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BARTON ZWIEBACH: Let's begin.

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00:00:24,380 --> 00:00:28,730
So today's lecture will
deal with the subject

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00:00:28,730 --> 00:00:33,620
of squeeze states
and photon states.

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00:00:33,620 --> 00:00:38,240
And it all builds up from
the ideas of coherent states

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00:00:38,240 --> 00:00:41,130
that we were talking
about last time.

13
00:00:41,130 --> 00:00:46,335
So let me begin by reminding
you about the few facts

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00:00:46,335 --> 00:00:49,350
that we had about
coherent states.

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00:00:49,350 --> 00:00:53,290
So a coherent state was
born by taking the ground

16
00:00:53,290 --> 00:00:56,870
state of the harmonic
oscillator and displacing it

17
00:00:56,870 --> 00:01:01,400
with a translation
operator some distance, x0.

18
00:01:01,400 --> 00:01:03,700
And then we let
it go, and we saw

19
00:01:03,700 --> 00:01:09,270
that this sort of wave function
would just move from the left

20
00:01:09,270 --> 00:01:13,480
and to the right coherently,
without spreading out,

21
00:01:13,480 --> 00:01:15,180
without changing shape.

22
00:01:15,180 --> 00:01:17,740
It would move in a nice way.

23
00:01:17,740 --> 00:01:23,050
Now, this was obtained with
a translation operator, which

24
00:01:23,050 --> 00:01:27,980
was an exponential that had
on the exponent the momentum

25
00:01:27,980 --> 00:01:29,900
operator.

26
00:01:29,900 --> 00:01:34,780
But we realized eventually that
the reason it all works out

27
00:01:34,780 --> 00:01:39,440
is because its exponential
of something that

28
00:01:39,440 --> 00:01:43,600
depends on creation or
annihilation operators,

29
00:01:43,600 --> 00:01:46,130
and we could do
something more general,

30
00:01:46,130 --> 00:01:50,880
which was to use a
complex number, alpha,

31
00:01:50,880 --> 00:01:54,430
and define a
displacement operator,

32
00:01:54,430 --> 00:02:00,110
a more general one, that is
some linear combination of a

33
00:02:00,110 --> 00:02:05,370
and a dagger with alpha and
the complex conjugate of alpha.

34
00:02:05,370 --> 00:02:12,230
So this is only proportional to
the momentum if alpha is real,

35
00:02:12,230 --> 00:02:15,900
but if alpha is not real,
that operator in the exponent

36
00:02:15,900 --> 00:02:17,700
is not quite the momentum.

37
00:02:17,700 --> 00:02:22,090
It's something that has a
bit of position as well.

38
00:02:22,090 --> 00:02:25,420
So this is a more
general operator,

39
00:02:25,420 --> 00:02:27,690
but on the other
hand, it's clear

40
00:02:27,690 --> 00:02:30,530
that it's anti-Hermitian,
because if you

41
00:02:30,530 --> 00:02:34,590
take the dagger of this
thing, this term becomes

42
00:02:34,590 --> 00:02:38,050
that and that term
becomes this one,

43
00:02:38,050 --> 00:02:40,270
each one with a change of sign.

44
00:02:40,270 --> 00:02:44,150
So you're really with an
anti-Hermitian operator.

45
00:02:44,150 --> 00:02:49,460
Therefore, the whole
operator is unitary

46
00:02:49,460 --> 00:02:53,960
and you're acting with a
unitary operator on the vacuum.

47
00:02:53,960 --> 00:02:56,930
And therefore, this state
is also well normalized

48
00:02:56,930 --> 00:03:00,570
and represents a state
with some expectation

49
00:03:00,570 --> 00:03:01,980
value of the position.

50
00:03:01,980 --> 00:03:04,850
Just like a coherent state,
we moved it to the right

51
00:03:04,850 --> 00:03:08,100
and it had some expectation
value of the position.

52
00:03:08,100 --> 00:03:11,720
But this one also has
some expectation value

53
00:03:11,720 --> 00:03:13,450
of the momentum.

54
00:03:13,450 --> 00:03:21,930
So in fact, we realize that the
real part of alpha in this axis

55
00:03:21,930 --> 00:03:26,170
was related to the expectation
value of the position divided

56
00:03:26,170 --> 00:03:27,580
by to the 0.

57
00:03:27,580 --> 00:03:29,250
So if you produce
a coherent state

58
00:03:29,250 --> 00:03:35,270
with this value of alpha
in the complex alpha plane,

59
00:03:35,270 --> 00:03:38,030
well, you go down and
that's the expectation value

60
00:03:38,030 --> 00:03:39,220
of the position.

61
00:03:39,220 --> 00:03:42,420
You go horizontally,
well, that's

62
00:03:42,420 --> 00:03:45,170
the expectation value
of the momentum scaled

63
00:03:45,170 --> 00:03:49,110
because this alpha is a
pure number, has no units.

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00:03:49,110 --> 00:03:55,560
So this x over square root
of 2 d0 and p d0 over h

65
00:03:55,560 --> 00:04:01,210
bar have no units, and
that's how it should be.

66
00:04:01,210 --> 00:04:06,520
So we learned also that the
annihilation operator acting

67
00:04:06,520 --> 00:04:10,560
on this coherent state was
alpha times the coherent state.

68
00:04:10,560 --> 00:04:13,030
So it's a very simple property.

69
00:04:13,030 --> 00:04:17,160
That number, alpha,
is the eigenvalue

70
00:04:17,160 --> 00:04:19,020
of the destruction operator.

71
00:04:22,250 --> 00:04:26,550
Now, that's a one line
computation based on this,

72
00:04:26,550 --> 00:04:29,820
or a two line computation maybe.

73
00:04:29,820 --> 00:04:34,360
But it should be a computation
that is easy for you to do.

74
00:04:34,360 --> 00:04:40,100
So make sure you
know how to get this

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00:04:40,100 --> 00:04:43,750
very quickly from
this definition.

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00:04:43,750 --> 00:04:45,570
So that's a coherent state.

77
00:04:45,570 --> 00:04:48,340
And then the thing we
finished the lecture

78
00:04:48,340 --> 00:04:53,080
with was with the time evolution
of this coherent state.

79
00:04:53,080 --> 00:05:00,360
And the time evolution was that
as the state, alpha, in time

80
00:05:00,360 --> 00:05:06,620
becomes the state alpha at time
t, it remains a coherent state

81
00:05:06,620 --> 00:05:09,720
but the value of
alpha is changed.

82
00:05:09,720 --> 00:05:12,160
In fact, the value
of alpha is changed

83
00:05:12,160 --> 00:05:16,890
in such a way that you can just
imagine this thing rotating,

84
00:05:16,890 --> 00:05:20,370
and rotating with
angular velocity, omega.

85
00:05:20,370 --> 00:05:25,870
So this thing was
the coherent state,

86
00:05:25,870 --> 00:05:30,130
e to the minus i omega t alpha.

87
00:05:30,130 --> 00:05:34,050
So this whole complex number,
instead of being alpha,

88
00:05:34,050 --> 00:05:35,030
is just this.

89
00:05:35,030 --> 00:05:37,650
There's no comma t.

90
00:05:37,650 --> 00:05:39,880
This is the time
development of the state.

91
00:05:39,880 --> 00:05:46,330
And there was a phase here, e
to the minus i omega t over 2,

92
00:05:46,330 --> 00:05:49,540
that was not very relevant.

93
00:05:49,540 --> 00:05:51,830
But that's what the
state was doing in time.

94
00:05:51,830 --> 00:05:56,890
So basically, that's
where we got last time.

95
00:05:56,890 --> 00:06:00,686
Before I push on, do
you have any questions?

96
00:06:05,710 --> 00:06:11,760
I did post the notes associated
to coherent states about half

97
00:06:11,760 --> 00:06:15,760
an hour ago, so you have them.

98
00:06:15,760 --> 00:06:19,830
You do have two problems
on coherent states

99
00:06:19,830 --> 00:06:24,300
in this homework, so the
notes should help you.

100
00:06:24,300 --> 00:06:27,260
But any questions
about this picture?

101
00:06:35,190 --> 00:06:35,690
OK.

102
00:06:35,690 --> 00:06:38,710
So I want to develop
it a little more

103
00:06:38,710 --> 00:06:45,730
before starting
with squeeze states.

104
00:06:45,730 --> 00:06:49,810
So here's what I
want to tell you.

105
00:06:49,810 --> 00:06:54,516
And this is an intuition that
people have about these states.

106
00:06:57,500 --> 00:06:59,620
Alpha is a complex number.

107
00:06:59,620 --> 00:07:02,970
And you know, it's a
well-defined complex number.

108
00:07:02,970 --> 00:07:06,400
But you know, this
is a coherent state,

109
00:07:06,400 --> 00:07:11,730
so it's not a
position eigenstate.

110
00:07:11,730 --> 00:07:14,755
It's not a momentum eigenstate.

111
00:07:14,755 --> 00:07:17,130
It's not an energy eigenstate.

112
00:07:17,130 --> 00:07:19,670
It has all kinds
of uncertainties.

113
00:07:19,670 --> 00:07:24,000
It has uncertainties in
position, in momentum,

114
00:07:24,000 --> 00:07:24,890
and in energy.

115
00:07:24,890 --> 00:07:26,226
Yes?

116
00:07:26,226 --> 00:07:30,242
AUDIENCE: On that complex plane
where you have the x and the p,

117
00:07:30,242 --> 00:07:33,865
are those expectation values
of the position of momentum?

118
00:07:33,865 --> 00:07:34,740
BARTON ZWIEBACH: Yes.

119
00:07:34,740 --> 00:07:37,220
I think I wrote them
with expectation values

120
00:07:37,220 --> 00:07:39,600
of the position of
momentum last time.

121
00:07:39,600 --> 00:07:40,710
Yes.

122
00:07:40,710 --> 00:07:43,400
So given alpha,
that number you get

123
00:07:43,400 --> 00:07:46,740
is the expectation value
of position or expectation

124
00:07:46,740 --> 00:07:48,336
value of momentum.

125
00:07:48,336 --> 00:07:48,835
Correct.

126
00:07:51,930 --> 00:07:56,940
So actually, this is
the expectation value

127
00:07:56,940 --> 00:07:59,810
of the position,
but the position

128
00:07:59,810 --> 00:08:03,726
is a little bit rounded.

129
00:08:03,726 --> 00:08:06,700
Yeah, this is the expectation
value of the position.

130
00:08:06,700 --> 00:08:07,460
It's a number.

131
00:08:07,460 --> 00:08:11,360
But intuitively, this
is spread out a little.

132
00:08:11,360 --> 00:08:16,020
The position is
not just one point.

133
00:08:16,020 --> 00:08:18,425
This is a coherent state.

134
00:08:18,425 --> 00:08:21,370
It looks like a
Gaussian wave function.

135
00:08:21,370 --> 00:08:25,650
The momentum is also
spread out a little.

136
00:08:25,650 --> 00:08:29,900
So in some sense, many people
draw this as a little blob.

137
00:08:38,419 --> 00:08:43,039
And that blob represents
your intuition

138
00:08:43,039 --> 00:08:46,275
that yes, the
expectation value is this

139
00:08:46,275 --> 00:08:48,210
and the expectation
value is this,

140
00:08:48,210 --> 00:08:53,680
but the position is, well,
somewhere around this thing

141
00:08:53,680 --> 00:08:57,890
and somewhere around that stuff.

142
00:08:57,890 --> 00:09:02,810
You can complain, this is very
hand wavy, but it's useful.

143
00:09:02,810 --> 00:09:06,660
It's good to have that physical
picture that the state really

144
00:09:06,660 --> 00:09:11,610
is some sort of blob here, not
that the expectation values are

145
00:09:11,610 --> 00:09:16,950
not well defined, but rather
that it's something like this.

146
00:09:16,950 --> 00:09:24,270
And I want to relate it to
an idea that comes along

147
00:09:24,270 --> 00:09:27,890
with waves, and it's
important for what

148
00:09:27,890 --> 00:09:30,890
we're going to be
doing later today.

149
00:09:30,890 --> 00:09:41,920
If you have a wave with energy
e, and suppose your wave

150
00:09:41,920 --> 00:09:45,740
is a light wave.

151
00:09:45,740 --> 00:09:49,090
Light wave with energy e.

152
00:09:49,090 --> 00:09:53,660
And it's described by,
say, A cosine omega t.

153
00:09:53,660 --> 00:09:56,450
That's some component
of the electric field

154
00:09:56,450 --> 00:10:00,910
or the magnetic
field for this wave.

155
00:10:00,910 --> 00:10:01,970
It is like that.

156
00:10:01,970 --> 00:10:06,190
Well, we've been talking
about energy time uncertainty,

157
00:10:06,190 --> 00:10:11,580
and we know that unless we
make things very precise.

158
00:10:11,580 --> 00:10:14,900
It's easy to get things wrong.

159
00:10:14,900 --> 00:10:20,630
So I will first do
something fairly imprecise

160
00:10:20,630 --> 00:10:24,550
to give you a feeling of
what people talk about,

161
00:10:24,550 --> 00:10:27,925
and then we'll use this picture
to do it more precisely.

162
00:10:30,650 --> 00:10:39,620
So if you have
this wave, the face

163
00:10:39,620 --> 00:10:44,526
of this wave-- we'll
call it phi-- is omega t.

164
00:10:44,526 --> 00:10:49,350
And if we are naive
there, the face

165
00:10:49,350 --> 00:10:54,280
is divided by omega
is the error in time.

166
00:10:58,090 --> 00:11:05,650
Now, this wave has energy E.
It has some number of photons.

167
00:11:05,650 --> 00:11:12,245
So the energy, E, is the number
of photons times h bar omega.

168
00:11:12,245 --> 00:11:14,380
N is equal to number of photons.

169
00:11:19,900 --> 00:11:29,430
And again, we could say delta
E is delta N h bar omega,

170
00:11:29,430 --> 00:11:36,270
and then substitute these
two relations into this

171
00:11:36,270 --> 00:11:37,590
to see what we get.

172
00:11:37,590 --> 00:11:47,540
Well, delta E is
delta N h bar omega.

173
00:11:47,540 --> 00:11:51,970
Delta t is delta phi over omega.

174
00:11:51,970 --> 00:11:55,660
This should be h bar over 2.

175
00:11:55,660 --> 00:11:59,180
The omegas cancel,
the h bar cancels,

176
00:11:59,180 --> 00:12:10,860
and you get delta N delta phi
is about 1, or 1 over 2 or 1

177
00:12:10,860 --> 00:12:12,170
over square root of 2.

178
00:12:14,680 --> 00:12:17,000
And that's, in
fact, the relation

179
00:12:17,000 --> 00:12:20,340
that people do take
somewhat seriously,

180
00:12:20,340 --> 00:12:24,820
if you have a wave
in quantum optics

181
00:12:24,820 --> 00:12:27,380
and say, well, the uncertainty
in the number of photons

182
00:12:27,380 --> 00:12:31,590
and the uncertainty in the
coherence of these photons,

183
00:12:31,590 --> 00:12:36,610
the phases, if they're out of
phase, they're not coherent,

184
00:12:36,610 --> 00:12:38,360
there's a relation of this kind.

185
00:12:38,360 --> 00:12:46,410
And this derivation is
certainly pretty bad.

186
00:12:46,410 --> 00:12:51,470
It's just not
precise because even

187
00:12:51,470 --> 00:12:55,210
we started with this that is
not precise unless you really

188
00:12:55,210 --> 00:12:58,590
explain what you
mean by delta t.

189
00:12:58,590 --> 00:13:02,605
So let's see if we can make
some sense of this picture here.

190
00:13:05,540 --> 00:13:06,880
So here we go.

191
00:13:06,880 --> 00:13:10,345
I want to do a small
calculation first.

192
00:13:15,190 --> 00:13:19,000
So let's see what we have.

193
00:13:19,000 --> 00:13:23,600
In this coherent state, what
is the expectation value

194
00:13:23,600 --> 00:13:26,860
of the number operator?

195
00:13:26,860 --> 00:13:31,720
Expectation value of the
number operator in alpha.

196
00:13:31,720 --> 00:13:34,210
Well, the number
operator in alpha,

197
00:13:34,210 --> 00:13:40,000
you would do this a
dagger a in alpha.

198
00:13:40,000 --> 00:13:47,420
These are easy to do because a
on alpha is alpha times alpha,

199
00:13:47,420 --> 00:13:51,160
and if you dagger that
equation, a dagger on alpha

200
00:13:51,160 --> 00:13:53,090
is alpha star.

201
00:13:53,090 --> 00:13:59,720
So you get-- I'll go slowly--
alpha alpha star here,

202
00:13:59,720 --> 00:14:02,055
and then you get alpha alpha.

203
00:14:04,990 --> 00:14:07,200
And alpha has unit norm.

204
00:14:07,200 --> 00:14:12,350
These are numbers, so
this is equal to length

205
00:14:12,350 --> 00:14:13,605
of alpha squared.

206
00:14:20,580 --> 00:14:23,990
So if this is a
harmonic oscillator,

207
00:14:23,990 --> 00:14:28,450
the expectation value of the
number operator, in fact,

208
00:14:28,450 --> 00:14:33,030
is the length squared
of this vector.

209
00:14:33,030 --> 00:14:35,480
Now, how about N squared?

210
00:14:39,030 --> 00:14:41,430
N squared is a little
more work because you

211
00:14:41,430 --> 00:14:47,410
have alpha a dagger
a a dagger a alpha.

212
00:14:51,090 --> 00:14:53,885
This one gives me
the factor you know,

213
00:14:53,885 --> 00:14:56,182
this one gives me
the factor you know.

214
00:14:56,182 --> 00:15:04,170
And therefore, we already
have alpha squared times alpha

215
00:15:04,170 --> 00:15:07,950
a a dagger alpha.

216
00:15:07,950 --> 00:15:10,760
And here, the a's
and the a daggers

217
00:15:10,760 --> 00:15:14,560
are kind of in the
wrong order because I

218
00:15:14,560 --> 00:15:21,110
know what a is on a ket,
but a is now on the bra.

219
00:15:21,110 --> 00:15:23,840
And I know what a
dagger is on the bra,

220
00:15:23,840 --> 00:15:27,490
but now a dagger is on the ket.

221
00:15:27,490 --> 00:15:29,130
But the answer is simple.

222
00:15:29,130 --> 00:15:34,020
You replace this
by the commutator

223
00:15:34,020 --> 00:15:37,790
plus the reverse order.

224
00:15:37,790 --> 00:15:41,580
So this is equal to the
commutator, which is 1,

225
00:15:41,580 --> 00:15:44,425
plus the thing in
the reverse order.

226
00:15:49,830 --> 00:15:53,380
And this is 1 plus
alpha squared.

227
00:15:53,380 --> 00:16:00,610
So you have alpha squared
times 1 plus alpha squared,

228
00:16:00,610 --> 00:16:03,860
and that's the expectation
value of N squared.

229
00:16:03,860 --> 00:16:07,130
All that, because we're
actually interested in what

230
00:16:07,130 --> 00:16:13,250
is delta N, the uncertainty
in N in the coherent state.

231
00:16:13,250 --> 00:16:17,060
And that would be
this, square root

232
00:16:17,060 --> 00:16:21,750
of this, which is alpha
to the fourth plus alpha

233
00:16:21,750 --> 00:16:28,550
squared minus the square of
the expectation value, which

234
00:16:28,550 --> 00:16:33,100
is minus alpha to the fourth.

235
00:16:33,100 --> 00:16:38,610
And this is length of alpha.

236
00:16:38,610 --> 00:16:47,876
So the uncertainty in N
is just length of alpha.

237
00:16:47,876 --> 00:16:55,710
It happens to be the square root
of the expectation value of N.

238
00:16:55,710 --> 00:16:59,840
So in fact, if you
think of this picture,

239
00:16:59,840 --> 00:17:03,240
you're tempted to say,
oh, this represents

240
00:17:03,240 --> 00:17:08,750
the number of excited
states that you have.

241
00:17:08,750 --> 00:17:14,050
This length represents the
expectation value of N. No.

242
00:17:14,050 --> 00:17:19,329
The expectation value of
N is this length squared.

243
00:17:19,329 --> 00:17:27,210
This length represents
delta N in the picture.

244
00:17:31,930 --> 00:17:36,430
So what else can we say?

245
00:17:36,430 --> 00:17:40,610
Well, this picture is
useful because now, I

246
00:17:40,610 --> 00:17:42,905
can be a little
more precise here.

247
00:17:46,760 --> 00:17:49,390
This thing is rotating.

248
00:17:49,390 --> 00:17:54,660
That is time evolution
of your coherent state.

249
00:17:54,660 --> 00:17:57,250
Now, this thing
this rotating, but I

250
00:17:57,250 --> 00:18:00,040
can ask now how wide this is.

251
00:18:02,720 --> 00:18:08,400
So what is the uncertainty
in x in a coherent state?

252
00:18:08,400 --> 00:18:11,190
Well, the uncertainty
in x in a coherent state

253
00:18:11,190 --> 00:18:14,650
is the same as the uncertainty
of the ground state

254
00:18:14,650 --> 00:18:17,710
because you just moved it.

255
00:18:17,710 --> 00:18:19,940
Uncertainty doesn't change.

256
00:18:19,940 --> 00:18:28,080
So the uncertainty,
delta x, is in fact

257
00:18:28,080 --> 00:18:36,460
this quantity that we call
d0 over square root of 2.

258
00:18:36,460 --> 00:18:42,370
That's the uncertainty delta
x, and the uncertainty delta p

259
00:18:42,370 --> 00:18:49,640
is h bar over d0
square root of 2.

260
00:18:49,640 --> 00:18:52,670
These are not hard to remember.

261
00:18:52,670 --> 00:18:57,990
d0 is the length scale of
the harmonic oscillator,

262
00:18:57,990 --> 00:19:01,080
so that's typically what
the uncertainty should be.

263
00:19:01,080 --> 00:19:04,620
The square root of 2, yes,
it's hard to remember.

264
00:19:04,620 --> 00:19:07,000
But delta p is this one.

265
00:19:07,000 --> 00:19:08,540
And then the other
thing that you

266
00:19:08,540 --> 00:19:12,100
know is that the product
should be h bar over 2,

267
00:19:12,100 --> 00:19:14,030
so that is correct.

268
00:19:16,640 --> 00:19:18,970
Now, look at this.

269
00:19:18,970 --> 00:19:21,715
How big is this thing?

270
00:19:21,715 --> 00:19:26,980
If the uncertainty in x is
d0 over square root of 2,

271
00:19:26,980 --> 00:19:36,540
this width is about
how much, roughly?

272
00:19:47,985 --> 00:19:48,485
Nobody?

273
00:19:51,170 --> 00:20:00,230
This is the uncertainty in
x, d0 over square root of 2

274
00:20:00,230 --> 00:20:04,100
in these units, if you move
the expectation value of x

275
00:20:04,100 --> 00:20:07,305
plus the uncertainty of x over
2 and the other uncertainty

276
00:20:07,305 --> 00:20:09,130
of x roughly.

277
00:20:09,130 --> 00:20:12,310
This thing is d0 over
square root of 2,

278
00:20:12,310 --> 00:20:17,420
so it represents
basically 1/2, because you

279
00:20:17,420 --> 00:20:21,440
change the expectation
value of x by this amount,

280
00:20:21,440 --> 00:20:26,330
and then this thing moves 1/2.

281
00:20:26,330 --> 00:20:30,560
The size of this
is 1/2, roughly.

282
00:20:30,560 --> 00:20:36,240
Could be 1/4 or could be
2, but it's roughly 1/2.

283
00:20:36,240 --> 00:20:43,440
And the vertical one corresponds
to the uncertainty in momentum.

284
00:20:43,440 --> 00:20:47,390
So intuitively, this is
h over square root v0,

285
00:20:47,390 --> 00:20:53,290
so if you plug it in there,
this amount, p plus delta p,

286
00:20:53,290 --> 00:20:57,460
you'll get 1/2 as well.

287
00:20:57,460 --> 00:21:01,872
So in this plot-- yes?

288
00:21:01,872 --> 00:21:03,800
AUDIENCE: Wouldn't
the width be 1

289
00:21:03,800 --> 00:21:05,728
because the uncertainty
is the width

290
00:21:05,728 --> 00:21:08,717
in one direction [INAUDIBLE]?

291
00:21:08,717 --> 00:21:10,300
BARTON ZWIEBACH:
Well, the uncertainty

292
00:21:10,300 --> 00:21:12,970
is neither the width in
one direction or not.

293
00:21:12,970 --> 00:21:16,445
It's a Gaussian, so I
don't know where it stops.

294
00:21:20,940 --> 00:21:26,900
This picture is not very
precise when I talk about this,

295
00:21:26,900 --> 00:21:32,150
so let me leave it with
1/2 or something like that.

296
00:21:32,150 --> 00:21:34,390
I don't think we can do better.

297
00:21:34,390 --> 00:21:37,450
Now, there's also 1/2 here.

298
00:21:37,450 --> 00:21:43,060
So finally, we get to something
that is kind of interesting.

299
00:21:43,060 --> 00:21:49,050
If really the state in some
sense, in terms of x and p,

300
00:21:49,050 --> 00:21:54,020
is spread here, and
this is moving around,

301
00:21:54,020 --> 00:21:58,000
the face is a little ambiguous.

302
00:21:58,000 --> 00:22:00,350
Because you would say,
well, the face is this one,

303
00:22:00,350 --> 00:22:08,170
but well, you could go the whole
uncertainty that you go here.

304
00:22:08,170 --> 00:22:12,120
The uncertainty in where
the coherent state is,

305
00:22:12,120 --> 00:22:21,370
we could call the face here
delta phi in this picture.

306
00:22:21,370 --> 00:22:26,640
We don't know where this state
is because it's a little blob.

307
00:22:26,640 --> 00:22:29,310
We know the expectation
values where they are,

308
00:22:29,310 --> 00:22:32,800
but the state itself
is a little imprecise.

309
00:22:32,800 --> 00:22:35,790
So there's an angle here
in this diagram that

310
00:22:35,790 --> 00:22:39,790
represents the face because this
is going with frequency omega

311
00:22:39,790 --> 00:22:40,460
t.

312
00:22:40,460 --> 00:22:42,670
So this is the face
as this goes around,

313
00:22:42,670 --> 00:22:46,865
so this angle, delta
phi, is how much.

314
00:22:50,120 --> 00:22:53,090
Well, if this is
1/2 and this is 1/2,

315
00:22:53,090 --> 00:22:57,750
I'm going to assume that
this is 1/2 as well, or 1,

316
00:22:57,750 --> 00:22:59,380
or something like that.

317
00:22:59,380 --> 00:23:02,510
So it's 1 over this length.

318
00:23:07,938 --> 00:23:08,896
That's the uncertainty.

319
00:23:11,530 --> 00:23:17,280
But delta N, we calculated.

320
00:23:17,280 --> 00:23:19,940
This is roughly.

321
00:23:19,940 --> 00:23:25,480
And delta N we calculated,
and it's exactly alpha.

322
00:23:25,480 --> 00:23:32,540
So delta phi delta N
is about 1 correctly.

323
00:23:36,800 --> 00:23:39,690
And here, there is
at least a picture

324
00:23:39,690 --> 00:23:42,890
of what the face uncertainty
is and why it originates.

325
00:23:46,120 --> 00:23:47,154
Yes?

326
00:23:47,154 --> 00:23:48,612
AUDIENCE: Can you
tell me again how

327
00:23:48,612 --> 00:23:50,370
the Gaussian relates
to the uncertainty?

328
00:23:50,370 --> 00:23:51,536
BARTON ZWIEBACH: One second.

329
00:23:51,536 --> 00:23:52,050
Let me see.

330
00:23:52,050 --> 00:23:53,216
I've got one question first.

331
00:23:53,216 --> 00:23:53,892
AUDIENCE: Yes.

332
00:23:53,892 --> 00:23:57,196
Can you explain one more time
where the 1/2's come from?

333
00:23:57,196 --> 00:23:58,140
[INAUDIBLE] the graph.

334
00:23:58,140 --> 00:24:00,500
I'm not sure why.

335
00:24:00,500 --> 00:24:01,597
BARTON ZWIEBACH: Yes.

336
00:24:01,597 --> 00:24:04,762
AUDIENCE: Are you saying
that the width is 1/2, or is

337
00:24:04,762 --> 00:24:06,670
that how high it is?

338
00:24:06,670 --> 00:24:09,118
BARTON ZWIEBACH: The
width of this little ball.

339
00:24:09,118 --> 00:24:11,926
AUDIENCE: So how does that
follow from the graph?

340
00:24:16,100 --> 00:24:19,070
BARTON ZWIEBACH: It's
a little hand wavy,

341
00:24:19,070 --> 00:24:22,410
but I'll say it like this.

342
00:24:25,630 --> 00:24:41,480
I expect the
position, if measured,

343
00:24:41,480 --> 00:24:52,535
to be between expectation
value of x plus minus delta x.

344
00:24:52,535 --> 00:24:56,470
So if I'm going to measure
the position of something

345
00:24:56,470 --> 00:25:01,440
in a state, the most likely
measurement that I will get

346
00:25:01,440 --> 00:25:04,950
is the expectation value
of x, statistically,

347
00:25:04,950 --> 00:25:06,770
after I repeat this many times.

348
00:25:06,770 --> 00:25:10,090
But if I just
measure, I'm probably

349
00:25:10,090 --> 00:25:15,130
going to get some number
between this and this.

350
00:25:15,130 --> 00:25:21,620
So if you think of this
diagram not as the expectation

351
00:25:21,620 --> 00:25:25,310
value of x in here,
but whatever you

352
00:25:25,310 --> 00:25:31,620
got for x as you measured,
if you do 1,000 measurements,

353
00:25:31,620 --> 00:25:36,860
you're going to get points
all over here in some region

354
00:25:36,860 --> 00:25:39,290
because you measure
x in one case,

355
00:25:39,290 --> 00:25:43,300
then you measure the momentum,
you get a plot of data,

356
00:25:43,300 --> 00:25:47,120
and you measure them all.

357
00:25:47,120 --> 00:25:51,450
And then suppose you're
doing it with x first.

358
00:25:51,450 --> 00:25:54,910
You measure x and you say,
well, I get all kinds of values.

359
00:25:54,910 --> 00:25:56,640
I don't know what
the momentum is,

360
00:25:56,640 --> 00:25:58,820
but I get all kinds of values.

361
00:25:58,820 --> 00:26:02,400
They're going to run all
over here between these two

362
00:26:02,400 --> 00:26:04,600
positions.

363
00:26:04,600 --> 00:26:10,910
So when I add to the expectation
value of x this thing,

364
00:26:10,910 --> 00:26:13,530
when I want to see in
this graph, what it is,

365
00:26:13,530 --> 00:26:17,540
I must divide by
square root of 2d.

366
00:26:17,540 --> 00:26:26,770
So I divide by 1 over square
root of 2 d0 to see how it goes

367
00:26:26,770 --> 00:26:29,330
and how I plot it in
this graph because these

368
00:26:29,330 --> 00:26:31,660
are the units in this graph.

369
00:26:31,660 --> 00:26:35,460
So if delta x is d0
over square root of 2,

370
00:26:35,460 --> 00:26:38,420
I'm going to get some values
that go from the expectation

371
00:26:38,420 --> 00:26:45,408
value of x up to 1/2
more and 1/2 less.

372
00:26:45,408 --> 00:26:47,256
AUDIENCE: So it's
not actually 1/2.

373
00:26:47,256 --> 00:26:51,570
It's actually 1/2 times
whatever that set amount.

374
00:26:51,570 --> 00:26:53,520
BARTON ZWIEBACH:
Well, if I say this,

375
00:26:53,520 --> 00:26:56,605
that you obtain between this and
that, then I should say it's 1.

376
00:27:00,290 --> 00:27:02,590
Maybe I had in mind
that you sort of get

377
00:27:02,590 --> 00:27:07,740
most things between
1/2 of delta x.

378
00:27:07,740 --> 00:27:11,250
It's not terribly
precise, but it's

379
00:27:11,250 --> 00:27:12,790
roughly this is the picture.

380
00:27:12,790 --> 00:27:15,680
You measure the position,
you're going to get that.

381
00:27:15,680 --> 00:27:18,700
Similarly, you decide
to measure momentum.

382
00:27:18,700 --> 00:27:20,760
You don't measure
position, measure momentum,

383
00:27:20,760 --> 00:27:24,440
and you're going to get
roughly the expectation value,

384
00:27:24,440 --> 00:27:28,270
but you're going to get a
little plus minus uncertainty.

385
00:27:28,270 --> 00:27:30,970
So you're going to all points
here in your measurement.

386
00:27:30,970 --> 00:27:34,740
So this dashed thing
is your histogram

387
00:27:34,740 --> 00:27:37,350
after doing lots of experiments.

388
00:27:37,350 --> 00:27:40,370
You have lots of dots in here.

389
00:27:40,370 --> 00:27:44,190
And roughly, this is
how it comes about.

390
00:27:44,190 --> 00:27:47,980
It's not terribly precise
because I cannot put a point

391
00:27:47,980 --> 00:27:52,520
either here, because if I
say the measurement was this,

392
00:27:52,520 --> 00:27:56,330
I'm suggesting that I also
measured the momentum on that

393
00:27:56,330 --> 00:28:00,150
state, or I could only
measure the position.

394
00:28:00,150 --> 00:28:02,730
But it's a rough
idea, rough picture,

395
00:28:02,730 --> 00:28:05,130
of how big the spread is here.

396
00:28:05,130 --> 00:28:07,370
There is a mathematical
theory to do

397
00:28:07,370 --> 00:28:11,400
this more precisely, although
physically not much clearer,

398
00:28:11,400 --> 00:28:15,540
which are called
Wigner distributions.

399
00:28:15,540 --> 00:28:18,680
I don't think it helps
too much to understand it,

400
00:28:18,680 --> 00:28:22,480
but the rough picture
is relatively clear.

401
00:28:22,480 --> 00:28:25,880
So if you divide by 1
over square root of 2,

402
00:28:25,880 --> 00:28:31,380
this quantity that was equal
to d over square root of 2,

403
00:28:31,380 --> 00:28:38,420
you get, in this scale plus
1/2 and minus 1/2, so 1, 1, 1,

404
00:28:38,420 --> 00:28:41,630
and this value there.

405
00:28:41,630 --> 00:28:42,840
There was a question there.

406
00:28:42,840 --> 00:28:44,880
Yes?

407
00:28:44,880 --> 00:28:47,838
AUDIENCE: Can you explain
again how the Gaussian relates

408
00:28:47,838 --> 00:28:49,372
to the uncertainty [INAUDIBLE]?

409
00:28:56,540 --> 00:28:58,180
BARTON ZWIEBACH:
So I don't know how

410
00:28:58,180 --> 00:29:00,520
the Gaussian relates
to uncertainty in x.

411
00:29:00,520 --> 00:29:05,500
So basically, we computed the
uncertainty in x for the ground

412
00:29:05,500 --> 00:29:08,820
state, and I claimed that
for a coherent state,

413
00:29:08,820 --> 00:29:12,120
the uncertainty in x cannot
change because you just took

414
00:29:12,120 --> 00:29:14,280
the state and you moved it away.

415
00:29:14,280 --> 00:29:17,980
And the uncertainty
of x doesn't talk

416
00:29:17,980 --> 00:29:21,390
about what the
expectation value of x is.

417
00:29:21,390 --> 00:29:23,540
That changes when
you move a state.

418
00:29:23,540 --> 00:29:26,980
But just how much it's
spread and how much

419
00:29:26,980 --> 00:29:30,740
the state is spread is not
changed by a translation.

420
00:29:30,740 --> 00:29:35,240
So this is the old result for
the ground state uncertainty,

421
00:29:35,240 --> 00:29:37,760
ground state uncertainty,
and neither is changed.

422
00:29:44,430 --> 00:29:47,055
Let's go now into
our squeezed states.

423
00:29:57,030 --> 00:30:01,300
So what are going to
be squeezed states?

424
00:30:01,300 --> 00:30:04,200
They're going to be
pretty useful states.

425
00:30:04,200 --> 00:30:06,840
They have lots of
applications nowadays.

426
00:30:06,840 --> 00:30:11,930
They've been constructed
over the last 10 years

427
00:30:11,930 --> 00:30:14,550
more and more often,
and people are now

428
00:30:14,550 --> 00:30:18,670
able to construct what is called
squeezed states experimentally.

429
00:30:18,670 --> 00:30:20,340
And the way we're
going to motivate it

430
00:30:20,340 --> 00:30:27,190
is by imagining that you have a
harmonic oscillator, a particle

431
00:30:27,190 --> 00:30:37,770
n, and some spring,
k, or an omega.

432
00:30:37,770 --> 00:30:39,400
And there's a Hamiltonian.

433
00:30:39,400 --> 00:30:51,220
H is equal to p squared over 2
m plus 1/2 m omega x squared.

434
00:30:51,220 --> 00:30:53,220
But this Hamiltonian
is going to be

435
00:30:53,220 --> 00:30:58,920
the Hamiltonian of a
particle that has mass m1,

436
00:30:58,920 --> 00:31:05,300
and the oscillator
has frequency w1,

437
00:31:05,300 --> 00:31:09,940
and that's what we're going
to call the first Hamiltonian.

438
00:31:09,940 --> 00:31:14,270
After a little while, you
observe this Hamiltonian.

439
00:31:17,320 --> 00:31:20,560
I will erase this thing.

440
00:31:20,560 --> 00:31:21,690
We don't need them anymore.

441
00:31:39,300 --> 00:31:46,380
We observe this thing and this
has an uncertainty, delta x,

442
00:31:46,380 --> 00:31:51,710
proportional to-- we know this v
over square root of 2, so h bar

443
00:31:51,710 --> 00:31:55,830
over 2 m1 omega 1.

444
00:31:55,830 --> 00:32:03,560
And an uncertainty in
p, which is delta p

445
00:32:03,560 --> 00:32:10,540
equals square root of h
bar m1 omega 1 over 2.

446
00:32:10,540 --> 00:32:14,410
And again, they
saturate the bound

447
00:32:14,410 --> 00:32:17,950
if you have your ground state.

448
00:32:17,950 --> 00:32:19,770
So ground state is here.

449
00:32:23,920 --> 00:32:27,220
These two uncertainties,
the bound is saturated,

450
00:32:27,220 --> 00:32:30,620
all is good.

451
00:32:30,620 --> 00:32:34,730
Nevertheless, suddenly,
at time equals

452
00:32:34,730 --> 00:32:40,000
0, this state, this particle,
is in the ground state.

453
00:32:40,000 --> 00:32:43,930
The Hamiltonian changes.

454
00:32:43,930 --> 00:32:47,810
There's an abrupt
change in the physics.

455
00:32:47,810 --> 00:32:52,270
Maybe the temperature
was changed

456
00:32:52,270 --> 00:32:59,400
and the spring constant
changed, or the particle, a drop

457
00:32:59,400 --> 00:33:02,320
was added to it and
its mass changed,

458
00:33:02,320 --> 00:33:06,550
but the Hamiltonian has
changed all of a sudden.

459
00:33:06,550 --> 00:33:16,470
So this Hamiltonian, H1,
is valid for H less than 0

460
00:33:16,470 --> 00:33:22,490
and a particle in
the ground state.

461
00:33:25,990 --> 00:33:27,720
So the particle's
in the ground state,

462
00:33:27,720 --> 00:33:33,760
the Hamiltonian is fine there,
but suddenly, the Hamiltonian

463
00:33:33,760 --> 00:33:36,020
changes.

464
00:33:36,020 --> 00:33:38,560
The particle identity
has not changed.

465
00:33:38,560 --> 00:33:44,860
The particle is there, but it
is the Hamilton that changes.

466
00:33:44,860 --> 00:33:49,480
So there's an H2,
p squared over 2 m2

467
00:33:49,480 --> 00:33:55,550
plus 1/2 m squared w
squared 2 x squared.

468
00:33:58,720 --> 00:34:00,580
The picture is physically clean.

469
00:34:00,580 --> 00:34:03,830
The particle is sitting
there in the ground state,

470
00:34:03,830 --> 00:34:08,480
and suddenly, the parameters
of the system change.

471
00:34:08,480 --> 00:34:12,350
So this particle was
having a good time,

472
00:34:12,350 --> 00:34:15,679
it was at the ground
state, relaxed.

473
00:34:15,679 --> 00:34:20,870
Then suddenly, the wave function
didn't change at time equals 0.

474
00:34:20,870 --> 00:34:23,110
It was spread over
some distance.

475
00:34:23,110 --> 00:34:25,690
No measurement
was done, nothing.

476
00:34:25,690 --> 00:34:29,179
And suddenly, this
particle finds itself

477
00:34:29,179 --> 00:34:33,590
with some wave function
but in another Hamiltonian.

478
00:34:33,590 --> 00:34:36,620
From now on, its
time evolution is

479
00:34:36,620 --> 00:34:39,080
going to be governed by
the second Hamiltonian.

480
00:34:42,080 --> 00:34:43,719
Now, since the
second Hamiltonian

481
00:34:43,719 --> 00:34:47,000
is different from the
first Hamiltonian,

482
00:34:47,000 --> 00:34:50,350
this particle is not going to
be any more in the ground state.

483
00:34:50,350 --> 00:34:53,900
Even though it was in the ground
state of the first Hamiltonian,

484
00:34:53,900 --> 00:34:57,490
it's not anymore in the ground
state of the second Hamiltonian

485
00:34:57,490 --> 00:35:00,620
as soon as the thing
gets turned on.

486
00:35:00,620 --> 00:35:06,250
So for t greater than 0,
this Hamilton is there.

487
00:35:06,250 --> 00:35:12,410
So actually, the wave
function does not change, so

488
00:35:12,410 --> 00:35:17,660
let me write delta x, and I'll
write it the following way,

489
00:35:17,660 --> 00:35:22,160
h bar over 2 m2 omega 1.

490
00:35:22,160 --> 00:35:24,600
But you say, no,
delta x didn't change.

491
00:35:24,600 --> 00:35:25,450
Correct.

492
00:35:25,450 --> 00:35:35,560
So I'll put the factor
m2 w2 over m1 w1,

493
00:35:35,560 --> 00:35:37,515
and now it's the same delta x.

494
00:35:40,200 --> 00:35:45,810
Similarly, for delta
p, I will write

495
00:35:45,810 --> 00:35:53,270
that this is square root
of h bar m2 omega 2 over 2,

496
00:35:53,270 --> 00:35:58,750
and put the factor m1
omega 1 over m2 omega

497
00:35:58,750 --> 00:36:01,280
2 in front in such
a way that it is

498
00:36:01,280 --> 00:36:05,430
the same delta x and
the same delta p.

499
00:36:08,140 --> 00:36:14,240
Now, delta x times delta p
multiply to be h bar over 2,

500
00:36:14,240 --> 00:36:16,560
and they still
multiply to that number

501
00:36:16,560 --> 00:36:18,860
because I didn't change them.

502
00:36:18,860 --> 00:36:28,091
But this is equal-- I'll call
this number e to the minus

503
00:36:28,091 --> 00:36:28,590
gamma.

504
00:36:34,460 --> 00:36:36,590
I'll go to another blackboard.

505
00:36:36,590 --> 00:36:44,020
Delta x is e to the minus
gamma times square root

506
00:36:44,020 --> 00:36:50,380
of h bar over 2 m2 omega 2.

507
00:36:50,380 --> 00:36:54,790
And delta p is e to
the gamma, because it's

508
00:36:54,790 --> 00:36:58,130
the inverse factor
on the one that we

509
00:36:58,130 --> 00:37:07,850
call gamma, square root of
h bar m2 omega 2 over 2,

510
00:37:07,850 --> 00:37:14,790
where e to the gamma is
the square root of m1 omega

511
00:37:14,790 --> 00:37:17,275
1 over m2 omega 2.

512
00:37:20,480 --> 00:37:22,870
Look, we've done
very simple things.

513
00:37:22,870 --> 00:37:25,510
We haven't done really much.

514
00:37:25,510 --> 00:37:29,560
But already, we start
to see what's happening.

515
00:37:29,560 --> 00:37:34,440
From the viewpoint of
the second Hamiltonian,

516
00:37:34,440 --> 00:37:38,210
these uncertainties
are not right.

517
00:37:38,210 --> 00:37:41,180
They are not the uncertainties
of the ground state,

518
00:37:41,180 --> 00:37:44,020
because from the viewpoint
of the second Hamiltonian,

519
00:37:44,020 --> 00:37:46,330
the ground state
uncertainty is this

520
00:37:46,330 --> 00:37:48,540
and the ground state
uncertainty is this.

521
00:37:48,540 --> 00:37:52,060
And indeed, this particle
was in the ground state,

522
00:37:52,060 --> 00:37:55,720
it had some Gaussian,
but that's not

523
00:37:55,720 --> 00:37:58,540
the right Gaussian for
the second Hamiltonian.

524
00:37:58,540 --> 00:38:01,420
It's the right Gaussian
for the first Hamiltonian.

525
00:38:01,420 --> 00:38:05,530
So it's not in the ground state
of the second Hamiltonian,

526
00:38:05,530 --> 00:38:13,465
but it's in a particular state
in which, if gamma is positive,

527
00:38:13,465 --> 00:38:19,020
the uncertainty in x is squeezed
from the lowest uncertainty

528
00:38:19,020 --> 00:38:22,660
that you get in an
energy eigenstate.

529
00:38:22,660 --> 00:38:24,160
And the uncertainty
and the momentum

530
00:38:24,160 --> 00:38:27,050
will be stretched
in that direction.

531
00:38:27,050 --> 00:38:32,910
So you see, in the ground state
of the harmonic oscillator,

532
00:38:32,910 --> 00:38:37,730
you get that uncertainty, and
that's a canonical uncertainty.

533
00:38:37,730 --> 00:38:40,690
But this uncertainty is
squeezed because it's

534
00:38:40,690 --> 00:38:43,370
different from what it should
be, and this is squeezed.

535
00:38:43,370 --> 00:38:48,420
So from the viewpoint of
the second Hamiltonian,

536
00:38:48,420 --> 00:38:52,290
the ground state of
the first Hamiltonian

537
00:38:52,290 --> 00:38:54,840
is a squeezed state.

538
00:38:54,840 --> 00:38:57,790
It's a staple
whose uncertainties

539
00:38:57,790 --> 00:38:59,900
have been squeezed.

540
00:38:59,900 --> 00:39:02,580
And those states
exist, and the purpose

541
00:39:02,580 --> 00:39:04,150
of what we're going
to do now is try

542
00:39:04,150 --> 00:39:07,610
to determine them, find
them, see what they are,

543
00:39:07,610 --> 00:39:08,290
how they behave.

544
00:39:11,900 --> 00:39:12,615
Any questions?

545
00:39:16,430 --> 00:39:18,154
Yes, Nicholas?

546
00:39:18,154 --> 00:39:19,740
AUDIENCE: I'm a
little confused why

547
00:39:19,740 --> 00:39:25,216
we can say that delta x
[INAUDIBLE] these new ones are

548
00:39:25,216 --> 00:39:28,920
just related to the old
ones by this factor.

549
00:39:28,920 --> 00:39:30,040
BARTON ZWIEBACH: OK.

550
00:39:30,040 --> 00:39:38,260
You see, what I assumed is
that before time equals 0,

551
00:39:38,260 --> 00:39:40,670
you had a Gaussian.

552
00:39:40,670 --> 00:39:42,860
That was the original Gaussian.

553
00:39:42,860 --> 00:39:44,770
That was the original
wave function,

554
00:39:44,770 --> 00:39:48,650
and you had some delta
x and some delta p that

555
00:39:48,650 --> 00:39:51,840
were given by this
one [INAUDIBLE].

556
00:39:51,840 --> 00:39:56,240
Now, I didn't do anything except
rewrite the same quantities

557
00:39:56,240 --> 00:39:59,100
here, because what
I said next was

558
00:39:59,100 --> 00:40:01,780
that even though
at time equals 0,

559
00:40:01,780 --> 00:40:05,360
the Hamiltonian changes,
at time equals 0,

560
00:40:05,360 --> 00:40:08,110
the wave function
doesn't change.

561
00:40:08,110 --> 00:40:09,840
The wave function
remains the same.

562
00:40:09,840 --> 00:40:11,960
After that time,
it's going to start

563
00:40:11,960 --> 00:40:14,780
changing because the new
Hamiltonian kicks in.

564
00:40:14,780 --> 00:40:18,230
But this delta x's
are the same as

565
00:40:18,230 --> 00:40:20,720
that I wrote, and
here are the same.

566
00:40:20,720 --> 00:40:26,070
But here, you see clearly
that this delta x with respect

567
00:40:26,070 --> 00:40:29,300
to the second Hamiltonian
is not the one

568
00:40:29,300 --> 00:40:31,270
that it would be if
it would be a ground

569
00:40:31,270 --> 00:40:34,680
state, nor the delta p.

570
00:40:34,680 --> 00:40:35,836
Yes?

571
00:40:35,836 --> 00:40:39,514
AUDIENCE: Just at the instant
you change the Hamiltonian,

572
00:40:39,514 --> 00:40:41,180
because they might
have all [INAUDIBLE],

573
00:40:41,180 --> 00:40:42,940
the uncertainties would change.

574
00:40:42,940 --> 00:40:44,943
BARTON ZWIEBACH: Sorry?

575
00:40:44,943 --> 00:40:47,058
AUDIENCE: Is this just
at the instant where

576
00:40:47,058 --> 00:40:49,182
we change the Hamiltonian,
because after some time,

577
00:40:49,182 --> 00:40:50,974
the wave function might
change [INAUDIBLE].

578
00:40:50,974 --> 00:40:52,223
BARTON ZWIEBACH: That's right.

579
00:40:52,223 --> 00:40:54,530
This is just after I
change the Hamiltonian.

580
00:40:54,530 --> 00:40:57,180
The time evolution of
this state is something

581
00:40:57,180 --> 00:40:59,040
that we have to
figure out later.

582
00:40:59,040 --> 00:41:01,800
But after I've changed
the Hamiltonian,

583
00:41:01,800 --> 00:41:04,710
the state looks squeezed.

584
00:41:04,710 --> 00:41:10,130
So how can we calculate and
understand these things?

585
00:41:10,130 --> 00:41:17,310
So the way to think of
this is the following.

586
00:41:17,310 --> 00:41:21,520
You see, you have this
system of two Hamiltonians.

587
00:41:21,520 --> 00:41:26,260
There's an x and a p operator,
and the second Hamiltonian

588
00:41:26,260 --> 00:41:28,510
has an x and a p operator.

589
00:41:28,510 --> 00:41:31,960
These are the properties
of the particles.

590
00:41:31,960 --> 00:41:35,000
Therefore, what that
I'm going to think of

591
00:41:35,000 --> 00:41:38,230
is that the x and
the p operators

592
00:41:38,230 --> 00:41:42,700
are the operators that
describe the particle.

593
00:41:42,700 --> 00:41:48,210
They are unchanged
because we're talking

594
00:41:48,210 --> 00:41:52,020
about this same
object, same particle.

595
00:41:52,020 --> 00:42:01,895
So if I have the x operator,
which is equal to this formula,

596
00:42:01,895 --> 00:42:14,470
h bar over 2 m1 omega 1 a1
dagger plus a1 like this.

597
00:42:14,470 --> 00:42:18,700
From the first
Hamiltonian, the x's

598
00:42:18,700 --> 00:42:21,600
are related to a1's
and a1 daggers,

599
00:42:21,600 --> 00:42:25,780
but this is the same x
describing the same position

600
00:42:25,780 --> 00:42:28,650
as you would do in the
second Hamiltonian.

601
00:42:28,650 --> 00:42:39,810
So m2 w2 a2 hat
plus a2 hat dagger.

602
00:42:39,810 --> 00:42:43,400
It's a very strong physical
assumption I'm making here.

603
00:42:43,400 --> 00:42:46,700
It's an assumption that's
so strong that in many ways,

604
00:42:46,700 --> 00:42:48,860
you could almost say,
well, I'll buy it,

605
00:42:48,860 --> 00:42:52,560
but we'll see if it gives
something reasonable.

606
00:42:52,560 --> 00:42:58,030
I'm saying the x operator
is really the same thing,

607
00:42:58,030 --> 00:43:00,420
and you could view
it as constructed

608
00:43:00,420 --> 00:43:02,930
from ingredients of
the first Hamiltonian

609
00:43:02,930 --> 00:43:05,220
or the second Hamiltonian.

610
00:43:05,220 --> 00:43:07,610
So is the p operator.

611
00:43:07,610 --> 00:43:14,060
p, which is-- well, I
have a formula here--

612
00:43:14,060 --> 00:43:24,420
minus i m1 omega 1 h bar
over 2 a1 minus a1 dagger--

613
00:43:24,420 --> 00:43:30,460
should be the same as
minus i m2 omega 2 h bar

614
00:43:30,460 --> 00:43:34,000
over 2 a2 minus a2 dagger.

615
00:43:42,150 --> 00:43:48,940
So x and p are not changing.

616
00:43:48,940 --> 00:43:51,960
We're not talking about
two particles that

617
00:43:51,960 --> 00:43:55,500
have an x1 and a p1, and
the second particle, an x2

618
00:43:55,500 --> 00:43:56,710
and a p2.

619
00:43:56,710 --> 00:44:00,920
It's just one particle
has an x and a p

620
00:44:00,920 --> 00:44:03,235
is what you observe when
you measure position

621
00:44:03,235 --> 00:44:06,020
and you observe when
you measure momentum.

622
00:44:06,020 --> 00:44:09,510
Nevertheless, x and p
are related in this way

623
00:44:09,510 --> 00:44:13,070
to the creation and
annihilation operators.

624
00:44:13,070 --> 00:44:18,450
So we're going to find from
this some very strange relation

625
00:44:18,450 --> 00:44:21,620
between the creation
operators, the annihilation

626
00:44:21,620 --> 00:44:25,310
operators of the first
system and the second system.

627
00:44:25,310 --> 00:44:27,020
So what do we get, in fact?

628
00:44:27,020 --> 00:44:30,170
Well, the constants disappear
from the first equation

629
00:44:30,170 --> 00:44:35,500
roughly, and you get a1
dagger is equal to-- you

630
00:44:35,500 --> 00:44:40,100
get the ratio of m1
omega 1 over m2 omega 2,

631
00:44:40,100 --> 00:44:48,390
so you get e to the gamma
a1 a2 plus a2 dagger.

632
00:44:51,270 --> 00:44:58,590
From the bottom one,
a1 minus a1 dagger

633
00:44:58,590 --> 00:45:07,830
is equal to e to the minus
gamma a2 minus a2 dagger.

634
00:45:07,830 --> 00:45:09,650
These two equations
give you that.

635
00:45:09,650 --> 00:45:11,520
It should be clear.

636
00:45:11,520 --> 00:45:14,190
You just cancel the
constants and remember

637
00:45:14,190 --> 00:45:16,050
the definition of
e to the gamma.

638
00:45:18,620 --> 00:45:25,550
And now we can solve
for a1 and a1 dagger

639
00:45:25,550 --> 00:45:28,860
in terms of a2 and a2 dagger.

640
00:45:28,860 --> 00:45:31,790
And what do we find?

641
00:45:31,790 --> 00:45:49,600
a1 is equal to a2 cosh gamma
plus a2 dagger sinch gamma,

642
00:45:49,600 --> 00:45:52,090
and the dagger is what
you would imagine.

643
00:45:52,090 --> 00:45:59,760
So a1 dagger is equal
to a2 dagger cosh

644
00:45:59,760 --> 00:46:04,675
gamma plus a2 sinch gamma.

645
00:46:10,250 --> 00:46:15,920
The second equation
that you can calculate

646
00:46:15,920 --> 00:46:17,920
is the dagger of the first.

647
00:46:17,920 --> 00:46:19,890
It should be that.

648
00:46:19,890 --> 00:46:25,820
And now you've
found the scrambling

649
00:46:25,820 --> 00:46:28,935
of the creation,
annihilation operators.

650
00:46:34,030 --> 00:46:38,040
The old annihilation
operator is a mixture

651
00:46:38,040 --> 00:46:43,320
of the new annihilation operator
and a creation operator.

652
00:46:43,320 --> 00:46:44,820
They're mixed.

653
00:46:44,820 --> 00:46:49,750
It's a very strange
thing that has happened,

654
00:46:49,750 --> 00:46:54,930
a mixture between creation
and annihilation operators.

655
00:46:54,930 --> 00:46:57,910
This is so famous in
physics, it has a name.

656
00:46:57,910 --> 00:47:02,470
It's called the
Bogoliubov transformation.

657
00:47:02,470 --> 00:47:07,610
It appears in the analysis
of black hole radiation.

658
00:47:07,610 --> 00:47:09,490
There's a Bogoliubov
transformation

659
00:47:09,490 --> 00:47:11,900
between the fields far
away of the black hole

660
00:47:11,900 --> 00:47:13,580
and the fields near
the black hole.

661
00:47:13,580 --> 00:47:14,890
It appears everywhere.

662
00:47:17,590 --> 00:47:20,210
And here it has
appeared, so we're

663
00:47:20,210 --> 00:47:24,550
going to try to understand
what it does for us.

664
00:47:24,550 --> 00:47:29,540
Similarly, you can
find what a2 is

665
00:47:29,540 --> 00:47:35,390
in terms of a1's by the
symmetry of these equations.

666
00:47:35,390 --> 00:47:39,020
This corresponds to
actually letting gamma

667
00:47:39,020 --> 00:47:44,200
go to minus gamma, because
if you pass these gammas

668
00:47:44,200 --> 00:47:48,510
to the other side, the
equations are of the same form.

669
00:47:48,510 --> 00:47:52,350
By letting 1 become 2,
2 becomes 1 and gamma

670
00:47:52,350 --> 00:47:55,480
goes to minus gamma.

671
00:47:55,480 --> 00:47:59,090
So we don't need it right
now, but in case you

672
00:47:59,090 --> 00:48:03,870
want to find the other ones,
the 2's in terms of the 1's, you

673
00:48:03,870 --> 00:48:10,830
would just change the sign of
gamma and it would work out.

674
00:48:10,830 --> 00:48:19,380
So this relation is the key to
allow you to calculate things.

675
00:48:19,380 --> 00:48:21,135
So what do we want to calculate?

676
00:48:25,010 --> 00:48:32,710
Well, here is what I
would like to calculate.

677
00:48:32,710 --> 00:48:37,930
The ground state of
the first oscillator

678
00:48:37,930 --> 00:48:39,470
is this thing we had.

679
00:48:39,470 --> 00:48:42,990
It's the thing that
has the wave function.

680
00:48:42,990 --> 00:48:48,190
But I want to express it as
a superposition of states

681
00:48:48,190 --> 00:48:52,320
of the second oscillator
because the second oscillator is

682
00:48:52,320 --> 00:48:54,300
what gives you the
new Hamiltonian

683
00:48:54,300 --> 00:48:57,250
and what's going to tell
you how the state is

684
00:48:57,250 --> 00:48:58,770
going to evolve later.

685
00:48:58,770 --> 00:49:06,830
So presumably, this state is
some number times the ground

686
00:49:06,830 --> 00:49:13,980
state of the second
oscillator, plus maybe

687
00:49:13,980 --> 00:49:21,120
some creation operator on
the second vacuum as well

688
00:49:21,120 --> 00:49:23,660
with a constant.

689
00:49:23,660 --> 00:49:34,630
Now, this wave function of
the ground state is even,

690
00:49:34,630 --> 00:49:39,005
and I would expect that
it's a superposition of even

691
00:49:39,005 --> 00:49:43,990
eigenstates of the second
oscillator as well.

692
00:49:43,990 --> 00:49:46,710
And even eigenstates
are things that

693
00:49:46,710 --> 00:49:48,530
have even occupation numbers.

694
00:49:48,530 --> 00:49:51,030
Those are the even
Hermite polynomials.

695
00:49:51,030 --> 00:49:56,070
So presumably, it goes
like this and things

696
00:49:56,070 --> 00:49:58,510
with four oscillators
and things like that.

697
00:49:58,510 --> 00:50:02,970
So what that after is
this sort of expression

698
00:50:02,970 --> 00:50:08,670
of the original state in terms
of energy eigenstates in terms

699
00:50:08,670 --> 00:50:11,365
of anything of the
second oscillator.

700
00:50:14,230 --> 00:50:16,610
So how can we do that?

701
00:50:16,610 --> 00:50:20,270
Well, one thing we
know about this state

702
00:50:20,270 --> 00:50:30,415
is that a1 [? on it ?]
is equal to 0.

703
00:50:35,040 --> 00:50:44,910
It's killed by a1, but that
a1 is an interesting thing.

704
00:50:44,910 --> 00:50:54,240
It's a2 cosh gamma plus
a2 dagger sinch gamma,

705
00:50:54,240 --> 00:50:57,055
and that thing must
kill that state.

706
00:51:00,590 --> 00:51:08,430
So I could at least, if I had
infinite time, put a few terms

707
00:51:08,430 --> 00:51:11,860
and try to calculate more
or less what kind of state

708
00:51:11,860 --> 00:51:16,590
is killed by this strange
combination of creation

709
00:51:16,590 --> 00:51:19,180
and annihilation operators.

710
00:51:19,180 --> 00:51:23,700
You see, we know
a ground state is

711
00:51:23,700 --> 00:51:26,450
killed by the normal
annihilation operator.

712
00:51:26,450 --> 00:51:28,700
That's what this is.

713
00:51:28,700 --> 00:51:31,680
But this operator,
now we know it's

714
00:51:31,680 --> 00:51:34,610
given by this
formula over there,

715
00:51:34,610 --> 00:51:38,940
and then it must kill all that.

716
00:51:38,940 --> 00:51:43,360
So we're faced with a problem
that is in principle fairly

717
00:51:43,360 --> 00:51:49,350
difficult, and you could not
hope for an except solution

718
00:51:49,350 --> 00:51:52,810
unless there's something
very nice going on.

719
00:51:52,810 --> 00:51:55,830
Happily, squeezed states
are still very nice

720
00:51:55,830 --> 00:52:03,550
and tractable states, so
let's see what we can do.

721
00:52:03,550 --> 00:52:06,660
Well, what I'm going to
do is to put an ansatz

722
00:52:06,660 --> 00:52:12,580
for this state based on this
expansion that I had there.

723
00:52:12,580 --> 00:52:21,290
I would say, look, there's going
to be a normalization constant,

724
00:52:21,290 --> 00:52:25,000
but at the end of the
day, we have things acting

725
00:52:25,000 --> 00:52:27,980
on the vacuum, so there's
going to be something very

726
00:52:27,980 --> 00:52:31,560
messy acting on the vacuum of 2.

727
00:52:36,099 --> 00:52:39,960
And what is that going to be?

728
00:52:39,960 --> 00:52:42,310
Well, we've learned about
coherent states that

729
00:52:42,310 --> 00:52:46,540
are exponentials of
oscillators, exponentials

730
00:52:46,540 --> 00:52:50,640
of a's and a daggers added.

731
00:52:50,640 --> 00:52:52,950
So here, we're going
to attempt something

732
00:52:52,950 --> 00:52:55,040
a little more general.

733
00:52:55,040 --> 00:53:04,620
I'll put an exponential minus
1/2, and what should I put?

734
00:53:04,620 --> 00:53:09,700
Well, let's try to be
simple minded still.

735
00:53:09,700 --> 00:53:14,690
It seems to go in even power,
so if we're very lucky,

736
00:53:14,690 --> 00:53:20,170
maybe we can put just an
a2 dagger a2 dagger here,

737
00:53:20,170 --> 00:53:23,510
an exponential something
quadratic in oscillators.

738
00:53:26,160 --> 00:53:28,745
And I don't know what the
coefficient is in front,

739
00:53:28,745 --> 00:53:31,810
and it may depend
on gamma because I

740
00:53:31,810 --> 00:53:33,550
have to solve an
equation with gamma.

741
00:53:33,550 --> 00:53:41,370
So I'll put minus 1/2
f of gamma times that.

742
00:53:43,880 --> 00:53:49,870
And we'll see if
we can solve this.

743
00:53:49,870 --> 00:53:52,000
So what does it
mean to solve it?

744
00:53:52,000 --> 00:53:57,960
Well, it means that it must be
annihilated by this operator.

745
00:53:57,960 --> 00:54:01,485
So our computations with the
creation and annihilation

746
00:54:01,485 --> 00:54:03,845
operators are becoming
more and more complicated.

747
00:54:06,920 --> 00:54:08,960
They look more and
more complicated.

748
00:54:08,960 --> 00:54:11,580
They're really not harder.

749
00:54:11,580 --> 00:54:12,850
Let's see what happens.

750
00:54:12,850 --> 00:54:23,260
So I need now that a2 cosh
gamma plus a2 dagger sinch gamma

751
00:54:23,260 --> 00:54:26,030
kill this state.

752
00:54:26,030 --> 00:54:28,470
So the N is going to go outside.

753
00:54:28,470 --> 00:54:29,210
It's a number.

754
00:54:29,210 --> 00:54:35,190
So acting on e to the
minus 1/2 f of gamma a2

755
00:54:35,190 --> 00:54:41,750
dagger a2 dagger on the
vacuum sub 2, that must be 0.

756
00:54:48,220 --> 00:54:49,805
How does one solve this?

757
00:54:52,360 --> 00:54:56,830
Well, let's see what we have.

758
00:54:56,830 --> 00:54:59,860
Let's see this term.

759
00:54:59,860 --> 00:55:03,700
a2 dagger, good.

760
00:55:03,700 --> 00:55:06,270
a2 dagger commutes
with a2 dagger,

761
00:55:06,270 --> 00:55:10,990
so I can bring the a2 dagger
all the way to the right

762
00:55:10,990 --> 00:55:16,450
and it doesn't kill the vacuum,
so I don't gain anything.

763
00:55:16,450 --> 00:55:19,080
Can be to the right or
to the left because it

764
00:55:19,080 --> 00:55:22,250
commutes with this whole thing,
so I haven't gained anything

765
00:55:22,250 --> 00:55:25,710
if I move it, so false start.

766
00:55:25,710 --> 00:55:28,200
I don't want to move that one.

767
00:55:28,200 --> 00:55:30,760
This one, I want
to leave it here,

768
00:55:30,760 --> 00:55:33,390
and this one
somehow must produce

769
00:55:33,390 --> 00:55:35,230
something that cancels this one.

770
00:55:39,190 --> 00:55:43,340
Now, a2, on the other
hand, is the kind of thing

771
00:55:43,340 --> 00:55:46,860
that always should be dealt with
because this is an annihilator

772
00:55:46,860 --> 00:55:49,200
and that does kill that.

773
00:55:49,200 --> 00:55:52,960
So as it moves along,
it encounters obstacles,

774
00:55:52,960 --> 00:55:58,420
but obstacles are opportunities
because an obstacle means

775
00:55:58,420 --> 00:56:03,530
we're going to get something
that maybe cancels that.

776
00:56:03,530 --> 00:56:06,360
So if it also went through
and killed the vacuum,

777
00:56:06,360 --> 00:56:07,090
we're finished.

778
00:56:07,090 --> 00:56:09,020
This doesn't kill the vacuum.

779
00:56:09,020 --> 00:56:11,300
Happily, it gets stuck here.

780
00:56:11,300 --> 00:56:12,990
Now the thing that
we have to hope

781
00:56:12,990 --> 00:56:17,670
is that we can disentangle
that commutator.

782
00:56:17,670 --> 00:56:20,050
Now, here is a universal thing.

783
00:56:25,000 --> 00:56:28,270
How do I want to write this?

784
00:56:28,270 --> 00:56:31,910
I'm going to write it like this.

785
00:56:31,910 --> 00:56:35,720
I have an a2, a
number, I don't care

786
00:56:35,720 --> 00:56:38,200
about the number, and
a complicated thing,

787
00:56:38,200 --> 00:56:39,640
and a vacuum.

788
00:56:39,640 --> 00:56:45,640
Whenever you have an a,
any operator, and a vacuum,

789
00:56:45,640 --> 00:56:49,380
this is equal to a
commutator with the operator

790
00:56:49,380 --> 00:56:51,290
on the vacuum.

791
00:56:51,290 --> 00:56:57,200
That should be second
nature because this is even

792
00:56:57,200 --> 00:57:02,980
given to that minus oa, but
oa, the a is near to the vacuum

793
00:57:02,980 --> 00:57:04,030
and it kills it.

794
00:57:04,030 --> 00:57:08,020
So whenever you
have an a o vacuum,

795
00:57:08,020 --> 00:57:11,310
you can put the commutator,
so I'll do that here.

796
00:57:11,310 --> 00:57:19,980
So I put a2, the cosh
gamma, I take it out.

797
00:57:19,980 --> 00:57:25,110
I put this whole thing minus
1/2 f a dagger a dagger 2.

798
00:57:27,810 --> 00:57:33,830
This whole thing and the vacuum.

799
00:57:33,830 --> 00:57:36,860
That's the first term.

800
00:57:36,860 --> 00:57:40,970
And the second term, I
have to just copy it.

801
00:57:40,970 --> 00:57:49,270
Sinch gamma a2 dagger
e to the minus 1/2 fa

802
00:57:49,270 --> 00:57:52,800
squared dagger on the vacuum.

803
00:57:52,800 --> 00:57:54,230
All that should be 0.

804
00:57:59,580 --> 00:58:04,460
So what do we get?

805
00:58:04,460 --> 00:58:07,273
Is that commutator
doable or undoable?

806
00:58:10,740 --> 00:58:13,410
It's happily a
simple commutator,

807
00:58:13,410 --> 00:58:16,990
even if it doesn't look
like it, because whenever

808
00:58:16,990 --> 00:58:22,270
you see a commutator like
that, you think A to the B,

809
00:58:22,270 --> 00:58:25,020
and then you know
if you're in luck,

810
00:58:25,020 --> 00:58:30,240
this is just AB e to
the B, and this is true

811
00:58:30,240 --> 00:58:37,800
if AB commutes with
B. So that's what

812
00:58:37,800 --> 00:58:42,350
you must think whenever
you see these things.

813
00:58:42,350 --> 00:58:46,800
Do mind this lucky situation.

814
00:58:46,800 --> 00:58:53,140
Yes, you are, because
with this commutator,

815
00:58:53,140 --> 00:58:55,520
one a will kill an
a dagger, so you

816
00:58:55,520 --> 00:58:58,250
will be left with an dagger.

817
00:58:58,250 --> 00:59:04,340
But a dagger commutes with b,
which is a dagger a dagger.

818
00:59:04,340 --> 00:59:12,360
So AB, A with B is
just add an a dagger up

819
00:59:12,360 --> 00:59:16,030
to a function or a number, and
then a dagger commutes with B

820
00:59:16,030 --> 00:59:17,830
so you are in good shape.

821
00:59:17,830 --> 00:59:18,990
This is true.

822
00:59:18,990 --> 00:59:20,570
So what do we get here?

823
00:59:20,570 --> 00:59:25,600
We get cosh gamma,
and then we just

824
00:59:25,600 --> 00:59:32,465
get the commutator
of a2 with minus 1/2

825
00:59:32,465 --> 00:59:41,512
f a2 dagger a2 dagger times
the whole exponential--

826
00:59:41,512 --> 00:59:47,980
I won't write it-- times
the vacuum plus sinch times

827
00:59:47,980 --> 00:59:52,670
a2 dagger times the whole
exponential times the vacuum.

828
01:00:00,520 --> 01:00:04,210
We have to do this commutator,
but the f doesn't matter.

829
01:00:04,210 --> 01:00:05,220
It's a constant.

830
01:00:05,220 --> 01:00:06,560
It's a function.

831
01:00:06,560 --> 01:00:11,460
No operator in there. a2
with a2 daggers are 1.

832
01:00:11,460 --> 01:00:13,690
There are two of
them, so you get a 2,

833
01:00:13,690 --> 01:00:21,372
and the 1/2 cancels this,
so you get minus cosh gamma

834
01:00:21,372 --> 01:00:32,730
f a2 dagger times
the exponential plus,

835
01:00:32,730 --> 01:00:42,490
from the other term, sinch gamma
a2 dagger times the exponential

836
01:00:42,490 --> 01:00:46,070
on the vacuum equals 0.

837
01:00:46,070 --> 01:00:49,570
And, as promised, we were good.

838
01:00:49,570 --> 01:00:52,450
We get an a2 dagger, a2 dagger.

839
01:00:52,450 --> 01:01:00,176
These two terms cancel if f
is equal to tan hyperbolic

840
01:01:00,176 --> 01:01:05,640
of gamma, which is
sine over cosine

841
01:01:05,640 --> 01:01:08,790
so that these two things cancel.

842
01:01:08,790 --> 01:01:18,590
I can write this, of course, as
minus cosh gamma f plus sinch

843
01:01:18,590 --> 01:01:24,970
gamma a2 dagger, the
exponential, and the vacuum,

844
01:01:24,970 --> 01:01:25,820
equals 0.

845
01:01:25,820 --> 01:01:30,580
So it's just a simple
relation, but there we go.

846
01:01:30,580 --> 01:01:34,060
Tanh gamma is the thing.

847
01:01:34,060 --> 01:01:38,590
Tanh gamma gives you
the answer, and let

848
01:01:38,590 --> 01:01:43,210
me write this state
so that you enjoy it.

849
01:01:47,660 --> 01:01:49,540
Let's see.

850
01:01:49,540 --> 01:01:55,410
The state is just a
fairly interesting thing,

851
01:01:55,410 --> 01:02:03,990
this 01 expressed in the new
Hilbert space of the second

852
01:02:03,990 --> 01:02:10,480
oscillator is some n of gamma
times the exponential of minus

853
01:02:10,480 --> 01:02:17,220
1/2 tangent hyperbolic
of gamma a2 dagger,

854
01:02:17,220 --> 01:02:21,965
a2 dagger on the vacuum sub 2.

855
01:02:25,360 --> 01:02:40,580
And you need the
normalization, n of gamma,

856
01:02:40,580 --> 01:02:42,520
and it will be done.

857
01:02:42,520 --> 01:02:44,990
Now, the normalization,
you may say well, look,

858
01:02:44,990 --> 01:02:47,560
normalizations are good things.

859
01:02:47,560 --> 01:02:51,580
Sometimes, you work without
normalizations and you're OK,

860
01:02:51,580 --> 01:02:55,350
but it turns out that these
normalizations are pretty

861
01:02:55,350 --> 01:02:59,080
useful, and unless you get
them, some calculations

862
01:02:59,080 --> 01:03:03,150
are kind of undoable.

863
01:03:03,150 --> 01:03:06,610
So it's a little
bit of a challenge

864
01:03:06,610 --> 01:03:09,480
to get that normalization.

865
01:03:09,480 --> 01:03:12,220
You can try in several ways.

866
01:03:12,220 --> 01:03:16,860
The most naive way
is to say, well, this

867
01:03:16,860 --> 01:03:21,040
must have unit
norms, so n squared,

868
01:03:21,040 --> 01:03:25,300
and then I take the bra of
this and the ket of that,

869
01:03:25,300 --> 01:03:29,430
so it would be a vacuum,
an exponential of minus 1/2

870
01:03:29,430 --> 01:03:34,430
tangent a a, and an
exponential of minus 1/2

871
01:03:34,430 --> 01:03:37,890
tangent a dagger, a dagger.

872
01:03:37,890 --> 01:03:40,820
Must be 1.

873
01:03:40,820 --> 01:03:43,260
n squared times that.

874
01:03:43,260 --> 01:03:48,240
The problem is that I've never
been able to compute this.

875
01:03:48,240 --> 01:03:51,020
At least it takes a
long time and you get it

876
01:03:51,020 --> 01:03:54,870
by indirect methods, but
getting a number out of this

877
01:03:54,870 --> 01:03:57,170
is painful.

878
01:03:57,170 --> 01:04:01,820
So there's one way of getting
the normalization here

879
01:04:01,820 --> 01:04:04,220
that is not so bad.

880
01:04:04,220 --> 01:04:07,935
It's a little
surprising what you do.

881
01:04:07,935 --> 01:04:09,980
You do the following.

882
01:04:09,980 --> 01:04:11,840
You declare, I'm
going to compute

883
01:04:11,840 --> 01:04:17,825
the overlap of 2, the vacuum
of 2, with the vacuum of 1.

884
01:04:26,930 --> 01:04:36,460
And now, what is this,
n gamma vacuum of 2

885
01:04:36,460 --> 01:04:45,270
here, e to the minus 1/2 tanh
gamma a2 dagger a2 dagger

886
01:04:45,270 --> 01:04:47,800
vacuum of 2.

887
01:04:47,800 --> 01:04:53,690
How difficult is it to
compute this inner product?

888
01:04:53,690 --> 01:04:54,950
AUDIENCE: [INAUDIBLE].

889
01:04:54,950 --> 01:04:56,270
BARTON ZWIEBACH: Sorry?

890
01:04:56,270 --> 01:04:57,270
AUDIENCE: Not difficult.

891
01:04:57,270 --> 01:04:58,561
BARTON ZWIEBACH: Not difficult.

892
01:04:58,561 --> 01:05:00,534
What is it?

893
01:05:00,534 --> 01:05:01,520
AUDIENCE: [INAUDIBLE]?

894
01:05:01,520 --> 01:05:03,694
BARTON ZWIEBACH:
Yeah, that thing.

895
01:05:03,694 --> 01:05:06,283
AUDIENCE: e to the
negative 1/2 tanh gamma.

896
01:05:09,061 --> 01:05:10,450
It's 1.

897
01:05:10,450 --> 01:05:11,923
BARTON ZWIEBACH: Sorry?

898
01:05:11,923 --> 01:05:14,445
AUDIENCE: I mean, you
multiply the a2 dagger right

899
01:05:14,445 --> 01:05:17,126
across to the left
hand side of the ket.

900
01:05:17,126 --> 01:05:19,740
BARTON ZWIEBACH: Yeah,
you're saying it, indeed.

901
01:05:19,740 --> 01:05:23,240
Look, this thing is
as simple as can be.

902
01:05:23,240 --> 01:05:25,660
This is just 1.

903
01:05:25,660 --> 01:05:27,580
Why is that so?

904
01:05:27,580 --> 01:05:32,320
You expand the exponential,
and you have 1 plus things,

905
01:05:32,320 --> 01:05:34,580
but all the things
have a daggers.

906
01:05:34,580 --> 01:05:36,960
Now, a daggers
don't kill this 1,

907
01:05:36,960 --> 01:05:39,440
but they killed the
other 1 on the left,

908
01:05:39,440 --> 01:05:43,380
and there's nothing obstructing
them from reaching the left,

909
01:05:43,380 --> 01:05:46,810
so this is 1.

910
01:05:46,810 --> 01:05:48,960
It's completely
different from this one

911
01:05:48,960 --> 01:05:52,790
because if you expand this one,
the a daggers kill the thing

912
01:05:52,790 --> 01:05:55,780
but there's lots
of a's to the left.

913
01:05:55,780 --> 01:05:57,770
And the a's want to
get here, but there's

914
01:05:57,770 --> 01:06:00,980
lots of a daggers to the
right, so this is hard,

915
01:06:00,980 --> 01:06:02,170
but this is easy.

916
01:06:02,170 --> 01:06:12,880
So n of gamma is 0 2 0 1.

917
01:06:12,880 --> 01:06:15,060
But what is that?

918
01:06:15,060 --> 01:06:20,300
If you introduce a complete
set of position states, zx,

919
01:06:20,300 --> 01:06:29,040
This is 0 2 x x 0 1.

920
01:06:29,040 --> 01:06:38,050
This one is the ground
state wave function

921
01:06:38,050 --> 01:06:41,980
of the first
Hamiltonian, and this

922
01:06:41,980 --> 01:06:44,330
is the start of the
ground state wave

923
01:06:44,330 --> 01:06:47,280
function of the
second Hamiltonian.

924
01:06:52,580 --> 01:06:56,082
And those you know
because you know m, omega.

925
01:06:56,082 --> 01:06:57,790
You know the ground
state wave functions,

926
01:06:57,790 --> 01:06:59,570
so this integral can be done.

927
01:06:59,570 --> 01:07:04,510
So this whole normalization
is given by this integral,

928
01:07:04,510 --> 01:07:11,370
and this integral gives you 1
over square root of cosh gamma.

929
01:07:11,370 --> 01:07:13,900
That interval takes
a few lines to make,

930
01:07:13,900 --> 01:07:17,115
but the end result is there.

931
01:07:19,670 --> 01:07:22,540
So you got your coherent states.

932
01:07:28,480 --> 01:07:32,340
You got now the squeezed
state completely normalized,

933
01:07:32,340 --> 01:07:34,550
so let's write it out.

934
01:07:34,550 --> 01:07:44,690
0 1 is equal to 1 over square
root of cosh gamma exponential

935
01:07:44,690 --> 01:07:55,710
of minus 1/2 tanh gamma a2
dagger a2 dagger on the vacuum

936
01:07:55,710 --> 01:07:56,400
sub 2.

937
01:08:00,030 --> 01:08:00,730
Wow.

938
01:08:00,730 --> 01:08:01,650
That's it.

939
01:08:01,650 --> 01:08:07,300
That's a squeeze state that
has been squeezed in such a way

940
01:08:07,300 --> 01:08:10,570
that the squeezing
parameter appears here

941
01:08:10,570 --> 01:08:13,660
in the exponential.

942
01:08:13,660 --> 01:08:15,710
Now, this is the
way we got to it,

943
01:08:15,710 --> 01:08:20,359
but now I wanted to just think
of it independently, just

944
01:08:20,359 --> 01:08:21,319
from the beginning.

945
01:08:21,319 --> 01:08:29,120
If you had a Hamiltonian,
this is an interesting state

946
01:08:29,120 --> 01:08:33,910
all in itself because
it is a squeezed state.

947
01:08:33,910 --> 01:08:39,850
It's a Gaussian, but of the
wrong shape for this system.

948
01:08:39,850 --> 01:08:43,770
This is a Gaussian of the
right shape for system two.

949
01:08:43,770 --> 01:08:45,830
But once you put all
these oscillators,

950
01:08:45,830 --> 01:08:49,350
it's not anymore a
Gaussian of the right type.

951
01:08:49,350 --> 01:08:51,439
It's a squeezed Gaussian.

952
01:08:51,439 --> 01:08:55,430
So if we forget about
this system one,

953
01:08:55,430 --> 01:09:00,229
let me write this thing from
the beginning and say like this.

954
01:09:00,229 --> 01:09:04,710
We have a Hamiltonian,
we have a ground state,

955
01:09:04,710 --> 01:09:12,240
we have m and omega, and
we have a and a dagger.

956
01:09:12,240 --> 01:09:19,380
Let's just define what we call
the squeezed vacuum, vacuum sub

957
01:09:19,380 --> 01:09:23,470
gamma, to be
precisely this thing.

958
01:09:23,470 --> 01:09:31,380
1 over square root of cosh
gamma exponential of minus 1/2

959
01:09:31,380 --> 01:09:39,620
tanh gamma a dagger
a dagger, not 2

960
01:09:39,620 --> 01:09:43,670
anymore because we have
just a single system.

961
01:09:43,670 --> 01:09:47,100
A single system, the
ground state, and now we've

962
01:09:47,100 --> 01:09:51,359
defined this state, which
is what we had there before,

963
01:09:51,359 --> 01:09:53,790
but we don't think of
it anymore as, oh, it

964
01:09:53,790 --> 01:09:55,770
came from some
other Hamiltonian,

965
01:09:55,770 --> 01:09:59,650
but rather, this is
a state on its own.

966
01:09:59,650 --> 01:10:03,060
It's a squeezed vacuum state.

967
01:10:03,060 --> 01:10:08,220
And from the computations
that we did here,

968
01:10:08,220 --> 01:10:14,340
the delta x for this state would
be e to the minus gamma h bar m

969
01:10:14,340 --> 01:10:15,850
omega.

970
01:10:15,850 --> 01:10:21,810
and m omega over here.

971
01:10:21,810 --> 01:10:25,940
So these are these,
and you don't

972
01:10:25,940 --> 01:10:27,860
need to know what gamma is.

973
01:10:27,860 --> 01:10:31,320
That's a number that
somebody chose for you.

974
01:10:31,320 --> 01:10:34,730
Any number that
you want is gamma,

975
01:10:34,730 --> 01:10:38,230
and therefore, you use
it to squeeze the state.

976
01:10:38,230 --> 01:10:40,280
And that's what you've achieved.

977
01:10:40,280 --> 01:10:43,720
So you have a Hamiltonian
of a harmonic oscillator.

978
01:10:43,720 --> 01:10:45,690
You can construct the vacuum.

979
01:10:45,690 --> 01:10:48,030
You know how to
construct coherent states

980
01:10:48,030 --> 01:10:49,740
by acting on the vacuum.

981
01:10:49,740 --> 01:10:53,430
Now you know how to construct
squeezed states, states

982
01:10:53,430 --> 01:10:56,170
in which the expectation
values do those things.

983
01:11:01,350 --> 01:11:06,000
We had a very nice formula
where we began the lecture today

984
01:11:06,000 --> 01:11:14,110
in which the coherent
state was just

985
01:11:14,110 --> 01:11:18,080
a unitary operator
acting on the vacuum.

986
01:11:18,080 --> 01:11:21,340
Now, we made sure
to normalize this,

987
01:11:21,340 --> 01:11:24,760
so we did check in
this calculation

988
01:11:24,760 --> 01:11:30,450
that o gamma 0
gamma is equal to 1.

989
01:11:30,450 --> 01:11:35,830
So this thing must
come from the action

990
01:11:35,830 --> 01:11:41,080
of some unitary operator
acting on the vacuum.

991
01:11:41,080 --> 01:11:46,420
Which is that unitary operator
that acts on the vacuum

992
01:11:46,420 --> 01:11:49,250
and gives you that?

993
01:11:49,250 --> 01:11:51,075
Not so easy to find.

994
01:11:51,075 --> 01:11:53,470
All the computations here
are a little challenging,

995
01:11:53,470 --> 01:11:54,750
as you've seen.

996
01:11:54,750 --> 01:11:58,570
But here's the answer.

997
01:11:58,570 --> 01:12:09,300
Cosh gamma e to the exponential
of minus 1/2 tanh gamma

998
01:12:09,300 --> 01:12:14,380
a dagger a dagger
should be something

999
01:12:14,380 --> 01:12:19,820
like an e to the what?

1000
01:12:19,820 --> 01:12:23,630
Should be something
like e to the a dagger

1001
01:12:23,630 --> 01:12:30,290
a dagger minus aa
acting on the vacuum.

1002
01:12:30,290 --> 01:12:31,260
Why?

1003
01:12:31,260 --> 01:12:36,280
Because certainly, the aa's
are going to disappear,

1004
01:12:36,280 --> 01:12:41,820
and you're going to get
products of this one squared.

1005
01:12:41,820 --> 01:12:48,010
And this is anti-Hermitian,
so that operator is unitary,

1006
01:12:48,010 --> 01:12:52,510
but I now must put the
gamma somewhere there.

1007
01:12:52,510 --> 01:12:58,635
So what should I put here in
order to get that to work?

1008
01:13:06,710 --> 01:13:11,790
Well, it's maybe
something you can

1009
01:13:11,790 --> 01:13:17,700
try by assuming gamma is very
small and expanding both sides,

1010
01:13:17,700 --> 01:13:21,270
or finding a differential
equation, or doing things,

1011
01:13:21,270 --> 01:13:25,930
but the answer is
incredibly simple.

1012
01:13:25,930 --> 01:13:32,026
It's e to the minus
just gamma over 2.

1013
01:13:32,026 --> 01:13:33,360
That's it.

1014
01:13:33,360 --> 01:13:37,220
Gamma appears here,
and by the time

1015
01:13:37,220 --> 01:13:41,730
you reorder this
quadratic form--

1016
01:13:41,730 --> 01:13:46,090
you see, what you have
to do here is expand,

1017
01:13:46,090 --> 01:13:47,830
and then you have
powers of these,

1018
01:13:47,830 --> 01:13:51,300
and then you have to bring all
the annihilators to the right

1019
01:13:51,300 --> 01:13:52,370
and kill them.

1020
01:13:52,370 --> 01:13:56,330
And then you have a power
series in squares of this thing.

1021
01:13:56,330 --> 01:13:59,780
That will reassemble
into this exponential.

1022
01:13:59,780 --> 01:14:01,540
It's almost a miracle
that something

1023
01:14:01,540 --> 01:14:07,120
like that could happen,
but it does happen.

1024
01:14:07,120 --> 01:14:10,110
And it's a very interesting
calculation, actually,

1025
01:14:10,110 --> 01:14:10,740
to do that.

1026
01:14:10,740 --> 01:14:13,340
We don't do it in the course.

1027
01:14:13,340 --> 01:14:16,770
I may post some pages that
I did once this computation.

1028
01:14:20,260 --> 01:14:26,350
And that is a nice operator.

1029
01:14:26,350 --> 01:14:29,056
We call it the
squeezing operator.

1030
01:14:33,980 --> 01:14:40,120
So s of gamma is a unitary
operator, s of gamma.

1031
01:14:40,120 --> 01:14:44,600
The squeezed state of
0 gamma is equal to s

1032
01:14:44,600 --> 01:14:49,510
of gamma on the vacuum
where s of gamma

1033
01:14:49,510 --> 01:14:54,560
is equal to e to the minus
gamma over 2 a dagger a dagger

1034
01:14:54,560 --> 01:14:58,370
minus aa, that operator.

1035
01:14:58,370 --> 01:15:02,585
It's a unitary operator
and it does the squeezing.

1036
01:15:05,590 --> 01:15:14,660
Actually, once you have squeezed
states, you can do more things,

1037
01:15:14,660 --> 01:15:18,330
and you can squeeze
and then translate.

1038
01:15:18,330 --> 01:15:20,280
Those are the most
general states

1039
01:15:20,280 --> 01:15:24,280
that people use
in quantum optics.

1040
01:15:24,280 --> 01:15:31,830
So you take a vacuum, you
squeeze it with s of gamma,

1041
01:15:31,830 --> 01:15:36,920
and then you translate
it with v of alpha.

1042
01:15:36,920 --> 01:15:43,540
And this is the state,
alpha gamma, squeeze factor,

1043
01:15:43,540 --> 01:15:46,090
translation factor.

1044
01:15:46,090 --> 01:15:50,490
One picture of that
is in our alpha plane.

1045
01:15:50,490 --> 01:15:54,680
You take the vacuum that
is some spherical ball here

1046
01:15:54,680 --> 01:15:59,860
in the x expectation
value, p expectation value.

1047
01:15:59,860 --> 01:16:00,980
You squeeze it.

1048
01:16:00,980 --> 01:16:06,130
You might decide, I don't
want to have too much delta x,

1049
01:16:06,130 --> 01:16:08,950
so you squeeze it and you
produce something like this.

1050
01:16:13,530 --> 01:16:17,630
That's a squeezed vacuum
by the time you apply this.

1051
01:16:17,630 --> 01:16:22,910
And then you do the alpha,
and you translate it out,

1052
01:16:22,910 --> 01:16:25,140
and this state is now
going to start rotating

1053
01:16:25,140 --> 01:16:28,781
and doing all kinds of motion.

1054
01:16:28,781 --> 01:16:31,410
It's pretty practical stuff.

1055
01:16:31,410 --> 01:16:37,050
Actually, some of you
are taking junior lab,

1056
01:16:37,050 --> 01:16:40,320
and the person that works
a lot there in junior lab

1057
01:16:40,320 --> 01:16:45,060
is Nergis Mavalvala, and she
does gravity wave detection,

1058
01:16:45,060 --> 01:16:47,850
and squeezed states has
been exactly what she's

1059
01:16:47,850 --> 01:16:49,350
been working.

1060
01:16:49,350 --> 01:16:51,890
In order to minimize
displacements

1061
01:16:51,890 --> 01:16:54,960
in the gravity wave
detectors, they

1062
01:16:54,960 --> 01:16:59,200
have a squeeze vacuum state
injected into the detector

1063
01:16:59,200 --> 01:17:02,950
to make the harmonic
oscillator that

1064
01:17:02,950 --> 01:17:07,320
represents the mirror stabilize
its uncertainty in position

1065
01:17:07,320 --> 01:17:09,400
to the maximum possible.

1066
01:17:09,400 --> 01:17:11,570
There's a whole
fabulous technique

1067
01:17:11,570 --> 01:17:14,770
that people use with
the squeezed states.

1068
01:17:14,770 --> 01:17:16,350
Now, the squeezed
states allow you

1069
01:17:16,350 --> 01:17:22,350
to construct some states that
seemed to us that they were

1070
01:17:22,350 --> 01:17:27,710
pretty strange and that we never
had good formulas for them.

1071
01:17:27,710 --> 01:17:30,040
So that's how I want to
conclude the lecture.

1072
01:17:30,040 --> 01:17:34,290
I will leave photon
states for next time,

1073
01:17:34,290 --> 01:17:37,470
but I want to discuss
one more application

1074
01:17:37,470 --> 01:17:43,080
of the squeezed states,
and this comes from limits.

1075
01:17:47,410 --> 01:17:54,740
So here is your squeezed
state, e to the minus gamma.

1076
01:17:54,740 --> 01:17:59,110
So let's squeeze the
state to the end.

1077
01:17:59,110 --> 01:18:01,066
Take gamma to go to infinity.

1078
01:18:04,690 --> 01:18:06,730
What happens to
the squeezed state?

1079
01:18:06,730 --> 01:18:11,420
So you're narrowing out the
ground state in position space

1080
01:18:11,420 --> 01:18:13,310
to the maximum possible.

1081
01:18:13,310 --> 01:18:15,420
What happens to the state?

1082
01:18:15,420 --> 01:18:18,725
Well, it goes a little singular,
but not terribly singular.

1083
01:18:23,960 --> 01:18:27,170
Gamma is going to
infinity, so cosh

1084
01:18:27,170 --> 01:18:30,150
is going to infinity as well.

1085
01:18:30,150 --> 01:18:36,230
So the state is going kind
of to 0, but 0 sub infinity.

1086
01:18:36,230 --> 01:18:39,690
It's proportional, but
the exponential is good.

1087
01:18:39,690 --> 01:18:45,480
Exponential of minus
1/2 tangent of gamma

1088
01:18:45,480 --> 01:18:49,850
as gamma goes to
infinity is just 1.

1089
01:18:49,850 --> 01:18:58,540
And this is a dagger a
dagger on the vacuum.

1090
01:18:58,540 --> 01:19:04,135
This state is in almost
terrible danger to be infinite.

1091
01:19:04,135 --> 01:19:07,260
If you try to find
its wave function,

1092
01:19:07,260 --> 01:19:09,560
you're not going to be
able to normalize it.

1093
01:19:09,560 --> 01:19:11,890
You've reached the
end of the road

1094
01:19:11,890 --> 01:19:15,340
kind of thing because of this.

1095
01:19:15,340 --> 01:19:16,900
Gamma goes to infinity.

1096
01:19:16,900 --> 01:19:19,440
This is going to
be infinite here

1097
01:19:19,440 --> 01:19:21,270
because this state,
if you compute

1098
01:19:21,270 --> 01:19:24,650
its overlap with
itself, is blowing up.

1099
01:19:24,650 --> 01:19:26,670
And here, you see
the niceness of this.

1100
01:19:26,670 --> 01:19:30,000
It also suggests that gamma
can go from plus infinity

1101
01:19:30,000 --> 01:19:33,310
to minus infinity, and
that's a natural thing.

1102
01:19:33,310 --> 01:19:37,630
Nevertheless, here, it goes
from plus 1 to minus 1.

1103
01:19:37,630 --> 01:19:40,900
If you had a number 3
here, this is a state

1104
01:19:40,900 --> 01:19:43,600
that blows much worse
than the worst delta

1105
01:19:43,600 --> 01:19:47,570
function or derivative or
square that you've ever had.

1106
01:19:47,570 --> 01:19:51,290
It's just unbelievably
divergent because it just

1107
01:19:51,290 --> 01:19:53,160
can't exist, this state.

1108
01:19:53,160 --> 01:19:58,100
You're going beyond infinity
here to go behind this thing.

1109
01:19:58,100 --> 01:20:00,430
So it's just pretty
much impossible.

1110
01:20:00,430 --> 01:20:03,250
So the limit is
states are reasonable

1111
01:20:03,250 --> 01:20:07,400
as long as this quadratic
form goes from minus 1 to 1.

1112
01:20:07,400 --> 01:20:11,660
And when you go to 1, you get
this, and what should this be?

1113
01:20:11,660 --> 01:20:14,940
This should be the
wave function I

1114
01:20:14,940 --> 01:20:17,240
associated to a delta function.

1115
01:20:17,240 --> 01:20:24,490
This would be the position
state, x equals 0.

1116
01:20:24,490 --> 01:20:27,250
Roughly, it's a delta function.

1117
01:20:27,250 --> 01:20:31,040
And indeed, if you
act with x on it,

1118
01:20:31,040 --> 01:20:33,620
x, remember, is a plus a dagger.

1119
01:20:37,640 --> 01:20:38,750
Act on this exponential.

1120
01:20:42,330 --> 01:20:45,180
Now, do you remember
how to do that?

1121
01:20:45,180 --> 01:20:50,330
This a dagger doesn't do
anything but the a goes here,

1122
01:20:50,330 --> 01:20:52,060
and it's a trivial commutator.

1123
01:20:52,060 --> 01:20:55,400
You get minus a dagger.

1124
01:20:55,400 --> 01:20:58,800
So it actually kills
this and gives you 0.

1125
01:20:58,800 --> 01:21:03,455
So in fact, the exposition
operator acting on here

1126
01:21:03,455 --> 01:21:06,690
gives you 0.

1127
01:21:06,690 --> 01:21:11,590
It looks like it is really
the state x equals 0.

1128
01:21:11,590 --> 01:21:17,210
If you go the other way
around, and you take gamma

1129
01:21:17,210 --> 01:21:23,380
to be minus infinity, the
only thing that changes here

1130
01:21:23,380 --> 01:21:24,110
is the sign.

1131
01:21:24,110 --> 01:21:32,430
So this is like the delta of
x, or the x equals 0 state.

1132
01:21:32,430 --> 01:21:40,410
And if you take 0 minus
infinity, goes like x minus 1/2

1133
01:21:40,410 --> 01:21:47,510
plus 1/2 a dagger a
dagger on the vacuum.

1134
01:21:47,510 --> 01:21:51,430
And this state is a delta
function in momentum.

1135
01:21:51,430 --> 01:21:55,950
It's the momentum
state p equals 0.

1136
01:21:55,950 --> 01:21:57,210
Why?

1137
01:21:57,210 --> 01:22:01,560
Because gamma is going
to minus infinity.

1138
01:22:01,560 --> 01:22:05,380
The uncertainty of
momentum is going to 0.

1139
01:22:05,380 --> 01:22:09,220
And therefore, indeed, if
you act with the momentum

1140
01:22:09,220 --> 01:22:11,570
operator on this
state, it's like acting

1141
01:22:11,570 --> 01:22:15,130
with a minus a
dagger, and you've

1142
01:22:15,130 --> 01:22:18,160
changed the sign of this, but
you've changed the sign here,

1143
01:22:18,160 --> 01:22:20,950
so it also kills this state.

1144
01:22:20,950 --> 01:22:26,240
So it looks like we can really
construct position and momentum

1145
01:22:26,240 --> 01:22:29,000
eigenstates now with
squeezed states,

1146
01:22:29,000 --> 01:22:31,200
and that's what they
are supposed to be.

1147
01:22:31,200 --> 01:22:34,690
A squeezed state is something
that has been squeezed enough

1148
01:22:34,690 --> 01:22:37,060
that you can get
a delta function.

1149
01:22:37,060 --> 01:22:39,490
So how do you finish
that construction?

1150
01:22:39,490 --> 01:22:40,240
Here is the claim.

1151
01:22:43,860 --> 01:22:53,390
Square root of 2 m omega over
h bar x a dagger minus 1/2

1152
01:22:53,390 --> 01:22:59,240
a dagger a dagger
acting on the vacuum.

1153
01:22:59,240 --> 01:23:03,770
This is the claim, that this
is the x position state.

1154
01:23:03,770 --> 01:23:08,150
So basically, you have
to squeeze first and then

1155
01:23:08,150 --> 01:23:13,420
translate this thing
to the exposition.

1156
01:23:13,420 --> 01:23:16,180
So how do you check this?

1157
01:23:16,180 --> 01:23:19,320
Well, you should check
that the x operator, which

1158
01:23:19,320 --> 01:23:27,440
is something times a plus a
dagger acting on this thing

1159
01:23:27,440 --> 01:23:29,420
gives you little x.

1160
01:23:29,420 --> 01:23:32,670
So you should have that
x operator on this thing

1161
01:23:32,670 --> 01:23:35,910
gives you x x.

1162
01:23:35,910 --> 01:23:39,720
And that is going to work out
because the a dagger is going

1163
01:23:39,720 --> 01:23:42,616
to sit here and it's going
to get canceled with the a

1164
01:23:42,616 --> 01:23:46,700
with this, but the
a with this part

1165
01:23:46,700 --> 01:23:49,810
is just going to bring down
an x with the right factor.

1166
01:23:49,810 --> 01:23:54,180
So this state, which is a
squeezed state and a little bit

1167
01:23:54,180 --> 01:24:00,110
of a coherent state as well,
is producing the position

1168
01:24:00,110 --> 01:24:01,830
eigenstate.

1169
01:24:01,830 --> 01:24:04,100
In the harmonic
oscillator, you can really

1170
01:24:04,100 --> 01:24:05,900
construct the
position eigenstate

1171
01:24:05,900 --> 01:24:08,900
and you can calculate
the normalization.

1172
01:24:08,900 --> 01:24:13,290
The normalization comes out
to be a rather simple thing.

1173
01:24:13,290 --> 01:24:17,190
So at the end of the day,
the position eigenstate

1174
01:24:17,190 --> 01:24:27,720
is m omega over pi h bar to the
1/4 e to the minus m omega x

1175
01:24:27,720 --> 01:24:30,580
squared over 2h bar.

1176
01:24:30,580 --> 01:24:33,570
And this whole
exponential, square root

1177
01:24:33,570 --> 01:24:41,150
of 2 m omega over h bar,
x a dagger minus 1/2

1178
01:24:41,150 --> 01:24:45,050
a dagger a dagger
acting on the vacuum.

1179
01:24:50,850 --> 01:24:55,610
So your basis of creation
and annihilation operators

1180
01:24:55,610 --> 01:24:58,930
on the harmonic oscillator
is flexible enough

1181
01:24:58,930 --> 01:25:02,320
to allow for a concrete
description of your position

1182
01:25:02,320 --> 01:25:06,470
eigenstates, and a
tractable one as well.

1183
01:25:06,470 --> 01:25:16,320
And that's the extreme
limit of squeezing, together

1184
01:25:16,320 --> 01:25:21,090
with some little bit of
coherent displacement.

1185
01:25:21,090 --> 01:25:23,210
Next time, we'll do
our photon states

1186
01:25:23,210 --> 01:25:25,790
and we'll illustrate the
ideas of both coherent

1187
01:25:25,790 --> 01:25:28,620
and squeezed states
at the same time.