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PROFESSOR: All right.

9
00:00:22,310 --> 00:00:30,750
So today, we'll continue our
kind of review that included,

10
00:00:30,750 --> 00:00:34,130
of course, the last lecture,
the variational principle that's

11
00:00:34,130 --> 00:00:38,560
supposed to be new stuff
you didn't see in 804.

12
00:00:38,560 --> 00:00:42,690
And today, as we
continue, we'll talk

13
00:00:42,690 --> 00:00:47,930
about position and momentum for
about 30 minutes or 40 minutes,

14
00:00:47,930 --> 00:00:54,430
and then begin
the study of spin.

15
00:00:54,430 --> 00:00:59,110
That will be spin-1/2 with
a Stern-Gerlach experiment

16
00:00:59,110 --> 00:01:02,430
and the mathematics
that comes out of it.

17
00:01:02,430 --> 00:01:05,310
Now, we will talk about the
Stern-Gerlach experiment

18
00:01:05,310 --> 00:01:07,440
in quite some detail
so that you can

19
00:01:07,440 --> 00:01:10,770
appreciate what
was going on there.

20
00:01:10,770 --> 00:01:18,060
And then we will extract a few
of the mathematical lessons

21
00:01:18,060 --> 00:01:23,410
that this experiment tells
us about quantum mechanics.

22
00:01:23,410 --> 00:01:25,530
Immediately after
that, which will

23
00:01:25,530 --> 00:01:30,030
be probably middle of next
lecture, we will pivot.

24
00:01:30,030 --> 00:01:32,580
And as we learn this
mathematics that

25
00:01:32,580 --> 00:01:37,790
the Stern-Gerlach experiment
is telling us or asking us for,

26
00:01:37,790 --> 00:01:42,700
we will go in some detail
on the necessary mathematics

27
00:01:42,700 --> 00:01:44,190
for quantum mechanics.

28
00:01:44,190 --> 00:01:48,790
We'll talk about vector
spaces, linear operators,

29
00:01:48,790 --> 00:01:54,200
Hermitian operators, unitary
operators, [INAUDIBLE],

30
00:01:54,200 --> 00:01:57,740
matrix representations,
all kinds of things.

31
00:01:57,740 --> 00:02:02,120
That probably will be about two
weeks, three lectures at least.

32
00:02:02,120 --> 00:02:05,940
So it will be a nice study.

33
00:02:05,940 --> 00:02:10,190
And in that way,
people that don't

34
00:02:10,190 --> 00:02:12,370
have a background
in linear algebra

35
00:02:12,370 --> 00:02:15,360
will feel more comfortable with
what we're going to be doing.

36
00:02:15,360 --> 00:02:16,880
And I think even
for the people that

37
00:02:16,880 --> 00:02:19,140
have a background
in linear algebra,

38
00:02:19,140 --> 00:02:22,470
you will gain a new
appreciation about the concepts

39
00:02:22,470 --> 00:02:23,950
that we meet here.

40
00:02:23,950 --> 00:02:28,790
So today, we begin, therefore,
with position and momentum,

41
00:02:28,790 --> 00:02:31,640
and these are operators
in quantum mechanics.

42
00:02:31,640 --> 00:02:36,260
And they have letters
to denote them. x,

43
00:02:36,260 --> 00:02:39,900
we put a hat with it,
that's a position operator.

44
00:02:39,900 --> 00:02:42,730
p, we put a hat on it.

45
00:02:42,730 --> 00:02:52,460
And the position and momentum
operators don't commute.

46
00:02:52,460 --> 00:02:58,250
And the commutator
is given by ih bar.

47
00:02:58,250 --> 00:03:02,990
Now, we have been dealing
so far with wave functions.

48
00:03:02,990 --> 00:03:07,920
Our wave functions, where
these functions of x and t,

49
00:03:07,920 --> 00:03:11,210
they represent the dynamics
of your system, the dynamics

50
00:03:11,210 --> 00:03:13,410
of your particle as
it moves in time.

51
00:03:13,410 --> 00:03:18,140
But time, as you are seeing
in quantum mechanics,

52
00:03:18,140 --> 00:03:19,602
is a little bit of a spectator.

53
00:03:19,602 --> 00:03:21,980
It's an arena where
things happen.

54
00:03:21,980 --> 00:03:25,670
But the operators, and most
of the interesting things,

55
00:03:25,670 --> 00:03:30,570
are going on without
reference to time.

56
00:03:30,570 --> 00:03:32,780
Time evolution, you
have an expansion

57
00:03:32,780 --> 00:03:35,140
of a wave function
in terms of energy,

58
00:03:35,140 --> 00:03:38,340
eigenstates, at a given time.

59
00:03:38,340 --> 00:03:40,650
And then you can
evolve it easily

60
00:03:40,650 --> 00:03:43,700
with the way we've
learned, adding e

61
00:03:43,700 --> 00:03:50,340
to the minus i et over h bar
for each energy eigenstate.

62
00:03:50,340 --> 00:03:52,710
So time will play no role here.

63
00:03:52,710 --> 00:03:55,380
So when I talk about the
wave function, at this moment

64
00:03:55,380 --> 00:03:57,630
you could put the
time, but we will

65
00:03:57,630 --> 00:04:01,400
talk about the wave functions
that have no time dependence.

66
00:04:01,400 --> 00:04:04,700
So, say, a psi of
x wave function.

67
00:04:11,010 --> 00:04:16,970
So this psi of x may be the true
wave function at time equals 0,

68
00:04:16,970 --> 00:04:23,510
or you could just simply
think of it as the psi of x.

69
00:04:23,510 --> 00:04:26,770
Now, this wave function
means that we're

70
00:04:26,770 --> 00:04:29,420
treating x in a
particular way, and we

71
00:04:29,420 --> 00:04:33,310
say that we're working in the
x representation, the position

72
00:04:33,310 --> 00:04:35,570
representation.

73
00:04:35,570 --> 00:04:39,870
Now, this means that
we have an easy way

74
00:04:39,870 --> 00:04:43,800
to figure out what
this operator does

75
00:04:43,800 --> 00:04:46,700
when it acts on this function.

76
00:04:46,700 --> 00:04:48,320
So what it acts
on this function,

77
00:04:48,320 --> 00:04:50,840
it will give you
another function,

78
00:04:50,840 --> 00:04:53,730
and the definition of
this is that the position

79
00:04:53,730 --> 00:04:58,810
operator acting on
the function psi of x

80
00:04:58,810 --> 00:05:03,070
is defined to be
another function,

81
00:05:03,070 --> 00:05:06,845
which is the function
x times psi of x.

82
00:05:16,390 --> 00:05:20,025
Well, we're talking about these
wave functions and operators

83
00:05:20,025 --> 00:05:21,380
on wave functions.

84
00:05:21,380 --> 00:05:25,550
And a recurrent theme
in quantum mechanics

85
00:05:25,550 --> 00:05:28,660
is that we will think
of wave functions,

86
00:05:28,660 --> 00:05:31,160
sometimes we call them states.

87
00:05:31,160 --> 00:05:34,200
Sometimes we call them vectors.

88
00:05:34,200 --> 00:05:39,640
And we basically think of
wave functions as vectors.

89
00:05:39,640 --> 00:05:42,550
And things that act
on wave functions

90
00:05:42,550 --> 00:05:44,480
are the things that
act on vectors.

91
00:05:44,480 --> 00:05:46,390
And the things that
act on vectors,

92
00:05:46,390 --> 00:05:50,110
as you know in
mathematics, is matrices.

93
00:05:50,110 --> 00:05:54,530
So we're compelled, even
at this early stage,

94
00:05:54,530 --> 00:05:57,600
to get a picture of
how that language would

95
00:05:57,600 --> 00:06:02,410
go if we're talking
about these things.

96
00:06:02,410 --> 00:06:07,020
So how do we think of a
wave function as a vector?

97
00:06:07,020 --> 00:06:12,000
And how do we think
of x as a matrix?

98
00:06:12,000 --> 00:06:15,550
So there's a way to do that.

99
00:06:15,550 --> 00:06:21,580
It will not be totally
precise, but it's clear enough.

100
00:06:21,580 --> 00:06:24,340
So suppose you have
a wave function,

101
00:06:24,340 --> 00:06:28,545
and we're interested in
its values from 0 up to a.

102
00:06:33,120 --> 00:06:38,230
This wave function is a
function of x between 0 and a.

103
00:06:38,230 --> 00:06:45,250
So it's the psi of x
for x between a and 0.

104
00:06:45,250 --> 00:06:47,420
That's all the information.

105
00:06:47,420 --> 00:06:50,410
What we're going to
do is we're going

106
00:06:50,410 --> 00:06:55,540
to divide this thing,
this line, this segment,

107
00:06:55,540 --> 00:06:56,940
into a lot of pieces.

108
00:06:56,940 --> 00:06:58,610
And we're going to
say, look, instead

109
00:06:58,610 --> 00:07:03,040
of writing a function like
sine of x or cosine of x,

110
00:07:03,040 --> 00:07:05,910
let's just give the
values and organize them

111
00:07:05,910 --> 00:07:09,780
as if this will be a
vector of many components.

112
00:07:09,780 --> 00:07:15,800
So let's divide this
in sizes epsilon,

113
00:07:15,800 --> 00:07:20,510
such that N times
epsilon is equal to a.

114
00:07:20,510 --> 00:07:23,700
So there are N of
these intervals.

115
00:07:23,700 --> 00:07:29,696
So we think of psi as a
vector whose first component

116
00:07:29,696 --> 00:07:31,390
is psi at 0.

117
00:07:31,390 --> 00:07:34,190
The second is psi at epsilon.

118
00:07:34,190 --> 00:07:38,500
The third is psi at 2 epsilon.

119
00:07:38,500 --> 00:07:43,560
And the last one is
psi at N epsilon.

120
00:07:43,560 --> 00:07:47,370
And depending on how much
accuracy you want to work with,

121
00:07:47,370 --> 00:07:54,390
you take epsilon smaller and
larger, keeping a constant.

122
00:07:54,390 --> 00:07:59,110
And this would be like
summarizing all the information

123
00:07:59,110 --> 00:08:00,705
of a function in a vector.

124
00:08:03,870 --> 00:08:08,800
Now, that's intuitively a
nice way to think of it.

125
00:08:08,800 --> 00:08:13,960
May look, with your background
in classical physics,

126
00:08:13,960 --> 00:08:18,910
a little strange that we
sort of put the value at 0

127
00:08:18,910 --> 00:08:22,740
along the x-axis, first
component, the value at epsilon

128
00:08:22,740 --> 00:08:26,260
along the y, the value
of 2 epsilon along the z.

129
00:08:26,260 --> 00:08:27,630
But we need more axes.

130
00:08:27,630 --> 00:08:30,270
So you need many axes here.

131
00:08:30,270 --> 00:08:39,299
In this case, this is a
N plus 1 column vector.

132
00:08:39,299 --> 00:08:43,480
It has N plus 1
entries, because 0 up

133
00:08:43,480 --> 00:08:45,740
to N, that's N plus 1 entries.

134
00:08:49,060 --> 00:08:53,900
But that's a fine way
of thinking of it.

135
00:08:53,900 --> 00:08:56,700
Not exact because
we have an epsilon.

136
00:08:56,700 --> 00:09:00,020
In this way of thinking
about the wave function,

137
00:09:00,020 --> 00:09:08,110
we can then ask, what does
the matrix x hat look like?

138
00:09:08,110 --> 00:09:13,110
So x hat is an operator,
and it acts this way.

139
00:09:13,110 --> 00:09:15,650
So here is how it looks like.

140
00:09:15,650 --> 00:09:25,180
We would think of x hat as an
N plus 1 times N plus 1 matrix.

141
00:09:30,110 --> 00:09:33,150
And its entries
are 0 everywhere,

142
00:09:33,150 --> 00:09:35,190
except in the
diagonal, where they

143
00:09:35,190 --> 00:09:41,980
are 0 epsilon, 2
epsilon, up to N epsilon.

144
00:09:44,685 --> 00:09:47,265
And here is a big 0 and a big 0.

145
00:09:49,980 --> 00:09:55,700
This, I claim, is the way you
should think of the x operator

146
00:09:55,700 --> 00:10:01,640
if you thought of the wave
function the way we wrote it.

147
00:10:01,640 --> 00:10:03,380
And how do we check that?

148
00:10:03,380 --> 00:10:09,500
Well, x operator acting on psi
should be this acting on that.

149
00:10:09,500 --> 00:10:13,200
And then, indeed,
we see that if x hat

150
00:10:13,200 --> 00:10:18,560
is acting on psi of
x, what do we get?

151
00:10:18,560 --> 00:10:25,350
Well, it's easy to multiply a
diagonal matrix times a vector.

152
00:10:25,350 --> 00:10:31,190
Here you get 0 times psi of 0.

153
00:10:31,190 --> 00:10:36,390
You get a vector, so let
me make this thinner.

154
00:10:36,390 --> 00:10:45,180
Then I get epsilon times psi
of epsilon, 2 epsilon times

155
00:10:45,180 --> 00:10:53,090
psi of 2 epsilon, up to N
epsilon times psi of N epsilon.

156
00:10:53,090 --> 00:10:58,490
And indeed, that matrix looks
like the matrix associated

157
00:10:58,490 --> 00:11:02,890
with this wave
function because here

158
00:11:02,890 --> 00:11:05,950
is the value at 0 of
this wave function.

159
00:11:05,950 --> 00:11:11,040
Here is the value at epsilon of
this wave function, and so on.

160
00:11:11,040 --> 00:11:15,410
So this has worked
out all right.

161
00:11:15,410 --> 00:11:20,560
We can think of the wave
function as a column vector,

162
00:11:20,560 --> 00:11:24,540
and then the position operator
as this vector as well.

163
00:11:27,270 --> 00:11:32,670
Now, given that we know how
the x operator is defined,

164
00:11:32,670 --> 00:11:37,250
we can also think
easily about what

165
00:11:37,250 --> 00:11:40,810
is the expectation value
of x on a wave function.

166
00:11:40,810 --> 00:11:43,800
Something that you
really know, but now

167
00:11:43,800 --> 00:11:46,370
maybe becomes a little clearer.

168
00:11:46,370 --> 00:11:49,850
Here you're supposed
to do psi star

169
00:11:49,850 --> 00:11:56,590
of x times the x operator
acting on psi of x.

170
00:11:56,590 --> 00:11:59,220
But we have the
definition of this,

171
00:11:59,220 --> 00:12:05,040
so this is, as you imagine, dx--
and I should put primes maybe,

172
00:12:05,040 --> 00:12:09,430
well, I don't have to
put primes-- dx psi

173
00:12:09,430 --> 00:12:14,910
star of x x psi of
x, which is what

174
00:12:14,910 --> 00:12:16,575
you would have done anyway.

175
00:12:20,960 --> 00:12:26,200
Well, given that we've started
with this, we can ask also,

176
00:12:26,200 --> 00:12:29,960
is there eigenstates
of the x operator?

177
00:12:39,360 --> 00:12:40,970
Yes, there are.

178
00:12:40,970 --> 00:12:44,790
but then fortunately,
are a bit singular.

179
00:12:44,790 --> 00:12:48,150
So what should be
an eigenstate of x?

180
00:12:48,150 --> 00:12:51,860
It's some sort of state.

181
00:12:51,860 --> 00:12:55,480
Intuitively, it has a definite
value of the position.

182
00:12:55,480 --> 00:12:57,970
So it just exists
for some value of x.

183
00:12:57,970 --> 00:13:01,390
So it's naturally thought
as a delta function.

184
00:13:01,390 --> 00:13:09,370
So let me define a
function, psi sub x0 of x.

185
00:13:09,370 --> 00:13:13,530
So it's a function
of x labeled by x0,

186
00:13:13,530 --> 00:13:17,320
and define it to be
delta of x minus x0.

187
00:13:21,050 --> 00:13:24,790
So I claim that is an
eigenstate of x hat.

188
00:13:24,790 --> 00:13:31,750
x hat on psi x0 of x is
equal, by definition,

189
00:13:31,750 --> 00:13:42,990
to x times psi x0 of x, which
is x times delta of x minus x0.

190
00:13:42,990 --> 00:13:45,220
And when you multiply
a function of x

191
00:13:45,220 --> 00:13:48,250
times a delta
function in x, it is

192
00:13:48,250 --> 00:13:50,660
possible to evaluate
the function that

193
00:13:50,660 --> 00:13:53,550
is being multiplied
by the delta function

194
00:13:53,550 --> 00:13:57,350
at the place where the
delta function fires.

195
00:13:57,350 --> 00:14:00,880
It has the same effect
on integrals or anything

196
00:14:00,880 --> 00:14:01,730
that you would do.

197
00:14:01,730 --> 00:14:08,230
So here, this is equal to
x0 times delta x minus x0.

198
00:14:08,230 --> 00:14:11,950
You evaluate the x at x0.

199
00:14:11,950 --> 00:14:18,550
And finally, this is x0 times
that function psi x0 of x.

200
00:14:18,550 --> 00:14:23,490
And therefore, you've shown
that this operator acting

201
00:14:23,490 --> 00:14:27,510
on this function reproduces
the function-- that's

202
00:14:27,510 --> 00:14:30,880
the definition of
eigenstate as an operator--

203
00:14:30,880 --> 00:14:36,170
and the eigenvalue is the number
that appears here, and it's x0.

204
00:14:36,170 --> 00:14:44,150
So this function
is an eigenstate

205
00:14:44,150 --> 00:14:51,950
of x hat with
eigenvalue, e.v., x0.

206
00:14:59,010 --> 00:15:03,260
The only complication
with this eigenfunction

207
00:15:03,260 --> 00:15:05,780
is that it's not normalizable.

208
00:15:05,780 --> 00:15:09,060
So it doesn't
represent the particle.

209
00:15:09,060 --> 00:15:13,690
It can be used to
represent the particle,

210
00:15:13,690 --> 00:15:16,490
but it's a useful function.

211
00:15:16,490 --> 00:15:19,000
You can think of it
as something that

212
00:15:19,000 --> 00:15:22,360
can help you do physics,
and don't insist

213
00:15:22,360 --> 00:15:26,000
that it represents a particle.

214
00:15:26,000 --> 00:15:28,640
So this is the
story for position.

215
00:15:28,640 --> 00:15:32,680
And the position gets actually
more interesting as soon

216
00:15:32,680 --> 00:15:37,380
as you introduce the
dual quantity, momentum.

217
00:15:37,380 --> 00:15:40,660
So what is momentum here?

218
00:15:40,660 --> 00:15:55,020
So momentum is an operator, and
this operator must be defined.

219
00:15:55,020 --> 00:15:59,890
Now, you had a
shorthand for it in 804,

220
00:15:59,890 --> 00:16:04,790
which is p hat equal
h bar over i d dx.

221
00:16:08,070 --> 00:16:10,910
And this shorthand
means actually

222
00:16:10,910 --> 00:16:17,220
that, in what we call the
position representation where

223
00:16:17,220 --> 00:16:22,420
we're using wave functions
that depend on x, well,

224
00:16:22,420 --> 00:16:25,190
the momentum is given
by this operator.

225
00:16:25,190 --> 00:16:27,880
And the story of why
this was the case

226
00:16:27,880 --> 00:16:30,150
was sort of something
that was elaborated

227
00:16:30,150 --> 00:16:33,870
on in 804, the work
of de Broglie, that

228
00:16:33,870 --> 00:16:38,430
saw that the
wavelength of the wave

229
00:16:38,430 --> 00:16:40,620
has to do with the
momentum of a wave.

230
00:16:40,620 --> 00:16:42,850
And finally, people
understood that this

231
00:16:42,850 --> 00:16:46,080
would measure the
momentum of the wave.

232
00:16:46,080 --> 00:16:47,800
So this is the operator.

233
00:16:47,800 --> 00:16:51,570
And therefore, in the
representation that

234
00:16:51,570 --> 00:16:56,750
we're working-- representation
is a word that has a lot

235
00:16:56,750 --> 00:17:00,250
of precise meaning, but now
I'm just using it in the sense

236
00:17:00,250 --> 00:17:03,430
that, well, we're working
either with x's or with p's.

237
00:17:03,430 --> 00:17:05,140
And we're working with x's.

238
00:17:05,140 --> 00:17:09,440
That's why p looks like
something to do with x.

239
00:17:09,440 --> 00:17:13,369
So what is p hat
on a wave function?

240
00:17:13,369 --> 00:17:17,739
Well, that's what this means.

241
00:17:17,739 --> 00:17:20,119
It's another wave
function obtained

242
00:17:20,119 --> 00:17:23,990
by taking the x derivative.

243
00:17:23,990 --> 00:17:28,620
So that's the definition of
it acting on a wave function.

244
00:17:28,620 --> 00:17:33,310
The one thing that must
be verified, of course,

245
00:17:33,310 --> 00:17:38,340
is that this definition
is consistent or implies

246
00:17:38,340 --> 00:17:39,780
this commutation relation.

247
00:17:43,330 --> 00:17:46,840
So you've defined
it as an operator.

248
00:17:46,840 --> 00:17:49,770
x, we've defined
it as an operator.

249
00:17:49,770 --> 00:17:51,330
But most of us think
that it doesn't

250
00:17:51,330 --> 00:17:53,850
look like an operator
is multiplying.

251
00:17:53,850 --> 00:17:56,580
But it is an operator.

252
00:17:56,580 --> 00:17:59,230
So this one does look
like an operator.

253
00:17:59,230 --> 00:18:02,000
It's a differential operator.

254
00:18:02,000 --> 00:18:08,250
And you can try to see
if this equation is true.

255
00:18:08,250 --> 00:18:10,900
And the way to test
these commutators

256
00:18:10,900 --> 00:18:13,110
is something that,
again, I don't think

257
00:18:13,110 --> 00:18:17,630
is unfamiliar to you,
but let's go through it,

258
00:18:17,630 --> 00:18:24,070
is that you try to evaluate
this product of operators

259
00:18:24,070 --> 00:18:27,510
acting on a wave function.

260
00:18:27,510 --> 00:18:32,400
And if things work
out well, we'll

261
00:18:32,400 --> 00:18:40,174
see you should get ih bar
times that wave function.

262
00:18:40,174 --> 00:18:43,090
If that is the case,
you say, OK, I've

263
00:18:43,090 --> 00:18:47,670
proven that equation, because
it's an operator equation.

264
00:18:47,670 --> 00:18:50,440
The left-hand side
of that equation

265
00:18:50,440 --> 00:18:53,510
is the product in different
orders of two operators,

266
00:18:53,510 --> 00:18:55,290
therefore it's an operator.

267
00:18:55,290 --> 00:18:57,710
The right-hand side
is another operator.

268
00:18:57,710 --> 00:19:02,270
It's the operator multiplied by
ih, anything that you'll get.

269
00:19:02,270 --> 00:19:07,530
Well, if this is an
operator identity,

270
00:19:07,530 --> 00:19:10,250
the operator on the left
must be equal to the operator

271
00:19:10,250 --> 00:19:14,440
on the right, which just means
that, acting on anything,

272
00:19:14,440 --> 00:19:16,080
they must give the same answer.

273
00:19:16,080 --> 00:19:20,000
So if I managed to prove
that this is equal to this,

274
00:19:20,000 --> 00:19:24,550
I've proven that for
anything that is the answer.

275
00:19:24,550 --> 00:19:28,330
And therefore, I can
write the top one.

276
00:19:28,330 --> 00:19:31,890
And let me just do
it, even though this

277
00:19:31,890 --> 00:19:37,240
may be kind of familiar
to many of you.

278
00:19:37,240 --> 00:19:42,510
It's good to do this
slowly once in your life.

279
00:19:42,510 --> 00:19:44,800
So let's go through this.

280
00:19:44,800 --> 00:19:50,360
So this says x operator
p operator on psi

281
00:19:50,360 --> 00:19:54,840
minus p operator of
x operator on psi.

282
00:19:54,840 --> 00:20:01,460
When you have several operators,
like ABC acting on psi,

283
00:20:01,460 --> 00:20:04,990
this really means
let C act on psi,

284
00:20:04,990 --> 00:20:11,960
and then let B act on C psi,
and then let A act on that.

285
00:20:11,960 --> 00:20:13,970
The operators act one by one.

286
00:20:13,970 --> 00:20:16,360
The closest one acts first.

287
00:20:16,360 --> 00:20:21,540
So here I'm supposed
to let B act on psi,

288
00:20:21,540 --> 00:20:23,160
but that means that thing.

289
00:20:23,160 --> 00:20:30,180
So now x is acting
on h over i d psi dx.

290
00:20:32,750 --> 00:20:40,090
On this one, I have
p acting on x psi,

291
00:20:40,090 --> 00:20:43,900
because that's
what x hat psi is.

292
00:20:43,900 --> 00:20:47,770
Here, this is multiplication
by x of a function of x.

293
00:20:47,770 --> 00:20:52,120
So this is just h
over i x d psi dx.

294
00:20:55,310 --> 00:21:01,920
And here, I have h over i d
dx of this whole thing x psi.

295
00:21:04,950 --> 00:21:07,750
And you can see that
when you act here,

296
00:21:07,750 --> 00:21:12,080
you act first on the x,
and you get something.

297
00:21:12,080 --> 00:21:16,580
And then you act on the psi,
and you get this same term.

298
00:21:16,580 --> 00:21:24,130
So the only contribution
here is equal to minus h over

299
00:21:24,130 --> 00:21:28,330
i, the d dx on x
times psi, which

300
00:21:28,330 --> 00:21:32,530
is ih bar psi, which is
what I wanted to show.

301
00:21:36,050 --> 00:21:41,170
So this is true.

302
00:21:41,170 --> 00:21:44,880
And therefore, you could
say that this definition is

303
00:21:44,880 --> 00:21:47,800
consistent with your
definition of x,

304
00:21:47,800 --> 00:21:51,710
and they represent
this operator.

305
00:21:51,710 --> 00:21:57,010
One more thing you could try
to do, and it's fun to do it,

306
00:21:57,010 --> 00:22:02,930
is we had a matrix
representation for x.

307
00:22:02,930 --> 00:22:05,353
Can I think of p as a matrix?

308
00:22:07,980 --> 00:22:09,180
How would you do it?

309
00:22:09,180 --> 00:22:11,880
What kind of matrix
would p look like?

310
00:22:15,680 --> 00:22:17,386
Well, yes?

311
00:22:17,386 --> 00:22:20,410
AUDIENCE: You just generate
a finite difference equation.

312
00:22:20,410 --> 00:22:22,620
PROFESSOR: You could
do it, exactly,

313
00:22:22,620 --> 00:22:25,920
with taking finite differences.

314
00:22:25,920 --> 00:22:28,590
So for example, if
you think that you

315
00:22:28,590 --> 00:22:35,555
want to produce the wave
function psi prime at 0,

316
00:22:35,555 --> 00:22:39,200
psi prime at epsilon,
psi prime, that's

317
00:22:39,200 --> 00:22:41,240
what the derivative
gives you, you'll

318
00:22:41,240 --> 00:22:47,970
write this as 1 over
epsilon, say, psi at epsilon

319
00:22:47,970 --> 00:22:50,040
minus psi at 0.

320
00:22:50,040 --> 00:22:53,950
That's the derivative
at 0 roughly.

321
00:22:53,950 --> 00:23:00,720
It would be psi at 2 epsilon
minus psi at 0 over 2.

322
00:23:00,720 --> 00:23:03,759
And you could build it.

323
00:23:03,759 --> 00:23:04,550
You could build it.

324
00:23:04,550 --> 00:23:05,600
I'm not going to do it.

325
00:23:05,600 --> 00:23:08,920
You may want to do
it and try and see

326
00:23:08,920 --> 00:23:13,550
how the derivative
operator looks as a matrix.

327
00:23:13,550 --> 00:23:18,100
And then if you really want
to spend some time thinking

328
00:23:18,100 --> 00:23:21,770
about it, you could try
to see if this matrix

329
00:23:21,770 --> 00:23:27,000
and this matrix commute
to give the right answer.

330
00:23:27,000 --> 00:23:28,800
And as you try it,
you will figure out

331
00:23:28,800 --> 00:23:32,430
all kinds of funny
things that we

332
00:23:32,430 --> 00:23:34,580
will talk about
later in the course.

333
00:23:34,580 --> 00:23:39,010
So you can represent the
momentum operator as a matrix

334
00:23:39,010 --> 00:23:41,580
indeed, and there are
interesting things

335
00:23:41,580 --> 00:23:47,960
to say about it, and
it's a good subject.

336
00:23:47,960 --> 00:23:52,090
So let's continue
with the momentum

337
00:23:52,090 --> 00:23:55,380
and ask for eigenstates
of the momentum.

338
00:23:55,380 --> 00:24:06,796
So eigenstates of
p, you know them.

339
00:24:06,796 --> 00:24:09,250
They're e to the ipx things.

340
00:24:09,250 --> 00:24:13,650
So let's write them with some
convenient normalization.

341
00:24:16,180 --> 00:24:18,780
This is an [INAUDIBLE]
wave function

342
00:24:18,780 --> 00:24:21,880
that depends on x
with momentum p.

343
00:24:21,880 --> 00:24:24,250
And we'll write it,
as a definition,

344
00:24:24,250 --> 00:24:32,000
as e to the ipx over h bar, and
I'll put it a 2 pi h bar here.

345
00:24:34,540 --> 00:24:38,440
It's kind of a
useful normalization.

346
00:24:38,440 --> 00:24:46,530
Then p hat on psi p of x,
well, p hat is supposed

347
00:24:46,530 --> 00:24:56,615
to take h over i d dx, and
take h over i d dx of psi p.

348
00:24:59,290 --> 00:25:03,330
And h over i cancels
the i over h.

349
00:25:03,330 --> 00:25:05,680
When you take the d
dx, you get p out,

350
00:25:05,680 --> 00:25:07,520
and you get the
same wave function.

351
00:25:07,520 --> 00:25:12,590
So indeed, you get
p times psi p of x.

352
00:25:12,590 --> 00:25:19,010
So indeed, this is the
eigenstate of the momentum

353
00:25:19,010 --> 00:25:21,655
operator, and it has momentum p.

354
00:25:24,980 --> 00:25:28,150
Well, what is the use of this?

355
00:25:28,150 --> 00:25:33,370
Well, say you have a
representation, what

356
00:25:33,370 --> 00:25:36,180
we call the position
representation,

357
00:25:36,180 --> 00:25:38,880
of the wave function
and operators.

358
00:25:38,880 --> 00:25:42,450
Let us think now of the
momentum representation.

359
00:25:42,450 --> 00:25:45,200
So what does all that mean?

360
00:25:45,200 --> 00:25:49,760
Well, there is the Fourier
transform operation

361
00:25:49,760 --> 00:25:54,780
in which we have psi of p.

362
00:25:54,780 --> 00:26:00,780
Well, let me write it
this way, actually.

363
00:26:00,780 --> 00:26:05,930
I'll write any psi
of x physically

364
00:26:05,930 --> 00:26:11,340
can be represented as a sum
of momentum eigenstates.

365
00:26:11,340 --> 00:26:15,040
Therefore, that's
Fourier's theorem,

366
00:26:15,040 --> 00:26:23,600
minus infinity to infinity
dp e to the ipx over h bar

367
00:26:23,600 --> 00:26:29,060
square root of 2 pi
h psi tilde of p.

368
00:26:31,880 --> 00:26:38,870
That's Fourier transformation,
defines psi tilde of p.

369
00:26:38,870 --> 00:26:44,450
And Fourier's theorem is
the fact that not only you

370
00:26:44,450 --> 00:26:49,020
can do that, but you can invert
it so that psi tilde of p

371
00:26:49,020 --> 00:26:52,935
can also be written as an
integral, this time over x

372
00:26:52,935 --> 00:27:00,620
from minus infinity to infinity
e to the minus ipx over h

373
00:27:00,620 --> 00:27:09,190
bar, also 2 pi h bar psi of x.

374
00:27:09,190 --> 00:27:12,690
So let's ponder this equation
for a couple of minutes.

375
00:27:15,640 --> 00:27:21,820
Well, as a physicist,
you think of this, well,

376
00:27:21,820 --> 00:27:25,910
this is telling me that
any wave function could

377
00:27:25,910 --> 00:27:30,380
be written as a superposition
of momentum eigenstates.

378
00:27:30,380 --> 00:27:33,050
Here are the
momentum eigenstates.

379
00:27:33,050 --> 00:27:35,140
And for each value
of momentum, you

380
00:27:35,140 --> 00:27:39,040
have some coefficient here
that tells me how much of that

381
00:27:39,040 --> 00:27:40,640
momentum eigenstate I have.

382
00:27:44,340 --> 00:27:46,600
Now, here is the opposite one.

383
00:27:46,600 --> 00:27:52,050
Psi tilde of p and psi of
x are related in this way.

384
00:27:52,050 --> 00:27:55,340
So these coefficients, if
you want to calculate them,

385
00:27:55,340 --> 00:27:57,900
you calculate them this way.

386
00:27:57,900 --> 00:28:03,100
But now let's think of it as
a change of representation.

387
00:28:06,810 --> 00:28:10,210
The physics is
contained in psi of x.

388
00:28:10,210 --> 00:28:15,060
All what you wish to know about
this physical system in quantum

389
00:28:15,060 --> 00:28:18,040
mechanics is there in psi of x.

390
00:28:18,040 --> 00:28:21,600
But it's also there in
psi of p, because they

391
00:28:21,600 --> 00:28:24,270
contain the same information.

392
00:28:24,270 --> 00:28:28,370
So there are different ways of
encoding the same information.

393
00:28:31,540 --> 00:28:36,150
What is the relation
between them?

394
00:28:36,150 --> 00:28:39,240
This, we thought
of it as a vector,

395
00:28:39,240 --> 00:28:44,210
vector in position space, an
infinite dimensional space

396
00:28:44,210 --> 00:28:46,880
that is talking about positions.

397
00:28:46,880 --> 00:28:51,580
This is another vector
in momentum space.

398
00:28:51,580 --> 00:28:53,720
Think of it now
the infinite line.

399
00:28:53,720 --> 00:28:58,390
So this is an infinite vector
with all those points little

400
00:28:58,390 --> 00:29:02,030
by little, from minus infinity
to plus infinity, all of them

401
00:29:02,030 --> 00:29:05,300
there, gigantic vector.

402
00:29:05,300 --> 00:29:08,410
And here is another
gigantic vector with p

403
00:29:08,410 --> 00:29:11,250
from minus infinity to infinity.

404
00:29:11,250 --> 00:29:13,420
And in between,
there's an integral.

405
00:29:13,420 --> 00:29:17,530
But now, with your picture
of quantum mechanics,

406
00:29:17,530 --> 00:29:21,740
you see an integral, but
you also see a matrix.

407
00:29:21,740 --> 00:29:23,580
And what is this matrix?

408
00:29:23,580 --> 00:29:28,650
Think of this as some
sort of psi sub p.

409
00:29:31,180 --> 00:29:38,590
And this as some
sort of matrix, px

410
00:29:38,590 --> 00:29:47,020
psi x, in which if you have a
product-- you'll remember when

411
00:29:47,020 --> 00:29:51,230
you multiply matrices,
a matrix on a vector,

412
00:29:51,230 --> 00:29:54,520
you sum over the second index.

413
00:29:54,520 --> 00:29:56,360
That's the product for matrix.

414
00:29:56,360 --> 00:29:59,400
And then the first
index is the index here.

415
00:29:59,400 --> 00:30:04,330
So here is what it,
more or less, is like.

416
00:30:04,330 --> 00:30:07,040
Psi tilde of p
[? subtend ?] by this,

417
00:30:07,040 --> 00:30:11,750
and this matrix depends on two
labels, p and x, and it's that.

418
00:30:11,750 --> 00:30:16,340
So it's a matrix full of phases.

419
00:30:16,340 --> 00:30:20,020
So how do you pass from the
coordinate representation

420
00:30:20,020 --> 00:30:23,490
of the information, a vector
of all values of the wave

421
00:30:23,490 --> 00:30:25,340
function in all positions?

422
00:30:25,340 --> 00:30:31,140
By multiplying with this
matrix of phases that is here,

423
00:30:31,140 --> 00:30:33,070
and it gives you
this representation.

424
00:30:33,070 --> 00:30:38,670
So different representations
means using different vectors

425
00:30:38,670 --> 00:30:42,140
to represent the physics.

426
00:30:42,140 --> 00:30:45,610
And this vector is
a very nice one.

427
00:30:45,610 --> 00:30:49,070
And because of these properties
of the momentum operator

428
00:30:49,070 --> 00:30:52,940
and all these things, this
vector is also a very nice one.

429
00:30:52,940 --> 00:30:55,690
And there's an
integral transform

430
00:30:55,690 --> 00:30:59,480
or some sort of infinite matrix
product that relates them.

431
00:31:02,450 --> 00:31:05,700
And we shouldn't be
uncomfortable about it.

432
00:31:05,700 --> 00:31:07,260
That's all fine.

433
00:31:07,260 --> 00:31:12,190
So we say that we
have, for example,

434
00:31:12,190 --> 00:31:22,260
psi of x as one representation
of the state and psi tilde of p

435
00:31:22,260 --> 00:31:25,635
as another representation
of the same physics.

436
00:31:30,950 --> 00:31:38,780
We can do one more thing
here, If I continue.

437
00:31:38,780 --> 00:31:46,160
We can take that boxed equation
on the blackboard up there

438
00:31:46,160 --> 00:31:54,430
and act with h bar over
i d dx on psi of x.

439
00:31:57,120 --> 00:32:02,160
So that is equal to
h i d dx, and I'll

440
00:32:02,160 --> 00:32:08,070
write what psi of x is, is
minus infinity to infinity dp e

441
00:32:08,070 --> 00:32:12,680
to the ipx over
h bar square root

442
00:32:12,680 --> 00:32:17,950
of 2 pi h bar psi tilde of p.

443
00:32:22,580 --> 00:32:28,190
Now, when we act on this, as
you know, h bar over i d dx

444
00:32:28,190 --> 00:32:32,020
just acts on this and
produces the factor of p.

445
00:32:32,020 --> 00:32:38,410
So this is equal to minus
infinity to infinity dp e

446
00:32:38,410 --> 00:32:45,890
to the ipx over h bar over
square root of 2 pi h bar p

447
00:32:45,890 --> 00:32:48,140
times psi tilde of p.

448
00:32:54,720 --> 00:32:59,240
So look at this equation again.

449
00:32:59,240 --> 00:33:02,940
This double arrow is
to mean that there

450
00:33:02,940 --> 00:33:05,080
are equivalent physics in them.

451
00:33:05,080 --> 00:33:06,790
They have the same information.

452
00:33:06,790 --> 00:33:10,300
It's the same data encoded
in a different way.

453
00:33:10,300 --> 00:33:13,329
And that different
way, this arrow

454
00:33:13,329 --> 00:33:14,412
is Fourier transformation.

455
00:33:17,420 --> 00:33:20,750
And this Fourier transformation
is explained here.

456
00:33:20,750 --> 00:33:26,920
So now you have Fourier
transformation the same way.

457
00:33:26,920 --> 00:33:30,270
So here we have--
what we've learned

458
00:33:30,270 --> 00:33:41,120
is that h over i d dx of psi is
represented in momentum space

459
00:33:41,120 --> 00:33:47,060
by p psi tilde of p.

460
00:33:47,060 --> 00:33:52,620
And this was p hat
acting on psi of x.

461
00:33:55,520 --> 00:34:00,900
So the corresponding thing
in momentum space of p hat

462
00:34:00,900 --> 00:34:08,530
acting on psi of x is p
multiplying psi tilde of p,

463
00:34:08,530 --> 00:34:15,590
which is to say that we can
think of the abstract operator

464
00:34:15,590 --> 00:34:25,620
p hat acting on psi tilde of
p as just p psi tilde of p.

465
00:34:33,040 --> 00:34:39,639
So in momentum space,
the operator p hat

466
00:34:39,639 --> 00:34:41,260
acts in a very easy way.

467
00:34:44,280 --> 00:34:48,300
In coordinate space,
it takes derivatives.

468
00:34:48,300 --> 00:34:51,940
In momentum space,
it's multiplicative.

469
00:34:51,940 --> 00:34:58,620
So in position space,
x is multiplicative.

470
00:34:58,620 --> 00:35:05,380
But in momentum space, x
would not be multiplicative.

471
00:35:05,380 --> 00:35:07,780
x would also be a derivative.

472
00:35:07,780 --> 00:35:11,870
So I leave it for
you as an exercise

473
00:35:11,870 --> 00:35:19,540
to show that or convince
yourself in several ways,

474
00:35:19,540 --> 00:35:34,270
that x hat is really i h bar d
dp in p space, in i h bar d dp.

475
00:35:36,960 --> 00:35:37,760
All right.

476
00:35:37,760 --> 00:35:43,590
So that's really all I wanted to
say about position and momentum

477
00:35:43,590 --> 00:35:45,850
operators at this moment.

478
00:35:45,850 --> 00:35:49,960
They will come back when we'll
introduce bra-ket notation

479
00:35:49,960 --> 00:35:50,460
in detail.

480
00:35:50,460 --> 00:35:52,550
We'll revisit this a little.

481
00:35:52,550 --> 00:35:56,690
But the main concepts
have been illustrated.

482
00:35:56,690 --> 00:35:57,980
Are there questions?

483
00:35:57,980 --> 00:36:01,330
We're about to leave
this, so if you

484
00:36:01,330 --> 00:36:02,750
have any questions
at this moment.

485
00:36:02,750 --> 00:36:03,310
Yes?

486
00:36:03,310 --> 00:36:04,795
AUDIENCE: Could
you explain again

487
00:36:04,795 --> 00:36:10,240
how you used this
[INAUDIBLE] h bar over i d dx

488
00:36:10,240 --> 00:36:12,730
assign to [INAUDIBLE]?

489
00:36:12,730 --> 00:36:13,920
PROFESSOR: Right.

490
00:36:13,920 --> 00:36:18,760
So the question was, why did
I associate these things?

491
00:36:18,760 --> 00:36:24,340
So it really goes back here to
what the meaning of this arrow

492
00:36:24,340 --> 00:36:25,420
is.

493
00:36:25,420 --> 00:36:28,510
The meaning of this arrow
is Fourier transformation.

494
00:36:28,510 --> 00:36:34,420
So this psi tilde and psi of
x are related in this way.

495
00:36:34,420 --> 00:36:36,540
That's Fourier
transformation, and that's

496
00:36:36,540 --> 00:36:38,850
what we mean by this arrow.

497
00:36:38,850 --> 00:36:43,950
It also means that whatever
physics you have here,

498
00:36:43,950 --> 00:36:45,760
you have it there.

499
00:36:45,760 --> 00:36:52,570
So really, when you have
something acting on a state,

500
00:36:52,570 --> 00:36:56,050
for example, if you have some
operator acting in here, well,

501
00:36:56,050 --> 00:36:58,270
you get a new wave function.

502
00:36:58,270 --> 00:37:00,410
And there should be
one on the right that

503
00:37:00,410 --> 00:37:03,810
corresponds to it, that has
the same information as the one

504
00:37:03,810 --> 00:37:06,030
in which you've
acted with something.

505
00:37:06,030 --> 00:37:10,330
So what we claim here is that,
also in the sense of Fourier

506
00:37:10,330 --> 00:37:15,420
transformation or having the
same information, h bar over i,

507
00:37:15,420 --> 00:37:20,130
the derivative of psi,
is encoded by this.

508
00:37:20,130 --> 00:37:24,660
So we say, thinking
abstractly, what is this?

509
00:37:24,660 --> 00:37:28,970
This is the momentum operator.

510
00:37:28,970 --> 00:37:35,350
Therefore, I'm going to say that
the momentum operator really

511
00:37:35,350 --> 00:37:39,000
is the same momentum
operator, whether it acts

512
00:37:39,000 --> 00:37:42,810
on wave functions that you show
them to mean this way or wave

513
00:37:42,810 --> 00:37:45,680
functions that, because
you're in another mood,

514
00:37:45,680 --> 00:37:48,660
you decide to give them
to me in momentum space.

515
00:37:48,660 --> 00:37:53,690
So as you change your mood, the
operator takes different forms

516
00:37:53,690 --> 00:37:56,380
but is doing the same thing.

517
00:37:56,380 --> 00:37:58,030
It's totally reversible.

518
00:37:58,030 --> 00:38:02,460
It's acting on that-- you
see, the operator is always

519
00:38:02,460 --> 00:38:06,370
the same, but you give me the
data in two different ways,

520
00:38:06,370 --> 00:38:10,170
then the operator has to do
the thing in a different way.

521
00:38:10,170 --> 00:38:12,770
So that's what it means
that the operator has

522
00:38:12,770 --> 00:38:15,030
different representations.

523
00:38:15,030 --> 00:38:16,770
In the [INAUDIBLE]
representation,

524
00:38:16,770 --> 00:38:18,790
it looks like a derivative.

525
00:38:18,790 --> 00:38:21,915
In the momentum representation,
it looks like multiplying.

526
00:38:25,130 --> 00:38:27,828
Other questions?

527
00:38:27,828 --> 00:38:28,806
Yes?

528
00:38:28,806 --> 00:38:31,904
AUDIENCE: So by saying
that they sort of represent

529
00:38:31,904 --> 00:38:33,320
[INAUDIBLE] to the
same positions,

530
00:38:33,320 --> 00:38:36,605
does that mean that h bar over
i p e to the xi and p psi p

531
00:38:36,605 --> 00:38:38,620
are like the same [INAUDIBLE]?

532
00:38:38,620 --> 00:38:43,690
PROFESSOR: That h bar over
d dx psi and p-- yeah.

533
00:38:43,690 --> 00:38:47,400
They are the same
data, the same state

534
00:38:47,400 --> 00:38:49,345
represented in different ways.

535
00:38:49,345 --> 00:38:49,845
Yeah.

536
00:38:54,800 --> 00:38:55,470
All right.

537
00:38:55,470 --> 00:38:58,470
So time for a change.

538
00:38:58,470 --> 00:39:03,460
We're going to talk about
Stern-Gerlach and spin.

539
00:39:03,460 --> 00:39:09,580
Now, spin will keep us busy the
biggest chunk of this semester.

540
00:39:09,580 --> 00:39:14,310
So it will be spin-1/2, and
we're really going to go

541
00:39:14,310 --> 00:39:17,200
into enormous detail on it.

542
00:39:17,200 --> 00:39:20,830
So this is just the
beginning of the story that

543
00:39:20,830 --> 00:39:25,030
will be elaborated
at various stages.

544
00:39:25,030 --> 00:39:30,420
So at this moment, I will
talk about this experiment

545
00:39:30,420 --> 00:39:33,090
that led to the
discovery of spin,

546
00:39:33,090 --> 00:39:37,200
and if you try to invent
the theory that describes

547
00:39:37,200 --> 00:39:42,050
this experiment, what you
would possibly begin doing.

548
00:39:42,050 --> 00:39:44,340
And then we go through
the mathematics,

549
00:39:44,340 --> 00:39:48,720
as I mentioned to you, for maybe
a week and a half or two weeks,

550
00:39:48,720 --> 00:39:52,260
and then return to the
spin with more tools

551
00:39:52,260 --> 00:39:55,960
to understand it well.

552
00:39:55,960 --> 00:40:00,420
So the subject is the
Stern-Gerlach experiment,

553
00:40:00,420 --> 00:40:06,050
Stern-Gerlach experiment.

554
00:40:12,650 --> 00:40:21,080
So the Stern-Gerlach experiment
was done in Frankfurt, 1922.

555
00:40:21,080 --> 00:40:23,930
It was an experiment
that, in fact, people

556
00:40:23,930 --> 00:40:25,672
were extraordinarily confused.

557
00:40:25,672 --> 00:40:29,080
It was not clear why
they were doing it.

558
00:40:29,080 --> 00:40:33,590
And for quite a
while, people didn't

559
00:40:33,590 --> 00:40:37,170
understand what
they were getting,

560
00:40:37,170 --> 00:40:39,400
what was happening with it.

561
00:40:39,400 --> 00:40:46,760
In fact, Pauli had thought
that the electron has

562
00:40:46,760 --> 00:40:48,890
like two degrees of
freedom and didn't

563
00:40:48,890 --> 00:40:53,410
know what it was, those
two degrees of freedom.

564
00:40:53,410 --> 00:40:57,605
Kronig suggested that
it had to do somehow

565
00:40:57,605 --> 00:41:01,350
with the rotation
of the electron.

566
00:41:01,350 --> 00:41:05,620
Now, Pauli said that's nonsense.

567
00:41:05,620 --> 00:41:09,950
How can an electron rotate
and have angular momentum

568
00:41:09,950 --> 00:41:11,770
because it has a rotation?

569
00:41:11,770 --> 00:41:15,450
It would have to rotate
so fast, even faster

570
00:41:15,450 --> 00:41:18,200
than the speed of light to
have the angular momentum,

571
00:41:18,200 --> 00:41:21,510
and then this little ball
that would be the electron

572
00:41:21,510 --> 00:41:22,790
would disintegrate.

573
00:41:22,790 --> 00:41:27,060
And it made no sense to him that
there would be such a thing.

574
00:41:27,060 --> 00:41:30,890
So Kronig didn't publish this.

575
00:41:30,890 --> 00:41:35,986
Then there were another two
people, Uhlenbeck and Goudsmit,

576
00:41:35,986 --> 00:41:40,190
at the same time,
around 1925, had

577
00:41:40,190 --> 00:41:44,870
the same idea, angular
momentum of this particle.

578
00:41:44,870 --> 00:41:48,860
And their advisor was
Ehrenfest, and said

579
00:41:48,860 --> 00:41:52,285
it doesn't make too much sense,
but you should publish it.

580
00:41:52,285 --> 00:41:54,390
[LAUGHTER]

581
00:41:54,390 --> 00:41:57,080
And thanks to their
publishing, they

582
00:41:57,080 --> 00:42:00,430
are given credit for discovering
the spin of the electron.

583
00:42:00,430 --> 00:42:03,390
And Pauli, a couple
of years later,

584
00:42:03,390 --> 00:42:05,260
decided, after all, I was wrong.

585
00:42:05,260 --> 00:42:08,380
Yes, it is spin, and
it's all working out.

586
00:42:08,380 --> 00:42:14,160
And 1927, five years after
the experiment basically,

587
00:42:14,160 --> 00:42:17,110
people understood
what was going on.

588
00:42:17,110 --> 00:42:20,040
So what were these
people trying to do?

589
00:42:20,040 --> 00:42:23,400
First, Stern and Gerlach
were atomic physicists,

590
00:42:23,400 --> 00:42:26,550
and they were just
interested in measuring

591
00:42:26,550 --> 00:42:34,220
speeds of thermal
motion of ions.

592
00:42:34,220 --> 00:42:37,460
So they would send
beams of these ions

593
00:42:37,460 --> 00:42:41,530
and put magnetic fields and
deflect them and measure

594
00:42:41,530 --> 00:42:43,550
their velocities.

595
00:42:43,550 --> 00:42:46,170
And eventually,
they were experts

596
00:42:46,170 --> 00:42:47,580
doing this kind of thing.

597
00:42:47,580 --> 00:42:51,630
And they heard of Bohr, that
said that the electron has

598
00:42:51,630 --> 00:42:56,440
angular momentum and is going
around the proton in circles,

599
00:42:56,440 --> 00:42:58,280
so it might have
angular momentum.

600
00:42:58,280 --> 00:43:01,120
They said, oh, if it has
angular momentum because it's

601
00:43:01,120 --> 00:43:05,330
going around the proton,
maybe we can detect it.

602
00:43:05,330 --> 00:43:09,080
And when they did the
experiment, they got something.

603
00:43:09,080 --> 00:43:11,510
And they said, well,
we're seeing it.

604
00:43:11,510 --> 00:43:14,520
But it was not that.

605
00:43:14,520 --> 00:43:17,700
They were not seeing the
orbital angular momentum

606
00:43:17,700 --> 00:43:24,650
of the electron because that
electron in these silver atoms

607
00:43:24,650 --> 00:43:27,310
actually has no
angular momentum,

608
00:43:27,310 --> 00:43:30,340
as we will see, no
orbital angular momentum.

609
00:43:30,340 --> 00:43:31,370
It only has spin.

610
00:43:31,370 --> 00:43:34,220
So they were actually
seeing the spin.

611
00:43:34,220 --> 00:43:36,090
So it was a big confusion.

612
00:43:36,090 --> 00:43:37,650
It took some time.

613
00:43:37,650 --> 00:43:41,660
Basically, they took the
beam, and they split it

614
00:43:41,660 --> 00:43:45,770
with a magnetic field,
and the clean split

615
00:43:45,770 --> 00:43:47,510
was something nobody understood.

616
00:43:47,510 --> 00:43:51,440
So they called it
space quantization,

617
00:43:51,440 --> 00:43:54,500
as of it's separated in space.

618
00:43:54,500 --> 00:43:56,710
Space is quantized.

619
00:43:56,710 --> 00:43:59,570
A pretty awful name, of course.

620
00:43:59,570 --> 00:44:03,200
There's nothing quantized
about space here.

621
00:44:03,200 --> 00:44:07,600
But it reflects that when
you don't know what's really

622
00:44:07,600 --> 00:44:12,500
happening, your names
don't come out too well.

623
00:44:12,500 --> 00:44:17,460
So what we have to
understand here,

624
00:44:17,460 --> 00:44:20,540
our goal today is
to just see what's

625
00:44:20,540 --> 00:44:27,310
happening in that experiment,
quantify a bit the results,

626
00:44:27,310 --> 00:44:33,050
and then extract the quantum
mechanical lessons from it.

627
00:44:33,050 --> 00:44:35,900
So let us begin with
the important thing.

628
00:44:35,900 --> 00:44:39,380
You don't see the spin directly.

629
00:44:39,380 --> 00:44:43,930
What you see is
magnetic moments.

630
00:44:43,930 --> 00:44:45,600
So what's that?

631
00:44:45,600 --> 00:44:47,680
So what are magnetic moments?

632
00:44:47,680 --> 00:45:00,340
Magnetic moments, mu, is the
analog, the magnetic analog

633
00:45:00,340 --> 00:45:01,780
of an electric dipole.

634
00:45:01,780 --> 00:45:03,740
A mu is called a
magnetic dipole.

635
00:45:03,740 --> 00:45:05,740
You say it has a
magnetic moment.

636
00:45:08,680 --> 00:45:14,160
And the magnetic moment is
given by I times the area.

637
00:45:14,160 --> 00:45:15,490
What does that mean?

638
00:45:15,490 --> 00:45:20,360
Well, a precise discussion
would take some time.

639
00:45:20,360 --> 00:45:24,110
But roughly, you
can simplify when

640
00:45:24,110 --> 00:45:28,150
you think of a loop that is
in a plane, in which case

641
00:45:28,150 --> 00:45:31,270
there's an area
associated to it.

642
00:45:31,270 --> 00:45:33,570
And if the loop is
this one, the area

643
00:45:33,570 --> 00:45:38,280
is defined as the normal
vector to the oriented loop.

644
00:45:38,280 --> 00:45:42,140
So an oriented loop
has an area vector.

645
00:45:42,140 --> 00:45:43,960
And the orientation
could be focused

646
00:45:43,960 --> 00:45:45,500
the direction of the current.

647
00:45:45,500 --> 00:45:46,450
There is some area.

648
00:45:46,450 --> 00:45:49,970
And the magnetic moment
is given by this thing.

649
00:45:49,970 --> 00:45:52,520
It points up in
the circumstances

650
00:45:52,520 --> 00:45:55,580
when this current
goes like that.

651
00:45:55,580 --> 00:45:58,630
So that's a magnetic moment.

652
00:45:58,630 --> 00:46:01,930
A little bit of units.

653
00:46:01,930 --> 00:46:06,020
The way units work out is
that mu B-- magnetic moments

654
00:46:06,020 --> 00:46:11,190
and magnetic fields
have units of energy.

655
00:46:14,680 --> 00:46:19,140
So magnetic moments
you could define

656
00:46:19,140 --> 00:46:26,450
as energy, which is
joules, divided by tesla,

657
00:46:26,450 --> 00:46:32,520
or ergs divided by
gauss, because mu B has

658
00:46:32,520 --> 00:46:35,490
units of energy.

659
00:46:35,490 --> 00:46:39,300
So how do magnetic
moments originate

660
00:46:39,300 --> 00:46:43,340
in a charge configuration?

661
00:46:43,340 --> 00:46:46,230
Well, you can simply have
a little current like that.

662
00:46:46,230 --> 00:46:49,630
But let's consider a
different situation

663
00:46:49,630 --> 00:46:58,040
in which you have
a ring of charge,

664
00:46:58,040 --> 00:47:08,945
a ring of charge of some radius
R. It has a total charge Q,

665
00:47:08,945 --> 00:47:12,350
and it has a linear
charge density lambda.

666
00:47:12,350 --> 00:47:17,980
It's uniform, and it's
rotating with some velocity

667
00:47:17,980 --> 00:47:24,220
v. If you wish, it also
has a mass M. There are all

668
00:47:24,220 --> 00:47:25,720
kinds of [? parameters. ?]
How many?

669
00:47:25,720 --> 00:47:29,490
Mass, charge,
radius, and velocity.

670
00:47:29,490 --> 00:47:30,100
Here we go.

671
00:47:30,100 --> 00:47:35,250
We have our solid ring
of charge rotating,

672
00:47:35,250 --> 00:47:39,030
and we want to figure out
something quite fundamental,

673
00:47:39,030 --> 00:47:41,470
which is the origin
of this principle.

674
00:47:41,470 --> 00:47:46,300
We said, you really
never see spins directly.

675
00:47:46,300 --> 00:47:49,960
You never see this intrinsic
angular momentum directly.

676
00:47:49,960 --> 00:47:53,280
You see magnetic moments.

677
00:47:53,280 --> 00:47:55,800
But then actually
what happens is

678
00:47:55,800 --> 00:47:58,310
that there's a
universal relation

679
00:47:58,310 --> 00:48:00,615
between magnetic moments
and angular momentum.

680
00:48:00,615 --> 00:48:04,370
This is a key
concept in physics.

681
00:48:04,370 --> 00:48:05,890
Maybe you've seen it before.

682
00:48:05,890 --> 00:48:07,080
Maybe you haven't.

683
00:48:07,080 --> 00:48:11,540
Probably you might
have seen that in 802.

684
00:48:11,540 --> 00:48:13,040
So how does that go?

685
00:48:13,040 --> 00:48:16,020
Let's calculate the
magnetic moment.

686
00:48:16,020 --> 00:48:22,830
So the current is the
linear charge density

687
00:48:22,830 --> 00:48:23,650
times the velocity.

688
00:48:28,440 --> 00:48:32,760
The linear charge density
is the total charge

689
00:48:32,760 --> 00:48:35,705
divided by 2 pi R
times the velocity.

690
00:48:38,360 --> 00:48:41,060
Now the area, to give
the magnetic moment,

691
00:48:41,060 --> 00:48:44,120
we'll have mu is equal
to I times the area.

692
00:48:44,120 --> 00:48:49,300
So it would be this
Q times 2 pi R v

693
00:48:49,300 --> 00:48:54,700
times the area, which
would be pi R squared.

694
00:48:54,700 --> 00:49:00,605
So the pi's cancel,
and we get 1/2 QvR.

695
00:49:08,520 --> 00:49:10,550
OK.

696
00:49:10,550 --> 00:49:14,510
1/2 QvR, and that's
fine and interesting.

697
00:49:14,510 --> 00:49:20,230
But OK, depends on the radius,
depends on the velocity.

698
00:49:20,230 --> 00:49:26,200
So here is the magnetic moment
is supposed to be going up.

699
00:49:26,200 --> 00:49:27,730
But what else is going up?

700
00:49:27,730 --> 00:49:31,020
The angular momentum of
this thing is also going up.

701
00:49:31,020 --> 00:49:35,026
So what is the magnitude
of the angular momentum L?

702
00:49:35,026 --> 00:49:36,476
L is angular momentum.

703
00:49:39,700 --> 00:49:43,690
Well, it's the mass
times the momentum--

704
00:49:43,690 --> 00:49:47,970
it's the mass momentum
cross R, so MvR.

705
00:49:53,790 --> 00:49:58,270
The momentum of R
cross p for each piece,

706
00:49:58,270 --> 00:50:02,910
contributes the same, so you
just take the total momentum.

707
00:50:02,910 --> 00:50:06,260
This really is 0, but add
them up little by little,

708
00:50:06,260 --> 00:50:08,270
and you've got your MvR.

709
00:50:11,050 --> 00:50:22,780
So here you have vR, so here
you put 1/2 Q over M MvR.

710
00:50:25,330 --> 00:50:38,600
And you discover that mu
is equal to 1/2 Q over M L.

711
00:50:38,600 --> 00:50:49,430
So maybe write it better--
Q over 2M L. I'm sorry,

712
00:50:49,430 --> 00:50:50,835
this is the normal.

713
00:50:50,835 --> 00:50:54,710
The M shouldn't change, M.

714
00:50:54,710 --> 00:50:59,680
And I box this relation
because an interesting thing

715
00:50:59,680 --> 00:51:00,440
has happened.

716
00:51:00,440 --> 00:51:04,510
All kinds of incidentals
have dropped out.

717
00:51:04,510 --> 00:51:08,090
Like the velocity
has dropped out.

718
00:51:08,090 --> 00:51:11,140
The radius has
dropped out as well.

719
00:51:11,140 --> 00:51:14,140
So if I have one
ring with this radius

720
00:51:14,140 --> 00:51:16,390
and another ring
with a bigger radius,

721
00:51:16,390 --> 00:51:19,740
the relation between
mu and L is the same,

722
00:51:19,740 --> 00:51:23,090
as long as it's rotating
with the same speed.

723
00:51:23,090 --> 00:51:26,900
So this is actually
a universal relation.

724
00:51:26,900 --> 00:51:29,330
It is not just true
for a little ring.

725
00:51:29,330 --> 00:51:33,790
It's true for a solid sphere
or any solid object axially

726
00:51:33,790 --> 00:51:35,880
symmetric.

727
00:51:35,880 --> 00:51:36,830
It would be true.

728
00:51:36,830 --> 00:51:43,030
You could consider any object
that is axially symmetric,

729
00:51:43,030 --> 00:51:45,870
and then you start considering
all the little rings that

730
00:51:45,870 --> 00:51:46,710
can be built.

731
00:51:46,710 --> 00:51:49,340
And for every ring,
mu over L is the same,

732
00:51:49,340 --> 00:51:51,880
and they all point in
the same direction.

733
00:51:51,880 --> 00:51:56,220
Therefore, it's true under
very general grounds.

734
00:51:56,220 --> 00:51:59,790
And that is a very
famous relation.

735
00:51:59,790 --> 00:52:03,770
So now you could
speculate that, indeed,

736
00:52:03,770 --> 00:52:12,390
the reason that a particle may
have a magnetic moment if it's

737
00:52:12,390 --> 00:52:15,610
made by a little ball of
charge that is rotating.

738
00:52:15,610 --> 00:52:18,900
But that was exactly what
Pauli didn't like, of course.

739
00:52:21,928 --> 00:52:24,600
And you would like
to see what's really

740
00:52:24,600 --> 00:52:25,980
happening with particles.

741
00:52:25,980 --> 00:52:31,770
So when you think of a true
quantum mechanical particle,

742
00:52:31,770 --> 00:52:37,550
let's think of a particle
in general, a solid particle

743
00:52:37,550 --> 00:52:38,110
rotating.

744
00:52:38,110 --> 00:52:43,160
We'll change the name to S
for spin angular momentum.

745
00:52:43,160 --> 00:52:46,480
Because that little part,
this is just one particle.

746
00:52:46,480 --> 00:52:48,920
We're not thinking of
that little particle

747
00:52:48,920 --> 00:52:51,700
going around a nucleus.

748
00:52:51,700 --> 00:52:55,310
We're thinking of that
little particle rotating.

749
00:52:55,310 --> 00:52:57,620
So this is a little piece
of that little particle

750
00:52:57,620 --> 00:52:59,270
that is rotating.

751
00:52:59,270 --> 00:53:08,640
So you could ask,
if, for the electron,

752
00:53:08,640 --> 00:53:12,150
for example, is
it true that mu is

753
00:53:12,150 --> 00:53:17,595
equal to e over 2 mass of
the electron times its spin?

754
00:53:22,030 --> 00:53:28,040
So this would be a vindication
of this classical analysis.

755
00:53:28,040 --> 00:53:34,770
It might be that it's
related in this way.

756
00:53:34,770 --> 00:53:48,910
So actually, it's
not quite true,

757
00:53:48,910 --> 00:53:53,080
but let's still improve
this a little bit.

758
00:53:53,080 --> 00:54:00,590
In terms of units, we like to
put an h bar here and a 2Me.

759
00:54:00,590 --> 00:54:05,700
And put spin here, angular
momentum, divided by h.

760
00:54:05,700 --> 00:54:11,960
Because this has no
units, h bar has the units

761
00:54:11,960 --> 00:54:17,000
of angular momentum, x times p.

762
00:54:17,000 --> 00:54:19,390
It's the same units, so
units of angular momentum.

763
00:54:19,390 --> 00:54:24,430
So h bar would be convenient.

764
00:54:24,430 --> 00:54:33,270
So that over here, you would
have units of a dipole moment,

765
00:54:33,270 --> 00:54:38,300
or magnetic moment,
magnetic moment units.

766
00:54:43,300 --> 00:54:47,810
So what does happen
for the electron?

767
00:54:47,810 --> 00:54:52,810
Well, it's almost
true, but not quite.

768
00:54:52,810 --> 00:54:57,680
In fact, what you get is
that you need a fudge factor.

769
00:54:57,680 --> 00:55:00,700
The fudge factor
is that, actually,

770
00:55:00,700 --> 00:55:02,800
for elementary
particles, you have

771
00:55:02,800 --> 00:55:06,980
a g, which is a constant,
which is the fudge factor,

772
00:55:06,980 --> 00:55:15,850
e h bar 2 over M of the
particle S over h bar.

773
00:55:15,850 --> 00:55:19,410
Sometimes called
the Lande factor.

774
00:55:19,410 --> 00:55:22,410
You must put a number there.

775
00:55:22,410 --> 00:55:28,140
Now, the good thing is
that the number sometimes

776
00:55:28,140 --> 00:55:31,030
can be calculated and predicted.

777
00:55:31,030 --> 00:55:34,100
So when people did
this, they figured out

778
00:55:34,100 --> 00:55:38,950
that for the electron the
number is actually a 2.

779
00:55:38,950 --> 00:55:46,210
So for the electron, g of
the electron is equal to 2.

780
00:55:46,210 --> 00:55:49,770
Now that, you would say,
cannot be an accident.

781
00:55:49,770 --> 00:55:54,720
It's twice what you would
predict sort of classically.

782
00:55:54,720 --> 00:55:59,250
And the Dirac equation,
the relativistic equation

783
00:55:59,250 --> 00:56:02,010
of the electron that
you have not studied yet

784
00:56:02,010 --> 00:56:07,200
but you will study soon,
predicts this g equal to 2.

785
00:56:07,200 --> 00:56:08,880
It was considered
a great success

786
00:56:08,880 --> 00:56:12,670
that that equation gave the
right answer, that people

787
00:56:12,670 --> 00:56:16,400
understood that this
number was going to be 2.

788
00:56:16,400 --> 00:56:19,670
So for the electron, this is 2.

789
00:56:19,670 --> 00:56:25,210
So this quantity is called--
it's a magnetic dipole

790
00:56:25,210 --> 00:56:32,210
moment-- is called mu
B for Bohr magneton.

791
00:56:39,900 --> 00:56:44,330
So how big is a mu B?

792
00:56:44,330 --> 00:56:51,880
It's about 9.3 times 10 to
the minus 24 joules per tesla.

793
00:56:55,454 --> 00:56:56,380
AUDIENCE: Professor.

794
00:56:56,380 --> 00:56:57,520
PROFESSOR: Yes?

795
00:56:57,520 --> 00:56:58,966
AUDIENCE: [INAUDIBLE].

796
00:56:58,966 --> 00:57:01,376
So where exactly does
the fudge factor come in?

797
00:57:01,376 --> 00:57:05,112
Is it just merely
because [INAUDIBLE]?

798
00:57:05,112 --> 00:57:05,820
PROFESSOR: Right.

799
00:57:05,820 --> 00:57:11,590
So the classical
analysis is not valid.

800
00:57:11,590 --> 00:57:14,800
So it's pretty invalid, in fact.

801
00:57:14,800 --> 00:57:19,530
You see, the picture
of an electron,

802
00:57:19,530 --> 00:57:22,480
as of today, is that
it's a point particle.

803
00:57:22,480 --> 00:57:27,200
And a point particle
literally means no size.

804
00:57:27,200 --> 00:57:30,880
The electron is not a
little ball of charge.

805
00:57:30,880 --> 00:57:32,740
Otherwise, it would have parts.

806
00:57:32,740 --> 00:57:34,970
So an electron is
a point particle.

807
00:57:34,970 --> 00:57:38,250
Therefore, a point particle
cannot be rotating and have

808
00:57:38,250 --> 00:57:39,020
a spin.

809
00:57:39,020 --> 00:57:43,670
So how does the electron
manage to have spin?

810
00:57:43,670 --> 00:57:45,720
That you can't
answer in physics.

811
00:57:45,720 --> 00:57:47,590
It just has it.

812
00:57:47,590 --> 00:57:51,640
Just like a point particle
that has no size can have mass.

813
00:57:51,640 --> 00:57:54,670
How do you have mass
if you have no size?

814
00:57:54,670 --> 00:57:56,130
You get accustomed to the idea.

815
00:57:56,130 --> 00:57:58,550
The mathematics
says it's possible.

816
00:57:58,550 --> 00:58:00,420
You don't run into trouble.

817
00:58:00,420 --> 00:58:06,510
So this particle has no size,
but it has an angular spin,

818
00:58:06,510 --> 00:58:09,070
angular momentum, as if
it would be rotating.

819
00:58:09,070 --> 00:58:12,520
But it's definitely not the
case that it's rotating.

820
00:58:12,520 --> 00:58:17,420
And therefore, this 2 confirms
that it was a pointless idea

821
00:58:17,420 --> 00:58:19,260
to believe that
it would be true.

822
00:58:19,260 --> 00:58:22,720
Nevertheless, kind
of unit analyses

823
00:58:22,720 --> 00:58:26,770
or maybe some truth to the
fact that quantum mechanics

824
00:58:26,770 --> 00:58:29,290
changes classical mechanics.

825
00:58:29,290 --> 00:58:32,580
Turns out that it's
closely related.

826
00:58:32,580 --> 00:58:36,910
For the proton, for example, the
magnetic moment of the proton

827
00:58:36,910 --> 00:58:40,510
is quite complicated as well
because the proton is made out

828
00:58:40,510 --> 00:58:43,110
of quarks that are
rotating inside.

829
00:58:43,110 --> 00:58:47,070
And how do you get
the spin of the proton

830
00:58:47,070 --> 00:58:48,790
and the magnetic
moment of the proton?

831
00:58:48,790 --> 00:58:50,000
It's complicated.

832
00:58:50,000 --> 00:58:54,560
The neutron, that has no
charge, has a magnetic moment,

833
00:58:54,560 --> 00:58:58,450
because somehow the quarks
inside arrange in a way

834
00:58:58,450 --> 00:59:01,900
that their angular momentum
doesn't quite cancel.

835
00:59:05,580 --> 00:59:08,690
So for example, the value
for a neutron, I believe,

836
00:59:08,690 --> 00:59:12,390
is minus 2.78 or
something like that.

837
00:59:12,390 --> 00:59:14,770
It's a strange number.

838
00:59:14,770 --> 00:59:19,070
Another thing that is sort of
interesting that is also true

839
00:59:19,070 --> 00:59:22,390
is that this mass is
the mass of a particle.

840
00:59:22,390 --> 00:59:25,250
So if you're talking
about the magnetic moment

841
00:59:25,250 --> 00:59:28,530
of the proton or
the neutron, it's

842
00:59:28,530 --> 00:59:31,740
suppressed with respect to
the one of the electron.

843
00:59:31,740 --> 00:59:34,580
The electron one is much bigger
because, actually, the mass

844
00:59:34,580 --> 00:59:36,440
shows up here.

845
00:59:36,440 --> 00:59:40,860
So for a neutron or a
proton, the magnetic moment

846
00:59:40,860 --> 00:59:44,210
is much, much smaller.

847
00:59:44,210 --> 00:59:48,560
So, in fact, for an
electron then, you

848
00:59:48,560 --> 00:59:49,870
would have the following.

849
00:59:49,870 --> 01:00:01,720
Mu is equal to minus g, which
is 2, mu B S bar over h.

850
01:00:01,720 --> 01:00:04,280
And actually, we
put the minus sign

851
01:00:04,280 --> 01:00:07,820
because the electron
has negative charge.

852
01:00:07,820 --> 01:00:12,270
So the magnetic moment
actually points opposite.

853
01:00:12,270 --> 01:00:16,120
If you rotate this way, the
angular momentum is always up.

854
01:00:16,120 --> 01:00:18,770
But if you rotate this
way and you're negative,

855
01:00:18,770 --> 01:00:21,740
it's as if the current goes
in the other direction.

856
01:00:21,740 --> 01:00:25,871
So this is due to the fact
that the electron is negatively

857
01:00:25,871 --> 01:00:26,370
charged.

858
01:00:26,370 --> 01:00:28,870
And that's the final expression.

859
01:00:31,740 --> 01:00:38,765
So OK, so that's the general
story with magnetic moments.

860
01:00:42,390 --> 01:00:45,360
So the next thing is,
how do magnetic moments

861
01:00:45,360 --> 01:00:49,530
react when you have
magnetic fields?

862
01:00:49,530 --> 01:00:54,710
So that is something
that you can calculate,

863
01:00:54,710 --> 01:00:57,790
or you can decide if
you have a picture.

864
01:00:57,790 --> 01:01:03,650
For example, if you have a
loop of charge like this,

865
01:01:03,650 --> 01:01:10,050
and you have magnetic field
lines that go like this,

866
01:01:10,050 --> 01:01:11,795
they diverge a bit.

867
01:01:15,830 --> 01:01:18,220
Let me see you use
your right-hand rule

868
01:01:18,220 --> 01:01:20,950
and tell me whether
that loop of current

869
01:01:20,950 --> 01:01:22,925
will feel a force up or down.

870
01:01:26,936 --> 01:01:31,090
I'll give you 30 seconds,
and I take a vote.

871
01:01:31,090 --> 01:01:32,880
Let's see how we're
doing with that.

872
01:01:39,547 --> 01:01:43,280
And I'll prepare these
blackboards in the meantime.

873
01:01:59,420 --> 01:02:00,310
All right.

874
01:02:00,310 --> 01:02:03,060
Who votes up?

875
01:02:03,060 --> 01:02:03,560
Nobody.

876
01:02:03,560 --> 01:02:05,503
Who votes down?

877
01:02:05,503 --> 01:02:06,405
Yeah, [INAUDIBLE].

878
01:02:06,405 --> 01:02:08,640
Down, exactly.

879
01:02:08,640 --> 01:02:10,150
How do you see down?

880
01:02:10,150 --> 01:02:14,250
Well, one way to see this,
look at the cross-section.

881
01:02:14,250 --> 01:02:18,980
You would have this
wire here like that.

882
01:02:18,980 --> 01:02:22,870
The current is coming in on this
side and going out this way.

883
01:02:22,870 --> 01:02:26,470
Here you have the field lines
that go through those two

884
01:02:26,470 --> 01:02:33,570
edges, and the magnetic
field is like that.

885
01:02:33,570 --> 01:02:38,550
And the force goes like I cross
B. So I goes in, B goes out.

886
01:02:38,550 --> 01:02:42,720
The force must be like
that, a little bit of force.

887
01:02:42,720 --> 01:02:46,700
In this one, I cross
B would be like that,

888
01:02:46,700 --> 01:02:48,160
a little bit of force.

889
01:02:48,160 --> 01:02:49,070
Yep.

890
01:02:49,070 --> 01:02:52,105
Has a component down because
the field lines are diverging.

891
01:02:56,080 --> 01:02:59,920
So what is the force
really given by?

892
01:02:59,920 --> 01:03:05,540
The force is given
by the gradient

893
01:03:05,540 --> 01:03:14,490
of mu dot B. This is derived in
E&M. I will not derive it here.

894
01:03:14,490 --> 01:03:17,490
This is not really the
point of this course.

895
01:03:17,490 --> 01:03:21,310
But you can see that
it's consistent.

896
01:03:21,310 --> 01:03:26,080
This is saying that the force
goes in the direction that

897
01:03:26,080 --> 01:03:29,810
makes mu dot B grow the fastest.

898
01:03:29,810 --> 01:03:34,830
Now mu, in this case, is up.

899
01:03:34,830 --> 01:03:40,170
So mu dot B is positive, because
mu and the magnetic field

900
01:03:40,170 --> 01:03:41,550
go in the same direction.

901
01:03:41,550 --> 01:03:43,610
So mu dot b is positive.

902
01:03:43,610 --> 01:03:47,720
So the force will be towards
the direction-- that's

903
01:03:47,720 --> 01:03:51,220
what the gradient is--
that this becomes bigger.

904
01:03:51,220 --> 01:03:55,210
So it becomes bigger here,
because as the field lines come

905
01:03:55,210 --> 01:03:58,570
together, that means
stronger magnetic field.

906
01:03:58,570 --> 01:04:03,550
And therefore, mu dot B would be
larger, so it's pointing down.

907
01:04:03,550 --> 01:04:07,790
If you have a magnetic
field that is roughly

908
01:04:07,790 --> 01:04:11,010
in the z direction, there
will be a simplification,

909
01:04:11,010 --> 01:04:13,020
as we will see very soon.

910
01:04:13,020 --> 01:04:16,480
So what did Stern
and Gerlach do?

911
01:04:16,480 --> 01:04:21,110
Well, they were working
with silver atoms.

912
01:04:21,110 --> 01:04:25,410
And silver atoms
have 47 electrons,

913
01:04:25,410 --> 01:04:30,070
out of which 46 fill up
the levels and equal 1,

914
01:04:30,070 --> 01:04:33,620
2, 3, and 4.

915
01:04:33,620 --> 01:04:41,010
Just one lone electron, a 5s
electron, the 47th electron,

916
01:04:41,010 --> 01:04:45,810
it's a lonely electron that
is out in a spherical shell,

917
01:04:45,810 --> 01:04:49,760
we know now with zero
orbital angular momentum.

918
01:04:49,760 --> 01:04:52,130
It's an S state.

919
01:04:52,130 --> 01:04:58,110
And therefore, throwing silver
atoms through your apparatus

920
01:04:58,110 --> 01:05:03,320
was pretty much the same
thing as throwing electrons,

921
01:05:03,320 --> 01:05:05,810
because all these
other electrons

922
01:05:05,810 --> 01:05:08,440
are tied up with each other.

923
01:05:08,440 --> 01:05:11,970
We know now one has
spin up, one spin down.

924
01:05:11,970 --> 01:05:15,710
Nothing contributes, no
angular momentum as a whole.

925
01:05:15,710 --> 01:05:19,090
And then you have this
last electron unpaired.

926
01:05:19,090 --> 01:05:21,060
It has a spin.

927
01:05:21,060 --> 01:05:23,066
So it's like throwing spins.

928
01:05:26,130 --> 01:05:32,820
So moreover, throwing spins,
as far as we're concerned,

929
01:05:32,820 --> 01:05:34,510
Stern and Gerlach wouldn't care.

930
01:05:34,510 --> 01:05:39,370
Because of these relations,
it's throwing in dipole moments.

931
01:05:39,370 --> 01:05:42,910
And they would care about
that because magnetic fields

932
01:05:42,910 --> 01:05:46,740
push dipole moments up or down.

933
01:05:46,740 --> 01:05:52,690
Therefore, what is the
apparatus these people had?

934
01:05:52,690 --> 01:06:01,760
It was sort of like
this, with an oven,

935
01:06:01,760 --> 01:06:04,580
and you produce
some silver atoms

936
01:06:04,580 --> 01:06:10,450
that come out as a gas,
a collimating slit.

937
01:06:14,490 --> 01:06:19,490
Then you put axes
here-- we put axes just

938
01:06:19,490 --> 01:06:22,060
to know the components
we're talking about.

939
01:06:22,060 --> 01:06:39,850
And then there's magnets,
some sort of magnet like this,

940
01:06:39,850 --> 01:06:41,780
and the screen over there.

941
01:06:45,320 --> 01:06:49,285
So basically, this
form of this magnet

942
01:06:49,285 --> 01:06:53,330
that I've tried to draw there,
although it's not so easy, if I

943
01:06:53,330 --> 01:06:56,945
would take a cross-section
it would look like this.

944
01:07:01,740 --> 01:07:03,866
So the magnetic
field has a gradient.

945
01:07:06,430 --> 01:07:09,330
The lines bend a bit,
so there's a gradient

946
01:07:09,330 --> 01:07:10,445
of the magnetic field.

947
01:07:10,445 --> 01:07:16,870
And it's mostly in the z
direction, so z direction

948
01:07:16,870 --> 01:07:18,280
being pointed out here.

949
01:07:18,280 --> 01:07:19,590
So there's the magnetic field.

950
01:07:19,590 --> 01:07:21,800
The beam then comes here.

951
01:07:21,800 --> 01:07:24,045
And the question is, what
do you get on this screen?

952
01:07:27,570 --> 01:07:33,278
Now, I have it a little too low.

953
01:07:33,278 --> 01:07:37,110
The beam comes there
and goes through there.

954
01:07:37,110 --> 01:07:41,840
So the analysis that
we would have to do

955
01:07:41,840 --> 01:07:47,100
is basically an
analysis of the forces.

956
01:07:47,100 --> 01:07:52,020
And relatively, we
don't care too much.

957
01:07:52,020 --> 01:07:55,870
The fact is that
there's basically,

958
01:07:55,870 --> 01:07:59,190
because the magnetic field
is mostly in the z direction

959
01:07:59,190 --> 01:08:01,570
and varies in z
direction, there will

960
01:08:01,570 --> 01:08:04,940
be a force basically
in the z direction.

961
01:08:04,940 --> 01:08:05,790
Why is that?

962
01:08:05,790 --> 01:08:07,720
Because you take
this, and you say,

963
01:08:07,720 --> 01:08:15,870
well, that's roughly mu
z Bz, because it's mostly

964
01:08:15,870 --> 01:08:18,120
a magnetic field
in the z direction.

965
01:08:18,120 --> 01:08:24,670
And mu is a constant, so it's
basically gradient of Bz.

966
01:08:24,670 --> 01:08:26,029
Now, that's a vector.

967
01:08:26,029 --> 01:08:30,160
But we're saying also
most of the gradient of Bz

968
01:08:30,160 --> 01:08:37,109
is in the z direction,
so it's basically dBz dz.

969
01:08:45,569 --> 01:08:47,830
Now, there is some
bending of the lines,

970
01:08:47,830 --> 01:08:50,750
so there's a little bit of
gradient in other directions.

971
01:08:50,750 --> 01:08:54,220
But people have gone
through the analysis,

972
01:08:54,220 --> 01:08:57,790
and they don't matter for
any calculation that you do.

973
01:08:57,790 --> 01:09:00,830
They actually average out.

974
01:09:00,830 --> 01:09:04,850
So roughly, this gradient
is in the z direction.

975
01:09:04,850 --> 01:09:09,040
I'm sorry, the gradient is
supposed to be a vector.

976
01:09:09,040 --> 01:09:11,600
So you get a force
in the z direction.

977
01:09:11,600 --> 01:09:16,350
And therefore, the thing
that people expected

978
01:09:16,350 --> 01:09:17,870
was the following.

979
01:09:17,870 --> 01:09:24,200
You know, here comes one atom,
and it has its magnetic moment.

980
01:09:24,200 --> 01:09:28,420
Well, they've all been boiling
in this oven for a while.

981
01:09:28,420 --> 01:09:29,740
They're very disordered.

982
01:09:29,740 --> 01:09:33,055
Some have a z component of
magnetic-- the magnetic moment

983
01:09:33,055 --> 01:09:37,029
is pointing like that, so they
have some component, some down.

984
01:09:37,029 --> 01:09:37,880
Some are here.

985
01:09:37,880 --> 01:09:39,569
They have no component.

986
01:09:39,569 --> 01:09:44,260
It's all Boltzmann distributed
all over the directions.

987
01:09:44,260 --> 01:09:49,870
Therefore, you're going
to get a smudge like this.

988
01:09:49,870 --> 01:09:53,240
Some ones are going to be
deflected a lot because they

989
01:09:53,240 --> 01:09:57,030
have lots of z
component of angular

990
01:09:57,030 --> 01:10:00,240
momentum or z magnetic moment.

991
01:10:00,240 --> 01:10:04,720
Others are going to
be deflected little.

992
01:10:04,720 --> 01:10:07,590
So this was the
classical expectation.

993
01:10:07,590 --> 01:10:11,080
And the shock was that
you got, actually,

994
01:10:11,080 --> 01:10:16,010
one peak here, an empty
space, and another peak there.

995
01:10:16,010 --> 01:10:17,950
That was called
space quantization.

996
01:10:22,720 --> 01:10:26,790
Stern and Gerlach worked
with a magnetic field

997
01:10:26,790 --> 01:10:35,190
that was of about 0.1
tesla, a tenth of a tesla.

998
01:10:35,190 --> 01:10:38,600
And in their experiment,
the space quantization,

999
01:10:38,600 --> 01:10:42,680
this difference, was
1/5 of a millimeter.

1000
01:10:45,740 --> 01:10:51,110
So not that big, but
it was a clear thing.

1001
01:10:51,110 --> 01:10:53,700
It was there.

1002
01:10:53,700 --> 01:11:00,350
So everybody was confused.

1003
01:11:00,350 --> 01:11:04,340
They thought it was the orbital
spin, angular momentum that

1004
01:11:04,340 --> 01:11:06,600
somehow had been measured.

1005
01:11:06,600 --> 01:11:08,880
At the end of the
day, that was wrong.

1006
01:11:08,880 --> 01:11:10,370
It couldn't have been that.

1007
01:11:10,370 --> 01:11:13,400
People understood the Bohr
atom, realized, no, there's

1008
01:11:13,400 --> 01:11:15,220
no angular momentum there.

1009
01:11:15,220 --> 01:11:17,810
The idea of the spin
came back, and you

1010
01:11:17,810 --> 01:11:20,820
would have to do a
calculation to determine

1011
01:11:20,820 --> 01:11:23,030
what is the value of the spin.

1012
01:11:23,030 --> 01:11:28,960
So the exact factor took
a while to get it right.

1013
01:11:28,960 --> 01:11:36,660
But with the idea that mu z is
equal to minus 2 Bohr magenton

1014
01:11:36,660 --> 01:11:39,250
Sz over h bar, which
we wrote before.

1015
01:11:42,330 --> 01:11:46,900
Well, mu z, if you
know the strength

1016
01:11:46,900 --> 01:11:51,350
of your magnetic field, you
can calculate the deflections.

1017
01:11:51,350 --> 01:11:53,030
You know what mu B is.

1018
01:11:53,030 --> 01:11:57,040
So therefore, you get
the value for Sz over h.

1019
01:11:57,040 --> 01:12:00,540
And experiments
suggested that Sz over h

1020
01:12:00,540 --> 01:12:04,165
was either plus or minus 1/2.

1021
01:12:08,000 --> 01:12:15,280
And this kind of particle, it
has Sz over h bar equal plus

1022
01:12:15,280 --> 01:12:20,970
or minus 1/2, is called
the spin-1/2 particle.

1023
01:12:20,970 --> 01:12:26,400
So again, from this equation,
this can be measured.

1024
01:12:29,930 --> 01:12:34,725
And you then use this,
and you get this value.

1025
01:12:39,800 --> 01:12:43,160
So the experiment is
a little confusing.

1026
01:12:43,160 --> 01:12:44,760
Why did this happen?

1027
01:12:44,760 --> 01:12:48,020
And how do we think of
it quantum mechanically?

1028
01:12:48,020 --> 01:12:53,340
Now 804 sort of began
with these kind of things.

1029
01:12:53,340 --> 01:12:56,680
And you know by now
that what's happening

1030
01:12:56,680 --> 01:13:01,090
is the following, that somehow,
mathematically, every state is

1031
01:13:01,090 --> 01:13:04,000
a superposition of a
spin up and a spin down.

1032
01:13:04,000 --> 01:13:06,410
So every particle
that goes there

1033
01:13:06,410 --> 01:13:10,230
has half of its brain in the
spin up and half of its brain

1034
01:13:10,230 --> 01:13:11,630
in the spin down.

1035
01:13:11,630 --> 01:13:14,170
And then as it goes
through the magnetic field,

1036
01:13:14,170 --> 01:13:17,990
this thing splits, but each
particle is in both beams

1037
01:13:17,990 --> 01:13:19,880
still.

1038
01:13:19,880 --> 01:13:22,140
And they just have
this dual existence

1039
01:13:22,140 --> 01:13:25,130
until there's a screen
and there's detectors.

1040
01:13:25,130 --> 01:13:27,460
So they have to decide
what happens, and then

1041
01:13:27,460 --> 01:13:33,260
either collapses in the
top beam or lower beam.

1042
01:13:33,260 --> 01:13:35,630
Nothing happens until
you put the screen.

1043
01:13:35,630 --> 01:13:38,790
That's what we think now
is the interpretation

1044
01:13:38,790 --> 01:13:39,980
of this experiment.

1045
01:13:39,980 --> 01:13:43,720
But let's use the
last few minutes

1046
01:13:43,720 --> 01:13:50,440
to just write this in terms of
boxes and get the right ideas.

1047
01:13:50,440 --> 01:13:57,590
So instead of drawing
all that stuff,

1048
01:13:57,590 --> 01:14:01,620
we'll draw a little
box called a z hat

1049
01:14:01,620 --> 01:14:04,690
box, a Stern-Gerlach apparatus.

1050
01:14:04,690 --> 01:14:10,350
In comes a beam, out
would come two beams,

1051
01:14:10,350 --> 01:14:17,150
Sz equal h bar over 2 and
Sz equal minus h bar over 2.

1052
01:14:17,150 --> 01:14:19,980
And the convention is
that the plus goes up

1053
01:14:19,980 --> 01:14:21,840
and the minus goes
down, which I think

1054
01:14:21,840 --> 01:14:24,555
is probably consistent
with that drawing.

1055
01:14:27,470 --> 01:14:29,980
And that's the
Stern-Gerlach apparatus.

1056
01:14:29,980 --> 01:14:34,150
It measures Sz, and
it splits the beam.

1057
01:14:34,150 --> 01:14:36,510
Each particle goes
into both beams

1058
01:14:36,510 --> 01:14:38,520
until there's a
device that measures

1059
01:14:38,520 --> 01:14:41,270
and decides where you go.

1060
01:14:41,270 --> 01:14:44,440
So you can do the
following arrangements.

1061
01:14:44,440 --> 01:14:50,590
So here's arrangement number 1,
a Stern-Gerlach device with z.

1062
01:14:50,590 --> 01:14:54,000
Then you block the
lower one and let

1063
01:14:54,000 --> 01:15:03,650
the top one go as Sz
equal h bar over 2.

1064
01:15:03,650 --> 01:15:08,710
And then you put another
Stern-Gerlach machine, z hat,

1065
01:15:08,710 --> 01:15:12,220
that has two outputs.

1066
01:15:12,220 --> 01:15:15,270
And then you ask,
what's going to happen?

1067
01:15:15,270 --> 01:15:19,680
And the experiment can be
done and, actually, there's

1068
01:15:19,680 --> 01:15:23,720
nothing here coming out, and
all the particles come out here

1069
01:15:23,720 --> 01:15:26,250
with Sz equal h bar over 2.

1070
01:15:30,230 --> 01:15:34,280
What are we going
to learn from this?

1071
01:15:34,280 --> 01:15:36,710
In our picture of
quantum mechanics,

1072
01:15:36,710 --> 01:15:39,450
we're going to think
of this as there

1073
01:15:39,450 --> 01:15:43,700
are states of the
electron that have--

1074
01:15:43,700 --> 01:15:46,610
and I will write
them with respect

1075
01:15:46,610 --> 01:15:51,950
to z-- they have
plus h bar over 2

1076
01:15:51,950 --> 01:15:58,060
and states that have
minus h bar over 2.

1077
01:15:58,060 --> 01:16:03,590
And what we will think is that
these are really old basis

1078
01:16:03,590 --> 01:16:09,080
states, that any other state,
even one that points along x,

1079
01:16:09,080 --> 01:16:11,950
is a superposition of those two.

1080
01:16:11,950 --> 01:16:15,030
This is a very incredible
physical assumption.

1081
01:16:15,030 --> 01:16:20,310
It's saying this system is a
2-dimensional complex vector

1082
01:16:20,310 --> 01:16:24,850
space, two vectors, two
unit, two basis vectors.

1083
01:16:24,850 --> 01:16:28,860
And from those two, all linear
combinations that are infinite

1084
01:16:28,860 --> 01:16:33,860
represent all possible
spin configurations.

1085
01:16:33,860 --> 01:16:36,090
And what is this saying?

1086
01:16:36,090 --> 01:16:41,060
Well, as we will
translate it into algebra,

1087
01:16:41,060 --> 01:16:45,480
we will say that, look,
here is a state plus.

1088
01:16:45,480 --> 01:16:49,380
And when you try to measure,
if it had any minus component,

1089
01:16:49,380 --> 01:16:50,740
it had nothing.

1090
01:16:50,740 --> 01:16:55,410
So we will state that as
saying that these states are

1091
01:16:55,410 --> 01:16:56,330
orthogonal.

1092
01:16:56,330 --> 01:17:00,590
The minus state and the plus
state have zero overlap.

1093
01:17:03,330 --> 01:17:05,950
They are orthogonal
basis states.

1094
01:17:05,950 --> 01:17:10,910
And, for example, well, you
could also do it this way.

1095
01:17:10,910 --> 01:17:12,800
That would also be 0.

1096
01:17:12,800 --> 01:17:17,540
And you could also
say that z plus

1097
01:17:17,540 --> 01:17:23,370
and z plus is 1, because
every state that came in

1098
01:17:23,370 --> 01:17:26,040
as a plus came out as a plus.

1099
01:17:26,040 --> 01:17:27,910
They had perfect overlap.

1100
01:17:27,910 --> 01:17:32,130
So these are two
orthonormal basis vectors.

1101
01:17:32,130 --> 01:17:34,395
That's what this
seems to suggest.

1102
01:17:34,395 --> 01:17:36,940
And it's a little
strange, if you think,

1103
01:17:36,940 --> 01:17:41,950
because there's a
clash between arrows

1104
01:17:41,950 --> 01:17:45,300
and the notion of
orthonormality.

1105
01:17:45,300 --> 01:17:48,110
In 3-dimensional
vectors, you think

1106
01:17:48,110 --> 01:17:50,489
of this vector being
orthogonal to this.

1107
01:17:50,489 --> 01:17:52,030
But you wouldn't
think of this vector

1108
01:17:52,030 --> 01:17:54,280
as being orthogonal to that one.

1109
01:17:54,280 --> 01:18:00,130
And here is the spin is up,
and this is the spin down.

1110
01:18:00,130 --> 01:18:01,600
And those two are orthogonal.

1111
01:18:01,600 --> 01:18:03,450
You say, no, they're
anti-parallel.

1112
01:18:03,450 --> 01:18:04,780
They're not orthogonal.

1113
01:18:04,780 --> 01:18:06,820
No, they are orthogonal.

1114
01:18:06,820 --> 01:18:12,070
And that's the endlessly
confusing thing about spin-1/2.

1115
01:18:12,070 --> 01:18:16,830
So these states, their pictures
of the spins are arrows.

1116
01:18:16,830 --> 01:18:20,020
But don't think that those
arrows and the dot product

1117
01:18:20,020 --> 01:18:23,370
give you the orthogonality,
because this is up and down.

1118
01:18:23,370 --> 01:18:26,280
If you would be doing the
dot product of an up and down

1119
01:18:26,280 --> 01:18:28,000
vector, you would not get 0.

1120
01:18:28,000 --> 01:18:29,870
But this is 0.

1121
01:18:29,870 --> 01:18:31,860
Then you do the
following experiment.

1122
01:18:36,550 --> 01:18:38,390
So let's do the next one.

1123
01:18:42,820 --> 01:18:47,680
And the next one is,
again, the z filter.

1124
01:18:47,680 --> 01:18:50,920
Take this one, block it.

1125
01:18:50,920 --> 01:18:53,025
Then you put an x filter.

1126
01:18:55,610 --> 01:18:58,550
And what actually happens
is that you would get states

1127
01:18:58,550 --> 01:19:05,570
with Sx, now, h bar over 2 and
Sx equal minus h bar over 2,

1128
01:19:05,570 --> 01:19:07,070
because it's an x filter.

1129
01:19:07,070 --> 01:19:10,490
The magnetic field is a
line in the x direction.

1130
01:19:10,490 --> 01:19:16,630
Now, all these things have
Sz equal h bar over 2.

1131
01:19:16,630 --> 01:19:18,860
And what happens
in the experiment

1132
01:19:18,860 --> 01:19:21,860
is that 50% of the
particles come out here

1133
01:19:21,860 --> 01:19:23,790
and 50% come out there.

1134
01:19:23,790 --> 01:19:29,280
So a spin state
along the x direction

1135
01:19:29,280 --> 01:19:33,340
has some overlap with a spin
state along the z direction.

1136
01:19:33,340 --> 01:19:38,190
Normal vectors, a z vector and
an x vector, are orthogonal.

1137
01:19:38,190 --> 01:19:40,470
Not here for spins.

1138
01:19:40,470 --> 01:19:44,350
The spin pointing in the z
and the spin pointing in the x

1139
01:19:44,350 --> 01:19:46,100
are not orthogonal states.

1140
01:19:46,100 --> 01:19:47,890
They have overlaps.

1141
01:19:47,890 --> 01:19:53,690
So this means that, for example,
the x plus state and the z

1142
01:19:53,690 --> 01:19:58,990
plus state have an overlap.

1143
01:19:58,990 --> 01:20:02,840
This is notations that-- we're
going to be precise later.

1144
01:20:02,840 --> 01:20:11,170
But the same thing with the x
minus state, it has an overlap,

1145
01:20:11,170 --> 01:20:14,140
and somehow they're
about the same.

1146
01:20:14,140 --> 01:20:21,920
Finally, the last experiment
is this, z hat, block again,

1147
01:20:21,920 --> 01:20:26,640
x hat, but this time block one.

1148
01:20:26,640 --> 01:20:32,040
So here is a state with Sx
equals minus h bar over 2.

1149
01:20:32,040 --> 01:20:35,830
Here is a state with
Sz equal h bar over 2.

1150
01:20:35,830 --> 01:20:38,400
And now you put the
z machine again.

1151
01:20:40,980 --> 01:20:44,160
And what happens?

1152
01:20:44,160 --> 01:20:46,330
Well, there's two options.

1153
01:20:46,330 --> 01:20:48,520
People who were inventing
quantum mechanics

1154
01:20:48,520 --> 01:20:50,190
no wonder thought about them.

1155
01:20:50,190 --> 01:20:55,290
Here they could say, look,
I filtered this thing,

1156
01:20:55,290 --> 01:21:01,100
and now all these electrons
have Sz equal h bar over 2.

1157
01:21:01,100 --> 01:21:06,340
And now all these electrons have
Sx equal minus h bar over 2.

1158
01:21:06,340 --> 01:21:08,830
Maybe, actually,
all these electrons

1159
01:21:08,830 --> 01:21:13,420
have both Sz equal
h over 2 and that

1160
01:21:13,420 --> 01:21:15,090
because I filtered it twice.

1161
01:21:15,090 --> 01:21:16,850
So it maybe satisfies both.

1162
01:21:16,850 --> 01:21:20,530
So if all these electrons
would have Sz equals

1163
01:21:20,530 --> 01:21:23,690
h over 2 and this,
then you would only

1164
01:21:23,690 --> 01:21:26,080
get something from the top one.

1165
01:21:26,080 --> 01:21:28,350
But no, that's not what happens.

1166
01:21:28,350 --> 01:21:30,330
You get in both.

1167
01:21:30,330 --> 01:21:36,870
So somehow, the memory of
these states coming from Sz

1168
01:21:36,870 --> 01:21:39,260
equals h over 2
has been destroyed

1169
01:21:39,260 --> 01:21:41,020
by the time it
turned into a state

1170
01:21:41,020 --> 01:21:44,130
with Sx equal minus h over 2.

1171
01:21:44,130 --> 01:21:48,970
And a state cannot have
simultaneously this and that.

1172
01:21:48,970 --> 01:21:54,840
That's two properties, because
you get 50% here and 50% there.

1173
01:21:54,840 --> 01:21:57,760
So we'll discuss next
time a little more

1174
01:21:57,760 --> 01:22:00,500
about these relations
and how can the states

1175
01:22:00,500 --> 01:22:04,480
be related, the ones
that we use as the basis

1176
01:22:04,480 --> 01:22:07,590
vectors and all the
others along x and others

1177
01:22:07,590 --> 01:22:09,790
that we could build
some other way.

1178
01:22:09,790 --> 01:22:10,430
All right.

1179
01:22:10,430 --> 01:22:11,340
See you next week.

1180
01:22:11,340 --> 01:22:18,110
There's office hours today, 5:00
to 6:00, Monday, 4:30 to 5:30.