134
Or is something [INAUDIBLE]?

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00:02:06,134 --> 00:02:07,300
PROFESSOR: It absolutely is.

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00:02:07,300 --> 00:02:09,591
So the question is-- and this
is a very good question--

46
00:02:09,591 --> 00:02:12,590
the question is, look,
if you look on Wikipedia

47
00:02:12,590 --> 00:02:15,880
about-- seriously, this
is a rational thing to do.

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00:02:15,880 --> 00:02:17,940
If you look at Wikipedia
to learn something

49
00:02:17,940 --> 00:02:20,290
about hydrogen,
what you discover

50
00:02:20,290 --> 00:02:21,860
is everything
we've talked about,

51
00:02:21,860 --> 00:02:24,500
including the fine structure,
including the Zeeman effect.

52
00:02:24,500 --> 00:02:26,708
But you see that there are
a couple of other effects.

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00:02:26,708 --> 00:02:29,460
So one effect, for example,
is when you look at the Zeeman

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00:02:29,460 --> 00:02:31,300
effect a little more
carefully than we did,

55
00:02:31,300 --> 00:02:34,100
there's a second term in
the Zeeman effect, which

56
00:02:34,100 --> 00:02:36,760
is an induced magnetic moment.

57
00:02:36,760 --> 00:02:38,660
So even when the
angular momentum

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00:02:38,660 --> 00:02:43,280
is smaller or even
zero, there's still

59
00:02:43,280 --> 00:02:46,740
a contribution to the energy
from the externally imposed

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00:02:46,740 --> 00:02:48,136
magnetic field.

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00:02:48,136 --> 00:02:49,760
So there are lots of
other corrections.

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00:02:49,760 --> 00:02:51,556
One of them in
particular is something

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00:02:51,556 --> 00:02:52,680
called spin orbit coupling.

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00:02:56,990 --> 00:02:59,180
And so the question
was, what is that?

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00:02:59,180 --> 00:03:02,450
And do we know enough to explain
what spin orbit coupling is?

66
00:03:02,450 --> 00:03:05,230
So let me give you a very
short answer to that question.

67
00:03:05,230 --> 00:03:08,320
So we know that if we
write down the energy

68
00:03:08,320 --> 00:03:14,230
operator for our system, we
have the energy operator--

69
00:03:14,230 --> 00:03:20,230
so the full energy operator
for hydrogen, I'll say,

70
00:03:20,230 --> 00:03:23,780
is the energy
operator for coulomb

71
00:03:23,780 --> 00:03:25,240
plus a bunch of corrections.

72
00:03:25,240 --> 00:03:27,670
So for example, we had the
relativistic correction

73
00:03:27,670 --> 00:03:29,350
plus a term that was
a coefficient times

74
00:03:29,350 --> 00:03:30,420
p to the fourth.

75
00:03:30,420 --> 00:03:32,567
And this is on your problem set.

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00:03:32,567 --> 00:03:34,400
And this came from
relativistic corrections.

77
00:03:34,400 --> 00:03:35,400
I shouldn't call this c.

78
00:03:35,400 --> 00:03:37,590
I'll call it beta, maybe.

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00:03:37,590 --> 00:03:39,566
Some coefficient.

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00:03:39,566 --> 00:03:41,690
And there are a whole bunch
of other contributions.

81
00:03:41,690 --> 00:03:43,170
For example, there's
the term we studied

82
00:03:43,170 --> 00:03:44,870
that contributed to
the Zeeman effect.

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00:03:44,870 --> 00:03:46,700
If there's an external
magnetic field,

84
00:03:46,700 --> 00:03:52,780
there's a B dot-- magnetic
moment of the system.

85
00:03:52,780 --> 00:03:55,800
And if the electron is in
a state with finite angular

86
00:03:55,800 --> 00:03:59,410
momentum, the magnetic moment is
some dimensionful coefficient,

87
00:03:59,410 --> 00:04:01,290
which we usually call
the Bohr magneton just

88
00:04:01,290 --> 00:04:03,390
because-- whatever--
we liked the guy.

89
00:04:03,390 --> 00:04:06,210
Times the angular momentum.

90
00:04:06,210 --> 00:04:08,930
But there can be
additional contributions.

91
00:04:08,930 --> 00:04:11,310
There are a whole bunch
of additional terms

92
00:04:11,310 --> 00:04:14,910
in the true potential
for hydrogen.

93
00:04:14,910 --> 00:04:16,810
One of them takes
the following form.

94
00:04:16,810 --> 00:04:25,280
It's a constant plus-- maybe
I'll call it kappa times

95
00:04:25,280 --> 00:04:33,200
spin dotted into the angular
momentum of the electron,

96
00:04:33,200 --> 00:04:36,360
of the electron bound
to the hydrogen system.

97
00:04:36,360 --> 00:04:39,997
So where this comes from is
a sort of beautiful story.

98
00:04:39,997 --> 00:04:41,330
So first off, could it be there?

99
00:04:41,330 --> 00:04:41,980
Sure.

100
00:04:41,980 --> 00:04:45,200
This is a term that could
exist with some coefficient.

101
00:04:45,200 --> 00:04:48,134
Why not?

102
00:04:48,134 --> 00:04:50,050
It turns out that you
can derive its existence

103
00:04:50,050 --> 00:04:55,280
from a study of the relativistic
version of the hydrogen system.

104
00:04:55,280 --> 00:04:58,790
We're not going to study
that in any detail in 804,

105
00:04:58,790 --> 00:05:01,560
but the basic idea
and what this does

106
00:05:01,560 --> 00:05:04,860
is totally amenable
to 804-level analysis.

107
00:05:04,860 --> 00:05:08,050
This is just saying that when
you have some orbital momentum,

108
00:05:08,050 --> 00:05:10,640
or when you have some spin
angular momentum, both of those

109
00:05:10,640 --> 00:05:13,000
corresponds to angular
momentum of an object that

110
00:05:13,000 --> 00:05:14,562
carries charge.

111
00:05:14,562 --> 00:05:16,270
And so it's not
unreasonable that there's

112
00:05:16,270 --> 00:05:18,103
going to be some
interaction between the two

113
00:05:18,103 --> 00:05:21,466
where the magnetic moments--
those two magnetic moments

114
00:05:21,466 --> 00:05:23,590
either want to be aligned
or anti-aligned depending

115
00:05:23,590 --> 00:05:25,930
on the sign of this coefficient.

116
00:05:25,930 --> 00:05:28,661
However, we haven't talked about
spin in any great detail yet.

117
00:05:28,661 --> 00:05:30,910
We're going to do that in
the last week of the course.

118
00:05:30,910 --> 00:05:33,641
So I'm going to defer talking
about this in any detail.

119
00:05:33,641 --> 00:05:35,390
And we're only going
to discuss it briefly

120
00:05:35,390 --> 00:05:37,310
at the end for a
couple of weeks.

121
00:05:37,310 --> 00:05:39,799
But absolutely, these
additional couplings

122
00:05:39,799 --> 00:05:41,590
are important for
hydrogen and they're also

123
00:05:41,590 --> 00:05:44,964
things you can study at
this level, at 804 level.

124
00:05:44,964 --> 00:05:46,380
I should also
emphasize that there

125
00:05:46,380 --> 00:05:48,004
are a whole bunch of
other corrections.

126
00:05:48,004 --> 00:05:50,980
There are lots of terms.

127
00:05:50,980 --> 00:05:53,020
One set of terms, an
infinite number of terms,

128
00:05:53,020 --> 00:05:55,930
are the further sub-leading
relativistic corrections

129
00:05:55,930 --> 00:05:57,400
of kinetic energy.

130
00:05:57,400 --> 00:05:58,610
And there are a whole
bunch of other corrections,

131
00:05:58,610 --> 00:06:00,568
so I'm going to leave
those out for the moment.

132
00:06:00,568 --> 00:06:04,440
But I want to emphasize
to everyone that we're

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00:06:04,440 --> 00:06:07,280
building models of hydrogen,
and they're all approximations.

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00:06:07,280 --> 00:06:08,562
Yeah.

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00:06:08,562 --> 00:06:10,895
AUDIENCE: So it's a question
regarding this [INAUDIBLE].

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00:06:13,330 --> 00:06:14,755
PROFESSOR: Yes.

137
00:06:14,755 --> 00:06:16,250
There is.

138
00:06:16,250 --> 00:06:19,360
So on the problem
set, you're asked

139
00:06:19,360 --> 00:06:22,440
to estimate the
correction to the energy.

140
00:06:22,440 --> 00:06:25,340
I wrote down the answer
in lecture the other day.

141
00:06:25,340 --> 00:06:30,220
You're asked to compute, in
particular, the l dependence.

142
00:06:30,220 --> 00:06:33,630
But to estimate the
magnitude or to estimate

143
00:06:33,630 --> 00:06:38,590
the value of the shift
in the energy eigenvalues

144
00:06:38,590 --> 00:06:42,145
to hydrogen from the first
relativistic correction

145
00:06:42,145 --> 00:06:44,670
is p to the fourth correction.

146
00:06:44,670 --> 00:06:47,260
And I expect that
this will be covered

147
00:06:47,260 --> 00:06:50,960
in your recitation in some
detail, but there the question

148
00:06:50,960 --> 00:06:54,994
was, is there a trick to
make this a little easier?

149
00:06:54,994 --> 00:06:56,910
There are a couple of
good physical arguments,

150
00:06:56,910 --> 00:06:57,980
which I'm not going to tell you.

151
00:06:57,980 --> 00:06:59,510
But there are nice
ways to do this.

152
00:06:59,510 --> 00:07:01,360
But at the end of
the day, you do not

153
00:07:01,360 --> 00:07:03,432
want to end up doing the
following computation.

154
00:07:03,432 --> 00:07:05,473
You don't want to take
the expectation value of p

155
00:07:05,473 --> 00:07:07,980
to the fourth and
write this as the sum

156
00:07:07,980 --> 00:07:11,660
integral of psi
complex conjugate

157
00:07:11,660 --> 00:07:14,540
derivative to the fourth psi.

158
00:07:14,540 --> 00:07:17,807
If you attempt to do this
integral, you will weep.

159
00:07:17,807 --> 00:07:19,390
This is not the way
you want to do it.

160
00:07:19,390 --> 00:07:20,880
And here's what you want to do.

161
00:07:20,880 --> 00:07:22,490
And I'm not going to
tell you how to get here,

162
00:07:22,490 --> 00:07:24,906
but you want to reduce the
calculation of this expectation

163
00:07:24,906 --> 00:07:28,920
value to the calculation of
the expectation value of 1

164
00:07:28,920 --> 00:07:33,570
over r and of 1 over r squared.

165
00:07:33,570 --> 00:07:36,100
And it turns out that if you
know these expectation values,

166
00:07:36,100 --> 00:07:38,730
that entirely suffices to
compute the expectation

167
00:07:38,730 --> 00:07:41,960
value of p to the fourth if
you take into account the fact

168
00:07:41,960 --> 00:07:46,130
that the energy eigenfunctions
satisfy the original energy

169
00:07:46,130 --> 00:07:47,660
eigenvalue equation.

170
00:07:47,660 --> 00:07:51,540
So p squared upon 2m
plus v of x of v of r

171
00:07:51,540 --> 00:07:55,720
is equal to e when acting
on those wave functions.

172
00:07:55,720 --> 00:07:57,930
So for computing
these guys, you could

173
00:07:57,930 --> 00:08:01,840
try to do this brute force,
not unrelated to problem,

174
00:08:01,840 --> 00:08:05,170
I believe, two on Rydberg
atoms on the problem set.

175
00:08:05,170 --> 00:08:06,670
But there are
actually sneakier ways

176
00:08:06,670 --> 00:08:11,580
to do these expectation
values as well.

177
00:08:11,580 --> 00:08:16,690
And I will leave that to you.

178
00:08:16,690 --> 00:08:19,810
But let me emphasize that you
can do a direct brute force

179
00:08:19,810 --> 00:08:22,780
calculation, but
you don't need to.

180
00:08:22,780 --> 00:08:24,580
And I would encourage
you to try to find

181
00:08:24,580 --> 00:08:29,375
an efficient, indirect way
to do these calculations.

182
00:08:29,375 --> 00:08:30,790
Did that answer your question?

183
00:08:30,790 --> 00:08:32,500
OK.

184
00:08:32,500 --> 00:08:34,409
Anything else?

185
00:08:34,409 --> 00:08:36,255
Yeah.

186
00:08:36,255 --> 00:08:38,183
AUDIENCE: So is [INAUDIBLE]
a time dependence

187
00:08:38,183 --> 00:08:45,368
to an operator with the
[INAUDIBLE] being [INAUDIBLE]?

188
00:08:45,368 --> 00:08:47,600
PROFESSOR: Does
adding-- well, it

189
00:08:47,600 --> 00:08:51,309
depends on how you introduce
the time dependents.

190
00:08:51,309 --> 00:08:52,850
AUDIENCE: Can a
operator [INAUDIBLE]?

191
00:08:56,780 --> 00:08:57,860
PROFESSOR: Yes.

192
00:08:57,860 --> 00:08:59,910
Yeah, absolutely.

193
00:08:59,910 --> 00:09:02,492
What's the reason
for the question?

194
00:09:02,492 --> 00:09:04,239
AUDIENCE: One of
the problems here.

195
00:09:04,239 --> 00:09:06,322
PROFESSOR: One of the
problems in the problem set?

196
00:09:06,322 --> 00:09:06,947
AUDIENCE: Yeah.

197
00:09:06,947 --> 00:09:08,740
It's like the last part of four.

198
00:09:08,740 --> 00:09:10,365
PROFESSOR: Remind me
which problem this

199
00:09:10,365 --> 00:09:11,445
is, I don't remember.

200
00:09:11,445 --> 00:09:13,007
AUDIENCE: [INAUDIBLE].

201
00:09:13,007 --> 00:09:13,590
PROFESSOR: Oh.

202
00:09:13,590 --> 00:09:14,727
Oh, oh!

203
00:09:14,727 --> 00:09:15,227
Ha, ha, ha.

204
00:09:18,640 --> 00:09:20,880
Sorry, I really
like that problem.

205
00:09:27,010 --> 00:09:29,210
Let me rephrase your question
in the following way.

206
00:09:34,640 --> 00:09:37,090
Is it intrinsically
non-Hermitian

207
00:09:37,090 --> 00:09:39,040
to have time
dependence in a system?

208
00:09:41,700 --> 00:09:44,280
So no, you can have-- so
what does Hermitian mean?

209
00:09:44,280 --> 00:09:46,380
Physically, what does
it mean for the energy

210
00:09:46,380 --> 00:09:47,579
to be Hermitian for example?

211
00:09:47,579 --> 00:09:49,120
What it means for
the energy operator

212
00:09:49,120 --> 00:09:54,160
to be Hermitian is that time
evolution, which is represented

213
00:09:54,160 --> 00:09:56,644
by the oper-- so let
me phrase it this way.

214
00:09:56,644 --> 00:09:58,810
So if we know that the
energy operator is Hermitian,

215
00:09:58,810 --> 00:09:59,720
what is that telling us?

216
00:09:59,720 --> 00:10:00,550
Well, it tells us
a lot of things.

217
00:10:00,550 --> 00:10:02,581
It tells us the energy
eigenvalues are real.

218
00:10:02,581 --> 00:10:03,080
That's good.

219
00:10:03,080 --> 00:10:06,267
So E dagger is equal to E.
But it tell us something else.

220
00:10:06,267 --> 00:10:07,850
Remember that the
Schrodinger equation

221
00:10:07,850 --> 00:10:14,110
is that ih bar dt on psi
is equal to E on psi.

222
00:10:14,110 --> 00:10:18,171
And we use this to argue
that the general solution

223
00:10:18,171 --> 00:10:19,920
to the Schrodinger
equation can be written

224
00:10:19,920 --> 00:10:22,960
in the following
elegant way, psi of t

225
00:10:22,960 --> 00:10:37,010
is equal to e to the minus
i upon h bar tE on psi at 0,

226
00:10:37,010 --> 00:10:41,730
where this was the
evolution operator u sub t.

227
00:10:41,730 --> 00:10:42,600
Yeah?

228
00:10:42,600 --> 00:10:44,560
And this is a unitary operator.

229
00:10:44,560 --> 00:10:47,122
And the way they say its
unitary goes back to the exam.

230
00:10:47,122 --> 00:10:48,580
The way to say it's
unitary is this

231
00:10:48,580 --> 00:10:51,200
is e to the i
Hermitian operator.

232
00:10:51,200 --> 00:10:53,400
t is a real number,
h is a real number.

233
00:10:53,400 --> 00:10:56,160
So this is e to the
i Hermitian operator.

234
00:10:56,160 --> 00:10:58,990
And anything of the form e
to the i Hermitian operator

235
00:10:58,990 --> 00:10:59,640
is unitary.

236
00:10:59,640 --> 00:11:01,450
It's adjoint to its inverse.

237
00:11:01,450 --> 00:11:04,930
What that tells you is
that since this is unitary,

238
00:11:04,930 --> 00:11:08,005
it preserves the magnitude, or
the norm, of the wave function.

239
00:11:08,005 --> 00:11:09,860
So probability is conserved.

240
00:11:09,860 --> 00:11:12,360
What would it mean for
the energy operator

241
00:11:12,360 --> 00:11:14,740
to be not Hermitian?

242
00:11:14,740 --> 00:11:16,240
Well, it would mean
a lot of things.

243
00:11:16,240 --> 00:11:17,880
One thing it would mean is
that the energy eigenvalues

244
00:11:17,880 --> 00:11:19,114
are no longer real.

245
00:11:19,114 --> 00:11:20,030
That's a little weird.

246
00:11:20,030 --> 00:11:21,430
But the much more
troubling thing

247
00:11:21,430 --> 00:11:24,000
is that the energy would
no longer be-- sorry,

248
00:11:24,000 --> 00:11:26,490
the probability would
no longer be conserved.

249
00:11:26,490 --> 00:11:29,430
The evolution operator, which is
the solution to the Schrodinger

250
00:11:29,430 --> 00:11:32,020
equation, would no longer
be a unitary operator.

251
00:11:32,020 --> 00:11:34,810
And the probability, the
norm or the wave function,

252
00:11:34,810 --> 00:11:37,210
would no longer be conserved.

253
00:11:37,210 --> 00:11:40,310
So the question at the
end of-- the last question

254
00:11:40,310 --> 00:11:42,600
in problem four
is really asking,

255
00:11:42,600 --> 00:11:44,670
in the system you're
thinking about

256
00:11:44,670 --> 00:11:46,040
is, probability conserved?

257
00:11:49,080 --> 00:11:53,190
And that's the question that
you should be asking yourself

258
00:11:53,190 --> 00:11:56,350
when you finish up problem four.

259
00:11:56,350 --> 00:11:57,990
Good?

260
00:11:57,990 --> 00:11:59,410
Yeah.

261
00:11:59,410 --> 00:12:01,810
AUDIENCE: When we solve
the Schrodinger equation

262
00:12:01,810 --> 00:12:04,795
in that way, does this
solution come from the fact

263
00:12:04,795 --> 00:12:08,971
that E is Hermitian or that
E is like that nice little

264
00:12:08,971 --> 00:12:10,839
[INAUDIBLE], that E
is time dependent.

265
00:12:10,839 --> 00:12:11,889
And that if E was
time dependent,

266
00:12:11,889 --> 00:12:14,305
why couldn't the coefficients
for the probability also be?

267
00:12:14,305 --> 00:12:15,910
PROFESSOR: Absolutely.

268
00:12:15,910 --> 00:12:18,070
So here I was just focusing
on the Hermiticity.

269
00:12:18,070 --> 00:12:19,880
And in solving
this, I'm assuming

270
00:12:19,880 --> 00:12:21,380
that the energy is
time independent.

271
00:12:21,380 --> 00:12:23,000
If the energy is not
time independent,

272
00:12:23,000 --> 00:12:26,117
then this is not the right
answer as you're pointing out.

273
00:12:26,117 --> 00:12:27,700
So in fact, you have
to do an integral

274
00:12:27,700 --> 00:12:29,116
and you have to
time order things,

275
00:12:29,116 --> 00:12:30,980
and it's a complicated story.

276
00:12:30,980 --> 00:12:32,380
But this is not the solution.

277
00:12:32,380 --> 00:12:36,150
But even if we have a
time independent system,

278
00:12:36,150 --> 00:12:40,980
if the energy operator is not
Hermitian, this is not unitary.

279
00:12:40,980 --> 00:12:43,200
So indeed, you're absolutely
right that if the energy

280
00:12:43,200 --> 00:12:44,460
operator is time
dependent, the story's

281
00:12:44,460 --> 00:12:46,010
more complicated than just this.

282
00:12:46,010 --> 00:12:49,410
But we already see the problem
that's salient for problem four

283
00:12:49,410 --> 00:12:52,070
at this stage with a time
independent operator.

284
00:12:52,070 --> 00:12:52,692
Yeah.

285
00:12:52,692 --> 00:12:54,150
AUDIENCE: Just one
question please.

286
00:12:54,150 --> 00:12:57,380
So if I had some sort of
system which is [INAUDIBLE],

287
00:12:57,380 --> 00:12:59,534
some of my particles are
weak in the other process.

288
00:12:59,534 --> 00:13:00,470
PROFESSOR: Yeah.

289
00:13:00,470 --> 00:13:02,836
AUDIENCE: And now the
probability is not conserved.

290
00:13:02,836 --> 00:13:05,850
Saying that this thing is not
[INAUDIBLE], the issue is,

291
00:13:05,850 --> 00:13:08,548
I think, which is also extremely
disturbing for [INAUDIBLE]

292
00:13:08,548 --> 00:13:11,880
complex in energy,
[INAUDIBLE], what is that?

293
00:13:11,880 --> 00:13:12,957
PROFESSOR: Yeah.

294
00:13:12,957 --> 00:13:14,290
AUDIENCE: That seems even worse.

295
00:13:14,290 --> 00:13:16,289
PROFESSOR: Well, I guess
it's a matter of taste.

296
00:13:19,367 --> 00:13:21,700
I would say that they're the
same thing in the following

297
00:13:21,700 --> 00:13:22,080
sense.

298
00:13:22,080 --> 00:13:23,650
Suppose I have complex
energy eigenvalues.

299
00:13:23,650 --> 00:13:25,700
Well, I know that if I
have a stationary state,

300
00:13:25,700 --> 00:13:28,430
then that stationary state
as a function of time

301
00:13:28,430 --> 00:13:31,630
is equal to the stationary
state at time 0 times

302
00:13:31,630 --> 00:13:35,370
e-- so this is an energy
eigenfunction rather than-- is

303
00:13:35,370 --> 00:13:39,830
e to the minus i the
energy times t over h bar?

304
00:13:39,830 --> 00:13:42,750
Or this is the
energy eigenvalue.

305
00:13:42,750 --> 00:13:44,780
Now, imagine e is complex.

306
00:13:44,780 --> 00:13:47,390
Sorry, first imagine e is real.

307
00:13:47,390 --> 00:13:50,110
That's not hard to imagine
because it's usually the case.

308
00:13:50,110 --> 00:13:53,510
And if that's true, what
happens to the wave function

309
00:13:53,510 --> 00:13:54,670
as we evolve through time?

310
00:13:54,670 --> 00:13:56,090
It rotates by phase.

311
00:13:56,090 --> 00:14:01,171
Now, if e is complex, imagine
e is of the form e-- let e

312
00:14:01,171 --> 00:14:01,670
be complex.

313
00:14:01,670 --> 00:14:10,740
So e is going to be E real minus
i gamma, some imaginary piece.

314
00:14:10,740 --> 00:14:15,370
So what that's going to give me
is an e to the minus gamma t.

315
00:14:15,370 --> 00:14:18,800
And that's a decaying thing that
when you get to norm squared

316
00:14:18,800 --> 00:14:20,640
gives you a decaying
envelope over time.

317
00:14:20,640 --> 00:14:22,610
It's going to give you
an exponential decay.

318
00:14:22,610 --> 00:14:25,310
This is going to be equal
to e to the minus ie

319
00:14:25,310 --> 00:14:31,490
real t over h bar times
e to be minus gamma

320
00:14:31,490 --> 00:14:34,849
t over h bar phi of 0.

321
00:14:34,849 --> 00:14:37,140
And when we take the norm
squared, the phase goes away,

322
00:14:37,140 --> 00:14:38,570
but this doesn't.

323
00:14:38,570 --> 00:14:44,110
So having loss of probability
is having a complex energy

324
00:14:44,110 --> 00:14:45,586
eigenvalue.

325
00:14:45,586 --> 00:14:46,570
Yeah.

326
00:14:46,570 --> 00:14:50,506
AUDIENCE: Why are we
talking about time

327
00:14:50,506 --> 00:14:53,950
independent operators if
there's an electromagnetic field

328
00:14:53,950 --> 00:14:56,642
and stuff is clearly going
on between the avenue

329
00:14:56,642 --> 00:14:56,950
of the electromagnetic field?

330
00:14:56,950 --> 00:14:58,080
PROFESSOR: This is a
very good question.

331
00:14:58,080 --> 00:15:00,070
And that actually
leads me into the thing

332
00:15:00,070 --> 00:15:01,375
I wanted to talk about first.

333
00:15:01,375 --> 00:15:03,750
So I got a bunch of questions--
thanks for that question.

334
00:15:03,750 --> 00:15:05,791
I got a bunch of questions
over the past few days

335
00:15:05,791 --> 00:15:11,180
about the magnetic moment of
the hydrogen system and what

336
00:15:11,180 --> 00:15:12,260
a strange idea that is.

337
00:15:12,260 --> 00:15:13,980
So let me talk about
that for a second.

338
00:15:13,980 --> 00:15:16,390
And when I'm done with
this little spiel,

339
00:15:16,390 --> 00:15:19,730
tell me if I've
answered your question.

340
00:15:19,730 --> 00:15:22,140
Well, let's just leave this up.

341
00:15:22,140 --> 00:15:22,640
OK.

342
00:15:22,640 --> 00:15:26,370
So in particular, let's think
about this term for a moment.

343
00:15:28,880 --> 00:15:31,130
Where this term came from
last time was we said,

344
00:15:31,130 --> 00:15:34,160
look, we turn on an
external magnetic field.

345
00:15:34,160 --> 00:15:36,500
Someone turns on the
switch and current runs

346
00:15:36,500 --> 00:15:39,870
through an electromagnet and
we get a uniform magnetic field

347
00:15:39,870 --> 00:15:42,360
in our fiducial volume, where
we're doing the experiment.

348
00:15:42,360 --> 00:15:48,760
And the electron system, if it
carries some angular momentum,

349
00:15:48,760 --> 00:15:50,747
it also has a charge,
angular momentum charge.

350
00:15:50,747 --> 00:15:52,330
That means it's got
a magnetic moment.

351
00:15:52,330 --> 00:15:54,950
That's how much magnetic moment.

352
00:15:54,950 --> 00:15:58,180
So this is saying that the
magnetic moment of the electron

353
00:15:58,180 --> 00:16:01,270
bound to the proton
with angular momentum

354
00:16:01,270 --> 00:16:03,600
wants to anti-align
with the magnetic field.

355
00:16:03,600 --> 00:16:05,943
Or align in this case
because I put the wrong sign.

356
00:16:09,330 --> 00:16:14,123
But here's something really
upsetting about that.

357
00:16:14,123 --> 00:16:16,230
So what I just said
sounds crazy if you

358
00:16:16,230 --> 00:16:18,050
think about it in
the following way.

359
00:16:18,050 --> 00:16:21,180
Take an electron in
the coulomb system,

360
00:16:21,180 --> 00:16:22,530
just straight up coulomb.

361
00:16:22,530 --> 00:16:24,160
Take an electron in
the coulomb system,

362
00:16:24,160 --> 00:16:28,860
put it in, say, the ground
state, lowest energy state.

363
00:16:28,860 --> 00:16:30,350
Is it moving?

364
00:16:30,350 --> 00:16:32,500
This is problem four.

365
00:16:32,500 --> 00:16:34,850
So is it moving?

366
00:16:34,850 --> 00:16:36,490
The expectation
value of position

367
00:16:36,490 --> 00:16:38,360
doesn't change in time.

368
00:16:38,360 --> 00:16:40,180
It's a stationary state.

369
00:16:40,180 --> 00:16:44,677
Expectation value of
nothing changes in time.

370
00:16:44,677 --> 00:16:46,135
OK, fine, but it's
the ground state

371
00:16:46,135 --> 00:16:47,170
that carries your
angular momentum.

372
00:16:47,170 --> 00:16:48,170
That's not so upsetting.

373
00:16:48,170 --> 00:16:50,950
Go to the first excited
state with angular momentum.

374
00:16:50,950 --> 00:16:54,200
The n equals 2, l equals
1, m equals 1 state.

375
00:16:54,200 --> 00:16:58,390
So it's got as much angular
momentum in lz as possible.

376
00:16:58,390 --> 00:17:00,651
Is that thing moving?

377
00:17:00,651 --> 00:17:01,150
Yeah.

378
00:17:01,150 --> 00:17:02,870
It's not moving at all.

379
00:17:02,870 --> 00:17:05,670
And yet, we're saying
there's a current associated

380
00:17:05,670 --> 00:17:07,760
with it, an electric current.

381
00:17:07,760 --> 00:17:10,550
And that electric current is
inducing a magnetic moment,

382
00:17:10,550 --> 00:17:12,859
a la right-hand rule.

383
00:17:12,859 --> 00:17:15,780
Maybe via Biot-Savart,
if you want to be fancy.

384
00:17:15,780 --> 00:17:19,540
And that magnetic moment--
a little current loop--

385
00:17:19,540 --> 00:17:23,325
is leading to this interaction,
the Zeeman's interaction.

386
00:17:25,829 --> 00:17:28,750
We know that it's true because
experiments show the Zeeman

387
00:17:28,750 --> 00:17:30,070
splitting.

388
00:17:30,070 --> 00:17:32,300
So it is definitely
true that there

389
00:17:32,300 --> 00:17:34,050
is a magnetic moment
of this thing.

390
00:17:34,050 --> 00:17:37,805
But it's not moving,
how can that be?

391
00:17:37,805 --> 00:17:39,930
So how can there be a
current if nothing is moving?

392
00:17:44,394 --> 00:17:47,900
AUDIENCE: It is, in some
sense moving, [INAUDIBLE].

393
00:17:47,900 --> 00:17:51,539
Why is the expectation value of
the [INAUDIBLE] to stop moving?

394
00:17:53,806 --> 00:17:55,180
PROFESSOR: What's
the expectation

395
00:17:55,180 --> 00:17:55,900
value of the momentum?

396
00:17:55,900 --> 00:17:56,680
AUDIENCE: I think
it's also probability.

397
00:17:56,680 --> 00:17:57,680
PROFESSOR: Yeah, it's 0.

398
00:18:01,757 --> 00:18:05,813
AUDIENCE: I mean, first of
all, the world isn't classical.

399
00:18:05,813 --> 00:18:08,440
You can't use
classical intuition.

400
00:18:08,440 --> 00:18:09,590
PROFESSOR: OK, true.

401
00:18:09,590 --> 00:18:13,520
AUDIENCE: Second of
all, let's suppose

402
00:18:13,520 --> 00:18:18,753
you have a uniform ring
of classical charge

403
00:18:18,753 --> 00:18:19,785
and set it spinning.

404
00:18:19,785 --> 00:18:20,410
PROFESSOR: Yes.

405
00:18:20,410 --> 00:18:23,372
AUDIENCE: That ring is
as a ring, not moving,

406
00:18:23,372 --> 00:18:25,330
but there's still a
current associated with it.

407
00:18:25,330 --> 00:18:28,274
PROFESSOR: So you're saying
that the electron is a ring?

408
00:18:28,274 --> 00:18:31,060
AUDIENCE: It might make sense
if you give it that [INAUDIBLE].

409
00:18:31,060 --> 00:18:31,590
PROFESSOR: I like this.

410
00:18:31,590 --> 00:18:32,340
No, this is good.

411
00:18:32,340 --> 00:18:33,520
It's wrong, but it's good.

412
00:18:33,520 --> 00:18:35,610
And the reason it's good
is that you're really

413
00:18:35,610 --> 00:18:37,620
pushing your assumptions
to try to figure out

414
00:18:37,620 --> 00:18:39,205
how the experimental
data can possibly match.

415
00:18:39,205 --> 00:18:41,630
And you're saying, look, we have
to just reject our intuition.

416
00:18:41,630 --> 00:18:43,454
Our intuition is clearly
leading us astray.

417
00:18:43,454 --> 00:18:44,120
And I like that.

418
00:18:44,120 --> 00:18:45,000
That's correct.

419
00:18:45,000 --> 00:18:46,360
So what we're going to do
in the next few minutes

420
00:18:46,360 --> 00:18:48,950
is work through that and try
to find the best way to phrase

421
00:18:48,950 --> 00:18:49,680
that.

422
00:18:49,680 --> 00:18:52,030
Your strategy is
the correct one.

423
00:18:52,030 --> 00:18:54,420
So let me rephrase
that slightly.

424
00:18:54,420 --> 00:18:58,796
Look, if you have a classical
distribution of charge,

425
00:18:58,796 --> 00:19:00,170
that distribution
of charge could

426
00:19:00,170 --> 00:19:05,217
be a stationary
distribution-- a distribution,

427
00:19:05,217 --> 00:19:07,550
which as a distribution of
charge doesn't change a dime.

428
00:19:07,550 --> 00:19:09,508
But each individual charge
in that distribution

429
00:19:09,508 --> 00:19:10,559
is itself moving.

430
00:19:10,559 --> 00:19:12,850
The problem here is that we
just have the one electron.

431
00:19:12,850 --> 00:19:16,590
If you ever look, you will
find the electron at a spot.

432
00:19:16,590 --> 00:19:19,119
But what you're really
saying is, look,

433
00:19:19,119 --> 00:19:21,160
it's a mistake to think
about the electron having

434
00:19:21,160 --> 00:19:22,460
a definite position
in the first place.

435
00:19:22,460 --> 00:19:23,340
You just shouldn't
think about that.

436
00:19:23,340 --> 00:19:24,460
The best you can
say is that it has

437
00:19:24,460 --> 00:19:26,370
some probability of
being in any spot.

438
00:19:26,370 --> 00:19:27,746
So let's work with that.

439
00:19:27,746 --> 00:19:29,370
Let's take that idea
and let's push it.

440
00:19:29,370 --> 00:19:31,911
Let's see how far we can take
this idea that the electron has

441
00:19:31,911 --> 00:19:33,310
some probability [INAUDIBLE].

442
00:19:33,310 --> 00:19:36,280
So suppose we have an
electron in a stationary state

443
00:19:36,280 --> 00:19:37,476
of the coulomb potential.

444
00:19:37,476 --> 00:19:38,850
The stationary
states are labeled

445
00:19:38,850 --> 00:19:40,687
by three integers-- n, l, and m.

446
00:19:40,687 --> 00:19:42,270
And we'd written
their wave functions.

447
00:19:42,270 --> 00:19:44,270
There are, of course, an
infinite number of ways

448
00:19:44,270 --> 00:19:46,522
to write in different notation.

449
00:19:46,522 --> 00:19:48,230
There are an infinite
number of fonts one

450
00:19:48,230 --> 00:19:55,025
could use for the normalization
constant 1/r R nl of little

451
00:19:55,025 --> 00:19:59,500
r, Y lm of theta and phi.

452
00:20:03,070 --> 00:20:04,350
And I'm going to rewrite this.

453
00:20:04,350 --> 00:20:05,933
I'm going to expand
this out slightly.

454
00:20:05,933 --> 00:20:12,620
This is N 1/r R of R and l.

455
00:20:12,620 --> 00:20:20,310
And Ylm, remember, was of the
form Pl of cosine theta times

456
00:20:20,310 --> 00:20:23,442
e to the im phi.

457
00:20:23,442 --> 00:20:25,420
Everyone cool with that?

458
00:20:25,420 --> 00:20:27,790
So I just wrote the spherical
harmonic as a polynomial

459
00:20:27,790 --> 00:20:30,935
in cosine theta times
an exponential in phi.

460
00:20:30,935 --> 00:20:33,060
AUDIENCE: Where did the
1/r come from, on the left?

461
00:20:33,060 --> 00:20:36,010
PROFESSOR: The 1/r
was-- so most people

462
00:20:36,010 --> 00:20:38,430
call capital R the whole thing.

463
00:20:38,430 --> 00:20:40,060
But this is the
pulling out to 1/r

464
00:20:40,060 --> 00:20:41,604
to simplify the
radio wave equation.

465
00:20:41,604 --> 00:20:43,020
And the reason I
prefer writing it

466
00:20:43,020 --> 00:20:44,760
this way is just that
this guy satisfies

467
00:20:44,760 --> 00:20:47,200
a simple 1d
Schrodinger equation.

468
00:20:47,200 --> 00:20:48,450
AUDIENCE: That was the u of r?

469
00:20:48,450 --> 00:20:49,950
PROFESSOR: That was a thing
that we called u of r,

470
00:20:49,950 --> 00:20:51,190
and then I confused the
hell out of everyone

471
00:20:51,190 --> 00:20:53,106
by calling three different
things on the board

472
00:20:53,106 --> 00:20:55,160
u, which was sort
of unnecessary.

473
00:20:55,160 --> 00:20:58,540
So this is the artist
previously known as u.

474
00:21:03,320 --> 00:21:05,050
The dude can sing.

475
00:21:05,050 --> 00:21:09,980
OK, so the first
question I want to ask

476
00:21:09,980 --> 00:21:11,730
is, so this is a
stationary state.

477
00:21:11,730 --> 00:21:14,850
Is the electron moving?

478
00:21:14,850 --> 00:21:16,640
No, not in any
conventional sense.

479
00:21:16,640 --> 00:21:19,040
If you compute the expectation
value of the position

480
00:21:19,040 --> 00:21:21,700
and you take its time
derivative, this is zero.

481
00:21:21,700 --> 00:21:23,677
We could do this either
by calculating it

482
00:21:23,677 --> 00:21:25,760
or just by observing that
it's a stationary state.

483
00:21:25,760 --> 00:21:30,520
And on principle, it
can't change in time.

484
00:21:30,520 --> 00:21:34,740
So this guy is not moving
in any conventional sense.

485
00:21:39,191 --> 00:21:40,690
So why is there an
electric current?

486
00:21:40,690 --> 00:21:43,300
Why do we get the Zeeman
magnetic moment, the Bohr

487
00:21:43,300 --> 00:21:45,420
magneton?

488
00:21:45,420 --> 00:21:48,750
So as was pointed out, look,
this is quantum mechanics.

489
00:21:48,750 --> 00:21:50,480
It's not classical mechanics.

490
00:21:50,480 --> 00:21:54,720
And in quantum mechanics, the
electron isn't at any point.

491
00:21:58,020 --> 00:22:00,500
Rather, there's some
probability density.

492
00:22:00,500 --> 00:22:02,130
And the probability
density that we

493
00:22:02,130 --> 00:22:06,800
find the electron at
some point r is psi

494
00:22:06,800 --> 00:22:15,630
squared l m or r squared.

495
00:22:15,630 --> 00:22:18,620
A familiar beast.

496
00:22:18,620 --> 00:22:21,730
And meanwhile, a wonderful
thing about the probability

497
00:22:21,730 --> 00:22:23,270
distribution in
quantum mechanics

498
00:22:23,270 --> 00:22:26,890
that we've already discussed
here is that it's conserved.

499
00:22:26,890 --> 00:22:31,350
The time derivative of the
probability density or r--

500
00:22:31,350 --> 00:22:33,850
remember, this is a
density, not a probability.

501
00:22:33,850 --> 00:22:37,920
The time rate of change of
the probability distribution

502
00:22:37,920 --> 00:22:45,070
is minus the gradient-- the
divergence of the probability

503
00:22:45,070 --> 00:22:49,250
current, where the
probability current j

504
00:22:49,250 --> 00:22:55,520
is equal to h bar
over the mass, which

505
00:22:55,520 --> 00:22:56,770
I'm going to call the mass mu.

506
00:22:56,770 --> 00:22:58,144
Oh, gee, I don't
want-- I'm going

507
00:22:58,144 --> 00:23:00,900
to call the mass
capital M. That's

508
00:23:00,900 --> 00:23:03,364
the mass of the
electron, which is

509
00:23:03,364 --> 00:23:05,530
a little strange because
it's a very small quantity.

510
00:23:05,530 --> 00:23:10,020
But anyway, h over capital
M of the imaginary part

511
00:23:10,020 --> 00:23:16,370
of psi complex
conjugate gradient psi.

512
00:23:16,370 --> 00:23:21,600
And you showed on a problem
set long ago that this is true

513
00:23:21,600 --> 00:23:25,180
if j takes this form by virtue
of the Schrodinger equation.

514
00:23:25,180 --> 00:23:28,610
Now, we usually write this
imaginary 1 over 2i times psi

515
00:23:28,610 --> 00:23:32,280
gradient-- or psi star gradient
psi minus psi gradient psi

516
00:23:32,280 --> 00:23:34,980
star, but that's equal
to the imaginary part

517
00:23:34,980 --> 00:23:38,030
of the first term.

518
00:23:38,030 --> 00:23:40,280
The thing you want to emphasize
is the imaginary part.

519
00:23:44,690 --> 00:23:46,570
So we have this current.

520
00:23:46,570 --> 00:23:51,290
In our system, the
position expectation value

521
00:23:51,290 --> 00:23:52,660
is time independent.

522
00:23:52,660 --> 00:23:54,620
And indeed, it's a
stationary state.

523
00:23:54,620 --> 00:23:56,930
So beyond the position
expectation value being time

524
00:23:56,930 --> 00:23:58,720
independent, the
probability density

525
00:23:58,720 --> 00:24:01,020
itself is time independent
because the wave function

526
00:24:01,020 --> 00:24:03,977
evolves by an overall phase
and the probability density

527
00:24:03,977 --> 00:24:04,810
is the norm squared.

528
00:24:04,810 --> 00:24:06,850
So the phase goes away.

529
00:24:06,850 --> 00:24:10,630
So in our system, in an
electron in this state

530
00:24:10,630 --> 00:24:12,380
in the coulomb
potential, the time rate

531
00:24:12,380 --> 00:24:15,690
of change of the
probability density--

532
00:24:15,690 --> 00:24:17,265
let me actually
do this over here.

533
00:24:20,830 --> 00:24:30,980
So in the stationary
state, psi l, n, m,

534
00:24:30,980 --> 00:24:33,950
the time rate of change
of the density is 0.

535
00:24:33,950 --> 00:24:36,920
And this tells us by the
conservation equation

536
00:24:36,920 --> 00:24:41,530
that the divergence of
the current is also 0.

537
00:24:41,530 --> 00:24:43,609
Does that tell us
that the current is 0?

538
00:24:43,609 --> 00:24:44,150
AUDIENCE: No.

539
00:24:44,150 --> 00:24:45,174
PROFESSOR: Right.

540
00:24:45,174 --> 00:24:46,090
So what's the current?

541
00:24:51,410 --> 00:24:54,170
Well, we've written everything
here in spherical coordinates.

542
00:24:54,170 --> 00:25:00,080
And the current is given in
terms of the gradient operator.

543
00:25:00,080 --> 00:25:03,880
So let me remind you quickly of
what the gradient operator is

544
00:25:03,880 --> 00:25:05,080
in spherical coordinates.

545
00:25:05,080 --> 00:25:07,430
It has three components--
a radial component,

546
00:25:07,430 --> 00:25:11,680
a theta component, and
then a phi component.

547
00:25:11,680 --> 00:25:17,130
And the radial part is just D r.

548
00:25:17,130 --> 00:25:22,360
The theta component
is 1 over r D theta.

549
00:25:22,360 --> 00:25:26,310
And the phi around
the equator component

550
00:25:26,310 --> 00:25:33,460
is equal to 1 over
r sine theta D phi.

551
00:25:33,460 --> 00:25:36,360
Everyone cool with that?

552
00:25:36,360 --> 00:25:37,570
So what's j?

553
00:25:37,570 --> 00:25:39,450
Well, j is going to
have a component, j

554
00:25:39,450 --> 00:25:40,500
in the radial direction.

555
00:25:40,500 --> 00:25:42,680
The current in the
radial direction.

556
00:25:42,680 --> 00:25:45,660
And intuitively,
what should that be?

557
00:25:45,660 --> 00:25:50,560
Is there stuff going out or in?

558
00:25:50,560 --> 00:25:51,650
There shouldn't be.

559
00:25:51,650 --> 00:25:54,990
It's hydrogen.

560
00:25:54,990 --> 00:25:56,570
It's not doing this.

561
00:25:56,570 --> 00:25:57,729
So this should be 0.

562
00:25:57,729 --> 00:25:59,270
And we can just
quickly see that this

563
00:25:59,270 --> 00:26:01,187
is h bar upon M imaginary
part of-- well, this

564
00:26:01,187 --> 00:26:03,311
is the r component, it's
going to be the derivative

565
00:26:03,311 --> 00:26:04,610
in the radial direction.

566
00:26:04,610 --> 00:26:08,585
But the derivative in the radial
direction is going to be real.

567
00:26:11,132 --> 00:26:13,590
The derivative in the radial
direction is going to be real.

568
00:26:13,590 --> 00:26:14,810
So when we take
the norm squared,

569
00:26:14,810 --> 00:26:15,600
we don't pick up anything.

570
00:26:15,600 --> 00:26:16,740
We pick up an
overall coefficient.

571
00:26:16,740 --> 00:26:18,031
It's going to be strictly real.

572
00:26:18,031 --> 00:26:20,647
The phase, e to the
i m phi cancels out

573
00:26:20,647 --> 00:26:22,230
because this is the
complex conjugate.

574
00:26:22,230 --> 00:26:26,850
So the imaginary part of
this thing is going to be 0.

575
00:26:26,850 --> 00:26:28,020
OK?

576
00:26:28,020 --> 00:26:31,020
So J r is 0.

577
00:26:31,020 --> 00:26:36,935
And similarly, J theta
p is a real function.

578
00:26:36,935 --> 00:26:39,660
When we take a derivative,
we get a real function.

579
00:26:39,660 --> 00:26:41,850
And then when we multiply
by its complex conjugate,

580
00:26:41,850 --> 00:26:43,516
again, the phase
cancels out and we just

581
00:26:43,516 --> 00:26:47,650
get a whole bunch of real stuff
whose imaginary piece is 0.

582
00:26:47,650 --> 00:26:50,390
So J theta is 0.

583
00:26:50,390 --> 00:26:52,115
But J phi is a cool one.

584
00:26:55,680 --> 00:26:59,650
In my head, I was
just thinking J 5.

585
00:26:59,650 --> 00:27:01,115
OK, at least someone got that.

586
00:27:05,600 --> 00:27:09,655
So J phi, however,
is not going to be

587
00:27:09,655 --> 00:27:10,780
0 for the following reason.

588
00:27:10,780 --> 00:27:11,740
So what is it equal to?

589
00:27:11,740 --> 00:27:15,360
It's equal to h bar
upon M-- the mass,

590
00:27:15,360 --> 00:27:19,800
M-- times the imaginary
part of-- well,

591
00:27:19,800 --> 00:27:23,780
psi star, which is
psi complex conjugate.

592
00:27:23,780 --> 00:27:29,440
And then the gradient
with respect to phi.

593
00:27:29,440 --> 00:27:32,604
The phi component is
1 over r sine theta.

594
00:27:32,604 --> 00:27:34,270
And then derivative
with respect to phi.

595
00:27:34,270 --> 00:27:39,280
The derivative with respect
to phi pulls down a i m.

596
00:27:39,280 --> 00:27:45,640
Oh, look-- times psi, which
is equal to h bar upon m,

597
00:27:45,640 --> 00:27:48,000
times-- well, psi
is norm squared.

598
00:27:48,000 --> 00:27:49,820
That's real.

599
00:27:49,820 --> 00:27:51,040
Psi squared.

600
00:27:51,040 --> 00:27:53,020
1/r sine theta.

601
00:27:53,020 --> 00:27:54,560
That's real, so I
can pull that out.

602
00:27:54,560 --> 00:27:56,610
And we are left
with imaginary part

603
00:27:56,610 --> 00:28:02,510
of im, which is just
m, which is not 0.

604
00:28:02,510 --> 00:28:05,320
And in particular, it's
proportional to h bar m.

605
00:28:11,050 --> 00:28:13,075
Everyone see that?

606
00:28:13,075 --> 00:28:19,540
So what this is saying is that
the current, the probability

607
00:28:19,540 --> 00:28:24,130
current, for an electron
in the stationary state psi

608
00:28:24,130 --> 00:28:31,710
nlm of the Coulomb potential
is equal to norm psi squared

609
00:28:31,710 --> 00:28:41,390
over m r sine theta h bar
m in the phi direction.

610
00:28:41,390 --> 00:28:45,220
The phi uni-vector.

611
00:28:45,220 --> 00:28:47,230
Cool?

612
00:28:47,230 --> 00:28:49,660
So nothing is moving,
but there's a current.

613
00:28:52,350 --> 00:28:53,240
What is moving?

614
00:28:58,130 --> 00:29:00,260
What is the thing of
whose current-- of whom

615
00:29:00,260 --> 00:29:01,220
this is the current?

616
00:29:01,220 --> 00:29:02,250
AUDIENCE: Probability density.

617
00:29:02,250 --> 00:29:03,708
PROFESSOR: The
probability density.

618
00:29:03,708 --> 00:29:08,450
The probability
density is rotating.

619
00:29:08,450 --> 00:29:13,500
So what this is telling
us is that it's true

620
00:29:13,500 --> 00:29:15,680
the system is stationary.

621
00:29:15,680 --> 00:29:20,400
But I want to know-- I'm looking
down at the equatorial plane,

622
00:29:20,400 --> 00:29:20,900
OK.

623
00:29:20,900 --> 00:29:22,566
So this plot is going
to be looking down

624
00:29:22,566 --> 00:29:25,200
on the equatorial
plane of hydrogen.

625
00:29:25,200 --> 00:29:27,265
Here is the origin, the
center of the potential.

626
00:29:27,265 --> 00:29:28,640
And what this is
telling-- so I'm

627
00:29:28,640 --> 00:29:30,000
going to draw these vectors.

628
00:29:30,000 --> 00:29:31,740
They're in e phi direction.

629
00:29:41,770 --> 00:29:42,270
OK.

630
00:29:42,270 --> 00:29:44,650
But their magnitude
falls off with 1

631
00:29:44,650 --> 00:29:46,970
over r and the norm square
of the wave function, which

632
00:29:46,970 --> 00:29:48,700
also is falling
off exponentially.

633
00:29:48,700 --> 00:29:52,280
And it goes to 0 at the
origin as r to the l plus 1.

634
00:29:52,280 --> 00:29:56,530
And m must be no bigger than l.

635
00:29:56,530 --> 00:29:58,910
So in order for m to be
non-zero, l must be non-zero.

636
00:29:58,910 --> 00:30:02,550
Which means this must go like r
to the something greater than m

637
00:30:02,550 --> 00:30:03,450
plus 1.

638
00:30:03,450 --> 00:30:04,540
So the r's cancel out.

639
00:30:04,540 --> 00:30:06,230
And we have that it vanishes.

640
00:30:06,230 --> 00:30:09,290
So the contribution
vanishes at the origin,

641
00:30:09,290 --> 00:30:15,730
is small near the origin,
is largest at some radius,

642
00:30:15,730 --> 00:30:17,690
and then falls off again.

643
00:30:17,690 --> 00:30:19,106
So the magnitude
of the arrow here

644
00:30:19,106 --> 00:30:21,064
is meant to indicate the
magnitude of the flux.

645
00:30:25,190 --> 00:30:29,470
So what we see is that we have
a probability distribution where

646
00:30:29,470 --> 00:30:33,800
the probability,
as a distribution,

647
00:30:33,800 --> 00:30:34,680
is time independent.

648
00:30:37,110 --> 00:30:37,610
Right?

649
00:30:37,610 --> 00:30:40,090
The probability distribution
is, in fact, time independent.

650
00:30:40,090 --> 00:30:44,355
But there's a current which is a
persistent, stationary current.

651
00:30:47,370 --> 00:30:48,980
Everyone cool with that?

652
00:30:48,980 --> 00:30:51,520
So if I ask you,
where is the particle?

653
00:30:51,520 --> 00:30:54,320
Well, the probability
density tells you that.

654
00:30:54,320 --> 00:30:56,670
And if I ask you,
what's the current?

655
00:30:56,670 --> 00:31:03,780
The electric current,
minus e times j.

656
00:31:03,780 --> 00:31:04,360
At a point.

657
00:31:04,360 --> 00:31:06,850
And this is really the current
density and the probability

658
00:31:06,850 --> 00:31:08,740
current density.

659
00:31:08,740 --> 00:31:13,650
Now what is the consequence of
the statement, the probability

660
00:31:13,650 --> 00:31:15,300
density itself is
not changing in time.

661
00:31:15,300 --> 00:31:20,220
It's that there is zero
divergence of this current.

662
00:31:20,220 --> 00:31:22,814
And indeed, there
is zero divergence.

663
00:31:22,814 --> 00:31:24,230
Every little bit
going into a spot

664
00:31:24,230 --> 00:31:27,460
is matched by some
equivalent amount going out.

665
00:31:27,460 --> 00:31:29,000
There's zero divergence.

666
00:31:29,000 --> 00:31:29,500
Cool?

667
00:31:33,690 --> 00:31:40,110
And so we see that while it's
true that nothing is moving,

668
00:31:40,110 --> 00:31:44,070
it is also true that there is
a non-trivial electric current.

669
00:31:44,070 --> 00:31:46,020
And when you use
the Biot-Savart law

670
00:31:46,020 --> 00:31:49,090
to sum up the contributions
from each little bit

671
00:31:49,090 --> 00:31:53,790
of this current, what you
get is-- right hand rule--

672
00:31:53,790 --> 00:31:55,790
a magnetic field,
a magnetic moment.

673
00:31:55,790 --> 00:31:58,540
And following nothing
but the Biot-Savart law

674
00:31:58,540 --> 00:32:01,710
and using what you know
about the wave functions,

675
00:32:01,710 --> 00:32:06,250
this gives us that the magnetic
moment is equal to the Bohr

676
00:32:06,250 --> 00:32:17,290
magneton, b times m, z.

677
00:32:17,290 --> 00:32:18,990
OK?

678
00:32:18,990 --> 00:32:20,390
Everyone cool with that?

679
00:32:20,390 --> 00:32:21,580
Yeah?

680
00:32:21,580 --> 00:32:26,480
AUDIENCE: So when the perimeter
intuitively is [INAUDIBLE],

681
00:32:26,480 --> 00:32:29,920
still make all these
other expectation values?

682
00:32:29,920 --> 00:32:30,796
PROFESSOR: Yeah.

683
00:32:30,796 --> 00:32:32,504
AUDIENCE: Why is that
an intuitive force?

684
00:32:32,504 --> 00:32:33,410
Or maybe it's not.

685
00:32:33,410 --> 00:32:35,160
PROFESSOR: Well, here's
one way to say it.

686
00:32:35,160 --> 00:32:37,517
There are two-- let me
rephrase that question.

687
00:32:37,517 --> 00:32:39,350
Let me give you a
different question to ask.

688
00:32:39,350 --> 00:32:41,630
And let's think about the
answers to that question

689
00:32:41,630 --> 00:32:43,050
and I hope that will
answer your question,

690
00:32:43,050 --> 00:32:44,174
if it doesn't ask it again.

691
00:32:44,174 --> 00:32:47,780
So here's the question I
would suggest that you ask.

692
00:32:47,780 --> 00:32:49,060
Or that someone ask.

693
00:32:49,060 --> 00:32:49,850
I'll ask it.

694
00:32:55,357 --> 00:32:57,190
And I just totally lost
my train of thought.

695
00:32:57,190 --> 00:32:58,480
That's totally me.

696
00:32:58,480 --> 00:33:00,820
What was the question
I wanted you to ask?

697
00:33:00,820 --> 00:33:01,680
Wow, that's amazing.

698
00:33:01,680 --> 00:33:05,230
I completely just in the blink
of an eye totally lost my train

699
00:33:05,230 --> 00:33:07,170
of -- ah, yes.

700
00:33:07,170 --> 00:33:10,330
So here, it was crucial
in this calculation

701
00:33:10,330 --> 00:33:12,840
that the current was given
by the imaginary part

702
00:33:12,840 --> 00:33:13,730
of the gradient.

703
00:33:13,730 --> 00:33:16,720
And as you showed in a problem
set a while back, anytime

704
00:33:16,720 --> 00:33:21,130
you have a wave
function which is real,

705
00:33:21,130 --> 00:33:23,490
then the current will vanish.

706
00:33:23,490 --> 00:33:25,344
Here it was crucial--
in order to get

707
00:33:25,344 --> 00:33:27,010
a non-vanishing
current-- it was crucial

708
00:33:27,010 --> 00:33:28,510
that the wave
function was not real.

709
00:33:28,510 --> 00:33:31,160
It had a phase.

710
00:33:31,160 --> 00:33:32,920
But you also showed
on a problem set

711
00:33:32,920 --> 00:33:35,182
that you can always take
your energy eigenfunctions--

712
00:33:35,182 --> 00:33:37,640
for bounce-- you can always
take your energy eigenfunctions

713
00:33:37,640 --> 00:33:38,245
and make them real.

714
00:33:38,245 --> 00:33:40,790
You can always construct a
basis of energy eigenfunctions

715
00:33:40,790 --> 00:33:43,740
which are strictly real.

716
00:33:43,740 --> 00:33:44,534
Sorry?

717
00:33:44,534 --> 00:33:45,950
AUDIENCE: In one
dimension, right?

718
00:33:45,950 --> 00:33:46,872
PROFESSOR: Yeah well--

719
00:33:46,872 --> 00:33:48,330
AUDIENCE: Aren't
you in three also?

720
00:33:48,330 --> 00:33:50,431
PROFESSOR: What did
you use for that proof?

721
00:33:50,431 --> 00:33:52,540
AUDIENCE: Just used linear
combination of the two.

722
00:33:52,540 --> 00:33:52,606
PROFESSOR: Yeah.

723
00:33:52,606 --> 00:33:54,439
You just use the energy
eigenvalue equation,

724
00:33:54,439 --> 00:33:58,230
hermiticity, and the
linear, the energy operator.

725
00:33:58,230 --> 00:34:00,500
And that's perfectly true
in any number of dimensions.

726
00:34:00,500 --> 00:34:02,020
AUDIENCE: OK.

727
00:34:02,020 --> 00:34:04,290
PROFESSOR: So that sounds crazy.

728
00:34:04,290 --> 00:34:09,225
So first off, why do we have a
not real energy eigenfunction?

729
00:34:09,225 --> 00:34:11,350
Can we construct purely
real energy eigenfunctions?

730
00:34:11,350 --> 00:34:14,110
If we can, those
energy eigenfunctions

731
00:34:14,110 --> 00:34:17,340
would appear to have no current.

732
00:34:17,340 --> 00:34:19,620
So they would have
no magnetic moment.

733
00:34:19,620 --> 00:34:20,740
How do these fit together?

734
00:34:25,850 --> 00:34:27,522
Can you construct
energy eigenfunctions

735
00:34:27,522 --> 00:34:28,980
that are pure real
for this system,

736
00:34:28,980 --> 00:34:30,063
for the Coulomb potential?

737
00:34:36,451 --> 00:34:37,409
Let me give you a hint.

738
00:34:37,409 --> 00:34:41,530
In 1d for a free particle, what
are the energy eigenfunctions?

739
00:34:41,530 --> 00:34:42,565
1d,

740
00:34:42,565 --> 00:34:43,870
AUDIENCE: [INTERPOSING VOICES].

741
00:34:43,870 --> 00:34:45,780
PROFESSOR: E to the ikx, right?

742
00:34:45,780 --> 00:34:48,053
That's not real.

743
00:34:48,053 --> 00:34:49,469
But there's another
eigenfunction,

744
00:34:49,469 --> 00:34:50,750
you get the minus ikx.

745
00:34:50,750 --> 00:34:52,525
And if you take
the sum of those,

746
00:34:52,525 --> 00:34:54,250
you get e to the ikx
plus e to the minus

747
00:34:54,250 --> 00:34:57,990
ikx-- divide by 2 for fun-- and
that gives you cosine of kx.

748
00:34:57,990 --> 00:34:58,494
That's real.

749
00:34:58,494 --> 00:34:59,910
But there's a
second one, which is

750
00:34:59,910 --> 00:35:01,477
sine of kx, which is also real.

751
00:35:01,477 --> 00:35:03,310
Now you can take linear
combinations of them

752
00:35:03,310 --> 00:35:05,500
with i's and get the
exponentials back.

753
00:35:05,500 --> 00:35:08,290
But let's just take
the real part, right?

754
00:35:08,290 --> 00:35:10,360
Now, do those
carry any momentum?

755
00:35:10,360 --> 00:35:14,420
What's the momentum expectation
value for cosine of kx?

756
00:35:14,420 --> 00:35:15,120
AUDIENCE: 0.

757
00:35:15,120 --> 00:35:16,020
PROFESSOR: 0, because
you get confirmation

758
00:35:16,020 --> 00:35:17,870
from plus k and
one from minus k.

759
00:35:17,870 --> 00:35:19,830
They exactly cancel.

760
00:35:19,830 --> 00:35:21,970
Now look at this solution.

761
00:35:21,970 --> 00:35:25,030
Can you construct a real
energy eigenfunction

762
00:35:25,030 --> 00:35:26,359
of the Coulomb potential?

763
00:35:26,359 --> 00:35:26,984
AUDIENCE: Sure.

764
00:35:26,984 --> 00:35:27,878
PROFESSOR: How.

765
00:35:27,878 --> 00:35:30,011
AUDIENCE: [INAUDIBLE].

766
00:35:30,011 --> 00:35:30,760
PROFESSOR: Great..

767
00:35:30,760 --> 00:35:32,690
Let's take this and
its complex conjugate.

768
00:35:32,690 --> 00:35:33,930
So what that would be?

769
00:35:33,930 --> 00:35:36,710
Well, if we take this, psi nlm.

770
00:35:36,710 --> 00:35:39,390
And its complex conjugate, well,
what is its complex conjugate?

771
00:35:39,390 --> 00:35:40,230
These are real.

772
00:35:40,230 --> 00:35:41,520
That just gives me a minus.

773
00:35:41,520 --> 00:35:44,420
So that's the state
plus psi and l minus

774
00:35:44,420 --> 00:35:47,810
m, which is also an allowed
state with the same energy

775
00:35:47,810 --> 00:35:50,450
by rotational invariance.

776
00:35:50,450 --> 00:35:55,650
And this is thus
the state psi nl.

777
00:35:55,650 --> 00:36:00,890
And it doesn't have a definite
lz angular momentum, right?

778
00:36:00,890 --> 00:36:06,220
So I'll just call it
psi sub nl unhappy face.

779
00:36:06,220 --> 00:36:07,470
It is not an lz eigenfunction.

780
00:36:07,470 --> 00:36:09,170
But it is an energy
eigenfunction.

781
00:36:09,170 --> 00:36:12,940
And it could also have
built a minus with divide

782
00:36:12,940 --> 00:36:13,770
by 2i for fun.

783
00:36:13,770 --> 00:36:16,400
I could have built the
sine or the cosine of phi.

784
00:36:16,400 --> 00:36:20,460
One of which showed
up on your exam.

785
00:36:20,460 --> 00:36:23,480
So I can find a basis of
these states, sine and cosine,

786
00:36:23,480 --> 00:36:26,112
instead of, give me the im
phi, I need the minus im phi.

787
00:36:26,112 --> 00:36:27,820
And those would have
been perfectly real.

788
00:36:27,820 --> 00:36:30,445
And in those situations, what
would the current have been?

789
00:36:30,445 --> 00:36:31,600
AUDIENCE: [INAUDIBLE].

790
00:36:31,600 --> 00:36:33,966
PROFESSOR: If I put the system
in this stationary state,

791
00:36:33,966 --> 00:36:35,340
what would the
current have been?

792
00:36:35,340 --> 00:36:36,120
AUDIENCE: 0.

793
00:36:36,120 --> 00:36:38,680
PROFESSOR: Identically 0.

794
00:36:38,680 --> 00:36:43,160
However, that wouldn't be a
state with a definite angular

795
00:36:43,160 --> 00:36:43,660
momentum.

796
00:36:43,660 --> 00:36:46,436
So it wouldn't be surprising
that it has zero current.

797
00:36:46,436 --> 00:36:47,560
It's got some contribution.

798
00:36:47,560 --> 00:36:49,100
It's a super position of
having some angular momentum

799
00:36:49,100 --> 00:36:50,808
and having the opposite
angular momentum.

800
00:36:50,808 --> 00:36:54,840
And the expectation value,
the expected value, is zero.

801
00:36:54,840 --> 00:36:57,540
I can also study states with a
definite value of the angular

802
00:36:57,540 --> 00:36:58,040
momentum.

803
00:36:58,040 --> 00:36:59,520
Those are not real.

804
00:36:59,520 --> 00:37:01,190
There's nothing wrong with that.

805
00:37:01,190 --> 00:37:02,810
I could have
constructed states which

806
00:37:02,810 --> 00:37:05,126
don't have a definite angular
momentum but are real.

807
00:37:05,126 --> 00:37:06,750
But instead, I want
to work with states

808
00:37:06,750 --> 00:37:09,790
that have a definite
angular momentum.

809
00:37:09,790 --> 00:37:11,094
That cool?

810
00:37:11,094 --> 00:37:12,010
So there's no tension.

811
00:37:12,010 --> 00:37:13,051
There's no contradiction.

812
00:37:15,610 --> 00:37:18,260
It's just that, if we're
interested in finding states

813
00:37:18,260 --> 00:37:22,030
that have a definite energy and
a definite lz angular momentum,

814
00:37:22,030 --> 00:37:24,240
that is not going to
be a real function.

815
00:37:24,240 --> 00:37:26,640
And nothing tells us
that it has to be.

816
00:37:26,640 --> 00:37:28,170
And when you have
a non-trivial lz,

817
00:37:28,170 --> 00:37:30,419
when you have a non-trivial
angular momentum, just

818
00:37:30,419 --> 00:37:32,460
like a classical particle
with nontrivial angular

819
00:37:32,460 --> 00:37:35,940
momentum that has charge, we
find that there's a current.

820
00:37:35,940 --> 00:37:37,590
And thus a magnetic moment.

821
00:37:37,590 --> 00:37:40,820
So the important thing with
having a definite nonzero

822
00:37:40,820 --> 00:37:43,360
momentum, or in this
case angular momentum,

823
00:37:43,360 --> 00:37:45,467
to give us the current.

824
00:37:45,467 --> 00:37:46,960
Yeah?

825
00:37:46,960 --> 00:37:49,115
Questions about this?

826
00:37:49,115 --> 00:37:49,615
Yeah?

827
00:37:49,615 --> 00:37:51,842
AUDIENCE: Using the same
logic, it kind of looks

828
00:37:51,842 --> 00:37:54,317
like we're taking it
a charged particle

829
00:37:54,317 --> 00:37:55,750
and moving it around in circles.

830
00:37:55,750 --> 00:37:57,178
Like, where we
have a distribution

831
00:37:57,178 --> 00:37:57,654
and we're spinning.

832
00:37:57,654 --> 00:37:58,130
PROFESSOR: Yeah.

833
00:37:58,130 --> 00:37:59,082
AUDIENCE: With spin a
distribution of charge,

834
00:37:59,082 --> 00:38:01,325
we're accelerating the
charged particles in it.

835
00:38:01,325 --> 00:38:01,950
PROFESSOR: Yes.

836
00:38:01,950 --> 00:38:02,290
AUDIENCE: Individually.

837
00:38:02,290 --> 00:38:02,630
PROFESSOR: Yes.

838
00:38:02,630 --> 00:38:04,460
AUDIENCE: If we accelerate
them, they emit radiation.

839
00:38:04,460 --> 00:38:04,855
PROFESSOR: Yes.

840
00:38:04,855 --> 00:38:06,520
AUDIENCE: Hydrogen atoms
don't emit radiation.

841
00:38:06,520 --> 00:38:07,145
PROFESSOR: Yes.

842
00:38:07,145 --> 00:38:08,960
AUDIENCE: How does that go?

843
00:38:08,960 --> 00:38:11,300
PROFESSOR: That sounds
a lot like problem four

844
00:38:11,300 --> 00:38:13,494
on your problem set.

845
00:38:13,494 --> 00:38:14,630
AUDIENCE: [INAUDIBLE]

846
00:38:14,630 --> 00:38:15,380
PROFESSOR: Ah, OK.

847
00:38:15,380 --> 00:38:18,480
Well, that's a great question.

848
00:38:18,480 --> 00:38:21,960
And you should think about it.

849
00:38:21,960 --> 00:38:25,380
How could I say--
you read my mind.

850
00:38:25,380 --> 00:38:26,021
Yes.

851
00:38:26,021 --> 00:38:28,020
Struggling with this is
exactly the point of one

852
00:38:28,020 --> 00:38:29,850
of the problems on
your problem set.

853
00:38:29,850 --> 00:38:32,840
And ask me that again after the
problem set has been turned in,

854
00:38:32,840 --> 00:38:35,372
and I'll give you a
happy disquisition on it.

855
00:38:35,372 --> 00:38:36,830
But I want you to
struggle with it.

856
00:38:36,830 --> 00:38:39,411
Because it's hard and
interesting question.

857
00:38:39,411 --> 00:38:39,910
Yeah?

858
00:38:39,910 --> 00:38:43,330
AUDIENCE: So I don't think this
answers my question necessarily

859
00:38:43,330 --> 00:38:44,850
about the radiation.

860
00:38:44,850 --> 00:38:48,320
Because, analogously,
the classical enm,

861
00:38:48,320 --> 00:38:50,402
this would be the
magnetostatic case,

862
00:38:50,402 --> 00:38:50,890
where you just have a static--

863
00:38:50,890 --> 00:38:51,330
PROFESSOR: Precisely.

864
00:38:51,330 --> 00:38:51,755
Precisely.

865
00:38:51,755 --> 00:38:53,000
AUDIENCE: But with
radiation, you have, quickly

866
00:38:53,000 --> 00:38:54,791
oscillating fields and
the transition times

867
00:38:54,791 --> 00:38:57,477
are really small and the
frequencies of the wave

868
00:38:57,477 --> 00:38:58,560
functions are really fast.

869
00:38:58,560 --> 00:39:00,487
So I still don't know
if that's the same.

870
00:39:00,487 --> 00:39:02,320
PROFESSOR: So this
question is unfortunately

871
00:39:02,320 --> 00:39:05,260
a linear combination of a
really good independent question

872
00:39:05,260 --> 00:39:06,690
and that question.

873
00:39:06,690 --> 00:39:08,520
So ask me after the
lecture and I'll

874
00:39:08,520 --> 00:39:11,417
talk to you about the part
that's linearly independent.

875
00:39:11,417 --> 00:39:13,370
OK.

876
00:39:13,370 --> 00:39:14,700
I'm on thin ice here.

877
00:39:14,700 --> 00:39:15,230
OK.

878
00:39:15,230 --> 00:39:19,160
Anything else before we
dispense with hydrogen?

879
00:39:19,160 --> 00:39:19,730
OK.

880
00:39:19,730 --> 00:39:22,640
The question that
you all keep coming

881
00:39:22,640 --> 00:39:26,689
to about the-- well,
problem four on the-- I

882
00:39:26,689 --> 00:39:28,480
think it's problem four
on the problem set.

883
00:39:28,480 --> 00:39:31,190
Maybe it's problem two.

884
00:39:31,190 --> 00:39:32,890
It involves almost
no computation,

885
00:39:32,890 --> 00:39:35,056
but it's the most intellectually
challenging problem

886
00:39:35,056 --> 00:39:36,540
on the exam-- on
the problem set.

887
00:39:36,540 --> 00:39:38,873
So take it seriously, even
though it's calculation-free.

888
00:39:41,500 --> 00:39:43,830
OK.

889
00:39:43,830 --> 00:39:44,922
Yeah?

890
00:39:44,922 --> 00:39:49,104
AUDIENCE: Due to this
proton in the [INAUDIBLE].

891
00:39:49,104 --> 00:39:49,770
PROFESSOR: Yeah.

892
00:39:49,770 --> 00:39:50,269
Yeah.

893
00:39:50,269 --> 00:39:52,669
It's so that the--
it's not by symmetry,

894
00:39:52,669 --> 00:39:53,960
because there isn't a symmetry.

895
00:39:53,960 --> 00:39:56,930
The proton is 2,000 times
heavier than the electron.

896
00:39:56,930 --> 00:40:04,740
But-- let me make sure I'm
understanding your question.

897
00:40:04,740 --> 00:40:08,955
Your question is, we
found that the electron is

898
00:40:08,955 --> 00:40:11,500
in bound states in
the Coulomb potential.

899
00:40:11,500 --> 00:40:13,000
But in hydrogen,
you have two parts,

900
00:40:13,000 --> 00:40:14,416
you have an electron
and a proton.

901
00:40:14,416 --> 00:40:16,980
So is the electron in
that state and the proton

902
00:40:16,980 --> 00:40:19,270
is just free and cruising
on its own thing?

903
00:40:19,270 --> 00:40:22,330
That's exactly the topic
of the next 45 minutes.

904
00:40:22,330 --> 00:40:23,150
OK.

905
00:40:23,150 --> 00:40:23,760
Good question.

906
00:40:23,760 --> 00:40:27,900
So with that insight,
let me turn now

907
00:40:27,900 --> 00:40:30,630
to the question of
identical particles

908
00:40:30,630 --> 00:40:33,570
or multiple particles.

909
00:40:33,570 --> 00:40:34,070
OK.

910
00:40:34,070 --> 00:40:36,026
So we're done with
Coulomb for the moment.

911
00:40:36,026 --> 00:40:37,650
Pretty much for the
rest of the course.

912
00:40:37,650 --> 00:40:40,270
So I want to move on to
the following question.

913
00:40:40,270 --> 00:40:42,769
Suppose I have a system-- we've
spent a lot of time thinking

914
00:40:42,769 --> 00:40:44,810
about a particle in a potential.

915
00:40:44,810 --> 00:40:46,860
I would like to think
about multiple particles

916
00:40:46,860 --> 00:40:47,560
for a minute.

917
00:40:51,320 --> 00:40:54,840
And neat things happen
for multiple particles

918
00:40:54,840 --> 00:41:00,450
that don't happen for
individual, isolated particles.

919
00:41:00,450 --> 00:41:02,104
So let's think
about what those--

920
00:41:02,104 --> 00:41:04,270
let's think about the physics
of multiple particles.

921
00:41:04,270 --> 00:41:08,570
So in particular, classically--
in classical mechanics--

922
00:41:08,570 --> 00:41:11,000
if I have two particles,
what is the information

923
00:41:11,000 --> 00:41:11,164
I have to specify?

924
00:41:11,164 --> 00:41:12,410
I have to specify the state.

925
00:41:12,410 --> 00:41:15,950
I have to specify the position
of the first particle.

926
00:41:15,950 --> 00:41:18,980
X1 and its momentum, p1.

927
00:41:18,980 --> 00:41:21,480
And then the position of the
second particle x2 and p2.

928
00:41:21,480 --> 00:41:23,670
I'm going to omit
vectors over everything,

929
00:41:23,670 --> 00:41:25,461
but everything is in
the appropriate number

930
00:41:25,461 --> 00:41:26,560
of dimensions.

931
00:41:26,560 --> 00:41:30,130
So in classical mechanics--
in quantum mechanics,

932
00:41:30,130 --> 00:41:32,130
the state of the
system is specified

933
00:41:32,130 --> 00:41:35,700
by a wave function, which is
a function of the positions--

934
00:41:35,700 --> 00:41:41,190
or the degrees of freedom,
let's say x1 and x2.

935
00:41:41,190 --> 00:41:47,840
And x1 and x2 and p1
and p2 are promoted

936
00:41:47,840 --> 00:41:51,739
to operators representing
these observables.

937
00:41:51,739 --> 00:41:53,280
And the wave function
is a function--

938
00:41:53,280 --> 00:41:55,760
now let me just quickly tell
you what this notation means.

939
00:41:55,760 --> 00:42:00,150
What this notation means is
x1 is some number, like 7.

940
00:42:00,150 --> 00:42:02,470
And the quantity
in the first spot,

941
00:42:02,470 --> 00:42:08,150
this means that the
first particle indicated

942
00:42:08,150 --> 00:42:12,450
by the first slot, is at x1.

943
00:42:12,450 --> 00:42:15,665
And the second
particle is at x2.

944
00:42:18,430 --> 00:42:20,430
So whenever I write a
wave function, what I mean

945
00:42:20,430 --> 00:42:23,010
is this is the probability
amplitude-- the thing

946
00:42:23,010 --> 00:42:24,967
whose norm squared is
probability-- to find

947
00:42:24,967 --> 00:42:27,300
the first particle at this
value and the second particle

948
00:42:27,300 --> 00:42:28,160
of that value.

949
00:42:28,160 --> 00:42:30,920
So the 1 and 2 label the
points, not the particles.

950
00:42:30,920 --> 00:42:33,270
The particle is labeled by
the position inside this wave

951
00:42:33,270 --> 00:42:33,770
function.

952
00:42:33,770 --> 00:42:35,438
Everyone cool with that?

953
00:42:35,438 --> 00:42:36,767
OK.

954
00:42:36,767 --> 00:42:37,850
So that's just a notation.

955
00:42:37,850 --> 00:42:40,290
So the quantum mechanical
description of two particles

956
00:42:40,290 --> 00:42:43,920
is a wave function of both
positions and operators

957
00:42:43,920 --> 00:42:48,510
representing the position
and the coordinates

958
00:42:48,510 --> 00:42:55,933
of the particles.

959
00:42:59,450 --> 00:43:04,010
And the probability--
actually, let's

960
00:43:04,010 --> 00:43:09,180
go ahead and finish up here--
and the probability density

961
00:43:09,180 --> 00:43:13,110
to find the first particle.

962
00:43:13,110 --> 00:43:22,500
First at x1, and
the second at x2.

963
00:43:22,500 --> 00:43:27,460
It's just the norm squared
of the amplitude, psi x1, x2,

964
00:43:27,460 --> 00:43:29,240
norm squared.

965
00:43:29,240 --> 00:43:30,480
Just as usual.

966
00:43:30,480 --> 00:43:31,790
That's the probability density.

967
00:43:34,718 --> 00:43:35,700
All right?

968
00:43:35,700 --> 00:43:37,770
Everyone cool with that?

969
00:43:37,770 --> 00:43:43,170
And what are x1, x2, commutator?

970
00:43:43,170 --> 00:43:44,170
What is this commutator?

971
00:43:47,937 --> 00:43:49,410
AUDIENCE: [INAUDIBLE].

972
00:43:49,410 --> 00:43:51,826
PROFESSOR: Can you know the
position of the first particle

973
00:43:51,826 --> 00:43:54,290
and the position of
the second particle?

974
00:43:54,290 --> 00:43:55,860
Simultaneously?

975
00:43:55,860 --> 00:43:56,450
Sure.

976
00:43:56,450 --> 00:43:56,990
I'm here.

977
00:43:56,990 --> 00:43:59,510
You're there.

978
00:43:59,510 --> 00:44:00,610
0.

979
00:44:00,610 --> 00:44:06,175
And x1 with p1-- just work in
one dimension for the moment--

980
00:44:06,175 --> 00:44:07,598
is equal to?

981
00:44:07,598 --> 00:44:09,510
AUDIENCE: [INAUDIBLE].

982
00:44:09,510 --> 00:44:10,840
PROFESSOR: Our h bar.

983
00:44:10,840 --> 00:44:13,630
And the same would have
been true of x2 and p2.

984
00:44:13,630 --> 00:44:18,621
And what about x1 with p2?

985
00:44:18,621 --> 00:44:19,120
0.

986
00:44:19,120 --> 00:44:21,420
These are independent
quantities.

987
00:44:21,420 --> 00:44:22,939
OK.

988
00:44:22,939 --> 00:44:24,480
So it's more or less
as you'd expect.

989
00:44:24,480 --> 00:44:28,900
So just to be explicit about
this, let's do an example.

990
00:44:28,900 --> 00:44:29,740
Two free particles.

991
00:44:37,640 --> 00:44:39,120
So what's the energy operator?

992
00:44:39,120 --> 00:44:42,620
Well, the energy operator is
equal to p squared upon 2m

993
00:44:42,620 --> 00:44:49,330
for the first particle, 1,
plus p squared 2 upon 2m 2.

994
00:44:49,330 --> 00:44:51,010
Second particle, its mass is m2.

995
00:44:51,010 --> 00:44:52,660
The first particle has mass m1.

996
00:44:52,660 --> 00:44:56,800
And the system is free,
so this is plus 0.

997
00:44:56,800 --> 00:44:57,687
Yeah?

998
00:44:57,687 --> 00:44:59,645
AUDIENCE: So for the
[INAUDIBLE] exponentially,

999
00:44:59,645 --> 00:45:05,124
is that true [INAUDIBLE]
particles [INAUDIBLE]?

1000
00:45:05,124 --> 00:45:05,790
PROFESSOR: Yeah.

1001
00:45:05,790 --> 00:45:06,290
OK.

1002
00:45:06,290 --> 00:45:10,970
So the question is, x1,
x2, commutator equals 0?

1003
00:45:10,970 --> 00:45:15,320
Is that true even if there are
forces between the particles?

1004
00:45:15,320 --> 00:45:17,430
So what are the forces
between particles--

1005
00:45:17,430 --> 00:45:19,466
what are they going
to contribute to?

1006
00:45:19,466 --> 00:45:20,550
Yeah, the potential.

1007
00:45:20,550 --> 00:45:22,280
And that's going to show
up in the energy operator.

1008
00:45:22,280 --> 00:45:23,520
So that will certainly
matter and it

1009
00:45:23,520 --> 00:45:25,353
will change what the
energy eigenvalues are.

1010
00:45:25,353 --> 00:45:27,610
But it won't tell what
can or can't be measured.

1011
00:45:27,610 --> 00:45:31,260
It won't tell you what
properties the system can have

1012
00:45:31,260 --> 00:45:32,431
or not.

1013
00:45:32,431 --> 00:45:32,930
Good.

1014
00:45:32,930 --> 00:45:33,718
Yeah?

1015
00:45:33,718 --> 00:45:35,176
AUDIENCE: What's
with the particles

1016
00:45:35,176 --> 00:45:37,303
in the state in which
measuring the position of one

1017
00:45:37,303 --> 00:45:39,460
makes the position of the
other one in certainty?

1018
00:45:39,460 --> 00:45:41,110
PROFESSOR: We'll talk about
things like that in a minute.

1019
00:45:41,110 --> 00:45:41,750
AUDIENCE: Does it screw this up?

1020
00:45:41,750 --> 00:45:42,333
PROFESSOR: No.

1021
00:45:42,333 --> 00:45:46,780
This is-- do the
commutation relations

1022
00:45:46,780 --> 00:45:49,080
care what state you're in?

1023
00:45:49,080 --> 00:45:50,990
They're relations
amongst operators.

1024
00:45:50,990 --> 00:45:54,550
So they're independent
of the state always.

1025
00:45:54,550 --> 00:45:55,990
OK.

1026
00:45:55,990 --> 00:45:56,849
Yeah?

1027
00:45:56,849 --> 00:45:58,390
AUDIENCE: For the
two free particles,

1028
00:45:58,390 --> 00:46:00,310
are we assuming that they're
not charged particles?

1029
00:46:00,310 --> 00:46:00,600
PROFESSOR: Yeah.

1030
00:46:00,600 --> 00:46:02,540
I'm assuming that there are
no interactions whatsoever.

1031
00:46:02,540 --> 00:46:04,000
Just totally free particles.

1032
00:46:04,000 --> 00:46:05,490
Potential is zero.

1033
00:46:05,490 --> 00:46:10,162
V of x1, x2 equals 0.

1034
00:46:10,162 --> 00:46:11,120
So they're not charged.

1035
00:46:11,120 --> 00:46:12,930
They're just totally
uninteresting.

1036
00:46:12,930 --> 00:46:15,210
No interactions, no
forces, no potential.

1037
00:46:15,210 --> 00:46:15,910
OK.

1038
00:46:15,910 --> 00:46:18,310
So we know how to
solve this problem

1039
00:46:18,310 --> 00:46:19,600
because we can use separation.

1040
00:46:19,600 --> 00:46:25,550
Psi of x1, x2 can be
written as-- well,

1041
00:46:25,550 --> 00:46:31,430
we know what the solutions
of p1 upon 2m is equal to er.

1042
00:46:31,430 --> 00:46:35,200
So we can write
this as chi of x1.

1043
00:46:35,200 --> 00:46:37,810
And actually, I'm going
to call the positions

1044
00:46:37,810 --> 00:46:40,419
of these guys a and b.

1045
00:46:40,419 --> 00:46:42,710
Because the subscriptions
are just terribly misleading.

1046
00:46:42,710 --> 00:46:46,170
So the first point is called a,
the second point is called b.

1047
00:46:46,170 --> 00:46:49,960
So a and b.

1048
00:46:49,960 --> 00:46:56,430
And this is p1 and p2 the
momentum of the two guys.

1049
00:46:56,430 --> 00:46:59,580
Chi of a and phi of b.

1050
00:46:59,580 --> 00:47:01,605
So all I'm doing here
is I'm using separation.

1051
00:47:05,310 --> 00:47:07,840
Pa, pb.

1052
00:47:07,840 --> 00:47:10,091
So here's, instead of calling
the positions x1 and x2,

1053
00:47:10,091 --> 00:47:12,090
I'm going to call them a
and b just because it's

1054
00:47:12,090 --> 00:47:14,340
going to be easier to write
and make things clear.

1055
00:47:14,340 --> 00:47:15,560
So I'm using separation.

1056
00:47:15,560 --> 00:47:21,640
And then p1 squared, or
pa squared, upon 2m on cih

1057
00:47:21,640 --> 00:47:26,870
of a is equal to ea, chi a.

1058
00:47:26,870 --> 00:47:29,240
And ditto for b.

1059
00:47:29,240 --> 00:47:33,850
This tells us that
chi of a is equal to e

1060
00:47:33,850 --> 00:47:40,680
to the sub-coefficient
c, e to the i k a.

1061
00:47:40,680 --> 00:47:42,900
Everyone cool with that?

1062
00:47:42,900 --> 00:47:46,370
A is just replacing x1.

1063
00:47:46,370 --> 00:47:47,504
So it's just the position.

1064
00:47:47,504 --> 00:47:49,170
So this is just saying
that the solution

1065
00:47:49,170 --> 00:47:52,370
to a single free
particle is a plane wave.

1066
00:47:52,370 --> 00:47:55,200
And because it's just the
sum of the momentum squared,

1067
00:47:55,200 --> 00:47:58,760
we can separate the equation and
the same thing obtains for psi.

1068
00:47:58,760 --> 00:48:02,200
So similarly for phi.

1069
00:48:02,200 --> 00:48:04,550
Ditto.

1070
00:48:04,550 --> 00:48:09,820
Phi of b is equal to
the e to the i k sub b.

1071
00:48:09,820 --> 00:48:13,030
I'll call this k sub
a to distinguish them.

1072
00:48:13,030 --> 00:48:18,240
So the wave function's a
basis for the wave functions

1073
00:48:18,240 --> 00:48:23,250
for two particles, psi
of a b with energy e

1074
00:48:23,250 --> 00:48:29,590
is equal to e to
the i ka A plus kb

1075
00:48:29,590 --> 00:48:39,390
B, Where E is equal to h
bar squared on 2m K sub

1076
00:48:39,390 --> 00:48:43,820
a squared plus k sub b squared.

1077
00:48:43,820 --> 00:48:45,820
Yeah?

1078
00:48:45,820 --> 00:48:47,640
For a system of
two free particles,

1079
00:48:47,640 --> 00:48:52,330
is every wave function of the
form chi of a times phi of b?

1080
00:48:56,917 --> 00:48:58,500
If you have a system
that's separable,

1081
00:48:58,500 --> 00:49:01,010
is every wave function, is every
solution itself, separated?

1082
00:49:01,010 --> 00:49:01,750
AUDIENCE: No.

1083
00:49:01,750 --> 00:49:02,333
PROFESSOR: No.

1084
00:49:02,333 --> 00:49:04,180
Because we can have
arbitrary superpositions

1085
00:49:04,180 --> 00:49:06,666
of forms of this type.

1086
00:49:06,666 --> 00:49:08,290
So we get superpositions
of plane waves

1087
00:49:08,290 --> 00:49:11,260
as long as the energies of each
plane wave in the superposition

1088
00:49:11,260 --> 00:49:13,440
are equal to e-- the total
energy is equal to e-- I

1089
00:49:13,440 --> 00:49:14,336
can just superpose
them and I still

1090
00:49:14,336 --> 00:49:15,650
have an energy eigenfunction.

1091
00:49:15,650 --> 00:49:17,900
Everyone cool with that?

1092
00:49:17,900 --> 00:49:18,420
OK.

1093
00:49:18,420 --> 00:49:20,040
So nothing shocking here.

1094
00:49:20,040 --> 00:49:22,020
But I did this
example so that we'd

1095
00:49:22,020 --> 00:49:27,540
have the notation of chi
and phi for the two states.

1096
00:49:27,540 --> 00:49:30,170
OK.

1097
00:49:30,170 --> 00:49:33,660
So now let me come to
this question of what

1098
00:49:33,660 --> 00:49:34,370
about the proton?

1099
00:49:34,370 --> 00:49:38,640
Well suppose I have a system now
which is not a free particle.

1100
00:49:38,640 --> 00:49:40,330
It's two particles, a and b.

1101
00:49:40,330 --> 00:49:46,530
And e is equal to pa squared
over 2a plus pb squared

1102
00:49:46,530 --> 00:49:49,000
upon 2mb.

1103
00:49:49,000 --> 00:49:52,510
Plus a potential that
depends only on a minus b.

1104
00:49:56,378 --> 00:49:57,280
OK?

1105
00:49:57,280 --> 00:50:00,004
So this is, for example,
what happens in the Coulomb

1106
00:50:00,004 --> 00:50:02,420
potential when you include the
proton having a finite mass

1107
00:50:02,420 --> 00:50:04,586
instead of being infinitely
massive and stuck still.

1108
00:50:07,250 --> 00:50:09,330
So now we can do
exactly the same thing

1109
00:50:09,330 --> 00:50:12,531
we did in classical mechanics
when you have a potential that

1110
00:50:12,531 --> 00:50:14,530
only depends on the
distance between two things.

1111
00:50:14,530 --> 00:50:16,730
I can reorganize
degrees of freedom

1112
00:50:16,730 --> 00:50:20,190
into the center
of mass position.

1113
00:50:20,190 --> 00:50:32,360
R is equal to 1 over ma plus
mb of ma, a, plus mb, b.

1114
00:50:32,360 --> 00:50:34,130
So that's the center
of mass motion.

1115
00:50:34,130 --> 00:50:41,860
And the relative distance
is equal to 1 over-- whoops.

1116
00:50:41,860 --> 00:50:43,130
I don't need that.

1117
00:50:43,130 --> 00:50:43,860
A minus b.

1118
00:50:47,370 --> 00:50:50,130
And then if you do this and you
write out the energy operator,

1119
00:50:50,130 --> 00:50:52,250
e is equal to
minus h bar squared

1120
00:50:52,250 --> 00:50:56,620
upon 2 capital M, total mass.

1121
00:50:56,620 --> 00:51:07,310
Dr squared Plus 1 minus h bar
squared over 2 mu d little r

1122
00:51:07,310 --> 00:51:11,675
squared plus v or r.

1123
00:51:11,675 --> 00:51:13,149
OK?

1124
00:51:13,149 --> 00:51:15,440
So this is exactly what
happens in classical mechanics.

1125
00:51:15,440 --> 00:51:17,550
You work in terms of the
center of mass coordinate

1126
00:51:17,550 --> 00:51:19,150
and the relative coordinate.

1127
00:51:19,150 --> 00:51:21,820
The relative coordinate
becomes effectively

1128
00:51:21,820 --> 00:51:24,159
an independent degree of
freedom with a potential,

1129
00:51:24,159 --> 00:51:25,450
which is the central potential.

1130
00:51:25,450 --> 00:51:28,766
And the center of mass
coordinate is a free particle.

1131
00:51:28,766 --> 00:51:30,390
So if we have a proton
and our electron

1132
00:51:30,390 --> 00:51:32,000
and they're attracted
to each other by Coulomb

1133
00:51:32,000 --> 00:51:33,364
there's a center of mass motion.

1134
00:51:33,364 --> 00:51:34,780
And then they do
together whatever

1135
00:51:34,780 --> 00:51:37,115
they do together, a
la Coulomb potential.

1136
00:51:37,115 --> 00:51:38,540
Everyone cool with that?

1137
00:51:38,540 --> 00:51:43,050
The only difference is that the
mass in the Coulomb potential

1138
00:51:43,050 --> 00:51:44,960
is not the mass of
the bare electron.

1139
00:51:44,960 --> 00:51:46,720
But it's the
geometric mean mu is

1140
00:51:46,720 --> 00:51:50,520
equal to ma and b
over ma plus mb.

1141
00:51:54,130 --> 00:51:56,280
Now for a proton
and an electron,

1142
00:51:56,280 --> 00:51:59,320
if ma is the proton
and mb is the electron,

1143
00:51:59,320 --> 00:52:03,030
then this is proton electron
over proton plus electron.

1144
00:52:03,030 --> 00:52:07,094
But proton is about 2,000
times the mass of the electron.

1145
00:52:07,094 --> 00:52:09,760
So this is basically the mass of
the proton and they factor out.

1146
00:52:09,760 --> 00:52:13,970
So the effective reduced mass
is roughly equal to the electron

1147
00:52:13,970 --> 00:52:16,130
mass.

1148
00:52:16,130 --> 00:52:20,242
Corrections of a part in 1,000.

1149
00:52:20,242 --> 00:52:21,200
OK?

1150
00:52:21,200 --> 00:52:23,840
So to answer your
question, is the proton

1151
00:52:23,840 --> 00:52:26,900
also in some complicated
state described it?

1152
00:52:26,900 --> 00:52:30,470
Well in fact, neither the
electron nor the proton

1153
00:52:30,470 --> 00:52:32,790
are described by the
Coulomb potential.

1154
00:52:32,790 --> 00:52:38,900
But the relative position,
the relative radial distance

1155
00:52:38,900 --> 00:52:42,060
between them, is controlled
by the Coulomb potential

1156
00:52:42,060 --> 00:52:46,371
and the center of mass degree
of freedom is a free particle.

1157
00:52:46,371 --> 00:52:47,680
Does that make sense?

1158
00:52:47,680 --> 00:52:48,180
OK.

1159
00:52:48,180 --> 00:52:48,680
Good.

1160
00:52:51,050 --> 00:52:51,680
OK.

1161
00:52:51,680 --> 00:52:58,770
So, so much for that example.

1162
00:52:58,770 --> 00:53:01,260
The free particle and
the central potential.

1163
00:53:04,239 --> 00:53:05,780
Here's the much more
interesting that

1164
00:53:05,780 --> 00:53:08,180
happens when we have
multiple particles.

1165
00:53:08,180 --> 00:53:08,750
Yeah?

1166
00:53:08,750 --> 00:53:11,426
AUDIENCE: Can you explain what
you mean by the [INAUDIBLE]?

1167
00:53:11,426 --> 00:53:12,550
PROFESSOR: Oh, yeah, sorry.

1168
00:53:12,550 --> 00:53:15,030
This is the-- I generated the
gradient with respect to r.

1169
00:53:15,030 --> 00:53:16,970
So if r is a vector,
this is the gradient

1170
00:53:16,970 --> 00:53:18,579
with respect to r, norm squared.

1171
00:53:18,579 --> 00:53:20,120
And this is the
gradient with respect

1172
00:53:20,120 --> 00:53:22,720
to the relative coordinate.

1173
00:53:22,720 --> 00:53:27,540
So for example, this is--
if we're in one dimension,

1174
00:53:27,540 --> 00:53:29,470
so this is strictly
one dimensional.

1175
00:53:29,470 --> 00:53:34,610
Then this is just the derivative
with respect to r squared.

1176
00:53:34,610 --> 00:53:40,260
And this is derivative with
respect to little r squared.

1177
00:53:40,260 --> 00:53:42,320
Just the gradient squared.

1178
00:53:42,320 --> 00:53:43,983
Did that answer your question?

1179
00:53:43,983 --> 00:53:44,608
AUDIENCE: Yeah.

1180
00:53:44,608 --> 00:53:46,520
What about in hydrogen?

1181
00:53:46,520 --> 00:53:48,830
PROFESSOR: Well then
it's gradient operator.

1182
00:53:48,830 --> 00:53:50,621
The thing that takes
the function gives you

1183
00:53:50,621 --> 00:53:53,740
a vector which is the
directional derivative

1184
00:53:53,740 --> 00:53:54,360
of that--

1185
00:53:54,360 --> 00:53:55,280
AUDIENCE: [INAUDIBLE].

1186
00:53:55,280 --> 00:53:55,740
PROFESSOR: Yeah.

1187
00:53:55,740 --> 00:53:56,360
Exactly.

1188
00:53:56,360 --> 00:53:59,395
In the r direction.

1189
00:53:59,395 --> 00:54:01,260
AUDIENCE: Is that
subraction right there?

1190
00:54:01,260 --> 00:54:02,230
PROFESSOR: Sorry?

1191
00:54:02,230 --> 00:54:04,070
AUDIENCE: Is that a
subtraction between--

1192
00:54:04,070 --> 00:54:05,211
PROFESSOR: A subtraction.

1193
00:54:05,211 --> 00:54:05,710
Where?

1194
00:54:05,710 --> 00:54:08,510
AUDIENCE: Between the first
and second [INAUDIBLE].

1195
00:54:08,510 --> 00:54:10,730
PROFESSOR: Minus h bar
squared upon 2m, dr squared.

1196
00:54:10,730 --> 00:54:13,091
Minus h bar squared
over 2 mu, dr-- this?

1197
00:54:13,091 --> 00:54:15,215
AUDIENCE: OK, so that's--
those are separate terms,

1198
00:54:15,215 --> 00:54:15,670
not multiplied?

1199
00:54:15,670 --> 00:54:15,770
PROFESSOR: Yeah.

1200
00:54:15,770 --> 00:54:17,140
They're not multiplied, right?

1201
00:54:17,140 --> 00:54:17,959
It's just the sum.

1202
00:54:17,959 --> 00:54:20,000
So the energy is kinetic
energy of the relative--

1203
00:54:20,000 --> 00:54:21,440
of the center of mass.

1204
00:54:21,440 --> 00:54:23,600
Kinetic energy of the
relative degree of freedom.

1205
00:54:23,600 --> 00:54:26,250
And then potential energy.

1206
00:54:26,250 --> 00:54:29,670
Which is exactly what happens
in classical mechanics.

1207
00:54:29,670 --> 00:54:30,170
Yeah?

1208
00:54:30,170 --> 00:54:33,747
AUDIENCE: What's that
circle under the first term?

1209
00:54:33,747 --> 00:54:34,580
PROFESSOR: This one?

1210
00:54:34,580 --> 00:54:35,357
M?

1211
00:54:35,357 --> 00:54:36,190
It's the total mass.

1212
00:54:36,190 --> 00:54:37,380
Ma plus mb.

1213
00:54:37,380 --> 00:54:41,160
And this one is the reduced
mass ma mbu upon ma [INAUDIBLE].

1214
00:54:44,760 --> 00:54:46,220
OK.

1215
00:54:46,220 --> 00:54:47,760
So here's a cool
thing that happens

1216
00:54:47,760 --> 00:54:51,080
with multiple particles that
didn't happen previously.

1217
00:54:51,080 --> 00:54:52,615
Suppose we have
identical particles.

1218
00:55:00,210 --> 00:55:04,070
So in particular, imagine
I have two billiard balls.

1219
00:55:04,070 --> 00:55:06,580
So I have two billiard balls
and I shoot-- I send one

1220
00:55:06,580 --> 00:55:07,732
in from one side.

1221
00:55:07,732 --> 00:55:09,690
And I send in the other
in from the other side.

1222
00:55:09,690 --> 00:55:11,126
And then they
collide and there's

1223
00:55:11,126 --> 00:55:12,500
some horrible
chaos that happens.

1224
00:55:12,500 --> 00:55:15,210
And one goes flying out
to this position, a.

1225
00:55:15,210 --> 00:55:17,865
And the other goes flying
out to this position, b.

1226
00:55:17,865 --> 00:55:18,365
OK?

1227
00:55:20,772 --> 00:55:21,730
Now here's my question.

1228
00:55:21,730 --> 00:55:24,609
Which ball went to a and
which ball went to b?

1229
00:55:24,609 --> 00:55:27,150
Well if we did this experiment,
that would be easy to answer.

1230
00:55:27,150 --> 00:55:30,490
Because we could paint a little
1 on this one and a little 2

1231
00:55:30,490 --> 00:55:32,680
on this one and
they'd go flying out.

1232
00:55:32,680 --> 00:55:35,726
And then at the end, when
you catch the ball at a

1233
00:55:35,726 --> 00:55:38,350
and you catch the ball at b, you
can grab them and look at them

1234
00:55:38,350 --> 00:55:40,800
and say aha, this one in my
left hand which I got from a

1235
00:55:40,800 --> 00:55:44,380
has 1 on it, and this one
has 2 on it, and I'm done.

1236
00:55:44,380 --> 00:55:47,570
The other way we could have
observed which ball went where

1237
00:55:47,570 --> 00:55:49,670
is we could've taken a
high speed film of this

1238
00:55:49,670 --> 00:55:52,042
and watched frame by frame
and said aha, particle 1,

1239
00:55:52,042 --> 00:55:53,930
particle 1, particle
1, particle 1.

1240
00:55:53,930 --> 00:55:54,920
Particle 2, 2, 2, 2.

1241
00:55:54,920 --> 00:55:55,420
Right?

1242
00:55:55,420 --> 00:55:57,322
We could have just
followed the paths.

1243
00:55:57,322 --> 00:55:59,280
And we haven't done
anything to the experiment.

1244
00:55:59,280 --> 00:56:00,390
We just took a film.

1245
00:56:00,390 --> 00:56:02,660
We haven't messed with it.

1246
00:56:02,660 --> 00:56:04,740
We don't change the
results of the experiment.

1247
00:56:04,740 --> 00:56:06,410
We just watch.

1248
00:56:06,410 --> 00:56:06,970
Right?

1249
00:56:06,970 --> 00:56:08,444
Perfectly doable classically.

1250
00:56:08,444 --> 00:56:09,985
Quantum mechanically,
is this doable?

1251
00:56:09,985 --> 00:56:11,457
AUDIENCE: No.

1252
00:56:11,457 --> 00:56:12,040
PROFESSOR: No.

1253
00:56:12,040 --> 00:56:15,325
Because first off, if you watch
carefully along and figure out

1254
00:56:15,325 --> 00:56:17,700
did it go through this slit,
did it go through that slit?

1255
00:56:17,700 --> 00:56:20,270
You know you change the results.

1256
00:56:20,270 --> 00:56:22,800
100% white versus 50-50.

1257
00:56:22,800 --> 00:56:26,000
If you go back to the boxes.

1258
00:56:26,000 --> 00:56:29,600
And meanwhile, if they're
truly identical particles

1259
00:56:29,600 --> 00:56:32,520
like electrons, there's
no way to paint anything

1260
00:56:32,520 --> 00:56:34,410
on the damn particle.

1261
00:56:34,410 --> 00:56:35,601
They're just electrons.

1262
00:56:35,601 --> 00:56:37,350
And they're completely,
as far as anyone's

1263
00:56:37,350 --> 00:56:40,160
ever been able to tell,
completely and utterly

1264
00:56:40,160 --> 00:56:40,700
identical.

1265
00:56:40,700 --> 00:56:43,529
They cannot be distinguished
in any way whatsoever.

1266
00:56:43,529 --> 00:56:45,820
So you can't do the thing
where you grab the one from a

1267
00:56:45,820 --> 00:56:48,403
and grab the one from b and say
aha, this one had the 1 on it.

1268
00:56:48,403 --> 00:56:49,920
They're indistinguishable.

1269
00:56:49,920 --> 00:56:50,420
Yeah.

1270
00:56:50,420 --> 00:56:51,295
AUDIENCE: [INAUDIBLE]

1271
00:56:56,202 --> 00:56:58,410
PROFESSOR: We're going to
come to the Pauli exclusion

1272
00:56:58,410 --> 00:56:59,430
principle.

1273
00:56:59,430 --> 00:57:00,110
Hold on to that.

1274
00:57:00,110 --> 00:57:00,860
Hold on to that question.

1275
00:57:00,860 --> 00:57:02,398
We're going to come--
we're going to get there.

1276
00:57:02,398 --> 00:57:02,898
OK.

1277
00:57:07,080 --> 00:57:09,780
So we can't-- if we have
truly identical particles,

1278
00:57:09,780 --> 00:57:13,840
what we mean by that is there's
no way to run this experiment

1279
00:57:13,840 --> 00:57:16,170
and determine which
particle ended up--

1280
00:57:16,170 --> 00:57:19,210
which of these two particles
went to a and which went to b.

1281
00:57:19,210 --> 00:57:20,390
Everyone cool with that?

1282
00:57:20,390 --> 00:57:21,500
They're identical.

1283
00:57:21,500 --> 00:57:24,280
And what I want to understand
is, what are the consequences?

1284
00:57:24,280 --> 00:57:26,340
So the first and basic
consequence of this

1285
00:57:26,340 --> 00:57:30,660
is that the probability that
the first particle ends up at a

1286
00:57:30,660 --> 00:57:33,250
and the second
particle ends up at b

1287
00:57:33,250 --> 00:57:39,590
must be equal to the probability
that the first particle ends up

1288
00:57:39,590 --> 00:57:42,020
at b and the second
particle ends up at a.

1289
00:57:42,020 --> 00:57:45,331
Because you can't
tell which is which.

1290
00:57:45,331 --> 00:57:47,580
If you can't tell, it must
be that those probabilities

1291
00:57:47,580 --> 00:57:49,144
are equal.

1292
00:57:49,144 --> 00:57:51,060
Because if they weren't
equal, you effectively

1293
00:57:51,060 --> 00:57:55,110
have skewed the results and
they're distinguishable.

1294
00:57:55,110 --> 00:57:56,610
These are totally
indistinguishable.

1295
00:57:56,610 --> 00:57:58,170
So the probability that
the first particle ends up

1296
00:57:58,170 --> 00:58:00,430
at a. second at b, must be equal
to the probability the first

1297
00:58:00,430 --> 00:58:02,263
particle ends up at b
and the second ends up

1298
00:58:02,263 --> 00:58:04,660
at a, because you cannot
tell the difference.

1299
00:58:04,660 --> 00:58:07,862
This is what it means
to be identical.

1300
00:58:07,862 --> 00:58:09,570
That equals sign is
what I mean by saying

1301
00:58:09,570 --> 00:58:11,585
I have identical particles.

1302
00:58:11,585 --> 00:58:12,085
Cool?

1303
00:58:15,110 --> 00:58:18,300
So let's find out what the
consequences of this are.

1304
00:58:18,300 --> 00:58:19,750
Define the following operator.

1305
00:58:19,750 --> 00:58:22,590
And Dave Larson,
if you're watching,

1306
00:58:22,590 --> 00:58:24,540
this is the [INAUDIBLE]
[INAUDIBLE] p.

1307
00:58:24,540 --> 00:58:27,530
So I'm going to call this
operator script-y p sub 1,2.

1308
00:58:27,530 --> 00:58:31,140
It's the operator-- so it's got
a little hat on it, one more

1309
00:58:31,140 --> 00:58:36,960
offense-- which takes
the first particle

1310
00:58:36,960 --> 00:58:40,110
and the second particle
and swaps them.

1311
00:58:40,110 --> 00:58:41,460
OK?

1312
00:58:41,460 --> 00:58:43,550
So-- and I guess I don't
even need the 1, 2.

1313
00:58:43,550 --> 00:58:48,210
This p operator swaps
particle 1 and particle 2.

1314
00:58:48,210 --> 00:58:52,170
So for example, it
takes probability

1315
00:58:52,170 --> 00:58:58,820
of a to b to
probability of b to a.

1316
00:59:06,090 --> 00:59:10,490
But more importantly,
this swapping operation

1317
00:59:10,490 --> 00:59:13,490
takes the wave function,
the amplitude, of a and b,

1318
00:59:13,490 --> 00:59:17,090
and it swaps a and b, psi of ba.

1319
00:59:21,600 --> 00:59:23,330
Now it's clear that
the probability--

1320
00:59:23,330 --> 00:59:27,099
that the swapping operation
does nothing to the probability,

1321
00:59:27,099 --> 00:59:29,140
because the fact that
they're identical particles

1322
00:59:29,140 --> 00:59:30,848
means that these are
equal to each other.

1323
00:59:30,848 --> 00:59:33,610
So it hasn't changed the answer.

1324
00:59:33,610 --> 00:59:35,040
But just because
the probabilities

1325
00:59:35,040 --> 00:59:36,490
are equal to each
other, does that

1326
00:59:36,490 --> 00:59:38,510
tell you that the
wave function is

1327
00:59:38,510 --> 00:59:40,592
invariant under
swapping the particles?

1328
00:59:40,592 --> 00:59:41,747
AUDIENCE: No.

1329
00:59:41,747 --> 00:59:42,330
PROFESSOR: No.

1330
00:59:42,330 --> 00:59:43,725
It doesn't have to be invariant.

1331
00:59:43,725 --> 00:59:46,640
The important thing is that
the norm squared of psi.

1332
00:59:46,640 --> 00:59:50,190
So in principle, this could
be equal to some phase,

1333
00:59:50,190 --> 00:59:54,184
e to the i of phi sub ab.

1334
00:59:54,184 --> 00:59:55,350
Let me call it theta sub ab.

1335
00:59:58,430 --> 01:00:00,470
Psi of a,b.

1336
01:00:00,470 --> 01:00:03,630
And if this was the case,
that there was a phase that we

1337
01:00:03,630 --> 01:00:05,800
got here, when we
take the norm squared,

1338
01:00:05,800 --> 01:00:08,670
the probability
remains the same.

1339
01:00:08,670 --> 01:00:09,170
OK?

1340
01:00:11,910 --> 01:00:13,660
On the other hand, we
know something else.

1341
01:00:13,660 --> 01:00:18,540
We know that if we take
the wave function psi ab.

1342
01:00:18,540 --> 01:00:23,650
And we swap it and then we
swap it again, what do we get?

1343
01:00:23,650 --> 01:00:25,710
I have ab.

1344
01:00:25,710 --> 01:00:31,312
But that means this is equal
to e to the 2i theta ab.

1345
01:00:34,760 --> 01:00:37,400
So swapping twice
had better give me 1,

1346
01:00:37,400 --> 01:00:39,675
so this had better
be equal to 1.

1347
01:00:39,675 --> 01:00:41,300
Let me write this
slightly differently.

1348
01:00:41,300 --> 01:00:46,446
This is e to the i theta
squared is equal to 1.

1349
01:00:46,446 --> 01:00:48,695
So what must be true of e
to the i theta of our phase?

1350
01:00:51,230 --> 01:00:53,120
It's a number that squares to 1.

1351
01:00:53,120 --> 01:00:54,860
So it could be
one of two values.

1352
01:00:54,860 --> 01:00:57,030
It could be 1 or it
could be minus 1.

1353
01:00:57,030 --> 01:00:59,850
That's it.

1354
01:00:59,850 --> 01:01:02,620
So what that tells
us is-- psi ab.

1355
01:01:05,390 --> 01:01:08,470
So that tells us
that p-- whoops.

1356
01:01:08,470 --> 01:01:19,661
P on psi ab is equal
plus or minus psi ba.

1357
01:01:19,661 --> 01:01:20,160
Sorry, ab.

1358
01:01:32,900 --> 01:01:35,540
Another way to say this
is just that-- another way

1359
01:01:35,540 --> 01:01:39,850
to say this is that p
squared acting on psi

1360
01:01:39,850 --> 01:01:41,360
is just psi again.

1361
01:01:41,360 --> 01:01:44,210
So the eigenvalues of p
have to be plus or minus 1.

1362
01:01:47,020 --> 01:01:47,860
Here they are.

1363
01:01:51,780 --> 01:01:53,955
So in fact, this
is-- let me phrase

1364
01:01:53,955 --> 01:01:56,760
this in a little
more correct way.

1365
01:01:56,760 --> 01:02:04,805
This tells us that
the eigenvalues of p

1366
01:02:04,805 --> 01:02:05,680
are plus and minus 1.

1367
01:02:16,980 --> 01:02:18,390
Yeah?

1368
01:02:18,390 --> 01:02:22,190
AUDIENCE: I'm trying to figure
out how it can be minus 1?

1369
01:02:22,190 --> 01:02:25,515
If p squared is psi
then it [INAUDIBLE]

1370
01:02:25,515 --> 01:02:26,950
has to be psi [INAUDIBLE]?

1371
01:02:26,950 --> 01:02:27,965
PROFESSOR: Yes.

1372
01:02:27,965 --> 01:02:30,074
AUDIENCE: How could we
add that other [? p? ?]

1373
01:02:30,074 --> 01:02:30,740
PROFESSOR: Good.

1374
01:02:30,740 --> 01:02:31,240
OK.

1375
01:02:31,240 --> 01:02:33,250
So let's check this quickly.

1376
01:02:39,230 --> 01:02:42,190
So the question is
how can it be minus 1.

1377
01:02:42,190 --> 01:02:45,212
How can that-- that doesn't--
that would seem to violate

1378
01:02:45,212 --> 01:02:45,920
our calculations.

1379
01:02:45,920 --> 01:02:48,010
So what this is
saying is that, we

1380
01:02:48,010 --> 01:02:52,120
know if we take p on, let's
say, p along psi of ab.

1381
01:02:52,120 --> 01:02:56,700
Let's say-- let psi be
an eigenfunction of p.

1382
01:02:56,700 --> 01:02:59,350
OK?

1383
01:02:59,350 --> 01:03:04,010
So if psi if an eigenfunction
of p, with eigenvalue minus 1,

1384
01:03:04,010 --> 01:03:07,040
then this p on psi is
equal to minus psi.

1385
01:03:07,040 --> 01:03:09,760
Yeah?

1386
01:03:09,760 --> 01:03:15,240
Now the probability is
equal to psi squared.

1387
01:03:15,240 --> 01:03:24,100
And so p on psi--
sorry, p on psi squared

1388
01:03:24,100 --> 01:03:31,240
is equal to minus psi squared.

1389
01:03:31,240 --> 01:03:34,349
We take each side,
it goes to minus psi.

1390
01:03:34,349 --> 01:03:35,890
So this is just
equal to psi squared.

1391
01:03:40,111 --> 01:03:41,050
Yeah?

1392
01:03:41,050 --> 01:03:42,170
Who asked the question?

1393
01:03:42,170 --> 01:03:42,800
Sorry.

1394
01:03:42,800 --> 01:03:43,130
Right.

1395
01:03:43,130 --> 01:03:43,629
OK.

1396
01:03:43,629 --> 01:03:48,410
So it leaves the norm
squared invariant.

1397
01:03:48,410 --> 01:03:51,750
So it's OK to have a minus
1 eigenvalue under p,

1398
01:03:51,750 --> 01:03:54,310
because that doesn't change
the probability distribution.

1399
01:03:54,310 --> 01:03:56,760
The probability distribution
is left invariant.

1400
01:03:56,760 --> 01:04:01,280
However, if we take p
squared on psi, that's

1401
01:04:01,280 --> 01:04:08,230
equal to p on p on psi, is equal
to the p on minus psi, which

1402
01:04:08,230 --> 01:04:13,584
is equal to minus minus
psi, which is equal to psi.

1403
01:04:13,584 --> 01:04:14,540
OK?

1404
01:04:14,540 --> 01:04:17,000
So this is the statement
that the square

1405
01:04:17,000 --> 01:04:17,875
acts as the identity.

1406
01:04:21,060 --> 01:04:23,004
Did that answer your question?

1407
01:04:23,004 --> 01:04:23,920
AUDIENCE: [INAUDIBLE].

1408
01:04:23,920 --> 01:04:24,880
PROFESSOR: OK.

1409
01:04:24,880 --> 01:04:26,320
OK.

1410
01:04:26,320 --> 01:04:30,538
AUDIENCE: Along the [INAUDIBLE],
we need p squared on side B

1411
01:04:30,538 --> 01:04:33,140
to go back to side
B. It would preserve

1412
01:04:33,140 --> 01:04:36,244
the probability regardless if
we had had the two [INAUDIBLE].

1413
01:04:36,244 --> 01:04:36,910
PROFESSOR: Good.

1414
01:04:36,910 --> 01:04:39,960
Why did p squared have
to be the identity?

1415
01:04:39,960 --> 01:04:41,390
Because what is p doing?

1416
01:04:41,390 --> 01:04:43,240
p takes two particles
and it swaps them.

1417
01:04:43,240 --> 01:04:44,340
AUDIENCE: Oh.

1418
01:04:44,340 --> 01:04:47,180
PROFESSOR: And if it swaps
them again, what do you get?

1419
01:04:47,180 --> 01:04:48,992
The original configuration.

1420
01:04:48,992 --> 01:04:50,470
Right?

1421
01:04:50,470 --> 01:04:52,840
If by swapping,
if by p, you mean

1422
01:04:52,840 --> 01:04:55,510
the thing that swaps those
particles, then doing it twice

1423
01:04:55,510 --> 01:04:56,980
is like not doing
anything at all.

1424
01:04:56,980 --> 01:04:58,250
You can define a
different quantity

1425
01:04:58,250 --> 01:05:00,510
which isn't this swapping
operation which does this

1426
01:05:00,510 --> 01:05:02,170
twice and gives
you something else.

1427
01:05:02,170 --> 01:05:03,410
That's perfectly reasonable.

1428
01:05:03,410 --> 01:05:04,770
But I'm going to be interested
in the operator, which

1429
01:05:04,770 --> 01:05:05,420
is just swap.

1430
01:05:05,420 --> 01:05:07,410
And if you do it twice, you
get back to the identity.

1431
01:05:07,410 --> 01:05:07,634
AUDIENCE: Oh.

1432
01:05:07,634 --> 01:05:08,420
OK.

1433
01:05:08,420 --> 01:05:09,346
PROFESSOR: Cool?

1434
01:05:09,346 --> 01:05:09,846
OK.

1435
01:05:09,846 --> 01:05:10,346
Yeah?

1436
01:05:10,346 --> 01:05:13,790
AUDIENCE: So regarding
the defaults,

1437
01:05:13,790 --> 01:05:17,241
is what the operation is doing
is changed particle number

1438
01:05:17,241 --> 01:05:21,190
1, particle number
2, [INAUDIBLE]?

1439
01:05:21,190 --> 01:05:22,210
PROFESSOR: Well--

1440
01:05:22,210 --> 01:05:24,162
AUDIENCE: What does it
do with [INAUDIBLE]?

1441
01:05:24,162 --> 01:05:24,870
PROFESSOR: Right.

1442
01:05:24,870 --> 01:05:28,230
So at any-- given any
configuration, at any moment

1443
01:05:28,230 --> 01:05:30,877
time, right, pick your wave
function, pick your state.

1444
01:05:30,877 --> 01:05:32,210
For example, two particles here.

1445
01:05:32,210 --> 01:05:34,765
What the script p operator
with swapping operator does,

1446
01:05:34,765 --> 01:05:35,916
it just swaps them.

1447
01:05:35,916 --> 01:05:39,350
So it swaps the position of one
and the position of the other.

1448
01:05:39,350 --> 01:05:40,330
AUDIENCE: [INAUDIBLE].

1449
01:05:40,330 --> 01:05:40,650
PROFESSOR: Right.

1450
01:05:40,650 --> 01:05:41,110
Exactly.

1451
01:05:41,110 --> 01:05:42,760
And that-- you do that,
at some moment in time.

1452
01:05:42,760 --> 01:05:43,920
You do that to a state.

1453
01:05:43,920 --> 01:05:46,330
So in that experiment,
what-- I mean,

1454
01:05:46,330 --> 01:05:48,390
there's no answer
the question, you

1455
01:05:48,390 --> 01:05:51,025
know, does p swap them before
or does it swap them after.

1456
01:05:51,025 --> 01:05:51,650
It's up to you.

1457
01:05:51,650 --> 01:05:54,740
You can apply the p
operator anytime you want.

1458
01:05:54,740 --> 01:05:55,522
Cool?

1459
01:05:55,522 --> 01:05:56,506
OK.

1460
01:05:56,506 --> 01:05:57,490
Yeah?

1461
01:05:57,490 --> 01:06:01,426
AUDIENCE: So when you
extracted the probabilities,

1462
01:06:01,426 --> 01:06:04,870
obviously if you have a case
where it's really far apart

1463
01:06:04,870 --> 01:06:08,314
and the two particles
end up [INAUDIBLE],

1464
01:06:08,314 --> 01:06:10,774
aren't they more likely
that they have not

1465
01:06:10,774 --> 01:06:13,240
moved the entire
distance in between?

1466
01:06:13,240 --> 01:06:15,960
PROFESSOR: Yeah, that
sounds reasonable.

1467
01:06:15,960 --> 01:06:17,904
AUDIENCE: It sounds like--

1468
01:06:17,904 --> 01:06:18,570
PROFESSOR: Yeah.

1469
01:06:18,570 --> 01:06:19,682
This is disconcerting.

1470
01:06:19,682 --> 01:06:22,015
AUDIENCE: --kind of like the
wave function. [INAUDIBLE].

1471
01:06:24,727 --> 01:06:26,060
PROFESSOR: That's exactly right.

1472
01:06:26,060 --> 01:06:27,640
So that's an
excellent observation.

1473
01:06:27,640 --> 01:06:28,700
Let me rephrase that slightly.

1474
01:06:28,700 --> 01:06:30,210
So here's the observation
that she's making.

1475
01:06:30,210 --> 01:06:31,668
It's exactly correct
and it's where

1476
01:06:31,668 --> 01:06:33,520
we're going to get
in a few minutes.

1477
01:06:33,520 --> 01:06:35,020
So the observation is this.

1478
01:06:35,020 --> 01:06:38,161
Look, imagine I take two
electrons, which for the moment

1479
01:06:38,161 --> 01:06:39,660
we'll just call
identical particles,

1480
01:06:39,660 --> 01:06:41,000
so we take two
identical particles.

1481
01:06:41,000 --> 01:06:42,916
Put one in my right hand,
one in my left hand.

1482
01:06:42,916 --> 01:06:45,940
And I just hold them there
and I wait for a while.

1483
01:06:45,940 --> 01:06:50,570
A while later, is it the same
electron in my right hand?

1484
01:06:50,570 --> 01:06:53,070
AUDIENCE: Probably.

1485
01:06:53,070 --> 01:06:54,160
PROFESSOR: I don't know.

1486
01:06:54,160 --> 01:06:55,930
They're identical.

1487
01:06:55,930 --> 01:06:57,460
I can't tell.

1488
01:06:57,460 --> 01:06:59,000
They're completely identical.

1489
01:06:59,000 --> 01:07:00,625
And so if you think
about this like you

1490
01:07:00,625 --> 01:07:03,250
do some like weak scattering--
if you did some weak scattering

1491
01:07:03,250 --> 01:07:06,050
process between these where you
pick them very far apart, very

1492
01:07:06,050 --> 01:07:07,730
slowly moving
along-- but they're

1493
01:07:07,730 --> 01:07:09,771
very far away so the
electrostatic interaction is

1494
01:07:09,771 --> 01:07:13,440
very small and so they repel
each other just a little bit.

1495
01:07:13,440 --> 01:07:15,264
If this system is,
in fact, identical

1496
01:07:15,264 --> 01:07:17,555
and if the wave function is,
let's say for the moment--

1497
01:07:17,555 --> 01:07:18,920
and we'll talk about
whether this is correct

1498
01:07:18,920 --> 01:07:21,680
or not-- if the wave function
is invariant under swapping

1499
01:07:21,680 --> 01:07:24,402
the particles-- let's just
imagine that it's invariant

1500
01:07:24,402 --> 01:07:26,110
and they're swapping
the particles-- then

1501
01:07:26,110 --> 01:07:27,530
there are two things
that could have happened.

1502
01:07:27,530 --> 01:07:29,340
The particles could
have done this.

1503
01:07:29,340 --> 01:07:32,590
Or there's also a contribution
where they do this.

1504
01:07:32,590 --> 01:07:34,700
Which kind of hurts.

1505
01:07:34,700 --> 01:07:37,110
And in order for the
system to be symmetric,

1506
01:07:37,110 --> 01:07:38,610
you have to have
both contributions.

1507
01:07:38,610 --> 01:07:39,600
So let's come to that.

1508
01:07:39,600 --> 01:07:43,970
But indeed, it's as if there's
some additional interactions,

1509
01:07:43,970 --> 01:07:45,370
or some additional correlations.

1510
01:07:45,370 --> 01:07:47,078
And that's exactly
what we want to study.

1511
01:07:47,078 --> 01:07:48,480
So let's get to that.

1512
01:07:48,480 --> 01:07:49,890
Very good observation.

1513
01:07:49,890 --> 01:07:52,570
So what I'd like to do
is make that precise.

1514
01:07:52,570 --> 01:07:56,120
So there are two
kinds of particles--

1515
01:07:56,120 --> 01:07:59,020
or three kinds of particles,
I should say-- in the world

1516
01:07:59,020 --> 01:08:00,880
from this point of view.

1517
01:08:00,880 --> 01:08:04,210
The first kind of particle
are distinguishable particles.

1518
01:08:04,210 --> 01:08:06,570
Suppose I have two
particles, one with a mass m

1519
01:08:06,570 --> 01:08:09,930
and one with a mass 2000m.

1520
01:08:09,930 --> 01:08:12,151
Say just to pick
randomly a number.

1521
01:08:12,151 --> 01:08:12,650
Right?

1522
01:08:12,650 --> 01:08:15,920
Those are distinguishable
because you can weigh them.

1523
01:08:15,920 --> 01:08:18,170
So you can tell which one
is the heavy one, which

1524
01:08:18,170 --> 01:08:19,080
one is the light one.

1525
01:08:19,080 --> 01:08:20,700
And you can tell.

1526
01:08:20,700 --> 01:08:21,500
Cool?

1527
01:08:21,500 --> 01:08:23,170
So there are
distinguishable particles.

1528
01:08:23,170 --> 01:08:24,919
And if we have
distinguishable particles--

1529
01:08:24,919 --> 01:08:27,439
I'll call psi sub d
for distinguishable--

1530
01:08:27,439 --> 01:08:30,670
then it's OK to have
the following thing.

1531
01:08:30,670 --> 01:08:32,250
Psi distinguishable
first particle

1532
01:08:32,250 --> 01:08:34,338
is-- the amplitude for
the first particle would

1533
01:08:34,338 --> 01:08:36,629
be an a and the amplitude
for the second particle would

1534
01:08:36,629 --> 01:08:41,880
be a b, could be chi of a, some
function of a, and phi of b.

1535
01:08:41,880 --> 01:08:45,579
This is not invariant
under a goes to b.

1536
01:08:45,579 --> 01:08:47,120
Because under a goes
to b, it becomes

1537
01:08:47,120 --> 01:08:49,139
chi of b, some
function of b, phi

1538
01:08:49,139 --> 01:08:51,330
of a, some different
function of a.

1539
01:08:51,330 --> 01:08:52,494
That's distinct.

1540
01:08:52,494 --> 01:08:53,410
But it doesn't matter.

1541
01:08:53,410 --> 01:08:54,810
They're distinguishable.

1542
01:08:54,810 --> 01:08:56,939
So that's perfectly fine.

1543
01:08:56,939 --> 01:09:05,040
It's not true p of ab
is not equal to p of ba.

1544
01:09:05,040 --> 01:09:08,979
But that's OK, because
they're distinguishable.

1545
01:09:08,979 --> 01:09:11,609
Everyone cool with that?

1546
01:09:11,609 --> 01:09:14,144
AUDIENCE: So p of ba is i of p?

1547
01:09:14,144 --> 01:09:14,810
PROFESSOR: Yeah.

1548
01:09:14,810 --> 01:09:15,309
Exactly.

1549
01:09:15,309 --> 01:09:17,090
So p of ab, this
is by definition

1550
01:09:17,090 --> 01:09:20,544
equal to norm squared of
psi d-- and I should say d.

1551
01:09:20,544 --> 01:09:26,450
D. Norm squared of psi
d of ab squared, which

1552
01:09:26,450 --> 01:09:32,330
is equal to norm squared of
chi of a, phi of b squared.

1553
01:09:32,330 --> 01:09:36,080
Whereas this guy would have
been chi of b, phi of a, norm

1554
01:09:36,080 --> 01:09:36,600
squared.

1555
01:09:36,600 --> 01:09:38,600
And since those are just
some stupid functions--

1556
01:09:38,600 --> 01:09:39,529
I haven't told
you what they are,

1557
01:09:39,529 --> 01:09:40,903
just some random
functions-- then

1558
01:09:40,903 --> 01:09:43,096
they're just different
probability distributions.

1559
01:09:46,439 --> 01:09:49,490
So on the other hand, if we
have indistinguishable wave

1560
01:09:49,490 --> 01:09:56,040
functions, then psi
indistinguishable,

1561
01:09:56,040 --> 01:10:04,895
we know that psi squared of a,b
squared is equal to psi of ba

1562
01:10:04,895 --> 01:10:05,710
norm squared.

1563
01:10:08,690 --> 01:10:12,690
And this is not of that form.

1564
01:10:12,690 --> 01:10:14,990
So we have two possibilities.

1565
01:10:14,990 --> 01:10:16,617
I'll write this
as psi plus minus.

1566
01:10:16,617 --> 01:10:18,950
If I know one of the particles
is in the state described

1567
01:10:18,950 --> 01:10:22,110
by chi, and the other particle
is in this state described

1568
01:10:22,110 --> 01:10:26,500
by phi, this would be
an example of a wave

1569
01:10:26,500 --> 01:10:28,170
function with that property.

1570
01:10:28,170 --> 01:10:30,960
However, it's not invariant
under swapping a and b.

1571
01:10:30,960 --> 01:10:34,141
So how could I make it invariant
under swapping a and b?

1572
01:10:34,141 --> 01:10:35,390
Well I could do the following.

1573
01:10:35,390 --> 01:10:39,660
1 over root 2, chi
of a, phi of b.

1574
01:10:39,660 --> 01:10:41,980
And if I want to
make it invariant,

1575
01:10:41,980 --> 01:10:46,050
I could add plus
chi of b, phi of a.

1576
01:10:46,050 --> 01:10:49,412
And now if I swap a and b, here
this becomes chi of b, phi a,

1577
01:10:49,412 --> 01:10:50,620
but that's exactly this term.

1578
01:10:50,620 --> 01:10:52,000
And this becomes
chi of a, phi b.

1579
01:10:52,000 --> 01:10:52,708
That's this term.

1580
01:10:52,708 --> 01:10:54,136
So just swap them.

1581
01:10:54,136 --> 01:10:55,642
Yeah?

1582
01:10:55,642 --> 01:10:57,100
But we don't need
the wave function

1583
01:10:57,100 --> 01:10:59,850
to be invariant under swapping.

1584
01:10:59,850 --> 01:11:03,780
We just need it to be
invariant under-- up to a sine.

1585
01:11:03,780 --> 01:11:05,870
So the other option is to
have a minus sign here.

1586
01:11:09,700 --> 01:11:12,720
And this gives us that
the swapping operation, p,

1587
01:11:12,720 --> 01:11:21,540
acting on psi plus minus of
a,b is equal to plus minus psi

1588
01:11:21,540 --> 01:11:22,040
of a,b.

1589
01:11:27,210 --> 01:11:31,190
And the plus is generally called
the symmetric, and the minus,

1590
01:11:31,190 --> 01:11:33,270
the anti-symmetric combination.

1591
01:11:37,540 --> 01:11:38,040
OK.

1592
01:11:38,040 --> 01:11:40,790
So distinguishable particles can
just be in some random state,

1593
01:11:40,790 --> 01:11:42,420
but there are
constraints on what

1594
01:11:42,420 --> 01:11:44,620
states, what combinations
of states are allowed,

1595
01:11:44,620 --> 01:11:46,110
for indistinguishable particles.

1596
01:11:46,110 --> 01:11:48,710
If you can't tell the difference
between two indistinguishable

1597
01:11:48,710 --> 01:11:51,220
particles and you know one is
in the state chi and the other

1598
01:11:51,220 --> 01:11:54,580
in the state phi, this
cannot be the wave function.

1599
01:11:54,580 --> 01:11:58,270
It must be either chi phi
plus chi phi in this fashion,

1600
01:11:58,270 --> 01:11:59,146
or minus.

1601
01:11:59,146 --> 01:12:01,740
Everyone cool with that?

1602
01:12:01,740 --> 01:12:02,684
Yeah?

1603
01:12:02,684 --> 01:12:07,624
AUDIENCE: I have a question
about what we mean by p of ab.

1604
01:12:07,624 --> 01:12:10,588
So normally, when we
talk about probabilities

1605
01:12:10,588 --> 01:12:15,231
we say that, yes, if you
measure a system what's

1606
01:12:15,231 --> 01:12:17,480
the probability that the
metric value will equal that?

1607
01:12:17,480 --> 01:12:18,170
PROFESSOR: Yes.

1608
01:12:18,170 --> 01:12:20,211
AUDIENCE: But if we can't
even determine anything

1609
01:12:20,211 --> 01:12:23,654
from that type of system, what
do we have a probability of?

1610
01:12:23,654 --> 01:12:24,320
PROFESSOR: Good.

1611
01:12:24,320 --> 01:12:26,110
So what this
probability means is,

1612
01:12:26,110 --> 01:12:29,900
what's the probability that
if-- that upon observation

1613
01:12:29,900 --> 01:12:32,760
in the system, I find
the first particle

1614
01:12:32,760 --> 01:12:35,720
to be at a and the
second part to be at b.

1615
01:12:35,720 --> 01:12:36,220
Right?

1616
01:12:36,220 --> 01:12:38,026
And I can check that
by saying like, look,

1617
01:12:38,026 --> 01:12:39,150
I catch the first particle.

1618
01:12:39,150 --> 01:12:40,640
I catch the second particle.

1619
01:12:40,640 --> 01:12:41,260
Is this a?

1620
01:12:41,260 --> 01:12:41,760
No.

1621
01:12:41,760 --> 01:12:42,260
OK.

1622
01:12:42,260 --> 01:12:45,099
Then that gives zero to the
probability distribution.

1623
01:12:45,099 --> 01:12:47,265
I do that a billion times
and I build up statistics.

1624
01:12:47,265 --> 01:12:50,262
And if I'm a fourth--
one out of four times,

1625
01:12:50,262 --> 01:12:52,345
I'll find a particle, the
first particle-- or I'll

1626
01:12:52,345 --> 01:12:53,550
find a particle at a.

1627
01:12:53,550 --> 01:12:55,589
One out of four times,
I'll find a partial at b.

1628
01:12:55,589 --> 01:12:57,880
And the probability that I
find the first particle at a

1629
01:12:57,880 --> 01:13:00,658
and the second particle
at b is one tenth, say.

1630
01:13:00,658 --> 01:13:01,158
OK.

1631
01:13:01,158 --> 01:13:04,010
AUDIENCE: So we can never make
that if they're identical,

1632
01:13:04,010 --> 01:13:04,510
right?

1633
01:13:04,510 --> 01:13:06,551
PROFESSOR: What you can't
do if they're identical

1634
01:13:06,551 --> 01:13:08,580
is you can't say which
particle you caught at a.

1635
01:13:08,580 --> 01:13:12,850
This is saying a particle at
a or a particle at b, right?

1636
01:13:12,850 --> 01:13:16,380
But it's-- but whether this is
the same or not of probability

1637
01:13:16,380 --> 01:13:19,473
that I find a particle at
b-- the first particle a b

1638
01:13:19,473 --> 01:13:21,499
and the second particle at a.

1639
01:13:21,499 --> 01:13:23,040
If you can't tell
the difference then

1640
01:13:23,040 --> 01:13:25,200
they're just a particle
at a and a particle at b.

1641
01:13:29,682 --> 01:13:32,172
OK.

1642
01:13:32,172 --> 01:13:33,670
OK.

1643
01:13:33,670 --> 01:13:37,010
So what does this give us?

1644
01:13:37,010 --> 01:13:39,360
So this gives us a
couple of nice facts.

1645
01:13:39,360 --> 01:13:48,010
So imagine-- that's
an exciting sound.

1646
01:13:48,010 --> 01:13:50,650
So this gives us a
couple of nice facts

1647
01:13:50,650 --> 01:13:54,840
with which we can find
awesomeness in the world.

1648
01:13:54,840 --> 01:13:57,440
The first is the following.

1649
01:14:00,320 --> 01:14:03,710
If you have identical
particles, then the energy

1650
01:14:03,710 --> 01:14:05,150
can't depend on the order.

1651
01:14:05,150 --> 01:14:07,650
If you have identical particles
and they're truly identical,

1652
01:14:07,650 --> 01:14:09,690
you swap them, then the
energy will be the same.

1653
01:14:09,690 --> 01:14:11,148
If it wasn't the
same, then they're

1654
01:14:11,148 --> 01:14:13,470
distinguishable by figuring
out what the energy is.

1655
01:14:13,470 --> 01:14:15,220
So in order that
they're identical,

1656
01:14:15,220 --> 01:14:19,730
it must be true that if
you swap the particles,

1657
01:14:19,730 --> 01:14:21,710
and then compute
the energy, this

1658
01:14:21,710 --> 01:14:23,757
should be the same as
what you get if you first

1659
01:14:23,757 --> 01:14:25,340
compute the energy
and then swap them.

1660
01:14:28,180 --> 01:14:32,020
Which is to say that
the commutator of e

1661
01:14:32,020 --> 01:14:36,695
with the swapping
operator, p, is 0.

1662
01:14:36,695 --> 01:14:37,660
OK?

1663
01:14:37,660 --> 01:14:43,967
But what that tells you is
that the expectation value of p

1664
01:14:43,967 --> 01:14:44,925
doesn't change in time.

1665
01:14:52,620 --> 01:14:56,110
In particular, if it's
some initial state--

1666
01:14:56,110 --> 01:15:10,310
if you were initially p on psi
is equal to plus psi at time 0,

1667
01:15:10,310 --> 01:15:15,200
then psi-- then p
at psi, p psi, is

1668
01:15:15,200 --> 01:15:18,875
equal to plus psi
for all future times.

1669
01:15:18,875 --> 01:15:23,251
P psi of p is equal
to plus psi of t.

1670
01:15:23,251 --> 01:15:23,750
OK.

1671
01:15:23,750 --> 01:15:25,810
So if you have two
identical particles

1672
01:15:25,810 --> 01:15:28,460
and the wave function is
invariant under swapping them

1673
01:15:28,460 --> 01:15:30,410
at some moment in time,
then it will always

1674
01:15:30,410 --> 01:15:32,610
be invariant under
swapping them.

1675
01:15:32,610 --> 01:15:35,230
It's a persistent
property of particles

1676
01:15:35,230 --> 01:15:38,730
that the wave function is
invariant under swapping them.

1677
01:15:38,730 --> 01:15:40,480
Yeah?

1678
01:15:40,480 --> 01:15:41,563
Yeah.

1679
01:15:41,563 --> 01:15:44,229
AUDIENCE: What about things that
like, become indistinguishable.

1680
01:15:44,229 --> 01:15:47,638
For example, you have atomic
nuclei like, for uranium.

1681
01:15:47,638 --> 01:15:50,560
And one of them is in
like a heavier isotope.

1682
01:15:50,560 --> 01:15:52,749
And during the time
that you're like,

1683
01:15:52,749 --> 01:15:54,332
holding them in your
hands one of them

1684
01:15:54,332 --> 01:15:55,874
decays, now they're
the same isotope.

1685
01:15:55,874 --> 01:15:56,540
PROFESSOR: Yeah.

1686
01:15:56,540 --> 01:15:57,090
That's-- OK.

1687
01:15:57,090 --> 01:15:58,200
AUDIENCE: [INAUDIBLE].

1688
01:15:58,200 --> 01:15:59,700
PROFESSOR: This
tells you something.

1689
01:15:59,700 --> 01:16:00,908
This is a very good question.

1690
01:16:00,908 --> 01:16:02,830
So the question is,
suppose I have an excited

1691
01:16:02,830 --> 01:16:04,620
isotope of uranium
that' s distinguishable

1692
01:16:04,620 --> 01:16:07,450
from some other
isotope of uranium.

1693
01:16:07,450 --> 01:16:10,230
I wait for a while
and then this thing

1694
01:16:10,230 --> 01:16:12,050
decays down to the
state-- it won't

1695
01:16:12,050 --> 01:16:13,290
decay to the stabilized
isotope of uranium,

1696
01:16:13,290 --> 01:16:15,081
but whatever-- it decays
down to-- we could

1697
01:16:15,081 --> 01:16:16,990
imagine a universe
in which it did.

1698
01:16:16,990 --> 01:16:19,400
It decays down to
a stable state.

1699
01:16:19,400 --> 01:16:21,390
And-- I mean,
uranium's never stable.

1700
01:16:21,390 --> 01:16:23,880
But anyway, you get the idea.

1701
01:16:23,880 --> 01:16:28,059
At which point they're
indistinguishable.

1702
01:16:28,059 --> 01:16:28,850
This sounds better.

1703
01:16:28,850 --> 01:16:32,370
Because now the wave
function should be invariant.

1704
01:16:32,370 --> 01:16:34,350
But it started out
not being invariant.

1705
01:16:34,350 --> 01:16:37,950
What's the problem
in this argument?

1706
01:16:37,950 --> 01:16:39,000
The system has changed.

1707
01:16:39,000 --> 01:16:40,957
In particular, something
went flying out.

1708
01:16:40,957 --> 01:16:42,540
So this is actually
kind of a nice way

1709
01:16:42,540 --> 01:16:45,010
to argue that there must
have been something else.

1710
01:16:45,010 --> 01:16:47,180
The wave function
describes a full system.

1711
01:16:47,180 --> 01:16:51,421
But if something leaves,
then it's not the same system

1712
01:16:51,421 --> 01:16:52,920
anymore and the
wave functions isn't

1713
01:16:52,920 --> 01:16:55,690
describing the same
degrees of freedom.

1714
01:16:55,690 --> 01:16:56,315
Something left.

1715
01:16:56,315 --> 01:16:58,315
AUDIENCE: But suppose we
keep track of that one,

1716
01:16:58,315 --> 01:17:00,139
we still can't swap
the two uraniums.

1717
01:17:00,139 --> 01:17:00,930
PROFESSOR: Exactly.

1718
01:17:00,930 --> 01:17:02,760
Then-- well, then there's some
additional constraint, right?

1719
01:17:02,760 --> 01:17:04,320
It must be invariant
under swapping that.

1720
01:17:04,320 --> 01:17:05,986
But it must-- but the
wave function also

1721
01:17:05,986 --> 01:17:08,039
knows about that extra
bit that went flying off.

1722
01:17:08,039 --> 01:17:09,455
And so the whole
wave function has

1723
01:17:09,455 --> 01:17:11,750
to be invariant under
swapping the identical parts,

1724
01:17:11,750 --> 01:17:15,950
but not invariant under
swap-- the invariance is not

1725
01:17:15,950 --> 01:17:17,450
just those two things.

1726
01:17:17,450 --> 01:17:20,434
They're correlated with that
thing that went flying away.

1727
01:17:20,434 --> 01:17:21,850
So another way to
think about this

1728
01:17:21,850 --> 01:17:24,540
is imagine two different
hydrogen atoms.

1729
01:17:24,540 --> 01:17:25,850
Here I've got a hydrogen atom.

1730
01:17:25,850 --> 01:17:27,130
It's an electron and a proton.

1731
01:17:27,130 --> 01:17:29,110
Here's a deuterium atom.

1732
01:17:29,110 --> 01:17:32,550
It's an electron bound to a
proton and a neutron glued

1733
01:17:32,550 --> 01:17:34,400
together, deuteron.

1734
01:17:34,400 --> 01:17:40,270
So are the electrons identical?

1735
01:17:40,270 --> 01:17:42,410
Yeah, they're totally identical.

1736
01:17:42,410 --> 01:17:43,910
So is the wave
function invariant--

1737
01:17:43,910 --> 01:17:45,680
does the wave function-- or
the probability distribution

1738
01:17:45,680 --> 01:17:47,832
have to be invariant under
swapping the electrons?

1739
01:17:47,832 --> 01:17:48,790
Yes, they're identical.

1740
01:17:48,790 --> 01:17:50,000
So the probability
distribution must

1741
01:17:50,000 --> 01:17:51,610
be invariant under
swapping the electrons.

1742
01:17:51,610 --> 01:17:53,870
However, is an identical
under swapping the hydrogen

1743
01:17:53,870 --> 01:17:55,129
with the deuterium?

1744
01:17:55,129 --> 01:17:55,670
AUDIENCE: No.

1745
01:17:55,670 --> 01:17:56,600
PROFESSOR: No.

1746
01:17:56,600 --> 01:17:59,990
So it's invariant under swapping
the identical parts and not

1747
01:17:59,990 --> 01:18:01,650
the non-identical parts.

1748
01:18:01,650 --> 01:18:02,160
Cool?

1749
01:18:02,160 --> 01:18:02,785
AUDIENCE: Yeah.

1750
01:18:02,785 --> 01:18:03,970
PROFESSOR: OK.

1751
01:18:03,970 --> 01:18:09,870
So this tells us that there
are two kinds of particles.

1752
01:18:09,870 --> 01:18:12,060
There are persistent--
or sorry, there

1753
01:18:12,060 --> 01:18:13,450
are three kinds of particles.

1754
01:18:13,450 --> 01:18:15,640
And these properties
are persistent.

1755
01:18:15,640 --> 01:18:18,171
The first kind of particle--
sets of particles--

1756
01:18:18,171 --> 01:18:19,420
are distinguishable particles.

1757
01:18:23,977 --> 01:18:26,310
If you have two particles
which are distinguishable then

1758
01:18:26,310 --> 01:18:26,810
you're done.

1759
01:18:26,810 --> 01:18:27,809
They're distinguishable.

1760
01:18:27,809 --> 01:18:28,650
Nothing else to say.

1761
01:18:28,650 --> 01:18:34,140
Two, you can have identical
particles with the property

1762
01:18:34,140 --> 01:18:39,750
that if you take p on psi
a,b-- sorry, p on psi.

1763
01:18:39,750 --> 01:18:43,450
If you swap the particles,
this is equal to plus psi.

1764
01:18:43,450 --> 01:18:47,840
And then you have-- so
identical with plus.

1765
01:18:47,840 --> 01:18:51,230
And you have three
identical particles

1766
01:18:51,230 --> 01:18:55,280
where if you swap the particles,
you get a minus sign on psi.

1767
01:18:58,250 --> 01:18:59,820
These particles are
called-- we have

1768
01:18:59,820 --> 01:19:01,236
a name for particles
of this kind.

1769
01:19:01,236 --> 01:19:02,907
Bosons.

1770
01:19:02,907 --> 01:19:03,990
These are called fermions.

1771
01:19:11,760 --> 01:19:14,990
My TA did a bad thing to me when
I was taking quantum mechanics.

1772
01:19:14,990 --> 01:19:17,110
And said, just imagine
them as little tiny Fermis.

1773
01:19:17,110 --> 01:19:18,943
So just take a picture
of Fermi in your head

1774
01:19:18,943 --> 01:19:21,200
and imagine little
tiny-- and this is cruel,

1775
01:19:21,200 --> 01:19:22,200
because I can't help it.

1776
01:19:22,200 --> 01:19:24,090
Every time someone in a seminar
is like, blah, blah, blah,

1777
01:19:24,090 --> 01:19:24,360
Fermi.

1778
01:19:24,360 --> 01:19:24,970
And I'm like, damn it.

1779
01:19:24,970 --> 01:19:25,511
Little Fermi.

1780
01:19:28,890 --> 01:19:30,880
It's really quite annoying.

1781
01:19:30,880 --> 01:19:32,090
So now you have it too.

1782
01:19:32,090 --> 01:19:34,390
Great.

1783
01:19:34,390 --> 01:19:37,735
So what are the consequences
of the fact that there

1784
01:19:37,735 --> 01:19:40,160
are two kinds of
particles in the universe?

1785
01:19:40,160 --> 01:19:44,390
These fermions and bosons?

1786
01:19:44,390 --> 01:19:46,000
This has a really
lovely consequence.

1787
01:19:46,000 --> 01:19:49,710
The first is, suppose
we have two fermions.

1788
01:19:49,710 --> 01:19:50,280
OK?

1789
01:19:50,280 --> 01:19:51,800
Examples of fermions
are electrons.

1790
01:19:51,800 --> 01:19:55,330
Suppose we have a wave
function for two fermions.

1791
01:19:55,330 --> 01:19:58,000
The first might-- what's
that probability amplitude

1792
01:19:58,000 --> 01:20:00,180
that the first is at a
and the second is t b?

1793
01:20:00,180 --> 01:20:03,789
Well it's this psi of a,b.

1794
01:20:03,789 --> 01:20:05,330
And the statement
that it's a fermion

1795
01:20:05,330 --> 01:20:09,310
is the statement that this is
equal to minus psi of b, a.

1796
01:20:09,310 --> 01:20:12,220
If we swap the positions
of the two particles,

1797
01:20:12,220 --> 01:20:14,300
we must pick up a minus sign.

1798
01:20:14,300 --> 01:20:17,250
This tells us in particular
that the probability amplitude

1799
01:20:17,250 --> 01:20:19,625
for the first particle to be
at a and the second particle

1800
01:20:19,625 --> 01:20:22,700
to be at a is equal
to minus itself.

1801
01:20:22,700 --> 01:20:24,200
Because upon swapping
the particles,

1802
01:20:24,200 --> 01:20:27,190
we get minus psi of a,a.

1803
01:20:27,190 --> 01:20:29,820
So the probability amplitude
to find two fermions

1804
01:20:29,820 --> 01:20:33,390
at the same place is equal to 0.

1805
01:20:33,390 --> 01:20:38,046
Two fermions cannot
occupy the same state.

1806
01:20:38,046 --> 01:20:39,670
This was the Pauli
exclusion principle,

1807
01:20:39,670 --> 01:20:41,420
which we needed to get
the periodic table.

1808
01:20:44,800 --> 01:20:47,620
Pauli.

1809
01:20:47,620 --> 01:20:49,500
Two.

1810
01:20:49,500 --> 01:20:53,400
If we have-- so this is
fermions-- if we have bosons,

1811
01:20:53,400 --> 01:20:57,590
psi of a,b-- let
me write this out.

1812
01:20:57,590 --> 01:21:02,180
So psi fermion or
boson-- so fermion

1813
01:21:02,180 --> 01:21:03,862
is going to be with
a minus and boson

1814
01:21:03,862 --> 01:21:05,570
is going to be with
a plus-- is equal to,

1815
01:21:05,570 --> 01:21:08,260
suppose I have two
particles 1 over root 2.

1816
01:21:08,260 --> 01:21:10,550
And one particle
is in the state chi

1817
01:21:10,550 --> 01:21:12,654
and the other particle
is in the state phi.

1818
01:21:12,654 --> 01:21:15,070
But in order to be fermionic
or bosonic, in order for this

1819
01:21:15,070 --> 01:21:16,695
to be invariant under
swapping a and b,

1820
01:21:16,695 --> 01:21:24,150
we have to have a plus or
minus chi of b, phi of a.

1821
01:21:24,150 --> 01:21:26,594
And here we immediately see
this Pauli principle at work.

1822
01:21:26,594 --> 01:21:28,760
If I could take the fermionic
example with the minus

1823
01:21:28,760 --> 01:21:34,370
sign, then psi evaluated at a,
a must be chi at a, phi at b,

1824
01:21:34,370 --> 01:21:36,770
minus chi at b, phi at a.

1825
01:21:36,770 --> 01:21:40,600
But if a is b, this is chi
a, phi a, minus chi a, phi a.

1826
01:21:40,600 --> 01:21:42,950
That's zero.

1827
01:21:42,950 --> 01:21:44,624
But let's think about
the bosonic case.

1828
01:21:44,624 --> 01:21:46,790
If we have a bosonic field--
or, if we have-- sorry.

1829
01:21:46,790 --> 01:21:50,260
If we have bosonic
identical particles, then

1830
01:21:50,260 --> 01:21:55,540
psi b at a with a-- for
our fermion, it was zero.

1831
01:21:55,540 --> 01:22:01,410
But psi b of the amplitude to
be at two at the same place

1832
01:22:01,410 --> 01:22:04,500
is equal to-- well, if b
is a, then these two terms

1833
01:22:04,500 --> 01:22:05,930
are identical and
we have a plus.

1834
01:22:05,930 --> 01:22:10,380
This is root 2 chi
at a, phi at a.

1835
01:22:10,380 --> 01:22:11,880
Which is greater
than what you might

1836
01:22:11,880 --> 01:22:13,255
have naively
guessed, which would

1837
01:22:13,255 --> 01:22:15,410
have been just chi a, phi a.

1838
01:22:15,410 --> 01:22:18,660
For bosons, they really like
being next to each other.

1839
01:22:18,660 --> 01:22:20,800
They really like being
in the same place.

1840
01:22:20,800 --> 01:22:23,990
And this will eventually
lead to lasers.

1841
01:22:23,990 --> 01:22:26,540
So from this simple
statistical property

1842
01:22:26,540 --> 01:22:28,440
under swapping, picking
up a minus sign,

1843
01:22:28,440 --> 01:22:29,814
we get the Pauli
principle, which

1844
01:22:29,814 --> 01:22:30,970
gave us the periodic table.

1845
01:22:30,970 --> 01:22:33,670
And is going to give us
in the next lecture bands

1846
01:22:33,670 --> 01:22:35,910
and solids in conductivity.

1847
01:22:35,910 --> 01:22:39,670
From the same principle but
with a plus, and the persistence

1848
01:22:39,670 --> 01:22:43,300
of this sine, from the
persistence of the statistics,

1849
01:22:43,300 --> 01:22:46,040
from the fact that we
have two identical bosons,

1850
01:22:46,040 --> 01:22:47,930
we get that they like
to be in the same spot.

1851
01:22:47,930 --> 01:22:50,361
And we'll get lasers and
boson [INAUDIBLE] condensates.

1852
01:22:50,361 --> 01:22:52,110
And next time, we'll
pick up with fermions

1853
01:22:52,110 --> 01:22:53,750
in a periodic
potential and we'll

1854
01:22:53,750 --> 01:22:56,900
study solids and get to diamond.