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PROFESSOR: All right.

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00:00:22,890 --> 00:00:27,800
So, this homework
that is due on Friday

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00:00:27,800 --> 00:00:31,000
contains some questions on
the harmonic oscillator.

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00:00:31,000 --> 00:00:34,630
And the harmonic oscillator
is awfully important.

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00:00:34,630 --> 00:00:36,240
I gave you notes on that.

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00:00:36,240 --> 00:00:39,930
And I want to use about
half of the lecture,

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00:00:39,930 --> 00:00:42,440
perhaps a little
less, to go over

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00:00:42,440 --> 00:00:45,470
some of those points
in the notes concerning

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00:00:45,470 --> 00:00:47,490
the harmonic oscillator.

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00:00:47,490 --> 00:00:52,150
After that, we're going
to begin, essentially,

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00:00:52,150 --> 00:00:55,140
our study of dynamics.

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00:00:55,140 --> 00:00:58,560
And we will give the revision,
today, of the Schrodinger

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00:00:58,560 --> 00:01:00,990
equation.

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00:01:00,990 --> 00:01:06,410
It's the way Dirac, in his
textbook on quantum mechanics,

22
00:01:06,410 --> 00:01:08,510
presents the
Schrodinger equation.

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00:01:08,510 --> 00:01:12,000
I think it's actually,
extremely insightful.

24
00:01:12,000 --> 00:01:13,890
It's probably not
the way you should

25
00:01:13,890 --> 00:01:17,180
see it the first
time in your life.

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00:01:17,180 --> 00:01:19,290
But it's a good way
to think about it.

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00:01:19,290 --> 00:01:23,280
And it will give
you a nice feeling

28
00:01:23,280 --> 00:01:25,540
that this Schrodinger
equation is something

29
00:01:25,540 --> 00:01:28,190
so fundamental and
so basic that it

30
00:01:28,190 --> 00:01:32,563
would be very hard to
change or do anything to it

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00:01:32,563 --> 00:01:34,600
and tinker with it.

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00:01:34,600 --> 00:01:40,280
It's a rather complete theory
and quite beautiful [? idea. ?]

33
00:01:40,280 --> 00:01:42,370
So we begin with the
harmonic oscillator.

34
00:01:53,240 --> 00:01:56,150
And this will be a bit quick.

35
00:01:56,150 --> 00:01:59,190
I won't go over every detail.

36
00:01:59,190 --> 00:02:00,190
You have the notes.

37
00:02:00,190 --> 00:02:03,640
I think that's pretty
much all you need to know.

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00:02:03,640 --> 00:02:07,880
So we'll leave it at that.

39
00:02:07,880 --> 00:02:11,740
So the harmonic oscillator
is a quantum system.

40
00:02:11,740 --> 00:02:14,740
And as quantum
systems go, they're

41
00:02:14,740 --> 00:02:17,480
inspired by classical systems.

42
00:02:17,480 --> 00:02:21,030
And the classical system
is very famous here.

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00:02:21,030 --> 00:02:24,460
It's the system in
which, for example, you

44
00:02:24,460 --> 00:02:26,130
have a mass and a spring.

45
00:02:26,130 --> 00:02:29,950
And it does an oscillation for
which the energy is written

46
00:02:29,950 --> 00:02:37,730
as p squared over 2m plus 1/2
m, omega squared, x squared.

47
00:02:37,730 --> 00:02:42,220
And m omega squared is
sometimes called k squared,

48
00:02:42,220 --> 00:02:43,200
the spring constant.

49
00:02:46,350 --> 00:02:50,010
And you are supposed to do
quantum mechanics with this.

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00:02:50,010 --> 00:02:55,460
So nobody can tell you this is
what the harmonic oscillators

51
00:02:55,460 --> 00:02:56,580
in quantum mechanics.

52
00:02:56,580 --> 00:02:58,240
You have to define it.

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00:02:58,240 --> 00:03:00,920
But since there's
only one logical way

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00:03:00,920 --> 00:03:03,790
to define the quantum
system, everybody

55
00:03:03,790 --> 00:03:08,130
agrees on what the harmonic
oscillator quantum system is.

56
00:03:08,130 --> 00:03:10,770
Basically, you use
the inspiration

57
00:03:10,770 --> 00:03:13,430
of the classical system
and declare, well,

58
00:03:13,430 --> 00:03:16,320
energy will be the
Hamiltonian operator.

59
00:03:16,320 --> 00:03:20,350
p will be the momentum operator.

60
00:03:20,350 --> 00:03:26,410
And x will be the
position operator.

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00:03:26,410 --> 00:03:29,270
And given that
these are operators,

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00:03:29,270 --> 00:03:34,980
will have a basic commutation
relation between x and p

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00:03:34,980 --> 00:03:37,190
being equal to i h-bar.

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00:03:37,190 --> 00:03:38,690
And that's it.

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00:03:38,690 --> 00:03:40,790
This is your quantum system.

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00:03:48,840 --> 00:03:52,180
Hamiltonian is--
the set of operators

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00:03:52,180 --> 00:03:56,720
that are relevant for this are
the x the p, and the energy

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00:03:56,720 --> 00:04:01,450
operator that will
control the dynamics.

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00:04:01,450 --> 00:04:06,990
You know also you should specify
a vector space, the vector

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00:04:06,990 --> 00:04:08,490
space where this acts.

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00:04:08,490 --> 00:04:13,420
And this will be complex
functions on the real line.

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00:04:13,420 --> 00:04:21,430
So this will act
in wave functions

73
00:04:21,430 --> 00:04:26,715
that define the vector space,
sometimes called Hilbert space.

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00:04:31,136 --> 00:04:37,790
It will be the set of integrable
functions on the real line,

75
00:04:37,790 --> 00:04:46,675
so complex functions
on the real line.

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00:04:50,640 --> 00:04:54,140
These are your wave functions,
a set of states of the theory.

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00:04:54,140 --> 00:04:57,640
All these complex functions
on the real line work.

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00:04:57,640 --> 00:05:00,080
I won't try to be more precise.

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00:05:00,080 --> 00:05:02,300
You could say they're
square integrable.

80
00:05:02,300 --> 00:05:04,233
That for sure is necessary.

81
00:05:08,070 --> 00:05:10,210
And we'll leave it at that.

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00:05:10,210 --> 00:05:12,720
Now you have to
solve this problem.

83
00:05:12,720 --> 00:05:17,730
And in 804, we discussed this by
using the differential equation

84
00:05:17,730 --> 00:05:22,070
and then through the creation
annihilation operators.

85
00:05:22,070 --> 00:05:24,440
And we're going
do it, today, just

86
00:05:24,440 --> 00:05:26,880
through creation and
annihilation operators.

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00:05:26,880 --> 00:05:30,940
But we want to
emphasize something

88
00:05:30,940 --> 00:05:35,460
about this Hamiltonian and
something very general, which

89
00:05:35,460 --> 00:05:38,500
is that you can right
the Hamiltonian as say

90
00:05:38,500 --> 00:05:44,580
1/2m, omega squared, x squared.

91
00:05:44,580 --> 00:05:50,530
And then you have plus
p squared, m squared,

92
00:05:50,530 --> 00:05:51,525
omega squared.

93
00:05:55,060 --> 00:05:58,760
And a great solution
to the problem

94
00:05:58,760 --> 00:06:03,910
of solving the Hamiltonian--
and it's the best you could ever

95
00:06:03,910 --> 00:06:08,040
hope-- is what is
called the factorization

96
00:06:08,040 --> 00:06:11,720
of the Hamiltonian, in
which you would manage

97
00:06:11,720 --> 00:06:17,001
to write this Hamiltonian as
some operator times the dagger

98
00:06:17,001 --> 00:06:17,500
operator.

99
00:06:21,410 --> 00:06:24,190
So this is the ideal situation.

100
00:06:24,190 --> 00:06:27,240
It's just wonderful,
as you will see,

101
00:06:27,240 --> 00:06:28,975
if you can manage to do that.

102
00:06:33,130 --> 00:06:36,460
If you could manage to
do this factorization,

103
00:06:36,460 --> 00:06:39,400
you would know immediately
what is the ground state

104
00:06:39,400 --> 00:06:42,930
energy, how low can
it go, something

105
00:06:42,930 --> 00:06:44,310
about the Hamiltonian.

106
00:06:44,310 --> 00:06:47,060
You're way on your way
of solving the problem.

107
00:06:47,060 --> 00:06:48,920
If you could just factorize it.

108
00:06:48,920 --> 00:06:49,969
Yes?

109
00:06:49,969 --> 00:06:52,548
AUDIENCE: [INAUDIBLE] if
you could just factorize it

110
00:06:52,548 --> 00:06:57,540
in terms of v and v
instead of v dagger and v?

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00:06:57,540 --> 00:07:01,561
PROFESSOR: You want to factorize
in which way instead of that?

112
00:07:01,561 --> 00:07:03,686
AUDIENCE: Would it be
helpful, if it were possible,

113
00:07:03,686 --> 00:07:07,530
to factor it in terms of v
times v instead of v dagger?

114
00:07:07,530 --> 00:07:09,730
PROFESSOR: No, no, I
want, really, v dagger.

115
00:07:09,730 --> 00:07:14,010
I don't want v v. That
that's not so good.

116
00:07:14,010 --> 00:07:17,490
I want that this factorization
has a v dagger there.

117
00:07:17,490 --> 00:07:21,350
It will make things
much, much better.

118
00:07:21,350 --> 00:07:23,860
So how can you achieve that?

119
00:07:23,860 --> 00:07:26,750
Well, it almost looks possible.

120
00:07:26,750 --> 00:07:31,860
If you have something like this,
like a squared plus b squared,

121
00:07:31,860 --> 00:07:36,640
you write it as a minus
ib times a plus ib.

122
00:07:41,150 --> 00:07:43,180
And that works out.

123
00:07:43,180 --> 00:07:49,390
So you try here, 1/2
m, omega squared,

124
00:07:49,390 --> 00:07:59,053
x minus ip over m omega,
x plus ip over m omega.

125
00:08:02,320 --> 00:08:07,870
And beware that's
not quite right.

126
00:08:07,870 --> 00:08:13,080
Because here, you have
cross terms that cancel.

127
00:08:13,080 --> 00:08:17,540
You have aib b and minus iba.

128
00:08:17,540 --> 00:08:20,380
And they would only
cancel if a and b commute.

129
00:08:20,380 --> 00:08:22,220
And here they don't commute.

130
00:08:22,220 --> 00:08:24,940
So it's almost perfect.

131
00:08:24,940 --> 00:08:29,710
But if you expand this out,
you get the x squared for sure.

132
00:08:29,710 --> 00:08:30,750
You get this term.

133
00:08:30,750 --> 00:08:36,006
But then you get an extra term
coming from the cross terms.

134
00:08:36,006 --> 00:08:37,750
And please calculate it.

135
00:08:37,750 --> 00:08:40,400
Happily, it's just a number,
because the commutator

136
00:08:40,400 --> 00:08:42,919
of x and b is just a number.

137
00:08:42,919 --> 00:08:48,790
So the answer for
this thing is that you

138
00:08:48,790 --> 00:08:52,980
get, here, x
squared plus this is

139
00:08:52,980 --> 00:09:00,425
equal to this, plus h-bar over m
omega, times the unit operator.

140
00:09:05,060 --> 00:09:15,760
So here is what you
could call v dagger.

141
00:09:15,760 --> 00:09:21,080
And this is what we'd call v.

142
00:09:21,080 --> 00:09:24,530
So what is your Hamiltonian?

143
00:09:24,530 --> 00:09:31,400
Your Hamiltonian has become
1/2 m, omega squared, v dagger

144
00:09:31,400 --> 00:09:37,460
v, plus, if you multiply out,
H omega times the identity.

145
00:09:41,300 --> 00:09:44,250
So we basically succeeded.

146
00:09:44,250 --> 00:09:48,350
And it's as good as what we
could hope or want, actually.

147
00:09:52,130 --> 00:09:56,670
I multiply this
out, so h-bar omega

148
00:09:56,670 --> 00:09:59,160
was the only thing
that was left.

149
00:09:59,160 --> 00:10:00,610
And there's your Hamiltonian.

150
00:10:00,610 --> 00:10:07,490
Now, in order to see
what this tells you,

151
00:10:07,490 --> 00:10:10,670
just sandwich it
between any two states.

152
00:10:13,220 --> 00:10:21,700
Well, this is 1/2 m, omega
squared, psi, v dagger, v,

153
00:10:21,700 --> 00:10:26,920
psi, plus 1/2 half h, omega.

154
00:10:26,920 --> 00:10:29,750
And assume it's a
normalized state,

155
00:10:29,750 --> 00:10:32,560
so it just gives you that.

156
00:10:32,560 --> 00:10:41,100
So this thing is the
norm of the state, v psi.

157
00:10:41,100 --> 00:10:45,080
You'd think it's
dagger and it's this.

158
00:10:45,080 --> 00:10:48,370
So this is the norm
squared of v psi.

159
00:10:52,920 --> 00:10:55,340
And therefore that's positive.

160
00:10:55,340 --> 00:11:01,540
So H, between any
normalized state,

161
00:11:01,540 --> 00:11:05,920
is greater than or equal
to 1/2 h-bar omega.

162
00:11:08,810 --> 00:11:16,760
In particular, if psi
is an energy eigenstate,

163
00:11:16,760 --> 00:11:21,420
so that H psi is equal to E psi.

164
00:11:24,580 --> 00:11:28,440
If psi is an energy
eigenstate, then you have this.

165
00:11:28,440 --> 00:11:33,060
And back here, you
get that the energy

166
00:11:33,060 --> 00:11:38,070
must be greater than or
equal to 1/2 h omega,

167
00:11:38,070 --> 00:11:42,980
because H and psi gives
you an E. The E goes out.

168
00:11:42,980 --> 00:11:46,530
And you're left with
psi, psi, which is 1.

169
00:11:46,530 --> 00:11:48,685
So you already know
that the energy

170
00:11:48,685 --> 00:11:53,140
is at least greater than
or equal to 1/2 h omega.

171
00:11:53,140 --> 00:11:56,650
So this factorization
has been very powerful.

172
00:11:56,650 --> 00:11:58,830
It has taught you
something extremely

173
00:11:58,830 --> 00:12:02,790
nontrivial about the
spectrum of the Hamiltonian.

174
00:12:02,790 --> 00:12:06,210
All energy eigenstates
must be greater than

175
00:12:06,210 --> 00:12:09,940
or equal to 1/2 h omega.

176
00:12:09,940 --> 00:12:11,860
In fact, this is
so good that people

177
00:12:11,860 --> 00:12:14,520
try to do this for
almost any problem.

178
00:12:14,520 --> 00:12:17,585
Any Hamiltonian, probably
the first thing you can try

179
00:12:17,585 --> 00:12:23,150
is to establish a
factorization of this kind.

180
00:12:23,150 --> 00:12:28,140
For the hydrogen atom, that
factorization is also possible.

181
00:12:28,140 --> 00:12:32,690
There will be some
homework sometime later on.

182
00:12:32,690 --> 00:12:38,420
It's less well known and
doesn't lead to useful creation

183
00:12:38,420 --> 00:12:39,940
and annihilation operators.

184
00:12:39,940 --> 00:12:42,580
But you can get the ground
state energy in a proof

185
00:12:42,580 --> 00:12:46,730
that you kind of go below
that energy very quickly.

186
00:12:46,730 --> 00:12:52,780
So a few things are done
now to clean up this system.

187
00:12:52,780 --> 00:12:59,380
And basically, here I have the
definition of v and v dagger.

188
00:12:59,380 --> 00:13:09,470
Then you define a to be square
root of m omega over 2 h-bar,

189
00:13:09,470 --> 00:13:20,420
v. And a dagger must be m
omega over 2 h-bar v dagger.

190
00:13:20,420 --> 00:13:25,540
And I have not written for
you the commutator of v and v

191
00:13:25,540 --> 00:13:26,040
dagger.

192
00:13:26,040 --> 00:13:30,880
We might as well do the
commutator of a and a dagger.

193
00:13:30,880 --> 00:13:34,600
And that commutator turns
out to be extremely simple.

194
00:13:34,600 --> 00:13:39,135
a with a dagger is
just equal to 1.

195
00:13:41,890 --> 00:13:45,360
Now things that are useful,
relations that are useful

196
00:13:45,360 --> 00:13:49,300
is-- just write
what v is in here

197
00:13:49,300 --> 00:13:52,770
so that you have a
formula for a and a dagger

198
00:13:52,770 --> 00:13:54,820
in terms of x and p.

199
00:13:54,820 --> 00:13:59,360
So I will not bother writing it.

200
00:13:59,360 --> 00:14:01,210
But it's here already.

201
00:14:01,210 --> 00:14:03,300
Maybe I'll do the first one.

202
00:14:03,300 --> 00:14:08,080
m omega over 2 h-bar.

203
00:14:08,080 --> 00:14:16,330
v is here would be x,
plus ip over m omega.

204
00:14:16,330 --> 00:14:19,460
And you can write
the other one there.

205
00:14:19,460 --> 00:14:22,730
So you have an expression
for a and a dagger

206
00:14:22,730 --> 00:14:25,040
in terms of x and p.

207
00:14:25,040 --> 00:14:27,390
And that can be
inverted as well.

208
00:14:27,390 --> 00:14:29,620
And it's pretty useful.

209
00:14:29,620 --> 00:14:32,200
And it's an example
of formulas that you

210
00:14:32,200 --> 00:14:33,860
don't need to know by heart.

211
00:14:33,860 --> 00:14:35,900
And they would be in
any formula sheet.

212
00:14:41,130 --> 00:14:48,670
And the units and
all those constants

213
00:14:48,670 --> 00:14:50,405
make it hard to remember.

214
00:14:54,460 --> 00:14:55,570
But here they are.

215
00:14:58,330 --> 00:15:08,250
So you should know that x is a
plus a dagger up to a constant.

216
00:15:08,250 --> 00:15:11,750
And p is a dagger minus a.

217
00:15:11,750 --> 00:15:15,610
Now p is Hermitian, that's
why there is an i here.

218
00:15:15,610 --> 00:15:18,115
So that this, this
anti-Hermitian,

219
00:15:18,115 --> 00:15:20,990
the i becomes a
Hermitian operator.

220
00:15:20,990 --> 00:15:28,310
x is manifestly Hermitian,
because a plus a dagger is.

221
00:15:28,310 --> 00:15:31,980
Finally, you want to
write the Hamiltonian.

222
00:15:31,980 --> 00:15:40,500
And the Hamiltonian is given
by the following formula.

223
00:15:40,500 --> 00:15:44,380
You know you just have to put
the v and v dagger, what they

224
00:15:44,380 --> 00:15:47,970
are in terms of the creation,
annihilation operators.

225
00:15:47,970 --> 00:15:51,630
So v dagger, you
substitute a dagger.

226
00:15:51,630 --> 00:15:55,290
v, you go back here
and just calculate it.

227
00:15:58,150 --> 00:16:01,740
And these calculations
really should be done.

228
00:16:01,740 --> 00:16:07,080
It's something that is
good practice and make sure

229
00:16:07,080 --> 00:16:11,520
you don't make silly mistakes.

230
00:16:11,520 --> 00:16:17,100
So this operator is so important
it has been given a name.

231
00:16:17,100 --> 00:16:28,810
It's called the
number operator, N.

232
00:16:28,810 --> 00:16:37,010
And its eigenvalues are numbers,
0, 1, 2, 3, all these things.

233
00:16:37,010 --> 00:16:39,950
And the good thing about
it is that, once you

234
00:16:39,950 --> 00:16:46,430
are with a's and a
daggers, all this m omega,

235
00:16:46,430 --> 00:16:48,480
h-bar are all gone.

236
00:16:48,480 --> 00:16:51,910
This is all that
is happening here.

237
00:16:51,910 --> 00:16:55,510
The basic energy is h-bar omega.

238
00:16:55,510 --> 00:17:00,410
Ground state energies, what
we'll see is 1/2 h-bar omega.

239
00:17:00,410 --> 00:17:02,300
And this is the number operator.

240
00:17:02,300 --> 00:17:08,430
So this is written as h-bar
omega, number operator--

241
00:17:08,430 --> 00:17:15,300
probably with a hat-- like that.

242
00:17:15,300 --> 00:17:26,520
So when you're talking
about eigenvalues,

243
00:17:26,520 --> 00:17:30,320
as we will talk soon, or
states for which these thing's

244
00:17:30,320 --> 00:17:33,430
are numbers, saying that
you have a state that

245
00:17:33,430 --> 00:17:35,550
is an eigenstate
of the Hamiltonian

246
00:17:35,550 --> 00:17:38,730
is exactly the same
thing as saying

247
00:17:38,730 --> 00:17:42,360
that it's an eigenstate
of the number operator.

248
00:17:42,360 --> 00:17:45,790
Because that's the only thing
that is an operator here.

249
00:17:45,790 --> 00:17:47,440
There's this plus this number.

250
00:17:47,440 --> 00:17:49,750
So this number
causes no problem.

251
00:17:49,750 --> 00:17:54,750
Any state multiplied by a number
is proportional to itself.

252
00:17:54,750 --> 00:17:58,160
But it's not true that every
state multiplied by a dagger a

253
00:17:58,160 --> 00:18:00,320
is proportional to itself.

254
00:18:00,320 --> 00:18:03,580
So being an
eigenstate of N means

255
00:18:03,580 --> 00:18:06,660
that acting on a state,
N, gives you a number.

256
00:18:06,660 --> 00:18:09,430
But then H is just
N times the number.

257
00:18:09,430 --> 00:18:12,450
So H is also an eigenstate.

258
00:18:12,450 --> 00:18:15,470
So eigenstates of N
or eigenstates of H

259
00:18:15,470 --> 00:18:19,970
are exactly the same thing.

260
00:18:19,970 --> 00:18:23,020
Now there's a couple
more properties

261
00:18:23,020 --> 00:18:29,060
that maybe need to be mentioned.

262
00:18:29,060 --> 00:18:32,010
So I wanted to talk in
terms of eigenvalues.

263
00:18:32,010 --> 00:18:35,330
I would just simply write
the energy eigenvalue

264
00:18:35,330 --> 00:18:41,130
is therefore equal h-bar
omega, the number eigenvalue--

265
00:18:41,130 --> 00:18:46,420
so the operator is
with a hat-- plus 1/2.

266
00:18:46,420 --> 00:18:51,130
So in terms of
eigenvalues, you have that.

267
00:18:51,130 --> 00:18:57,140
From here, the energy is
greater than 1/2 h omega.

268
00:18:57,140 --> 00:19:07,130
So the number must be greater
or equal than 0 on any state.

269
00:19:07,130 --> 00:19:12,080
And that's also clear from the
definition of this operator.

270
00:19:12,080 --> 00:19:16,770
On any state, the expectation
value of this operator

271
00:19:16,770 --> 00:19:19,200
has to be positive.

272
00:19:19,200 --> 00:19:23,070
And therefore, you have this.

273
00:19:23,070 --> 00:19:29,520
So two more properties
that are crucial here

274
00:19:29,520 --> 00:19:34,320
are that the
Hamiltonian commuted

275
00:19:34,320 --> 00:19:40,530
with a is equal
to minus h omega a

276
00:19:40,530 --> 00:19:46,100
and that the Hamiltonian
committed with a dagger

277
00:19:46,100 --> 00:19:52,350
is plus h omega a dagger.

278
00:19:52,350 --> 00:19:59,160
Now there is a
reasonably precise way

279
00:19:59,160 --> 00:20:01,250
of going through
the whole spectrum

280
00:20:01,250 --> 00:20:05,700
of the harmonic oscillator
without solving differential

281
00:20:05,700 --> 00:20:10,560
equations, almost to
any degree, and trying

282
00:20:10,560 --> 00:20:14,680
to be just very
logical about it.

283
00:20:14,680 --> 00:20:18,810
It's possible to deduce the
properties of the spectrum.

284
00:20:18,810 --> 00:20:24,040
So I will do that right now.

285
00:20:24,040 --> 00:20:26,630
And we begin with the
following statement.

286
00:20:26,630 --> 00:20:33,850
We assume there is
some energy eigenstate.

287
00:20:33,850 --> 00:20:47,046
So assume there is a state E
such that the Hamiltonian--

288
00:20:47,046 --> 00:20:48,670
for some reason in
the notes apparently

289
00:20:48,670 --> 00:20:53,190
I put hats on the Hamiltonian,
so I'll start putting hats

290
00:20:53,190 --> 00:21:03,890
here-- so that the states
are labeled by the energy.

291
00:21:03,890 --> 00:21:08,280
And this begins a tiny bit of
confusion about the notation.

292
00:21:08,280 --> 00:21:11,500
Many times you want to label
the states by the energy.

293
00:21:11,500 --> 00:21:17,340
We'll end up labeling them
with the number operator.

294
00:21:17,340 --> 00:21:19,300
And then, I said,
it will turn out,

295
00:21:19,300 --> 00:21:23,640
when the number operator is
0, we'll put a 0 in here.

296
00:21:23,640 --> 00:21:25,385
And that doesn't mean 0 energy.

297
00:21:25,385 --> 00:21:31,040
It means energy equal
1/2 h-bar omega.

298
00:21:31,040 --> 00:21:37,470
So if you assume there
is an energy eigenstate,

299
00:21:37,470 --> 00:21:40,190
that's the first step
in the construction.

300
00:21:40,190 --> 00:21:42,400
You assume there is one.

301
00:21:42,400 --> 00:21:44,920
And what does that mean?

302
00:21:44,920 --> 00:21:46,980
It means that this
is a good state.

303
00:21:46,980 --> 00:21:49,680
So it may be normalized.

304
00:21:49,680 --> 00:21:51,760
It may not be normalized.

305
00:21:51,760 --> 00:21:55,105
In any case, it
should be positive.

306
00:21:57,680 --> 00:22:01,860
I put first the equal, but
I shouldn't put the equal.

307
00:22:01,860 --> 00:22:05,840
Because we know in a
complex vector space,

308
00:22:05,840 --> 00:22:08,630
if a state has 0 norm, it's 0.

309
00:22:08,630 --> 00:22:10,930
And I want to say
that there's really

310
00:22:10,930 --> 00:22:16,470
some state that is non-0,
that has this energy.

311
00:22:16,470 --> 00:22:19,650
If the state would be 0, this
would become a triviality.

312
00:22:19,650 --> 00:22:21,710
So this state is good.

313
00:22:21,710 --> 00:22:27,410
It's all good.

314
00:22:27,410 --> 00:22:34,430
Now with this state, you can
define, now, two other states,

315
00:22:34,430 --> 00:22:38,620
acting with the creation,
annihilation operators.

316
00:22:38,620 --> 00:22:40,700
I didn't mention that name.

317
00:22:40,700 --> 00:22:44,910
But a dagger is going to be
called the creation operator.

318
00:22:44,910 --> 00:22:48,770
And this is the destruction
or annihilation operator.

319
00:22:48,770 --> 00:23:00,190
And we built two states, E
plus is a dagger acting on E.

320
00:23:00,190 --> 00:23:07,730
And E minus is a
acting on E. Now

321
00:23:07,730 --> 00:23:10,310
you could fairly
ask a this moment

322
00:23:10,310 --> 00:23:14,240
and say, well, how do you
know these states are good?

323
00:23:14,240 --> 00:23:16,010
How do you know they even exist?

324
00:23:16,010 --> 00:23:18,230
How do you know that
if you act with this,

325
00:23:18,230 --> 00:23:20,810
don't you get an
inconsistent state?

326
00:23:20,810 --> 00:23:23,540
How do you know
this makes sense?

327
00:23:23,540 --> 00:23:25,560
And these are perfectly
good questions.

328
00:23:25,560 --> 00:23:29,590
And in fact, this is exactly
what you have to understand.

329
00:23:29,590 --> 00:23:34,040
This procedure can
give some funny things.

330
00:23:34,040 --> 00:23:37,700
And we want to
discuss algebraically

331
00:23:37,700 --> 00:23:42,670
why some things are safe and
why some things may not quite

332
00:23:42,670 --> 00:23:44,410
be safe.

333
00:23:44,410 --> 00:23:49,090
And adding an a dagger,
we will see it's safe.

334
00:23:49,090 --> 00:23:54,920
While adding a's to the
state could be fairly unsafe.

335
00:23:54,920 --> 00:23:59,040
So what can be bad
about the state?

336
00:23:59,040 --> 00:24:04,750
It could be a 0 state, or it
could be an inconsistent state.

337
00:24:04,750 --> 00:24:07,650
And what this an
inconsistent state?

338
00:24:07,650 --> 00:24:13,030
Well, all our states are
represented by wave functions.

339
00:24:13,030 --> 00:24:16,070
And they should be normalizable.

340
00:24:16,070 --> 00:24:20,910
And therefore they have
norms that are positive,

341
00:24:20,910 --> 00:24:22,990
norms squared that are positive.

342
00:24:22,990 --> 00:24:26,050
Well you may find,
here, that you

343
00:24:26,050 --> 00:24:29,280
have states that have
norms that are negative,

344
00:24:29,280 --> 00:24:30,940
norm squareds that are negative.

345
00:24:30,940 --> 00:24:34,580
So this thing that
should be positive,

346
00:24:34,580 --> 00:24:37,420
algebraically you may
show that actually you

347
00:24:37,420 --> 00:24:39,830
can get into trouble.

348
00:24:39,830 --> 00:24:42,730
And trouble, of course,
is very interesting.

349
00:24:42,730 --> 00:24:51,020
So I want to skip this
calculation and state something

350
00:24:51,020 --> 00:24:54,350
that you probably checked
in 804, several times,

351
00:24:54,350 --> 00:24:57,380
that this state has
more energy than E

352
00:24:57,380 --> 00:25:03,930
and, in fact, has as much
energy as E plus h-bar omega.

353
00:25:03,930 --> 00:25:07,090
Because a dagger, the
creation operator,

354
00:25:07,090 --> 00:25:10,050
adds energy, h-bar omega.

355
00:25:10,050 --> 00:25:13,395
And this subtracts
energy, h-bar omega.

356
00:25:18,010 --> 00:25:21,890
This state has an
energy, E plus,

357
00:25:21,890 --> 00:25:26,410
which is equal to
E plus h-bar omega.

358
00:25:26,410 --> 00:25:31,080
And E minus is equal
to E minus h-bar omega.

359
00:25:31,080 --> 00:25:33,650
Now how do you check that?

360
00:25:33,650 --> 00:25:37,980
You're supposed to act
with a Hamiltonian on this,

361
00:25:37,980 --> 00:25:42,000
use the commutation relation
that we wrote up there,

362
00:25:42,000 --> 00:25:44,720
and prove that those are
the energy eigenvalues.

363
00:25:47,930 --> 00:25:54,350
So at this moment, you
can do the following.

364
00:25:54,350 --> 00:25:59,460
So these states have energies,
they have number operators,

365
00:25:59,460 --> 00:26:01,170
they have number eigenvalues.

366
00:26:01,170 --> 00:26:08,070
So we can test, if
these states are good,

367
00:26:08,070 --> 00:26:10,630
by computing their norms.

368
00:26:10,630 --> 00:26:17,300
So let's compute the
norm, a dagger on E,

369
00:26:17,300 --> 00:26:21,050
a dagger on E for the first one.

370
00:26:21,050 --> 00:26:29,780
And we'll compute a E, a E.
We'll do this computation.

371
00:26:29,780 --> 00:26:33,640
We just want to
see what this is.

372
00:26:33,640 --> 00:26:37,630
Now remember how you do this.

373
00:26:37,630 --> 00:26:40,550
An operator acting
here goes with a dagger

374
00:26:40,550 --> 00:26:41,960
into the other side.

375
00:26:41,960 --> 00:26:54,550
So this is equal to
E a, a dagger, E.

376
00:26:54,550 --> 00:26:59,700
Now a, a dagger is
not quite perfect.

377
00:26:59,700 --> 00:27:03,040
It differs from the
one that we know

378
00:27:03,040 --> 00:27:07,350
is an eigenvalue for this state,
which is the number operator.

379
00:27:07,350 --> 00:27:12,460
So what is a, a
dagger in terms of N?

380
00:27:12,460 --> 00:27:15,180
Well, a, a dagger--
it's something

381
00:27:15,180 --> 00:27:18,280
you will use many,
many times-- is

382
00:27:18,280 --> 00:27:25,140
equal to a commutator with
a dagger plus a dagger a.

383
00:27:25,140 --> 00:27:32,870
So that's 1 plus
the number operator.

384
00:27:32,870 --> 00:27:39,340
So this thing is E
1 plus the number

385
00:27:39,340 --> 00:27:44,510
operator acting on the state E.

386
00:27:44,510 --> 00:27:49,380
Well, the 1 is clear what it is.

387
00:27:49,380 --> 00:27:51,510
And the number operate is clear.

388
00:27:51,510 --> 00:27:55,330
If this has some
energy E, well, I

389
00:27:55,330 --> 00:27:58,380
can now what is the eigenvalue
of the number operator

390
00:27:58,380 --> 00:28:03,650
because the energy on
the number eigenvalues

391
00:28:03,650 --> 00:28:05,520
are related that way.

392
00:28:05,520 --> 00:28:10,740
So I will simply call
it the number of E

393
00:28:10,740 --> 00:28:12,100
and leave it at that.

394
00:28:12,100 --> 00:28:12,820
Times EE.

395
00:28:19,230 --> 00:28:24,080
So in here, the
computation is easier

396
00:28:24,080 --> 00:28:29,480
because it's just E a dagger
a E. That's the number,

397
00:28:29,480 --> 00:28:33,150
so that's just NE times EE.

398
00:28:39,578 --> 00:28:44,380
OK, so these are
the key equations

399
00:28:44,380 --> 00:28:46,290
we're going to be
using to understand

400
00:28:46,290 --> 00:28:49,700
the spectrum quickly.

401
00:28:49,700 --> 00:28:58,910
And let me say a couple
of things about them.

402
00:28:58,910 --> 00:29:03,500
So I'll repeat what we
have there, a dagger

403
00:29:03,500 --> 00:29:15,160
E a dagger E is equal
to 1 plus NE EE.

404
00:29:15,160 --> 00:29:23,870
On the other hand, 888
aE aE is equal to NE EE.

405
00:29:27,340 --> 00:29:31,550
OK, so here it goes.

406
00:29:31,550 --> 00:29:35,020
Here is the main thing that
you have to think about.

407
00:29:35,020 --> 00:29:39,800
Suppose this state
was good, which

408
00:29:39,800 --> 00:29:46,500
means this state has
a good norm here.

409
00:29:46,500 --> 00:29:49,240
And moreover, we've
already learned

410
00:29:49,240 --> 00:29:52,180
that the energy is
greater than some value.

411
00:29:52,180 --> 00:29:55,620
So the number
operator of this state

412
00:29:55,620 --> 00:30:00,350
could be 0-- could
take eigenvalue 0.

413
00:30:00,350 --> 00:30:07,040
But it could be bigger
than 0, so that's all good.

414
00:30:07,040 --> 00:30:18,240
Now, at this stage, we have
that-- for example, this state,

415
00:30:18,240 --> 00:30:23,290
a dagger E has number
one higher than this one,

416
00:30:23,290 --> 00:30:29,810
than the state E because it has
an extra factor of the a dagger

417
00:30:29,810 --> 00:30:33,100
which adds an energy of h omega.

418
00:30:33,100 --> 00:30:36,050
Which means that it
adds number of 1,

419
00:30:36,050 --> 00:30:39,950
So if this state
has some number,

420
00:30:39,950 --> 00:30:43,160
this state has a
number which is bigger.

421
00:30:43,160 --> 00:30:47,150
So suppose you keep adding.

422
00:30:47,150 --> 00:30:49,170
Now, look at the
norm of this state.

423
00:30:49,170 --> 00:30:52,760
The norm of this state is pretty
good because this is positive

424
00:30:52,760 --> 00:30:54,380
and this is positive.

425
00:30:54,380 --> 00:30:58,970
If you keep adding
a daggers here,

426
00:30:58,970 --> 00:31:05,520
you always have that this state,
the state with two a daggers,

427
00:31:05,520 --> 00:31:08,030
you could use that
to find its norm.

428
00:31:08,030 --> 00:31:10,575
You could use this formula,
put in the states with one

429
00:31:10,575 --> 00:31:12,140
a dagger here.

430
00:31:12,140 --> 00:31:15,980
But the states with one a
dagger already has a good norm.

431
00:31:15,980 --> 00:31:20,440
So this state with two a daggers
would have also good norm.

432
00:31:20,440 --> 00:31:24,610
So you can go on step by
step using this equation

433
00:31:24,610 --> 00:31:28,530
to show that as long as
you keep adding a daggers,

434
00:31:28,530 --> 00:31:32,820
all these states will
have positive norms.

435
00:31:32,820 --> 00:31:37,260
And they have positive norms
because their number eigenvalue

436
00:31:37,260 --> 00:31:39,320
is bigger and bigger.

437
00:31:39,320 --> 00:31:41,250
And therefore,
the recursion says

438
00:31:41,250 --> 00:31:43,210
that when you add
one a dagger, you

439
00:31:43,210 --> 00:31:47,540
don't change the sign of this
norm because this is positive

440
00:31:47,540 --> 00:31:50,900
and this is positive,
and this keeps happening.

441
00:31:50,900 --> 00:31:53,390
On the other hand,
this is an equation

442
00:31:53,390 --> 00:31:55,940
that's a lot more dangerous.

443
00:31:55,940 --> 00:32:02,370
Because this says that in this
equation, a lowers the number.

444
00:32:02,370 --> 00:32:09,870
So if this has some number,
NE, this has NE minus 1.

445
00:32:09,870 --> 00:32:13,050
And if you added
another a here, you

446
00:32:13,050 --> 00:32:15,960
would use this
equation again and try

447
00:32:15,960 --> 00:32:19,980
to find, what is the norm
of things with two a's here?

448
00:32:19,980 --> 00:32:23,320
And put in the one
with one a here

449
00:32:23,320 --> 00:32:26,230
and the number of that state.

450
00:32:26,230 --> 00:32:32,080
But eventually, the number can
turn into a negative number.

451
00:32:32,080 --> 00:32:36,850
And as soon as the number turns
negative, you run into trouble.

452
00:32:36,850 --> 00:32:40,350
So this is the equation
that is problematic

453
00:32:40,350 --> 00:32:43,610
and the equation that
you need to understand.

454
00:32:43,610 --> 00:32:47,810
So let me do it in two stages.

455
00:32:47,810 --> 00:32:51,030
Here are the numbers.

456
00:32:51,030 --> 00:32:57,390
And here is 5 4, 3, 2, 1, 0.

457
00:32:57,390 --> 00:33:04,010
Possibly minus 1, minus
2, and all these numbers.

458
00:33:04,010 --> 00:33:11,335
Now, suppose you start with
a number that is an integer.

459
00:33:14,320 --> 00:33:16,160
Well, you go with this equation.

460
00:33:16,160 --> 00:33:18,090
This has number 4.

461
00:33:18,090 --> 00:33:19,780
Well, you put an a.

462
00:33:19,780 --> 00:33:24,310
Now it's a state with
number 3, but its norm

463
00:33:24,310 --> 00:33:26,050
is given 4 times that.

464
00:33:26,050 --> 00:33:27,690
So it's good.

465
00:33:27,690 --> 00:33:32,930
Now you go down another 1, you
have a state with number 3,

466
00:33:32,930 --> 00:33:36,310
with number 2, with
number 1, with number 0.

467
00:33:36,310 --> 00:33:39,320
And then if you keep lowering,
you will get minus 1,

468
00:33:39,320 --> 00:33:41,430
which is not so good.

469
00:33:41,430 --> 00:33:42,810
We'll see what happens.

470
00:33:42,810 --> 00:33:46,360
Well, here you go
on and you start

471
00:33:46,360 --> 00:33:49,050
producing the states--
the state with number 4,

472
00:33:49,050 --> 00:33:53,220
state with number 3, state with
number 2, state with number 1.

473
00:33:53,220 --> 00:33:59,710
And state here, let's call
it has an energy E prime.

474
00:33:59,710 --> 00:34:05,540
And it has number equal 0.

475
00:34:05,540 --> 00:34:10,130
Number of E prime equals 0.

476
00:34:10,130 --> 00:34:12,360
So you look at this
equation and it

477
00:34:12,360 --> 00:34:24,169
says aE prime times aE prime is
equal N E prime times E prime E

478
00:34:24,169 --> 00:34:24,669
prime.

479
00:34:30,010 --> 00:34:33,980
Well, you obtain this
state at E prime,

480
00:34:33,980 --> 00:34:36,929
and it was a good
state because it

481
00:34:36,929 --> 00:34:39,969
came from a state
that was good before.

482
00:34:39,969 --> 00:34:42,250
And therefore, when
you did the last step,

483
00:34:42,250 --> 00:34:46,850
you had the state at 1
here, with n equals to 1,

484
00:34:46,850 --> 00:34:49,230
and then that was the
norm of this state.

485
00:34:49,230 --> 00:34:54,639
So this E E prime is a
fine number positive.

486
00:34:54,639 --> 00:34:57,285
But the number E prime is 0.

487
00:35:01,060 --> 00:35:11,005
So this equation says that aE
prime aE prime is equal to 0.

488
00:35:11,005 --> 00:35:14,612
And if that's equal
to 0, the state

489
00:35:14,612 --> 00:35:22,010
aE prime must be equal to 0.

490
00:35:22,010 --> 00:35:27,190
And 0 doesn't mean the
vacuum state or anything.

491
00:35:27,190 --> 00:35:28,660
It's just not there.

492
00:35:28,660 --> 00:35:30,150
There's no such state.

493
00:35:30,150 --> 00:35:32,210
You can't create it.

494
00:35:32,210 --> 00:35:37,650
You see, aE prime would be a
state here with number minus 1.

495
00:35:37,650 --> 00:35:42,730
And everything suggests to
us that that's not possible.

496
00:35:42,730 --> 00:35:44,550
It's an inconsistent state.

497
00:35:44,550 --> 00:35:47,750
The number must be less than 1.

498
00:35:47,750 --> 00:35:53,310
And we avoided the inconsistency
because this procedure

499
00:35:53,310 --> 00:35:56,660
said that as you go ahead
and do these things,

500
00:35:56,660 --> 00:36:03,370
you eventually run into this
state E prime at 0 number.

501
00:36:03,370 --> 00:36:08,150
But then, you get that
the next state is 0

502
00:36:08,150 --> 00:36:09,560
and there's no inconsistency.

503
00:36:12,710 --> 00:36:15,920
Now, that's one possibility.

504
00:36:15,920 --> 00:36:19,910
The other possibility
that could happen

505
00:36:19,910 --> 00:36:31,710
is that there are
energy eigenstates that

506
00:36:31,710 --> 00:36:38,430
have numbers which are not--
well, I'll put it here.

507
00:36:38,430 --> 00:36:40,230
That are not integer.

508
00:36:40,230 --> 00:36:47,910
So maybe you have a state
here with some number E

509
00:36:47,910 --> 00:36:49,710
which is not an integer.

510
00:36:49,710 --> 00:36:54,320
It doesn't belong
to the integers.

511
00:36:54,320 --> 00:36:56,350
OK, so what happens now?

512
00:36:59,120 --> 00:37:03,300
Well, this number is positive.

513
00:37:03,300 --> 00:37:07,470
So you can lower it and you can
put another state with number

514
00:37:07,470 --> 00:37:08,530
1 less.

515
00:37:08,530 --> 00:37:13,300
Also, not integer
and it has good norm.

516
00:37:13,300 --> 00:37:16,830
And this thing has
number 2.5, say.

517
00:37:16,830 --> 00:37:19,620
Well, if I use the
equation again,

518
00:37:19,620 --> 00:37:23,390
I put the 2.5 state
with its number 2.5

519
00:37:23,390 --> 00:37:26,516
and now I get the
state with number 1.5

520
00:37:26,516 --> 00:37:29,990
and it still has positive norm.

521
00:37:29,990 --> 00:37:35,300
Do it again, you find
the state with 0.5 number

522
00:37:35,300 --> 00:37:39,340
and still positive norm.

523
00:37:39,340 --> 00:37:41,940
And looking at this,
you start with a state

524
00:37:41,940 --> 00:37:45,060
with 0.5, with 0.5 here.

525
00:37:45,060 --> 00:37:50,202
And oops, you get a
state that minus 0.5.

526
00:37:50,202 --> 00:37:56,110
And it seems to be
good, positive norm.

527
00:37:56,110 --> 00:37:59,640
But then, if this
is possible, you

528
00:37:59,640 --> 00:38:03,760
could also build another
state acting with another a.

529
00:38:03,760 --> 00:38:09,410
And this state is now very bad
because the N for this state

530
00:38:09,410 --> 00:38:11,200
was minus 1/2.

531
00:38:11,200 --> 00:38:13,370
And therefore, if
you put that state,

532
00:38:13,370 --> 00:38:15,985
that state at the minus
1/2, you get the norm

533
00:38:15,985 --> 00:38:21,020
of the next one
that has one less.

534
00:38:21,020 --> 00:38:23,010
And this state now
is inconsistent.

535
00:38:30,640 --> 00:38:33,940
So you run into a difficulty.

536
00:38:33,940 --> 00:38:37,730
So what are the ways in
which this difficulty

537
00:38:37,730 --> 00:38:40,240
could be avoided?

538
00:38:40,240 --> 00:38:43,260
What are the escape hatches?

539
00:38:43,260 --> 00:38:46,900
There are two possibilities.

540
00:38:46,900 --> 00:38:52,170
Well, the simplest one would
be that the assumption is bad.

541
00:38:52,170 --> 00:38:56,090
There's no state with
fractional number

542
00:38:56,090 --> 00:38:59,550
because it leads to
inconsistent states.

543
00:38:59,550 --> 00:39:02,580
You can build them and
they should be good,

544
00:39:02,580 --> 00:39:05,850
but they're bad.

545
00:39:05,850 --> 00:39:07,950
The other possibility
is that just

546
00:39:07,950 --> 00:39:11,050
like this one sort
of terminated,

547
00:39:11,050 --> 00:39:15,160
and when you hit 0--
boom, the state became 0.

548
00:39:15,160 --> 00:39:18,990
Maybe this one with
a fractional one,

549
00:39:18,990 --> 00:39:24,260
before you run into trouble
you hit a 0 and the state

550
00:39:24,260 --> 00:39:26,240
becomes 0.

551
00:39:26,240 --> 00:39:29,760
So basically, what you
really need to know now

552
00:39:29,760 --> 00:39:38,160
on the algebraic method cannot
tell you is how many states are

553
00:39:38,160 --> 00:39:41,040
killed by a.

554
00:39:41,040 --> 00:39:45,450
If maybe the state of
1/2 is also killed by a,

555
00:39:45,450 --> 00:39:49,710
then we would have trouble.

556
00:39:49,710 --> 00:39:54,524
Now, as we will see now,
that's a simple problem.

557
00:39:54,524 --> 00:39:55,940
And it's the only
place where it's

558
00:39:55,940 --> 00:39:58,340
interesting to
solve some equation.

559
00:39:58,340 --> 00:40:00,760
So the equation that
we want to solve

560
00:40:00,760 --> 00:40:05,105
is the equation a on
some state is equal to 0.

561
00:40:09,360 --> 00:40:14,070
Now, that equation already says
that this possibility is not

562
00:40:14,070 --> 00:40:15,520
going to happen.

563
00:40:15,520 --> 00:40:16,020
Why?

564
00:40:16,020 --> 00:40:20,100
Because from this equation, you
can put an a dagger on this.

565
00:40:25,340 --> 00:40:31,400
And therefore, you get
that NE is equal to 0.

566
00:40:31,400 --> 00:40:35,320
This is the number operator,
so the eigenvalue of the number

567
00:40:35,320 --> 00:40:37,660
operator, we call it NE.

568
00:40:37,660 --> 00:40:43,470
So in order to be killed by a,
you have to have NE equals 0.

569
00:40:43,470 --> 00:40:48,470
So in the fractional case,
no state will be killed

570
00:40:48,470 --> 00:40:51,260
and you would arrive
to an inconsistency.

571
00:40:51,260 --> 00:40:54,585
So the only possibility is that
there's no fractional states.

572
00:40:57,500 --> 00:41:03,610
So it's still interesting to
figure out this differential

573
00:41:03,610 --> 00:41:05,850
equation, what it gives you.

574
00:41:05,850 --> 00:41:08,780
And why do we call it a
differential equation?

575
00:41:08,780 --> 00:41:13,260
Because a is this
operator over there.

576
00:41:13,260 --> 00:41:15,030
It has x and ip.

577
00:41:15,030 --> 00:41:22,840
So the equation
is x a E equals 0,

578
00:41:22,840 --> 00:41:29,390
which is square root of
m omega over 2 h bar x

579
00:41:29,390 --> 00:41:36,635
x plus ip over m
omega on E equals 0.

580
00:41:39,980 --> 00:41:45,050
And you've translated
these kind of things.

581
00:41:45,050 --> 00:41:49,990
The first term is an x
multiplying the wave function.

582
00:41:49,990 --> 00:41:55,290
We can call it psi E of x.

583
00:41:55,290 --> 00:41:58,200
The next term, the
coefficient in front

584
00:41:58,200 --> 00:42:01,020
is something you don't
have to worry, of course.

585
00:42:01,020 --> 00:42:02,800
It's just multiplying
everything,

586
00:42:02,800 --> 00:42:04,990
so it's just irrelevant.

587
00:42:04,990 --> 00:42:08,620
So have i over m omega.

588
00:42:08,620 --> 00:42:18,250
And p, as you remember, is h bar
over i d dx of psi E of x zero.

589
00:42:20,990 --> 00:42:26,600
So it's so simple
differential equation,

590
00:42:26,600 --> 00:42:39,380
x plus h bar over m omega d dx
on psi E of x is equal to 0.

591
00:42:39,380 --> 00:42:42,070
Just one solution
up to a constant

592
00:42:42,070 --> 00:42:46,160
is the Gaussian that
you know represents

593
00:42:46,160 --> 00:42:47,810
a simple harmonic oscillator.

594
00:42:56,460 --> 00:43:00,500
So that's pretty
much the end of it.

595
00:43:00,500 --> 00:43:05,310
This ground state wave
function is a number

596
00:43:05,310 --> 00:43:11,720
times the exponential of minus
m omega over 2 h bar x squared.

597
00:43:15,840 --> 00:43:21,954
And that's that.

598
00:43:21,954 --> 00:43:24,070
This is called the ground state.

599
00:43:24,070 --> 00:43:27,790
It has N equals 0
represented as a state.

600
00:43:31,130 --> 00:43:36,720
We say this number
is N equals 0.

601
00:43:36,720 --> 00:43:41,870
So this state is the thing
that represents this psi

602
00:43:41,870 --> 00:43:50,290
E. In other words, psi
E of x is x with 0.

603
00:43:50,290 --> 00:43:54,680
And that 0 is a
little confusing.

604
00:43:54,680 --> 00:43:57,000
Some people think
it's the 0 vector.

605
00:43:57,000 --> 00:43:59,580
That's not good.

606
00:43:59,580 --> 00:44:01,700
This is not the 0 vector.

607
00:44:01,700 --> 00:44:04,290
The 0 vector is not a state.

608
00:44:04,290 --> 00:44:06,430
It's not in the Hilbert space.

609
00:44:06,430 --> 00:44:08,410
This is the ground state.

610
00:44:08,410 --> 00:44:12,070
Then, the worst confusion is
to think it's the 0 vector.

611
00:44:12,070 --> 00:44:16,540
The next confusion is
to think it's 0 energy.

612
00:44:16,540 --> 00:44:20,350
That's not 0 energy,
it's number equals 0.

613
00:44:20,350 --> 00:44:25,050
The energy is, therefore,
1/2 h bar omega.

614
00:44:28,510 --> 00:44:32,030
And now, given
our discussion, we

615
00:44:32,030 --> 00:44:35,880
can start building states
with more oscillators.

616
00:44:35,880 --> 00:44:40,050
So we build a state
with number equal 1,

617
00:44:40,050 --> 00:44:44,970
which is constructed by
an a dagger on the vacuum.

618
00:44:44,970 --> 00:44:48,530
This has energy 1
h bar omega more.

619
00:44:48,530 --> 00:44:51,630
It has number equal to 1.

620
00:44:51,630 --> 00:44:53,550
And that's sometimes
useful to just

621
00:44:53,550 --> 00:45:00,150
make sure you understand why
N on a dagger on the vacuum

622
00:45:00,150 --> 00:45:06,680
is a dagger a a
dagger on the vacuum.

623
00:45:06,680 --> 00:45:10,140
Now, a kills the
vacuum, so this can

624
00:45:10,140 --> 00:45:14,190
be replaced by the
commutator, which is 1.

625
00:45:14,190 --> 00:45:16,890
And therefore, you're left
with a dagger on the vacuum.

626
00:45:20,300 --> 00:45:23,410
And that means that
the eigenvalue of n hat

627
00:45:23,410 --> 00:45:26,300
is 1 for this state.

628
00:45:26,300 --> 00:45:32,010
Moreover, this state is
where normalized 1 with 1

629
00:45:32,010 --> 00:45:35,270
actually gives you
a good normalization

630
00:45:35,270 --> 00:45:37,120
if 0 is well-normalized.

631
00:45:37,120 --> 00:45:42,940
So we'll take 0 with 0
to be 1, the number 1.

632
00:45:42,940 --> 00:45:48,660
And that requires fixing
that N0 over here.

633
00:45:48,660 --> 00:45:52,470
Now, these are things
that you've mostly seen,

634
00:45:52,470 --> 00:45:57,920
so I don't want to say
much more about them.

635
00:45:57,920 --> 00:46:02,040
I'd rather go through
the Schrodinger thing

636
00:46:02,040 --> 00:46:03,180
that we have later.

637
00:46:03,180 --> 00:46:11,610
So let me conclude by just
listing the general states,

638
00:46:11,610 --> 00:46:15,220
and then leaving for you to read
what is left there in the notes

639
00:46:15,220 --> 00:46:18,960
so that you can just get an
appreciation of how you use it.

640
00:46:18,960 --> 00:46:21,800
And with the practice
problems, you'll be done.

641
00:46:21,800 --> 00:46:25,000
So here it is.

642
00:46:25,000 --> 00:46:26,430
Here is the answer.

643
00:46:26,430 --> 00:46:32,160
The n state is given by 1 over
square root of n factorial

644
00:46:32,160 --> 00:46:35,960
a dagger to the n
acting on the vacuum.

645
00:46:38,790 --> 00:46:47,400
And these n states are such
that m with n is delta mn.

646
00:46:47,400 --> 00:46:51,360
So here we're using
all kinds of things.

647
00:46:51,360 --> 00:46:55,270
First, you should check
this is well normalized,

648
00:46:55,270 --> 00:46:57,980
or read it and do
the calculations.

649
00:46:57,980 --> 00:47:02,350
And these are, in
fact, orthogonal

650
00:47:02,350 --> 00:47:06,980
unless they have the same
number of creation operators

651
00:47:06,980 --> 00:47:08,790
are the same number.

652
00:47:08,790 --> 00:47:11,300
Now, that had to be expected.

653
00:47:11,300 --> 00:47:16,145
These are eigenstates
of a Hermitian operator.

654
00:47:16,145 --> 00:47:19,330
The N operator is Hermitian.

655
00:47:19,330 --> 00:47:22,010
Eigenstates of a
Hermitian operator

656
00:47:22,010 --> 00:47:25,320
with different
eigenvalues are always

657
00:47:25,320 --> 00:47:27,950
orthogonal to each other.

658
00:47:27,950 --> 00:47:30,330
If you have eigenstates
of a Hermitian operator

659
00:47:30,330 --> 00:47:33,850
with the same eigenvalue,
if you have a degeneracy,

660
00:47:33,850 --> 00:47:38,240
you can always arrange them
to make them orthogonal.

661
00:47:38,240 --> 00:47:42,280
But if the eigenvalues are
different, they are orthogonal.

662
00:47:42,280 --> 00:47:47,400
And there's no degeneracies
in this spectrum whatsoever.

663
00:47:47,400 --> 00:47:52,230
You will, in fact, argue that
because there's no degeneracy

664
00:47:52,230 --> 00:47:56,050
in the ground state, there
cannot be degeneracy anywhere

665
00:47:56,050 --> 00:47:58,210
else.

666
00:47:58,210 --> 00:48:00,740
So this result,
this orthonormality

667
00:48:00,740 --> 00:48:04,880
is really a consequence of
all the theorems we've proven.

668
00:48:04,880 --> 00:48:08,240
And you could check it
by doing the algebra

669
00:48:08,240 --> 00:48:10,840
and you would start
moving a and a daggers.

670
00:48:10,840 --> 00:48:13,570
And you would be left
with either some a's or

671
00:48:13,570 --> 00:48:14,610
some a daggers.

672
00:48:14,610 --> 00:48:16,850
If you're left
with some a's, they

673
00:48:16,850 --> 00:48:18,330
would kill the
thing on the right.

674
00:48:18,330 --> 00:48:19,800
If you're left with
some a daggers,

675
00:48:19,800 --> 00:48:22,350
it would kill the
thing on the left.

676
00:48:22,350 --> 00:48:23,840
So this can be proven.

677
00:48:23,840 --> 00:48:27,760
But this is just a consequence
that these are eigenstates

678
00:48:27,760 --> 00:48:34,620
of the Hermitian operator n
that have different eigenvalues.

679
00:48:34,620 --> 00:48:38,130
And therefore, you've
succeeded in constructing

680
00:48:38,130 --> 00:48:42,090
a full decomposition
of the state

681
00:48:42,090 --> 00:48:45,940
space of the
harmonic oscillator.

682
00:48:45,940 --> 00:48:49,680
We spoke about
the Hilbert space.

683
00:48:49,680 --> 00:48:52,000
Are now very
precisely, see we can

684
00:48:52,000 --> 00:48:59,690
say this is u0 plus
u1 plus u2 where

685
00:48:59,690 --> 00:49:13,030
uk is the states of the form
alpha k, where N on k-- maybe

686
00:49:13,030 --> 00:49:15,760
I should put n here.

687
00:49:15,760 --> 00:49:17,515
It looks nicer.

688
00:49:17,515 --> 00:49:18,878
n.

689
00:49:18,878 --> 00:49:25,000
Where N n equal n n.

690
00:49:25,000 --> 00:49:28,140
So every
one-dimensional subspace

691
00:49:28,140 --> 00:49:31,480
is spanned by that
state of number n.

692
00:49:31,480 --> 00:49:34,590
So you have the states of
number 0, states of number 1,

693
00:49:34,590 --> 00:49:36,220
states of number 2.

694
00:49:36,220 --> 00:49:39,050
These are all
orthogonal subspaces.

695
00:49:39,050 --> 00:49:40,910
They add up to form everything.

696
00:49:44,170 --> 00:49:46,640
It's a nice description.

697
00:49:46,640 --> 00:49:48,930
So the general
state in this system

698
00:49:48,930 --> 00:49:52,490
is a complex number times
the state with number 0

699
00:49:52,490 --> 00:49:56,240
plus the complex number states
of number 1, complex number,

700
00:49:56,240 --> 00:49:57,590
and that.

701
00:49:57,590 --> 00:50:03,350
Things couldn't have
been easier in a sense.

702
00:50:03,350 --> 00:50:07,350
The other thing that you
already know from 804

703
00:50:07,350 --> 00:50:11,060
is that if you try to compute
expectation values, most

704
00:50:11,060 --> 00:50:17,110
of the times you want to
use a's and a daggers.

705
00:50:17,110 --> 00:50:20,800
So the typical thing
that one wants to compete

706
00:50:20,800 --> 00:50:29,440
is on the state n, what is the
uncertainty in x on the state

707
00:50:29,440 --> 00:50:31,610
n?

708
00:50:31,610 --> 00:50:33,310
How much is it?

709
00:50:33,310 --> 00:50:36,260
What is the
uncertainty of momentum

710
00:50:36,260 --> 00:50:40,815
on the energy
eigenstate of number n?

711
00:50:43,800 --> 00:50:47,100
These are relatively
straightforward calculations.

712
00:50:47,100 --> 00:50:51,650
If you have to do the integrals,
each one-- by the time you

713
00:50:51,650 --> 00:50:56,200
organize all your constants--
half an hour, maybe 20 minutes.

714
00:50:56,200 --> 00:50:58,790
If you do it with
a and a daggers,

715
00:50:58,790 --> 00:51:01,430
this computation
should be five minutes,

716
00:51:01,430 --> 00:51:03,890
or something like that.

717
00:51:03,890 --> 00:51:06,460
We'll see that
done on the notes.

718
00:51:06,460 --> 00:51:09,780
You can also do them yourselves.

719
00:51:09,780 --> 00:51:13,000
You probably have
played with them a bit.

720
00:51:13,000 --> 00:51:19,860
So this was a brief review and
discussion of them spectrum.

721
00:51:19,860 --> 00:51:21,770
It was a little detailed.

722
00:51:21,770 --> 00:51:25,710
We had to argue things
carefully to make

723
00:51:25,710 --> 00:51:28,820
sure we don't assume things.

724
00:51:28,820 --> 00:51:31,110
And this is the
way we'll do also

725
00:51:31,110 --> 00:51:34,750
with angular momentum
in a few weeks from now.

726
00:51:34,750 --> 00:51:38,510
But now I want to leave that,
so I'm going to take questions.

727
00:51:38,510 --> 00:51:46,640
If there are any questions
on this logic, please ask.

728
00:51:46,640 --> 00:51:47,270
Yes.

729
00:51:47,270 --> 00:51:50,574
AUDIENCE: [INAUDIBLE] for
how you got a dagger, a,

730
00:51:50,574 --> 00:51:53,880
a dagger, 0, 2 dagger, 0?

731
00:51:53,880 --> 00:51:55,640
PROFESSOR: Yes,
that calculation.

732
00:51:55,640 --> 00:52:00,220
So let me do at the step
that I did in words.

733
00:52:00,220 --> 00:52:03,570
So at this place--
so the question was,

734
00:52:03,570 --> 00:52:07,560
how did I do this computation?

735
00:52:07,560 --> 00:52:10,060
Here I just copied what N is.

736
00:52:10,060 --> 00:52:11,910
So I just copied that.

737
00:52:11,910 --> 00:52:16,476
Then, the next step was to
say, since a kills this,

738
00:52:16,476 --> 00:52:26,280
this is equal to a dagger times
a a dagger minus a dagger a.

739
00:52:26,280 --> 00:52:28,140
Because a kills it.

740
00:52:28,140 --> 00:52:31,160
And I can add this, it
doesn't cost me anything.

741
00:52:31,160 --> 00:52:33,810
Now, I added something
that is convenient,

742
00:52:33,810 --> 00:52:37,430
so that this is a
dagger commutator of a

743
00:52:37,430 --> 00:52:41,270
with a dagger on 0.

744
00:52:41,270 --> 00:52:44,062
This is 1, so you get that.

745
00:52:47,290 --> 00:52:49,000
It's a little more
interesting when

746
00:52:49,000 --> 00:52:51,780
you have, for
example, the state 2,

747
00:52:51,780 --> 00:52:57,730
which is 1 over square root
of 2 a dagger a dagger on 0.

748
00:52:57,730 --> 00:53:01,330
I advise you to try to
calculate n on that.

749
00:53:01,330 --> 00:53:04,240
And in general,
convince yourselves

750
00:53:04,240 --> 00:53:07,750
that n is a number
operator, which means

751
00:53:07,750 --> 00:53:11,170
counts the number of a daggers.

752
00:53:11,170 --> 00:53:18,956
You'll have to use that
property if you have N with AB.

753
00:53:18,956 --> 00:53:25,610
It's N with A B and then A N
with B. The derivative property

754
00:53:25,610 --> 00:53:30,460
of the bracket has to
be used all the time.

755
00:53:30,460 --> 00:53:36,010
So Schrodinger dynamics,
let's spend the last 20

756
00:53:36,010 --> 00:53:37,780
minutes of our lecture on this.

757
00:53:46,100 --> 00:53:54,360
So basically, it's
a postulate of how

758
00:53:54,360 --> 00:53:57,040
evolution occurs in
quantum mechanics.

759
00:53:57,040 --> 00:53:59,935
So we'll state it as follows.

760
00:54:03,950 --> 00:54:06,140
What is time in
quantum mechanics?

761
00:54:06,140 --> 00:54:09,080
Well, you have a state space.

762
00:54:09,080 --> 00:54:12,090
And you see the
state space, you've

763
00:54:12,090 --> 00:54:14,040
seen it in the
harmonic oscillator

764
00:54:14,040 --> 00:54:15,680
is this sum of vectors.

765
00:54:15,680 --> 00:54:19,570
And these vectors were wave
functions, if you wish.

766
00:54:19,570 --> 00:54:22,090
There's no time anywhere there.

767
00:54:22,090 --> 00:54:24,820
There's no time on
this vector space.

768
00:54:24,820 --> 00:54:28,300
This vector space is an
abstract vector space

769
00:54:28,300 --> 00:54:37,150
of functions or states, but time
comes because you have clocks.

770
00:54:37,150 --> 00:54:39,300
And then you can ask,
where is my state?

771
00:54:39,300 --> 00:54:41,840
And that's that vector
on that state space.

772
00:54:41,840 --> 00:54:43,960
And you ask the
question a littler later

773
00:54:43,960 --> 00:54:45,270
and the state has moved.

774
00:54:45,270 --> 00:54:46,940
It's another vector.

775
00:54:46,940 --> 00:54:53,000
So these are vectors and
the vectors change in time.

776
00:54:53,000 --> 00:54:56,850
And that's all the dynamics
is in quantum mechanics.

777
00:54:56,850 --> 00:55:00,010
The time is sort of
auxiliary to all this.

778
00:55:02,570 --> 00:55:04,580
So we must have a
picture of that.

779
00:55:04,580 --> 00:55:13,610
And the way we do this is to
imagine that we have a vector

780
00:55:13,610 --> 00:55:16,870
space H. And here is a vector.

781
00:55:16,870 --> 00:55:20,960
And that H is for Hilbert space.

782
00:55:20,960 --> 00:55:24,490
We used to call it in our
math part of the course

783
00:55:24,490 --> 00:55:27,920
V, the complex vector space.

784
00:55:27,920 --> 00:55:32,280
And this state is the
state of the system.

785
00:55:32,280 --> 00:55:35,170
And we sometimes
put the time here

786
00:55:35,170 --> 00:55:36,985
to indicate that's what it is.

787
00:55:36,985 --> 00:55:39,400
At time t0, that's it.

788
00:55:39,400 --> 00:55:46,440
Well, at time t, some arbitrary
later time, it could be here.

789
00:55:46,440 --> 00:55:48,180
And the state moves.

790
00:55:48,180 --> 00:55:50,310
But one thing is clear.

791
00:55:50,310 --> 00:55:54,255
If it's a state of a
system, if we normalize it,

792
00:55:54,255 --> 00:55:57,780
it should be of unit length.

793
00:55:57,780 --> 00:56:03,970
And we can think of a sphere
in which this unit sphere is

794
00:56:03,970 --> 00:56:10,170
the set of all the tips of the
vectors that have unit norm.

795
00:56:10,170 --> 00:56:15,330
And this vector will move here
in time, trace a trajectory,

796
00:56:15,330 --> 00:56:18,340
and reach this one.

797
00:56:18,340 --> 00:56:22,870
And it should do it preserving
the length of the vector.

798
00:56:22,870 --> 00:56:25,735
And in fact, if you don't
use a normalized vector,

799
00:56:25,735 --> 00:56:28,320
it has a norm of 3.

800
00:56:28,320 --> 00:56:30,770
Well, it should preserve
that 3 because you'd

801
00:56:30,770 --> 00:56:34,060
normalize the state
once and forever.

802
00:56:34,060 --> 00:56:38,790
So we proved in our
math part of the subject

803
00:56:38,790 --> 00:56:43,020
that an operator
that always preserves

804
00:56:43,020 --> 00:56:46,620
the length of all vectors
is a unitary operator.

805
00:56:46,620 --> 00:56:50,060
So this is the fundamental
thing that we want.

806
00:56:50,060 --> 00:56:59,010
And the idea of quantum
mechanics is that psi at time t

807
00:56:59,010 --> 00:57:04,420
is obtained by the action
of a unitary operator

808
00:57:04,420 --> 00:57:07,130
from the state psi at time t0.

809
00:57:12,020 --> 00:57:17,090
And this is for all t and t0.

810
00:57:17,090 --> 00:57:18,583
And this being unitary.

811
00:57:22,760 --> 00:57:27,520
Now, I want to make
sure this is clear.

812
00:57:27,520 --> 00:57:32,570
It can be misinterpreted,
this equation.

813
00:57:32,570 --> 00:57:39,660
Here, psi at t0 is
an arbitrary state.

814
00:57:39,660 --> 00:57:42,765
If you had another
state, psi prime of t0,

815
00:57:42,765 --> 00:57:45,150
it would also evolve
with this formula.

816
00:57:45,150 --> 00:57:48,060
And this U is the same.

817
00:57:48,060 --> 00:57:51,120
So the postulate of
unitary time evolution

818
00:57:51,120 --> 00:57:56,350
is that there is this
magical U operator that

819
00:57:56,350 --> 00:58:00,230
can evolve any state.

820
00:58:00,230 --> 00:58:01,990
Any state that you
give me at time

821
00:58:01,990 --> 00:58:05,170
equal 0, any possible
state in the Hilbert space,

822
00:58:05,170 --> 00:58:07,580
you plug it in here.

823
00:58:07,580 --> 00:58:11,150
And by acting with
this unitary operator,

824
00:58:11,150 --> 00:58:13,230
you get the state
at the later time.

825
00:58:16,360 --> 00:58:22,550
Now, you've slipped an
extraordinary amount of physics

826
00:58:22,550 --> 00:58:24,570
into that statement.

827
00:58:24,570 --> 00:58:28,430
If you've bought it, you've
bought the Schrodinger equation

828
00:58:28,430 --> 00:58:29,440
already.

829
00:58:29,440 --> 00:58:31,920
That is going to
come out by just

830
00:58:31,920 --> 00:58:35,290
doing a little
calculation from this.

831
00:58:35,290 --> 00:58:40,070
So the Schrodinger equation
is really fundamentally,

832
00:58:40,070 --> 00:58:42,865
at the end of the
day, the statement

833
00:58:42,865 --> 00:58:45,520
that this unitary
time evolution, which

834
00:58:45,520 --> 00:58:48,510
is to mean there's a
unitary operator that

835
00:58:48,510 --> 00:58:51,700
evolves any physical state.

836
00:58:51,700 --> 00:58:55,200
So let's try to discuss this.

837
00:58:55,200 --> 00:58:57,540
Are there any questions?

838
00:58:57,540 --> 00:58:58,362
Yes.

839
00:58:58,362 --> 00:59:00,028
AUDIENCE: So you
mentioned at first that

840
00:59:00,028 --> 00:59:01,652
in the current
formulation [INAUDIBLE]?

841
00:59:04,590 --> 00:59:05,916
PROFESSOR: A little louder.

842
00:59:05,916 --> 00:59:08,005
We do what in our
current formulation?

843
00:59:08,005 --> 00:59:11,370
AUDIENCE: So if you don't
include time [INAUDIBLE].

844
00:59:11,370 --> 00:59:12,370
PROFESSOR: That's right.

845
00:59:12,370 --> 00:59:13,825
There's no start of
the vector space.

846
00:59:13,825 --> 00:59:14,491
AUDIENCE: Right.

847
00:59:14,491 --> 00:59:17,390
So is it possible to consider
a vector space with time?

848
00:59:20,970 --> 00:59:21,930
PROFESSOR: Unclear.

849
00:59:21,930 --> 00:59:25,250
I don't think so.

850
00:59:25,250 --> 00:59:28,720
It's just nowhere there.

851
00:59:28,720 --> 00:59:34,070
What would it mean, even, to
add time to the vector space?

852
00:59:34,070 --> 00:59:36,480
I think you would
have a hard time even

853
00:59:36,480 --> 00:59:37,915
imagining what it means.

854
00:59:41,500 --> 00:59:43,920
Now, people try to
change quantum mechanics

855
00:59:43,920 --> 00:59:46,510
in all kinds of ways.

856
00:59:46,510 --> 00:59:49,305
Nobody has succeeded in
changing quantum mechanics.

857
00:59:52,210 --> 00:59:55,770
That should not be a
deterrent for you to try,

858
00:59:55,770 --> 00:59:58,690
but should give you
a little caution

859
00:59:58,690 --> 01:00:03,300
that is not likely to be easy.

860
01:00:03,300 --> 01:00:07,770
So we'll not try to do that.

861
01:00:07,770 --> 01:00:11,960
Now, let me follow on this
and see what it gives us.

862
01:00:16,860 --> 01:00:20,120
Well, a few things.

863
01:00:20,120 --> 01:00:22,790
This operator is unique.

864
01:00:22,790 --> 01:00:24,650
If it exists, it's unique.

865
01:00:24,650 --> 01:00:28,220
If there's another operator that
evolves states the same way,

866
01:00:28,220 --> 01:00:30,520
it must be the same as that one.

867
01:00:30,520 --> 01:00:33,420
Easy to prove.

868
01:00:33,420 --> 01:00:36,410
Two operators that attack
the same way on every state

869
01:00:36,410 --> 01:00:39,060
are the same, so that's it.

870
01:00:39,060 --> 01:00:48,360
Unitary, what does it mean
that u t, t0 dagger times u t,

871
01:00:48,360 --> 01:00:54,390
t0 is equal to 1?

872
01:00:54,390 --> 01:00:58,660
Now, here these parentheses
are a little cumbersome.

873
01:00:58,660 --> 01:01:01,320
This is very clear,
you take this operator

874
01:01:01,320 --> 01:01:02,280
and you dagger it.

875
01:01:02,280 --> 01:01:05,980
But it's cumbersome, so
we write it like this.

876
01:01:12,740 --> 01:01:15,300
This means the dagger
of the whole operator.

877
01:01:15,300 --> 01:01:18,550
So this is just the same thing.

878
01:01:21,310 --> 01:01:24,065
OK, what else?

879
01:01:27,620 --> 01:01:36,180
u of t0, t0, it's
the unit operator.

880
01:01:36,180 --> 01:01:40,990
If the times are the same,
you get the unit operator

881
01:01:40,990 --> 01:01:48,160
for all t0 because you're
getting psi of t0 here

882
01:01:48,160 --> 01:01:49,210
and psi of t0 here.

883
01:01:49,210 --> 01:01:52,570
And the only operator that
leaves all states the same

884
01:01:52,570 --> 01:01:53,960
is the unit operator.

885
01:01:53,960 --> 01:01:59,760
So this unitary operator must
become the unit operator,

886
01:01:59,760 --> 01:02:05,100
in fact, for the two
arguments being equal.

887
01:02:05,100 --> 01:02:07,050
Composition.

888
01:02:07,050 --> 01:02:15,550
If you have psi t2, that can be
obtained as U of t2, t1 times

889
01:02:15,550 --> 01:02:19,350
the psi of t1.

890
01:02:19,350 --> 01:02:30,310
And it can be obtained as u of
t2, t1, u of t1, t0, psi of t0.

891
01:02:34,110 --> 01:02:36,310
So what do we learn from here?

892
01:02:36,310 --> 01:02:43,890
That this state itself is u of
t2, t0 on the original state.

893
01:02:43,890 --> 01:02:55,390
So u of t2, t0 is u of
t2, t1 times u of t1, t0.

894
01:02:59,500 --> 01:03:06,120
It's like time composition is
like matrix multiplication.

895
01:03:06,120 --> 01:03:10,020
You go from t0 to t1,
then from t1 to t2.

896
01:03:10,020 --> 01:03:14,520
It's like the second
index of this matrix.

897
01:03:14,520 --> 01:03:16,270
In the first index
of this matrix,

898
01:03:16,270 --> 01:03:21,280
you are multiplying them
and you get this thing.

899
01:03:21,280 --> 01:03:23,220
So that's composition.

900
01:03:23,220 --> 01:03:26,055
And then, you have
inverses as well.

901
01:03:33,640 --> 01:03:37,340
And here are the inverses.

902
01:03:37,340 --> 01:03:45,270
In that equation, you
take t2 equal to t0.

903
01:03:45,270 --> 01:03:47,635
So the left-hand side becomes 1.

904
01:03:50,400 --> 01:04:02,280
And t1 equal to t, so you get
u of t0, t be times u of t,

905
01:04:02,280 --> 01:04:07,610
t0 is equal to 1,
which makes sense.

906
01:04:07,610 --> 01:04:10,770
You propagate from t0 to t.

907
01:04:10,770 --> 01:04:14,210
And then from t to
t0, you get nothing.

908
01:04:14,210 --> 01:04:21,570
Or if it's to say that the
inverse of an operator--

909
01:04:21,570 --> 01:04:24,300
the inverse of this
operator is this one.

910
01:04:24,300 --> 01:04:29,190
So to take the inverse of a
u, you flip the arguments.

911
01:04:29,190 --> 01:04:39,170
So I'll write it like that,
the inverse minus 1 of t, t0.

912
01:04:39,170 --> 01:04:40,700
You just flip the arguments.

913
01:04:40,700 --> 01:04:42,120
It's u of t0, t.

914
01:04:45,040 --> 01:04:48,160
And since the
operator is Hermitian,

915
01:04:48,160 --> 01:04:51,380
the dagger is equal
to the inverse.

916
01:04:51,380 --> 01:04:56,660
So the inverse of an operator
is equal to the dagger.

917
01:04:56,660 --> 01:05:00,780
so t, t0 as well.

918
01:05:00,780 --> 01:05:03,930
So this one we got here.

919
01:05:03,930 --> 01:05:09,564
And Hermiticity says that the
dagger is equal to the inverse.

920
01:05:09,564 --> 01:05:12,120
Inverse and dagger are the same.

921
01:05:12,120 --> 01:05:16,090
So basically, you can
delete the word "inverse"

922
01:05:16,090 --> 01:05:18,170
by flipping the order
of the arguments.

923
01:05:18,170 --> 01:05:20,620
And since dagger is
the same as inverse,

924
01:05:20,620 --> 01:05:22,890
you can delete the
dagger by flipping

925
01:05:22,890 --> 01:05:24,383
the order of the arguments.

926
01:05:28,170 --> 01:05:32,065
All right, so let's try to
find the Schrodinger equation.

927
01:05:41,210 --> 01:05:44,185
So how c we c the
Schrodinger equation?

928
01:05:47,322 --> 01:05:51,940
Well, we try obtaining
the differential equation

929
01:05:51,940 --> 01:05:54,375
using that time
evolution over there.

930
01:05:57,120 --> 01:05:59,500
So the time evolution
is over there.

931
01:05:59,500 --> 01:06:05,860
Let's try to find
what is d dt of psi t.

932
01:06:10,240 --> 01:06:15,230
So d dt of psi of
t is just the d dt

933
01:06:15,230 --> 01:06:21,070
of this operator u
of t, t0 psi of t0.

934
01:06:24,170 --> 01:06:29,060
And I should only
differentiate that operate.

935
01:06:29,060 --> 01:06:33,610
Now, I want an
equation for psi of t.

936
01:06:33,610 --> 01:06:35,855
So I have here psi of t0.

937
01:06:35,855 --> 01:06:44,810
So I can write this
as du of t, t0 dt.

938
01:06:44,810 --> 01:06:48,990
And now put a psi at t.

939
01:06:48,990 --> 01:06:52,670
And then, I could
put a u from t to t0.

940
01:07:01,920 --> 01:07:10,640
Now, this u of t and t0 just
brings it back to time t0.

941
01:07:10,640 --> 01:07:15,880
And this is all good now, I have
this complicated operator here.

942
01:07:15,880 --> 01:07:18,410
But there's nothing too
complicated about it.

943
01:07:18,410 --> 01:07:21,560
Especially if I
reverse the order here,

944
01:07:21,560 --> 01:07:31,110
I'll have du dt of t, t0
and u dagger of t, t0.

945
01:07:31,110 --> 01:07:36,000
And I reverse the order there
in order that this operator

946
01:07:36,000 --> 01:07:38,980
is the same as that, the one
that is being [INAUDIBLE] that

947
01:07:38,980 --> 01:07:44,140
has the same order of
arguments, t and t0.

948
01:07:44,140 --> 01:07:47,510
So I've got something now.

949
01:07:47,510 --> 01:07:53,770
And I'll call this
lambda of t and t0.

950
01:07:56,530 --> 01:07:58,150
So what have I learned?

951
01:07:58,150 --> 01:08:07,860
That d dt of psi and t is equal
to lambda of t, t0 psi of t.

952
01:08:12,650 --> 01:08:13,770
Questions?

953
01:08:13,770 --> 01:08:16,950
I don't want to loose
you in their derivation.

954
01:08:16,950 --> 01:08:18,760
Look at it.

955
01:08:18,760 --> 01:08:23,250
Anything-- you got lost,
notation, anything.

956
01:08:23,250 --> 01:08:26,090
It's a good time to ask.

957
01:08:26,090 --> 01:08:26,734
Yes.

958
01:08:26,734 --> 01:08:29,275
AUDIENCE: Just to make sure when
you differentiated the state

959
01:08:29,275 --> 01:08:32,470
by t, the reason that you don't
put that in the derivative

960
01:08:32,470 --> 01:08:35,110
because it doesn't have a
time [INAUDIBLE] necessarily,

961
01:08:35,110 --> 01:08:38,240
or because-- oh, because
you're using the value at t0.

962
01:08:38,240 --> 01:08:39,180
PROFESSOR: Right.

963
01:08:39,180 --> 01:08:42,434
Here I looked at that
equation and the only part

964
01:08:42,434 --> 01:08:45,130
that has anything
to do with time t

965
01:08:45,130 --> 01:08:46,729
is the operator, not the state.

966
01:08:50,790 --> 01:08:52,555
Any other comments or questions?

967
01:08:56,029 --> 01:08:58,620
OK, so what have we learned?

968
01:08:58,620 --> 01:09:04,550
We want to know some important
things about this operator

969
01:09:04,550 --> 01:09:08,319
lambda because
somehow, it's almost

970
01:09:08,319 --> 01:09:10,210
looking like a
Schrodinger equation.

971
01:09:10,210 --> 01:09:12,660
So we want to see a
couple of things about it.

972
01:09:16,330 --> 01:09:20,060
So the first thing
that I will show to you

973
01:09:20,060 --> 01:09:26,680
is that lambda is, in
fact, anti-Hermitian.

974
01:09:31,510 --> 01:09:33,430
Here is lambda.

975
01:09:33,430 --> 01:09:36,790
I could figure out,
what is lambda dagger?

976
01:09:36,790 --> 01:09:40,500
Well, lambda dagger is you
take the dagger of this.

977
01:09:40,500 --> 01:09:43,609
You have to think when you
take the dagger of this thing.

978
01:09:43,609 --> 01:09:47,490
It looks a little worrisome,
but this is an operator.

979
01:09:47,490 --> 01:09:50,450
This is another operator,
which is a time derivative.

980
01:09:50,450 --> 01:09:54,810
So you take the dagger by
doing the reverse operators

981
01:09:54,810 --> 01:09:55,620
and daggers.

982
01:09:55,620 --> 01:10:01,370
So the first factor
is clearly u of t, t0.

983
01:10:01,370 --> 01:10:04,880
And then the dagger of this.

984
01:10:04,880 --> 01:10:09,830
Now, dagger doesn't interfere
at all with time derivatives.

985
01:10:09,830 --> 01:10:12,800
Think of the time derivative--
operator at one time,

986
01:10:12,800 --> 01:10:15,050
operator at another
slightly different time.

987
01:10:15,050 --> 01:10:16,590
Subtract it.

988
01:10:16,590 --> 01:10:19,210
You take the dagger
and the dagger

989
01:10:19,210 --> 01:10:20,730
goes through the derivative.

990
01:10:20,730 --> 01:10:28,430
So this is d u dagger t, t0 dt.

991
01:10:28,430 --> 01:10:32,220
So I wrote here what
lambda dagger is.

992
01:10:32,220 --> 01:10:34,840
You have here what lambda is.

993
01:10:34,840 --> 01:10:39,370
And the claim is that one
is minus the other one.

994
01:10:39,370 --> 01:10:41,150
It doesn't look
obvious because it's

995
01:10:41,150 --> 01:10:42,400
supposed to be anti-Hermitian.

996
01:10:45,810 --> 01:10:50,580
But you can show it is true by
doing the following-- u of t,

997
01:10:50,580 --> 01:11:00,470
t0 u dagger of t, t0
is a unitary operator.

998
01:11:00,470 --> 01:11:02,850
So this is 1.

999
01:11:02,850 --> 01:11:05,760
And now you differentiate
with respect to t.

1000
01:11:10,480 --> 01:11:12,600
If you differentiate
with respect to t,

1001
01:11:12,600 --> 01:11:24,390
you get du dt of t, t0 u dagger
of t, t0 plus u of t, t0 du

1002
01:11:24,390 --> 01:11:32,840
dagger of t, t0 equals 0 because
the right-hand side is 1.

1003
01:11:32,840 --> 01:11:38,000
And this term is lambda.

1004
01:11:38,000 --> 01:11:43,332
And the second term
is lambda dagger.

1005
01:11:43,332 --> 01:11:49,150
And they add up to 0, so
lambda dagger is minus lambda.

1006
01:11:49,150 --> 01:11:52,920
Lambda is, therefore,
anti-Hermitian as claimed.

1007
01:12:03,800 --> 01:12:04,990
Now, look.

1008
01:12:04,990 --> 01:12:09,470
This is starting to
look pretty good.

1009
01:12:09,470 --> 01:12:12,360
This lambda depends on t and t0.

1010
01:12:12,360 --> 01:12:14,663
That's a little nasty though.

1011
01:12:14,663 --> 01:12:15,162
Why?

1012
01:12:17,980 --> 01:12:19,570
Here is t.

1013
01:12:19,570 --> 01:12:24,270
What is t0 doing here?

1014
01:12:24,270 --> 01:12:27,400
It better not be there.

1015
01:12:27,400 --> 01:12:30,580
So what I want to show to
you is that even though this

1016
01:12:30,580 --> 01:12:35,740
looks like it has a t0
in there, there's no t0.

1017
01:12:35,740 --> 01:12:41,200
So we want to show this operator
is actually independent of t0.

1018
01:12:41,200 --> 01:12:49,570
So I will show that if
you have lambda of t, t0,

1019
01:12:49,570 --> 01:12:56,590
it's actually equal to
lambda of t, t1 for any t1.

1020
01:12:59,270 --> 01:13:00,250
We'll show that.

1021
01:13:03,240 --> 01:13:05,202
Sorry.

1022
01:13:05,202 --> 01:13:09,820
[LAUGHTER]

1023
01:13:09,820 --> 01:13:15,290
PROFESSOR: So this will
show that you could take t1

1024
01:13:15,290 --> 01:13:18,110
to be t0 plus epsilon.

1025
01:13:18,110 --> 01:13:20,440
And take the limit
and say the derivative

1026
01:13:20,440 --> 01:13:22,900
of this with respect of t0 is 0.

1027
01:13:22,900 --> 01:13:25,580
Or take this to
mean that it's just

1028
01:13:25,580 --> 01:13:30,900
absolutely independent of t0
and t0 is really not there.

1029
01:13:30,900 --> 01:13:33,960
So if you take t1 equal
t dot plus epsilon,

1030
01:13:33,960 --> 01:13:36,295
you could just
conclude from these

1031
01:13:36,295 --> 01:13:41,230
that this lambda with
respect to t0 is 0.

1032
01:13:41,230 --> 01:13:42,670
No dependence on t0.

1033
01:13:42,670 --> 01:13:43,980
So how do we do that?

1034
01:13:43,980 --> 01:13:46,370
Let's go a little quick.

1035
01:13:46,370 --> 01:13:54,170
This is du t, t0 dt
times u dagger of t, t0.

1036
01:13:56,960 --> 01:14:00,770
Complete set of states
said add something.

1037
01:14:00,770 --> 01:14:03,800
We want to put the t1 here.

1038
01:14:03,800 --> 01:14:08,090
So let's add something
that will help us do that.

1039
01:14:08,090 --> 01:14:14,680
So let's add t, t0 and
put here a u of t0,

1040
01:14:14,680 --> 01:14:18,160
t1 and a u dagger of t0, t1.

1041
01:14:22,600 --> 01:14:27,710
This thing is 1, and I've put
the u dagger of t, t0 here.

1042
01:14:30,260 --> 01:14:36,910
OK, look at this.

1043
01:14:36,910 --> 01:14:44,855
T0 and t1 here and t
dot t1 there like that.

1044
01:14:49,100 --> 01:14:53,545
So actually, we'll do
it the following way.

1045
01:14:56,240 --> 01:15:03,345
Think of this whole
thing, this d dt

1046
01:15:03,345 --> 01:15:06,150
is acting just on this factor.

1047
01:15:06,150 --> 01:15:09,070
But since it's time,
it might as well

1048
01:15:09,070 --> 01:15:13,680
be acting on all of this factor
because this has no time.

1049
01:15:13,680 --> 01:15:22,640
So this is d dt on
u t, t0 u t0, t1.

1050
01:15:28,690 --> 01:15:37,260
And this thing is u of t1m t0.

1051
01:15:37,260 --> 01:15:39,840
The dagger can be
compensated by this.

1052
01:15:39,840 --> 01:15:47,330
And this dagger is u of t0, t.

1053
01:15:47,330 --> 01:15:49,470
This at a t and that's a comma.

1054
01:15:53,325 --> 01:15:54,510
t0, t.

1055
01:15:54,510 --> 01:15:56,490
Yes.

1056
01:15:56,490 --> 01:16:00,391
OK, so should I go there?

1057
01:16:00,391 --> 01:16:00,890
Yes.

1058
01:16:04,190 --> 01:16:05,890
We're almost there.

1059
01:16:05,890 --> 01:16:11,120
You see that the first
derivative is already

1060
01:16:11,120 --> 01:16:20,210
d dt of u of t, t1.

1061
01:16:20,210 --> 01:16:23,370
And the second operator
by compensation

1062
01:16:23,370 --> 01:16:37,490
is u of t1, t, which is the
same as u dagger of t, t1.

1063
01:16:37,490 --> 01:16:43,970
And then, du of t, t1 u dagger
of t, t1 is lambda of t, t1.

1064
01:16:46,540 --> 01:16:50,680
So it's a little
sneaky, the proof,

1065
01:16:50,680 --> 01:16:53,590
but it's totally rigorous.

1066
01:16:53,590 --> 01:16:55,420
And I don't think
there's any step

1067
01:16:55,420 --> 01:16:56,830
you should be worried there.

1068
01:16:56,830 --> 01:17:00,420
They're all very
logical and reasonable.

1069
01:17:00,420 --> 01:17:03,100
So we have two things.

1070
01:17:03,100 --> 01:17:06,150
First of all, that
this quantity,

1071
01:17:06,150 --> 01:17:10,170
even though it looks
like it depends on t0,

1072
01:17:10,170 --> 01:17:14,790
we finally realized that
it does not depend on t0.

1073
01:17:14,790 --> 01:17:19,405
So I will rewrite this
equation as lambda of t.

1074
01:17:26,020 --> 01:17:31,620
And lambda of t
is anti-Hermitian,

1075
01:17:31,620 --> 01:17:39,400
so we will multiply by an
i to make it Hermitian.

1076
01:17:39,400 --> 01:17:45,490
And in fact, lambda has
units of 1 over time.

1077
01:17:45,490 --> 01:17:49,890
Unitary operators have no units.

1078
01:17:49,890 --> 01:17:54,060
They're like numbers,
like 1 or e to the i phi,

1079
01:17:54,060 --> 01:17:55,820
or something like
that-- have no units.

1080
01:17:55,820 --> 01:17:58,880
So this has units
of 1 over time.

1081
01:17:58,880 --> 01:18:07,110
So if I take i h
bar lambda of t,

1082
01:18:07,110 --> 01:18:13,410
this goes from lambda being
anti-Hermitian-- this operator

1083
01:18:13,410 --> 01:18:15,520
is now Hermitian.

1084
01:18:15,520 --> 01:18:19,190
This goes from lambda
having units of 1 over time

1085
01:18:19,190 --> 01:18:23,680
to this thing having
units of energy.

1086
01:18:23,680 --> 01:18:38,185
So this is a Hermitian
operator with units of energy.

1087
01:18:41,260 --> 01:18:45,390
Well, I guess not much
more needs to be said.

1088
01:18:45,390 --> 01:18:49,360
If that's a Hermitian
operator with units of energy,

1089
01:18:49,360 --> 01:18:55,450
we will give it a name
called H, or Hamiltonian.

1090
01:18:55,450 --> 01:19:00,040
i h bar lambda of t.

1091
01:19:00,040 --> 01:19:07,510
Take this equation and multiply
by i h bar to get i h bar

1092
01:19:07,510 --> 01:19:13,990
d dt of psi is equal
to this i h bar

1093
01:19:13,990 --> 01:19:19,581
lambda, which is
h of t psi of t.

1094
01:19:22,880 --> 01:19:25,680
Schrodinger equation.

1095
01:19:25,680 --> 01:19:28,820
So we really got it.

1096
01:19:28,820 --> 01:19:31,220
That's the Schrodinger equation.

1097
01:19:31,220 --> 01:19:33,440
That's the question
that must be satisfied

1098
01:19:33,440 --> 01:19:38,020
by any system governed by
unitary time evolution.

1099
01:19:38,020 --> 01:19:41,080
There's not more information
in the Schrodinger equation

1100
01:19:41,080 --> 01:19:44,580
than unitary time evolution.

1101
01:19:44,580 --> 01:19:49,290
But it allows you to
turn the problem around.

1102
01:19:49,290 --> 01:19:53,970
You see, when you went to
invent a quantum system,

1103
01:19:53,970 --> 01:19:58,580
you don't quite know how
to find this operator u.

1104
01:19:58,580 --> 01:20:03,210
If you knew u, you know
how to evolve anything.

1105
01:20:03,210 --> 01:20:06,550
And you don't have
any more questions.

1106
01:20:06,550 --> 01:20:10,820
All your questions in life
have been answered by that.

1107
01:20:10,820 --> 01:20:12,880
You know how to find the future.

1108
01:20:12,880 --> 01:20:14,900
You can invest in
the stock market.

1109
01:20:14,900 --> 01:20:16,380
You can do anything now.

1110
01:20:19,660 --> 01:20:24,860
Anyway, but the unitary operator
then gives you the Hamiltonian.

1111
01:20:24,860 --> 01:20:28,420
So if somebody tells you,
here's my unitary operator.

1112
01:20:28,420 --> 01:20:31,150
And they ask you, what
is the Hamiltonian?

1113
01:20:31,150 --> 01:20:36,167
You go here and calculate
I h bar lambda, where

1114
01:20:36,167 --> 01:20:37,250
lambda is this derivative.

1115
01:20:37,250 --> 01:20:38,375
And that's the Hamiltonian.

1116
01:20:41,360 --> 01:20:45,160
And we conversely,
if you are lucky--

1117
01:20:45,160 --> 01:20:47,560
and that's what we're
going to do next time.

1118
01:20:47,560 --> 01:20:50,390
If you have a
Hamiltonian, you try

1119
01:20:50,390 --> 01:20:52,870
to find the unitary
time evolution.

1120
01:20:52,870 --> 01:20:55,260
That's all you want to know.

1121
01:20:55,260 --> 01:20:59,640
But that's a harder problem
because you have a differential

1122
01:20:59,640 --> 01:21:00,920
equation.

1123
01:21:00,920 --> 01:21:06,490
You have h, which is here
, and you are to find u.

1124
01:21:06,490 --> 01:21:11,025
So it's a first-order matrix
differential equation.

1125
01:21:11,025 --> 01:21:13,860
So it's not a simple problem.

1126
01:21:13,860 --> 01:21:16,020
But why do we like Hamiltonians?

1127
01:21:16,020 --> 01:21:18,710
Because Hamiltonians
have to do with energy.

1128
01:21:18,710 --> 01:21:22,460
And we can get inspired
and write quantum systems

1129
01:21:22,460 --> 01:21:26,120
because we know the energy
functional of systems.

1130
01:21:26,120 --> 01:21:29,760
So we invent a Hamiltonian
and typically try

1131
01:21:29,760 --> 01:21:33,520
to find the unitary
time operator.

1132
01:21:33,520 --> 01:21:35,970
But logically
speaking, there's not

1133
01:21:35,970 --> 01:21:39,590
more and no less in the
Schrodinger equation

1134
01:21:39,590 --> 01:21:42,970
than the postulate of
unitary time evolution.

1135
01:21:42,970 --> 01:21:46,400
All right, we'll
see you next week.

1136
01:21:46,400 --> 01:21:46,900
In fact--

1137
01:21:46,900 --> 01:21:47,800
[APPLAUSE]

1138
01:21:47,800 --> 01:21:49,650
Thank you.